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Some antiferromagnetic materials exhibit a non-zero magnetic moment at a temperature near absolute zero . This effect is ascribed to spin canting , a phenomenon through which spins are tilted by a small angle about their axis rather than being exactly co-parallel.
Spin canting is due to two factors contrasting each other: isotropic exchange would align the spins exactly antiparallel, while antisymmetric exchange arising from relativistic effects ( spin–orbit coupling ) would align the spins at 90° to each other. The net result is a small perturbation, the extent of which depends on the relative strength of these effects. [ 1 ]
This effect is observable in many materials such as hematite . [ 2 ] | https://en.wikipedia.org/wiki/Spin_canting |
A spin chain is a type of model in statistical physics . Spin chains were originally formulated to model magnetic systems, which typically consist of particles with magnetic spin located at fixed sites on a lattice . A prototypical example is the quantum Heisenberg model . Interactions between the sites are modelled by operators which act on two different sites, often neighboring sites.
They can be seen as a quantum version of statistical lattice models , such as the Ising model , in the sense that the parameter describing the spin at each site is promoted from a variable taking values in a discrete set (typically { + 1 , − 1 } {\displaystyle \{+1,-1\}} , representing 'spin up' and 'spin down') to a variable taking values in a vector space (typically the spin-1/2 or two-dimensional representation of s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} ).
The prototypical example of a spin chain is the Heisenberg model, described by Werner Heisenberg in 1928. [ 1 ] This models a one-dimensional lattice of fixed particles with spin 1/2. A simple version (the antiferromagnetic XXX model) was solved, that is, the spectrum of the Hamiltonian of the Heisenberg model was determined, by Hans Bethe using the Bethe ansatz . [ 2 ] Now the term Bethe ansatz is used generally to refer to many ansatzes used to solve exactly solvable problems in spin chain theory such as for the other variations of the Heisenberg model (XXZ, XYZ), and even in statistical lattice theory, such as for the six-vertex model .
Another spin chain with physical applications is the Hubbard model , introduced by John Hubbard in 1963. [ 3 ] This model was shown to be exactly solvable by Elliott Lieb and Fa-Yueh Wu in 1968. [ 4 ]
Another example of (a class of) spin chains is the Gaudin model , described and solved by Michel Gaudin in 1976 [ 5 ]
The lattice is described by a graph G {\displaystyle G} with vertex set V {\displaystyle V} and edge set E {\displaystyle E} .
The model has an associated Lie algebra s l 2 := s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}_{2}:={\mathfrak {sl}}(2,\mathbb {C} )} . More generally, this Lie algebra can be taken to be any complex, finite-dimensional semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} . More generally still it can be taken to be an arbitrary Lie algebra.
Each vertex v ∈ V {\displaystyle v\in V} has an associated representation of the Lie algebra g {\displaystyle {\mathfrak {g}}} , labelled V v {\displaystyle V_{v}} . This is a quantum generalization
of statistical lattice models, where each vertex has an associated 'spin variable'.
The Hilbert space H {\displaystyle {\mathcal {H}}} for the whole system, which could be called the configuration space , is the tensor product of the representation spaces at each vertex: H = ⨂ v ∈ V V v . {\displaystyle {\mathcal {H}}=\bigotimes _{v\in V}V_{v}.}
A Hamiltonian is then an operator on the Hilbert space. In the theory of spin chains, there are possibly many Hamiltonians which mutually commute. This allows the operators to be simultaneously diagonalized.
There is a notion of exact solvability for spin chains, often stated as determining the spectrum of the model. In precise terms, this means determining the simultaneous eigenvectors of the Hilbert space for the Hamiltonians of the system as well as the eigenvalues of each eigenvector with respect to each Hamiltonian.
The prototypical example, and a particular example of the Heisenberg spin chain, is known as the spin 1/2 Heisenberg XXX model . [ 6 ]
The graph G {\displaystyle G} is the periodic 1-dimensional lattice with N {\displaystyle N} -sites. Explicitly, this is given by V = { 1 , ⋯ , N } {\displaystyle V=\{1,\cdots ,N\}} , and the elements of E {\displaystyle E} being { n , n + 1 } {\displaystyle \{n,n+1\}} with N + 1 {\displaystyle N+1} identified with 1 {\displaystyle 1} .
The associated Lie algebra is s l 2 {\displaystyle {\mathfrak {sl}}_{2}} .
At site n {\displaystyle n} there is an associated Hilbert space h n {\displaystyle h_{n}} which is isomorphic to the two dimensional representation of s l 2 {\displaystyle {\mathfrak {sl}}_{2}} (and therefore further isomorphic to C 2 {\displaystyle \mathbb {C} ^{2}} ). The Hilbert space of system configurations is H = ⨂ n = 1 N h n {\displaystyle {\mathcal {H}}=\bigotimes _{n=1}^{N}h_{n}} , of dimension 2 N {\displaystyle 2^{N}} .
Given an operator A {\displaystyle A} on the two-dimensional representation h {\displaystyle h} of s l 2 {\displaystyle {\mathfrak {sl}}_{2}} , denote by A ( n ) {\displaystyle A^{(n)}} the operator on H {\displaystyle {\mathcal {H}}} which acts as A {\displaystyle A} on h n {\displaystyle h_{n}} and as identity on the other h m {\displaystyle h_{m}} with m ≠ n {\displaystyle m\neq n} . Explicitly, it can be written A ( n ) = 1 ⊗ ⋯ ⊗ A ⏟ n ⊗ ⋯ ⊗ 1 , {\displaystyle A^{(n)}=1\otimes \cdots \otimes \underbrace {A} _{n}\otimes \cdots \otimes 1,} where the 1 denotes identity.
The Hamiltonian is essentially, up to an affine transformation, H = ∑ n = 1 N σ i ( n ) σ i ( n + 1 ) {\displaystyle H=\sum _{n=1}^{N}\sigma _{i}^{(n)}\sigma _{i}^{(n+1)}} with implied summation over index i {\displaystyle i} , and where σ i {\displaystyle \sigma _{i}} are the Pauli matrices . The Hamiltonian has s l 2 {\displaystyle {\mathfrak {sl}}_{2}} symmetry under the action of the three total spin operators σ i = ∑ n = 1 N σ i ( n ) {\displaystyle \sigma _{i}=\sum _{n=1}^{N}\sigma _{i}^{(n)}} .
The central problem is then to determine the spectrum (eigenvalues and eigenvectors in H {\displaystyle {\mathcal {H}}} ) of the Hamiltonian. This is solved by the method of an Algebraic Bethe ansatz, discovered by Hans Bethe and further explored by Ludwig Faddeev . | https://en.wikipedia.org/wiki/Spin_chain |
Spin chemistry is a sub-field of chemistry positioned at the intersection of chemical kinetics , photochemistry , magnetic resonance and free radical chemistry, that deals with magnetic and spin effects in chemical reactions. Spin chemistry concerns phenomena such as chemically induced dynamic nuclear polarization (CIDNP), chemically induced electron polarization (CIDEP), magnetic isotope effects in chemical reactions, and it is hypothesized to be key in the underlying mechanism for avian magnetoreception [ 1 ] and consciousness. [ 2 ]
The radical-pair mechanism explains how a magnetic field can affect reaction kinetics by affecting electron spin dynamics. Most commonly demonstrated in reactions of organic compounds involving radical intermediates, a magnetic field can speed up a reaction by decreasing the frequency of reverse reactions.
The radical-pair mechanism emerged as an explanation to CIDNP and CIDEP and was proposed in 1969 by Closs; Kaptein and Oosterhoff. [ 3 ]
A radical is a molecule with an odd number of electrons , and is induced in a variety of ways, including ultraviolet radiation. A sun burn is largely due to radical formation from this radiation. The radical-pair, however, is not simply two radicals. This is because radical-pairs (specifically singlets) are quantum entangled , even as separate molecules. [ 1 ] More fundamental to the radical-pair mechanism, however, is the fact that radical-pair electrons both have spin, short for spin angular momentum , which gives each separate radical a magnetic moment . Therefore, spin states can be altered by magnetic fields.
The radical-pair is characterized as triplet or singlet by the spin state of the two lone electrons, paired together. The spin relationship is such that the two unpaired electrons, one in each radical molecule, may have opposite spin (singlet; anticorrelated), or the same spin (triplet; correlated). The singlet state is called such because there is only one way for the electrons’ spins to anticorrelate (S), whereas the triplet state is called such because the electron's spin may be correlated in three different fashions, denoted T +1 , T 0 , and T −1 .
Spin states relate to chemical and biochemical reaction mechanisms because bonds can be formed only between two electrons of opposite spin ( Hund's rules ). Sometimes when a bond is broken in a particular manner, for example, when struck by photons, each electron in the bond relocates to each respective molecule, and a radical-pair is formed. Furthermore, the spin of each electron previously involved in the bond is conserved, [ 1 ] [ 3 ] which means that the radical-pair now formed is a singlet (each electron has opposite spin, as in the origin bond). As such, the reverse reaction, i.e. the reforming of a bond, called recombination, readily occurs. The radical-pair mechanism explains how external magnetic fields can prevent radical-pair recombination with Zeeman interactions , the interaction between spin and an external magnetic field, and shows how a higher occurrence of the triplet state accelerates radical reactions because triplets can proceed only to products, and singlets are in equilibrium with the reactants as well as with the products. [ 1 ] [ 3 ] [ 4 ]
Zeeman interactions can "flip" only one of the radical's electron's spin if the radical-pair is anisotropic , thereby converting singlet radical-pairs to triplets. [ 1 ]
The Zeeman interaction is an interaction between spin and external magnetic field, and is given by the equation
where Δ E {\displaystyle \Delta E} is the energy of the Zeeman interaction , ν L {\displaystyle \nu _{\text{L}}} is the Larmor frequency , B {\displaystyle B} is the external magnetic field, μ B {\displaystyle \mu _{\text{B}}} is the Bohr magneton , h {\displaystyle h} is the Planck constant , and g {\displaystyle g} is the g -factor of a free electron, −2.002 319 , [ 5 ] which is slightly different in different radicals. [ 1 ]
It is common to see the Zeeman interaction formulated in other ways. [ 4 ]
Hyperfine interactions , the internal magnetic fields of local magnetic isotopes, play a significant role as well in the spin dynamics of radical-pairs. [ 1 ] [ 3 ] [ 4 ]
Because the Zeeman interaction is a function of magnetic field and Larmor frequency, it can be obstructed or amplified by altering the external magnetic or the Larmor frequency with experimental instruments that generate oscillating fields. It has been observed that migratory birds lose their navigational abilities in such conditions where the Zeeman interaction is obstructed in radical-pairs. [ 1 ] | https://en.wikipedia.org/wiki/Spin_chemistry |
Spin column-based nucleic acid purification is a solid phase extraction method to quickly purify nucleic acids . This method relies on the fact that nucleic acid will bind to the solid phase of silica under certain conditions.
The different stages of the method are lyse, bind, wash, and elute. [ 1 ] [ 2 ] More specifically, this entails the lysis of target cells to release nucleic acids , selective binding of nucleic acid to a silica membrane, washing away particulates and inhibitors that are not bound to the silica membrane, and elution of the nucleic acid, with the end result being purified nucleic acid in an aqueous solution.
For lysis, the cells (blood, tissue, etc.) of the sample must undergo a treatment to break the cell membrane and free the nucleic acid. Depending on the target material, this can include the use of detergent or other buffers, proteinases or other enzymes, heating to various times/temperatures, or mechanical disruption such as cutting with a knife or homogenizer , using a mortar and pestle, or bead-beating with a bead mill.
For binding, a buffer solution is then added to the lysed sample along with ethanol or isopropanol . The sample in binding solution is then transferred to a spin column , and the column is put either in a centrifuge or attached to a vacuum. The centrifuge/vacuum forces the solution through a silica membrane that is inside the spin column, where under the right ionic conditions, nucleic acids will bind to the silica membrane, as the rest of the solution passes through. With the target material bound, the flow-through can be removed.
To wash, a new buffer is added onto the column, then centrifuged/vacuumed through the membrane. This buffer is intended to maintain binding conditions, while removing the binding salts and other remaining contaminants. Generally it takes several washes, often with increasing percentages of ethanol/isopropanol, until the nucleic acid on the silica membrane is free of contaminants. The last 'wash' is often a dry step to allow the alcohol to evaporate, leaving only purified nucleic acids bound to the column.
Finally, elution is the process of adding an aqueous solution to the column, allowing the hydrophilic nucleic acid to leave the column and return to solution. This step may be improved with salt, pH, time, or heat. Finally, to capture the eluate/eluent, the column is transferred into a clean microtube prior to a last centrifugation step.
Even prior to the nucleic acid methods employed today, it was known that in the presence of chaotropic agents , such as sodium iodide or sodium perchlorate , DNA binds to silica, glass particles or to unicellular algae called diatoms which shield their cell walls with silica . This property was used to purify nucleic acid using glass powder or silica beads under alkaline conditions. [ 3 ] This was later improved using guanidinium thiocyanate or guanidinium hydrochloride as the chaotropic agent. [ 4 ] For ease of handling, the use of glass beads was later changed to silica columns. And to enable use of automated extraction instruments, there was development of silica-coated paramagnetic beads, more commonly referred to as "magnetic bead" extraction. | https://en.wikipedia.org/wiki/Spin_column-based_nucleic_acid_purification |
In computational chemistry , spin contamination is the artificial mixing of different electronic spin -states. This can occur when an approximate orbital-based wave function is represented in an unrestricted form – that is, when the spatial parts of α and β spin-orbitals are permitted to differ. Approximate wave functions with a high degree of spin contamination are undesirable. In particular, they are not eigenfunctions of the total spin-squared operator, Ŝ 2 , but can formally be expanded in terms of pure spin states of higher multiplicities (the contaminants).
Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals. For an open-shell system, the mean-field approach of Hartree–Fock theory gives rise to different equations for the α and β orbitals. Consequently, there are two approaches that can be taken – either to force double occupation of the lowest orbitals by constraining the α and β spatial distributions to be the same ( restricted open-shell Hartree–Fock , ROHF) or permit complete variational freedom ( unrestricted Hartree–Fock UHF). In general, an N -electron Hartree–Fock wave function composed of N α α-spin orbitals and N β β-spin orbitals can be written as [ 1 ]
where A {\displaystyle {\mathcal {A}}} is the antisymmetrization operator . This wave function is an eigenfunction of the total spin projection operator, Ŝ z , with eigenvalue ( N α − N β )/2 (assuming N α ≥ N β ). For a ROHF wave function, the first 2 N β spin-orbitals are forced to have the same spatial distribution:
There is no such constraint in an UHF approach. [ 2 ]
The total spin-squared operator commutes with the nonrelativistic molecular Hamiltonian , so it is desirable that any approximate wave function is an eigenfunction of Ŝ 2 . The eigenvalues of Ŝ 2 are S ( S + 1), where S is the spin quantum number of the system and can take the values 0 ( singlet ), 1/2 ( doublet ), 1 ( triplet ), 3/2 (quartet), and so forth. The Ŝ 2 eigenvalues of the most common spin multiplicities are listed below.
The Ŝ² operator can be decomposed as: [ 3 ]
Slater determinants have well-defined spin projections:
⟨ S ^ z ⟩ = m s = N α − N β 2 {\displaystyle \langle {\hat {S}}_{z}\rangle =m_{s}={\frac {N_{\alpha }-N_{\beta }}{2}}\quad } ⟨ S ^ z 2 ⟩ = m s 2 = ( N α − N β 2 ) 2 {\displaystyle \quad \langle {\hat {S}}_{z}^{2}\rangle =m_{s}^{2}=\left({\frac {N_{\alpha }-N_{\beta }}{2}}\right)^{2}}
S ^ − S ^ + {\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}} can be expressed in terms of individual electron operators: S ^ − S ^ + = ∑ i s ^ − ( i ) ∑ j s ^ + ( j ) {\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}=\sum _{i}{\hat {s}}_{-}(i)\sum _{j}{\hat {s}}_{+}(j)} .
For an Unrestricted Hartree-Fock (UHF) wavefunction, the expectation value Ŝ 2 is [ 4 ]
where the first two terms follow directly from the decomposition of Ŝ², the third term corresponds to the diagonal contribution of S ^ − S ^ + {\displaystyle {\hat {S}}_{-}{\hat {S}}_{+}} and the last term involves the overlap between the α {\displaystyle \alpha } and β {\displaystyle \beta } spin-orbitals with the negative sign arising from the antisymmetry of the Hartree-Fock wavefunction. [ 3 ]
For Slater determinants constructed from restricted spin-orbitals , the spatial parts of corresponding α {\displaystyle \alpha } and β {\displaystyle \beta } orbitals are the same, thus the α {\displaystyle \alpha } and β {\displaystyle \beta } overlaps evaluate to 1 for i = j {\displaystyle i=j} and 0 otherwise.
This simplifies to:
In Restricted open-shell Hartree-Fock (ROHF) , all unpaired electrons have the same spin, yielding: [ 3 ]
making ROHF wavefunctions eigenfunctions of Ŝ².
For multi-configurational wavefunctions expressed as | Ψ ⟩ = ∑ I c I | Φ I ⟩ {\displaystyle |\Psi \rangle =\sum _{I}c_{I}|\Phi _{I}\rangle } with Φ I {\displaystyle \Phi _{I}} being Slater determinants, ⟨Ŝ²⟩ is :
⟨ Ψ | S ^ 2 | Ψ ⟩ = ∑ I , J c I ∗ c J ⟨ Φ I | S ^ 2 | Φ J ⟩ {\displaystyle \langle \Psi |{\hat {S}}^{2}|\Psi \rangle =\sum _{I,J}c_{I}^{*}c_{J}\langle \Phi _{I}|{\hat {S}}^{2}|\Phi _{J}\rangle } .
The diagonal terms ⟨ Φ I | S ^ 2 | Φ I ⟩ {\displaystyle \langle \Phi _{I}|{\hat {S}}^{2}|\Phi _{I}\rangle } are calculated as above, while cross-terms ⟨ Φ J | S ^ 2 | Φ I ⟩ {\displaystyle \langle \Phi _{J}|{\hat {S}}^{2}|\Phi _{I}\rangle } (where I≠J) require computing ⟨ Φ J | S ^ − S ^ + | Φ I ⟩ {\displaystyle \langle \Phi _{J}|{\hat {S}}_{-}{\hat {S}}_{+}|\Phi _{I}\rangle } using individual electron operators. ⟨ Φ J | S ^ z | Φ I ⟩ {\displaystyle \langle \Phi _{J}|{\hat {S}}_{z}|\Phi _{I}\rangle } and ⟨ Φ J | S ^ z 2 | Φ I ⟩ {\displaystyle \langle \Phi _{J}|{\hat {S}}_{z}^{2}|\Phi _{I}\rangle } vanish when I≠J.
The sum of the last two terms in the UHF equation is a measure of the extent of spin contamination in the unrestricted Hartree–Fock approach and is always non-negative – the wave function is usually contaminated to some extent by higher order spin eigenstates unless a ROHF approach is taken. Therefore, the deviation of the UHF expectation value of Ŝ 2 from the exact Ŝ 2 eigenvalue as would be expected from the spin multiplicity (see the table above) is usually taken as a measure of the severity of the spin contamination. Naturally, there is no contamination if all electrons are the same spin. Also, there is often (but not always, as in open-shell singlets) no contamination if the number of α and β electrons is the same. A small basis set could also constrain the wavefunction sufficiently to prevent spin contamination.
Such contamination is a manifestation of the different treatment of α and β electrons that would otherwise occupy the same molecular orbital. It is also present in Møller–Plesset perturbation theory calculations that employ an unrestricted wave function as a reference state (and even some that employ a restricted wave function) and, to a much lesser extent, in the unrestricted Kohn–Sham approach to density functional theory using approximate exchange-correlation functionals. [ 5 ]
Although the ROHF approach does not suffer from spin contamination, it is less commonly available in quantum chemistry computer programs . Given this, several approaches to remove or minimize spin contamination from UHF wave functions have been proposed.
The annihilated UHF (AUHF) approach involves the annihilation of first spin contaminant of the density matrix at each step in the self-consistent solution of the Hartree–Fock equations using a state-specific Löwdin annihilator . [ 6 ] The resulting wave function, while not completely free of contamination, dramatically improves upon the UHF approach especially in the absence of high order contamination. [ 7 ] [ 8 ]
Projected UHF (PUHF) annihilates all spin contaminants from the self-consistent UHF wave function. The projected energy is evaluated as the expectation of the projected wave function. [ 9 ]
The spin-constrained UHF (SUHF) introduces a constraint into the Hartree–Fock equations of the form λ( Ŝ 2 − S ( S + 1)), which as λ tends to infinity reproduces the ROHF solution. [ 10 ]
All of these approaches are readily applicable to unrestricted Møller–Plesset perturbation theory .
Although many density functional theory (DFT) codes simply calculate spin-contamination using the Kohn-Sham orbitals as if they were Hartree-Fock orbitals, this is not necessarily correct. [ 11 ] [ 12 ] [ 13 ] [ 14 ] The unrestricted Kohn-Sham (UKS) wavefunction of a spin-pure electronic state may have a Ŝ 2 expectation value that is inconsistent with a spin-pure state, if it is calculated using the UHF formula. Therefore, the benefits of using a restricted open-shell treatment over an unrestricted one is smaller in a DFT context than in HF. In particular, the exact spin density of the S z =S spin-component of an open-shell molecule generally adopts negative values in small parts of the real space due to spin polarization, but a restricted open-shell Kohn-Sham (ROKS) treatment necessarily gives non-negative spin density everywhere, which proves that, in general, only UKS can possibly give the exact spin density of a system. [ 15 ] In practical calculations, the energy error of an unrestricted DFT calculation due to spin contamination can sometimes be remedied by a spin projection approach, particularly if the spin contamination is large (e.g. when there is antiferromagnetic coupling in the system). [ 16 ] However, for moderate spin contamination, spin projection may actually be counterproductive, and create unphysical cusps (i.e. derivative discontinuities) on potential energy curves. [ 14 ]
On the other hand, in TDDFT calculations, one frequently observes severe spin contamination in the excited states, even if the ground state is described by ROKS (i.e. even when the ground state itself is not spin contaminated). This is because the single excitations out of an open-shell reference state (usually the ground state) do not span a spin-complete manifold, i.e. it is impossible to unitarily transform the set of all single excitations of an open-shell reference state, such that all of the transformed states are eigenstates of the Ŝ 2 operator. [ 17 ] The excited state 〈Ŝ 2 〉 values of a TDDFT calculation of an open-shell state can have a maximum error of 2 when the ground state is described by ROKS, and sometimes slightly larger than 2 when the ground state is described by UKS. [ 18 ] To eliminate the excited state spin contamination in TDDFT, spin-adapted TDDFT methods [ 19 ] must be used, which either explicitly or implicitly include some double excitations and thereby go beyond the adiabatic approximation of the TDDFT response kernel. Spin-adapted TDDFT methods not only improve the prediction of absorption spectra, [ 20 ] [ 21 ] but also improve the prediction of the fluorescence spectra [ 19 ] [ 22 ] and excited state relaxation pathways [ 22 ] of open-shell molecules. | https://en.wikipedia.org/wiki/Spin_contamination |
Spin crossover (SCO) is a phenomenon that occurs in some metal complexes wherein the spin state of the complex changes due to an external stimulus. The stimuli can include temperature, pressure or radiation. [ 1 ] Spin crossover is referred to as spin transition if takes place suddenly,or spin equilibrium , when is gradual. However, this terminology is not strict, the "cross-over" and "transition" being often considered equivalent. The change in spin state usually involves transformation from a low spin (LS) ground state (at low temperatures and/or high pressure) towards a high spin (HS) ground configuration (at high temperature or reduced/normal pressure). [ 2 ]
Spin crossover is commonly observed with first row transition metal complexes with a d 4 through d 7 electron configuration in an octahedral ligand geometry. [ 1 ] Spin transition curves typically plot the high-spin molar fraction against temperature. [ 3 ] The abruptness with hysteresis indicates cooperativity, or “communication”, between neighboring metal complexes, throughout the whole lattice. [ 4 ] A transparent account of cooperative lattice effects is the mechano-elastic model of spin transition, [ 5 ] developed in terms of volume changes of molecular units, along the spin state swap and intermolecular effects assumed as harmonic oscillators. The spin crossover, or spin transition is a neat example of bi-stability. A material is bistable when it exists in the two different states, with distinct properties, the form being tunable by external stimuli (e.g. temperature). The two-step transition is relatively rare but is observed, for example, with dinuclear SCO complexes for which the spin transition in one metal center renders the transition in the second metal center less favorable. Several types of spin crossover have been identified; some of them are light induced excited spin-state trapping (LIESST) , ligand-driven light induced spin change (LD-LISC), and charge transfer induced spin transition (CTIST). [ 2 ]
SCO was first observed in 1931 by Cambi et al. who discovered anomalous magnetic behavior for the tris(N,N-dialkyldithiocarbamatoiron(III) complexes . [ 7 ] The spin states of these complexes were sensitive to the nature of the amine substituents. In the 1960s, the first Co II SCO complex was reported. [ 8 ] Magnetic measurements and Mössbauer spectroscopic studies established the nature of the spin transition in iron(II) SCO complexes. [ 9 ] Building on those early studies, there is now interest in applications of SCO in electronic and optical displays. [ 10 ]
Due to the changes in magnetic properties that occur from a spin transition - the complex being less magnetic in a LS state and more magnetic in a HS state - magnetic susceptibility measurements are key to characterization of spin crossover compounds. The magnetic susceptibility as a function of temperature, (χT) is the principal technique used to characterize SCO complexes.
Another very useful technique for characterizing SCO complexes, especially iron complexes, is 57 Fe Mössbauer spectroscopy . [ 2 ] This technique is especially sensitive to magnetic effects. When spectra are recorded as a function of temperature, the areas under the curves of the absorption peaks are proportional to the fraction of HS and LS states in the sample.
SCO induces changes in metal-to-ligand bond distances due to the population or depopulation of the e g orbitals that have a slight antibonding character. Consequently X-ray crystallography above and below transition temperatures will generally reveal changes in metal-ligand bond lengths. Transitions from a HS to a LS state cause a decrease in and a strengthening of the metal-ligand bond. These changes are also manifested in FT-IR and Raman spectra.
The spin crossover phenomenon is very sensitive to grinding, milling and pressure, but Raman spectroscopy has the advantage that the sample does not require further preparation, in contrast to Fourier Transform Infrared spectroscopy, FT-IR , techniques; highly colored samples may affect the measurements however. [ 12 ] Raman spectroscopy is also advantageous because it allows perturbation of the sample with external stimuli to induce SCO. Thermally induced spin crossover is due to the higher electronic degeneracies of the LS form and lower vibrational frequencies of the HS form, thus increasing the entropy. The Raman spectrum of an iron(II) complex in the HS and LS state, emphasizing the changes in the M-L vibrational modes, where a shift from 2114 cm −1 to 2070 cm −1 corresponds to changes in the stretching vibrational modes of the thiocyanate ligand from a LS state to a HS state, respectively.
SCO behavior can be followed with UV-vis spectroscopy . In some cases, the absorption bands obscured due to the high intensity absorption bands caused by the Metal-to-Ligand Charge Transfer (MLCT) absorption bands. [ 13 ]
Thermal perturbations are the most common type of external stimulus used to induce SCO. [ 14 ] One example is [Fe II (tmphen) 2 ] 3 [Co III (CN) 6 ] 2 trigonal bipyramid (TBP), with the Fe II centers in the equatorial positions. The HS Fe II remains under 20% i the range of 4.2 K to 50 K, but at room temperature about two-thirds of the Fe II ions in the sample are HS, as shown by the absorption band at 2.1 mm/s, while the other third of the ions remain in the LS state. The thermally induced spin transition is an entropy driven process. Around 25% of the total entropy gain from the LS to HS transition originates from the increase in spin multiplicity according to the relationship:
Δ S m u l t = R ⋅ l n ( ( 2 S + 1 ) H S / ( 2 S + 1 ) L S ) {\displaystyle \Delta {S}_{mult}=R\cdot ln((2S+1)_{HS}/(2S+1)_{LS})}
and the larger contribution arises from vibrational effects, since the metal-ligand bond distances are larger in the HS state. [ 15 ]
SCO is also influenced by the application of pressure, which changes the population of the HS and LS states. Upon application of pressure, a conversion from the HS state to the LS state and a shift from T 1/2 , (the temperature at which half of the complex is in a LS state), to higher temperatures will occur. This effect results from an increase in the zero point energy difference, ΔE° HL , caused by an increase in the relative vertical displacement of the potential wells and a decrease in the activation energy , ΔW° HL , which favors the LS state. [ 16 ] The complex Fe(phen) 2 (SCN) 2 exhibits this effect. At high pressures the LS state predominates and the transition temperature increases. At high pressures the compound is almost entirely transformed to the LS state at room temperature. As a result of the application of pressure on the Fe(phen) 2 (SCN) 2 compound, the bond lengths are affected. The difference in M-L bond lengths in both HS and LS states changes the entropy of the system. The change in spin transition temperature, T 1/2 and pressure obeys the Clausius-Clapeyron relationship: [ 16 ]
∂ T 1 ╱ 2 ∂ P = Δ V Δ S H L {\displaystyle {\frac {\partial {{T}_{{}^{1}\!\!\diagup \!\!{}_{2}\;}}}{\partial P}}={\frac {{\text{ }}\!\!\Delta \!\!{\text{ V}}}{{\text{ }}\!\!\Delta \!\!{\text{ }}{{S}_{HL}}}}}
The increase in pressure will decrease the volume of the unit cell of the Fe(phen) 2 (SCN) 2 and increase the T 1/2 of the system. A linear relationship between T 1/2 and pressure for Fe(phen) 2 (SCN) 2 , where the slope of the line is ∂ T 1 ╱ 2 ∂ P {\displaystyle {\frac {\partial {{T}_{{}^{1}\!\!\diagup \!\!{}_{2}\;}}}{\partial P}}} .
In Light Induced Excited Spin State Trapping ( LIESST ), the HS-LS transition is triggered by irradiating the sample. At low temperatures it is possible to trap compounds in the HS state - a phenomenon known as the LIESST effect. The compound can be converted back to a LS state by irradiation with a photon of different energy. Irradiation of d-d transitions of the LS metal complex or MLCT absorption bands leads to population of HS states. [ 17 ] A good example to illustrate the LIESST effect is the complex [Fe(1-propyltetrazole) 6 ](BF4) 2 . The sample was irradiated with green light at temperatures below 50 K. By doing this, a spin allowed transition is promoted which is 1 A 1 → 1 T 1 . [ 3 ] However, the 1 T 1 excited state has a very short lifetime, decreasing the probability for the excited state to relax via a double intersystem crossing to reach the 5 T 2 HS state . [ 3 ] Since the HS state is spin forbidden the lifetime for this state is long, therefore it can be trapped at low temperatures.
Due to the aim to design photoswitchable materials that have higher working temperatures than those reported to date (~80 K), along with long-lifetime photoexcited states, another strategy for SCO called Ligand-Driven Light Induced Spin Change (LD-LISC) has been studied. [ 18 ] This method consists of using a ligand that is photosensitive in order to trigger the spin interconversion of the metal ion and exciting this ligand with light. The LD-LISC effect is followed by a structural change of the photoresponsive ligands in contrast to the SCO process where the structures of the ligands are essentially unaffected. The driving force behind the metal ion SCO in this photochemical transformation is cis-trans photoisomerization . The prerequisite for LD-LISC to be observed is that the two complexes formed with the ligand photoisomers, must exhibit different magnetic behaviors as a function of temperature. Upon successive irradiations of the system at two different wavelengths within a temperature range where the metal ion can either be LS or HS, a spin-state interconversion should occur. In order to achieve this, it is convenient to design a metal environment to where at least one of the complexes exhibits a thermally induced SCO. The LD-LISC has been observed in several iron(II) and iron(III) complexes.
The SCO phenomenon has potential uses as switches, data storage devices, and optical displays. These potential applications would exploit the bistability (HS and LS) which leads to changes in the colour and magnetism of samples. [ 2 ] Molecular switches, like electrical switches, require a mechanism that for turning ON and OFF, as is achieved with the abrupt spin transitions with hysteresis . In order for the size of data storage devices to be reduced while the capacity of them increase, smaller units (such as molecules) that exhibit a bistability and thermal hysteresis are required. [ 2 ] One research goal is to develop new materials where the SCO response time can be decreased from nanoseconds, as we know it, to femtoseconds. One of the advantages of SCO phenomena is the absence of fatigue, because there is an intraelectronic transition instead of an electron displacement through space. | https://en.wikipedia.org/wiki/Spin_crossover |
Spin-density wave (SDW) and charge-density wave (CDW) are names for two similar low-energy ordered states of solids. Both these states occur at low temperature in anisotropic , low-dimensional materials or in metals that have high densities of states at the Fermi level N ( E F ) {\displaystyle N(E_{F})} . Other low-temperature ground states that occur in such materials are superconductivity , ferromagnetism and antiferromagnetism . The transition to the ordered states is driven by the condensation energy which is approximately N ( E F ) Δ 2 {\displaystyle N(E_{F})\Delta ^{2}} where Δ {\displaystyle \Delta } is the magnitude of the energy gap opened by the transition.
Fundamentally SDWs and CDWs involve the development of a superstructure in the form of a periodic modulation in the density of the electronic spins and charges with a characteristic spatial frequency q {\displaystyle q} that does not transform according to the symmetry group that describes the ionic positions.
The new periodicity associated with CDWs can easily be observed using scanning tunneling microscopy or electron diffraction while the more elusive SDWs are typically observed via neutron diffraction or susceptibility measurements. If the new periodicity is a rational fraction or multiple of the lattice constant , the density wave is said to be commensurate ; otherwise the density wave is termed incommensurate .
Some solids with a high N ( E F ) {\displaystyle N(E_{F})} form density waves while others choose a superconducting or magnetic ground state at low temperatures, because of the existence of nesting vectors in the materials' Fermi surfaces . The concept of a nesting vector is illustrated in the Figure for the famous case of chromium , which transitions from a paramagnetic to SDW state at a Néel temperature of 311 K. Cr is a body-centered cubic metal whose Fermi surface features many parallel boundaries between electron pockets centered at Γ {\displaystyle \Gamma } and hole pockets at H. These large parallel regions can be spanned by the nesting wavevector q {\displaystyle q} shown in red. The real-space periodicity of the resulting spin-density wave is given by 2 π / q {\displaystyle 2\pi /q} . The formation of an SDW with a corresponding spatial frequency causes the opening of an energy gap that lowers the system's energy. The existence of the SDW in Cr was first posited in 1960 by Albert Overhauser of Purdue . The theory of CDWs was first put forth by Rudolf Peierls of Oxford University , who was trying to explain superconductivity.
Many low-dimensional solids have anisotropic Fermi surfaces that have prominent nesting vectors. Well-known examples include layered materials like NbSe 3 , [ 1 ] TaSe 2 [ 2 ] and K 0.3 MoO 3 (a Chevrel phase ) [ 3 ] and quasi-1D organic conductors like TMTSF or TTF-TCNQ. [ 4 ] CDWs are also common at the surface of solids where they are more commonly called surface reconstructions or even dimerization. Surfaces so often support CDWs because they can be described by two-dimensional Fermi surfaces like those of layered materials. Chains of Au and In on semiconducting substrates have been shown to exhibit CDWs. [ 5 ] More recently, monatomic chains of Co on a metallic substrate were experimentally shown to exhibit a CDW instability and was attributed to ferromagnetic correlations. [ 6 ]
The most intriguing properties of density waves are their dynamics. Under an appropriate electric field or magnetic field, a density wave will "slide" in the direction indicated by the field due to the electrostatic or magnetostatic force. Typically the sliding will not begin until a "depinning" threshold field is exceeded where the wave can escape from a potential well caused by a defect. The hysteretic motion of density waves is therefore not unlike that of dislocations or magnetic domains . The current-voltage curve of a CDW solid therefore shows a very high electrical resistance up to the depinning voltage, above which it shows a nearly ohmic behavior. Under the depinning voltage (which depends on the purity of the material), the crystal is an insulator . | https://en.wikipedia.org/wiki/Spin_density_wave |
Spin diffusion describes a situation wherein the individual nuclear spins undergo continuous exchange of energy. [ 1 ] This permits polarization differences within the sample to be reduced on a timescale much shorter than relaxation effects. [ 1 ]
Spin diffusion is a process by which magnetization can be exchanged spontaneously between spins. The process is driven by dipolar coupling, and is therefore related to internuclear distances. Spin diffusion has been used to study many structural problems in the past, ranging from domain sizes in polymers and disorder in glassy materials to high-resolution crystal structure determination of small molecules and proteins.
In solid-state nuclear magnetic resonance , spin diffusion plays a major role in Cross Polarization (CP) experiments. As mentioned before, by transferring the magnetization (and thus the population) from nuclei with different values for the spin-lattice relaxation ( T 1 ), the overall time for the experiment is reduced. Is a very common practice when the sample contains hydrogen. Another desirable effect is that the signal to noise ratio (S/N) is increased until a theoretical factor γ A /γ B , being γ the gyromagnetic ratio .
This nuclear magnetic resonance –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin_diffusion |
In magnetic resonance , a spin echo or Hahn echo is the refocusing of spin magnetisation by a pulse of resonant electromagnetic radiation . [ 1 ] Modern nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) make use of this effect.
The NMR signal observed following an initial excitation pulse decays with time due to both spin relaxation and any inhomogeneous effects which cause spins in the sample to precess at different rates. The first of these, relaxation, leads to an irreversible loss of magnetisation. But the inhomogeneous dephasing can be removed by applying a 180° inversion pulse that inverts the magnetisation vectors. [ 2 ] Examples of inhomogeneous effects include a magnetic field gradient and a distribution of chemical shifts . If the inversion pulse is applied after a period t of dephasing, the inhomogeneous evolution will rephase to form an echo at time 2 t . In simple cases, the intensity of the echo relative to the initial signal is given by e –2t/T 2 where T 2 is the time constant for spin–spin relaxation. The echo time ( TE ) is the time between the excitation pulse and the peak of the signal. [ 3 ]
Echo phenomena are important features of coherent spectroscopy which have been used in fields other than magnetic resonance including laser spectroscopy [ 4 ] and neutron scattering .
Echoes were first detected in nuclear magnetic resonance by Erwin Hahn in 1950, [ 5 ] and spin echoes are sometimes referred to as Hahn echoes . In nuclear magnetic resonance and magnetic resonance imaging , radiofrequency radiation is most commonly used.
In 1972 F. Mezei introduced spin-echo neutron scattering, a technique that can be used to study magnons and phonons in single crystals. [ 6 ] The technique is now applied in research facilities using triple axis spectrometers.
In 2020 two teams demonstrated [ 7 ] [ 8 ] that when strongly coupling an ensemble of spins to a resonator, the Hahn pulse sequence does not just lead to a single echo, but rather to a whole train of periodic echoes. In this process the first Hahn echo acts back on the spins as a refocusing pulse, leading to self-stimulated secondary echoes.
The spin-echo effect was discovered by Erwin Hahn when he applied two successive 90° pulses separated by short time period, but detected a signal, the echo, when no pulse was applied. This phenomenon of spin echo was explained by Erwin Hahn in his 1950 paper, [ 5 ] and further developed by Carr and Purcell who pointed out the advantages of using a 180° refocusing pulse for the second pulse. [ 9 ] The pulse sequence may be better understood by breaking it down into the following steps:
Several simplifications are used in this sequence: no decoherence is included and each spin experiences perfect pulses during which the environment provides no spreading. Six spins are shown above and these are not given the chance to dephase significantly. The spin-echo technique is more useful when the spins have dephased more significantly such as in the animation below:
A Hahn-echo decay experiment can be used to measure the spin–spin relaxation time, as shown in the animation below. The size of the echo is recorded for different spacings of the two pulses. This reveals the decoherence which is not refocused by the π pulse. In simple cases, an exponential decay is measured which is described by the T 2 time.
Hahn's 1950 paper [ 5 ] showed that another method for generating spin echoes is to apply three successive 90° pulses. After the first 90° pulse, the magnetization vector spreads out as described above, forming what can be thought of as a "pancake" in the x-y plane. The spreading continues for a time τ {\displaystyle \tau } , and then a second 90° pulse is applied such that the "pancake" is now in the x-z plane. After a further time T {\displaystyle T} a third pulse is applied and a stimulated echo is observed after waiting for a time τ {\displaystyle \tau } after the last pulse.
Hahn echos have also been observed at optical frequencies. [ 4 ] For this, resonant light is applied to a material with an inhomogeneously broadened absorption resonance. Instead of using two spin states in a magnetic field, photon echoes use two energy levels that are present in the material even in zero magnetic field.
Fast spin echo (RARE, FAISE or FSE [ 10 ] [ 11 ] [ 12 ] ), also called turbo spin echo (TSE) is an MRI sequence that results in fast scan times. In this sequence, several 180 refocusing radio-frequency pulses are delivered during each echo time (TR) interval, and the phase-encoding gradient is briefly switched on between echoes. [ 13 ] The FSE/TSE pulse sequence superficially resembles a conventional spin-echo (CSE) sequence in that it uses a series of 180º-refocusing pulses after a single 90º-pulse to generate a train of echoes. The FSE/TSE technique, however, changes the phase-encoding gradient for each of these echoes (a conventional multi-echo sequence collects all echoes in a train with the same phase encoding). As a result of changing the phase-encoding gradient between echoes, multiple lines of k-space (i.e., phase-encoding steps) can be acquired within a given repetition time (TR). As multiple phase-encoding lines are acquired during each TR interval, FSE/TSE techniques may significantly reduce imaging time. [ 14 ] | https://en.wikipedia.org/wiki/Spin_echo |
Spin echo small angle neutron scattering (SESANS) measures structures from around 20 to 2000 nm in size. The information is presented as a real-space (similar to g(r)) as opposed to a reciprocal space (q(r)) mapping. This can simplify the interpretation for some systems. [ 1 ] SESANS is useful for studying processes that occur over relatively long time scales, as data collection is often slow, but large length scales. Aggregation of colloids, [ 2 ] block copolymer micelles, [ 3 ] Stöber silica particles [ 4 ] being a prime examples.
The technique offers some advantages over SANS [ 5 ] but there are fewer SESANS instruments available than SANS instruments. Facilities for SESANS exist at TUDelft (Netherlands) [ 6 ] and Rutherford Appleton Laboratory (UK). [ 7 ]
This article about materials science is a stub . You can help Wikipedia by expanding it .
This article about analytical chemistry is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin_echo_small_angle_neutron_scattering |
Spin engineering describes the control and manipulation of quantum spin systems to develop devices and materials. This includes the use of the spin degrees of freedom as a probe for spin based phenomena.
Because of the basic importance of quantum spin for physical and chemical processes, spin engineering is relevant for a wide range of scientific and technological applications. Current examples range from Bose–Einstein condensation to spin-based data storage and reading in state-of-the-art hard disk drives, as well as from powerful analytical tools like nuclear magnetic resonance spectroscopy and electron paramagnetic resonance spectroscopy to the development of magnetic molecules as qubits and magnetic nanoparticles . In addition, spin engineering exploits the functionality of spin to design materials with novel properties as well as to provide a better understanding and advanced applications of conventional material systems. Many chemical reactions are devised to create bulk materials or single molecules with well defined spin properties, such as a single-molecule magnet .
The aim of this article is to provide an outline of fields of research and development where the focus is on the properties and applications of quantum spin.
As spin is one of the fundamental quantum properties of elementary particles it is relevant for a large range of physical and chemical phenomena. For instance, the spin of the electron plays a key role in the electron configuration of atoms which is the basis of the periodic table of elements. The origin of ferromagnetism is also closely related to the magnetic moment associated with the spin and the spin-dependent Pauli exclusion principle . Thus, the engineering of ferromagnetic materials like mu-metals or Alnico at the beginning of the last century can be considered as early examples of spin engineering, although the concept of spin was not yet known at that time. Spin engineering in its generic sense became possible only after the first experimental characterization of spin in the Stern–Gerlach experiment in 1922 followed by the development of relativistic quantum mechanics by Paul Dirac. This theory was the first to accommodate the spin of the electron and its magnetic moment.
Whereas the physics of spin engineering dates back to the groundbreaking findings of quantum chemistry and physics within the first decades of the 20th century, the chemical aspects of spin engineering have received attention especially within the last twenty years. Today, researchers focus on specialized topics, such as the design and synthesis of molecular magnets or other model systems in order to understand and harness the fundamental principles behind phenomena such as the relation between magnetism and chemical reactivity as well as microstructure related mechanical properties of metals and the biochemical impact of spin (e. g. photoreceptor proteins ) and spin transport.
Spintronics is the exploitation of both the intrinsic spin of the electron and its fundamental electronic charge in solid-state devices and is thus a part of spin engineering. Spintronics is probably one of the most advanced fields of spin engineering with many important inventions which can be found in end-user devices like the reading heads for magnetic hard disk drives. This section is divided in basic spintronic phenomena and their applications.
this section is devoted to current and possible future applications of spintronics which make use of one or the combination of several basic spintronic phenomena:
Materials which properties are determined or strongly influenced by quantum spin:
methods to characterize materials and physical or chemical processes via spin based phenomena: | https://en.wikipedia.org/wiki/Spin_engineering |
In physics , the topological structure of spinfoam or spin foam [ 1 ] consists of two-dimensional faces representing a configuration required by functional integration to obtain a Feynman's path integral description of quantum gravity . These structures are employed in loop quantum gravity as a version of quantum foam .
The covariant formulation of loop quantum gravity provides the best formulation of the dynamics of the theory of quantum gravity – a quantum field theory where the invariance under diffeomorphisms of general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam. [ how? ]
A spin network is a two-dimensional graph , together with labels on its vertices and edges which encode aspects of a spatial geometry.
A spin network is defined as a diagram like the Feynman diagram which makes a basis of connections between the elements of a differentiable manifold for the Hilbert spaces defined over them, and for computations of amplitudes between two different hypersurfaces of the manifold . Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network. [ clarification needed ] A spin foam is analogous to quantum history . [ why? ]
Spin networks provide a language to describe the quantum geometry of space. Spin foam does the same job for spacetime.
Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In topology this sort of space is called a 2- complex . A spin foam is a particular type of 2- complex , with labels for vertices , edges and faces . The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold.
In loop quantum gravity, the present spin foam theory has been inspired by the work of Ponzano–Regge model . The idea was introduced by Reisenberger and Rovelli in 1997, [ 2 ] and later developed into the Barrett–Crane model . The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers, [ 3 ] but the theory has also seen fundamental contributions from the work of many others, such as Laurent Freidel (FK model) and Jerzy Lewandowski (KKL model).
The summary partition function for a spin foam model is
Z := ∑ Γ w ( Γ ) [ ∑ j f , i e ∏ f A f ( j f ) ∏ e A e ( j f , i e ) ∏ v A v ( j f , i e ) ] {\displaystyle Z:=\sum _{\Gamma }w(\Gamma )\left[\sum _{j_{f},i_{e}}\prod _{f}A_{f}(j_{f})\prod _{e}A_{e}(j_{f},i_{e})\prod _{v}A_{v}(j_{f},i_{e})\right]}
with: | https://en.wikipedia.org/wiki/Spin_foam |
Spin gapless semiconductors are materials in which the electronic band structure has no band gap for one spin channel while but a finite gap for the other. [ 1 ]
In a spin gapless semiconductor, conduction and valence band edges touch, so that no threshold energy is required to move electrons from occupied (valence) states to empty (conduction) states. This gives spin-gapless semiconductors unique properties: namely that their band structures are extremely sensitive to external influences (e.g., pressure or magnetic field). [ 2 ]
Because very little energy is needed to excite electrons in an SGS, charge concentrations are very easily ‘tuneable’. For example, this can be done by introducing a new element (doping) or by application of a magnetic or electric field (gating).
A new type of SGS identified in 2017, known as Dirac-type linear spin-gapless semiconductors, has linear dispersion and is considered an ideal platform for massless and dissipationless spintronics because spin-orbital coupling opens a gap for the spin fully polarized conduction and valence band, and as a result, the interior of the sample becomes an insulator, however, an electrical current can flow without resistance at the sample edge. This effect, the quantum anomalous Hall effect has only previously been realised in magnetically doped topological insulators. [ 2 ]
As well as Dirac/linear SGSs, the other major category of SGS are parabolic spin gapless semiconductors. [ 3 ] [ 4 ]
Electron mobility in such materials is two to four orders of magnitude higher than in classical semiconductors. [ 5 ]
A convergence of topology and magnetism known as Chern magnetism makes SGSs ideal candidate materials for realizing room-temperature quantum anomalous Hall effect (QAHE). [ 6 ]
SGSs are topologically non-trivial . [ 3 ]
The spin gapless semiconductor was first proposed as a new spintronics concept and a new class of candidate spintronic materials in 2008 in a paper by Xiaolin Wang of the University of Wollongong in Australia. [ 7 ] [ 8 ] [ 9 ]
The dependence of bandgap on spin direction leads to high carrier-spin-polarization, and offers promising spin-controlled electronic and magnetic properties for spintronics applications. [ 10 ]
The spin gapless semiconductor is a promising candidate material for spintronics because its charged particles can be fully spin-polarised, so that spin can be controlled via only a small applied external energy. [ 2 ]
This condensed matter physics -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin_gapless_semiconductor |
In condensed matter physics , a spin glass is a magnetic state characterized by randomness, besides cooperative behavior in freezing of spins at a temperature called the "freezing temperature," T f . [ 1 ] In ferromagnetic solids, component atoms' magnetic spins all align in the same direction. Spin glass when contrasted with a ferromagnet is defined as " disordered " magnetic state in which spins are aligned randomly or without a regular pattern and the couplings too are random. [ 1 ] A spin glass should not be confused with a " spin-on glass ". The latter is a thin film, usually based on SiO 2 , which is applied via spin coating .
The term "glass" comes from an analogy between the magnetic disorder in a spin glass and the positional disorder of a conventional, chemical glass , e.g., a window glass. In window glass or any amorphous solid the atomic bond structure is highly irregular; in contrast, a crystal has a uniform pattern of atomic bonds. In ferromagnetic solids, magnetic spins all align in the same direction; this is analogous to a crystal's lattice-based structure .
The individual atomic bonds in a spin glass are a mixture of roughly equal numbers of ferromagnetic bonds (where neighbors have the same orientation) and antiferromagnetic bonds (where neighbors have exactly the opposite orientation: north and south poles are flipped 180 degrees). These patterns of aligned and misaligned atomic magnets create what are known as frustrated interactions – distortions in the geometry of atomic bonds compared to what would be seen in a regular, fully aligned solid. They may also create situations where more than one geometric arrangement of atoms is stable.
There are two main aspects of spin glass. On the physical side, spin glasses are real materials with distinctive properties, a review of which was published in 1982. [ 2 ] On the mathematical side, simple statistical mechanics models, inspired by real spin glasses, are widely studied and applied. [ 3 ]
Spin glasses and the complex internal structures that arise within them are termed " metastable " because they are "stuck" in stable configurations other than the lowest-energy configuration (which would be aligned and ferromagnetic). The mathematical complexity of these structures is difficult but fruitful to study experimentally or in simulations ; with applications to physics, chemistry, materials science and artificial neural networks in computer science .
It is the time dependence which distinguishes spin glasses from other magnetic systems.
Above the spin glass transition temperature , T c , [ note 1 ] the spin glass exhibits typical magnetic behaviour (such as paramagnetism ).
If a magnetic field is applied as the sample is cooled to the transition temperature, magnetization of the sample increases as described by the Curie law . Upon reaching T c , the sample becomes a spin glass, and further cooling results in little change in magnetization. This is referred to as the field-cooled magnetization.
When the external magnetic field is removed, the magnetization of the spin glass falls rapidly to a lower value known as the remanent magnetization.
Magnetization then decays slowly as it approaches zero (or some small fraction of the original value – this remains unknown ). This decay is non-exponential , and no simple function can fit the curve of magnetization versus time adequately. [ 4 ] This slow decay is particular to spin glasses. Experimental measurements on the order of days have shown continual changes above the noise level of instrumentation. [ 4 ]
Spin glasses differ from ferromagnetic materials by the fact that after the external magnetic field is removed from a ferromagnetic substance, the magnetization remains indefinitely at the remanent value. Paramagnetic materials differ from spin glasses by the fact that, after the external magnetic field is removed, the magnetization rapidly falls to zero, with no remanent magnetization. The decay is rapid and exponential. [ citation needed ]
If the sample is cooled below T c in the absence of an external magnetic field, and a magnetic field is applied after the transition to the spin glass phase, there is a rapid initial increase to a value called the zero-field-cooled magnetization. A slow upward drift then occurs toward the field-cooled magnetization.
Surprisingly, the sum of the two complicated functions of time (the zero-field-cooled and remanent magnetizations) is a constant, namely the field-cooled value, and thus both share identical functional forms with time, [ 5 ] at least in the limit of very small external fields.
This is similar to the Ising model . In this model, we have spins arranged on a d {\displaystyle d} -dimensional lattice with only nearest neighbor interactions. This model can be solved exactly for the critical temperatures and a glassy phase is observed to exist at low temperatures. [ 6 ] The Hamiltonian for this spin system is given by:
where S i {\displaystyle S_{i}} refers to the Pauli spin matrix for the spin-half particle at lattice point i {\displaystyle i} , and the sum over ⟨ i j ⟩ {\displaystyle \langle ij\rangle } refers to summing over neighboring lattice points i {\displaystyle i} and j {\displaystyle j} . A negative value of J i j {\displaystyle J_{ij}} denotes an antiferromagnetic type interaction between spins at points i {\displaystyle i} and j {\displaystyle j} . The sum runs over all nearest neighbor positions on a lattice, of any dimension. The variables J i j {\displaystyle J_{ij}} representing the magnetic nature of the spin-spin interactions are called bond or link variables.
In order to determine the partition function for this system, one needs to average the free energy f [ J i j ] = − 1 β ln Z [ J i j ] {\displaystyle f\left[J_{ij}\right]=-{\frac {1}{\beta }}\ln {\mathcal {Z}}\left[J_{ij}\right]} where Z [ J i j ] = Tr S ( e − β H ) {\displaystyle {\mathcal {Z}}\left[J_{ij}\right]=\operatorname {Tr} _{S}\left(e^{-\beta H}\right)} , over all possible values of J i j {\displaystyle J_{ij}} . The distribution of values of J i j {\displaystyle J_{ij}} is taken to be a Gaussian with a mean J 0 {\displaystyle J_{0}} and a variance J 2 {\displaystyle J^{2}} :
Solving for the free energy using the replica method , below a certain temperature, a new magnetic phase called the spin glass phase (or glassy phase) of the system is found to exist which is characterized by a vanishing magnetization m = 0 {\displaystyle m=0} along with a non-vanishing value of the two point correlation function between spins at the same lattice point but at two different replicas:
where α , β {\displaystyle \alpha ,\beta } are replica indices. The order parameter for the ferromagnetic to spin glass phase transition is therefore q {\displaystyle q} , and that for paramagnetic to spin glass is again q {\displaystyle q} . Hence the new set of order parameters describing the three magnetic phases consists of both m {\displaystyle m} and q {\displaystyle q} .
Under the assumption of replica symmetry, the mean-field free energy is given by the expression: [ 6 ]
In addition to unusual experimental properties, spin glasses are the subject of extensive theoretical and computational investigations. A substantial part of early theoretical work on spin glasses dealt with a form of mean-field theory based on a set of replicas of the partition function of the system.
An important, exactly solvable model of a spin glass was introduced by David Sherrington and Scott Kirkpatrick in 1975. It is an Ising model with long range frustrated ferro- as well as antiferromagnetic couplings. It corresponds to a mean-field approximation of spin glasses describing the slow dynamics of the magnetization and the complex non-ergodic equilibrium state.
Unlike the Edwards–Anderson (EA) model, in the system though only two-spin interactions are considered, the range of each interaction can be potentially infinite (of the order of the size of the lattice). Therefore, we see that any two spins can be linked with a ferromagnetic or an antiferromagnetic bond and the distribution of these is given exactly as in the case of Edwards–Anderson model. The Hamiltonian for SK model is very similar to the EA model:
where J i j , S i , S j {\displaystyle J_{ij},S_{i},S_{j}} have same meanings as in the EA model. The equilibrium solution of the model, after some initial attempts by Sherrington, Kirkpatrick and others, was found by Giorgio Parisi in 1979 with the replica method. The subsequent work of interpretation of the Parisi solution—by M. Mezard , G. Parisi , M.A. Virasoro and many others—revealed the complex nature of a glassy low temperature phase characterized by ergodicity breaking, ultrametricity and non-selfaverageness. Further developments led to the creation of the cavity method , which allowed study of the low temperature phase without replicas. A rigorous proof of the Parisi solution has been provided in the work of Francesco Guerra and Michel Talagrand . [ 7 ]
When there is a uniform external magnetic field of magnitude M {\displaystyle M} , the energy function becomes H = − 1 N ∑ i < j J i j S i S j − M ∑ i S i {\displaystyle H=-{\frac {1}{\sqrt {N}}}\sum _{i<j}J_{ij}S_{i}S_{j}-M\sum _{i}S_{i}} Let all couplings J i j {\displaystyle J_{ij}} are IID samples from the gaussian distribution of mean 0 and variance J 2 {\displaystyle J^{2}} . In 1979, J.R.L. de Almeida and David Thouless [ 8 ] found that, as in the case of the Ising model, the mean-field solution to the SK model becomes unstable when under low-temperature, low-magnetic field state.
The stability region on the phase diagram of the SK model is determined by two dimensionless parameters x := k T J , y := M J {\displaystyle x:={\frac {kT}{J}},\quad y:={\frac {M}{J}}} . Its phase diagram has two parts, divided by the de Almeida-Thouless curve , The curve is the solution set to the equations [ 8 ] x 2 = 1 ( 2 π ) 1 / 2 ∫ d z e − 1 2 z 2 sech 4 ( q 1 / 2 z + y x ) , q = 1 ( 2 π ) 1 / 2 ∫ d z e − 1 2 z 2 tanh 2 ( q 1 / 2 z + y x ) . {\displaystyle {\begin{aligned}&x^{2}={\frac {1}{(2\pi )^{1/2}}}\int \mathrm {d} z\;\mathrm {e} ^{-{\frac {1}{2}}z^{2}}\operatorname {sech} ^{4}\left({\frac {q^{1/2}z+y}{x}}\right),\\&q={\frac {1}{(2\pi )^{1/2}}}\int \mathrm {d} z\;\mathrm {e} ^{-{\frac {1}{2}}z^{2}}\tanh ^{2}\left({\frac {q^{1/2}z+y}{x}}\right).\end{aligned}}} The phase transition occurs at x = 1 {\displaystyle x=1} . Just below it, we have y 2 ≈ 4 3 ( 1 − x ) 3 . {\displaystyle y^{2}\approx {\frac {4}{3}}(1-x)^{3}.} At low temperature, high magnetic field limit, the line is x ≈ 4 3 2 π e − 1 2 y 2 {\displaystyle x\approx {\frac {4}{3{\sqrt {2\pi }}}}e^{-{\frac {1}{2}}y^{2}}}
This is also called the "p-spin model". [ 3 ] The infinite-range model is a generalization of the Sherrington–Kirkpatrick model where we not only consider two-spin interactions but p {\displaystyle p} -spin interactions, where p ≤ N {\displaystyle p\leq N} and N {\displaystyle N} is the total number of spins. Unlike the Edwards–Anderson model, but similar to the SK model, the interaction range is infinite. The Hamiltonian for this model is described by:
where J i 1 … i p , S i 1 , … , S i p {\displaystyle J_{i_{1}\dots i_{p}},S_{i_{1}},\dots ,S_{i_{p}}} have similar meanings as in the EA model. The p → ∞ {\displaystyle p\to \infty } limit of this model is known as the random energy model . In this limit, the probability of the spin glass existing in a particular state depends only on the energy of that state and not on the individual spin configurations in it.
A Gaussian distribution of magnetic bonds across the lattice is assumed usually to solve this model. Any other distribution is expected to give the same result, as a consequence of the central limit theorem . The Gaussian distribution function, with mean J 0 N {\displaystyle {\frac {J_{0}}{N}}} and variance J 2 N {\displaystyle {\frac {J^{2}}{N}}} , is given as:
The order parameters for this system are given by the magnetization m {\displaystyle m} and the two point spin correlation between spins at the same site q {\displaystyle q} , in two different replicas, which are the same as for the SK model. This infinite range model can be solved explicitly for the free energy [ 6 ] in terms of m {\displaystyle m} and q {\displaystyle q} , under the assumption of replica symmetry as well as 1-Replica Symmetry Breaking. [ 6 ]
A thermodynamic system is ergodic when, given any (equilibrium) instance of the system, it eventually visits every other possible (equilibrium) state (of the same energy). One characteristic of spin glass systems is that, below the freezing temperature T f {\displaystyle T_{\text{f}}} , instances are trapped in a "non-ergodic" set of states: the system may fluctuate between several states, but cannot transition to other states of equivalent energy. Intuitively, one can say that the system cannot escape from deep minima of the hierarchically disordered energy landscape ; the distances between minima are given by an ultrametric , with tall energy barriers between minima. [ note 2 ] The participation ratio counts the number of states that are accessible from a given instance, that is, the number of states that participate in the ground state . The ergodic aspect of spin glass was instrumental in the awarding of half the 2021 Nobel Prize in Physics to Giorgio Parisi . [ 9 ] [ 10 ] [ 11 ]
For physical systems, such as dilute manganese in copper, the freezing temperature is typically as low as 30 kelvins (−240 °C), and so the spin-glass magnetism appears to be practically without applications in daily life. The non-ergodic states and rugged energy landscapes are, however, quite useful in understanding the behavior of certain neural networks , including Hopfield networks , as well as many problems in computer science optimization and genetics .
Elemental crystalline neodymium is paramagnetic at room temperature and becomes an antiferromagnet with incommensurate order upon cooling below 19.9 K. [ 12 ] Below this transition temperature it exhibits a complex set of magnetic phases [ 13 ] [ 14 ] that have long spin relaxation times and spin-glass behavior that does not rely on structural disorder. [ 15 ]
A detailed account of the history of spin glasses from the early 1960s to the late 1980s can be found in a series of popular articles by Philip W. Anderson in Physics Today . [ 16 ] [ 17 ] [ 18 ] [ 19 ] [ 20 ] [ 21 ] [ 22 ] [ 23 ]
In 1930s, material scientists discovered the Kondo effect , where the resistivity of nominally pure gold reaches a minimum at 10 K, and similarly for nominally pure Cu at 2 K. It was later understood that the Kondo effect occurs when a nonmagnetic metal contains a very small fraction of magnetic atoms (i.e., at high dilution).
Unusual behavior was observed in iron-in-gold alloy (Au Fe ) and manganese-in-copper alloy (Cu Mn ) at around 1 to 10 atom percent . Cannella and Mydosh observed in 1972 [ 24 ] that Au Fe had an unexpected cusplike peak in the a.c. susceptibility at a well defined temperature, which would later be termed spin glass freezing temperature . [ 25 ]
It was also called "mictomagnet" (micto- is Greek for "mixed"). The term arose from the observation that these materials often contain a mix of ferromagnetic ( J > 0 {\displaystyle J>0} ) and antiferromagnetic ( J < 0 {\displaystyle J<0} ) interactions, leading to their disordered magnetic structure. This term fell out of favor as the theoretical understanding of spin glasses evolved, recognizing that the magnetic frustration arises not just from a simple mixture of ferro- and antiferromagnetic interactions, but from their randomness and frustration in the system.
Sherrington and Kirkpatrick proposed the SK model in 1975, and solved it by the replica method. [ 26 ] They discovered that at low temperatures, its entropy becomes negative, which they thought was because the replica method is a heuristic method that does not apply at low temperatures.
It was then discovered that the replica method was correct, but the problem lies in that the low-temperature broken symmetry in the SK model cannot be purely characterized by the Edwards-Anderson order parameter. Instead, further order parameters are necessary, which leads to replica breaking ansatz of Giorgio Parisi . At the full replica breaking ansatz, infinitely many order parameters are required to characterize a stable solution. [ 27 ]
The formalism of replica mean-field theory has also been applied in the study of neural networks , where it has enabled calculations of properties such as the storage capacity of simple neural network architectures without requiring a training algorithm (such as backpropagation ) to be designed or implemented. [ 28 ]
More realistic spin glass models with short range frustrated interactions and disorder, like the Gaussian model where the couplings between neighboring spins follow a Gaussian distribution , have been studied extensively as well, especially using Monte Carlo simulations . These models display spin glass phases bordered by sharp phase transitions .
Besides its relevance in condensed matter physics, spin glass theory has acquired a strongly interdisciplinary character, with applications to neural network theory, computer science, theoretical biology, econophysics etc.
Spin glass models were adapted to the folding funnel model of protein folding . | https://en.wikipedia.org/wiki/Spin_glass |
A spin ice is a magnetic substance that does not have a single minimal-energy state . It has magnetic moments (i.e. "spin" ) as elementary degrees of freedom which are subject to frustrated interactions . By their nature, these interactions prevent the moments from exhibiting a periodic pattern in their orientation down to a temperature much below the energy scale set by the said interactions. Spin ices show low-temperature properties, residual entropy in particular, closely related to those of common crystalline water ice . [ 1 ] The most prominent compounds with such properties are dysprosium titanate (Dy 2 Ti 2 O 7 ) and holmium titanate (Ho 2 Ti 2 O 7 ). The orientation of the magnetic moments in spin ice resembles the positional organization of hydrogen atoms (more accurately, ionized hydrogen, or protons ) in conventional water ice (see figure 1).
Experiments have found evidence for the existence of deconfined magnetic monopoles in these materials, [ 2 ] [ 3 ] [ 4 ] with properties resembling those of the hypothetical magnetic monopoles postulated to exist in vacuum.
In 1935, Linus Pauling noted that the hydrogen atoms in water ice would be expected to remain disordered even at absolute zero . That is, even upon cooling to zero temperature , water ice is expected to have residual entropy , i.e. , intrinsic randomness. This is due to the fact that the hexagonal crystalline structure of common water ice contains oxygen atoms with four neighboring hydrogen atoms. In ice, for each oxygen atom, two of the neighboring hydrogen atoms are near (forming the traditional H 2 O molecule ), and two are further away (being the hydrogen atoms of two neighboring water molecules). Pauling noted that the number of configurations conforming to this "two-near, two-far" ice rule grows exponentially with the system size, and, therefore, that the zero-temperature entropy of ice was expected to be extensive . [ 5 ] Pauling's findings were confirmed by specific heat measurements, though pure crystals of water ice are particularly hard to create.
Spin ices are materials that consist of regular corner-linked tetrahedra of magnetic ions , each of which has a non-zero magnetic moment , often abridged to " spin ", which must satisfy in their low-energy state a "two-in, two-out" rule on each tetrahedron making the crystalline structure (see figure 2). This is highly analogous to the two-near, two far rule in water ice (see figure 1). Just as Pauling showed that the ice rule leads to an extensive entropy in water ice, so does the two-in, two-out rule in the spin ice systems – these exhibit the same residual entropy properties as water ice. Be that as it may, depending on the specific spin ice material, it is generally much easier to create large single crystals of spin ice materials than water ice crystals. Additionally, the ease to induce interaction of the magnetic moments with an external magnetic field in a spin ice system makes the spin ices more suitable than water ice for exploring how the residual entropy can be affected by external influences.
While Philip Anderson had already noted in 1956 [ 6 ] the connection between the problem of the frustrated Ising antiferromagnet on a ( pyrochlore ) lattice of corner-shared tetrahedra and Pauling's water ice problem, real spin ice materials were only discovered forty years later. [ 7 ] The first materials identified as spin ices were the pyrochlores Dy 2 Ti 2 O 7 ( dysprosium titanate ), Ho 2 Ti 2 O 7 (holmium titanate). In addition, compelling evidence has been reported that Dy 2 Sn 2 O 7 ( dysprosium stannate ) and Ho 2 Sn 2 O 7 ( holmium stannate ) are spin ices. [ 8 ] These four compounds belong to the family of rare-earth pyrochlore oxides. CdEr 2 Se 4 , a spinel in which the magnetic Er 3+ ions sit on corner-linked tetrahedra, also displays spin ice behavior. [ 9 ]
Spin ice materials are characterized by a random disorder in the orientation of the moment of the magnetic ions , even when the material is at very low temperatures . Alternating current (AC) magnetic susceptibility measurements find evidence for a dynamic freezing of the magnetic moments as the temperature is lowered somewhat below the temperature at which the specific heat displays a maximum. The broad maximum in the heat capacity does not correspond to a phase transition. Rather, the temperature at which the maximum occurs, about 1 K in Dy 2 Ti 2 O 7 , signals a rapid change in the number of tetrahedra where the two-in, two-out rule is violated. Tetrahedra where the rule is violated are sites where the aforementioned monopoles reside. Mathematically, spin ice configurations can be described by closed Eulerian paths . [ 10 ] [ 11 ]
Spin ices are geometrically frustrated magnetic systems. While frustration is usually associated with triangular or tetrahedral arrangements of magnetic moments coupled via antiferromagnetic exchange interactions, as in Anderson's Ising model, [ 6 ] spin ices are frustrated ferromagnets. It is the very strong local magnetic anisotropy from the crystal field forcing the magnetic moments to point either in or out of a tetrahedron that renders ferromagnetic interactions frustrated in spin ices. Most importantly, it is the long-range magnetostatic dipole–dipole interaction, and not the nearest-neighbor exchange, that causes the frustration and the consequential two-in, two-out rule that leads to the spin ice phenomenology. [ 12 ] [ 13 ]
For a tetrahedron in a two-in, two-out state, the magnetization field is divergent-free ; there is as much "magnetization intensity" entering a tetrahedron as there is leaving (see figure 3). In such a divergent-free situation, there exists no source or sink for the field. According to Gauss' theorem (also known as Ostrogradsky's theorem), a nonzero divergence of a field is caused, and can be characterized, by a real number called "charge" . In the context of spin ice, such charges characterizing the violation of the two-in, two-out magnetic moment orientation rule are the aforementioned monopoles. [ 2 ] [ 3 ] [ 4 ]
In Autumn 2009, researchers reported experimental observation of low-energy quasiparticles resembling the predicted monopoles in spin ice. [ 2 ] A single crystal of the dysprosium titanate spin ice candidate was examined in the temperature range of 0.6–2.0 K. Using neutron scattering , the magnetic moments were shown to align in the spin ice material into interwoven tube-like bundles resembling Dirac strings . At the defect formed by the end of each tube, the magnetic field looks like that of a monopole. Using an applied magnetic field, the researchers were able to control the density and orientation of these strings. A description of the heat capacity of the material in terms of an effective gas of these quasiparticles was also presented. [ 14 ] [ 15 ]
The effective charge of a magnetic monopole, Q (see figure 3 ) in both the dysprosium and holmium titanate spin ice compounds is approximately Q = 5 μ B Å −1 ( Bohr magnetons per angstrom ). [ 2 ] The elementary magnetic constituents of spin ice are magnetic dipoles, so the emergence of monopoles is an example of the phenomenon of fractionalization .
The microscopic origin of the atomic magnetic moments in magnetic materials is quantum mechanical; the Planck constant enters explicitly in the equation defining the magnetic moment of an electron , along with its charge and its mass. Yet, the magnetic moments in the dysprosium titanate and the holmium titanate spin ice materials are effectively described by classical statistical mechanics , and not quantum statistical mechanics, over the experimentally relevant and reasonably accessible temperature range (between 0.05 K and 2 K) where the spin ice phenomena manifest themselves. Although the weakness of quantum effects in these two compounds is rather unusual, it is believed to be understood. [ 16 ] There is current interest in the search of quantum spin ices, [ 17 ] materials in which the laws of quantum mechanics now become needed to describe the behavior of the magnetic moments. Magnetic ions other than dysprosium (Dy) and holmium (Ho) are required to generate a quantum spin ice, with praseodymium (Pr), terbium (Tb) and ytterbium (Yb) being possible candidates. [ 17 ] [ 18 ] One reason for the interest in quantum spin ice is the belief that these systems may harbor a quantum spin liquid , [ 19 ] a state of matter where magnetic moments continue to wiggle (fluctuate) down to absolute zero temperature. The theory [ 20 ] describing the low-temperature and low-energy properties of quantum spin ice is akin to that of vacuum quantum electrodynamics , or QED. This constitutes an example of the idea of emergence . [ 21 ]
Artificial spin ices are metamaterials consisting of coupled nanomagnets arranged on periodic and aperiodic lattices. [ 22 ] These systems have enabled the experimental investigation of a variety of phenomena such as frustration, emergent magnetic monopoles, and phase transitions. In addition, artificial spin ices show potential as reprogrammable magnonic crystals and have been studied for their fast dynamics. A variety of geometries have been explored, including quasicrystalline systems and 3D structures, as well as different magnetic materials to modify anisotropies and blocking temperatures.
For example, polymer magnetic composites comprising 2D lattices of droplets of solid-liquid phase change material, with each droplet containing a single magnetic dipole particle, form an artificial spin ice above the droplet melting point, and, after cooling, a spin glass state with low bulk remanence. Spontaneous emergence of 2D magnetic vortices was observed in such spin ices, which vortex geometries were correlated with the external bulk remanence. [ 23 ]
Future work in this field includes further developments in fabrication and characterization methods, exploration of new geometries and material combinations, and potential applications in computation, [ 24 ] data storage, and reconfigurable microwave circuits. [ 25 ] In 2021 a study demonstrated neuromorphic reservoir computing using artificial spin ice, solving a range of computational tasks using the complex magnetic dynamics of the artificial spin ice. [ 26 ] In 2022, another studied achieved an artificial kagome spin ice which could potentially be used in the future for novel high-speed computers with low power consumption. [ 27 ] | https://en.wikipedia.org/wiki/Spin_ice |
A spin label (SL) is an organic molecule which possesses an unpaired electron , usually on a nitrogen atom, and the ability to bind to another molecule. Spin labels are normally used as tools for probing proteins or biological membrane -local dynamics using electron paramagnetic resonance spectroscopy . The site-directed spin labeling (SDSL) technique allows one to monitor a specific region within a protein. In protein structure examinations, amino acid -specific SLs can be used.
The goal of spin labeling is somewhat similar to that of isotopic substitution in NMR spectroscopy . There one replaces an atom lacking a nuclear spin (and so is NMR-silent) with an isotope having a spin I ≠ 0 (and so is NMR-active). This technique is useful for tracking the chemical environment around an atom when full substitution with an NMR-active isotope is not feasible. Recently, spin-labelling has also been used to probe chemical local environment in NMR itself, in a technique known as Paramagnetic Relaxation Enhancement (PRE).
Recent developments in the theory and experimental measurement of PREs have enabled the detection, characterization and visualization of sparsely populated states of proteins and their complexes. [ 1 ] Such states, which are invisible to conventional biophysical and structural techniques, play a key role in many biological processes including molecular recognition, allostery, macromolecular assembly and aggregation.
Spin labelled fatty acids have been extensively used to understand dynamic organization of lipids in bio-membranes and membrane biophysics . For example, stearic acid labelled with aminoxyl spin label moiety at various carbons (5, 7, 9, 12, 13, 14 and 16) with respect to first carbon of carbonyl group have been used to study the flexibility gradient of membrane lipids to understand membrane fluidity conditions at different depths of their lipid bilayer organization. [ 2 ] | https://en.wikipedia.org/wiki/Spin_label |
A spin model is a mathematical model used in physics primarily to explain magnetism . Spin models may either be classical or quantum mechanical in nature. Spin models have been studied in quantum field theory as examples of integrable models . Spin models are also used in quantum information theory and computability theory in theoretical computer science . The theory of spin models is a far reaching and unifying topic that cuts across many fields.
In ordinary materials, the magnetic dipole moments of individual atoms produce magnetic fields that cancel one another, because each dipole points in a random direction. Ferromagnetic materials below their Curie temperature , however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.
The study of the behavior of such "spin models" is a thriving area of research in condensed matter physics . For instance, the Ising model describes spins (dipoles) that have only two possible states, up and down, whereas in the Heisenberg model the spin vector is allowed to point in any direction. In certain magnets, the magnetic dipoles are only free to rotate in a 2D plane, a system which can be adequately described by the so-called xy-model .
The lack of a unified theory of magnetism [ 1 ] forces scientist to model magnetic systems theoretically with one, or a combination of these spin models in order to understand the intricate behavior of atomic magnetic interactions. Numerical implementation of these models has led to several interesting results, such as quantitative research in the theory of phase transitions .
A quantum spin model is a quantum Hamiltonian model that describes a system which consists of spins either interacting or not and are an active area of research in the fields of strongly correlated electron systems, quantum information theory , and quantum computing . [ 2 ] The physical observables in these quantum models are actually operators in a Hilbert space acting on state vectors as opposed to the physical observables in the corresponding classical spin models - like the Ising model - which are commutative variables. | https://en.wikipedia.org/wiki/Spin_model |
In particle physics , spin polarization is the degree to which the spin , i.e., the intrinsic angular momentum of elementary particles , is aligned with a given direction. [ 1 ] This property may pertain to the spin, hence to the magnetic moment , of conduction electrons in ferromagnetic metals, such as iron , giving rise to spin-polarized currents . It may refer to (static) spin waves , preferential correlation of spin orientation with ordered lattices ( semiconductors or insulators ).
It may also pertain to beams of particles, produced for particular aims, such as polarized neutron scattering or muon spin spectroscopy . Spin polarization of electrons or of nuclei , often called simply magnetization , is also produced by the application of a magnetic field . Curie law is used to produce an induction signal in electron spin resonance (ESR or EPR) and in nuclear magnetic resonance (NMR).
Spin polarization is also important for spintronics , a branch of electronics . Magnetic semiconductors are being researched as possible spintronic materials.
The spin of free electrons is measured either by a LEED image from a clean wolfram crystal (SPLEED) [ 2 ] [ 3 ] [ 4 ] or by an electron microscope composed purely of electrostatic lenses and a gold foil as a sample. Back scattered electrons are decelerated by annular optics and focused onto a ring shaped electron multiplier at about 15°. The position on the ring is recorded. This whole device is called a Mott detector . Depending on their spin the electrons have the chance to hit the ring at different positions. 1% of the electrons are scattered in the foil. Of these 1% are collected by the detector and then about 30% of the electrons hit the detector at the wrong position. Both devices work due to spin-orbit coupling.
The circular polarization of electromagnetic fields is due to spin polarization of their constituent photons .
In the most generic context, spin polarization is any alignment of the components of a non-scalar (vectorial, tensorial, spinorial) field with its arguments, i.e., with the nonrelativistic three spatial or relativistic four spatiotemporal regions over which it is defined. In this sense, it also includes gravitational waves and any field theory that couples its constituents with the differential operators of vector analysis. | https://en.wikipedia.org/wiki/Spin_polarization |
A spin probe is a molecule with stable free radical character that carries a functional group . This group can be used to couple the probe to another molecule, e.g. a biomolecule .
Electron spin resonance can be employed to quantify the probe's concentration. [ 1 ]
This molecular physics –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spin_probe |
Spin states when describing transition metal coordination complexes refers to the potential spin configurations of the central metal's d electrons. For several oxidation states, metals can adopt high-spin and low-spin configurations. The ambiguity only applies to first row metals, because second- and third-row metals are invariably low-spin. These configurations can be understood through the two major models used to describe coordination complexes; crystal field theory and ligand field theory (a more advanced version based on molecular orbital theory ). [ 1 ]
The Δ splitting of the d orbitals plays an important role in the electron spin state of a coordination complex. Three factors affect Δ: the period (row in periodic table) of the metal ion, the charge of the metal ion, and the field strength of the complex's ligands as described by the spectrochemical series . Only octahedral complexes of first row transition metals adopt high-spin states.
In order for low spin splitting to occur, the energy cost of placing an electron into an already singly occupied orbital must be less than the cost of placing the additional electron into an e g orbital at an energy cost of Δ. If the energy required to pair two electrons is greater than the energy cost of placing an electron in an e g , Δ, high spin splitting occurs.
If the separation between the orbitals is large, then the lower energy orbitals are completely filled before population of the higher orbitals according to the Aufbau principle . Complexes such as this are called "low-spin" since filling an orbital matches electrons and reduces the total electron spin. If the separation between the orbitals is small enough then it is easier to put electrons into the higher energy orbitals than it is to put two into the same low-energy orbital, because of the repulsion resulting from matching two electrons in the same orbital. So, one electron is put into each of the five d orbitals before any pairing occurs in accord with Hund's rule resulting in what is known as a "high-spin" complex. Complexes such as this are called "high-spin" since populating the upper orbital avoids matches between electrons with opposite spin.
The charge of the metal center plays a role in the ligand field and the Δ splitting. The higher the oxidation state of the metal, the stronger the ligand field that is created. In the event that there are two metals with the same d electron configuration, the one with the higher oxidation state is more likely to be low spin than the one with the lower oxidation state; for example, Fe 2+ and Co 3+ are both d 6 ; however, the higher charge of Co 3+ creates a stronger ligand field than Fe 2+ . All other things being equal, Fe 2+ is more likely to be high spin than Co 3+ .
Ligands also affect the magnitude of Δ splitting of the d orbitals according to their field strength as described by the spectrochemical series . Strong-field ligands, such as CN − and CO, increase the Δ splitting and are more likely to be low-spin. Weak-field ligands, such as I − and Br − cause a smaller Δ splitting and are more likely to be high-spin.
Some octahedral complexes exhibit spin crossover , where the high and low spin states exist in dynamic equilibrium.
The Δ splitting energy for tetrahedral metal complexes (four ligands), Δ tet is smaller than that for an octahedral complex. Consequently, tetrahedral complexes are almost always high spin [ 3 ] Examples of low spin tetrahedral complexes include Fe(2-norbornyl) 4 , [ 4 ] [Co(4-norbornyl) 4 ] + , and the nitrosyl complex Cr(NO)( (N(tms) 2 ) 3 .
Many d 8 complexes of the first row metals exist in tetrahedral or square planar geometry. In some cases these geometries exist in measurable equilibria. For example, dichlorobis(triphenylphosphine)nickel(II) has been crystallized in both tetrahedral and square planar geometries. [ 5 ]
In terms of d-orbital splitting, ligand field theory (LFT) and crystal field theory (CFT) give similar results. CFT is an older, simpler model that treats ligands as point charges. LFT is more chemical, emphasizes covalent bonding and accommodates pi-bonding explicitly.
In the case of octahedral complexes, the question of high spin vs low spin first arises for d 4 , since it has more than the 3 electrons to fill the non-bonding d orbitals according to ligand field theory or the stabilized d orbitals according to crystal field splitting.
All complexes of second and third row metals are low-spin.
The spin state of the complex affects an atom's ionic radius . For a given d-electron count, high-spin complexes are larger. [ 7 ]
Generally, the rates of ligand dissociation from low spin complexes are lower than dissociation rates from high spin complexes. In the case of octahedral complexes, electrons in the e g levels are anti-bonding with respect to the metal-ligand bonds. Famous "exchange inert" complexes are octahedral complexes of d 3 and low-spin d 6 metal ions, illustrated respectfully by Cr 3+ and Co 3+ . [ 8 ] | https://en.wikipedia.org/wiki/Spin_states_(d_electrons) |
The spin stiffness or spin rigidity is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in-plane twist of the spins. The importance of this constant is in its use as an indicator of quantum phase transitions —specifically in models with metal-insulator transitions such as Mott insulators . It is also related to other topological invariants such as the Berry phase and Chern numbers as in the Quantum Hall effect .
Mathematically it can be defined by the following equation:
where E 0 {\displaystyle E_{0}} is the ground state energy, θ {\displaystyle \theta } is the twisting angle, and N is the number of lattice sites.
Start off with the simple Heisenberg spin Hamiltonian:
Now we introduce a rotation in the system at site i by an angle θ i around the z-axis:
Plugging these back into the Heisenberg Hamiltonian:
now let θ ij = θ i - θ j and expand around θ ij = 0 via a MacLaurin expansion only keeping terms up to second order in θ ij
where the first term is independent of θ and the second term is a perturbation for small θ.
Consider now the case of identical twists, θ x only that exist along nearest neighbor bonds along the x-axis.
Then since the spin stiffness is related to the difference in the ground state energy by
then for small θ x and with the help of second order perturbation theory we get: | https://en.wikipedia.org/wiki/Spin_stiffness |
In mathematics , mathematical physics , and theoretical physics , the spin tensor is a quantity used to describe the rotational motion of particles in spacetime . The spin tensor has application in general relativity and special relativity , as well as quantum mechanics , relativistic quantum mechanics , and quantum field theory .
The special Euclidean group SE( d ) of direct isometries is generated by translations and rotations . Its Lie algebra is written s e ( d ) {\displaystyle {\mathfrak {se}}(d)} .
This article uses Cartesian coordinates and tensor index notation .
The Noether current for translations in space is momentum, while the current for increments in time is energy. These two statements combine into one in spacetime: translations in spacetime, i.e. a displacement between two events, is generated by the four-momentum P . Conservation of four-momentum is given by the continuity equation :
∂ ν T μ ν = 0 , {\displaystyle \partial _{\nu }T^{\mu \nu }=0\,,}
where T μ ν {\displaystyle T^{\mu \nu }\,} is the stress–energy tensor , and ∂ are partial derivatives that make up the four-gradient (in non-Cartesian coordinates this must be replaced by the covariant derivative ). Integrating over space:
∫ d 3 x T μ 0 ( x → , t ) = P μ {\displaystyle \int d^{3}xT^{\mu 0}\left({\vec {x}},t\right)=P^{\mu }}
gives the four-momentum vector at time t .
The Noether current for a rotation about the point y is given by a tensor of 3rd order, denoted M y α β μ {\displaystyle M_{y}^{\alpha \beta \mu }} . Because of the Lie algebra relations
M y α β μ ( x ) = M 0 α β μ ( x ) + y α T β μ ( x ) − y β T α μ ( x ) , {\displaystyle M_{y}^{\alpha \beta \mu }(x)=M_{0}^{\alpha \beta \mu }(x)+y^{\alpha }T^{\beta \mu }(x)-y^{\beta }T^{\alpha \mu }(x)\,,}
where the 0 subscript indicates the origin (unlike momentum, angular momentum depends on the origin), the integral:
∫ d 3 x M 0 μ ν ( x → , t ) {\displaystyle \int d^{3}xM_{0}^{\mu \nu }({\vec {x}},t)}
gives the angular momentum tensor M μ ν {\displaystyle M^{\mu \nu }\,} at time t .
The spin tensor is defined at a point x to be the value of the Noether current at x of a rotation about x ,
S α β μ ( x ) = d e f M x α β μ ( x ) = M 0 α β μ ( x ) + x α T β μ ( x ) − x β T α μ ( x ) {\displaystyle S^{\alpha \beta \mu }(\mathbf {x} )\mathrel {\stackrel {\mathrm {def} }{=}} M_{x}^{\alpha \beta \mu }(\mathbf {x} )=M_{0}^{\alpha \beta \mu }(\mathbf {x} )+x^{\alpha }T^{\beta \mu }(\mathbf {x} )-x^{\beta }T^{\alpha \mu }(\mathbf {x} )}
The continuity equation
∂ μ M 0 α β μ = 0 , {\displaystyle \partial _{\mu }M_{0}^{\alpha \beta \mu }=0\,,}
implies:
∂ μ S α β μ = T β α − T α β ≠ 0 {\displaystyle \partial _{\mu }S^{\alpha \beta \mu }=T^{\beta \alpha }-T^{\alpha \beta }\neq 0}
and therefore, the stress–energy tensor is not a symmetric tensor .
The quantity S is the density of spin angular momentum (spin in this case is not only for a point-like particle, but also for an extended body), and M is the density of orbital angular momentum. The total angular momentum is always the sum of spin and orbital contributions.
The relation:
T i j − T j i {\displaystyle T_{ij}-T_{ji}}
gives the torque density showing the rate of conversion between the orbital angular momentum and spin.
Examples of materials with a nonzero spin density are molecular fluids , the electromagnetic field and turbulent fluids . For molecular fluids, the individual molecules may be spinning. The electromagnetic field can have circularly polarized light . For turbulent fluids, we may arbitrarily make a distinction between long wavelength phenomena and short wavelength phenomena. A long wavelength vorticity may be converted via turbulence into tinier and tinier vortices transporting the angular momentum into smaller and smaller wavelengths while simultaneously reducing the vorticity . This can be approximated by the eddy viscosity . | https://en.wikipedia.org/wiki/Spin_tensor |
The spin transition is an example of transition between two electronic states in molecular chemistry . The ability of an electron to transit from a stable to another stable (or metastable ) electronic state in a reversible and detectable fashion, makes these molecular systems appealing in the field of molecular electronics .
When a transition metal ion of configuration d n {\displaystyle d^{n}} , n = 4 {\displaystyle n=4} to 7 {\displaystyle 7} , is in octahedral surroundings, its ground state may be low spin (LS) or high spin (HS), depending to a first approximation on the magnitude of the Δ {\displaystyle \Delta } energy gap between e g {\displaystyle e_{g}} and t 2 g {\displaystyle t_{2g}} metal orbitals relative to the mean spin pairing energy P {\displaystyle P} (see Crystal field theory ). More precisely, for Δ >> P {\displaystyle \Delta >>P} , the ground state arises from the configuration where the d {\displaystyle d} electrons occupy first the t 2 g {\displaystyle t_{2g}} orbitals of lower energy, and if there are more than six electrons, the e g {\displaystyle e_{g}} orbitals of higher energy. The ground state is then LS. On the other hand, for Δ << P {\displaystyle \Delta <<P} , Hund's rule is obeyed. The HS ground state has got the same multiplicity as the free metal ion . If the values of P {\displaystyle P} and Δ {\displaystyle \Delta } are comparable, a LS↔HS transition may occur.
Between all the possible d n {\displaystyle d^{n}} configurations of the metal ion, d 5 {\displaystyle d^{5}} and d 6 {\displaystyle d^{6}} are by far the most important. The spin transition phenomenon, in fact, was first observed in 1930 for tris (dithiocarbamato) iron(III) compounds. On the other hand, the iron(II) spin transition complexes were the most extensively studied: among these two of them may be considered as archetypes of spin transition systems, namely Fe(NCS) 2 (bipy) 2 and Fe(NCS) 2 (phen) 2 (bipy = 2,2'-bipyridine and phen = 1,10-phenanthroline).
We discuss the mechanism of the spin transition by focusing on the specific case of iron(II) complexes. At the molecular scale the spin transition corresponds to an interionic electron transfer with spin flip of the transferred electrons. For an iron(II) compound this transfer involves two electrons and the spin variations is Δ S = 2 {\displaystyle \Delta S=2} . The occupancy of the e g {\displaystyle e_{g}} orbitals is higher in the HS state than in the LS state and these orbitals are more antibonding than the t 2 g {\displaystyle t_{2g}} . It follows that the average metal- ligand bond length is longer in the HS state than in the LS state. This difference is in the range 1.4–2.4 pm for iron(II) compounds.
The most common way to induce a spin transition is to change the temperature of the system: the transition will be then characterized by a ρ H = f ( T ) {\displaystyle \rho _{H}=f(T)} , where ρ H {\displaystyle \rho _{H}} is the molar fraction of molecules in high-spin state. Several techniques are currently used to obtain such curves. The simplest method consists of measuring the temperature dependence of molar susceptibility. Any other technique that provides different responses according to whether the state is LS or HS may also be used to determine ρ H {\displaystyle \rho _{H}} . Among these techniques, Mössbauer spectroscopy has been particularly useful in the case of iron compounds , showing two well resolved quadrupole doublets. One of these is associated with LS molecules, the other with HS molecules: the high-spin molar fraction then may be deduced from the relative intensities of the doublets.
Various types of transition have been observed. This may be abrupt, occurring within a few kelvins range, or smooth, occurring within a large temperature range. It could also be incomplete both at low temperature and at high temperature, even if the latter is more often observed. Moreover, the ρ H = f ( T ) {\displaystyle \rho _{H}=f(T)} curves may be strictly identical in the cooling or heating modes, or exhibit a hysteresis : in this case the system could assume two different electronic states in a certain range of temperature. Finally the transition may occur in two steps. | https://en.wikipedia.org/wiki/Spin_transition |
Spin trapping is an analytical technique employed in chemistry [ 1 ] and biology [ 2 ] for the detection and identification of short-lived free radicals through the use of electron paramagnetic resonance (EPR) spectroscopy. EPR spectroscopy detects paramagnetic species such as the unpaired electrons of free radicals. However, when the half-life of radicals is too short to detect with EPR, compounds known as spin traps are used to react covalently with the radical products and form more stable adduct that will also have paramagnetic resonance spectra detectable by EPR spectroscopy. [ 3 ] The use of radical-addition reactions to detect short-lived radicals was developed by several independent groups by 1968. [ 4 ]
The most commonly used spin traps are alpha-phenyl N-tertiary-butyl nitrone (PBN) and 5,5-dimethyl-pyrroline N-oxide ( DMPO ). More rarely, C-nitroso spin traps such as 3,5-dibromo-4-nitrosobenzenesulfonic acid (DBNBS) can be used: often additional hyperfine information is derived, but at a cost of specificity (due to facile non-radical addition of many compounds to C-nitroso species, and subsequent oxidation of the resulting hydroxylamine).
5-Diisopropoxyphosphoryl-5-methyl-1-pyrroline-N-oxide ( DIPPMPO ) spin trapping has been used in measuring superoxide production in mitochondria.
A comprehensive list of Spin Trapping molecules is maintained by the IUPAC. [ 5 ] [ failed verification ]
A common method for spin-trapping involves the addition of radical to a nitrone spin trap resulting in the formation of a spin adduct, a nitroxide-based persistent radical, that can be detected using EPR. The spin adduct usually yields a distinctive EPR spectrum characteristic of a particular free radical that is trapped. The identity of the radical can be inferred based on the EPR spectral profile of their respective spin adducts such as the g value, but most importantly, the hyperfine-coupling constants of relevant nuclei. Unambiguous assignments of the identity of the trapped radical can often be made by using stable isotope substitution of the radicals parent compound, so that further hyperfine couplings are introduced or altered.
The radical adduct (or products such as the hydroxylamine ) can often be stable enough to allow non-EPR detection techniques. The groups of London, and Berliner & Khramtsov have used NMR to study such adducts and Timmins and co-workers used charge changes upon DBNBS trapping to isolate protein adducts for study. A major advance has been the development of anti-DMPO antibodies by Mason's group, allowing study of spin trapping reactions by a simple immuno-based techniques. | https://en.wikipedia.org/wiki/Spin_trapping |
In condensed matter physics , a spin wave is a propagating disturbance in the ordering of a magnetic material. These low-lying collective excitations occur in magnetic lattices with continuous symmetry . From the equivalent quasiparticle point of view, spin waves are known as magnons , which are bosonic modes of the spin lattice that correspond roughly to the phonon excitations of the nuclear lattice. As temperature is increased, the thermal excitation of spin waves reduces a ferromagnet 's spontaneous magnetization . The energies of spin waves are typically only μeV in keeping with typical Curie points at room temperature and below.
The simplest way of understanding spin waves is to consider the Hamiltonian H {\displaystyle {\mathcal {H}}} for the Heisenberg ferromagnet:
where J is the exchange energy , the operators S represent the spins at Bravais lattice points, g is the Landé g -factor , μ B is the Bohr magneton and H is the internal field which includes the external field plus any "molecular" field. Note that in the classical continuum case and in 1 + 1 dimensions the Heisenberg ferromagnet equation has the form
In 1 + 1, 2 + 1 and 3 + 1 dimensions this equation admits several integrable and non-integrable extensions like the Landau-Lifshitz equation , the Ishimori equation and so on. For a ferromagnet J > 0 and the ground state of the Hamiltonian | 0 ⟩ {\displaystyle |0\rangle } is that in which all spins are aligned parallel with the field H . That | 0 ⟩ {\displaystyle |0\rangle } is an eigenstate of H {\displaystyle {\mathcal {H}}} can be verified by rewriting it in terms of the spin-raising and spin-lowering operators given by:
resulting in
where z has been taken as the direction of the magnetic field. The spin-lowering operator S − annihilates the state with minimum projection of spin along the z -axis, while the spin-raising operator S + annihilates the ground state with maximum spin projection along the z -axis. Since
for the maximally aligned state, we find
where N is the total number of Bravais lattice sites. The proposition that the ground state is an eigenstate of the Hamiltonian is confirmed.
One might guess that the first excited state of the Hamiltonian has one randomly selected spin at position i rotated so that
but in fact this arrangement of spins is not an eigenstate. The reason is that such a state is transformed by the spin raising and lowering operators. The operator S i + {\displaystyle S_{i}^{+}} will increase the z -projection of the spin at position i back to its low-energy orientation, but the operator S j − {\displaystyle S_{j}^{-}} will lower the z -projection of the spin at position j . The combined effect of the two operators is therefore to propagate the rotated spin to a new position, which is a hint that the correct eigenstate is a spin wave , namely a superposition of states with one reduced spin. The exchange energy penalty associated with changing the orientation of one spin is reduced by spreading the disturbance over a long wavelength. The degree of misorientation of any two near-neighbor spins is thereby minimized. From this explanation one can see why the Ising model magnet with discrete symmetry has no spin waves: the notion of spreading a disturbance in the spin lattice over a long wavelength makes no sense when spins have only two possible orientations. The existence of low-energy excitations is related to the fact that in the absence of an external field, the spin system has an infinite number of degenerate ground states with infinitesimally different spin orientations. The existence of these ground states can be seen from the fact that the state | 0 ⟩ {\displaystyle |0\rangle } does not have the full rotational symmetry of the Hamiltonian H {\displaystyle {\mathcal {H}}} , a phenomenon which is called spontaneous symmetry breaking .
In this model the magnetization
where V is the volume. The propagation of spin waves is described by the Landau-Lifshitz equation of motion:
where γ is the gyromagnetic ratio and λ is the damping constant. The cross-products in this forbidding-looking equation show that the propagation of spin waves is governed by the torques generated by internal and external fields. (An equivalent form is the Landau-Lifshitz-Gilbert equation , which replaces the final term by a more "simple looking" equivalent one.)
The first term on the right hand side of the equation describes the precession of the magnetization under the influence of the applied field, while the above-mentioned final term describes how the magnetization vector "spirals in" towards the field direction as time progresses. In metals the damping forces described by the constant λ are in many cases dominated by the eddy currents.
One important difference between phonons and magnons lies in their dispersion relations . The dispersion relation for phonons is to first order linear in wavevector k , namely ώ = ck , where ω is frequency, and c is the velocity of sound. Magnons have a parabolic dispersion relation: ώ = Ak 2 where the parameter A represents a " spin stiffness ." The k 2 form is the third term of a Taylor expansion of a cosine term in the energy expression originating from the S i ⋅ S j dot product . The underlying reason for the difference in dispersion relation is that the order parameter (magnetization) for the ground-state in ferromagnets violates time-reversal symmetry . Two adjacent spins in a solid with lattice constant a that participate in a mode with wavevector k have an angle between them equal to ka .
Spin waves are observed through four experimental methods: inelastic neutron scattering , inelastic light scattering ( Brillouin scattering , Raman scattering and inelastic X-ray scattering), inelastic electron scattering (spin-resolved electron energy loss spectroscopy ), and spin-wave resonance ( ferromagnetic resonance ).
When magnetoelectronic devices are operated at high frequencies, the generation of spin waves can be an important energy loss mechanism. Spin wave generation limits the linewidths and therefore the quality factors Q of ferrite components used in microwave devices. The reciprocal of the lowest frequency of the characteristic spin waves of a magnetic material gives a time scale for the switching of a device based on that material. | https://en.wikipedia.org/wiki/Spin_wave |
Spinach is an open-source magnetic resonance simulation package initially released in 2011 [ 1 ] and continuously updated since. [ 2 ] The package is written in Matlab and makes use of the built-in parallel computing and GPU interfaces of Matlab . [ 3 ]
The name of the package whimsically refers to the physical concept of spin and to Popeye the Sailor who, in the eponymous comic books, becomes stronger after consuming spinach . [ 4 ]
Spinach implements magnetic resonance spectroscopy and imaging simulations by solving the equation of motion for the density matrix ρ ( t ) {\displaystyle \mathbf {\rho } \left(t\right)} in the time domain: [ 1 ]
∂ ∂ t ρ ( t ) = − i L ( t ) ρ ( t ) ⇓ ρ ( t + d t ) = exp [ − i L ( t ) d t ] ρ ( t ) {\displaystyle {\begin{matrix}{\frac {\partial }{\partial t}}\mathbf {\rho } \left(t\right)=-i\mathbf {L} \left(t\right)\mathbf {\rho } \left(t\right)\\\Downarrow \\\mathbf {\rho } \left(t+dt\right)=\exp \left[-i\mathbf {L} \left(t\right)dt\right]\mathbf {\rho } \left(t\right)\\\end{matrix}}}
where the Liouvillian superoperator L ( t ) {\displaystyle \mathbf {L} \left(t\right)} is a sum of the Hamiltonian commutation superoperator H ( t ) {\displaystyle \mathbf {H} \left(t\right)} , relaxation superoperator R {\displaystyle \mathbf {R} } , kinetics superoperator K {\displaystyle \mathbf {K} } , and potentially other terms that govern spatial dynamics and coupling to other degrees of freedom: [ 2 ]
Computational efficiency is achieved through the use of reduced state spaces , sparse matrix arithmetic, on-the-fly trajectory analysis, and dynamic parallelization . [ 5 ]
As of 2023, Spinach is cited in over 300 academic publications. [ 1 ] According to the documentation [ 2 ] and academic papers citing its features, the most recent version 2.8 of the package performs:
Common models of spin relaxation ( Redfield theory, stochastic Liouville equation , Lindblad theory ) and chemical kinetics are supported, and a library of powder averaging grids is included with the package. [ 2 ]
Spinach contains an implementation the gradient ascent pulse engineering (GRAPE) algorithm [ 16 ] for quantum optimal control . The documentation [ 2 ] and the book describing the optimal control module of the package [ 17 ] list the following features:
Dissipative background evolution generators and control operators are supported, as well as ensemble control over distributions in common instrument calibration parameters, such as control channel power and offset. [ 2 ] | https://en.wikipedia.org/wiki/Spinach_(software) |
The need for fluorescently tracking RNA rose as its roles in complex cellular functions has grown to not only include mRNA , rRNA , and tRNA , but also RNAi , siRNA , snoRNA , and lncRNA , among others. [ 1 ] [ 2 ] Spinach is a synthetically derived RNA aptamer born out of the need for a way of studying the role of RNAs at the cellular level. [ 3 ] This aptamer was created using Systematic Evolution for Ligands by EXponential enrichment, or SELEX , which is also known as in vitro evolution. [ 4 ]
The aptamer was designed to be an RNA mimic of green fluorescent protein (GFP); similar to GFP for proteins, Spinach can be used for the fluorescently labeling RNA and tracking it in vivo. A method for inserting the Spinach sequence after an RNA sequence of interest is readily available. [ 5 ] [ 6 ]
GFP’s fluorophore is made up of three cyclized amino acids within the beta-barrel structure: Serine65-Tyrosine66-Glycine67. This structure, 4-hydroxybenzylidene imidazolinone (HBI) was the basis for the synthetic analogue used in the SELEX studies. Many derivatives of this structure were screened using SELEX, but the chosen fluorophore, 3,5-difluoro-4-hydroxybenzylidene imidazolinone (DFHBI), showed the best selective fluorescence with high quantum yield (0.72) when bound to the RNA sequence 24-2, deemed Spinach.
It was determined that DFHBI only binds Spinach in the phenolate form. At pH < 6.0, both the phenolic and phenolate forms are detected. At pH = 6.0, only the phenolate peak is detected. DFHBI is also incredibly robust and resists photobleaching over a long period of time as compared to HBI and EGFP. It is believed that the free exchange of bound and unbound ligand allows for this persistence. As the fluorophore of GFP and its derivatives are covalently bound to/a part of the protein, free exchange cannot happen and thus photobleaching results.
Spinach is an 84-nucleotide-long structure with two helical strands and an internal bulge with a G-quadruplex motif. It is at this G-quadruplex that the fluorophore binds. Based on crystallographic data, massive rearrangement of the adjacent bases occurs once the fluorophore binds to accommodate the molecule. This binding is favorable, however, as it promotes base-stacking in a normally unstable region, i.e. the internal bulge. Similar to GFP, the DFHBI is also dehydrated, which would help with its high quantum yield.
Spinach has also been adapted for sensing proteins or molecules in vivo. An adapted structure, which includes two binding sites, limits fluorescence of the aptamer to (1) the fluorophore and (2) the protein or small molecule. | https://en.wikipedia.org/wiki/Spinach_aptamer |
Spinal cord injury research seeks new ways to cure or treat spinal cord injury in order to lessen the debilitating effects of the injury in the short or long term. There is no cure for SCI, and current treatments are mostly focused on spinal cord injury rehabilitation and management of the secondary effects of the condition. [ 1 ] Two major areas of research include neuroprotection , ways to prevent damage to cells caused by biological processes that take place in the body after the injury, and neuroregeneration , regrowing or replacing damaged neural circuits.
Secondary injury takes place minutes to weeks after the initial insult and includes a number of cascading processes that further harm tissues already damaged by the primary injury. [ 2 ] It results in formation of a glial scar, which impedes axonal growth. [ 2 ] Secondary injuries can occur from different forms of stress added to the spinal cord in forms such as additional contusions, compressions, kinking, or stretching of the spinal cord. [ 3 ]
Complications from a secondary SCI are a result of a homeostatic imbalance potentially leading to metabolic and hemostatic changes from an inflammatory response. Potential immediate effects of secondary SCI include neuronal injury, neuroinflammation, breakdown of blood-spinal cord barrier (BSCB), ischemic dysfunction, oxidative stress, and daily-life function complications. [ citation needed ]
Animals used as SCI model organisms in research include mice, rats, cats, dogs, pigs, and non-human primates; the latter are close to humans but raise ethical concerns about primate experimentation . [ 1 ] Special devices exist to deliver blows of specific, monitored force to the spinal cord of an experimental animal. [ 1 ] There are various mechanical impact classifications of these injuries that can be replicated in an animal model. This includes contusion, compression, collagenase and ischemia reperfusion, distraction, dislocation, and transection.
Limitations of these model experiments are common. For instance, ischemia-reperfusion SCI involves the interruption of blood flow to the spinal cord. Complications have been observed to arise in animal models from the need to cross clamp the aorta.
Epidural cooling saddles, surgically placed over acutely traumatized spinal cord tissue, have been used to evaluate potentially beneficial effects of localized hypothermia, with and without concomitant glucocorticoids . [ 4 ] [ 5 ]
Surgery is currently used to provide stability to the injured spinal column or to relieve pressure from the spinal cord. [ 1 ] [ 6 ] How soon after injury to perform decompressive surgery is a controversial topic, and it has been difficult to prove that earlier surgery provides better outcomes in human trials. [ 1 ] Some argue that early surgery might further deprive an already injured spinal cord of oxygen, but most studies show no difference in outcomes between early (within three days) and late surgery (after five days), and some show a benefit to earlier surgery. [ 7 ]
In 2014 Darek Fidyka underwent pioneering spinal surgery that used nerve grafts, from his ankle, to 'bridge the gap' in his severed spinal cord and olfactory ensheathing cells (OECs) to stimulate the spinal cord cells. The surgery was performed in Poland in collaboration with Prof. Geoff Raisman, chair of neural regeneration at University College London's Institute of Neurology, and his research team. The OECs were taken from the patient's olfactory bulbs in his brain and then grown in the lab, these cells were then injected above and below the impaired spinal tissue. [ 8 ]
Neuroprotection aims to prevent the harm that occurs from secondary injury. [ 2 ] One example is to target the protein calpain which appears to be involved in apoptosis ; inhibiting the protein has produced improved outcomes in animal trials. [ 2 ] Iron from blood damages the spinal cord through oxidative stress , so one option is to use a chelation agent to bind the iron; animals treated this way have shown improved outcomes. [ 2 ] Free radical damage by reactive oxygen species (ROS) is another therapeutic target that has shown improvement when targeted in animals. [ 2 ] One antibiotic, minocycline , is under investigation in human trials for its ability to reduce free radical damage, excitotoxicity , disruption of mitochondrial function, and apoptosis. [ 2 ] Riluzole, an anticonvulsant, is also being investigated in clinical trials for its ability to block sodium channels in neurons, which could prevent damage by excitotoxicity. [ 2 ] Other potentially neuroprotective agents under investigation in clinical trials include cethrin , erythropoietin , and dalfampridine . [ 2 ]
One experimental treatment, therapeutic hypothermia , is used in treatment but there is no evidence that it improves outcomes. [ 9 ] [ 10 ] Some experimental treatments, including systemic hypothermia, have been performed in isolated cases in order to draw attention to the need for further preclinical and clinical studies to help clarify the role of hypothermia in acute spinal cord injury. [ 11 ] Despite limited funding, a number of experimental treatments such as local spine cooling and oscillating field stimulation have reached controlled human trials. [ 12 ] [ 13 ]
Inflammation and glial scar are considered important inhibitory factors to neuroregeneration after SCI. However, aside from methylprednisolone , none of these developments have reached even limited use in the clinical care of human spinal cord injury in the US. [ 14 ] Methylprednisolone can be given shortly after the injury but evidence for harmful side effects outweighs that for a benefit. [ 6 ] Research is being done into more efficient delivery mechanisms for methylprednisolone that would reduce its harmful effects. [ 1 ]
Neuroregeneration aims to reconnect the broken circuits in the spinal cord to allow function to return. [ 2 ] One way is to regrow axons, which occurs spontaneously in the peripheral nervous system . However, myelin in the central nervous system contains molecules that impede axonal growth; thus, these factors are a target for therapies to create an environment conducive to growth. [ 2 ] One such molecule is Nogo-A , a protein associated with myelin. When this protein is targeted with inhibitory antibodies in animal models, axons grow better and functional recovery is improved. [ 2 ]
Stem cells are cells that can differentiate to become different types of cells. [ 15 ] The hope is that stem cells transplanted into an injured area of the spinal cord will allow neuroregeneration . [ 6 ] Types of cells being researched for use in SCI include embryonic stem cells , neural stem cells , mesenchymal stem cells , olfactory ensheathing cells , Schwann cells , activated macrophages , and induced pluripotent stem cells . [ 1 ] When stem cells are injected in the area of damage in the spinal cord, they secrete neurotrophic factors , and these factors help neurons and blood vessels to grow, thus helping repair the damage. [ 16 ] [ 17 ] [ 18 ] It is also necessary to recreate an environment in which stem cells will grow. [ 19 ]
An ongoing Phase 2 trial in 2016 presented data [ 20 ] showing that after 90 days of treatment with oligodendrocyte progenitor cells derived from embryonic stem cells, 4 out of 4 subjects with complete cervical injuries had improved motor levels, with 2 of 4 improving two motor levels (on at least one side, with one patient improving two motor levels on both sides). The trial's original endpoint had been 2/5 patients improving two levels on one side within 6–12 months. All 8 cervical subjects in this Phase 1–2 trial had exhibited improved upper extremity motor scores (UEMS) relative to baseline with no serious adverse side effects, and a 2010 Phase 1 trial in 5 thoracic patients has found no safety issues after 5–6 years of follow-up.
Six-month efficacy data is expected in January 2017; meanwhile, a higher dose is being investigated and the study is now also recruiting patients with incomplete injuries. [ 21 ]
In 2022, a team reported the first [ 22 ] engineered functional human (motor-)neuronal networks derived from induced pluripotent stem cells (iPSCs) from the patient for implantation to regenerate injured spinal cord that shows success in tests with mice. [ 23 ] [ 24 ]
Embryonic stem cells (ESCs) are pluripotent ; they can develop into every type of cell in an organism, such as oligodendrocytes . . [ 6 ] Oligodendrocytes and motor neurons have been predicted to be a favorable target for ESCs regarding treating neurological disorders and traumas. [ 25 ] After a SCI takes place there is evidence of oligodendrocytes degradation, eventually leading to cell death. This results in a lack of myelination , which enhances signals sent amongst neurons, causing disfunction in signaling. A potential solution could be hESC-derived oligodendrocyte transplantation; however, the success of this process depends off the cell’s ability to differentiate towards neural cell types in vitro. This is where more testing and research is being conducted using animal models.
Neural stem cells (NSCs) are multipotent ; they can differentiate into different kinds of neural cells, either neurons or glia , namely oligodendrocytes and astrocytes . [ 15 ] The hope is that these cells when injected into an injured spinal cord will replace killed neurons and oligodendrocytes and secrete factors that support growth. [ 1 ] However they may fail to differentiate into neurons when transplanted, either remaining undifferentiated or becoming glia. [ 15 ] A phase I/II clinical trials implanting NSCs into humans with SCI began in 2011 [ 1 ] and ended in June 2015. [ 26 ]
Mesenchymal stem cells do not need to come from fetuses, so avoid difficulties around ethics; they come from tissues including bone marrow, adipose tissue , the umbilical cord . [ 1 ] Unlike other types of stem cells, mesenchymal cells do not present the threat of tumor formation or triggering an immune system response. [ 1 ] Animal studies with injection of bone marrow stem cells have shown improvement in motor function; however not so in a human trial a year post-injury. [ 1 ] More trials are underway. [ 1 ] Adipose and umbilical tissue stem cells need further study before human trials can be performed, but two Korean studies were begun to investigate adipose cells in SCI patients. [ 1 ]
Transplantation of tissues such as olfactory ensheathing cells from the olfactory bulbs has been shown to produce beneficial effects in spinal cord injured rats. [ 27 ] Trials have also begun to show success when olfactory ensheathing cells are transplanted into humans with severed spinal cords. [ 28 ] People have recovered sensation, use of formerly paralysed muscles, and bladder and bowel function after the surgeries, [ 29 ] eg Darek Fidyka .
Japanese researchers in 2006 discovered that adding certain transcription factors to cells caused them to become pluripotent and able to differentiate into multiple cell types. [ 6 ] This way a patient's own tissues could be used, theoretically because of a reduced chance of transplant rejection . [ 6 ]
Recent approaches have used various engineering techniques to improve spinal cord injury repair.
Use of biomaterials is an engineering approach to SCI treatment that can be combined with stem cell transplantation. [ 6 ] They can help deliver cells to the injured area and create an environment that fosters their growth. [ 6 ] The general hypothesis behind engineered biomaterials is that bridging the lesion site using a growth permissive scaffold may help axons grow and thereby improve function. The biomaterials used must be strong enough to provide adequate support but soft enough not to compress the spinal cord. [ 2 ] They must degrade over time to make way for the body to regrow tissue. [ 2 ] Engineered treatments do not induce an immune response as biological treatments may, and they are easily tunable and reproducible. In-vivo administration of hydrogels or self-assembling nanofibers has been shown to promote axonal sprouting and partial functional recovery. [ 30 ] [ 31 ] In addition, administration of carbon nanotubes has shown to increase motor axon extension and decrease the lesion volume, without inducing neuropathic pain . [ 32 ] In addition, administration of poly-lactic acid microfibers has shown that topographical guidance cues alone can promote axonal regeneration into the injury site. [ 33 ] However, all of these approaches induced modest behavioral or functional recovery suggesting that further investigation is necessary.
Hydrogels are structures made of polymers that are designed to be similar to the natural extracellular matrix around cells. [ 2 ] They can be used to help deliver drugs more efficiently to the spinal cord and to support cells, and they can be injected into an injured area to fill a lesion. [ 2 ] They can be implanted into a lesion site with drugs or growth factors in them to give the chemicals the best access to the damaged area and to allow sustained release. [ 2 ]
In November 2021, a novel therapy for spinal cord injury was reported – an injectable gel of nanofibers that mimic the matrix around cells and contain molecules that were engineered to wiggle. These moving molecules connect with receptors of cells, causing repair signals inside – in particular, leading to relatively higher vascular growth, axonal regeneration, myelination, survival of motor neurons, reduced gliosis, and functional recovery – enabling paralyzed mice to walk again. [ 34 ] [ 35 ] [ 36 ]
The technology for creating powered exoskeletons , wearable machinery to assist with walking movements, is currently making significant advances. There are products available, such as the Ekso, which allows individuals with up to a C7 complete (or any level of incomplete) spinal injury to stand upright and make technologically assisted steps. [ 37 ] The initial purpose for this technology is for functional based rehabilitation, but as the technology develops, so will its uses. [ 37 ]
Functional electrical stimulation (FES) uses coordinated electric shocks to muscles to cause them to contract in a walking pattern. [ 38 ] While it can strengthen muscles, a significant downside for the users of FES is that their muscles tire after a short time and distance. [ 38 ] One research direction combines FES with exoskeletons to minimize the downsides of both technologies, supporting the person's joints and using the muscles to reduce the power needed from the machine, and thus its weight. [ 38 ] A research team at the McKelvey School of Engineering at Washington University in St. Louis , led by assistant professor of biomedical engineering Ismael Seáñez , is launching a clinical trial of electrical spinal cord stimulation for helping restore movement in movement-impaired or paralyzed patients. [ 39 ]
Recent research shows that combining brain–computer interface and functional electrical stimulation can restore voluntary control of paralyzed muscles. A study with monkeys showed that it is possible to directly use commands from the brain, bypassing the spinal cord and enable limited hand control and function. [ 40 ]
A 2016 study developed by the Walk Again Project with eight paraplegics demonstrated neurological recovernment with the use of therapies based upon BMI, virtual reality an the use of robots. One patient became able to walk with supports after a decade paralised and another became able to carry out a pregnancy. [ 41 ] [ 42 ] [ 43 ] [ 44 ] [ 45 ]
Spinal cord implants , such as e-dura implants, designed for implantation on the surface of the spinal cord, are being studied for paralysis following a spinal cord injury. [ 46 ]
E-dura implants are designed using methods of soft neurotechnology , in which electrodes and a microfluidic delivery system are distributed along the spinal implant. [ 47 ] Chemical stimulation of the spinal cord is administered through the microfluidic channel of the e-dura. The e-dura implants, unlike previous surface implants, closely mimic the physical properties of living tissue and can deliver electric impulses and pharmacological substances simultaneously. Artificial dura mater was constructed through the utilization of PDMS and gelatin hydrogel. [ 47 ] The hydrogel simulates spinal tissue and a silicone membrane simulates the dura mater. These properties allow the e-dura implants to sustain long-term application to the spinal cord and brain without leading to inflammation, scar tissue buildup, and rejection normally caused by surface implants rubbing against nerve tissue.
In 2018 two distinct research teams from Minnesota's Mayo Clinic and Kentucky's University of Louisville managed to restore some mobility to patients suffering from paraplegia with an electronic spinal cord stimulator. The theory behind the new spinal cord stimulator is that in certain cases of spinal cord injury the spinal nerves between the brain and the legs are still alive, but just dormant. [ 48 ] On 1 November 2018 a third distinct research team from the University of Lausanne published similar results with a similar stimulation technique in the journal Nature . [ 49 ] [ 50 ] In 2022, researchers demonstrated a spinal cord stimulator that enabled patients with spinal cord injury to walk again via epidural electrical stimulation (EES) with substantial neurorehabilitation-progress during the first day. [ 51 ] [ 52 ] In a study published in May 2023 in the journal Nature , researchers in Switzerland described implants which allowed a 40-year old man, paralyzed from the hips down for 12 years, to stand, walk and ascend a steep ramp with only the assistance of a walker. More than a year after the implant was inserted, he has retained these abilities and was walking with crutches even when the implant was switched off. [ 53 ] | https://en.wikipedia.org/wiki/Spinal_cord_injury_research |
SPINDLE ( Sub-glacial Polar Ice Navigation, Descent, and Lake Exploration ) is a 2-stage autonomous vehicle system consisting of a robotic ice-penetrating carrier vehicle ( cryobot ) and an autonomous submersible HAUV (hovering autonomous underwater vehicle). [ 1 ] The cryobot is designed to descend through an ice body into a sub-surface ocean and deploy the HAUV submersible to conduct long range reconnaissance, life search, and sample collection. [ 2 ] The HAUV submersible will return to, and auto-dock with, the cryobot at the conclusion of the mission for subsequent data uplink and sample return to the surface.
The SPINDLE designed is targeted at sub-glacial lakes such as Lake Vostok and South Pole Lake in Antarctica . SPINDLE would develop the technologies for a Flagship-class mission to either the shallow lakes of Jupiter's moon Europa , the sub-surface ocean of Ganymede , or the geyser sources on both Europa and Enceladus . [ 1 ] [ 3 ] The project is funded by NASA and is being designed at Stone Aerospace under the supervision of Principal Investigator Bill Stone . [ 4 ]
In 2011, NASA awarded Stone Aerospace $4 million to fund Phase 2 of project VALKYRIE (Very-Deep Autonomous Laser-Powered Kilowatt-Class Yo-Yoing Robotic Ice Explorer). [ 5 ] This project created an autonomous cryobot capable of melting through vast amounts of ice. [ 6 ] The 5 kW (6.7 hp) power source on the surface uses optic fiber to conduct a high-energy laser beam to produce hot water jets that melt the ice ahead. [ 5 ] [ 7 ] Some beam energy is converted to electricity via photovoltaic cells to power on-board electronics and jet pumps. [ 5 ] Phase 2 of project VALKYRIE consisted of testing a scaled-down version of the cryobot in Matanuska Glacier , Alaska in 2015. [ 8 ]
Stone Aerospace is now looking at designs integrating a scaled-down version of the HAUV submersible called ARTEMIS (4.3 m or 14 ft 1 in long, 1,270 kg or 2,800 lb) [ 9 ] with VALKYRIE-type technology to produce SPINDLE. [ 10 ] This goal is a full-scale cryobot which can melt its way to an Antarctic subglacial lake — Lake Vostok — to collect samples, and then resurface. [ 6 ] [ 8 ] SPINDLE does not use hot water jets as its predecessor VALKYRIE, but will beam its laser straight into the ice ahead. [ 9 ] The vehicle features a radar integrated to an intelligent algorithm for autonomous scientific sampling and navigation through ice and water. [ 11 ] [ 12 ] This phase of the project would be viewed as a precursor to possible future missions to an icy world such as Europa , Enceladus , or Ganymede to explore the liquid water oceans thought to be present below their ice, assess their potential habitability and seek biosignatures . [ 13 ] [ 14 ] If the system is flown to an icy moon, it would likely deposit radio receiver buoys into the ice from its rear as it makes its descent. [ 9 ]
SPINDLE is designed for a 1.5 to 4 km (0.93 to 2.49 mi) [ 15 ] penetration through a terrestrial ice sheet and the HAUV has been designed for exploration up to one kilometer (0.62 mi) radius from the cryobot. [ 1 ] The cryobot is bi-directional and vertically controllable both in an ice sheet as well as following breakthrough into a subglacial water cavity or ocean. The vehicle is designed for subsequent return to the surface at a much later date or subsequent season. [ 1 ]
The required power plant must be very powerful, so the engineers are working on preliminary designs of a compact fission power plant that would be used for actual ocean planet missions. [ 1 ] | https://en.wikipedia.org/wiki/Spindle_(vehicle) |
Spinlock is a technology based company specialized in the manufacture and development of nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) equipment.
Spinlock was founded in 2003 by Dr. Daniel J. Pusiol, a renowned physicist specialized in NMR and NQR, professor at the National University of Córdoba and member of the National Scientific and Technical Research Council (CONICET) of Argentina.
Mr. Pusiol lead a team of young researchers to build Spinlock, manufacturing NMR spectrometers and providing Research and Development services for different industries, engaging on the discovery of new applications for the NMR technology.
Closely related to the National University of Córdoba, Spinlock contributes with the University on projects and receives continuous support in the form of education and consultation with leading scientists. [ 1 ]
The organization employs scientist and technical staff from diverse areas: engineering, physics, chemistry and computer sciences.
Spinlock has shared its knowledge in international publications, forums and blogs (see Determinación del contenido de ácido oleico en semillas de maní por medio de la resonancia magnética nuclear (RMN)).
SLK - 100 : (discontinued product) This Sigmometer use NMR spectroscopy to carry out a variety of analysis in food industry processes [ 2 ] (e.g. moisture and fat content in food; oil and fatty acids in oily seeds [ 3 ] and olives; solid and liquid fat ratio in cheese, chocolate and other dairy products).
SLK - 200 : Desktop magnetic resonance equipment for measurement of oil or fat, moisture, fatty acid and protein content. Simultaneous, non-destructive and automated determination.
SLK-GOW Multiphase Water-Cut Meter: Resonance magnetic equipment for multiphase measurement Oil, water and gas. Deviates a multiphase sample to a secondary line (Bypass system). Water Cut Measurement (net oil and water), Gas Volume Fraction (GVF) and Oil Viscosity.
SLK-MFM-III Multiphase flow meter prototype equipment for magnetic resonance. Measures flow and cut of multiphase fluid, including water, gas and oil. Built for ø2´ lines with ø2´ Halbach magnet.
SLK-MRI-1400 Time domain magnetic resonance equipment (TD-NMR) suitable for imaging (MRI) with an ø 4´ circular Halbach magnet, with temperature control and compensation.
SLK-1000-PM Desktop magnetic resonance equipment for measurements of sidewall cores (core plugs) and other oil rock samples. Equipment for ambient pressure and temperature samples.
SLK-2000-PM Desktop magnetic resonance equipment for measurements of sidewall cores (core plugs) and other oil rock samples. Shale rock measurement capacity. It includes Z-gradient for diffusion experiments. Equipment suitable for high pressure and high temperature sample measurements | https://en.wikipedia.org/wiki/Spinlock_SRL |
Spinmechatronics / ˌ s p ɪ n əm ɛ k ə ˈ t r ɒ n ɪ k s / is neologism referring to an emerging field of research concerned with the exploitation of spin -dependent phenomena and established spintronic methodologies and technologies in conjunction with electro-mechanical, magno-mechanical, acousto-mechanical and opto-mechanical systems. Most especially, spinmechatronics (or spin mechatronics) concerns the integration of micro- and nano- mechatronic systems with spin physics and spintronics .
While spinmechatronics has been recognised only recently [ 1 ] (2008) as an independent field, hybrid spin -mechanical system development dates back to the early nineteen-nineties, [ 2 ] with devices combining spintronics and micromechanics emerging at the turn of the twenty-first century.
One of the longest established spinmechatronic systems is the Magnetic Resonance Force Microscope or MRFM. First proposed by J. A. Sidles in a seminal paper of 1991 [ 2 ] – and since extensively developed both theoretically and experimentally by a number of international research groups [ 3 ] [ 4 ] – the MRFM operates by coupling a magnetically loaded micro-mechanical cantilever to an excited nuclear, proton or electron spin system. The MRFM concept effectively combines scanning atomic force microscopy ( AFM ) with magnetic resonance spectroscopy to provide a spectroscopic tool of unparalleled sensitivity. Nanometre resolution is possible, and the technique potentially forms the basis for ultra-high sensitivity, ultra-high resolution magnetic, biochemical, biomedical, and clinical diagnostics.
The synergy of micromechanics and established spintronic technologies for sensing applications is one of the most significant spinmechatronic developments of the last decade. At the beginning of this century, strain sensors incorporating magnetoresistive technologies emerged [ 5 ] and a wide range of devices exploiting similar principles are likely to realize research and commercial potential by 2015.
Contemporary innovation in spinmechatronics drives forward the independent advancement of cutting-edge science in spin physics , spintronics and micro- and nano- mechatronics and catalyses the development of wholly new instrumentation, control and fabrication techniques to facilitate and exploit their integration.
MEMS : micro- electromechanical systems are the key ingredient of micro- mechatronics . Micro-electromechanical systems are – as the name suggests – devices with significant dimensions in the micrometre regime or less. [ 6 ] [ 7 ] Highly suited to integration with electronic and microwave circuitry, they provide the key to electro-mechanical functionalities unachievable with classical precision mechatronics . Commercialisation of mass-produced Microelectromechanical systems products is rapidly picking up pace and includes printer ink-jet technology, 3D accelerometers , integrated pressure sensors, and Digital Light Processing (DLP) displays. At the cutting edge of Microelectromechanical systems fabrication and integration technologies are nano- electromechanical systems [ 8 ] ( NEMS ). Typical examples are micrometres long, tens of nanometres thick, and have mechanical resonance frequencies approaching 100 MHz. Their small physical dimensions and mass (of order pico- grams ) makes them highly sensitive to changes in stiffness ; this, their synergy with mechanical and data processing systems, and the option of attaching chemical/ biological molecules, makes them ideal for ultra high-performance mechanical, chemical and biological sensing applications.
Spin physics is a broad and active area of condensed-matter physics research. ‘ Spin ’ in this context refers to a quantum mechanical property of certain elementary particles and nuclei , and should not be confused with the classical (and better-known) concept of rotation . Spin physics spans studies of nuclear , electron and proton magnetic resonance , magnetism , and certain areas of optics. Spintronics is a branch of spin physics. Perhaps the two best known applications of spin physics are Magnetic Resonance Imaging (or MRI ) and the spintronic giant-magnetoresistive ( GMR ) hard disk read head.
Spintronic magnetoresistance is a major scientific and commercial success story. Today, most families own a spintronic device: the giant-magnetoresistive ( GMR ) hard disk read head in their computer. The science that gave rise to this phenomenal business opportunity – and earned the 2007 Nobel Prize for Physics – was the recognition that electrical carriers are characterized by both charge and spin . [ 9 ] [ 10 ] [ 11 ] Today, tunnelling-magnetoresistance (TMR) – which uses the electron spin as a label to allow or forbid electron tunnelling [ 12 ] – dominates the hard disk market and is rapidly establishing itself in areas as diverse as magnetic logic devices and biosensors. [ 13 ] Ongoing development is pushing the frontiers of TMR devices towards the nanoscale . | https://en.wikipedia.org/wiki/Spinmechatronics |
Spinning is a manufacturing process for creating polymer fibers . It is a specialized form of extrusion that uses a spinneret to form multiple continuous filaments. [ 1 ]
If the polymer is a thermoplastic then it can undergo melt spinning. The molten polymer is extruded through a spinneret composed of capillaries where the resulting filament is solidified by cooling. Nylon , olefin , polyester , saran , and sulfar are produced via this process. [ 1 ]
Pellets or granules of the solid polymer are fed into an extruder . The pellets are compressed, heated and melted by an extrusion screw, then fed to a spinning pump and into the spinneret.
The direct spinning process avoids the stage of solid polymer pellets. The polymer melt is produced from the raw materials, and then from the polymer finisher directly pumped to the spinning mill. Direct spinning is mainly applied during production of polyester fibers and filaments and is dedicated to high production capacity (>100 ton/day).
If the melting point of the polymer is higher than its degradation temperature, the polymer must undergo solution spinning techniques for fiber formation. The polymer is first dissolved in a solvent , forming a spinning solution (sometimes called a " dope "). The spinning solution then undergoes dry, wet, dry-jet wet, gel, or electrospinning techniques.
A spinning solution consisting of polymer and a volatile solvent is extruded through a spinneret into an evaporating chamber. A stream of hot air impinges on the jets of spinning solution emerging from the spinneret, evaporating the solvent, and solidifying the filaments. Solution blow spinning is a similar technique where polymer solution is sprayed directly onto a target to produce a nonwoven fiber mat. [ 2 ]
Wet spinning is the oldest of the five processes. The polymer is dissolved in a spinning solvent where it is extruded out through a spinneret submerged in a coagulation bath composed of nonsolvents. The coagulation bath causes the polymer to precipitate in fiber form. Acrylic , rayon , aramid , modacrylic , and spandex are produced via this process. [ 1 ]
A variant of wet spinning is dry-jet wet spinning, where the spinning solution passes through an air-gap prior to being submerged into the coagulation bath. This method is used in Lyocell spinning of dissolved cellulose , and can lead to higher polymer orientation due to the higher stretchability of the spinning solution versus the precipitated fiber.
Gel spinning, also known as semi-melt spinning, is used to obtain high strength or other special properties in the fibers. Instead of wet spinning, which relies on precipitation as the main mechanism for solidification, gel spinning relies on temperature-induced physical gelation as the primary method for solidification. The resulting gelled fiber is then swollen with the spinning solvent (similar to gelatin desserts ) which keeps the polymer chains somewhat bound together, resisting relaxation which is prevalent in wet spinning. The high solvent retention allows for ultra-high drawing as with ultra high molecular weight polyethylene (UHMWPE) (e.g., Spectra ® ) to produce fibers with a high degree of orientation, which increases fiber strength. The fibers are first cooled either with air or in a liquid bath to induce gelation, then the solvent is removed through ageing in a nonsolvent, or during the drawing stage. Some high strength polyethylene and polyacrylonitrile fibers are produced via this process. [ 1 ]
Electrospinning uses an electrical charge to draw very fine (typically on the micro or nano scale) fibres from a liquid - either a polymer solution or a polymer melt. Electrospinning shares characteristics of both electrospraying and conventional solution dry spinning [ 3 ] of fibers. The process does not require the use of coagulation chemistry or high temperatures to produce solid threads from solution. This makes the process particularly suited to the production of fibers using large and complex molecules. Melt electrospinning is also practiced; this method ensures that no solvent can be carried over into the final product. [ 4 ] [ 5 ]
Finally, the fibers are drawn to increase strength and orientation. This may be done while the polymer is still solidifying or after it has completely cooled. [ 1 ] | https://en.wikipedia.org/wiki/Spinning_(polymers) |
Spinning is a twisting technique to form yarn from fibers . The fiber intended is drawn out, twisted, and wound onto a bobbin . A few popular fibers that are spun into yarn other than cotton , which is the most popular, are viscose (the most common form of rayon), animal fibers such as wool , and synthetic polyester . [ 1 ] Originally done by hand using a spindle whorl , starting in the 500s AD the spinning wheel became the predominant spinning tool across Asia and Europe. The spinning jenny and spinning mule , invented in the late 1700s, made mechanical spinning far more efficient than spinning by hand, and especially made cotton manufacturing one of the most important industries of the Industrial Revolution .
The yarn issuing from the drafting rollers passes through a thread-guide, round a traveller that is free to rotate around a ring, and then onto a tube or bobbin , which is carried on to a spindle , the axis of which passes through a center of the ring.
The spindle is driven (usually at an angular velocity that is either constant or changes only slowly), and the traveller is dragged around a ring by the loop of yarn passing round it.
If the drafting rollers were stationary, the angular velocity of the traveller would be the same as that of the spindle, and each revolution of the spindle would cause one turn of a twist to be inserted in the loop of yarn between the roller nip and the traveller.
In spinning, however, the yarn is continually issuing from the rollers of the drafting system and, under these circumstances, the angular velocity of the traveller is less than that of the spindle by an amount that is just sufficient to allow the yarn to be wound onto the bobbin at the same rate as that at which it issues from the drafting rollers.
Each revolution of the traveller now inserts one turn of twist into the loop of yarn between the roller nip and the traveller but, in equilibrium, the number of turns of twist in the loop of yarn remains constant as the twisted yarn is passing through the traveller at a corresponding rate. [ citation needed ]
Artificial fibres are made by extruding a polymer through a spinneret into a medium where it hardens. Wet spinning ( rayon ) uses a coagulating medium. In dry spinning ( acetate and triacetate), the polymer is contained in a solvent that evaporates in the heated exit chamber. In melt spinning (nylons and polyesters ) the extruded polymer is cooled in gas or air and sets. [ 2 ] All these fibres will be of great length, often kilometers long.
Natural fibres can be divided into three categories: animals (sheep, goat, rabbit , silkworm ), minerals ( asbestos , gold , silver [ 1 ] ), or plants (cotton, flax , sisal ). These vegetable fibres can come from the seed (cotton), the stem (known as bast fibres : they include flax , hemp , and jute ) or the leaf ( sisal ). [ 3 ] Many processes are needed before a clean even staple is obtained. With the exception of silk, each of these fibres is short, only centimetres in length, and each has a rough surface that enables it to bond with similar staples. [ 3 ]
Artificial fibres can be processed as long fibres or batched and cut so they can be processed like a natural fibre.
Ring spinning is one of the most common spinning methods in the world. [ 4 ] Other systems include air-jet and open-end spinning , a technique where the staple fiber is blown by air into a rotor and attaches to the tail of formed yarn that is continually being drawn out of the chamber. Other methods of break spinning use needles and electrostatic forces. [ 5 ]
The processes to make short-staple yarn (typically spun from fibers from 1.9 to 5.1 centimetres (0.75 to 2.0 in)) are blending, opening, carding , pin-drafting, roving , spinning, and—if desired—plying and dyeing . In long staple spinning, the process may start with stretch-break of tow, a continuous "rope" of synthetic fiber. In open-end and air-jet spinning, the roving operation is eliminated. The spinning frame winds yarn around a bobbin. [ 6 ] Generally, after this step the yarn is wound to a cone for knitting or weaving.
In a spinning mule , the roving is pulled off bobbins and sequentially fed through rollers operating at several different speeds, thinning the roving at a consistent rate. The yarn is twisted through the spinning of the bobbin as the carriage moves out, and is rolled onto a cop as the carriage returns. Mule spinning produces a finer thread than ring spinning. [ 7 ] Spinning by the mule machine is an intermittent process as the frame advances and returns. It is the descendant of a device invented in 1779 by Samuel Crompton , and produces a softer, less twisted thread that is favored for fines and for weft .
The ring was a descendant of the Arkwright water frame of 1769 and creates yarn in a continuous process. The yarn is coarser, has a greater twist, and is stronger, making it more suitable for warp . Ring spinning is slow due to the distance the thread must pass around the ring. Similar methods have improved on this including flyer and bobbin and cap spinning.
The pre-industrial techniques of hand spinning with a spindle or spinning wheel continue to be practiced as handicraft or hobby and enable wool or unusual vegetable and animal staples to be used.
The origins of hand spinning fibers is unknown, but is believed to have originated separately in several cultures around the world long before the common era. The oldest known twisted fiber was found in southern France, and archaeologists believe it was created around 50,000-40,000 BCE. [ 8 ] People are thought to have originally twisted fibers together by rolling them up the thigh or between the fingers, although soon a stick was used to maintain tension and hold the twist in the fibers. [ 9 ]
People eventually discovered that adding a weight to the stick, often made of stone, wood, or clay and known as a whorl , helped to maintain momentum and left the hands free to draft the fiber. Whorl spindles are still the predominant method of spinning fiber in some parts of the world. [ 10 ]
The cultivation of cotton as well as the knowledge of its spinning and weaving in Meroë reached a high level around the 4th century BC. The export of textiles was one of the sources of wealth for Meroë. [ 11 ]
Hand spinning was an important cottage industry in medieval Europe, where the wool spinners (most often women and children) would provide enough yarn to service the needs of the men who operated the looms or to sell on in the putting-out system . After the invention of the spinning jenny water frame the demand was greatly reduced by mechanization. Its technology was specialized and costly and employed water as motive power. Spinning and weaving as cottage industries were displaced by dedicated manufactories, developed by industrialists and their investors; the spinning and weaving industries, once widespread, were concentrated where the sources of water, raw materials, and manpower were most readily available, particularly West Yorkshire . The British government was very protective of the technology and restricted its export. [ when? ] After World War I the colonies where the cotton was grown started to purchase and manufacture significant quantities of cotton spinning machinery. The next breakthrough was with the move over to break or open-end spinning , and then the adoption of artificial fibres . By then [ when? ] most production had moved to Asia. | https://en.wikipedia.org/wiki/Spinning_(textiles) |
Spinning band distillation is a technique used to separate liquid mixtures which are similar in boiling points. When liquids with similar boiling points are distilled , the vapors are mixtures, and not pure compounds. Fractionating columns help separate the mixture by allowing the mixed vapors to cool, condense, and vaporize again in accordance with Raoult's law . With each condensation-vaporization cycles, the vapors are enriched in a certain component. A larger surface area allows more cycles, improving separation.
Spinning band distillation takes this concept one step further by using a spinning helical band made of an inert material such as metal or Teflon to push the rising vapors and descending condensate to the sides of the column, coming into close contact with each other. This speeds up equilibration and provides for a greater number of condensation-vaporization cycles.
Spinning band distillation may sometimes be used to recycle waste solvents which contain different solvents, and other chemical compounds . [ citation needed ] | https://en.wikipedia.org/wiki/Spinning_band_distillation |
Spinning cone columns are used in a form of low temperature vacuum steam distillation to gently extract volatile chemicals from liquid foodstuffs while minimising the effect on the taste of the product. For instance, the columns can be used to remove some of the alcohol from wine, to remove "off" smells from cream, and to capture aroma compounds that would otherwise be lost in coffee processing .
The columns are made of stainless steel. Conical vanes are attached alternately to the wall of the column and to a central rotating shaft. The product is poured in at the top under vacuum, and steam is pumped into the column from below. [ 1 ] The vanes provide a large surface area over which volatile compounds can evaporate into the steam, and the rotation ensures a thin layer of the product is constantly moved over the moving cone. It typically takes 20 seconds for the liquid to move through the column, and industrial columns might process 16–160 litres per minute (960–9,600 L/h; 4.2–42.3 US gal/min; 250–2,540 US gal/h). The temperature and pressure can be adjusted depending on the compounds targeted.
Improvements in viticulture and warmer vintages have led to increasing levels of sugar in wine grapes, which have translated to higher levels of alcohol - which can reach over 15% ABV in Zinfandels from California . Some producers feel that this unbalances their wine, and use spinning cones to reduce the alcohol by 1-2 percentage points. In this case the wine is passed through the column once to distill out the most volatile aroma compounds which are then put to one side while the wine goes through the column a second time at higher temperature to extract alcohol. The aroma compounds are then mixed back into the wine. Some producers such as Joel Peterson of Ravenswood argue that technological "fixes" such as spinning cones remove a sense of terroir from the wine; if the wine has the tannins and other components to balance 15% alcohol, Peterson argues that it should be accepted on its own terms. [ 2 ]
The use of spinning cones, and other technologies such as reverse osmosis, was banned in the EU until recently, although for many years they could freely be used in wines imported into the EU from certain New World wine producing countries such as Australia and the USA. [ 3 ] In November 2007, the Wine Standards Branch (WSB) of the UK's Food Standards Agency banned the sale of a wine called Sovio, [ 3 ] made from Spanish grapes that would normally produce wines of 14% ABV. [ 4 ] Sovio runs 40-50% of the wine over spinning cones to reduce the alcohol content to 8%, which means that under EU law it could not be sold as wine as it was below 8.5%; above that, under the rules prevailing at the time, it would be banned because spinning cones could not be used in EU winemaking. [ 4 ]
Subsequently, the EU legalized dealcoholization with a 2% adjustment limit in its Code of Winemaking Practices, publishing that in its Commission Regulation (EC) No 606/2009 [ 5 ] and stipulating that the dealcoholization must be accomplished by physical separation techniques which would embrace the spinning cone method.
More recently, in International Organisation of Vine and Wine Resolutions OIV-OENO 394A-2012 [ 6 ] and OIV-OENO 394B-2012 [ 7 ] of June 22, 2012 EU recommended winemaking procedures were modified to permit use of the spinning cone column and membrane techniques such as reverse osmosis on wine, subject to a 20% limitation on the adjustment. That limitation is currently under review following the proposal by some EU members that it be eliminated altogether. The limitation is applicable only to products formally labeled as "wine". | https://en.wikipedia.org/wiki/Spinning_cone |
The spinning pinwheel is a type of progress indicator and a variation of the mouse pointer used in Apple 's macOS to indicate that an application is busy. [ 1 ]
Officially, the macOS Human Interface Guidelines refer to it as the spinning wait cursor , [ 2 ] but it is also known by other names. These include, but are not limited to, the spinning beach ball , [ 3 ] the spinning wheel of death , [ 4 ] and the spinning beach ball of death . [ 5 ]
A wristwatch was used as the first wait cursor in early versions of the classic Mac OS . Apple's HyperCard first popularized animated cursors, including a black-and-white spinning quartered circle resembling a beach ball . The beach-ball cursor was also adopted to indicate running script code in the HyperTalk -like AppleScript . The cursors could be advanced by repeated HyperTalk invocations of "set cursor to busy".
Wait cursors are activated by applications performing lengthy operations. Some versions of the Apple Installer used an animated "counting hand" cursor. Other applications provided their own theme-appropriate custom cursors, such as a revolving Yin Yang symbol, Fetch 's running dog, Retrospect 's spinning tape, and Pro Tools ' tapping fingers. Apple provided the standard interfaces for animating cursors: originally the Cursor Utilities (SpinCursor, RotateCursor) [ 6 ] and, in Mac OS 8 and later, the Appearance Manager (SetAnimatedThemeCursor). [ 7 ]
NeXTStep 1.0 used a monochrome icon resembling a spinning magneto-optical disk . [ a ] Some NeXT computers included an optical drive, which was often slower than a magnetic hard drive. This made it a common reason for the wait cursor to appear.
When color support was added in NeXTStep 2.0, color versions of all icons were added. The wait cursor was updated to reflect the bright rainbow surface of these removable disks, and that icon remained, even when later machines began using hard disk drives as primary storage. Contemporary CD-ROM drives were even slower (at 1x, 150 kbit/s). [ b ]
With the arrival of Mac OS X, the wait cursor was often called the "spinning beach ball" in the press, [ 8 ] presumably by authors not knowing its NeXT history or relating it to the HyperCard wait cursor.
The two-dimensional appearance was kept essentially unchanged [ c ] from NeXT to Rhapsody / Mac OS X Server 1.0 which otherwise had a user interface design resembling Mac OS 8 / Platinum theme . This continued through Mac OS X 10.0/Cheetah and Mac OS X 10.1/Puma , which introduced the Aqua user interface theme.
Mac OS X 10.2/Jaguar gave the cursor a glossy rounded "gumdrop" look in keeping with other OS X interface elements. [ 9 ] In OS X 10.10 , the entire pinwheel rotates (previously only the overlaying translucent layer moved).
With OS X 10.11 El Capitan the spinning wait-cursor's design was updated. It now has less shadowing and has brighter, more solid colors to better match the design of the user interface and the colors also turn with the spinning, not just the texture.
In single-task operating systems like the original Macintosh operating system , the wait cursor might indicate that the computer was completely unresponsive to user input, or just indicate that response may temporarily be slower than usual due to disk access. This changed with multitasking operating systems such as System Software 5 , where it is possible to switch to another application and continue to work there. Individual applications could also choose to display the wait cursor during long operations (and were often able to cancel this display with a keyboard command).
After the transition to Mac OS X ( macOS ), the display of the wait cursor was only able to be controlled by the operating system, not by the application . This could indicate that the application was in an infinite loop , or just performing a lengthy operation and ignoring events. Each application has an event queue that receives events from the operating system (for example, key presses and mouse button clicks); and if an application takes longer than 2 seconds [ 10 ] to process the events in its event queue (regardless of the cause), the operating system displays the wait cursor whenever the cursor hovers over that application's windows.
The icon is meant to indicate that the application is temporarily unresponsive, a state from which it should recover. It may also indicate that all or part of the application has entered an unrecoverable state or an infinite loop. During this time the user may be prevented from closing, resizing, or even minimizing the windows of the affected application (although moving the window is still possible in OS X, as well as previously hidden parts of the window which are usually redrawn, even when the application is otherwise unresponsive). While one application is unresponsive, typically other applications are usable. A file system and network delays are another common cause.
By default, events (and any actions they initiate) are processed sequentially, intended to limit the trivial amount of processing from each event. The spinning wait cursor will appear until the operation is complete. If the operation takes too long, the application will appear unresponsive. Developers may prevent this by using separate threads for lengthy processing, allowing the application's main thread to continue responding to external events. However, this greatly increases the application's complexity. Another approach is to divide the work into smaller packets and use NSRunLoop or Grand Central Dispatch .
Instruments is an application that comes with the Mac OS X Developer Tools. Along with its other functions, it allows the user to monitor and sample applications that are either not responding or performing a lengthy operation. Each time an application does not respond and the spinning wait cursor is activated, Instruments can sample the process to determine which code is causing the application to stop responding. With this information, the developer can rewrite code to avoid the cursor being activated.
Apple's guidelines suggest that developers try to avoid invoking the spinning wait cursor, and instead suggest using other user interface indicators, such as an asynchronous progress indicator . | https://en.wikipedia.org/wiki/Spinning_pinwheel |
Spinnova Plc (natively Spinnova Oyj ) is a Finnish textile material innovation company that has developed a patented technology for making textile fibre from wood, pulp, or waste, without harmful dissolving chemicals. [ 1 ]
The company has developed a machine which can transform cellulosic pulp into fiber for the textile industry. [ 2 ] The company’s headquarters and pilot factory are located in Jyväskylä , Finland , and it has offices in Helsinki , Finland. In 2021, Spinnova and its partner, Suzano Papel e Celulose , announced plans to build the first commercial-scale fiber production facility in Jyväskylä. [ 3 ] The facility, called Woodspin, opened in May 2023, with a capacity to produce 1,000 tonnes of sustainable, recyclable and fully biodegradable textile fibre from responsibly-grown wood each year. [ 4 ]
In March 2024, Spinnova announced an update to its strategy and targets, focused on technology sales. [ 5 ]
Spinnova's technology, initially developed at the VTT Technical Research Centre of Finland , led to the formation of an independent company in 2014. [ 6 ] This technology is focused on mechanically converting cellulosic fiber into textile fibers using bio-based raw materials . Spinnova has incorporated various materials in its fiber production, including wood, textile waste, and agricultural by-products like wheat and barley straw . [ 7 ] In 2021, the company expanded its research and development efforts to include the creation of fibers derived from leather waste. [ 8 ] | https://en.wikipedia.org/wiki/Spinnova |
In thermodynamics , the limit of local stability against phase separation with respect to small fluctuations is clearly defined by the condition that the second derivative of Gibbs free energy is zero.
The locus of these points (the inflection point within a G-x or G-c curve, Gibbs free energy as a function of composition ) is known as the spinodal curve. [ 1 ] [ 2 ] [ 3 ] For compositions within this curve, infinitesimally small fluctuations in composition and density will lead to phase separation via spinodal decomposition . Outside of the curve, the solution will be at least metastable with respect to fluctuations. [ 3 ] In other words, outside the spinodal curve some careful process may obtain a single phase system. [ 3 ] Inside it, only processes far from thermodynamic equilibrium , such as physical vapor deposition , will enable one to prepare single phase compositions. [ 4 ] The local points of coexisting compositions, defined by the common tangent construction, are known as a binodal coexistence curve , which denotes the minimum-energy equilibrium state of the system. Increasing temperature results in a decreasing difference between mixing entropy and mixing enthalpy, and thus the coexisting compositions come closer. The binodal curve forms the basis for the miscibility gap in a phase diagram. The free energy of mixing changes with temperature and concentration, and the binodal and spinodal meet at the critical or consolute temperature and composition. [ 5 ]
For binary solutions, the thermodynamic criterion which defines the spinodal curve is that the second derivative of free energy with respect to density or some composition variable is zero. [ 3 ] [ 6 ] [ 7 ]
Extrema of the spinodal in a temperature vs composition plot coincide with those of the binodal curve, and are known as critical points . [ 7 ] The spinodal itself can be thought of as a line of pseudocritical points, with the correlation function taking a scaling form with non-classical critical exponents . [ 8 ] Strictly speaking, a spinodal is defined as a mean field theoretic object . As such, the spinodal does not exist in real systems, [ 9 ] but one can extrapolate to infer the existence of a pseudospinodal that exhibits critical-like behavior such as critical slowing down . [ 10 ]
In the case of ternary isothermal liquid-liquid equilibria, the spinodal curve (obtained from the Hessian matrix) and the corresponding critical point can be used to help the experimental data correlation process. [ 11 ] [ 12 ] [ 13 ] | https://en.wikipedia.org/wiki/Spinodal |
Spinodal decomposition is a mechanism by which a single thermodynamic phase spontaneously separates into two phases (without nucleation ). [ 1 ] Decomposition occurs when there is no thermodynamic barrier to phase separation. As a result, phase separation via decomposition does not require the nucleation events resulting from thermodynamic fluctuations, which normally trigger phase separation.
Spinodal decomposition is observed when mixtures of metals or polymers separate into two co-existing phases, each rich in one species and poor in the other. [ 2 ] When the two phases emerge in approximately equal proportion (each occupying about the same volume or area), characteristic intertwined structures are formed that gradually coarsen (see animation). The dynamics of spinodal decomposition is commonly modeled using the Cahn–Hilliard equation .
Spinodal decomposition is fundamentally different from nucleation and growth. When there is a nucleation barrier to the formation of a second phase, time is taken by the system to overcome that barrier. As there is no barrier (by definition) to spinodal decomposition, some fluctuations (in the order parameter that characterizes the phase) start growing instantly. Furthermore, in spinodal decomposition, the two distinct phases start growing in any location uniformly throughout the volume, whereas a nucleated phase change begins at a discrete number of points.
Spinodal decomposition occurs when a homogenous phase becomes thermodynamically unstable. An unstable phase lies at a maximum in free energy . In contrast, nucleation and growth occur when a homogenous phase becomes metastable . That is, another biphasic system becomes lower in free energy, but the homogenous phase remains at a local minimum in free energy , and so is resistant to small fluctuations. J. Willard Gibbs described two criteria for a metastable phase: that it must remain stable against a small change over a large area. [ 3 ]
In the early 1940s, Bradley reported the observation of sidebands around the Bragg peaks in the X-ray diffraction pattern of a Cu-Ni-Fe alloy that had been quenched and then annealed inside the miscibility gap . Further observations on the same alloy were made by Daniel and Lipson, who demonstrated that the sidebands could be explained by a periodic modulation of composition in the <100> directions. From the spacing of the sidebands, they were able to determine the wavelength of the modulation, which was of the order of 100 angstroms (10 nm).
The growth of a composition modulation in an initially homogeneous alloy implies uphill diffusion or a negative diffusion coefficient. Becker and Dehlinger had already predicted a negative diffusivity inside the spinodal region of a binary system, but their treatments could not account for the growth of a modulation of a particular wavelength, such as was observed in the Cu-Ni-Fe alloy. In fact, any model based on Fick's law yields a physically unacceptable solution when the diffusion coefficient is negative.
The first explanation of the periodicity was given by Mats Hillert in his 1955 Doctoral Dissertation at MIT . Starting with a regular solution model, he derived a flux equation for one-dimensional diffusion on a discrete lattice. This equation differed from the usual one by the inclusion of a term, which allowed for the effect of the interfacial energy on the driving force of adjacent interatomic planes that differed in composition. Hillert solved the flux equation numerically and found that inside the spinodal it yielded a periodic variation of composition with distance. Furthermore, the wavelength of the modulation was of the same order as that observed in the Cu-Ni-Fe alloys. [ 4 ] [ 5 ]
Building on Hillert's work, a more flexible continuum model was subsequently developed by John W. Cahn and John Hilliard, who included the effects of coherency strains as well as the gradient energy term. The strains are significant in that they dictate the ultimate morphology of the decomposition in anisotropic materials. [ 6 ] [ 7 ] [ 8 ]
Free energies in the presence of small amplitude fluctuations, e.g. in concentration, can be evaluated using an approximation introduced by Ginzburg and Landau to describe magnetic field gradients in superconductors. This approach allows one to approximate the free energy as an expansion in terms of the concentration gradient ∇ c {\displaystyle \nabla c} , a vector . Since free energy is a scalar and we are probing near its minima, the term proportional to ∇ c {\displaystyle \nabla c} is negligible. The lowest order term is the quadratic expression κ ( ∇ c ) 2 {\displaystyle \kappa (\nabla c)^{2}} , a scalar. Here κ {\displaystyle \kappa } is a parameter that controls the free energy cost of variations in concentration c {\displaystyle c} .
The Cahn–Hilliard free energy is then
where f b {\displaystyle f_{b}} is the bulk free energy per unit volume of the homogeneous solution, and the integral is over the volume of the system.
We now want to study the stability of the system with respect to small fluctuations in the concentration c {\displaystyle c} , for example a sine wave of amplitude a {\displaystyle a} and wavevector q = 2 π / λ {\displaystyle q=2\pi /\lambda } , for λ {\displaystyle \lambda } the wavelength of the concentration wave. To be thermodynamically stable, the free energy change δ F {\displaystyle \delta F} due to any small amplitude concentration fluctuation δ c = a sin ( q → . r → ) {\displaystyle \delta c=a\sin({\vec {q}}.{\vec {r}})} , must be positive.
We may expand f b {\displaystyle f_{b}} about the average composition c o as follows:
and for the perturbation δ c = a sin ( q → . r → ) {\displaystyle \delta c=a\sin({\vec {q}}.{\vec {r}})} the free energy change is
When this is integrated over the volume V {\displaystyle V} , the sin ( q → . r → ) {\displaystyle \sin({\vec {q}}.{\vec {r}})} gives zero, while sin 2 ( q → . r → ) {\displaystyle \sin ^{2}({\vec {q}}.{\vec {r}})} and cos 2 ( q → . r → ) {\displaystyle \cos ^{2}({\vec {q}}.{\vec {r}})} integrate to give V / 2 {\displaystyle V/2} . So, then [ 9 ]
As a 2 > 0 {\displaystyle a^{2}>0} , thermodynamic stability requires that the term in brackets be positive. The 2 κ q 2 {\displaystyle 2\kappa q^{2}} is always positive but tends to zero at small wavevectors, large wavelengths. Since we are interested in macroscopic fluctuations, q → 0 {\displaystyle q\to 0} , stability requires that the second derivative of the free energy be positive. When it is, there is no spinodal decomposition, but when it is negative, spinodal decomposition will occur. Then fluctuations with wavevectors q < q c {\displaystyle q<q_{c}} become spontaneously unstable, where the critical wave number q c {\displaystyle q_{c}} is given by:
which corresponds to a fluctuations above a critical wavelength
Spinodal decomposition can be modeled using a generalized diffusion equation : [ 10 ] [ 11 ] [ 12 ]
for μ {\displaystyle \mu } the chemical potential and M {\displaystyle M} the mobility. As pointed out by Cahn, this equation can be considered as a phenomenological definition of the mobility M, which must by definition be positive. [ 13 ] It consists of the ratio of the flux to the local gradient in chemical potential. The chemical potential is a variation of the free energy and when this is the Cahn–Hilliard free energy this is [ 10 ]
and so
and now we want to see what happens to a small concentration fluctuation δ c = a exp ( ω t ) sin ( q → . r → ) {\displaystyle \delta c=a\exp(\omega t)\sin({\vec {q}}.{\vec {r}})} - note that now it has time dependence as a wavevector dependence. Here ω {\displaystyle \omega } is a growth rate. If ω < 0 {\displaystyle \omega <0} then the perturbation shrinks to nothing, the system is stable with respect to small perturbations or fluctuations, and there is no spinodal decomposition. However, if ω > 0 {\displaystyle \omega >0} then the perturbation grows and the system is unstable with respect to small perturbations or fluctuations: There is spinodal decomposition.
Substituting in this concentration fluctuation, we get
This gives the same expressions for the stability as above, but it also gives an expression for the growth rate of concentration perturbations
which has a maximum at a wavevector
So, at least at the beginning of spinodal decomposition, we expect the growing concentrations to mostly have this wavevector.
This type of phase transformation is known as spinodal decomposition , and can be illustrated on a phase diagram exhibiting a miscibility gap. Thus, phase separation occurs whenever a material transition into the unstable region of the phase diagram. The boundary of the unstable region sometimes referred to as the binodal or coexistence curve, is found by performing a common tangent construction of the free-energy diagram. Inside the binodal is a region called the spinodal, which is found by determining where the curvature of the free-energy curve is negative. The binodal and spinodal meet at the critical point. It is when a material is moved into the spinodal region of the phase diagram that spinodal decomposition can occur. [ 14 ]
The free energy curve is plotted as a function of composition for a temperature below the convolute temperature, T. Equilibrium phase compositions are those corresponding to the free energy minima. Regions of negative curvature (∂ 2 f/∂c 2 < 0 ) lie within the inflection points of the curve (∂ 2 f/∂c 2 = 0 ) which are called the spinodes. Their locus as a function of temperature defines the spinodal curve. For compositions within the spinodal, a homogeneous solution is unstable against infinitesimal fluctuations in density or composition, and there is no thermodynamic barrier to the growth of a new phase. Thus, the spinodal represents the limit of physical and chemical stability.
To reach the spinodal region of the phase diagram, a transition must take the material through the binodal region or the critical point. Often phase separation will occur via nucleation during this transition, and spinodal decomposition will not be observed. To observe spinodal decomposition, a very fast transition, often called a quench , is required to move from the stable to the spinodal unstable region of the phase diagram.
In some systems, ordering of the material leads to a compositional instability and this is known as a conditional spinodal , e.g. in the feldspars . [ 15 ] [ 16 ] [ 17 ] [ 18 ] [ 19 ]
For most crystalline solid solutions, there is a variation of lattice parameters with the composition. If the lattice of such a solution is to remain coherent in the presence of a composition modulation, mechanical work has to be done to strain the rigid lattice structure. The maintenance of coherency thus affects the driving force for diffusion. [ 13 ] [ 20 ] [ 21 ] [ 22 ]
Consider a crystalline solid containing a one-dimensional composition modulation along the x-direction. We calculate the elastic strain energy for a cubic crystal by estimating the work required to deform a slice of material so that it can be added coherently to an existing slab of cross-sectional area. We will assume that the composition modulation is along the x' direction and, as indicated, a prime will be used to distinguish the reference axes from the standard axes of a cubic system (that is, along the <100>). [ 11 ]
Let the lattice spacing in the plane of the slab be a o and that of the undeformed slice a . If the slice is to be coherent after the addition of the slab, it must be subjected to a strain ε in the z' and y' directions which is given by:
In the first step, the slice is deformed hydrostatically in order to produce the required strains to the z' and y' directions. We use the linear compressibility of a cubic system 1 / ( c 11 + 2 c 12 ) where the c's are the elastic constants. The stresses required to produce a hydrostatic strain of δ are therefore given by:
The elastic work per unit volume is given by:
where the ε's are the strains. The work performed per unit volume of the slice during the first step is therefore given by:
In the second step, the sides of the slice parallel to the x' direction are clamped and the stress in this direction is relaxed reversibly. Thus, ε z' = ε y' = 0. The result is that:
The net work performed on the slice in order to achieve coherency is given by:
or
The final step is to express c 1'1' in terms of the constants referred to the standard axes. From the rotation of axes, we obtain the following:
where l, m, n are the direction cosines of the x' axis and, therefore the direction cosines of the composition modulation. Combining these, we obtain the following:
The existence of any shear strain has not been accounted for. Cahn considered this problem, and concluded that shear would be absent for modulations along <100>, <110>, <111> and that for other directions the effect of shear strains would be small. It then follows that the total elastic strain energy of a slab of cross-sectional area A is given by:
We next have to relate the strain δ to the composition variation. Let a o be the lattice parameter of the unstrained solid of the average composition c o . Using a Taylor series expansion about c o yields the following:
in which
where the derivatives are evaluated at c o . Thus, neglecting higher-order terms, we have:
Substituting, we obtain:
This simple result indicates that the strain energy of a composition modulation depends only on the amplitude and is independent of the wavelength. For a given amplitude, the strain energy W E is proportional to Y. Consider a few special cases.
For an isotropic material:
so that:
This equation can also be written in terms of Young's modulus E and Poisson's ratio υ using the standard relationships:
Substituting, we obtain the following:
For most metals, the left-hand side of this equation
is positive, so that the elastic energy will be a minimum for those directions that minimize the term: l 2 m 2 + m 2 n 2 + l 2 n 2 . By inspection, those are seen to be <100>. For this case:
the same as for an isotropic material. At least one metal (molybdenum) has an anisotropy of the opposite sign. In this case, the directions for minimum W E will be those that maximize the directional cosine function. These directions are <111>, and
As we will see, the growth rate of the modulations will be a maximum in the directions that minimize Y. These directions, therefore, determine the morphology and structural characteristics of the decomposition in cubic solid solutions.
Rewriting the diffusion equation and including the term derived for the elastic energy yields the following:
or
which can alternatively be written in terms of the diffusion coefficient D as:
The simplest way of solving this equation is by using the method of Fourier transforms.
The motivation for the Fourier transformation comes from the study of a Fourier series . In the study of a Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines . Due to the properties of sine and cosine, it is possible to recover the amount of each wave in the sum by an integral. In many cases it is desirable to use Euler's formula , which states that e 2 πiθ = cos 2 πθ + i sin 2 πθ , to write Fourier series in terms of the basic waves e 2 πiθ , with the distinct advantage of simplifying many unwieldy formulas.
The passage from sines and cosines to complex exponentials makes it necessary for the Fourier coefficients to be complex-valued. The usual interpretation of this complex number is that it gives both the amplitude (or size) of the wave present in the function and the phase (or the initial angle) of the wave. This passage also introduces the need for negative "frequencies". (E.G. If θ were measured in seconds then the waves e 2 πiθ and e −2 πiθ would both complete one cycle per second—but they represent different frequencies in the Fourier transform. Hence, frequency no longer measures the number of cycles per unit time, but is closely related.)
If A(β) is the amplitude of a Fourier component of wavelength λ and wavenumber β = 2π/λ the spatial variation in composition can be expressed by the Fourier integral: [ 13 ]
in which the coefficients are defined by the inverse relationship:
Substituting, we obtain on equating coefficients:
This is an ordinary differential equation that has the solution:
in which A(β) is the initial amplitude of the Fourier component of wave wavenumber β and R(β) defined by:
or, expressed in terms of the diffusion coefficient D:
In a similar manner, the new diffusion equation:
has a simple sine wave solution given by:
where R ( β ) {\displaystyle R(\beta )} is obtained by substituting this solution back into the diffusion equation as follows:
For solids, the elastic strains resulting from coherency add terms to the amplification factor R ( β ) {\displaystyle R(\beta )} as follows:
where, for isotropic solids:
where E is Young's modulus of elasticity, ν is Poisson's ratio, and η is the linear strain per unit composition difference. For anisotropic solids, the elastic term depends on the direction in a manner that can be predicted by elastic constants and how the lattice parameters vary with composition. For the cubic case, Y is a minimum for either (100) or (111) directions, depending only on the sign of the elastic anisotropy.
Thus, by describing any composition fluctuation in terms of its Fourier components, Cahn showed that a solution would be unstable concerning to the sinusoidal fluctuations of a critical wavelength. By relating the elastic strain energy to the amplitudes of such fluctuations, he formalized the wavelength or frequency dependence of the growth of such fluctuations, and thus introduced the principle of selective amplification of Fourier components of certain wavelengths. The treatment yields the expected mean particle size or wavelength of the most rapidly growing fluctuation.
Thus, the amplitude of composition fluctuations should grow continuously until a metastable equilibrium is reached with preferential amplification of components of particular wavelengths. The kinetic amplification factor R is negative when the solution is stable to the fluctuation, zero at the critical wavelength, and positive for longer wavelengths—exhibiting a maximum at exactly 2 {\displaystyle {\sqrt {2}}} times the critical wavelength.
Consider a homogeneous solution within the spinodal. It will initially have a certain amount of fluctuation from the average composition which may be written as a Fourier integral. Each Fourier component of that fluctuation will grow or diminish according to its wavelength.
Because of the maximum in R as a function of wavelength, those components of the fluctuation with 2 {\displaystyle {\sqrt {2}}} times the critical wavelength will grow fastest and will dominate. This "principle of selective amplification" depends on the initial presence of these wavelengths but does not critically depend on their exact amplitude relative to other wavelengths (if the time is large compared with (1/R). It does not depend on any additional assumptions, since different wavelengths can coexist and do not interfere with one another.
Limitations of this theory would appear to arise from this assumption and the absence of an expression formulated to account for irreversible processes during phase separation which may be associated with internal friction and entropy production . In practice, frictional damping is generally present and some of the energy is transformed into thermal energy. Thus, the amplitude and intensity of a one-dimensional wave decrease with distance from the source, and for a three-dimensional wave, the decrease will be greater.
In the spinodal region of the phase diagram, the free energy can be lowered by allowing the components to separate, thus increasing the relative concentration of a component material in a particular region of the material. The concentration will continue to increase until the material reaches the stable part of the phase diagram. Very large regions of material will change their concentration slowly due to the amount of material that must be moved. Very small regions will shrink away due to the energy cost of maintaining an interface between two dissimilar component materials. [ 23 ] [ 24 ] [ 25 ]
To initiate a homogeneous quench a control parameter, such as temperature, is abruptly and globally changed. For a binary mixture of A {\displaystyle A} -type and B {\displaystyle B} -type materials, the Landau free-energy
is a good approximation of the free energy near the critical point and is often used to study homogeneous quenches. The mixture concentration ϕ = ρ A − ρ B {\displaystyle \phi =\rho _{A}-\rho _{B}} is the density difference of the mixture components, the control parameters which determine the stability of the mixture are A {\displaystyle A} and B {\displaystyle B} , and the interfacial energy cost is determined by κ {\displaystyle \kappa } .
Diffusive motion often dominates at the length-scale of spinodal decomposition. The equation of motion for a diffusive system is
where m {\displaystyle m} is the diffusive mobility, ξ ( x ) {\displaystyle \xi (x)} is some random noise such that ⟨ ξ ( x ) ⟩ = 0 {\displaystyle \langle \xi (x)\rangle =0} , and the chemical potential μ {\displaystyle \mu } is derived from the Landau free-energy:
We see that if A < 0 {\displaystyle A<0} , small fluctuations around ϕ = 0 {\displaystyle \phi =0} have a negative effective diffusive mobility and will grow rather than shrink. To understand the growth dynamics, we disregard the fluctuating currents due to ξ {\displaystyle \xi } , linearize the equation of motion around ϕ = ϕ i n {\displaystyle \phi =\phi _{in}} and perform a Fourier transform into k {\displaystyle k} -space. This leads to
which has an exponential growth solution:
Since the growth rate R ( k ) {\displaystyle R(k)} is exponential, the fastest growing angular wavenumber
will quickly dominate the morphology. We now see that spinodal decomposition results in domains of the characteristic length scale called the spinodal length :
The growth rate of the fastest-growing angular wave number is
where t s p {\displaystyle t_{sp}} is known as the spinodal time .
The spinodal length and spinodal time can be used to nondimensionalize the equation of motion, resulting in universal scaling for spinodal decomposition.
Spinodal phase decomposition has been used to generate architected materials by interpreting one phase as solid, and the other phase as void. These spinodal architected materials present interesting mechanical properties, such as high energy absorption, [ 26 ] insensitivity to imperfections, [ 27 ] superior mechanical resilience, [ 28 ] and high stiffness-to-weight ratio. [ 29 ] Furthermore, by controlling the phase separation, i.e., controlling the proportion of materials, and/or imposing preferential directions in the decompositions, one can control the density, and preferential directions effectively tuning the strength, weight, and anisotropy of the resulting architected material. [ 30 ] Another interesting property of spinodal materials is the capability to seamlessly transition between different classes, orientations, and densities, [ 30 ] thereby enabling the manufacturing of effectively multi-material structures. [ 31 ] | https://en.wikipedia.org/wiki/Spinodal_decomposition |
In geometry and physics , spinors (pronounced "spinner" IPA / s p ɪ n ər / ) are elements of a complex vector space that can be associated with Euclidean space . [ b ] A spinor transforms linearly when the Euclidean space is subjected to a slight ( infinitesimal ) rotation, [ c ] but unlike geometric vectors and tensors , a spinor transforms to its negative when the
space rotates through 360° (see picture). It takes a rotation of 720° for a spinor to go back to its original state. This property characterizes spinors: spinors can be viewed as the "square roots" of vectors (although this is inaccurate and may be misleading; they are better viewed as "square roots" of sections of vector bundles – in the case of the exterior algebra bundle of the cotangent bundle , they thus become "square roots" of differential forms ).
It is also possible to associate a substantially similar notion of spinor to Minkowski space , in which case the Lorentz transformations of special relativity play the role of rotations. Spinors were introduced in geometry by Élie Cartan in 1913. [ 1 ] [ d ] In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum , or "spin", of the electron and other subatomic particles. [ e ]
Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group ). There are two topologically distinguishable classes ( homotopy classes ) of paths through rotations that result in the same overall rotation, as illustrated by the belt trick puzzle. These two inequivalent classes yield spinor transformations of opposite sign. The spin group is the group of all rotations keeping track of the class. [ f ] It doubly covers the rotation group, since each rotation can be obtained in two inequivalent ways as the endpoint of a path. The space of spinors by definition is equipped with a (complex) linear representation of the spin group, meaning that elements of the spin group act as linear transformations on the space of spinors, in a way that genuinely depends on the homotopy class. [ g ] In mathematical terms, spinors are described by a double-valued projective representation of the rotation group SO(3) .
Although spinors can be defined purely as elements of a representation space of the spin group (or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford algebra . The Clifford algebra is an associative algebra that can be constructed from Euclidean space and its inner product in a basis-independent way. Both the spin group and its Lie algebra are embedded inside the Clifford algebra in a natural way, and in applications the Clifford algebra is often the easiest to work with. [ h ] A Clifford space operates on a spinor space, and the elements of a spinor space are spinors. [ 3 ] After choosing an orthonormal basis of Euclidean space, a representation of the Clifford algebra is generated by gamma matrices , matrices that satisfy a set of canonical anti-commutation relations. The spinors are the column vectors on which these matrices act. In three Euclidean dimensions, for instance, the Pauli spin matrices are a set of gamma matrices, [ i ] and the two-component complex column vectors on which these matrices act are spinors. However, the particular matrix representation of the Clifford algebra, hence what precisely constitutes a "column vector" (or spinor), involves the choice of basis and gamma matrices in an essential way. As a representation of the spin group, this realization of spinors as (complex [ j ] ) column vectors will either be irreducible if the dimension is odd, or it will decompose into a pair of so-called "half-spin" or Weyl representations if the dimension is even. [ k ]
What characterizes spinors and distinguishes them from geometric vectors and other tensors is subtle. Consider applying a rotation to the coordinates of a system. No object in the system itself has moved, only the coordinates have, so there will always be a compensating change in those coordinate values when applied to any object of the system. Geometrical vectors, for example, have components that will undergo the same rotation as the coordinates. More broadly, any tensor associated with the system (for instance, the stress of some medium) also has coordinate descriptions that adjust to compensate for changes to the coordinate system itself.
Spinors do not appear at this level of the description of a physical system, when one is concerned only with the properties of a single isolated rotation of the coordinates. Rather, spinors appear when we imagine that instead of a single rotation, the coordinate system is gradually ( continuously ) rotated between some initial and final configuration. For any of the familiar and intuitive ("tensorial") quantities associated with the system, the transformation law does not depend on the precise details of how the coordinates arrived at their final configuration. Spinors, on the other hand, are constructed in such a way that makes them sensitive to how the gradual rotation of the coordinates arrived there: They exhibit path-dependence. It turns out that, for any final configuration of the coordinates, there are actually two (" topologically ") inequivalent gradual (continuous) rotations of the coordinate system that result in this same configuration. This ambiguity is called the homotopy class of the gradual rotation. The belt trick (shown, in which both ends of the rotated object are physically tethered to an external reference) demonstrates two different rotations, one through an angle of 2 π and the other through an angle of 4 π , having the same final configurations but different classes. Spinors actually exhibit a sign-reversal that genuinely depends on this homotopy class. This distinguishes them from vectors and other tensors, none of which can feel the class.
Spinors can be exhibited as concrete objects using a choice of Cartesian coordinates . In three Euclidean dimensions, for instance, spinors can be constructed by making a choice of Pauli spin matrices corresponding to ( angular momenta about) the three coordinate axes. These are 2×2 matrices with complex entries, and the two-component complex column vectors on which these matrices act by matrix multiplication are the spinors. In this case, the spin group is isomorphic to the group of 2×2 unitary matrices with determinant one, which naturally sits inside the matrix algebra. This group acts by conjugation on the real vector space spanned by the Pauli matrices themselves, [ m ] realizing it as a group of rotations among them, [ n ] but it also acts on the column vectors (that is, the spinors).
More generally, a Clifford algebra can be constructed from any vector space V equipped with a (nondegenerate) quadratic form , such as Euclidean space with its standard dot product or Minkowski space with its standard Lorentz metric. The space of spinors is the space of column vectors with 2 ⌊ dim V / 2 ⌋ {\displaystyle 2^{\lfloor \dim V/2\rfloor }} components. The orthogonal Lie algebra (i.e., the infinitesimal "rotations") and the spin group associated to the quadratic form are both (canonically) contained in the Clifford algebra, so every Clifford algebra representation also defines a representation of the Lie algebra and the spin group. [ o ] Depending on the dimension and metric signature , this realization of spinors as column vectors may be irreducible or it may decompose into a pair of so-called "half-spin" or Weyl representations. [ p ] When the vector space V is four-dimensional, the algebra is described by the gamma matrices .
The space of spinors is formally defined as the fundamental representation of the Clifford algebra . (This may or may not decompose into irreducible representations.) The space of spinors may also be defined as a spin representation of the orthogonal Lie algebra . These spin representations are also characterized as the finite-dimensional projective representations of the special orthogonal group that do not factor through linear representations. Equivalently, a spinor is an element of a finite-dimensional group representation of the spin group on which the center acts non-trivially.
There are essentially two frameworks for viewing the notion of a spinor: the representation theoretic point of view and the geometric point of view .
From a representation theoretic point of view, one knows beforehand that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations , and their constituents spinors . From this view, a spinor must belong to a representation of the double cover of the rotation group SO( n , R {\displaystyle \mathbb {R} } ) , or more generally of a double cover of the generalized special orthogonal group SO + ( p , q , R {\displaystyle \mathbb {R} } ) on spaces with a metric signature of ( p , q ) . These double covers are Lie groups , called the spin groups Spin( n ) or Spin( p , q ) . All the properties of spinors, and their applications and derived objects, are manifested first in the spin group. Representations of the double covers of these groups yield double-valued projective representations of the groups themselves. (This means that the action of a particular rotation on vectors in the quantum Hilbert space is only defined up to a sign.)
In summary, given a representation specified by the data ( V , Spin ( p , q ) , ρ ) {\displaystyle (V,{\text{Spin}}(p,q),\rho )} where V {\displaystyle V} is a vector space over K = R {\displaystyle K=\mathbb {R} } or C {\displaystyle \mathbb {C} } and ρ {\displaystyle \rho } is a homomorphism ρ : Spin ( p , q ) → GL ( V ) {\displaystyle \rho :{\text{Spin}}(p,q)\rightarrow {\text{GL}}(V)} , a spinor is an element of the vector space V {\displaystyle V} .
From a geometrical point of view, one can explicitly construct the spinors and then examine how they behave under the action of the relevant Lie groups. This latter approach has the advantage of providing a concrete and elementary description of what a spinor is. However, such a description becomes unwieldy when complicated properties of the spinors, such as Fierz identities , are needed.
The language of Clifford algebras [ 5 ] (sometimes called geometric algebras ) provides a complete picture of the spin representations of all the spin groups, and the various relationships between those representations, via the classification of Clifford algebras . It largely removes the need for ad hoc constructions.
In detail, let V be a finite-dimensional complex vector space with nondegenerate symmetric bilinear form g . The Clifford algebra Cℓ( V , g ) is the algebra generated by V along with the anticommutation relation xy + yx = 2 g ( x , y ) . It is an abstract version of the algebra generated by the gamma or Pauli matrices . If V = C n {\displaystyle \mathbb {C} ^{n}} , with the standard form g ( x , y ) = x T y = x 1 y 1 + ... + x n y n we denote the Clifford algebra by Cℓ n ( C {\displaystyle \mathbb {C} } ). Since by the choice of an orthonormal basis every complex vector space with non-degenerate form is isomorphic to this standard example, this notation is abused more generally if dim C {\displaystyle \mathbb {C} } ( V ) = n . If n = 2 k is even, Cℓ n ( C {\displaystyle \mathbb {C} } ) is isomorphic as an algebra (in a non-unique way) to the algebra Mat(2 k , C {\displaystyle \mathbb {C} } ) of 2 k × 2 k complex matrices (by the Artin–Wedderburn theorem and the easy to prove fact that the Clifford algebra is central simple ). If n = 2 k + 1 is odd, Cℓ 2 k +1 ( C {\displaystyle \mathbb {C} } ) is isomorphic to the algebra Mat(2 k , C {\displaystyle \mathbb {C} } ) ⊕ Mat(2 k , C {\displaystyle \mathbb {C} } ) of two copies of the 2 k × 2 k complex matrices. Therefore, in either case Cℓ( V , g ) has a unique (up to isomorphism) irreducible representation (also called simple Clifford module ), commonly denoted by Δ, of dimension 2 [ n /2] . Since the Lie algebra so ( V , g ) is embedded as a Lie subalgebra in Cℓ( V , g ) equipped with the Clifford algebra commutator as Lie bracket, the space Δ is also a Lie algebra representation of so ( V , g ) called a spin representation . If n is odd, this Lie algebra representation is irreducible. If n is even, it splits further [ clarification needed ] into two irreducible representations Δ = Δ + ⊕ Δ − called the Weyl or half-spin representations .
Irreducible representations over the reals in the case when V is a real vector space are much more intricate, and the reader is referred to the Clifford algebra article for more details.
Spinors form a vector space , usually over the complex numbers , equipped with a linear group representation of the spin group that does not factor through a representation of the group of rotations (see diagram). The spin group is the group of rotations keeping track of the homotopy class. Spinors are needed to encode basic information about the topology of the group of rotations because that group is not simply connected , but the simply connected spin group is its double cover . So for every rotation there are two elements of the spin group that represent it. Geometric vectors and other tensors cannot feel the difference between these two elements, but they produce opposite signs when they affect any spinor under the representation. Thinking of the elements of the spin group as homotopy classes of one-parameter families of rotations, each rotation is represented by two distinct homotopy classes of paths to the identity. If a one-parameter family of rotations is visualized as a ribbon in space, with the arc length parameter of that ribbon being the parameter (its tangent, normal, binormal frame actually gives the rotation), then these two distinct homotopy classes are visualized in the two states of the belt trick puzzle (above). The space of spinors is an auxiliary vector space that can be constructed explicitly in coordinates, but ultimately only exists up to isomorphism in that there is no "natural" construction of them that does not rely on arbitrary choices such as coordinate systems. A notion of spinors can be associated, as such an auxiliary mathematical object, with any vector space equipped with a quadratic form such as Euclidean space with its standard dot product , or Minkowski space with its Lorentz metric . In the latter case, the "rotations" include the Lorentz boosts , but otherwise the theory is substantially similar. [ citation needed ]
The constructions given above, in terms of Clifford algebra or representation theory, can be thought of as defining spinors as geometric objects in zero-dimensional space-time . To obtain the spinors of physics, such as the Dirac spinor , one extends the construction to obtain a spin structure on 4-dimensional space-time ( Minkowski space ). Effectively, one starts with the tangent manifold of space-time, each point of which is a 4-dimensional vector space with SO(3,1) symmetry, and then builds the spin group at each point. The neighborhoods of points are endowed with concepts of smoothness and differentiability: the standard construction is one of a fiber bundle , the fibers of which are affine spaces transforming under the spin group. After constructing the fiber bundle, one may then consider differential equations, such as the Dirac equation , or the Weyl equation on the fiber bundle. These equations (Dirac or Weyl) have solutions that are plane waves , having symmetries characteristic of the fibers, i.e. having the symmetries of spinors, as obtained from the (zero-dimensional) Clifford algebra/spin representation theory described above. Such plane-wave solutions (or other solutions) of the differential equations can then properly be called fermions ; fermions have the algebraic qualities of spinors. By general convention, the terms "fermion" and "spinor" are often used interchangeably in physics, as synonyms of one-another. [ citation needed ]
It appears that all fundamental particles in nature that are spin-1/2 are described by the Dirac equation, with the possible exception of the neutrino . There does not seem to be any a priori reason why this would be the case. A perfectly valid choice for spinors would be the non-complexified version of Cℓ 2,2 ( R {\displaystyle \mathbb {R} } ) , the Majorana spinor . [ 6 ] There also does not seem to be any particular prohibition to having Weyl spinors appear in nature as fundamental particles.
The Dirac, Weyl, and Majorana spinors are interrelated, and their relation can be elucidated on the basis of real geometric algebra. [ 7 ] Dirac and Weyl spinors are complex representations while Majorana spinors are real representations.
Weyl spinors are insufficient to describe massive particles, such as electrons , since the Weyl plane-wave solutions necessarily travel at the speed of light; for massive particles, the Dirac equation is needed. The initial construction of the Standard Model of particle physics starts with both the electron and the neutrino as massless Weyl spinors; the Higgs mechanism gives electrons a mass; the classical neutrino remained massless, and was thus an example of a Weyl spinor. [ q ] However, because of observed neutrino oscillation , it is now believed that they are not Weyl spinors, but perhaps instead Majorana spinors. [ 8 ] It is not known whether Weyl spinor fundamental particles exist in nature.
The situation for condensed matter physics is different: one can construct two and three-dimensional "spacetimes" in a large variety of different physical materials, ranging from semiconductors to far more exotic materials. In 2015, an international team led by Princeton University scientists announced that they had found a quasiparticle that behaves as a Weyl fermion. [ 9 ]
One major mathematical application of the construction of spinors is to make possible the explicit construction of linear representations of the Lie algebras of the special orthogonal groups , and consequently spinor representations of the groups themselves. At a more profound level, spinors have been found to be at the heart of approaches to the Atiyah–Singer index theorem , and to provide constructions in particular for discrete series representations of semisimple groups .
The spin representations of the special orthogonal Lie algebras are distinguished from the tensor representations given by Weyl's construction by the weights . Whereas the weights of the tensor representations are integer linear combinations of the roots of the Lie algebra, those of the spin representations are half-integer linear combinations thereof. Explicit details can be found in the spin representation article.
The spinor can be described, in simple terms, as "vectors of a space the transformations of which are related in a particular way to rotations in physical space". [ 10 ] Stated differently:
Spinors ... provide a linear representation of the group of rotations in a space with any number n {\displaystyle n} of dimensions, each spinor having 2 ν {\displaystyle 2^{\nu }} components where n = 2 ν + 1 {\displaystyle n=2\nu +1} or 2 ν {\displaystyle 2\nu } . [ 2 ]
Several ways of illustrating everyday analogies have been formulated in terms of the plate trick , tangloids and other examples of orientation entanglement .
Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah 's statement that is recounted by Dirac's biographer Graham Farmelo:
No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the "square root" of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors. [ 11 ]
The most general mathematical form of spinors was discovered by Élie Cartan in 1913. [ 12 ] The word "spinor" was coined by Paul Ehrenfest in his work on quantum physics . [ 13 ]
Spinors were first applied to mathematical physics by Wolfgang Pauli in 1927, when he introduced his spin matrices . [ 14 ] The following year, Paul Dirac discovered the fully relativistic theory of electron spin by showing the connection between spinors and the Lorentz group . [ 15 ] By the 1930s, Dirac, Piet Hein and others at the Niels Bohr Institute (then known as the Institute for Theoretical Physics of the University of Copenhagen) created toys such as Tangloids to teach and model the calculus of spinors.
Spinor spaces were represented as left ideals of a matrix algebra in 1930, by Gustave Juvett [ 16 ] and by Fritz Sauter . [ 17 ] [ 18 ] More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero. In this manner the spinor space became a minimal left ideal in Mat(2, C {\displaystyle \mathbb {C} } ) . [ r ] [ 20 ]
In 1947 Marcel Riesz constructed spinor spaces as elements of a minimal left ideal of Clifford algebras . In 1966/1967, David Hestenes [ 21 ] [ 22 ] replaced spinor spaces by the even subalgebra Cℓ 0 1,3 ( R {\displaystyle \mathbb {R} } ) of the spacetime algebra Cℓ 1,3 ( R {\displaystyle \mathbb {R} } ). [ 18 ] [ 20 ] As of the 1980s, the theoretical physics group at Birkbeck College around David Bohm and Basil Hiley has been developing algebraic approaches to quantum theory that build on Sauter and Riesz' identification of spinors with minimal left ideals.
Some simple examples of spinors in low dimensions arise from considering the even-graded subalgebras of the Clifford algebra Cℓ p , q ( R {\displaystyle \mathbb {R} } ) . This is an algebra built up from an orthonormal basis of n = p + q mutually orthogonal vectors under addition and multiplication, p of which have norm +1 and q of which have norm −1, with the product rule for the basis vectors e i e j = { + 1 i = j , i ∈ ( 1 , … , p ) − 1 i = j , i ∈ ( p + 1 , … , n ) − e j e i i ≠ j . {\displaystyle e_{i}e_{j}={\begin{cases}+1&i=j,\,i\in (1,\ldots ,p)\\-1&i=j,\,i\in (p+1,\ldots ,n)\\-e_{j}e_{i}&i\neq j.\end{cases}}}
The Clifford algebra Cℓ 2,0 ( R {\displaystyle \mathbb {R} } ) is built up from a basis of one unit scalar, 1, two orthogonal unit vectors, σ 1 and σ 2 , and one unit pseudoscalar i = σ 1 σ 2 . From the definitions above, it is evident that ( σ 1 ) 2 = ( σ 2 ) 2 = 1 , and ( σ 1 σ 2 )( σ 1 σ 2 ) = − σ 1 σ 1 σ 2 σ 2 = −1 .
The even subalgebra Cℓ 0 2,0 ( R {\displaystyle \mathbb {R} } ), spanned by even-graded basis elements of Cℓ 2,0 ( R {\displaystyle \mathbb {R} } ), determines the space of spinors via its representations. It is made up of real linear combinations of 1 and σ 1 σ 2 . As a real algebra, Cℓ 0 2,0 ( R {\displaystyle \mathbb {R} } ) is isomorphic to the field of complex numbers C {\displaystyle \mathbb {C} } . As a result, it admits a conjugation operation (analogous to complex conjugation ), sometimes called the reverse of a Clifford element, defined by ( a + b σ 1 σ 2 ) ∗ = a + b σ 2 σ 1 {\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}} which, by the Clifford relations, can be written ( a + b σ 1 σ 2 ) ∗ = a + b σ 2 σ 1 = a − b σ 1 σ 2 . {\displaystyle (a+b\sigma _{1}\sigma _{2})^{*}=a+b\sigma _{2}\sigma _{1}=a-b\sigma _{1}\sigma _{2}.}
The action of an even Clifford element γ ∈ Cℓ 0 2,0 ( R {\displaystyle \mathbb {R} } ) on vectors, regarded as 1-graded elements of Cℓ 2,0 ( R {\displaystyle \mathbb {R} } ), is determined by mapping a general vector u = a 1 σ 1 + a 2 σ 2 to the vector γ ( u ) = γ u γ ∗ , {\displaystyle \gamma (u)=\gamma u\gamma ^{*},} where γ ∗ {\displaystyle \gamma ^{*}} is the conjugate of γ {\displaystyle \gamma } , and the product is Clifford multiplication. In this situation, a spinor [ s ] is an ordinary complex number. The action of γ {\displaystyle \gamma } on a spinor ϕ {\displaystyle \phi } is given by ordinary complex multiplication: γ ( ϕ ) = γ ϕ . {\displaystyle \gamma (\phi )=\gamma \phi .}
An important feature of this definition is the distinction between ordinary vectors and spinors, manifested in how the even-graded elements act on each of them in different ways. In general, a quick check of the Clifford relations reveals that even-graded elements conjugate-commute with ordinary vectors: γ ( u ) = γ u γ ∗ = γ 2 u . {\displaystyle \gamma (u)=\gamma u\gamma ^{*}=\gamma ^{2}u.}
On the other hand, in comparison with its action on spinors γ ( ϕ ) = γ ϕ {\displaystyle \gamma (\phi )=\gamma \phi } , the action of γ {\displaystyle \gamma } on ordinary vectors appears as the square of its action on spinors.
Consider, for example, the implication this has for plane rotations. Rotating a vector through an angle of θ corresponds to γ 2 = exp( θ σ 1 σ 2 ) , so that the corresponding action on spinors is via γ = ± exp( θ σ 1 σ 2 /2) . In general, because of logarithmic branching , it is impossible to choose a sign in a consistent way. Thus the representation of plane rotations on spinors is two-valued.
In applications of spinors in two dimensions, it is common to exploit the fact that the algebra of even-graded elements (that is just the ring of complex numbers) is identical to the space of spinors. So, by abuse of language , the two are often conflated. One may then talk about "the action of a spinor on a vector". In a general setting, such statements are meaningless. But in dimensions 2 and 3 (as applied, for example, to computer graphics ) they make sense.
The Clifford algebra Cℓ 3,0 ( R {\displaystyle \mathbb {R} } ) is built up from a basis of one unit scalar, 1, three orthogonal unit vectors, σ 1 , σ 2 and σ 3 , the three unit bivectors σ 1 σ 2 , σ 2 σ 3 , σ 3 σ 1 and the pseudoscalar i = σ 1 σ 2 σ 3 . It is straightforward to show that ( σ 1 ) 2 = ( σ 2 ) 2 = ( σ 3 ) 2 = 1 , and ( σ 1 σ 2 ) 2 = ( σ 2 σ 3 ) 2 = ( σ 3 σ 1 ) 2 = ( σ 1 σ 2 σ 3 ) 2 = −1 .
The sub-algebra of even-graded elements is made up of scalar dilations, u ′ = ρ ( 1 2 ) u ρ ( 1 2 ) = ρ u , {\displaystyle u'=\rho ^{\left({\frac {1}{2}}\right)}u\rho ^{\left({\frac {1}{2}}\right)}=\rho u,} and vector rotations u ′ = γ u γ ∗ , {\displaystyle u'=\gamma u\gamma ^{*},} where
corresponds to a vector rotation through an angle θ about an axis defined by a unit vector v = a 1 σ 1 + a 2 σ 2 + a 3 σ 3 .
As a special case, it is easy to see that, if v = σ 3 , this reproduces the σ 1 σ 2 rotation considered in the previous section; and that such rotation leaves the coefficients of vectors in the σ 3 direction invariant, since
[ cos ( θ 2 ) − i σ 3 sin ( θ 2 ) ] σ 3 [ cos ( θ 2 ) + i σ 3 sin ( θ 2 ) ] = [ cos 2 ( θ 2 ) + sin 2 ( θ 2 ) ] σ 3 = σ 3 . {\displaystyle \left[\cos \left({\frac {\theta }{2}}\right)-i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]\sigma _{3}\left[\cos \left({\frac {\theta }{2}}\right)+i\sigma _{3}\sin \left({\frac {\theta }{2}}\right)\right]=\left[\cos ^{2}\left({\frac {\theta }{2}}\right)+\sin ^{2}\left({\frac {\theta }{2}}\right)\right]\sigma _{3}=\sigma _{3}.}
The bivectors σ 2 σ 3 , σ 3 σ 1 and σ 1 σ 2 are in fact Hamilton's quaternions i , j , and k , discovered in 1843:
i = − σ 2 σ 3 = − i σ 1 j = − σ 3 σ 1 = − i σ 2 k = − σ 1 σ 2 = − i σ 3 {\displaystyle {\begin{aligned}\mathbf {i} &=-\sigma _{2}\sigma _{3}=-i\sigma _{1}\\\mathbf {j} &=-\sigma _{3}\sigma _{1}=-i\sigma _{2}\\\mathbf {k} &=-\sigma _{1}\sigma _{2}=-i\sigma _{3}\end{aligned}}}
With the identification of the even-graded elements with the algebra H {\displaystyle \mathbb {H} } of quaternions, as in the case of two dimensions the only representation of the algebra of even-graded elements is on itself. [ t ] Thus the (real [ u ] ) spinors in three-dimensions are quaternions, and the action of an even-graded element on a spinor is given by ordinary quaternionic multiplication.
Note that the expression (1) for a vector rotation through an angle θ , the angle appearing in γ was halved . Thus the spinor rotation γ ( ψ ) = γψ (ordinary quaternionic multiplication) will rotate the spinor ψ through an angle one-half the measure of the angle of the corresponding vector rotation. Once again, the problem of lifting a vector rotation to a spinor rotation is two-valued: the expression (1) with (180° + θ /2) in place of θ /2 will produce the same vector rotation, but the negative of the spinor rotation.
The spinor/quaternion representation of rotations in 3D is becoming increasingly prevalent in computer geometry and other applications, because of the notable brevity of the corresponding spin matrix, and the simplicity with which they can be multiplied together to calculate the combined effect of successive rotations about different axes.
A space of spinors can be constructed explicitly with concrete and abstract constructions. The
equivalence of these constructions is a consequence of the uniqueness of the spinor representation of the complex Clifford algebra. For a complete example in dimension 3, see spinors in three dimensions .
Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ( V , g ) can be defined as follows. Choose an orthonormal basis e 1 ... e n for V i.e. g ( e μ e ν ) = η μν where η μμ = ±1 and η μν = 0 for μ ≠ ν . Let k = ⌊ n /2⌋ . Fix a set of 2 k × 2 k matrices γ 1 ... γ n such that γ μ γ ν + γ ν γ μ = 2 η μν 1 (i.e. fix a convention for the gamma matrices ). Then the assignment e μ → γ μ extends uniquely to an algebra homomorphism Cℓ( V , g ) → Mat(2 k , C {\displaystyle \mathbb {C} } ) by sending the monomial e μ 1 ⋅⋅⋅ e μ k in the Clifford algebra to the product γ μ 1 ⋅⋅⋅ γ μ k of matrices and extending linearly. The space Δ = C 2 k {\displaystyle \Delta =\mathbb {C} ^{2^{k}}} on which the gamma matrices act is now a space of spinors. One needs to construct such matrices explicitly, however. In dimension 3, defining the gamma matrices to be the Pauli sigma matrices gives rise to the familiar two component spinors used in non relativistic quantum mechanics . Likewise using the 4 × 4 Dirac gamma matrices gives rise to the 4 component Dirac spinors used in 3+1 dimensional relativistic quantum field theory . In general, in order to define gamma matrices of the required kind, one can use the Weyl–Brauer matrices .
In this construction the representation of the Clifford algebra Cℓ( V , g ) , the Lie algebra so ( V , g ) , and the Spin group Spin( V , g ) , all depend on the choice of the orthonormal basis and the choice of the gamma matrices. This can cause confusion over conventions, but invariants like traces are independent of choices. In particular, all physically observable quantities must be independent of such choices. In this construction a spinor can be represented as a vector of 2 k complex numbers and is denoted with spinor indices (usually α , β , γ ). In the physics literature, such indices are often used to denote spinors even when an abstract spinor construction is used.
There are at least two different, but essentially equivalent, ways to define spinors abstractly. One approach seeks to identify the minimal ideals for the left action of Cℓ( V , g ) on itself. These are subspaces of the Clifford algebra of the form Cℓ( V , g ) ω , admitting the evident action of Cℓ( V , g ) by left-multiplication: c : xω → cxω . There are two variations on this theme: one can either find a primitive element ω that is a nilpotent element of the Clifford algebra, or one that is an idempotent . The construction via nilpotent elements is more fundamental in the sense that an idempotent may then be produced from it. [ 23 ] In this way, the spinor representations are identified with certain subspaces of the Clifford algebra itself. The second approach is to construct a vector space using a distinguished subspace of V , and then specify the action of the Clifford algebra externally to that vector space.
In either approach, the fundamental notion is that of an isotropic subspace W . Each construction depends on an initial freedom in choosing this subspace. In physical terms, this corresponds to the fact that there is no measurement protocol that can specify a basis of the spin space, even if a preferred basis of V is given.
As above, we let ( V , g ) be an n -dimensional complex vector space equipped with a nondegenerate bilinear form. If V is a real vector space, then we replace V by its complexification V ⊗ R C {\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} } and let g denote the induced bilinear form on V ⊗ R C {\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} } . Let W be a maximal isotropic subspace, i.e. a maximal subspace of V such that g | W = 0 . If n = 2 k is even, then let W ′ be an isotropic subspace complementary to W . If n = 2 k + 1 is odd, let W ′ be a maximal isotropic subspace with W ∩ W ′ = 0 , and let U be the orthogonal complement of W ⊕ W ′ . In both the even- and odd-dimensional cases W and W ′ have dimension k . In the odd-dimensional case, U is one-dimensional, spanned by a unit vector u .
Since W ′ is isotropic, multiplication of elements of W ′ inside Cℓ( V , g ) is skew . Hence vectors in W ′ anti-commute, and Cℓ( W ′ , g | W ′ ) = Cℓ( W ′ , 0) is just the exterior algebra Λ ∗ W ′ . Consequently, the k -fold product of W ′ with itself, W ′ k , is one-dimensional. Let ω be a generator of W ′ k . In terms of a basis w ′ 1 , ..., w ′ k of in W ′ , one possibility is to set ω = w 1 ′ w 2 ′ ⋯ w k ′ . {\displaystyle \omega =w'_{1}w'_{2}\cdots w'_{k}.}
Note that ω 2 = 0 (i.e., ω is nilpotent of order 2), and moreover, w ′ ω = 0 for all w ′ ∈ W ′ . The following facts can be proven easily:
In detail, suppose for instance that n is even. Suppose that I is a non-zero left ideal contained in Cℓ( V , g ) ω . We shall show that I must be equal to Cℓ( V , g ) ω by proving that it contains a nonzero scalar multiple of ω .
Fix a basis w i of W and a complementary basis w i ′ of W ′ so that
Note that any element of I must have the form αω , by virtue of our assumption that I ⊂ Cℓ( V , g ) ω . Let αω ∈ I be any such element. Using the chosen basis, we may write α = ∑ i 1 < i 2 < ⋯ < i p a i 1 … i p w i 1 ⋯ w i p + ∑ j B j w j ′ {\displaystyle \alpha =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}+\sum _{j}B_{j}w'_{j}} where the a i 1 ... i p are scalars, and the B j are auxiliary elements of the Clifford algebra. Observe now that the product α ω = ∑ i 1 < i 2 < ⋯ < i p a i 1 … i p w i 1 ⋯ w i p ω . {\displaystyle \alpha \omega =\sum _{i_{1}<i_{2}<\cdots <i_{p}}a_{i_{1}\dots i_{p}}w_{i_{1}}\cdots w_{i_{p}}\omega .} Pick any nonzero monomial a in the expansion of α with maximal homogeneous degree in the elements w i : a = a i 1 … i max w i 1 … w i max {\displaystyle a=a_{i_{1}\dots i_{\text{max}}}w_{i_{1}}\dots w_{i_{\text{max}}}} (no summation implied),
then w i max ′ ⋯ w i 1 ′ α ω = a i 1 … i max ω {\displaystyle w'_{i_{\text{max}}}\cdots w'_{i_{1}}\alpha \omega =a_{i_{1}\dots i_{\text{max}}}\omega } is a nonzero scalar multiple of ω , as required.
Note that for n even, this computation also shows that Δ = C ℓ ( W ) ω = ( Λ ∗ W ) ω {\displaystyle \Delta =\mathrm {C} \ell (W)\omega =\left(\Lambda ^{*}W\right)\omega } as a vector space. In the last equality we again used that W is isotropic. In physics terms, this shows that Δ is built up like a Fock space by creating spinors using anti-commuting creation operators in W acting on a vacuum ω .
The computations with the minimal ideal construction suggest that a spinor representation can
also be defined directly using the exterior algebra Λ ∗ W = ⊕ j Λ j W of the isotropic subspace W .
Let Δ = Λ ∗ W denote the exterior algebra of W considered as vector space only. This will be the spin representation, and its elements will be referred to as spinors. [ 24 ] [ 25 ]
The action of the Clifford algebra on Δ is defined first by giving the action of an element of V on Δ, and then showing that this action respects the Clifford relation and so extends to a homomorphism of the full Clifford algebra into the endomorphism ring End(Δ) by the universal property of Clifford algebras . The details differ slightly according to whether the dimension of V is even or odd.
When dim( V ) is even, V = W ⊕ W ′ where W ′ is the chosen isotropic complement. Hence any v ∈ V decomposes uniquely as v = w + w ′ with w ∈ W and w ′ ∈ W ′ . The action of v on a spinor is given by c ( v ) w 1 ∧ ⋯ ∧ w n = ( ϵ ( w ) + i ( w ′ ) ) ( w 1 ∧ ⋯ ∧ w n ) {\displaystyle c(v)w_{1}\wedge \cdots \wedge w_{n}=\left(\epsilon (w)+i\left(w'\right)\right)\left(w_{1}\wedge \cdots \wedge w_{n}\right)} where i ( w ′ ) is interior product with w ′ using the nondegenerate quadratic form to identify V with V ∗ , and ε ( w ) denotes the exterior product . This action is sometimes called the Clifford product . It may be verified that c ( u ) c ( v ) + c ( v ) c ( u ) = 2 g ( u , v ) , {\displaystyle c(u)\,c(v)+c(v)\,c(u)=2\,g(u,v)\,,} and so c respects the Clifford relations and extends to a homomorphism from the Clifford algebra to End(Δ).
The spin representation Δ further decomposes into a pair of irreducible complex representations of the Spin group [ 26 ] (the half-spin representations, or Weyl spinors) via Δ + = Λ even W , Δ − = Λ odd W . {\displaystyle \Delta _{+}=\Lambda ^{\text{even}}W,\,\Delta _{-}=\Lambda ^{\text{odd}}W.}
When dim( V ) is odd, V = W ⊕ U ⊕ W ′ , where U is spanned by a unit vector u orthogonal to W . The Clifford action c is defined as before on W ⊕ W ′ , while the Clifford action of (multiples of) u is defined by c ( u ) α = { α if α ∈ Λ even W − α if α ∈ Λ odd W {\displaystyle c(u)\alpha ={\begin{cases}\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{even}}W\\-\alpha &{\hbox{if }}\alpha \in \Lambda ^{\text{odd}}W\end{cases}}} As before, one verifies that c respects the Clifford relations, and so induces a homomorphism.
If the vector space V has extra structure that provides a decomposition of its complexification into two maximal isotropic subspaces, then the definition of spinors (by either method) becomes natural.
The main example is the case that the real vector space V is a hermitian vector space ( V , g ) , i.e., V is equipped with a complex structure J that is an orthogonal transformation with respect to the inner product g on V . Then V ⊗ R C {\displaystyle V\otimes _{\mathbb {R} }\mathbb {C} } splits in the ± i eigenspaces of J . These eigenspaces are isotropic for the complexification of g and can be identified with the complex vector space ( V , J ) and its complex conjugate ( V , − J ) . Therefore, for a hermitian vector space ( V , g ) the vector space Λ C ⋅ V ¯ {\displaystyle \Lambda _{\mathbb {C} }^{\cdot }{\bar {V}}} (as well as its complex conjugate Λ C ⋅ V {\displaystyle \Lambda _{\mathbb {C} }^{\cdot }V} ) is a spinor space for the underlying real euclidean vector space.
With the Clifford action as above but with contraction using the hermitian form, this construction gives a spinor space at every point of an almost Hermitian manifold and is the reason why every almost complex manifold (in particular every symplectic manifold ) has a Spin c structure . Likewise, every complex vector bundle on a manifold carries a Spin c structure. [ 27 ]
A number of Clebsch–Gordan decompositions are possible on the tensor product of one spin representation with another. [ 28 ] These decompositions express the tensor product in terms of the alternating representations of the orthogonal group.
For the real or complex case, the alternating representations are
In addition, for the real orthogonal groups, there are three characters (one-dimensional representations)
The Clebsch–Gordan decomposition allows one to define, among other things:
If n = 2 k is even, then the tensor product of Δ with the contragredient representation decomposes as Δ ⊗ Δ ∗ ≅ ⨁ p = 0 n Γ p ≅ ⨁ p = 0 k − 1 ( Γ p ⊕ σ Γ p ) ⊕ Γ k {\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{n}\Gamma _{p}\cong \bigoplus _{p=0}^{k-1}\left(\Gamma _{p}\oplus \sigma \Gamma _{p}\right)\oplus \Gamma _{k}} which can be seen explicitly by considering (in the Explicit construction) the action of the Clifford algebra on decomposable elements αω ⊗ βω ′ . The rightmost formulation follows from the transformation properties of the Hodge star operator . Note that on restriction to the even Clifford algebra, the paired summands Γ p ⊕ σ Γ p are isomorphic, but under the full Clifford algebra they are not.
There is a natural identification of Δ with its contragredient representation via the conjugation in the Clifford algebra: ( α ω ) ∗ = ω ( α ∗ ) . {\displaystyle (\alpha \omega )^{*}=\omega \left(\alpha ^{*}\right).} So Δ ⊗ Δ also decomposes in the above manner. Furthermore, under the even Clifford algebra, the half-spin representations decompose Δ + ⊗ Δ + ∗ ≅ Δ − ⊗ Δ − ∗ ≅ ⨁ p = 0 k Γ 2 p Δ + ⊗ Δ − ∗ ≅ Δ − ⊗ Δ + ∗ ≅ ⨁ p = 0 k − 1 Γ 2 p + 1 {\displaystyle {\begin{aligned}\Delta _{+}\otimes \Delta _{+}^{*}\cong \Delta _{-}\otimes \Delta _{-}^{*}&\cong \bigoplus _{p=0}^{k}\Gamma _{2p}\\\Delta _{+}\otimes \Delta _{-}^{*}\cong \Delta _{-}\otimes \Delta _{+}^{*}&\cong \bigoplus _{p=0}^{k-1}\Gamma _{2p+1}\end{aligned}}}
For the complex representations of the real Clifford algebras, the associated reality structure on the complex Clifford algebra descends to the space of spinors (via the explicit construction in terms of minimal ideals, for instance). In this way, we obtain the complex conjugate Δ of the representation Δ, and the following isomorphism is seen to hold: Δ ¯ ≅ σ − Δ ∗ {\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}}
In particular, note that the representation Δ of the orthochronous spin group is a unitary representation . In general, there are Clebsch–Gordan decompositions Δ ⊗ Δ ¯ ≅ ⨁ p = 0 k ( σ − Γ p ⊕ σ + Γ p ) . {\displaystyle \Delta \otimes {\bar {\Delta }}\cong \bigoplus _{p=0}^{k}\left(\sigma _{-}\Gamma _{p}\oplus \sigma _{+}\Gamma _{p}\right).}
In metric signature ( p , q ) , the following isomorphisms hold for the conjugate half-spin representations
Using these isomorphisms, one can deduce analogous decompositions for the tensor products of the half-spin representations Δ ± ⊗ Δ ± .
If n = 2 k + 1 is odd, then Δ ⊗ Δ ∗ ≅ ⨁ p = 0 k Γ 2 p . {\displaystyle \Delta \otimes \Delta ^{*}\cong \bigoplus _{p=0}^{k}\Gamma _{2p}.} In the real case, once again the isomorphism holds Δ ¯ ≅ σ − Δ ∗ . {\displaystyle {\bar {\Delta }}\cong \sigma _{-}\Delta ^{*}.} Hence there is a Clebsch–Gordan decomposition (again using the Hodge star to dualize) given by Δ ⊗ Δ ¯ ≅ σ − Γ 0 ⊕ σ + Γ 1 ⊕ ⋯ ⊕ σ ± Γ k {\displaystyle \Delta \otimes {\bar {\Delta }}\cong \sigma _{-}\Gamma _{0}\oplus \sigma _{+}\Gamma _{1}\oplus \dots \oplus \sigma _{\pm }\Gamma _{k}}
There are many far-reaching consequences of the Clebsch–Gordan decompositions of the spinor spaces. The most fundamental of these pertain to Dirac's theory of the electron, among whose basic requirements are | https://en.wikipedia.org/wiki/Spinor |
Spinor condensates are degenerate Bose gases that have degrees of freedom arising from the internal spin of the constituent particles [ 1 ] . [ 2 ] They are described by a multi-component (spinor) order parameter .
Since their initial experimental realisation,
a wealth of studies have appeared, both
experimental and theoretical, focusing
on the physical properties of spinor condensates, including their ground states , non-equilibrium dynamics, and vortices .
The study of spinor condensates was initiated in 1998 by experimental groups at JILA [ 3 ] and MIT. [ 4 ] These experiments utilised 23 Na and 87 Rb atoms, respectively.
In contrast to most prior experiments on ultracold gases, these experiments utilised a purely
optical trap, which is spin-insensitive. Shortly thereafter, theoretical work appeared [ 5 ] [ 6 ] which described the possible mean-field phases of spin-one spinor condensates.
The Hamiltonian describing a spinor condensate is most frequently written using the language of second quantization . Here the field operator ψ ^ m † ( r ) {\displaystyle {\hat {\psi }}_{m}^{\dagger }({\bf {r}})} creates a boson in Zeeman level m {\displaystyle m} at position r {\displaystyle {\bf {r}}} . These
operators satisfy bosonic commutation relations :
[ ψ ^ m ( r ) , ψ ^ m ′ † ( r ′ ) ] = δ ( r − r ′ ) δ m m ′ . {\displaystyle [{\hat {\psi }}_{m}({\bf {r}}),{\hat {\psi }}_{m'}^{\dagger }({\bf {r}}')]=\delta ({\bf {r}}-{\bf {r}}')\delta _{mm'}.}
The free (non-interacting) part of the Hamiltonian is
H 0 = ∑ m ∫ d 3 r ψ ^ m † ( r ) ( − ℏ 2 2 m ∇ 2 + V e x t ( r ) ) ψ ^ m † ( r ) . {\displaystyle H_{0}=\sum _{m}\int d^{3}r{\hat {\psi }}_{m}^{\dagger }({\bf {r}})\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V_{\rm {ext}}({\bf {r}})\right){\hat {\psi }}_{m}^{\dagger }({\bf {r}}).}
where m {\displaystyle m} denotes the mass of the constituent particles and V e x t ( r ) {\displaystyle V_{\rm {ext}}({\bf {r}})} is an external potential.
For a spin-one spinor condensate, the interaction Hamiltonian is [ 5 ] [ 6 ]
H i n t = 1 2 ∫ d 3 r : ( c 0 ρ ^ ( r ) 2 + c 1 ( F ^ ( r ) ) 2 ) : . {\displaystyle H_{\rm {int}}={\frac {1}{2}}\int d^{3}r:\left(c_{0}{\hat {\rho }}({\bf {r}})^{2}+c_{1}({\hat {\bf {F}}}({\bf {r}}))^{2}\right):.}
In this expression, ρ ^ ( r ) = ∑ m ψ ^ m † ( r ) ψ ^ m ( r ) {\displaystyle {\hat {\rho }}({\bf {r}})=\sum _{m}{\hat {\psi }}_{m}^{\dagger }({\bf {r}}){\hat {\psi }}_{m}({\bf {r}})} is the operator corresponding to the density, F ^ ( r ) = ∑ m m ′ ψ ^ m † ( r ) S m m ′ ψ ^ m ′ ( r ) {\displaystyle {\hat {\bf {F}}}({\bf {r}})=\sum _{mm'}{\hat {\psi }}_{m}^{\dagger }({\bf {r}}){\bf {S}}_{mm'}{\hat {\psi }}_{m'}({\bf {r}})} is the local spin operator ( S m m ′ {\displaystyle {\bf {S}}_{mm'}} is a vector composed of the spin-one matrices ),
and :: denotes normal ordering . The parameters c 0 , c 1 {\displaystyle c_{0},c_{1}} can be expressed in terms of the s-wave scattering lengths of the constituent particles.
Higher spin versions of the
interaction Hamiltonian are slightly more involved, but
can generally be expressed by using Clebsch–Gordan coefficients .
The full Hamiltonian then is H = H 0 + H i n t {\displaystyle H=H_{0}+H_{\rm {int}}} .
In Gross-Pitaevskii mean field theory , one replaces the field operators with c-number functions: ψ ^ m ( r ) → ψ m ( r ) {\displaystyle {\hat {\psi }}_{m}({\bf {r}})\rightarrow {\psi }_{m}({\bf {r}})} . To find the mean-field
ground states, one then minimises the resulting energy with respect to these c-number functions.
For a spatially uniform system spin-one system, there are two possible mean-field ground states.
When c 1 > 0 {\displaystyle c_{1}>0} , the ground state is ψ p o l a r = ρ ¯ ( 0 , 1 , 0 ) {\displaystyle \psi _{\rm {polar}}={\sqrt {\bar {\rho }}}(0,1,0)} while for c 1 < 0 {\displaystyle c_{1}<0} the ground state is ψ f e r r o = ρ ¯ ( 1 , 0 , 0 ) . {\displaystyle \psi _{\rm {ferro}}={\sqrt {\bar {\rho }}}(1,0,0).} The former expression is referred to as the polar state while the latter is the
ferromagnetic state. [ 1 ] Both states are unique up to overall spin rotations. Importantly, ψ f e r r o {\displaystyle \psi _{\rm {ferro}}} cannot be rotated into ψ p o l a r {\displaystyle \psi _{\rm {polar}}} .
The Majorana stellar representation [ 7 ] provides a particularly insightful description of the mean-field phases of spinor
condensates with larger spin. [ 2 ]
Due to being described by a multi-component order parameter , numerous types of topological defects (vortices) can appear in spinor condensates
. [ 8 ] Homotopy theory provides a natural description of topological defects, [ 9 ] and is regularly employed to understand
vortices in spinor condensates. | https://en.wikipedia.org/wiki/Spinor_condensate |
In mathematics , the spinor concept as specialised to three dimensions can be treated by means of the traditional notions of dot product and cross product . This is part of the detailed algebraic discussion of the rotation group SO(3) .
The association of a spinor with a 2×2 complex traceless Hermitian matrix was formulated by Élie Cartan . [ 1 ]
In detail, given a vector x = ( x 1 , x 2 , x 3 ) of real (or complex) numbers, one can associate the complex matrix
In physics, this is often written as a dot product X ≡ σ → ⋅ x → {\displaystyle X\equiv {\vec {\sigma }}\cdot {\vec {x}}} , where σ → ≡ ( σ 1 , σ 2 , σ 3 ) {\displaystyle {\vec {\sigma }}\equiv (\sigma _{1},\sigma _{2},\sigma _{3})} is the vector form of Pauli matrices . Matrices of this form have the following properties, which relate them intrinsically to the geometry of 3-space:
The last property can be used to simplify rotational operations. It is an elementary fact from linear algebra that any rotation in 3-space factors as a composition of two reflections. (More generally, any orientation-reversing orthogonal transformation is either a reflection or the product of three reflections.) Thus if R is a rotation which decomposes as the reflection in the plane perpendicular to a unit vector u → 1 {\displaystyle {\vec {u}}_{1}} followed by the reflection in the plane perpendicular to u → 2 {\displaystyle {\vec {u}}_{2}} , then the matrix U 2 U 1 X U 1 U 2 {\displaystyle U_{2}U_{1}XU_{1}U_{2}} represents the rotation of the vector x → {\displaystyle {\vec {x}}} through R .
Having effectively encoded all the rotational linear geometry of 3-space into a set of complex 2×2 matrices, it is natural to ask what role, if any, the 2×1 matrices (i.e., the column vectors ) play. Provisionally, a spinor is a column vector
The space of spinors is evidently acted upon by complex 2×2 matrices. As shown above, the product of two reflections in a pair of unit vectors defines a 2×2 matrix whose action on euclidean vectors is a rotation. So there is an action of rotations on spinors. However, there is one important caveat: the factorization of a rotation is not unique. Clearly, if X ↦ R X R − 1 {\displaystyle X\mapsto RXR^{-1}} is a representation of a rotation, then replacing R by − R will yield the same rotation. In fact, one can easily show that this is the only ambiguity which arises. Thus the action of a rotation on a spinor is always double-valued .
There were some precursors to Cartan's work with 2×2 complex matrices: Wolfgang Pauli had used these matrices so intensively that elements of a certain basis of a four-dimensional subspace are called Pauli matrices σ i , so that the Hermitian matrix is written as a Pauli vector x → ⋅ σ → . {\displaystyle {\vec {x}}\cdot {\vec {\sigma }}.} [ 2 ] In the mid 19th century the algebraic operations of this algebra of four complex dimensions were studied as biquaternions .
Michael Stone and Paul Goldbar, in Mathematics for Physics , contest this, saying, "The spin representations were discovered by ´Elie Cartan in 1913, some years before they were needed in physics."
Spinors can be constructed directly from isotropic vectors in 3-space without using the quaternionic construction. To motivate this introduction of spinors, suppose that X is a matrix representing a vector x in complex 3-space. Suppose further that x is isotropic: i.e.,
Then since the determinant of X is zero there is a proportionality between its rows or columns. Thus the matrix may be written as an outer product of two complex 2-vectors:
This factorization yields an overdetermined system of equations in the coordinates of the vector x :
subject to the constraint
This system admits the solutions
Either choice of sign solves the system ( 1 ). Thus a spinor may be viewed as an isotropic vector, along with a choice of sign. Note that because of the logarithmic branching , it is impossible to choose a sign consistently so that ( 3 ) varies continuously along a full rotation among the coordinates x . In spite of this ambiguity of the representation of a rotation on a spinor, the rotations do act unambiguously by a fractional linear transformation on the ratio ξ 1 : ξ 2 since one choice of sign in the solution ( 3 ) forces the choice of the second sign. In particular, the space of spinors is a projective representation of the orthogonal group .
As a consequence of this point of view, spinors may be regarded as a kind of "square root" of isotropic vectors. Specifically, introducing the matrix
the system ( 1 ) is equivalent to solving X = 2 ξ t ξ C for the undetermined spinor ξ .
A fortiori , if the roles of ξ and x are now reversed, the form Q ( ξ ) = x defines, for each spinor ξ , a vector x quadratically in the components of ξ . If this quadratic form is polarized , it determines a bilinear vector-valued form on spinors Q ( μ , ξ ). This bilinear form then transforms tensorially under a reflection or a rotation.
The above considerations apply equally well whether the original euclidean space under consideration is real or complex. When the space is real, however, spinors possess some additional structure which in turn facilitates a complete description of the representation of the rotation group. Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:
With this convention, real vectors correspond to Hermitian matrices. Furthermore, real rotations preserving the form ( 4 ) correspond (in the double-valued sense) to unitary matrices of determinant one. In modern terms, this presents the special unitary group SU(2) as a double cover of SO(3). As a consequence, the spinor Hermitian product
is preserved by all rotations, and therefore is canonical.
If, however, the signature of the inner product on 3-space is indefinite (i.e., non-degenerate, but also not positive definite), then the foregoing analysis must be adjusted to reflect this. Suppose then that the length form on 3-space is given by:
Then the construction of spinors of the preceding sections proceeds, but with x 2 {\displaystyle x_{2}} replacing i {\displaystyle i} x 2 {\displaystyle x_{2}} in all the formulas. With this new convention, the matrix associated to a real vector ( x 1 , x 2 , x 3 ) {\displaystyle (x_{1},x_{2},x_{3})} is itself real:
The form ( 5 ) is no longer invariant under a real rotation (or reversal), since the group stabilizing ( 4′ ) is now a Lorentz group O(2,1). Instead, the anti-Hermitian form
defines the appropriate notion of inner product for spinors in this metric signature. This form is invariant under transformations in the connected component of the identity of O(2,1).
In either case, the quartic form
is fully invariant under O(3) (or O(2,1), respectively), where Q is the vector-valued bilinear form described in the previous section. The fact that this is a quartic invariant, rather than quadratic, has an important consequence. If one confines attention to the group of special orthogonal transformations, then it is possible unambiguously to take the square root of this form and obtain an identification of spinors with their duals. In the language of representation theory, this implies that there is only one irreducible spin representation of SO(3) (or SO(2,1)) up to isomorphism. If, however, reversals (e.g., reflections in a plane) are also allowed, then it is no longer possible to identify spinors with their duals owing to a change of sign on the application of a reflection. Thus there are two irreducible spin representations of O(3) (or O(2,1)), sometimes called the pin representations .
The differences between these two signatures can be codified by the notion of a reality structure on the space of spinors. Informally, this is a prescription for taking a complex conjugate of a spinor, but in such a way that this may not correspond to the usual conjugate per the components of a spinor. Specifically, a reality structure is specified by a Hermitian 2 × 2 matrix K whose product with itself is the identity matrix: K 2 = Id . The conjugate of a spinor with respect to a reality structure K is defined by
The particular form of the inner product on vectors (e.g., ( 4 ) or ( 4′ )) determines a reality structure (up to a factor of -1) by requiring
Thus K = i C is the reality structure in Euclidean signature ( 4 ), and K = Id is that for signature ( 4′ ). With a reality structure in hand, one has the following results:
Often, the first example of spinors that a student of physics
encounters are the 2×1 spinors used in Pauli's theory of electron spin.
The Pauli matrices are a vector of three 2×2 matrices that are used as spin operators .
Given a unit vector in 3 dimensions, for example ( a , b , c ), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for
spin in the direction of the unit vector.
The eigenvectors of that spin matrix are the spinors for
spin-1/2 oriented in the direction given by the vector.
Example: u = (0.8, -0.6, 0) is a unit vector. Dotting this with the Pauli
spin matrices gives the matrix:
The eigenvectors may be found by the usual methods of linear algebra , but a convenient trick
is to note that a Pauli spin matrix is an involutory matrix , that is, the square of the above matrix is the identity matrix .
Thus a (matrix) solution to the eigenvector problem with eigenvalues of
±1 is simply 1 ± S u . That is,
One can then choose either of the columns of the eigenvector matrix as the vector solution, provided that the column chosen
is not zero. Taking the first column of the above,
eigenvector solutions for the two eigenvalues are:
The trick used to find the eigenvectors is related to the concept of ideals , that is, the matrix eigenvectors (1 ± S u )/2 are projection operators or idempotents and therefore each generates an ideal in the Pauli algebra. The same trick
works in any Clifford algebra , in particular
the Dirac algebra that is discussed below. These projection
operators are also seen in density matrix theory
where they are examples of pure density matrices.
More generally, the projection operator for spin in the ( a , b , c ) direction
is given by
and any non zero column can be taken as the projection operator. While the
two columns appear different, one can use a 2 + b 2 + c 2 = 1 to show that they are multiples (possibly zero) of the same spinor.
In atomic physics and quantum mechanics , the property of spin plays a major role. In addition to their other properties all particles possess a non-classical property, i.e., which has no correspondence at all in conventional physics, namely the spin , which is a kind of intrinsic angular momentum . In the position representation, instead of a wavefunction without spin, ψ = ψ ( r ), one has with spin: ψ = ψ ( r , σ ), where σ takes the following discrete set of values:
The total angular momentum operator, J → {\displaystyle {\vec {\mathbb {J} }}} , of a particle corresponds to the sum of the orbital angular momentum (i.e., there only integers are allowed) and the intrinsic part , the spin . One distinguishes bosons (S = 0, ±1, ±2, ...) and fermions (S = ±1/2, ±3/2, ±5/2, ...). | https://en.wikipedia.org/wiki/Spinors_in_three_dimensions |
A spinthariscope ( / s p ɪ n ˈ θ ær ɪ s k oʊ p / ) [ 2 ] [ 3 ] is a device for observing individual nuclear disintegrations caused by the interaction of ionizing radiation with a phosphor (see radioluminescence ) or scintillator .
The spinthariscope was invented by William Crookes in 1903. [ 4 ] [ 5 ] While observing the apparently uniform fluorescence on a zinc sulfide screen created by the radioactive emissions (mostly alpha radiation ) of a sample of radium bromide , he spilled some of the sample, and, owing to its extreme rarity and cost, he was eager to find and recover it. [ 6 ] Upon inspecting the zinc sulfide screen under a microscope , he noticed separate flashes of light created by individual alpha particle collisions with the screen. Crookes took his discovery a step further and invented a device specifically intended to view these scintillations. It consisted of a small screen coated with zinc sulfide affixed to the end of a tube, with a tiny amount of radium salt suspended a short distance from the screen and a lens on the other end of the tube for viewing the screen. Crookes named his device from Ancient Greek : σπινθήρ ( spinthḗr ) "spark". [ 7 ]
Crookes debuted the spinthariscope at a meeting of the Royal Society , London on 15 May 1903. [ 8 ]
Spinthariscopes were quickly replaced with more accurate and quantitative devices for measuring radiation in scientific experiments, but enjoyed a modest revival in the mid 20th century as children's educational toys. [ 9 ] In 1947, Kix cereal offered a Lone Ranger atomic bomb ring that contained a small one, in exchange for a box top and US$0.15 (equivalent to $2.11 in 2024). [ 10 ] [ 11 ] [ 12 ] Spinthariscopes can still be bought today as instructional novelties, but they now use americium or thorium . Looking into a properly focused toy spinthariscope, one can see many flashes of light spread randomly across the screen. Almost all are circular, with a very bright pinpoint centre surrounded by a dimmer circle of emission. [ citation needed ]
The American History Museum of the Smithsonian has several spinthariscopes in its collections, and an article discussing them. [ 13 ] However, as of 2022 [update] none are currently on display. [ 14 ] | https://en.wikipedia.org/wiki/Spinthariscope |
Spintronics (a portmanteau meaning spin transport electronics [ 1 ] [ 2 ] [ 3 ] ), also known as spin electronics , is the study of the intrinsic spin of the electron and its associated magnetic moment , in addition to its fundamental electronic charge , in solid-state devices . [ 4 ] The field of spintronics concerns spin-charge coupling in metallic systems; the analogous effects in insulators fall into the field of multiferroics .
Spintronics fundamentally differs from traditional electronics in that, in addition to charge state, electron spins are used as a further degree of freedom, with implications in the efficiency of data storage and transfer. Spintronic systems are most often realised in dilute magnetic semiconductors (DMS) and Heusler alloys and are of particular interest in the field of quantum computing and neuromorphic computing , upon which leads to integrated research requirements around Hyperdimensional Computation .
Spintronics emerged from discoveries in the 1980s concerning spin-dependent electron transport phenomena in solid-state devices. This includes the observation of spin-polarized electron injection from a ferromagnetic metal to a normal metal by Johnson and Silsbee (1985) [ 5 ] and the discovery of giant magnetoresistance independently by Albert Fert et al. [ 6 ] and Peter Grünberg et al. (1988). [ 7 ] The origin of spintronics can be traced to the ferromagnet/superconductor tunneling experiments pioneered by Meservey and Tedrow and initial experiments on magnetic tunnel junctions by Julliere in the 1970s. [ 8 ] The use of semiconductors for spintronics began with the theoretical proposal of a spin field-effect-transistor by Datta and Das in 1990 [ 9 ] and of the electric dipole spin resonance by Rashba in 1960. [ 10 ]
The spin of the electron is an intrinsic angular momentum that is separate from the angular momentum due to its orbital motion. The magnitude of the projection of the electron's spin along an arbitrary axis is 1 2 ℏ {\displaystyle {\tfrac {1}{2}}\hbar } , implying that the electron acts as a fermion by the spin-statistics theorem . Like orbital angular momentum, the spin has an associated magnetic moment , the magnitude of which is expressed as
In a solid, the spins of many electrons can act together to affect the magnetic and electronic properties of a material, for example endowing it with a permanent magnetic moment as in a ferromagnet .
In many materials, electron spins are equally present in both the up and the down state, and no transport properties are dependent on spin. A spintronic device requires generation or manipulation of a spin-polarized population of electrons, resulting in an excess of spin up or spin down electrons. The polarization of any spin dependent property X can be written as
A net spin polarization can be achieved either through creating an equilibrium energy split between spin up and spin down. Methods include putting a material in a large magnetic field ( Zeeman effect ), the exchange energy present in a ferromagnet or forcing the system out of equilibrium. The period of time that such a non-equilibrium population can be maintained is known as the spin lifetime, τ {\displaystyle \tau } .
In a diffusive conductor, a spin diffusion length λ {\displaystyle \lambda } can be defined as the distance over which a non-equilibrium spin population can propagate. Spin lifetimes of conduction electrons in metals are relatively short (typically less than 1 nanosecond). An important research area is devoted to extending this lifetime to technologically relevant timescales.
The mechanisms of decay for a spin polarized population can be broadly classified as spin-flip scattering and spin dephasing. Spin-flip scattering is a process inside a solid that does not conserve spin, and can therefore switch an incoming spin up state into an outgoing spin down state. Spin dephasing is the process wherein a population of electrons with a common spin state becomes less polarized over time due to different rates of electron spin precession . In confined structures, spin dephasing can be suppressed, leading to spin lifetimes of milliseconds in semiconductor quantum dots at low temperatures.
Superconductors can enhance central effects in spintronics such as magnetoresistance effects, spin lifetimes and dissipationless spin-currents. [ 11 ] [ 12 ]
The simplest method of generating a spin-polarised current in a metal is to pass the current through a ferromagnetic material. The most common applications of this effect involve giant magnetoresistance (GMR) devices. A typical GMR device consists of at least two layers of ferromagnetic materials separated by a spacer layer. When the two magnetization vectors of the ferromagnetic layers are aligned, the electrical resistance will be lower (so a higher current flows at constant voltage) than if the ferromagnetic layers are anti-aligned. This constitutes a magnetic field sensor.
Two variants of GMR have been applied in devices: (1) current-in-plane (CIP), where the electric current flows parallel to the layers and (2) current-perpendicular-to-plane (CPP), where the electric current flows in a direction perpendicular to the layers.
Other metal-based spintronics devices:
Non-volatile spin-logic devices to enable scaling are being extensively studied. [ 13 ] Spin-transfer, torque-based logic devices that use spins and magnets for information processing have been proposed. [ 14 ] [ 15 ] These devices are part of the ITRS exploratory road map. Logic-in memory applications are already in the development stage. [ 16 ] [ 17 ] A 2017 review article can be found in Materials Today . [ 4 ]
A generalized circuit theory for spintronic integrated circuits has been proposed [ 18 ] so that the physics of spin transport can be utilized by SPICE developers and subsequently by circuit and system designers for the exploration of spintronics for "beyond CMOS computing".
Read heads of magnetic hard drives are based on the GMR or TMR effect.
Motorola developed a first-generation 256 kb magnetoresistive random-access memory (MRAM) based on a single magnetic tunnel junction and a single transistor that has a read/write cycle of under 50 nanoseconds. [ 19 ] Everspin has since developed a 4 Mb version. [ 20 ] Two second-generation MRAM techniques are in development: thermal-assisted switching (TAS) [ 21 ] and spin-transfer torque (STT). [ 22 ]
Another design, racetrack memory , a novel memory architecture proposed by Dr. Stuart S. P. Parkin , encodes information in the direction of magnetization between domain walls of a ferromagnetic wire.
In 2012, persistent spin helices of synchronized electrons were made to persist for more than a nanosecond, a 30-fold increase over earlier efforts, and longer than the duration of a modern processor clock cycle. [ 23 ]
Doped semiconductor materials display dilute ferromagnetism. In recent years, dilute magnetic oxides (DMOs) including ZnO based DMOs and TiO 2 -based DMOs have been the subject of numerous experimental and computational investigations. [ 24 ] [ 25 ] Non-oxide ferromagnetic semiconductor sources (like manganese-doped gallium arsenide (Ga,Mn)As ), [ 26 ] increase the interface resistance with a tunnel barrier, [ 27 ] or using hot-electron injection. [ 28 ]
Spin detection in semiconductors has been addressed with multiple techniques:
The latter technique was used to overcome the lack of spin-orbit interaction and materials issues to achieve spin transport in silicon . [ 33 ]
Because external magnetic fields (and stray fields from magnetic contacts) can cause large Hall effects and magnetoresistance in semiconductors (which mimic spin-valve effects), the only conclusive evidence of spin transport in semiconductors is demonstration of spin precession and dephasing in a magnetic field non-collinear to the injected spin orientation, called the Hanle effect .
Applications using spin-polarized electrical injection have shown threshold current reduction and controllable circularly polarized coherent light output. [ 34 ] Examples include semiconductor lasers. Future applications may include a spin-based transistor having advantages over MOSFET devices such as steeper sub-threshold slope.
Magnetic-tunnel transistor : The magnetic-tunnel transistor with a single base layer [ 35 ] has the following terminals:
The magnetocurrent (MC) is given as:
And the transfer ratio (TR) is
MTT promises a highly spin-polarized electron source at room temperature.
Antiferromagnetic storage media have been studied as an alternative to ferromagnetism , [ 36 ] especially since with antiferromagnetic material the bits can be stored as well as with ferromagnetic material. Instead of the usual definition 0 ↔ 'magnetisation upwards', 1 ↔ 'magnetisation downwards', the states can be, e.g., 0 ↔ 'vertically alternating spin configuration' and 1 ↔ 'horizontally-alternating spin configuration'. [ 37 ] ).
The main advantages of antiferromagnetic material are:
Research is being done into how to read and write information to antiferromagnetic spintronics as their net zero magnetization makes this difficult compared to conventional ferromagnetic spintronics. In modern MRAM, detection and manipulation of ferromagnetic order by magnetic fields has largely been abandoned in favor of more efficient and scalable reading and writing by electrical current. Methods of reading and writing information by current rather than fields are also being investigated in antiferromagnets as fields are ineffective anyway. Writing methods currently being investigated in antiferromagnets are through spin-transfer torque and spin-orbit torque from the spin Hall effect and the Rashba effect . Reading information in antiferromagnets via magnetoresistance effects such as tunnel magnetoresistance is also being explored. [ 40 ] | https://en.wikipedia.org/wiki/Spintronics |
In condensed matter physics , spin–charge separation is an unusual behavior of electrons in some materials in which they 'split' into three independent particles, the spinon , the orbiton and the holon (or chargon). The electron can always be theoretically considered as a bound state of the three, with the spinon carrying the spin of the electron, the orbiton carrying the orbital degree of freedom and the chargon carrying the charge , but in certain conditions they can behave as independent quasiparticles .
The theory of spin–charge separation originates with the work of Sin-Itiro Tomonaga who developed an approximate method for treating one-dimensional interacting quantum systems in 1950. [ 1 ] This was then developed by Joaquin Mazdak Luttinger in 1963 with an exactly solvable model which demonstrated spin–charge separation. [ 2 ] In 1981 F. Duncan M. Haldane generalized Luttinger's model to the Tomonaga– Luttinger liquid concept [ 3 ] whereby the physics of Luttinger's model was shown theoretically to be a general feature of all one-dimensional metallic systems. Although Haldane treated spinless fermions, the extension to spin-½ fermions and associated spin–charge separation was so clear that the promised follow-up paper did not appear.
Spin–charge separation is one of the most unusual manifestations of the concept of quasiparticles . This property is counterintuitive, because neither the spinon, with zero charge and spin half, nor the chargon, with charge minus one and zero spin, can be constructed as combinations of the electrons, holes , phonons and photons that are the constituents of the system. It is an example of fractionalization , the phenomenon in which the quantum numbers of the quasiparticles are not multiples of those of the elementary particles, but fractions. [ citation needed ]
The same theoretical ideas have been applied in the framework of ultracold atoms . In a two-component Bose gas in 1D, strong interactions can produce a maximal form of spin–charge separation. [ 4 ]
Building on physicist F. Duncan M. Haldane 's 1981 theory, experts from the Universities of Cambridge and Birmingham proved experimentally in 2009 that a mass of electrons artificially confined in a small space together will split into spinons and holons due to the intensity of their mutual repulsion (from having the same charge). [ 5 ] [ 6 ] A team of researchers working at the Advanced Light Source (ALS) of the U.S. Department of Energy's Lawrence Berkeley National Laboratory observed the peak spectral structures of spin–charge separation three years prior. [ 7 ] | https://en.wikipedia.org/wiki/Spin–charge_separation |
During nuclear magnetic resonance observations, spin–lattice relaxation is the mechanism by which the longitudinal component of the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium state to thermodynamic equilibrium with its surroundings (the "lattice"). It is characterized by the spin–lattice relaxation time , a time constant known as T 1 .
There is a different parameter, T 2 , the spin–spin relaxation time , which concerns the exponential relaxation of the transverse component of the nuclear magnetization vector ( perpendicular to the external magnetic field). Measuring the variation of T 1 and T 2 in different materials is the basis for some magnetic resonance imaging techniques. [ 1 ]
T 1 characterizes the rate at which the longitudinal M z component of the magnetization vector recovers exponentially towards its thermodynamic equilibrium, according to equation M z ( t ) = M z , e q − [ M z , e q − M z ( 0 ) ] e − t / T 1 {\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }-\left[M_{z,\mathrm {eq} }-M_{z}(0)\right]e^{-t/T_{1}}} Or, for the specific case that M z ( 0 ) = − M z , e q {\displaystyle M_{z}(0)=-M_{z,\mathrm {eq} }} M z ( t ) = M z , e q ( 1 − 2 e − t / T 1 ) {\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-2e^{-t/T_{1}}\right)}
It is thus the time it takes for the longitudinal magnetization to recover approximately 63% [1-(1/ e )] of its initial value after being flipped into the magnetic transverse plane by a 90° radiofrequency pulse.
Nuclei are contained within a molecular structure, and are in constant vibrational and rotational motion, creating a complex magnetic field. The magnetic field caused by thermal motion of nuclei within the lattice is called the lattice field. The lattice field of a nucleus in a lower energy state can interact with nuclei in a higher energy state, causing the energy of the higher energy state to distribute itself between the two nuclei. Therefore, the energy gained by nuclei from the RF pulse is dissipated as increased vibration and rotation within the lattice, which can slightly increase the temperature of the sample. The name spin–lattice relaxation refers to the process in which the spins give the energy they obtained from the RF pulse back to the surrounding lattice, thereby restoring their equilibrium state. The same process occurs after the spin energy has been altered by a change of the surrounding static magnetic field (e.g. pre-polarization by or insertion into high magnetic field) or if the nonequilibrium state has been achieved by other means (e.g., hyperpolarization by optical pumping). [ citation needed ]
The relaxation time, T 1 (the average lifetime of nuclei in the higher energy state) is dependent on the gyromagnetic ratio of the nucleus and the mobility of the lattice. As mobility increases, the vibrational and rotational frequencies increase, making it more likely for a component of the lattice field to be able to stimulate the transition from high to low energy states. However, at extremely high mobilities, the probability decreases as the vibrational and rotational frequencies no longer correspond to the energy gap between states.
Different tissues have different T 1 values. For example, fluids have long T 1 s (1500-2000 ms), and water-based tissues are in the 400-1200 ms range, while fat based tissues are in the shorter 100-150 ms range. The presence of strongly magnetic ions or particles (e.g., ferromagnetic or paramagnetic ) also strongly alter T 1 values and are widely used as MRI contrast agents .
Magnetic resonance imaging uses the resonance of the protons to generate images. Protons are excited by a radio frequency pulse at an appropriate frequency ( Larmor frequency ) and then the excess energy is released in the form of a minuscule amount of heat to the surroundings as the spins return to their thermal equilibrium. The magnetization of the proton ensemble goes back to its equilibrium value with an exponential curve characterized by a time constant T 1 (see Relaxation (NMR) ). [ citation needed ]
T 1 weighted images can be obtained by setting short repetition time (TR) such as < 750 ms and echo time (TE) such as < 40 ms in conventional spin echo sequences, while in Gradient Echo Sequences they can be obtained by using flip angles of larger than 50 o while setting TE values to less than 15 ms.
T 1 is significantly different between grey matter and white matter and is used when undertaking brain scans. A strong T 1 contrast is present between fluid and more solid anatomical structures, making T 1 contrast suitable for morphological assessment of the normal or pathological anatomy, e.g., for musculoskeletal applications.
Spin–lattice relaxation in the rotating frame is the mechanism by which M xy , the transverse component of the magnetization vector, exponentially decays towards its equilibrium value of zero, under the influence of a radio frequency (RF) field in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). It is characterized by the spin–lattice relaxation time constant in the rotating frame, T 1ρ . It is named in contrast to T 1 , the spin-lattice relaxation time . [ 2 ]
T 1ρ MRI is an alternative to conventional T 1 and T 2 MRI by its use of a long-duration, low-power radio frequency referred to as spin-lock (SL) pulse applied to the magnetization in the transverse plane. The magnetization is effectively spin-locked around an effective B 1 field created by the vector sum of the applied B 1 and any off-resonant component. The spin-locked magnetization will relax with a time constant T 1ρ , which is the time it takes for the magnetic resonance signal to reach 37% (1/e) of its initial value, M x y ( 0 ) {\displaystyle M_{xy}(0)} . Hence the relation: M x y ( t S L ) = M x y ( 0 ) e − t S L / T 1 ρ {\displaystyle M_{xy}(t_{\rm {SL}})=M_{xy}(0)e^{-t_{\rm {SL}}/T_{1\rho }}\,} , where t SL is the duration of the RF field.
T 1ρ can be quantified (relaxometry) by curve fitting the signal expression above as a function of the duration of the spin-lock pulse while the amplitude of spin-lock pulse ( γB 1 ~0.1-few kHz) is fixed. Quantitative T 1ρ MRI relaxation maps reflect the biochemical composition of tissues. [ 3 ]
T 1ρ MRI has been used to image tissues such as cartilage, [ 4 ] [ 5 ] intervertebral discs, [ 6 ] brain, [ 7 ] [ 8 ] and heart, [ 9 ] as well as certain types of cancers. [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Spin–lattice_relaxation |
During nuclear magnetic resonance observations, spin–lattice relaxation is the mechanism by which the longitudinal component of the total nuclear magnetic moment vector (parallel to the constant magnetic field) exponentially relaxes from a higher energy, non-equilibrium state to thermodynamic equilibrium with its surroundings (the "lattice"). It is characterized by the spin–lattice relaxation time , a time constant known as T 1 .
There is a different parameter, T 2 , the spin–spin relaxation time , which concerns the exponential relaxation of the transverse component of the nuclear magnetization vector ( perpendicular to the external magnetic field). Measuring the variation of T 1 and T 2 in different materials is the basis for some magnetic resonance imaging techniques. [ 1 ]
T 1 characterizes the rate at which the longitudinal M z component of the magnetization vector recovers exponentially towards its thermodynamic equilibrium, according to equation M z ( t ) = M z , e q − [ M z , e q − M z ( 0 ) ] e − t / T 1 {\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }-\left[M_{z,\mathrm {eq} }-M_{z}(0)\right]e^{-t/T_{1}}} Or, for the specific case that M z ( 0 ) = − M z , e q {\displaystyle M_{z}(0)=-M_{z,\mathrm {eq} }} M z ( t ) = M z , e q ( 1 − 2 e − t / T 1 ) {\displaystyle M_{z}(t)=M_{z,\mathrm {eq} }\left(1-2e^{-t/T_{1}}\right)}
It is thus the time it takes for the longitudinal magnetization to recover approximately 63% [1-(1/ e )] of its initial value after being flipped into the magnetic transverse plane by a 90° radiofrequency pulse.
Nuclei are contained within a molecular structure, and are in constant vibrational and rotational motion, creating a complex magnetic field. The magnetic field caused by thermal motion of nuclei within the lattice is called the lattice field. The lattice field of a nucleus in a lower energy state can interact with nuclei in a higher energy state, causing the energy of the higher energy state to distribute itself between the two nuclei. Therefore, the energy gained by nuclei from the RF pulse is dissipated as increased vibration and rotation within the lattice, which can slightly increase the temperature of the sample. The name spin–lattice relaxation refers to the process in which the spins give the energy they obtained from the RF pulse back to the surrounding lattice, thereby restoring their equilibrium state. The same process occurs after the spin energy has been altered by a change of the surrounding static magnetic field (e.g. pre-polarization by or insertion into high magnetic field) or if the nonequilibrium state has been achieved by other means (e.g., hyperpolarization by optical pumping). [ citation needed ]
The relaxation time, T 1 (the average lifetime of nuclei in the higher energy state) is dependent on the gyromagnetic ratio of the nucleus and the mobility of the lattice. As mobility increases, the vibrational and rotational frequencies increase, making it more likely for a component of the lattice field to be able to stimulate the transition from high to low energy states. However, at extremely high mobilities, the probability decreases as the vibrational and rotational frequencies no longer correspond to the energy gap between states.
Different tissues have different T 1 values. For example, fluids have long T 1 s (1500-2000 ms), and water-based tissues are in the 400-1200 ms range, while fat based tissues are in the shorter 100-150 ms range. The presence of strongly magnetic ions or particles (e.g., ferromagnetic or paramagnetic ) also strongly alter T 1 values and are widely used as MRI contrast agents .
Magnetic resonance imaging uses the resonance of the protons to generate images. Protons are excited by a radio frequency pulse at an appropriate frequency ( Larmor frequency ) and then the excess energy is released in the form of a minuscule amount of heat to the surroundings as the spins return to their thermal equilibrium. The magnetization of the proton ensemble goes back to its equilibrium value with an exponential curve characterized by a time constant T 1 (see Relaxation (NMR) ). [ citation needed ]
T 1 weighted images can be obtained by setting short repetition time (TR) such as < 750 ms and echo time (TE) such as < 40 ms in conventional spin echo sequences, while in Gradient Echo Sequences they can be obtained by using flip angles of larger than 50 o while setting TE values to less than 15 ms.
T 1 is significantly different between grey matter and white matter and is used when undertaking brain scans. A strong T 1 contrast is present between fluid and more solid anatomical structures, making T 1 contrast suitable for morphological assessment of the normal or pathological anatomy, e.g., for musculoskeletal applications.
Spin–lattice relaxation in the rotating frame is the mechanism by which M xy , the transverse component of the magnetization vector, exponentially decays towards its equilibrium value of zero, under the influence of a radio frequency (RF) field in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). It is characterized by the spin–lattice relaxation time constant in the rotating frame, T 1ρ . It is named in contrast to T 1 , the spin-lattice relaxation time . [ 2 ]
T 1ρ MRI is an alternative to conventional T 1 and T 2 MRI by its use of a long-duration, low-power radio frequency referred to as spin-lock (SL) pulse applied to the magnetization in the transverse plane. The magnetization is effectively spin-locked around an effective B 1 field created by the vector sum of the applied B 1 and any off-resonant component. The spin-locked magnetization will relax with a time constant T 1ρ , which is the time it takes for the magnetic resonance signal to reach 37% (1/e) of its initial value, M x y ( 0 ) {\displaystyle M_{xy}(0)} . Hence the relation: M x y ( t S L ) = M x y ( 0 ) e − t S L / T 1 ρ {\displaystyle M_{xy}(t_{\rm {SL}})=M_{xy}(0)e^{-t_{\rm {SL}}/T_{1\rho }}\,} , where t SL is the duration of the RF field.
T 1ρ can be quantified (relaxometry) by curve fitting the signal expression above as a function of the duration of the spin-lock pulse while the amplitude of spin-lock pulse ( γB 1 ~0.1-few kHz) is fixed. Quantitative T 1ρ MRI relaxation maps reflect the biochemical composition of tissues. [ 3 ]
T 1ρ MRI has been used to image tissues such as cartilage, [ 4 ] [ 5 ] intervertebral discs, [ 6 ] brain, [ 7 ] [ 8 ] and heart, [ 9 ] as well as certain types of cancers. [ 10 ] [ 11 ] | https://en.wikipedia.org/wiki/Spin–lattice_relaxation_in_the_rotating_frame |
In physics , the spin–spin relaxation is the mechanism by which M xy , the transverse component of the magnetization vector, exponentially decays towards its equilibrium value in nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). It is characterized by the spin–spin relaxation time , known as T 2 , a time constant characterizing the signal decay. [ 1 ] [ 2 ] [ 3 ] It is named in contrast to T 1 , the spin–lattice relaxation time. It is the time it takes for the magnetic resonance signal to irreversibly decay to 37% (1/ e ) of its initial value after its generation by tipping the longitudinal magnetization towards the magnetic transverse plane. [ 4 ] Hence the relation
T 2 relaxation generally proceeds more rapidly than T 1 recovery, and different samples and different biological tissues have different T 2 . For example, fluids have the longest T 2 , and water based tissues are in the 40–200 ms range, while fat based tissues are in the 10–100 ms range. Amorphous solids have T 2 in the range of milliseconds, while the transverse magnetization of crystalline samples decays in around 1/20 ms.
When excited nuclear spins—i.e., those lying partially in the transverse plane—interact with each other by sampling local magnetic field inhomogeneities on the micro- and nanoscales, their respective accumulated phases deviate from expected values. [ 4 ] While the slow- or non-varying component of this deviation is reversible, some net signal will inevitably be lost due to short-lived interactions such as collisions and random processes such as diffusion through heterogeneous space.
T 2 decay does not occur due to the tilting of the magnetization vector away from the transverse plane. Rather, it is observed due to the interactions of an ensemble of spins dephasing from each other. [ 5 ] Unlike spin-lattice relaxation , considering spin-spin relaxation using only a single isochromat is trivial and not informative.
Like spin-lattice relaxation, spin-spin relaxation can be studied using a molecular tumbling autocorrelation framework. [ 6 ] The resulting signal decays exponentially as the echo time (TE), i.e., the time after excitation at which readout occurs, increases. In more complicated experiments, multiple echoes can be acquired simultaneously in order to quantitatively evaluate one or more superimposed T 2 decay curves. [ 6 ] The relaxation rate experienced by a spin, which is the inverse of T 2 , is proportional to a spin's tumbling energy at the frequency difference between one spin and another; in less mathematical terms, energy is transferred between two spins when they rotate at a similar frequency to their beat frequency, ω 1 {\displaystyle \omega _{1}} in the figure at right. [ 6 ] In that the beat frequency range is very small relative to the average rotation rate ( 1 / τ c ) {\displaystyle (1/\tau _{c})} , spin-spin relaxation is not heavily dependent on magnetic field strength. This directly contrasts with spin-lattice relaxation, which occurs at tumbling frequencies equal to the Larmor frequency ω 0 {\displaystyle \omega _{0}} . [ 7 ] Some frequency shifts, such as the NMR chemical shift , occur at frequencies proportional to the Larmor frequency, and the related but distinct parameter T 2 * can be heavily dependent on field strength due to the difficulty of correcting for inhomogeneity in stronger magnet bores. [ 4 ]
Assuming isothermal conditions, spins tumbling faster through space will generally have a longer T 2 . Since slower tumbling displaces the spectral energy at high tumbling frequencies to lower frequencies, the relatively low beat frequency will experience a monotonically increasing amount of energy as τ c {\displaystyle \tau _{c}} increases, decreasing relaxation time. [ 6 ] The figure at the left illustrates this relationship. Fast tumbling spins, such as those in pure water, have similar T 1 and T 2 relaxation times, [ 6 ] while slow tumbling spins, such as those in crystal lattices, have very distinct relaxation times.
A spin echo experiment can be used to reverse time-invariant dephasing phenomena such as millimeter-scale magnetic inhomogeneities. [ 6 ] The resulting signal decays exponentially as the echo time (TE), i.e., the time after excitation at which readout occurs, increases. In more complicated experiments, multiple echoes can be acquired simultaneously in order to quantitatively evaluate one or more superimposed T 2 decay curves. [ 6 ] In MRI, T 2 -weighted images can be obtained by selecting an echo time on the order of the various tissues' T 2 s. [ 8 ] In order to reduce the amount of T 1 information and therefore contamination in the image, excited spins are allowed to return to near- equilibrium on a T 1 scale before being excited again. (In MRI parlance, this waiting time is called the "repetition time" and is abbreviated TR). Pulse sequences other than the conventional spin echo can also be used to measure T 2 ; gradient echo sequences such as steady-state free precession (SSFP) and multiple spin echo sequences can be used to accelerate image acquisition or inform on additional parameters. [ 6 ] [ 8 ] | https://en.wikipedia.org/wiki/Spin–spin_relaxation |
The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle ( angular momentum not due to the orbital motion) and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics . In units of the reduced Planck constant ħ , all particles that move in 3 dimensions have either integer spin and obey Bose–Einstein statistics or half-integer spin and obey Fermi–Dirac statistics . [ 1 ] [ 2 ]
All known particles obey either Fermi–Dirac statistics or Bose–Einstein statistics. A particle's intrinsic spin always predicts the statistics of a collection of such particles and conversely: [ 3 ]
A spin–statistics theorem shows that the mathematical logic of quantum mechanics predicts or explains this physical result. [ 4 ]
The statistics of indistinguishable particles is among the most fundamental of physical effects.
The Pauli exclusion principle – that every occupied quantum state contains at most one fermion – controls the formation of matter. The basic building blocks of matter such as protons , neutrons , and electrons are all fermions. Conversely, particles such as the photon , which mediate forces between matter particles, are all bosons. [ citation needed ] A spin–statistics theorem attempts to explain the origin of this fundamental dichotomy. [ 5 ] : 4
Naively, spin, an angular momentum property intrinsic to a particle, would be unrelated to fundamental properties of a collection of such particles. However, these are indistinguishable particles: any physical prediction relating multiple indistinguishable particles must not change when the particles are exchanged.
In a quantum system, a physical state is described by a state vector . A pair of distinct state vectors are physically equivalent if they differ only by an overall phase factor, ignoring other interactions. A pair of indistinguishable particles such as this have only one state. This means that if the positions of the particles are exchanged (i.e., they undergo a permutation), this does not identify a new physical state, but rather one matching the original physical state. In fact, one cannot tell which particle is in which position.
While the physical state does not change under the exchange of the particles' positions, it is possible for the state vector to change sign as a result of an exchange. Since this sign change is just an overall phase, this does not affect the physical state.
The essential ingredient in proving the spin-statistics relation is relativity, that the physical laws do not change under Lorentz transformations . The field operators transform under Lorentz transformations according to the spin of the particle that they create, by definition.
Additionally, the assumption (known as microcausality) that spacelike-separated fields either commute or anticommute can be made only for relativistic theories with a time direction. Otherwise, the notion of being spacelike is meaningless. However, the proof involves looking at a Euclidean version of spacetime, in which the time direction is treated as a spatial one, as will be now explained.
Lorentz transformations include 3-dimensional rotations and boosts . A boost transfers to a frame of reference with a different velocity and is mathematically like a rotation into time. By analytic continuation of the correlation functions of a quantum field theory, the time coordinate may become imaginary , and then boosts become rotations. The new "spacetime" has only spatial directions and is termed Euclidean .
Bosons are particles whose wavefunction is symmetric under such an exchange or permutation, so if we swap the particles, the wavefunction does not change. Fermions are particles whose wavefunction is antisymmetric, so under such a swap the wavefunction gets a minus sign, meaning that the amplitude for two identical fermions to occupy the same state must be zero. This is the Pauli exclusion principle : two identical fermions cannot occupy the same state. This rule does not hold for bosons.
In quantum field theory, a state or a wavefunction is described by field operators operating on some basic state called the vacuum . In order for the operators to project out the symmetric or antisymmetric component of the creating wavefunction, they must have the appropriate commutation law. The operator
(with ϕ {\displaystyle \phi } an operator and ψ ( x , y ) {\displaystyle \psi (x,y)} a numerical function with complex values) creates a two-particle state with wavefunction ψ ( x , y ) {\displaystyle \psi (x,y)} , and depending on the commutation properties of the fields, either only the antisymmetric parts or the symmetric parts matter.
Let us assume that x ≠ y {\displaystyle x\neq y} and the two operators take place at the same time; more generally, they may have spacelike separation, as is explained hereafter.
If the fields commute , meaning that the following holds:
then only the symmetric part of ψ {\displaystyle \psi } contributes, so that ψ ( x , y ) = ψ ( y , x ) {\displaystyle \psi (x,y)=\psi (y,x)} , and the field will create bosonic particles.
On the other hand, if the fields anti-commute , meaning that ϕ {\displaystyle \phi } has the property that
then only the antisymmetric part of ψ {\displaystyle \psi } contributes, so that ψ ( x , y ) = − ψ ( y , x ) {\displaystyle \psi (x,y)=-\psi (y,x)} , and the particles will be fermionic.
An elementary explanation for the spin–statistics theorem cannot be given despite the fact that the theorem is so simple to state. In The Feynman Lectures on Physics , Richard Feynman said that this probably means that we do not have a complete understanding of the fundamental principle involved. [ 3 ]
Numerous notable proofs have been published, with different kinds of limitations and assumptions. They are all "negative proofs", meaning that they establish that integer spin fields cannot result in fermion statistics while half-integral spin fields cannot result in boson statistics. [ 5 ] : 487
Proofs that avoid using any relativistic quantum field theory mechanism have defects. Many such proofs rely on a claim that | ψ ( α 1 , α 2 , α 3 , … ) | 2 = | P ^ ψ ( α 1 , α 2 , α 3 , … ) | 2 , {\displaystyle |\psi (\alpha _{1},\alpha _{2},\alpha _{3},\dots )|^{2}=|{\hat {P}}\psi (\alpha _{1},\alpha _{2},\alpha _{3},\dots )|^{2},} where the operator P ^ {\displaystyle {\hat {P}}} permutes the coordinates. However, the value on the left-hand side represents the probability of particle 1 at r 1 {\displaystyle r_{1}} , particle 2 at r 2 {\displaystyle r_{2}} , and so on, and is thus quantum-mechanically invalid for indistinguishable particles. [ 6 ] : 567
The first proof was formulated [ 7 ] in 1939 by Markus Fierz , a student of Wolfgang Pauli , and was rederived in a more systematic way by Pauli the following year. [ 8 ] In a later summary, Pauli listed three postulates within relativistic quantum field theory as required for these versions of the theorem:
Their analysis neglected particle interactions other than commutation/anti-commutation of the state. [ 9 ] [ 5 ] : 374
In 1949 Richard Feynman gave a completely different type of proof [ 10 ] based on vacuum polarization , which was later critiqued by Pauli. [ 9 ] [ 5 ] : 368 Pauli showed that Feynman's proof explicitly relied on the first two postulates he used and implicitly used the third one by first allowing negative probabilities but then rejecting field theory results with probabilities greater than one.
A proof by Julian Schwinger in 1950 based on time-reversal invariance [ 11 ] followed a proof by Frederik Belinfante in 1940 based on charge-conjugation invariance, leading to a connection to the CPT theorem more fully developed by Pauli in 1955. [ 12 ] These proofs were notably difficult to follow. [ 5 ] : 393
Work on the axiomatization of quantum field theory by Arthur Wightman lead to a theorem that stated that the expectation value of the product of two fields, ϕ ( x ) ϕ ( y ) {\displaystyle \phi (x)\phi (y)} , could be analytically continued to all separations ( x − y ) {\displaystyle (x-y)} . [ 5 ] : 425 (The first two postulates of the Pauli-era proofs involve the vacuum state and fields at separate locations.) The new result allowed more rigorous proofs of the spin–statistics theorems by Gerhart Lüders and Bruno Zumino [ 13 ] and by Peter Burgoyne. [ 5 ] : 393 In 1957 Res Jost derived the CPT theorem using the spin–statistics theorem, and Burgoyne's proof of the spin–statistics theorem in 1958 required no constraints on the interactions nor on the form of the field theories. These results are among the most rigorous practical theorems. [ 14 ] : 529
In spite of these successes, Feynman, in his 1963 undergraduate lecture that discussed the spin–statistics connection, says: "We apologize for the fact that we cannot give you an elementary explanation." [ 3 ] Neuenschwander echoed this in 1994, asking whether there was any progress, [ 15 ] spurring additional proofs and books. [ 5 ] Neuenschwander's 2013 popularization of the spin–statistics connection suggested that simple explanations remain elusive. [ 16 ]
In 1987 Greenberg and Mohapatra proposed that the spin–statistics theorem could have small violations. [ 17 ] [ 18 ] With the help of very precise calculations for states of the He atom that violate the Pauli exclusion principle , [ 19 ] Deilamian, Gillaspy and Kelleher [ 20 ] looked for the 1s2s 1 S 0 state of He using an atomic-beam spectrometer. The search was unsuccessful with an upper limit of 5×10 −6 .
The Lorentz group has no non-trivial unitary representations of finite dimension. Thus it seems impossible to construct a Hilbert space in which all states have finite, non-zero spin and positive, Lorentz-invariant norm. This problem is overcome in different ways depending on particle spin–statistics.
For a state of integer spin the negative norm states (known as "unphysical polarization") are set to zero, which makes the use of gauge symmetry necessary.
For a state of half-integer spin the argument can be circumvented by having fermionic statistics. [ 21 ]
In 1982, physicist Frank Wilczek published a research paper on the possibilities of possible fractional-spin particles, which he termed anyons from their ability to take on "any" spin. [ 22 ] He wrote that they were theoretically predicted to arise in low-dimensional systems where motion is restricted to fewer than three spatial dimensions. Wilczek described their spin statistics as "interpolating continuously between the usual boson and fermion cases". [ 22 ] The effect has become the basis for understanding the fractional quantum hall effect . [ 23 ] [ 24 ] | https://en.wikipedia.org/wiki/Spin–statistics_theorem |
A spiral antenna is a type of radio frequency antenna shaped as a spiral , [ 1 ] : 14‑2 first described in 1956. [ 2 ] Archimedean spiral antennas are the most popular, while logarithmic spiral antennas are independent of frequency: [ 3 ] the driving point impedance, radiation pattern and polarization of such antennas remain unchanged over a large bandwidth. [ 4 ] Spiral antennas are inherently circularly polarized with low gain ; antenna arrays can be used to increase the gain. Spiral antennas are reduced in size with its windings making it an extremely small structure. Lossy cavities [ 5 ] are usually placed at the back to eliminate back lobes, because a unidirectional pattern is usually preferred in such antennas. Spiral antennas are classified into different configurations: Archimedean spiral, logarithmic spiral, square spiral , etc.
In general, antennas may operate in three different modes: traveling wave, fast wave, and leaky wave. Spiral antennas use all three.
The traveling wave, formed on spiral arms, allows for broadband performance. Fast wave is due to mutual coupling phenomenon occurring between arms of spiral. Leaky wave “leaks” the energy during propagation through the spiral arms to produce radiation.
Ring theory (band theory) explains the working principle of spiral antenna. The theory states that spiral antenna radiate from an active region where the circumference of the spiral equals the wavelength. [ 6 ]
Different design parameters are to be considered while designing a square spiral antenna. The parameters include spacing between the turns s {\displaystyle s} , width of arm w {\displaystyle w} , inner radius r 1 {\displaystyle r_{1}} and outer radius r 2 {\displaystyle r_{2}} . The inner radius is measured from center of the spiral to center of the first turn while the outer radius is measured from center of the spiral to center of the outermost turn. Other than these design parameters, spiral antennas have lowest ( f low = c / 2 π r 2 ) {\displaystyle f_{\text{low}}=c/2\pi r_{2})} and highest ( f high = c / 2 π r 1 ) {\displaystyle (f_{\text{high}}=c/2\pi r_{1})} operating frequencies. Here c ≤ 299.79 Mm/s = c 0 {\displaystyle c\leq 299.79{\text{ Mm/s}}=c_{0}} corresponds to speed of light in the metal of the antenna, mainly determined by the electrical permittivity of the substrate the spiral lies on, and its over-coating (if any).
In a polar ( r , θ ) {\displaystyle (r,\theta )} coordinate system, the spiral grows along the r {\displaystyle r} -axis and θ {\displaystyle \theta } -axis simultaneously. Often-used Archemedian spirals satisfy a particularly simple equation r = a + b θ {\displaystyle r=a+b\,\theta } where a {\displaystyle a} corresponds to growth factor and b {\displaystyle b} corresponds to multiplication factor. The consequence is equal spacing between successive turns, which limits the width of the spiral arms, which is usually kept constant. Other choices of spiral shape can also be used, such as logarithmic spirals that satisfy r = a + b e m θ {\displaystyle r=a+b\,e^{m\theta }} ; the resulting spiral arms are more widely spaced in the outer turns, which can better accommodate arms that widen significantly.
Different designs of spiral antenna can be obtained by varying number of turns for each arm, the number of arms, the type of spiral, the spacing between its turns, the variation of the width of its arm(s), and the material(s) that surround it, such as the substrate it lies on.
The antenna usually has two conductive spiral arms, extending from the center outwards. The direction of rotation of the spiral defines the direction of antenna polarization. Additional spirals may be included as well, to form a multi-spiral structure. The antenna may be a flat disc, with conductors resembling a pair of loosely nested clock springs, or the spirals may extend in a three-dimensional shape like a screw thread.
The output of a two-arm or four-arm spiral antenna is a balanced line . If a single input or output line is desired – for example a grounded coaxial line – then a balun or other transformer is added to alter the signal's electrical mode.
Usually the spiral is cavity-backed – that is, there is a cavity of air or non-conductive material or vacuum, surrounded by conductive walls behind the spiral. A cavity with the proper shape and size changes the antenna pattern to receive and transmit in a single direction, away from the cavity.
The spiral can be printed or etched over a specifically chosen dielectric medium, whose permittivity can be used to alter the frequency for a given size. Dielectric mediums like Rogers RT Duroid help in reducing the physical size of antenna. Thin substrates with higher permittivity can achieve the same result as thick substrates with lower permittivity. The only problem with such materials is their less availability and high costs. [ 7 ]
Spiral antennas transmit circularly polarized radio waves, and will receive linearly polarized waves in any orientation, but will drastically attenuate circularly polarized signals received with the opposite-rotation. A spiral antenna will reject circularly polarized waves of one type, while receiving perfectly well waves having the other polarization.
One application of spiral antennas is wide-band communications. Another application of spiral antennas is monitoring of the frequency spectrum. One antenna can receive over a wide bandwidth, for example a ratio 5:1 between the maximum and minimum frequency. Usually a pair of spiral antennas are used in this application, having identical parameters except the polarization, which is opposite (one is right-hand, the other left-hand oriented). Spiral antennas are useful for microwave direction-finding. [ 8 ] | https://en.wikipedia.org/wiki/Spiral_antenna |
Spiral arms are a defining feature of spiral galaxies . They manifest as spiral -shaped regions of enhanced brightness within the galactic disc . Typically, spiral galaxies exhibit two or more spiral arms. The collective configuration of these arms is referred to as the spiral pattern or spiral structure of the galaxy.
The appearance of spiral sleeves is quite diverse. Grand design spiral galaxies exhibit a symmetrical and distinct pattern, comprising two spiral arms that extend throughout the galaxy. In contrast, the spiral structure of flocculent galaxies comprises numerous small fragments of arms that are not connected to each other. The appearance of spiral arms varies across the electromagnetic spectrum .
In addition to increased brightness, spiral arms are characterised by an increased concentration of interstellar gas and dust , bright stars and star clusters , active starburst , a bluer colour, and an enhanced magnetic field strength in galaxies. The contribution of spiral arms to the total galaxy luminosity can reach 40–50% for some galaxies. The characteristics of spiral arms are correlated with other properties of galaxies, for example, the twist angle of spiral arms is related to parameters such as the mass of the supermassive black hole at the centre and the contribution of the bulge to the total luminosity.
Two main theories have been proposed to explain the origin of spiral arms: the stochastic self-propagating star formation model and the density wave theory . These theories describe different variants of the spiral structure and do not exclude each other. In addition to these theories, there are other theories that can explain the appearance of spiral structure in some cases.
The spiral structure was first identified in 1850 by Lord Rosse in the galaxy M51 . The nature of spiral structure in galaxies remained unresolved for a considerable period of time.
Spiral arms [ 1 ] are a defining feature of the structural composition of spiral galaxies , which are situated within discs and exhibit heightened brightness relative to their surrounding environment. [ 2 ] Such structures take the form of spirals , which in unbarred galaxies usually originate from a region near the centre of the galaxy, whereas in barred galaxies they originate at the ends of the bar. [ 3 ] The spiral arms do not extend over the entire radius of the disc and cease at the distance at which the disc can still be discerned. [ 4 ] A galaxy typically comprises two or more spiral arms. [ 5 ] The collective configuration of these arms within a galaxy is referred to as a spiral pattern or spiral structure. [ 6 ]
Around two thirds of all massive galaxies are spiral galaxies. [ 7 ] Spiral arms have been observed in galaxies at redshifts up to z ≈ 1 {\displaystyle z\approx 1} , and on occasion even at greater distances, which corresponds to a time when the age of the Universe was less than half of the present one. This suggests that the spiral structure is a long-lived phenomenon. [ 8 ]
The spiral arms exhibit considerable variation in their appearance. [ 5 ] In general, they are characterized by an increased concentration of gas and dust , active starburst , and a greater prevalence of star clusters , H II regions , and bright stars than in the remainder of the disk. [ 2 ] While spiral arms are primarily identifiable due to their young stellar population, there also exists an increased concentration of old stars within them. [ 4 ] [ 7 ]
The appearance and expression of spiral branches in a galaxy may vary depending on the part of the electromagnetic spectrum in which it is observed. In the blue and ultraviolet parts of the spectrum, the spiral arms are well defined due to the presence of blue supergiants . In the red and near-infrared , older stars contribute more, which makes the spiral arms appear smoother, but less contrasted. Radiation from interstellar dust makes the spiral arms bright in the far infrared, while radiation from neutral hydrogen and molecules makes them bright at radio band . The greatest contrast and amount of fine detail in spiral arms can be seen when observed in emission spectral lines produced by emission nebulae , as well as in polyaromatic hydrocarbon lines produced by cold gas clouds. [ 9 ]
The appearance of spiral arms is one of the criteria for galaxy morphological classification . For example, in Hubble's classification scheme , spiral galaxies are divided into types Sa, Sb, Sc. Barred spiral galaxies are divided into types SBa, SBb and SBc. The spiral arms of early type Sa and SBa galaxies are tightly wound and smooth, while those of late type Sc and SBc galaxies are knotty and loosely wound. Types Sb and SBb exhibit intermediate characteristics. [ 10 ] [ 11 ]
The spiral structure of galaxies exhibits considerable diversity in appearance. Grand design spiral galaxies exhibit a symmetrical and clear pattern comprising two spiral arms that extend throughout the galaxy. They account for 10% of the total number of spiral galaxies. In contrast, the spiral structure of flocculent galaxies consists of numerous small fragments of arms that are not connected to each other. Among spiral galaxies, the fraction of such galaxies is equal to 30%. [ 4 ] [ 13 ]
The remainder of the galaxies are of an intermediate type, referred to as "multi-armed", [ 14 ] which exhibit the proberties of both the flocculent and grand design galaxies. For example, they may appear to be grand design galaxies, yet possess more than two arms. Alternatively, they may exhibit a more ordered two-arm structure in the interior, which becomes irregular at the periphery. [ 15 ] [ 16 ] [ 17 ] Nevertheless, in almost all cases, both types of structure are present in the spiral structure. Even grand design galaxies have details that do not fit into the spiral pattern. [ 4 ] Additionally, there are galaxies that exhibit different types of spiral structure when observed across different spectral ranges. [ 18 ] The distinction between the two main types of spiral arms appears to be related to fundamental physical differences between them. [ 19 ]
Additionally, spiral arms are subdivided into two categories: massive and filamentary. In the first instance, the spiral arms are wide, diffuse, and do not contrast significantly with the space between them. In contrast, in the second instance, the spiral arms are narrow and clearly defined. [ 21 ]
The shape of the arm is usually parameterised by the pitch angle μ {\displaystyle \mu } . The pitch angle is the angle between the tangent to spiral arm at a given point and the perpendicular to the radius drawn to that point. In the majority of spiral galaxies, the average pitch angle lies within the range of 5° to 30°. [ 13 ] [ 23 ] Spiral arms with a small pitch angle are called tightly wound, while those with a larger pitch angle are called open. [ 24 ]
The shape of spiral arms is often described in a simplified manner as a logarithmic spiral . However, spiral arms may also be described as an Archimedean or hyperbolic spiral . In the case of the logarithmic spiral, the pitch angle is constant. It decreases with increasing distance from the centre in the Archimedean spiral and increases in the hyperbolic spiral. The measurements of twist angles in galaxies indicate that only a minority of spiral galaxies have pitch angles of the arms that are close to constant. More than two-thirds of galaxies have pitch angles that vary by more than 20%. The average twist angle is found to correlate with a number of different galaxy parameters. For example, the spiral arms of galaxies with brighter bulges tend to be wound tighter. [ 24 ]
Spiral arms may additionally be categorized as either trailing or leading. In the case of trailing spiral arms, their outer tips point in the direction opposite to the direction of galaxy rotation. In the case of leading arms, their outer tips point in the same direction in which the galaxy rotates. In practice, it is challenging to ascertain whether the arms of a given galaxy are leading or trailing. To observe the spiral structure, the galaxy should not be tilted excessively towards the picture plane. However, a slight tilt is necessary to determine the direction of rotation. Additionally, the side of the galaxy closer to the observer needs to be identified. A review of the observational data indicates that the majority of galaxies exhibit trailing spiral arms, with leading arms being relatively uncommon. For instance, among the two hundred galaxies studied in this manner, only two may have leading arms. In some instances, galaxies exhibit both leading and trailing spiral arms, as exemplified by NGC 4622 . Numerical simulations have demonstrated that leading spiral arms can emerge in specific circumstances. One such instance is when the dark matter halo rotates in opposition to the galaxy disk. [ 26 ] [ 27 ]
The width of spiral arms in the majority of galaxies increases with increasing distance from the centre. Grand design galaxies exhibit the greatest width of spiral arms. [ 28 ]
The ratio of the luminosity of the spiral structure to the luminosity of the entire galaxy is greatest for grand design spiral galaxies. For these galaxies, this ratio is 21% on average, with some reaching as high as 40–50%. For flocculent and multi-arm galaxies, the ratio is 13% and 14%, respectively. Additionally, the proportion of spiral arms in the total luminosity increases in later morphological types. For Sa-type galaxies, this proportion averages 13%, while for Sc-type galaxies it averages 30%. [ 28 ]
The colour of the spiral arms becomes increasingly blue for galaxies of late morphological types. The colour index g-r for Sc-type galaxies is approximately 0.3–0.4 m , while for Sa-type galaxies it is 0.5–0.6 m . [ 28 ]
Additionally, there are anemic galaxies (anemic spirals). [ 29 ] These galaxies are distinguished by a diffuse, faint spiral pattern, which is attributed to a reduced quantity of gas and, consequently, a diminished star formation rate in comparison to normal spiral galaxies of the same morphological type. Anemic galaxies are more prevalent in galaxy clusters . Apparently, the galaxies in these clusters are subject to ram pressure , which results in the rapid loss of gas. It is hypothesized that this type of galaxy may be in-between spiral and lenticular galaxies. [ 30 ] [ 31 ]
Stronger magnetic fields are observed in the spiral arms than in the remainder of the galaxy. The average value of magnetic fields in spiral galaxies is 10 microgauss , while in their spiral arms it is 25 microgauss . In galaxies with a pronounced spiral pattern, the magnetic fields are orientated along the arms. However, in some cases, the magnetic field may form a separate spiral structure that runs in the space between the visible spiral arms. Conversely, magnetic fields can influence the movement of gas within the galaxy and contribute to the formation of spiral arms. [ 32 ] [ 33 ] However, they are insufficiently strong to play a dominant role in the formation of spiral arms. [ 34 ]
The parameters of spiral arms correlate with other galaxy properties. For instance, it is established that galaxies with a greater pitch angle typically exhibit a lower mass of the supermassive black hole at their centre [ 35 ] and a smaller galaxy mass in general. Additionally, their bulge contributes less to the total luminosity, they have a lower velocity dispersion in the centre, and their rotation curves appear to be more increasing. [ 36 ] However, these dependencies are not particularly pronounced. [ 37 ] Although the pitch angle of the spiral arms was originally introduced into the galaxy morphological classification as one of the classification criteria, subsequent analysis has revealed that this value correlates with the morphological type to a lesser extent than, for example, the indicator of the colour of the spiral arms. [ 28 ] The correlation between the pitch angle and the aforementioned parameters can be theoretically explained. The described quantities are related to the mass distribution within the galaxy, which affects the manner in which the density wave propagates within the galaxy disc. [ 38 ]
In more massive galaxies with a more ordered structure, spiral arms are observed to be more pronounced and contrasting. [ 28 ] Additionally, the contrast between spiral arms is more pronounced in galaxies with a pronounced bar , although this correlation is relatively weak. [ 39 ] In general, flocculent galaxies have a lower mass and a later morphological type than grand design galaxies. [ 40 ]
It is challenging to ascertain the presence of spiral arms in the disc of our galaxy through optical observation, given that the Sun is situated within the plane of the Milky Way disc, and the light is being absorbed by interstellar dust . Nevertheless, spiral arms can be observed, for instance, when mapping the distribution of neutral hydrogen or molecular clouds . [ 41 ]
The precise location, length, and number of spiral arms remain uncertain. [ 1 ] [ 42 ] However, the prevailing view is that the Milky Way contains four major spiral arms: two main ones – the Scutum–Centaurus and Perseus arms, and two secondary ones – the Norma and Sagittarius arms. [ 43 ] Their pitch angle is approximately 12°, and their width is estimated at 800 parsecs . [ 44 ] In addition to the large arms, smaller, similar formations, such as the Orion arm , are also distinguished. [ 45 ]
The prevalence of spiral galaxies indicates that spiral structure is a long-lived phenomenon. However, since galaxies themselves rotate differentially rather than as solid bodies, any structure in the disc should curve significantly and disappear in approximately one to two revolutions. The two most prevalent solutions to this issue are the stochastic self-propagating star formation model (SSPSF) and the density wave theory , which describe disparate variants of the spiral structure. The first explanation posits that spiral arms are perpetually forming and dissipating without sufficient time to undergo significant twisting – such spiral arms are designated as material arms. The density wave theory posits that the spiral pattern is a density wave, thereby rotating independently of the disc as a solid body. Consequently, spiral arms are designated as wave arms. It is possible for these types of spiral arms to occur simultaneously within the same galaxy. [ 19 ] [ 46 ]
Tidal tails observed in interacting galaxies are also considered material spiral arms. Due to the low velocity of matter at a distance from the galaxy, tidal tails appear to persist for an extended period of time. [ 47 ]
The SSPSF model posits that spiral arms emerge when starburst becomes active within a region of the galaxy. The presence of young, bright stars in this region has the effect of influencing the surrounding interstellar medium . For instance, a supernova explosion generates a shockwave in the gas, thereby facilitating the spread of star formation across the galactic disk. [ 48 ] In a period of less than 100 million years, the brightest stars in this region have time to extinguish. This is less than the time required for one revolution of the galaxy. The differential rotation of this region allows it to stretch into a short arc. Given that starburst is a continuous process occurring in different regions of the disc, there are numerous such arcs at different times throughout the disc, which can be observed as a flocculent spiral pattern. [ 49 ] [ 50 ] Given that such spiral arms are only visible due to young stars, they have a minimal impact on the mass distribution within the galaxy and are rarely observed in the infrared . [ 47 ]
In the context of density wave theory, spiral arms are understood to emerge when mechanical oscillations occur within a disc, giving rise to a density wave – the stars move within the disc in such a way that they converge in specific regions and become more concentrated. The density wave exerts a governing influence not only on the stars but also on the gas, thereby promoting a more active starburst in regions where the concentration of stars is higher. Concurrently, at various points in time, different stars emerge within the spiral arm, resulting in the density wave moving at a different speed than the stellar disc. Consequently, the density wave is not subject to twisting. The influence of this mechanism results in the formation of a large-scale, ordered spiral structure, which is also observed in the infrared. [ 51 ] [ 52 ] [ 53 ] The concentration of stars in the spiral arm increases by a mere 10–20%, yet this relatively modest change in gravitational potential has a profound impact on the gas dynamics. The gas accelerates, and shock waves can occur in it, appearing as dark dust lanes in the arms. [ 6 ]
It is challenging to confirm the presence of a density wave in practice. However, it is possible to do so, for instance, by detecting a specific corotation radius , which is a region where the spiral arm moves at the same speed as the stars. It can be identified by observing colour gradients within the arms. Since the stellar population forms within an arm and subsequently reddens over time, a colour gradient should be observed across the arm if its velocity differs from that of the arm. [ 54 ] [ 55 ] It is hypothesised that density waves are created and maintained by the bars of galaxies or by tidal force of their satellites . [ 6 ]
The density wave theory postulates that only trailing spiral arms are stable, and that any leading structure must at some point transition into a trailing one. Concurrently, the structure itself is amplified for a period following the transformation, which is called swing amplification. [ 56 ]
Some theories propose alternative mechanisms for the appearance of spiral arms that differ from the density wave theory and the SSPSF model. These theories are not intended to replace the aforementioned theories entirely, but rather to explain the appearance of spiral arms in specific cases. For instance, the manifold theory is applicable only to barred spiral galaxies . According to this theory, the gravitational influence of the bar causes the orbits of the stars to be arranged in a certain way, creating spiral arms and moving along them. The name of the theory is related to the fact that in this model the stars moving in spiral arms form a manifold in phase space . In contrast to the density wave theory, the manifold theory does not posit the emergence of colour gradients in spiral arms, which are in fact observed in numerous galaxies. The fact that in galaxies with a bar, spiral arms originate from a region proximate to the bar may suggest a correlation between these structures and the manifold theory. However, this is not the sole theory that explains the genesis of arms due to bars. [ 57 ] [ 58 ]
The spiral arms were first discovered in the Whirlpool Galaxy (M51), in which Lord Rosse identified the spiral structure in 1850. [ 42 ]
In 1896, the problem of twisting was formulated. If spiral arms were material entities, due to differential rotation, they would twist very rapidly to the point where they would be impossible to observe. Consequently, the question of the nature of the spiral structure remained unresolved for a considerable period of time. Since 1927, this question has been addressed by Bertil Lindblad , who in 1961 correctly concluded that the spiral arms arise due to gravitational interaction between the stars in the disc. Subsequently, in 1964, Chia-Chiao Lin and Frank Shu proposed the theory that spiral arms can be conceptualised as density waves. [ 52 ] [ 59 ] The SSPSF model was first proposed in 1978, although the concept of a supernova explosion stimulating starburst in neighbouring regions was first proposed by Ernst Opik as early as 1953. This observation formed the basis of the subsequent theory. [ 60 ] [ 61 ]
In 1953, the distances to the various stellar associations in our galaxy were measured with a high degree of accuracy. This enabled the discovery of a spiral structure in the Milky Way. [ 41 ]
The classification of galaxies into flocculent, multi-armed, and grand design categories is derived from a more complex morphological classification scheme involving 10 classes that describe the type of spiral pattern. The classification scheme was developed by Debra and Bruce Elmegreen in 1987. Subsequently, they proposed a simplified scheme, which is the one that is currently in use. [ 62 ] [ 63 ]
Despite the considerable successes of the density wave theory, the physical nature of spiral arms remains a topic of debate, with no clear consensus yet reached. [ 64 ] [ 65 ] | https://en.wikipedia.org/wiki/Spiral_arm |
In geometry , the spiral of Theodorus (also called the square root spiral , Pythagorean spiral , or Pythagoras's snail ) [ 1 ] is a spiral composed of right triangles , placed edge-to-edge. It was named after Theodorus of Cyrene .
The spiral is started with an isosceles right triangle, with each leg having unit length . Another right triangle (which is the only automedian right triangle ) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2 ) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3 . The process then repeats; the n {\displaystyle n} th triangle in the sequence is a right triangle with the side lengths n {\displaystyle {\sqrt {n}}} and 1, and with hypotenuse n + 1 {\displaystyle {\sqrt {n+1}}} . For example, the 16th triangle has sides measuring 4 = 16 {\displaystyle 4={\sqrt {16}}} , 1 and hypotenuse of 17 {\displaystyle {\sqrt {17}}} .
Although all of Theodorus' work has been lost, Plato put Theodorus into his dialogue Theaetetus , which tells of his work. It is assumed that Theodorus had proved that all of the square roots of non-square integers from 3 to 17 are irrational by means of the Spiral of Theodorus. [ 2 ]
Plato does not attribute the irrationality of the square root of 2 to Theodorus, because it was well known before him. Theodorus and Theaetetus split the rational numbers and irrational numbers into different categories. [ 3 ]
Each of the triangles' hypotenuses h n {\displaystyle h_{n}} gives the square root of the corresponding natural number , with h 1 = 2 {\displaystyle h_{1}={\sqrt {2}}} .
Plato, tutored by Theodorus, questioned why Theodorus stopped at 17 {\displaystyle {\sqrt {17}}} . The reason is commonly believed to be that the 17 {\displaystyle {\sqrt {17}}} hypotenuse belongs to the last triangle that does not overlap the figure. [ 4 ]
In 1958, Kaleb Williams proved that two hypotenuses will never overlap, regardless of how far the spiral is continued. Also, if the sides of unit length are extended into a line , they will never pass through any of the other vertices of the total figure. [ 4 ] [ 5 ]
Theodorus stopped his spiral at the triangle with a hypotenuse of 17 {\displaystyle {\sqrt {17}}} . If the spiral is continued to infinitely many triangles, many more interesting characteristics are found.
If φ n {\displaystyle \varphi _{n}} is the angle of the n {\displaystyle n} th triangle (or spiral segment), then: tan ( φ n ) = 1 n . {\displaystyle \tan \left(\varphi _{n}\right)={\frac {1}{\sqrt {n}}}.} Therefore, the growth of the angle φ n {\displaystyle \varphi _{n}} of the next triangle n {\displaystyle n} is: [ 1 ] φ n = arctan ( 1 n ) . {\displaystyle \varphi _{n}=\arctan \left({\frac {1}{\sqrt {n}}}\right).}
The sum of the angles of the first k {\displaystyle k} triangles is called the total angle φ ( k ) {\displaystyle \varphi (k)} for the k {\displaystyle k} th triangle. It grows proportionally to the square root of k {\displaystyle k} , with a bounded correction term c 2 {\displaystyle c_{2}} : [ 1 ] φ ( k ) = ∑ n = 1 k φ n = 2 k + c 2 ( k ) {\displaystyle \varphi \left(k\right)=\sum _{n=1}^{k}\varphi _{n}=2{\sqrt {k}}+c_{2}(k)} where lim k → ∞ c 2 ( k ) = − 2.157782996659 … {\displaystyle \lim _{k\to \infty }c_{2}(k)=-2.157782996659\ldots } ( OEIS : A105459 ).
The growth of the radius of the spiral at a certain triangle n {\displaystyle n} is Δ r = n + 1 − n . {\displaystyle \Delta r={\sqrt {n+1}}-{\sqrt {n}}.}
The Spiral of Theodorus approximates the Archimedean spiral . [ 1 ] Just as the distance between two windings of the Archimedean spiral equals mathematical constant π {\displaystyle \pi } , as the number of spins of the spiral of Theodorus approaches infinity , the distance between two consecutive windings quickly approaches π {\displaystyle \pi } . [ 6 ]
The following table shows successive windings of the spiral approaching pi:
As shown, after only the fifth winding, the distance is a 99.97% accurate approximation to π {\displaystyle \pi } . [ 1 ]
The question of how to interpolate the discrete points of the spiral of Theodorus by a smooth curve was proposed and answered by Philip J. Davis in 2001 by analogy with Euler's formula for the gamma function as an interpolant for the factorial function. Davis found the function [ 7 ] T ( x ) = ∏ k = 1 ∞ 1 + i / k 1 + i / x + k ( − 1 < x < ∞ ) {\displaystyle T(x)=\prod _{k=1}^{\infty }{\frac {1+i/{\sqrt {k}}}{1+i/{\sqrt {x+k}}}}\qquad (-1<x<\infty )} which was further studied by his student Leader [ 8 ] and by Iserles . [ 9 ] This function can be characterized axiomatically as the unique function that satisfies the functional equation f ( x + 1 ) = ( 1 + i x + 1 ) ⋅ f ( x ) , {\displaystyle f(x+1)=\left(1+{\frac {i}{\sqrt {x+1}}}\right)\cdot f(x),} the initial condition f ( 0 ) = 1 , {\displaystyle f(0)=1,} and monotonicity in both argument and modulus . [ 10 ]
An analytic continuation of Davis' continuous form of the Spiral of Theodorus extends in the opposite direction from the origin. [ 11 ]
In the figure the nodes of the original (discrete) Theodorus spiral are shown as small green circles. The blue ones are those, added in the opposite direction of the spiral.
Only nodes n {\displaystyle n} with the integer value of the polar radius r n = ± | n | {\displaystyle r_{n}=\pm {\sqrt {|n|}}} are numbered in the figure.
The dashed circle in the coordinate origin O {\displaystyle O} is the circle of curvature at O {\displaystyle O} . | https://en.wikipedia.org/wiki/Spiral_of_Theodorus |
A spiral plater is an instrument used to dispense a liquid sample onto a Petri dish in a spiral pattern. Commonly used as part of a CFU count procedure for the purpose of determining the number of microbes in the sample. [ 1 ] In this setting, after spiral plating, the Petri dish is incubated for several hours after which the number of colony forming microbes (CFU) is determined. Spiral platers are also used for research, clinical diagnostics and as a method for covering a Petri dish with bacteria before placing antibiotic discs for AST .
The spiral plater rotates the dish while simultaneously dispensing the liquid and either linearly moving the dish or the dispensing tip. This creates the common spiral pattern. If all movements are done in constant speed, the spiral created would have a lower concentration on the outside of the plate than on the inside. More advanced spiral platers provide different options for spiral patterns such as constant concentration (by slowing down the spinning and / or the lateral movements) or exponential concentration (by speeding up the spinning and / or the lateral movements). [ citation needed ]
Spiral plating is used extensively for microbiological testing of food, milk and milk products and cosmetics. It is an approved method by the FDA. [ 2 ] The advantage of spiral plating is less plates used versus plating manually because different concentrations are present on each plate. [ 3 ] [ 1 ] This also makes it harder to count the colonies and requires special techniques and equipment. [ 2 ]
Spiral platers are either available as stand-alone instruments that are fed manually with plates and samples or fed automatically using dedicated stackers. Alternatively spiral platers are available as integrated devices as part of larger automated platforms. In this case a larger workflow is often automated, e.g. plating, incubation and counting . [ citation needed ] | https://en.wikipedia.org/wiki/Spiral_plater |
The term spiral separator can refer to either a device for separating slurry components by density ( wet spiral separators ), or for a device for sorting particles by shape ( dry spiral separators ).
Spiral separators of the wet type, also called spiral concentrators , are devices to separate solid components in a slurry, based upon a combination of the solid particle density as well as the particle's hydrodynamic properties (e.g. drag ). The device consists of a tower, around which is wound a sluice , from which slots or channels are placed in the base of the sluice to extract solid particles that have come out of suspension .
As larger and heavier particles sink to the bottom of the sluice faster and experience more drag from the bottom, they travel slower, and so move towards the center of the spiral. Conversely, light particles stay towards the outside of the spiral, with the water, and quickly reach the bottom. At the bottom, a "cut" is made with a set of adjustable bars, channels, or slots, separating the low and high density parts.
Typical spiral concentrators will use a slurry from about 20%-40% solids by weight, with a particle size somewhere between 0.75—1.5mm (17-340 mesh), though somewhat larger particle sizes are sometimes used. The spiral separator is less efficient at the particle sizes of 0.1—0.074mm however. [ citation needed ] For efficient separation, the density difference between the heavy minerals and the light minerals in the feedstock should be at least 1 g/cm 3 ; [ 1 ] and because the separation is dependent upon size and density, spiral separators are most effective at purifying ore if its particles are of uniform size and shape. A spiral separator may process a couple tons per hour of ore, per flight, and multiple flights may be stacked in the same space as one, to improve capacity. [ 2 ]
Many things can be done to improve the separation efficiency, including:
Dry spiral separators, capable of distinguishing round particles from nonrounds, are used to sort the feed by shape. The device consists of a tower, around which is wound an inwardly inclined flight. A catchment funnel is placed around this inner flight. Round particles roll at a higher speed than other objects, and so are flung off the inner flight and into the collection funnel. Shapes which are not round enough are collected at the bottom of the flight.
Separators of this type may be used for removing weed seeds from the intended harvest, or to remove deformed lead shot . | https://en.wikipedia.org/wiki/Spiral_separator |
In geometry , a spirangle is a spiral polygonal chain . Spirangles are similar to spirals in that they expand from a center point as they grow larger, but they are made out of straight line segments , instead of curves. Spirangle vectographs are used in vision therapy to promote stereopsis and help resolve problems with hand–eye coordination .
A two-dimensional spirangle is an open figure consisting of a line bent into angles similar to a corresponding polygon. The spirangle can start at a center point, or a distance from the center, and has some number of turns around the center point.
Three-dimensional spirangles have layers that slant upward, progressively gaining height from the previous segment. This is similar to staircases in large buildings that turn at the top of each flight. The segments also may progressively lose an amount of length and resemble a pyramid. | https://en.wikipedia.org/wiki/Spirangle |
Spirapril , sold under the brand name Renormax among others, is an ACE inhibitor antihypertensive drug used to treat hypertension . It belongs to dicarboxy group of ACE inhibitors. [ citation needed ]
It was patented in 1980 and approved for medical use in 1995. [ 1 ]
Like many ACE inhibitors, this prodrug is converted to the active metabolite spiraprilat following oral administration. Unlike other members of the group, it is eliminated both by renal and hepatic routes, which may allow for greater use in patients with renal impairment. [ 2 ] However, data on its effect upon the renal function are conflicting. [ 3 ]
This drug article relating to the cardiovascular system is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spirapril |
Spiridoula Christos Matsika (born 1971) is a Greek theoretical chemist . She was elected as a fellow of the American Physical Society in 2014.
Spiridoula Christos Matsika was born in 1971 in Greece; [ 1 ] [ 2 ] she attended the National and Kapodistrian University of Athens for her bachelor's degree in chemistry, graduating in 1994. She completed her PhD at the Ohio State University , graduating in 2000 under the advisorship of Russell M. Pitzer . [ 3 ] Following the completion of her PhD, she was a postdoctoral researcher at Johns Hopkins University under David Yarkony for three years.
In 2003 she was hired at Temple University as an assistant professor in its College of Science and Technology . [ 2 ] She was promoted to associate professor in 2009 and full professor in 2014. [ 3 ]
In 2005 she was awarded the National Science Foundation CAREER Award . [ 2 ] She was awarded a Alexander von Humboldt Foundation fellowship in 2013. [ 3 ] In 2014 she was elected as a fellow of the American Physical Society "for her contributions to understanding the dynamics of excited molecules around conical intersections and method development to calculate such at the highest levels of theory". [ 4 ]
This article about a Greek scientist is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spiridoula_Matsika |
A spirit safe or intermediate spirit receiver is an enclosed device used in the distillation of Scotch whisky . The distillate from the still passes into it, and can be seen through the glass sides or windows, but cannot be directly accessed. The distiller can analyse the spirit inside the device, and decide where it should be sent. [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ]
This whisky -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spirit_safe |
In organic chemistry , spiro compounds are compounds that have at least two molecular rings sharing one common atom. Simple spiro compounds are bicyclic (having just two rings). [ 2 ] : SP-0 [ 3 ] : 653, 839 The presence of only one common atom connecting the two rings distinguishes spiro compounds from other bicyclics. [ 4 ] [ 3 ] : 653ff : 839ff Spiro compounds may be fully carbocyclic (all carbon) or heterocyclic (having one or more non-carbon atom). One common type of spiro compound encountered in educational settings is a heterocyclic one— the acetal formed by reaction of a diol with a cyclic ketone .
The common atom that connects the two (or sometimes three) rings is called the spiro atom . [ 2 ] : SP-0 In carbocyclic spiro compounds like spiro[5.5]undecane, the spiro-atom is a quaternary carbon , and as the -ane ending implies, these are the types of molecules to which the name spirane was first applied (though it is now used general of all spiro compounds). [ 5 ] : 1138ff The two rings sharing the spiro atom are most often different, although they can be identical [e.g., spiro[5.5]undecane and spiropentadiene , at right]. [ 3 ] : 319f.846f
Bicyclic ring structures in organic chemistry that have two fully carbocyclic (all carbon) rings connected through a carbon atom are the usual focus of the topic of spirocycles. Simple parent spirocycles include spiropentane, spirohexane, etc. up to spiroundecane. Several exist as isomers. Lower members of the class are strained. The symmetric isomer of spiroundecane is not.
Some spirocyclic compounds occur as natural products . [ 6 ]
The spirocyclic core is usually prepared by dialkylation of an activated carbon center. The dialkylating group is often a 1,3-, 1,4-, etc. dihalide. [ 9 ] In some cases the dialkylating group is a dilithio reagent, such as 1,5-dilithiopentane. [ 10 ] For generating spirocycles containing a cyclopropane ring, cyclopropanation with cyclic carbenoids has been demonstrated. [ 11 ]
Spiro compounds are often prepared by diverse rearrangement reactions. For example, the pinacol-pinacolone rearrangement is illustrated below. [ 3 ] : 985 is employed in the preparation of aspiro[4.5]decane. [ 12 ] ].
Spiro compounds are considered heterocyclic if the spiro atom or any atom in either ring are not carbon atoms. Cases with a spiro heteroatom such as boron, silicon, and nitrogen (but also other Group IVA [14] are often trivial to prepare. Many borate esters derived from glycols illustrate this case. [ 14 ] Likewise, a tetravalent neutral silicon and quaternary nitrogen atom ( ammonium cation) can be the spiro center. Many such compounds have been described. [ 5 ] : 1139f
Particularly common spiro compounds are ketal (acetal) formed by condensation of cyclic ketones and diols and dithiols . [ 15 ] [ 16 ] [ 17 ] A simple case is the acetal 1,4-dioxaspiro[4.5]decane from cyclohexanone and glycol . Cases of such ketals and dithioketals are common.
Spiranes can be chiral , [ 18 ] in various ways. [ 5 ] : 1138ff First, while nevertheless appearing to be twisted, they yet may have a chiral center making them analogous to any simple chiral compound , and second, while again appearing twisted, the specific location of substituents, as with alkylidenecycloalkanes, may make a spiro compound display central chirality (rather than axial chirality resulting from the twist); third, the substituents of the rings of the spiro compound may be such that the only reason they are chiral arises solely from the twist of their rings, e.g., in the simplest bicyclic case, where two structurally identical rings are attached via their spiro atom, resulting in a twisted presentation of the two rings. [ 5 ] : 1138ff, 1119ff [ 3 ] : 319f.846f Hence, in the third case, the lack of planarity described above gives rise to what is termed axial chirality in otherwise identical isomeric pair of spiro compounds, because they differ only in the right- versus left-handed "twist" of structurally identical rings (as seen in allenes , sterically hindered biaryls , and alkylidenecycloalkanes as well). [ 5 ] : 1119f Assignment of absolute configuration of spiro compounds has been challenging, but a number of each type have been unequivocally assigned. [ 5 ] : 1139ff
Some spiro compounds exhibit axial chirality . Spiroatoms can be the origin of chirality even when they lack the required four different substituents normally observed in chirality. When two rings are identical the priority is determined by a slight modification of the CIP system assigning a higher priority to one ring extension and a lower priority to an extension in the other ring. When rings are dissimilar the regular rules apply. [ clarification needed ]
Nomenclature for spiro compounds was first discussed by Adolf von Baeyer in 1900. [ 19 ] IUPAC provides advice on naming of spiro compounds. [ 20 ]
The prefix spiro denotes two rings with a spiro junction. The main method of systematic nomenclature is to follow with square brackets containing the number of atoms in the smaller ring then the number of atoms in the larger ring, separated by a period, in each case excluding the spiroatom (the atom by which the two rings are bonded) itself. Position-numbering starts with an atom of the smaller ring adjacent to the spiroatom around the atoms of that ring, then the spiroatom itself, then around the atoms of the larger ring. [ 21 ] For example, compound A in Image #4 above ( Selected Spiro Compounds ) is called 1-bromo-3-chlorospiro[4.5]decan-7-ol , and compound B is called 1-bromo-3-chlorospiro[3.6]decan-7-ol .
A spiro compound , or spirane , from the Latin spīra , meaning a twist or coil, [ 22 ] [ 5 ] : 1138 [ 23 ] is a chemical compound , typically an organic compound , that presents a twisted structure of two or more rings (a ring system), in which 2 or 3 rings are linked together by one common atom, [ 2 ] : SP-0 | https://en.wikipedia.org/wiki/Spiroaromaticity |
Spiroligomer molecules (also known as bis-peptides ) are synthetic oligomers made by coupling pairs of bis-amino acids into a fused ring system. [ 1 ] Spiroligomer molecules are rich in stereochemistry and functionality because of the variety of bis-amino acids that are capable of being incorporated during synthesis. [ 2 ] Due to the rigidity of the fused ring system, [ 3 ] the three-dimensional shape of a Spiroligomer molecule – as well as the display of any functional groups – can be predicted, allowing for molecular modeling and dynamics.
Spiroligomer molecules are synthesized in a step-wise approach by adding a single bis-amino acid at each stage of the synthesis. This stepwise elongation allows for complete control of the stereochemistry, as any bis-amino acid can be incorporated to allow for elongation; or any mono-amino acid can be added to terminate a chain. This can be accomplished using either solution-phase or solid-phase reactions. [ 4 ] The original synthesis of Spiroligomer molecule allowed for functionalization on the ends of the oligomers, but it did not allow for the incorporation of functionality on the interior diketopiperazine (DKP) nitrogens. [ 5 ] Much work has been done to allow for the functionalization of the entire Spiroligomer molecule, as opposed to just the ends. [ 2 ] By exploiting a neighboring group effect, [ 6 ] Spiroligomer molecule can be synthesized with a variety of functional groups along the length of the molecule.
Spiroligomer molecules can be synthesized in any direction, and between any pair of bis-amino acids.
Spiroligomer diketopiperazines can be created between either end of a bis-amino acid.
Spiroligomer molecules are known to be conformationally rigid, due to the fused-ring backbone. [ 3 ]
Spiroligomer molecules are peptidomimetics , completely resistant to proteases , and not likely to raise an immune response.
Spiroligomer molecules have been utilized for a variety of applications which include catalysis, protein binding, metal-binding, molecular scaffolds, and charge-transfer studies, et al.
Two unique types of Spiroligomer catalysts (spiroligozymes) have been developed, an esterase mimic and a Claisen catalyst .
The first Spiroligomer catalyst was an esterase -mimic, which catalyzed the transfer of a trifluoroacetate group. [ 7 ]
The second Spiroligomer catalyst accelerated an aromatic Claisen rearrangement with a catalytic dyad similar to that found in ketosteroid isomerase . [ 8 ]
A Spiroligomer peptidomimetic was designed to mimic P53 and bind HDM2 . The molecule enters cells through passive diffusion, and this mimic was shown to stabilize HDM2 in cell culture. [ 9 ]
Binuclear metal binding [ 10 ]
Rods used for distance measuring with spin probes . [ 3 ]
Donor-Bridge-Acceptor [ 11 ]
Possible applications that are currently investigated include the binding and inactivation of cholera toxin and the cross linkage of surface proteins of various viruses ( HIV , Ebola virus ). Further the group of Christian Schafmeister developed molecular hinges, which can be used for the construction of molecular machines , such as nano- valves or data storage systems. [ 12 ] | https://en.wikipedia.org/wiki/Spiroligomer |
A spiropyran is a type of photochromic organic chemical compound , characterized by their ability to reversibly switch between two structural forms—spiropyran and merocyanine—upon exposure to light or other external stimuli. This reversible transformation alters their optical and electronic properties, making them valuable in various applications, including molecular switches, optical data storage, sensors, and smart materials. [ 1 ]
Spiropyrans were discovered in the early twentieth century, but it was not until 1952 that their photochromic properties were formally documented by chemists Fischer and Gerhard Hirshberg. [ 1 ] Their pioneering work demonstrated that spiropyrans undergo reversible structural and color changes when exposed to ultraviolet light, a phenomenon that sparked widespread interest in photoresponsive organic compounds. Throughout the latter half of the twentieth century, advancements in synthetic methods enabled the development of a wide range of spiropyran derivatives with enhanced stability and responsiveness. By the 1990s and 2000s, the integration of spiropyrans into polymers, nanomaterials, and biological systems had established them as key components in emerging technologies such as molecular electronics, smart coatings, and environmental sensors. Today, spiropyrans continue to be actively investigated for their potential in dynamic and multifunctional materials. [ 2 ] [ 3 ] [ 4 ] [ 5 ]
There are two methods for the production of spiropyrans. The first one can be by condensation of methylene bases with o-hydroxy aromatic aldehydes (or the condensation of the precursor of methylene bases). Spiropyrans generally could be obtained by boiling the aldehyde and the respective benzazolium salts in presence of pyridine or piperidine :
A second route involves condensation of o-hydroxy aromatic aldehydes with the salts of heterocyclic cations which contains active methylene groups and isolation of the intermediate styryl salts. This second procedure is followed by the removal of the elements of the acid from the obtained styryl salt, such as perchloric acid, with organic bases (gaseous ammonia or amines).
A spiropyran is a 2H- pyran isomer that has the hydrogen atom at position two replaced by a second ring system linked to the carbon atom at position two of the pyran molecule in a spiro way . So there is a carbon atom which is common on both rings, the pyran ring and the replaced ring. The second ring, the replaced one, is usually heterocyclic but there are exceptions.
A solution of the spiropyran in polar solvents upon heating ( thermochromism ) or radiation ( photochromism ) becomes coloured owing to formation of the merocyanine isomer. The structural differences between spiropyran and merocyanine form is that, while in the first one the ring is in the closed form, in the other one the ring is opened. The photochromism is arises from electrocyclic cleavage of the C-spiro-O bond.
Photochromism is the phenomenon that produces a change of colour in a substance by incident radiation. In other words, Photochromism is a light-induced change of colour of a chemical substance. The spiropyrans are one of the photochromatic molecules that have raised more interest lately. These molecules consist of two heterocyclic functional groups in orthogonal planes bound by a carbon atom. Spiropyrans are one of the oldest families of photochromism. As solids, the spiropyrans do not present photochromism. It is possible in solution and in the dry state that radiation between 250 nm and 380 nm (approximately) is able to, by breaking the C-O binding, transform the spiropyrans into its colour emitting merocyanin-form . The structure of the colourless molecules, the substrate of the reaction (N), is more thermodynamically stable than the product – depending on the solvent in which it is stored. For example in NMP the equilibrium could be switched more toward the merocyanin form (solvatochromic effects). The photoisomers of the spiropyrans have a structure similar to cyanines , even though it is not symmetric about the center of the polymethine chain, and it is classified as a merocyanine (Figure 2).
Once the irradiation has stopped, the merocyanine in solution starts to discolour and to revert to its original form, the spiropyran (SP).
Procedure:
Spiropyrans are widely studied for their photochromic properties, which enable reversible transformations between structurally distinct forms in response to external stimuli such as light, heat, pH, or metal ions. This unique behavior has led to their application in a diverse range of fields.
In materials science, spiropyrans are incorporated into smart materials and coatings that respond dynamically to environmental changes, offering potential for use in sensors, actuators, and light-responsive surfaces.
In electronics, they serve as molecular switches and components in optical data storage systems due to their reversible and controllable optical properties.
Spiropyrans also play a significant role in biomedical research, particularly in the development of light-activated drug delivery systems and biosensors.
The versatility and tunability of spiropyran derivatives continue to drive research into their integration in emerging technologies across chemistry, physics, and engineering disciplines.
Some more detailed examples of the applications of spiropyrans are listed below: | https://en.wikipedia.org/wiki/Spiropyran |
Spirotryprostatin B is an indolic alkaloid found in the Aspergillus fumigatus fungus that belongs to a class of naturally occurring 2,5-diketopiperazines . [1] Spirotryprostatin B and several other indolic alkaloids (including Spirotryprostatin A , as well as other tryprostatins and cyclotryprostatins) have been found to have anti- mitotic properties, and as such they have become of great interest as anti- cancer drugs. [2] Because of this, the total syntheses of these compounds is a major pursuit of organic chemists, and a number of different syntheses have been published in the chemical literature.
The first total synthesis was accomplished in 2000 by the Danishefsky group at Columbia University , [3] with a number of other syntheses following shortly thereafter by Williams, [4] Ganesan, [5] Fuji, [6] Carreira, [7] Horne, [8] Overman, [9] and most recently Trost. [10]
From a synthetic point of view, the most challenging structural features of the molecule are the C3 spirocyclic ring juncture and the adjacent prenyl-substituted carbon. Approaches toward preparing the skeleton of spirotryprostatin B have varied considerably.
Danishefsky spirotryprostatin B synthesis
In the Danishefsky synthesis, an amine derived from tryptophan was condensed with an aldehyde , triggering a Mannich-type reaction wherein the pendant oxindole acted as a nucleophile toward the intermediate iminium species.
Williams spirotryprostatin B synthesis
The synthesis by the Williams group utilized a 3-component coupling reaction. A secondary amine was combined with an aldehyde to form an intermediate azomethine ylide , which underwent a 1,3-dipolar cycloaddition with an unsaturated oxindole also present in the reaction mixture.
Ganesan spirotryprostatin B synthesis
Ganesan made use of a biomimetic strategy in his synthesis of spirotryprostatin B. An indole was treated with N -bromosuccinimide to trigger an oxidative rearrangement, forming the quaternary stereocenter in a diastereoselective manner.
Fuji spirotryprostatin B synthesis
In the synthesis developed by the Fuji group, the stereochemistry at the spirocyclic carbon was established by a nitroolefination reaction. An oxindole with pendant prenyl group was reacted with a nitroolefin bearing a chiral leaving group .
Carreira spirotryprostatin B synthesis
The Carreira group made use of a magnesium iodide promoted annulation reaction in their approach toward spirotryprostatin B. An oxindole bearing a cyclopropane was reacted with an imine in the presence of the magnesium iodide, triggering the ring-expansion reaction.
Horne spirotryprostatin B synthesis
Horne's synthesis of spirotryprostatin B also made use of a Mannich-type process, wherein a chloro- indole served as the pro- nucleophile . The cyclization was triggered by treating the pendant imine with the acyl chloride derived from proline . The resulting iminium species was attacked by the chloro- indole , forming the spirocyclic bond.
Overman spirotryprostatin B synthesis
The Overman group utilized a Heck reaction to prepare the molecule. An iodo- aniline with a tethered alkene was subjected to palladium catalysis . The intermediate palladium-allyl species was intercepted by the pendant amide nitrogen to generate the prenyl stereocenter in the same reaction.
Trost spirotryprostatin B synthesis
In the synthesis developed by the Trost group, the stereochemistry at the spirocyclic ring juncture is established by a decarboxylation-prenylation sequence, reminiscent of the Carroll reaction . Here, a prenyl ester serves as both the nucleophile and electrophile precursor. Upon treatment with a chiral palladium catalyst the prenyl group ionizes and decarboxylates. The resulting ion pair subsequently recombines to generate the prenylated product. Notably, double bond migration occurs and the prenyl group is attacked at the oxindole carbon. | https://en.wikipedia.org/wiki/Spirotryprostatin_B |
In fair division problems, spite is a phenomenon that occurs when a player's value of an allocation decreases when one or more other players' valuation increases. Thus, other things being equal, a player exhibiting spite will prefer an allocation in which other players receive less than more (if more of the good is desirable).
In this language, spite is difficult to analyze because one has to assess two sets of preferences. For example, in the divide and choose method, a spiteful player would have to make a trade-off between depriving his opponent of cake, and getting more himself.
Within the field of sociobiology , spite is used to describe those social behaviors that have a negative impact on both the actor and recipient(s). Spite can be favored by kin selection when: (a) it leads to an indirect benefit to some third party that is sufficiently related to the actor (Wilsonian spite); or (b) when it is directed primarily at negatively related individuals ( Hamiltonian spite ). Negative relatedness occurs when two individuals are less related than average.
The iterated prisoner's dilemma provides an example where players may "punish" each other for failing to cooperate in previous rounds, even if doing so would cause negative consequences for both players. For example, the simple " tit for tat " strategy has been shown to be effective in round-robin tournaments of iterated prisoner's dilemma.
There is always difficulty in fairly dividing the proceeds of a business between the business owners and the employees.
When a trade union decides to call a strike , both employer and the union members lose money (and may damage the national economy ). The unionists hope that the employer will give in to their demands before such losses have destroyed the business.
In the reverse direction, an employer may terminate the employment of certain productive workers who are agitating for higher wages or organising a trade union. Losing productive workers is a setback to both the business and the employees but this can serve as an example to others and thus maximise employer power.
Polyembryonic wasps, including C. floridanum , exhibit spite through instances of precocious larval development. [ 1 ] Spite provides an explanation for how natural selection can favor harmful behaviors that are costly to both the actor and the recipient; [ 1 ] spite is typically considered a form of altruism that benefits a secondary recipient. Two criteria demonstrate that spite is truly occurring: (i) the behavior is truly costly to the actor and does not provide a long-term direct benefit; and (ii) harming behaviors are directed toward relatively unrelated individuals. [ 2 ] | https://en.wikipedia.org/wiki/Spite_(game_theory) |
Spitting is the act of forcibly ejecting saliva , sputum , nasal mucus and/or other substances from the mouth . The act is often done to get rid of unwanted or foul-tasting substances in the mouth, or to get rid of a large buildup of mucus . Spitting of small saliva droplets can also happen unintentionally during talking , especially when articulating ejective and implosive consonants .
Spitting in public is considered rude and a social taboo in many parts of the world including the West , while in some other parts of the world it is considered more socially acceptable.
Spitting upon another person, especially onto the face , is a global sign of anger , hatred , disrespect or contempt . It can represent a "symbolical regurgitation" or an act of intentional contamination. [ 1 ]
Social attitudes towards spitting have changed greatly in Western Europe since the Middle Ages . Then, frequent spitting was part of everyday life, and at all levels of society, it was thought ill-mannered to suck back saliva to avoid spitting. [ citation needed ] By the early 18th century, spitting had become seen as something which should be concealed, and by 1859 it had progressed to being described by at least one etiquette guide as "at all times a disgusting habit." Sentiments against spitting gradually transitioned from being included in adult conduct books to so obvious as to only appear in guides for children to not be included in conduct literature even for children "because most [Western] children have the spitting ban internalized well before learning how to read." [ 2 ]
Spittoons (also known as cuspidors ) were used openly during the 19th century to provide an acceptable outlet for spitters. Spittoons became far less common after the influenza epidemic of 1918 , and their use has since virtually disappeared, though each justice of the Supreme Court of the United States continues to be provided with a personal one. [ 3 ]
In the first half of the 20th century the National Association for the Study and Prevention of Tuberculosis, the precursor to the American Lung Association , and state affiliates had educational campaigns against spitting to reduce the chance of spreading tuberculosis . [ 4 ] According to the World Health Organization coughing, sneezing, or spitting, can spread tuberculosis. [ 5 ] The chance of catching a contagious disease by being spit on is low. [ 6 ]
After coffee cupping , tea tasting , and wine tasting , the sample is spit into a 'spit bucket' or spittoon . [ citation needed ] Spitting is commonplace among athletes. [ 7 ] There are multiple explanations for this behavior, including getting rid of the MUC5B secreted during intense exercise, as well as carb-rinsing to provide a performance boost. [ 8 ]
There have been instances of spitting reported in the US, particularly from American men. [ 9 ] In Minnesota , instances have been reported from some young people. [ 10 ] [ 11 ] In Canada, spitting has been reported for cities such as Ottawa and Winnipeg . [ 12 ] [ 13 ]
In certain nations, spitting is an accepted part of one's lifestyle.
Spitting has been attributed to some people [ further explanation needed ] from Asia-Pacific countries such as Bangladesh, [ 14 ] China, [ 15 ] [ 16 ] [ 17 ] India, [ 18 ] [ 19 ] Indonesia, [ 20 ] Myanmar, [ 21 ] [ 22 ] [ 23 ] Papua New Guinea, [ 24 ] Philippines, [ 25 ] [ 26 ] [ 27 ] South Korea, [ 28 ] [ 29 ] United Arab Emirates, [ 30 ] [ 31 ] and Vietnam. [ 32 ] [ 33 ] [ 34 ] The practice is often linked to betel chewing in many of those regions. [ 35 ] Spitting has also been reported in some parts of Africa, such as Ghana. [ 36 ]
In India and Indonesia, spitting is often associated with various forms of chewing juices. [ 37 ]
According to Ross Coomber, a professor of sociology at Plymouth University , spitting is perceived as a cleansing practice for the body by many individuals in China. [ 37 ]
There are some places where spitting is a competitive sport, with or without a projectile in the mouth. For example, there is a Guinness World Record for cherry pit spitting and cricket spitting , and there are world championships in Kudu dung spitting .
In rural parts of North India , it was customary in olden days for mothers to lightly spit at their children (usually to the side of the children rather than directly at them) to imply a sense of disparagement and imperfection that protects them from evil eye (or nazar ). [ 38 ] Excessive admiration, even from well-meaning people, is believed to attract the evil eye, so this is believed to protect children from nazar that could be caused by their own mothers' "excessive" love of them. [ 38 ] However, because of hygiene, transmission of disease and social taboos, this practice has waned and instead a black mark of kohl or kajal is put on the forehead or cheek of the child to ward off the evil eye. Adults use an amulet containing alum or chillies and worn on the body for this purpose. Sometimes, this is also done with brides and others by their loved ones to protect them from nazar .
Shopkeepers in the region used to sometimes make a spitting gesture on the cash proceeds from the first sale of the day (called bohni ), which is a custom believed to ward-off nazar from the business. [ 39 ]
Such a habit also existed in some Eastern European countries like Romania , and Moldova , although it is no longer widely practiced. People would gently spit in the face of younger people (often younger relatives such as grandchildren or nephews) they admire in order to avoid deochi , [ 40 ] an involuntary curse on the individual being admired or "strangely looked upon", [ 40 ] which is claimed to be the cause of bad fortune and sometimes malaise or various illnesses. [ 41 ] In Greece , it is customary to "spit" three times after making a compliment to someone, the spitting is done to protect from the evil eye. [ 42 ] This applies to all people, not just between mothers and children.
A similar-sounding expression for verbal spitting occurs in modern Hebrew as "Tfu, tfu" (here, only twice), which some say that Hebrew-speakers borrowed from Russian. [ 43 ]
When a suspect in a criminal case is arrested, they will sometimes try to spit at their captors, which often causes a fear of infection by Hepatitis C and other diseases. Spit hoods are meant to prevent this.
Gleeking is the projection of saliva from the submandibular gland under the tongue. It can happen deliberately or accidentally, particularly when yawning . [ 44 ] [ 45 ] | https://en.wikipedia.org/wiki/Spitting |
The Spitzenkörper ( German for 'pointed body', SPK ) is a structure found in fungal hyphae that is the organizing center for hyphal growth and morphogenesis . It consists of many small vesicles and is present in growing hyphal tips, during spore germination , and where branch formation occurs. Its position in the hyphal tip correlates with the direction of hyphal growth. The Spitzenkörper is a part of the endomembrane system in fungi. [ 1 ]
The vesicles are organized around a central area that contains a dense meshwork of microfilaments . Polysomes are often found closely to the posterior boundary of the Spitzenkörper core within the Ascomycota , microtubules extend into and often through the Spitzenkörper and within the Ascomycota Woronin bodies are found in the apical region near the Spitzenkörper. [ 2 ]
The cytoplasm of the extreme apex is occupied almost exclusively by secretory vesicles. In the higher fungi ( Ascomycota and Basidiomycota ), secretory and endocytic vesicles are arranged into a dense, spherical aggregation called the Spitzenkörper or ‘apical body’. [ 1 ] The Spitzenkörper may be seen in growing hyphae even with a light microscope. Hyphae of the Oomycota and some lower Eumycota (notably the Zygomycota ) do not contain a recognizable Spitzenkörper, and the vesicles are instead distributed more loosely often in a crescent-shaped arrangement beneath the apical plasma membrane . [ 3 ]
This structure is most commonly found in Dikarya and was at first thought to only occur among them. [ 4 ] [ 5 ] [ 6 ] [ 7 ] [ 8 ] Vargas et al 1993 however were the first to find a Spitzenkörper in another clade, specifically the Allomyces ( Blastocladiomycota ), [ 5 ] [ 9 ] [ 6 ] [ 4 ] [ 7 ] [ 8 ] then subsequently Basidiobolus ranarum – which has been placed in several different phyla – was also found to have an SPK. [ 4 ] As of 2020 [update] these and the Blastocladiella (also in Blastocladiomycota) are the only known taxa to bear this structure. [ 8 ]
This cell biology article is a stub . You can help Wikipedia by expanding it .
This mycology -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spitzenkörper |
In probability theory , Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published by Frank Spitzer in 1956. [ 1 ] The formula is regarded as "a stepping stone in the theory of sums of independent random variables". [ 2 ]
Let X 1 , X 2 , . . . {\displaystyle X_{1},X_{2},...} be independent and identically distributed random variables and define the partial sums S n = X 1 + X 2 + . . . + X n {\displaystyle S_{n}=X_{1}+X_{2}+...+X_{n}} . Define R n = max ( 0 , S 1 , S 2 , . . . S n ) {\displaystyle R_{n}={\text{max}}(0,S_{1},S_{2},...S_{n})} . Then [ 3 ]
where
and S ± denotes (| S | ± S )/2.
Two proofs are known, due to Spitzer [ 1 ] and Wendel. [ 3 ]
This probability -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spitzer's_formula |
The Spitzer Space Telescope , formerly the Space Infrared Telescope Facility ( SIRTF ), was an infrared space telescope launched in 2003, that was deactivated when operations ended on 30 January 2020. [ 5 ] [ 9 ] Spitzer was the third space telescope dedicated to infrared astronomy, following IRAS (1983) and ISO (1995–1998). It was the first spacecraft to use an Earth-trailing orbit , later used by the Kepler planet-finder.
The planned mission period was to be 2.5 years with a pre-launch expectation that the mission could extend to five or slightly more years until the onboard liquid helium supply was exhausted. This occurred on 15 May 2009. [ 10 ] Without liquid helium to cool the telescope to the very low temperatures needed to operate, most of the instruments were no longer usable. However, the two shortest-wavelength modules of the IRAC camera continued to operate with the same sensitivity as before the helium was exhausted, and continued to be used into early 2020 in the Spitzer Warm Mission . [ 11 ] [ 12 ]
During the warm mission, the two short wavelength channels of IRAC operated at 28.7 K and were predicted to experience little to no degradation at this temperature compared to the nominal mission. The Spitzer data, from both the primary and warm phases, are archived at the Infrared Science Archive (IRSA).
In keeping with NASA tradition, the telescope was renamed after its successful demonstration of operation, on 18 December 2003. Unlike most telescopes that are named by a board of scientists, typically after famous deceased astronomers, the new name for SIRTF was obtained from a contest open to the general public.
The contest led to the telescope being named in honor of astronomer Lyman Spitzer , who had promoted the concept of space telescopes in the 1940s. [ 13 ] Spitzer wrote a 1946 report for RAND Corporation describing the advantages of an extraterrestrial observatory and how it could be realized with available or upcoming technology. [ 14 ] [ 15 ] He has been cited for his pioneering contributions to rocketry and astronomy , as well as "his vision and leadership in articulating the advantages and benefits to be realized from the Space Telescope Program." [ 13 ]
The US$776 million [ 16 ] Spitzer was launched on 25 August 2003 at 05:35:39 UTC from Cape Canaveral SLC-17B aboard a Delta II 7920H rocket. [ 3 ] It was placed into a heliocentric (as opposed to a geocentric ) orbit trailing and drifting away from Earth's orbit at approximately 0.1 astronomical units per year (an "Earth-trailing" orbit [ 1 ] ).
The primary mirror is 85 centimeters (33 in) in diameter, f /12 , made of beryllium and was cooled to 5.5 K (−268 °C; −450 °F). The satellite contains three instruments that allowed it to perform astronomical imaging and photometry from 3.6 to 160 micrometers, spectroscopy from 5.2 to 38 micrometers, and spectrophotometry from 55 to 95 micrometers. [ 8 ]
By the early 1970s, astronomers began to consider the possibility of placing an infrared telescope above the obscuring effects of Earth's atmosphere.
In 1979, a report from the National Research Council of the National Academy of Sciences , A Strategy for Space Astronomy and Astrophysics for the 1980s , identified a Shuttle Infrared Telescope Facility (SIRTF) [ 17 ] as "one of two major astrophysics facilities [to be developed] for Spacelab ", a shuttle-borne platform. Anticipating the major results from an upcoming Explorer satellite and from the Shuttle mission, the report also favored the "study and development of ... long-duration spaceflights of infrared telescopes cooled to cryogenic temperatures [ 18 ] ."
The launch in January 1983 of the Infrared Astronomical Satellite , jointly developed by the United States, the Netherlands, and the United Kingdom, to conduct the first infrared survey of the sky, whetted the appetites of scientists worldwide for follow-up space missions capitalizing on the rapid improvements in infrared detector technology.
Earlier infrared observations had been made by both space-based and ground-based observatories . Ground-based observatories have the drawback that at infrared wavelengths or frequencies , both the Earth's atmosphere and the telescope itself will radiate (glow) brightly. Additionally, the atmosphere is opaque at most infrared wavelengths. This necessitates lengthy exposure times and greatly decreases the ability to detect faint objects. It could be compared to trying to observe the stars in the optical at noon from a telescope built out of light bulbs. Previous space observatories (such as IRAS , the Infrared Astronomical Satellite, and ISO , the Infrared Space Observatory) were launched during the 1980s and 1990s and great advances in astronomical technology have been made since then.
Most of the early concepts envisioned repeated flights aboard the NASA Space Shuttle. This approach was developed in an era when the Shuttle program was expected to support weekly flights of up to 30 days duration. A May 1983 NASA proposal described SIRTF as a Shuttle-attached mission, with an evolving scientific instrument payload. Several flights were anticipated with a probable transition into a more extended mode of operation, possibly in association with a future space platform or space station. SIRTF would be a 1-meter class, cryogenically cooled, multi-user facility consisting of a telescope and associated focal plane instruments. It would be launched on the Space Shuttle and remain attached to the Shuttle as a Spacelab payload during astronomical observations, after which it would be returned to Earth for refurbishment prior to re-flight. The first flight was expected to occur about 1990, with the succeeding flights anticipated beginning approximately one year later. However, the Spacelab-2 flight aboard STS-51-F showed that the Shuttle environment was poorly suited to an onboard infrared telescope due to contamination from the relatively "dirty" vacuum associated with the orbiters. By September 1983, NASA was considering the "possibility of a long duration [free-flyer] SIRTF mission". [ 19 ] [ 20 ]
Spitzer is the only one of the Great Observatories not launched by the Space Shuttle , as was originally intended [ citation needed ] . However, after the 1986 Challenger disaster , the Shuttle-Centaur upper stage, which would have been required to place it into its final orbit, was abandoned. The mission underwent a series of redesigns during the 1990s, primarily due to budget considerations. This resulted in a much smaller but still fully capable mission that could use the smaller Delta II expendable launch vehicle. [ 21 ]
One of the most important advances of this redesign was an Earth-trailing orbit . [ 1 ] Cryogenic satellites that require liquid helium (LHe, T ≈ 4 K) temperatures in near-Earth orbit are typically exposed to a large heat load from Earth, and consequently require large amounts of LHe coolant, which then tends to dominate the total payload mass and limits mission life. Placing the satellite in solar orbit far from Earth allowed innovative passive cooling. The sun shield protected the rest of the spacecraft from the Sun's heat, the far side of the spacecraft was painted black to enhance passive radiation of heat, and the spacecraft bus was thermally isolated from the telescope. All of these design choices combined to drastically reduce the total mass of helium needed, resulting in an overall smaller and lighter payload, resulting in major cost savings, but with a mirror the same diameter as originally designed. This orbit also simplified telescope pointing, but did require the NASA Deep Space Network for communications. [ citation needed ]
The primary instrument package (telescope and cryogenic chamber) was developed by Ball Aerospace & Technologies , in Boulder, Colorado . The individual instruments were developed jointly by industrial, academic, and government institutions. The principals were Cornell University , the University of Arizona , the Smithsonian Astrophysical Observatory , Ball Aerospace , and Goddard Spaceflight Center . The shorter-wavelength infrared detectors were developed by Raytheon in Goleta, California . Raytheon used indium antimonide and a doped silicon detector in the creation of the infrared detectors. These detectors are 100 times more sensitive than what was available at the beginning of the project during the 1980s. [ 22 ] The far-infrared detectors (70–160 micrometers) were developed jointly by the University of Arizona and Lawrence Berkeley National Laboratory using gallium -doped germanium . The spacecraft was built by Lockheed Martin . The mission was operated and managed by the Jet Propulsion Laboratory and the Spitzer Science Center , [ 23 ] located at IPAC on the Caltech campus in Pasadena, California. [ citation needed ]
Spitzer ran out of liquid helium coolant on 15 May 2009, which stopped far-IR observations. Only the IRAC instrument remained in use, and only at the two shorter wavelength bands (3.6 μm and 4.5 μm). The telescope equilibrium temperature was then around 30 K (−243 °C; −406 °F), and IRAC continued to produce valuable images at those wavelengths as the "Spitzer Warm Mission". [ 24 ]
Late in the mission, ~2016, Spitzer's distance to Earth and the shape of its orbit meant the spacecraft had to pitch over at an extreme angle to aim its antenna at Earth. [ 25 ] The solar panels were not fully illuminated at this angle, and this limited those communications to 2.5 hours due to the battery drain. [ 26 ] The telescope was retired on 30 January 2020 [ 5 ] when NASA sent a shutdown signal to the telescope from the Goldstone Deep Space Communications Complex (GDSCC) instructing the telescope to go into safe mode. [ 27 ] After receiving confirmation that the command was successful, Spitzer Project Manager Joseph Hunt officially declared that the mission had ended. [ 28 ]
Spitzer carries three instruments on board: [ 29 ] [ 30 ] [ 31 ] [ 32 ]
All three instruments used liquid helium for cooling the sensors. Once the helium was exhausted, only the two shorter wavelengths in IRAC were used in the "warm mission".
While some time on the telescope was reserved for participating institutions and crucial projects, astronomers around the world also had the opportunity to submit proposals for observing time. Prior to launch, there was a proposal call for large, coherent investigations using Spitzer. If the telescope failed early and/or ran out of cryogen very quickly, these so-called Legacy Projects would ensure the best possible science could be obtained quickly in the early months of the mission. As a requirement tied to the funding these Legacy teams received, the teams had to deliver high-level data products back to the Spitzer Science Center (and the NASA/IPAC Infrared Science Archive ) for use by the community, again ensuring the rapid scientific return of the mission. The international scientific community quickly realized the value of delivering products for others to use, and even though Legacy projects were no longer explicitly solicited in subsequent proposal calls, teams continued to deliver products to the community. The Spitzer Science Center later reinstated named "Legacy" projects (and later still "Exploration Science" projects) in response to this community-driven effort. [ 36 ]
Important targets included forming stars ( young stellar objects , or YSOs), planets, and other galaxies. Images are freely available for educational and journalistic purposes. [ 37 ] [ 38 ]
The first released images from Spitzer were designed to show off the abilities of the telescope and showed a glowing stellar nursery, a big swirling, dusty galaxy , a disc of planet-forming debris, and organic material in the distant universe. Since then, many monthly press releases have highlighted Spitzer 's capabilities, as the NASA and ESA images do for the Hubble Space Telescope .
As one of its most noteworthy observations, in 2005, Spitzer became one of the first telescopes to directly capture light from exoplanets , namely the "hot Jupiters" HD 209458 b and TrES-1b , although it did not resolve that light into actual images. [ 39 ] This was one of the first times the light from extrasolar planets had been directly detected; earlier observations had been indirectly made by drawing conclusions from behaviors of the stars the planets were orbiting. The telescope also discovered in April 2005 that Cohen-kuhi Tau/4 had a planetary disk that was vastly younger and contained less mass than previously theorized, leading to new understandings of how planets are formed.
In 2004, it was reported that Spitzer had spotted a faintly glowing body that may be the youngest star ever seen. The telescope was trained on a core of gas and dust known as L1014 which had previously appeared completely dark to ground-based observatories and to ISO ( Infrared Space Observatory ), a predecessor to Spitzer. The advanced technology of Spitzer revealed a bright red hot spot in the middle of L1014.
Scientists from the University of Texas at Austin , who discovered the object, believe the hot spot to be an example of early star development, with the young star collecting gas and dust from the cloud around it. Early speculation about the hot spot was that it might have been the faint light of another core that lies 10 times further from Earth but along the same line of sight as L1014. Follow-up observation from ground-based near-infrared observatories detected a faint fan-shaped glow in the same location as the object found by Spitzer. That glow is too feeble to have come from the more distant core, leading to the conclusion that the object is located within L1014. (Young et al. , 2004)
In 2005, astronomers from the University of Wisconsin at Madison and Whitewater determined, on the basis of 400 hours of observation on the Spitzer Space Telescope, that the Milky Way galaxy has a more substantial bar structure across its core than previously recognized.
Also in 2005, astronomers Alexander Kashlinsky and John Mather of NASA's Goddard Space Flight Center reported that one of Spitzer 's earliest images may have captured the light of the first stars in the universe. An image of a quasar in the Draco constellation , intended only to help calibrate the telescope, was found to contain an infrared glow after the light of known objects was removed. Kashlinsky and Mather are convinced that the numerous blobs in this glow are the light of stars that formed as early as 100 million years after the Big Bang , redshifted by cosmic expansion . [ 40 ]
In March 2006, astronomers reported an 80- light-year long (25 pc ) nebula near the center of the Milky Way Galaxy, the Double Helix Nebula , which is, as the name implies, twisted into a double spiral shape. This is thought to be evidence of massive magnetic fields generated by the gas disc orbiting the supermassive black hole at the galaxy's center, 300 light-years (92 pc) from the nebula and 25,000 light-years (7,700 pc) from Earth. This nebula was discovered by Spitzer and published in the magazine Nature on 16 March 2006.
In May 2007, astronomers successfully mapped the atmospheric temperature of HD 189733 b , thus obtaining the first map of some kind of an extrasolar planet.
Starting in September 2006, the telescope participated in a series of surveys called the Gould Belt Survey , observing the Gould's Belt region in multiple wavelengths. The first set of observations by the Spitzer Space Telescope was completed from 21 September 2006 through 27 September. Resulting from these observations, the team of astronomers led by Dr. Robert Gutermuth, of the Center for Astrophysics | Harvard & Smithsonian reported the discovery of Serpens South , a cluster of 50 young stars in the Serpens constellation.
Scientists have long wondered how tiny silicate crystals, which need high temperatures to form, have found their way into frozen comets, born in the very cold environment of the Solar System's outer edges. The crystals would have begun as non-crystallized, amorphous silicate particles, part of the mix of gas and dust from which the Solar System developed. This mystery has deepened with the results of the Stardust sample return mission, which captured particles from Comet Wild 2 . Many of the Stardust particles were found to have formed at temperatures in excess of 1,000 K.
In May 2009, Spitzer researchers from Germany, Hungary, and the Netherlands found that amorphous silicate appears to have been transformed into crystalline form by an outburst from a star. They detected the infrared signature of forsterite silicate crystals on the disk of dust and gas surrounding the star EX Lupi during one of its frequent flare-ups, or outbursts, seen by Spitzer in April 2008. These crystals were not present in Spitzer 's previous observations of the star's disk during one of its quiet periods. These crystals appear to have formed by radiative heating of the dust within 0.5 AU of EX Lupi. [ 41 ] [ 42 ]
In August 2009, the telescope found evidence of a high-speed collision between two burgeoning planets orbiting a young star. [ 43 ]
In October 2009, astronomers Anne J. Verbiscer, Michael F. Skrutskie, and Douglas P. Hamilton published findings of the " Phoebe ring " of Saturn , which was found with the telescope; the ring is a huge, tenuous disc of material extending from 128 to 207 times the radius of Saturn. [ 44 ]
GLIMPSE, the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire , was a series of surveys that spanned 360° of the inner region of the Milky Way galaxy, which provided the first large-scale mapping of the galaxy. [ 45 ] [ 46 ] It consists of more than 2 million snapshots taken in four separate wavelengths using the Infrared Array Camera. [ 47 ] The images were taken over a 10-year period beginning in 2003 when Spitzer launched. [ 48 ]
MIPSGAL, a similar survey that complements GLIMPSE, covers 248° of the galactic disk [ 49 ] using the 24 and 70 μm channels of the MIPS instrument. [ 50 ]
On 3 June 2008, scientists unveiled the largest, most detailed infrared portrait of the Milky Way , created by stitching together more than 800,000 snapshots, at the 212th meeting of the American Astronomical Society in St. Louis , Missouri . [ 51 ] [ 52 ] This composite survey is now viewable with the GLIMPSE/MIPSGAL Viewer. [ 53 ]
Spitzer observations, announced in May 2011, indicate that tiny forsterite crystals might be falling down like rain on to the protostar HOPS-68. The discovery of the forsterite crystals in the outer collapsing cloud of the protostar is surprising because the crystals form at lava-like high temperatures, yet they are found in the molecular cloud where the temperatures are about −170 °C (103 K; −274 °F). This led the team of astronomers to speculate that the bipolar outflow from the young star may be transporting the forsterite crystals from near the star's surface to the chilly outer cloud. [ 54 ] [ 55 ]
In January 2012, it was reported that further analysis of the Spitzer observations of EX Lupi can be understood if the forsterite crystalline dust was moving away from the protostar at a remarkable average speed of 38 kilometres per second (24 mi/s). It would appear that such high speeds can arise only if the dust grains had been ejected by a bipolar outflow close to the star. [ 56 ] Such observations are consistent with an astrophysical theory, developed in the early 1990s, where it was suggested that bipolar outflows garden or transform the disks of gas and dust that surround protostars by continually ejecting reprocessed, highly heated material from the inner disk, adjacent to the protostar, to regions of the accretion disk further away from the protostar. [ 57 ]
In April 2015, Spitzer and the Optical Gravitational Lensing Experiment were reported as co-discovering one of the most distant planets ever identified: a gas giant about 13,000 light-years (4,000 pc) away from Earth. [ 58 ]
In June and July 2015, the brown dwarf OGLE-2015-BLG-1319 was discovered using the gravitational microlensing detection method in a joint effort between Swift , Spitzer, and the ground-based Optical Gravitational Lensing Experiment , the first time two space telescopes have observed the same microlensing event. This method was possible because of the large separation between the two spacecraft: Swift is in low-Earth orbit while Spitzer is more than one AU distant in an Earth-trailing heliocentric orbit. [ 1 ] This separation provided significantly different perspectives of the brown dwarf, allowing for constraints to be placed on some of the object's physical characteristics. [ 59 ]
Reported in March 2016, Spitzer and Hubble were used to discover the most distant-known galaxy, GN-z11 . This object was seen as it appeared 13.4 billion years ago. [ 60 ] [ 25 ]
On 1 October 2016, Spitzer began its Observation Cycle 13, a 2 + 1 ⁄ 2 year extended mission nicknamed Beyond . One of the goals of this extended mission was to help prepare for the James Webb Space Telescope , also an infrared telescope, by identifying candidates for more detailed observations. [ 25 ]
Another aspect of the Beyond mission was the engineering challenges of operating Spitzer in its progressing orbital phase. As the spacecraft moved farther from Earth on the same orbital path from the Sun, its antenna had to point at increasingly higher angles to communicate with ground stations; this change in angle imparted more and more solar heating on the vehicle while its solar panels received less sunlight. [ 25 ]
Spitzer was also put to work studying exoplanets thanks to creatively tweaking its hardware. This included doubling its stability by modifying its heating cycle, finding a new use for the "peak-up" camera, and analyzing the sensor at a sub-pixel level. Although in its "warm" mission, the spacecraft's passive cooling system kept the sensors at 29 K (−244 °C; −407 °F). [ 61 ] Spitzer used the transit photometry and gravitational microlensing techniques to perform these observations. [ 25 ] According to NASA's Sean Carey, "We never even considered using Spitzer for studying exoplanets when it launched. ... It would have seemed ludicrous back then, but now it's an important part of what Spitzer does." [ 25 ]
Examples of exoplanets discovered using Spitzer include HD 219134 b in 2015, which was shown to be a rocky planet about 1.5 times as large as Earth in a three-day orbit around its star; [ 62 ] and an unnamed planet found using microlensing located about 13,000 light-years (4,000 pc) from Earth. [ 63 ]
In September–October 2016, Spitzer was used to discover five of a total of seven known planets around the star TRAPPIST-1 , all of which are approximately Earth-sized and likely rocky. [ 64 ] [ 65 ] Three of the discovered planets are located in the habitable zone , which means they are capable of supporting liquid water given sufficient parameters. [ 66 ] Using the transit method , Spitzer helped measure the sizes of the seven planets and estimate the mass and density of the inner six. Further observations will help determine if there is liquid water on any of the planets. [ 64 ] | https://en.wikipedia.org/wiki/Spitzer_Space_Telescope |
In fluid mechanics , a splash is a sudden disturbance to the otherwise quiescent free surface of a liquid (usually water ). The disturbance is typically caused by a solid object suddenly hitting the surface, although splashes can occur in which moving liquid supplies the energy. This use of the word is onomatopoeic ; in the past, the term " plash " has also been used.
Splash also happens when a liquid droplet impacts on a liquid or a solid surface; [ citation needed ] in this case, a symmetric corona (resembling a coronet ) is usually formed as shown in Harold Edgerton 's famous milk splash photography, as milk is opaque. [ citation needed ] Historically, Worthington (1908) was the first one who systematically investigated the splash dynamics using photographs. [ citation needed ]
Splashes are characterized by transient ballistic flow, and are governed by the Reynolds number and the Weber number . In the image of a brick splashing into water, one can identify freely moving airborne water droplets, a phenomenon typical of high Reynolds number flows; the intricate non-spherical shapes of the droplets show that the Weber number is high. [ citation needed ] Also seen are entrained air bubbles in the body of the water, and an expanding ring of disturbance propagating away from the impact site.
Sand is said to splash if hit sufficiently hard (see dry quicksand ) and sometimes the impact of a meteorite is referred to as splashing, if small bits of ejecta are formed.
Physicist Lei Xu and coworkers at the University of Chicago discovered that the splash due to the impact of a small drop of ethanol onto a dry solid surface could be suppressed by reducing the pressure below a specific threshold. For drops of diameter 3.4 mm falling through air , this pressure was about 20 kilopascals , or 0.2 atmosphere . [ 1 ]
A plate made of a hard material on which a stream of liquid is designed to fall is called a "splash plate". It may serve to protect the ground from erosion by falling water, such as beneath an artificial waterfall or water outlet in soft ground. Splash plates are also part of spray nozzles , such as in irrigation sprinkler systems. [ citation needed ] | https://en.wikipedia.org/wiki/Splash_(fluid_mechanics) |
Splash lubrication is a rudimentary form of lubrication found in early engines and transmissions. [ 1 ] Such engines could be external combustion engines (such as stationary steam engines ), or internal combustion engines (such as petrol, diesel or paraffin engines). [ 2 ]
An engine that uses splash lubrication requires neither oil pump nor oil filter. Splash lubrication is an antique system whereby scoops on the big-ends of the connecting rods dip into the oil sump and splash the lubricant upwards towards the cylinders, creating an oil mist which settles into droplets. The oil droplets then pass through drillings to the bearings and thereby lubricate the moving parts. [ 3 ] Provided that the bearing is a ball bearing or a roller bearing , splash lubrication would usually be sufficient; however, plain bearings typically need a pressure feed to maintain the oil film, loss of which leads to overheating and seizure. In a transmission, some of the gears are partially submerged in fluid, which transfers lubricant to the other gears, as long as the transmission is spinning.
The splash lubrication system has simplicity, reliability, and cheapness within its virtues. [ 4 ] However, splash lubrication can work only on very low-revving engines, as otherwise the sump oil would become a frothy mousse. The Norwegian firm, Sabb Motor , produced a number of small marine diesel engines, mostly single-cylinder or twin-cylinder units, that used splash lubrication.
Splash lubrication is still used in modern engines and mechanisms, [ 5 ] [ 6 ] such as:
This article about an automotive technology is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Splash_lubrication |
In offshore construction , the splash zone is the transition from air to water when lowering heavy burdens into the sea. The overall efforts applied on the crane change dramatically when the load starts touching water, up to the point where it is completely submerged. Its buoyancy reduces the static mass that the crane has to support, but contact with the waves creates widely fluctuating dynamic forces.
Simulation of these changing efforts are necessary to correctly dimension cranes and lifting equipment. See for example DNV-RP-H103 (Det Norske Veritas recommended practices) for a mention of the piston effect created in the splash zone between two walls.
Special made Access Tools are often made for doing inspections or maintenance in the splash zone, typical down to 15 m depth. This zone is very difficult to access for divers or remotely operated vehicles (ROV's) due to waves and current. Rigging of equipment in this zone also needs special precautions do to the same. By using Remotely Operated Equipment (Robots) that holds on to the structures, work and inspections can be done. Earlier this Zone was looked at as unaccessible .
This engineering-related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Splash_zone |
Splat quenching is a metallurgical , metal morphing technique used for forming metals with a particular crystal structure by means of extremely rapid quenching , or cooling.
A typical technique for splat quenching involves casting molten metal by pouring it between two massive, cooled copper rollers that are constantly chilled by the circulation of water. These provide a near-instant quench because of the large surface area in close contact with the melt. The thin sheet formed has a low ratio of volume relative to the area used for cooling.
Products that are formed through this process have a crystal structure that is near- amorphous , or non-crystalline. They are commonly used for their valuable magnetic properties, specifically high magnetic permeability . This makes them useful for magnetic shielding and for low-loss transformer cores in electrical grids .
The process of splat quenching involves rapid quenching or cooling of molten metal. A typical procedure for splat quenching involves pouring the molten metal between two cooled copper rollers that are circulated with water to transfer the heat away from the metal, causing it to almost instantaneously solidify. [ 1 ]
A more efficient splat quenching technique is Duwez's and Willen's gun technique. Their technique produces higher rates of cooling of the droplet of metal because the sample is propelled at high velocities and hits a quencher plate causing its surface area to increase which immediately solidifies the metal. This allows for a wider range of metals that can be quenched and be given amorphous-like features instead of the general iron alloy. [ 2 ]
Another technique involves the consecutive spraying of the molten metal onto a chemical vapor deposition surface. However, the layers do not fuse together as desired and this causes oxides to be contained in the structure and pores to form around the structure. Manufacturing companies take an interest in the resultant products because of their near-net shaping capabilities. [ 3 ]
Some varying factors in splat quenching are the drop size and velocity of the metal in ensuring the complete solidification of the metal. In cases where the volume of the drop is too large or the velocity is too slow, the metal will not solidify past equilibrium causing it to remelt. [ 4 ] Therefore, experiments are carried out to determine the precise volume and velocity of the droplet that will ensure complete solidification of a certain metal. [ 5 ] Intrinsic and extrinsic factors influencing the glass-forming ability of metallic alloys were analyzed and classified. [ 6 ]
The near-instantaneous quenching of the metal causes the metal to have a near- amorphous crystalline structure, which is very uncharacteristic of a typical crystal. This structure is very similar to liquids, and the only difference between liquids and amorphous solids is the high viscosity of the solid. Solids in general have a crystalline structure instead of an amorphous structure because the crystalline structure has a stronger binding energy. The way a solid can have the irregular spacing between its atoms is when a liquid is cooled below its melting temperature. The reason for this is the molecules do not have enough time to rearrange themselves in a crystalline structure, and therefore stay in the liquid-like structure. [ 7 ]
Amorphous solids in general have a unique magnetic property because of their atomic disorder as explained above. They are rather soft metals and each has its own specific magnetic property depending on the means of production. In the splat quenching process, the metals are very soft and have superparamagnetic properties or shifting polarity behavior caused by the rapid and intense heat transfer. [ 8 ] | https://en.wikipedia.org/wiki/Splat_quenching |
Splenocytes are white blood cells that reside in the spleen and are involved in functions of the spleen, such as filtering blood and the immune response. [ 1 ]
Splenocytes consist of a variety of cell populations such as T and B lymphocytes, dendritic cells and macrophages, which have different immune functions. [ 2 ]
Splenocytes are spleen cells and consist of leukocytes like B and T cells , dendritic cells , and macrophages . [ 2 ] The spleen is split into red and white pulp regions with the marginal zone separating the two areas. The red pulp is involved with filtering blood and recycling iron, while the white pulp is involved in the immune response. [ 2 ]
The red pulp contains macrophages that phagocytose old or damaged red blood cells. [ 1 ]
The white pulp contains separate compartments for B and T cells called the B cell zone (BCZ) and the T cell zone (TCZ). [ 3 ] B cells make antibodies to fight off bacterial, viral, and fungal infections, and T cells are activated in response to antigens. [ 1 ] [ 2 ] [ 3 ]
The marginal zone (MZ) separates the red and white pulp regions and contains macrophages, B cells, and dendritic cells. MZ macrophages remove some types of blood-borne bacteria and viruses. [ 2 ] MZ B and dendritic cells are involved in antigen processing and presentation to lymphocytes in the white pulp. [ 2 ]
This cell biology article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Splenocyte |
Splenogonadal fusion is a rare congenital malformation that results from an abnormal connection between the primitive spleen and gonad during gestation . A portion of the splenic tissue then descends with the gonad. Splenogonadal fusion has been classified into two types: continuous, where there remains a connection between the main spleen and gonad; and discontinuous, where ectopic splenic tissue is attached to the gonad, but there is no connection to the orthotopic spleen. Patients can also have an accessory spleen . Patients with continuous splenogonadal fusion frequently have additional congenital abnormalities including limb defects, micrognathia , skull anomalies, Spina bifida , cardiac defects, anorectal abnormalities, and most commonly cryptorchidism . [ 1 ] Terminal limb defects have been documented in at least 25 cases which makes up a separate diagnosis of splenogonadal fusion limb defect (SGFLD) syndrome .
The anomaly was first described in 1883 by Bostroem . [ 2 ] Since then more than 150 cases of splenogonadal fusion have been documented, predominantly in males. [ 3 ] The condition is considered benign. [ 4 ] A few cases of testicular neoplasm have been reported in association with splenogonadal fusion. [ 5 ] [ 6 ] The reported cases have occurred in patients with a history of cryptorchidism , which is associated with an elevated risk of neoplasm. [ 6 ]
Splenogonadal fusion occurs with a male-to-female ratio of 16:1, and is seen nearly exclusively on the left side. [ 3 ] The condition remains a diagnostic challenge, but preoperative consideration of the diagnosis and use of ultrasound may help avoid unnecessary orchiectomy . The presence of splenic tissue may be confirmed with a technetium-99m sulfur colloid scan. [ 7 ]
Splenogonadal fusion is separated into two types:
The cause of splenogonadal fusion is still unclear, and there are several proposed mechanisms. A developmental field defect that occurs during blastogenesis is the current explanation of pathogenesis. The spleen derives from mesenchymal tissue and an inappropriate fusion can happen between the gonadal ridge during gut rotation, which occurs between weeks 5 and 8 of fetal life. [ 8 ] Splenogonadal fusion may result from an unknown teratogenic insult. The timing of this insult may correlate to the severity of associated defects. There is also postulation that fusion may occur due to adhesion after an inflammatory response or a lack of apoptosis between the structures. Siblings documented to have splenogonadal fusion and an accessory spleen provides additional evidence of a possible genetic component. [ 8 ]
Additional congenital abnormalities are most often found associated with the continuous type of splenogonadal fusion. These abnormalities include micrognathia , macroglossia , anal atresia , and pulmonary hypoplasia . [ 9 ] The most commonly associated malformation is cryptorchidism . When limb abnormalities occur, Splenogonadal Fusion with Limb Defects is made as a separate diagnosis. Splenogonadal Fusion with Limb Defects is also of unknown cause but some literature suggests that the condition may be related to Hanhart syndrome and facial femoral syndrome . [ 9 ] Splenogonadal fusion has also been identified in infants with Möbius syndrome and Poland syndrome . [ 10 ]
Diagnosis can be challenging pre-operatively, as patients are commonly asymptomatic. Physical examination can aid in diagnosis if the mass is palpated, but further confirmatory tests are necessary. Females are reportedly less affected by splenogonadal fusion, though it is possible that the condition is underdiagnosed due to the difficulty of internal gonad examination in females. [ 8 ] On scrotal ultrasound , ectopic splenic tissue may appear as an encapsulated homogeneous extratesticular mass, isoechoic with the normal testis. Subtle hypoechoic nodules may be present in the mass. [ 7 ] Limitations of doppler ultrasonography include visualizing the nonspecific paratesticular masses which can mimic malignancies such as rhabdomyosarcoma or embryonal sarcoma. [ 11 ] Technetium-99m sulfur colloid scan is sensitive to target the liver, spleen, and bone marrow, therefore the scans can be used to identify ectopic splenic tissue when clinical suspicion is high. [ 12 ] When technetium-99m sulfur colloid scans are not included in a patient's diagnostic workup, the diagnosis is often not made until the surgeon performs an abdominal exploration using laparoscopy which can visualize the splenic tissue grossly. Evaluation by biopsy including frozen section procedure is confirmatory for splenogonadal fusion. Histological examination will show normal splenic tissue which is made up of red and white pulp . [ 13 ] Many people with splenogonadal fusion go undiagnosed without complication during their lifetime. Many cases have been diagnosed at autopsy or incidentally after orchiectomy. [ 8 ]
Treatment remains controversial given the benign nature of splenogonadal fusion. Splenogonadal fusion does not have known clinical manifestations that require intervention. Clinical observation is considered when the mass is diagnosed pre-operatively. Surgery is often required to confirm the diagnosis and exclude a mimic of a malignant mass. Surgical approach should attempt to divide the mass and the gonad at the connecting plane for removal of the splenic tissue. Orchiectomy is generally not indicated. Causality between splenogonadal fusion and future malignant transformation has not been established, but literature has highlighted infrequent cases of testicular neoplasms and splenogonadal fusion. These cases were found in patients who also had a history of cryptorchidism , which is a known risk factor for testicular cancer . [ 14 ] | https://en.wikipedia.org/wiki/Splenogonadal_fusion |
SpliceInfo is a database for the four major alternative-splicing modes (exon skipping, 5'-alternative splicing, 3'-alternative splicing and intron retention) in the human genome . [ 1 ]
This resource appears to be no longer available. [ 2 ]
This Biological database -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/SpliceInfo |
Splice is a cloud -based music creation platform founded by Matt Aimonetti and Steve Martocci which includes a sample library, audio plug-ins on a subscription basis, and integration with several digital audio workstations (DAWs) . The program is available for MacOS, Windows, iOS and Android .
The site and the MacOS version of Studio Splice was launched in private beta in October 2013. A Windows version was released a few months later and the offer became available to the public in September 2014. [ 1 ] [ 2 ] Splice Studio allows musicians to remotely collaborate through the cloud. The technology is compatible with several popular digital audio workstations (DAWs) programs including Ableton Live , Logic Pro X , FL Studio , Garageband , and Studio One .
Its subscription-based sample marketplace, Splice Sounds, was launched in 2015. [ 3 ] [ 4 ]
In 2016, Splice introduced Rent-To-Own, which allowed users to pay a monthly fee to access premium products like synths. The model enabled users to put their rental balance toward the ownership cost and own the product outright or move on to a new program without paying the full market price. The Rent-To-Own program expanded in 2017 to products like Ozone 8 and Neutron 2, prompting The Verge to proclaim: "Splice is making it much easier to buy expensive music production plugins". [ 5 ]
In 2019, Splice launched its proprietary AI-driven feature Similar Sounds, [ 6 ] giving users machine-learning based similarity search across the entire marketplace of audio samples. In November, the company introduced a mobile version of its app on iOS and Android .
In 2020 Splice launched the inaugural Splice Awards, [ 7 ] recognizing artists in multiple categories, including "Splice Producer of the Year", "Splice Breakout Artist of the Year", "Splice New Standard", "Best Use of Splice Sounds" and "Best Sound Design". The Awards [ 8 ] also recognized Splice's "Most Downloaded One Shot", "Most Downloaded Loop", and "Most Downloaded Pack & Vocal Pack".
In June 2021, Splice launched Skills and Connected Instruments with their new Creator Plans. [ 9 ] In May 2022, Steve Martocci stepped down as CEO and transitioned to the company's executive chairman and chief strategy officer. Kakul Srivastava was named Splice's new CEO. [ 10 ]
In March 2023 Splice announced it was closing its Studio collaboration platform. [ 11 ]
In April 2025 Splice acquired British music technology company, Spitfire Audio . [ 12 ]
In October 2013, Splice raised $2.75 million in a seed funding round led by Union Square Ventures with participation from True Ventures, Lerer Ventures, SV Angel, First Round Capital , Code Advisors, Rob Wiesenthal , David Tisch and Seth Goldstein . Splice raised an additional $4.5 in funding from a Series A round led by True Ventures with participation from Scooter Braun , Tiësto , Steve Angello , AM Only, WME, Plus Eight Equity Fund LP and existing investors including Union Square Ventures in 2014. Splice raised $35 million in a Series B round of funding led by Draper Fisher Jurvetson in 2017. A$57.7 million Series C round followed in 2019, co-led by Union Square Ventures and True Ventures with participation from DFJ Growth , Flybridge, Lerer Hippeau, LionTree , Founders Circle Capital and Matt Pincus . In February 2021, Splice raised a $55 million series D round led by Goldman Sachs , with additional participation from the venture firm Music. The latest round brought the company's total funding to more than $150 million and estimated valuation to nearly $500 million.
In 2018, Zedd & Grey's single "The Middle" (feat. Maren Morris) featured sounds from Grey's Singular Sounds sample label. The song was nominated for three Grammy Awards and was certified 3× Platinum by the RIAA in 2019. Other notable Top 40 hits to feature Splice samples include Ariana Grande 's “ Break Up With Your Girlfriend, I’m Bored ,” Lil Nas X's “ Panini ,” Dua Lipa 's “ Don’t Start Now ,” Bad Bunny 's “ Dakiti ” and Shane Codd's "Get Out My Head". [ citation needed ]
In February 2020, the song "Running Over" was released by Justin Bieber . His producers had used a sample from Splice created by UK producer Laxcity. [ 13 ] Musician Asher Monroe accused Bieber of stealing the melody from his 2019 song "Synergy". However, the sound came from the same Laxcity sample pack used by Bieber. The Verge stated, "Justin Bieber was accused of stealing a melody, but it’s actually a royalty-free sample you can buy online." [ 14 ]
On April 11th, 2024, Sabrina Carpenter released the critically acclaimed song, "Espresso". The song features multiple loops from the Splice sample pack "Oliver: Power Tools Sample Pack III", created by Oliver . [ 15 ] [ 16 ] | https://en.wikipedia.org/wiki/Splice_(platform) |
A splice site mutation is a genetic mutation that inserts , deletes or changes a number of nucleotides in the specific site at which splicing takes place during the processing of precursor messenger RNA into mature messenger RNA . Splice site consensus sequences that drive exon recognition are located at the very termini of introns . [ 1 ] The deletion of the splicing site results in one or more introns remaining in mature mRNA and may lead to the production of abnormal proteins . When a splice site mutation occurs, the mRNA transcript possesses information from these introns that normally should not be included. Introns are supposed to be removed, while the exons are expressed.
The mutation must occur at the specific site at which intron splicing occurs: within non-coding sites in a gene, directly next to the location of the exon. The mutation can be an insertion, deletion, frameshift , etc. The splicing process itself is controlled by the given sequences, known as splice-donor and splice-acceptor sequences, which surround each exon. Mutations in these sequences may lead to retention of large segments of intronic DNA by the mRNA, or to entire exons being spliced out of the mRNA. These changes could result in production of a nonfunctional protein. [ 2 ] An intron is separated from its exon by means of the splice site. Acceptor-site and donor-site relating to the splice sites signal to the spliceosome where the actual cut should be made. These donor sites, or recognition sites, are essential in the processing of mRNA. The average vertebrate gene consists of multiple small exons (average size, 137 nucleotides) separated by introns that are considerably larger. [ 1 ]
In 1993, Richard J. Roberts and Phillip Allen Sharp received the Nobel Prize in Physiology or Medicine for their discovery of " split genes ". [ 4 ] Using the model adenovirus in their research, they were able to discover splicing—the fact that pre-mRNA is processed into mRNA once introns were removed from the RNA segment. These two scientists discovered the existence of splice sites, thereby changing the face of genomics research. They also discovered that the splicing of the messenger RNA can occur in different ways, opening up the possibility for a mutation to occur.
Today, many different types of technologies exist in which splice sites can be located and analyzed for more information. The Human Splicing Finder is an online database stemming from the Human Genome Project data. The genome database identifies thousands of mutations related to medical and health fields, as well as providing critical research information regarding splice site mutations. The tool specifically searches for pre-mRNA splicing errors, the calculation of potential splice sites using complex algorithms, and correlation with several other online genomic databases, such as the Ensembl genome browser. [ 5 ]
Due to the sensitive location of splice sites, mutations in the acceptor or donor areas of splice sites can become detrimental to a human individual. In fact, many different types of diseases stem from anomalies within the splice sites.
A study researching the role of splice site mutations in cancer supported that a splice site mutation was common in a set of women who were positive for breast and ovarian cancer. These women had the same mutation, according to the findings. An intronic single base-pair substitution destroys an acceptor site, thus activating a cryptic splice site, leading to a 59 base-pair insertion and chain termination. The four families with both breast and ovarian cancer had chain termination mutations in the N-terminal half of the protein. [ 6 ] The mutation in this research example was located within the splice-site.
Splice-site mutations are recurrently found in key lymphoma genes [ 7 ] like BCL7A [ 8 ] or CD79B [ 7 ] due to aberrant somatic hypermutation as the sequence targeted by AID overlaps with the sequences of the splice-sites. [ 9 ]
According to a research study conducted Hutton, M et al, a missense mutation occurring on the 5' region of the RNA associated with the tau protein was found to be correlated with inherited dementia (known as FTDP-17). The splice-site mutations all destabilize a potential stem–loop structure which is most likely involved in regulating the alternative splicing of exon10 in chromosome 17. Consequently, more usage occurs on the 5' splice site and an increased proportion of tau transcripts that include exon 10 are created. Such drastic increase in mRNA will increase the proportion of Tau containing four microtubule-binding repeats, which is consistent with the neuropathology described in several families with FTDP-17, a type inherited dementia. [ 10 ]
Some types of epilepsy may be brought on due to a splice site mutation.
In addition to a mutation in a stop codon , a splice site mutation on the 3' strand was found in a gene coding for cystatin B in Progressive Myoclonus Epilepsy [ 11 ] patients. This combination of mutations was not found in unaffected individuals. By comparing sequences with and without the splice site mutation, investigators were able to determine that a G-to-C nucleotide transversion occurs at the last position of the first intron. This transversion occurs in the region that codes for the cystatin B gene. Individuals suffering from Progressive Myoclonus Epilepsy possess a mutated form of this gene, which results in decreased output of mature mRNA, and subsequently decreases in protein expression.
A study has also shown that a type of Childhood Absence Epilepsy (CAE) causing febrile seizures may be linked to a splice site mutation in the sixth intron of the GABRG2 gene . This splice site mutation was found to cause a nonfunctional GABRG2 subunit in affected individuals. [ 12 ] According to this study, a point mutation was the culprit for the splice-donor site mutation, which occurred in intron 6. A nonfunctional protein product is produced, leading to the also nonfunctional subunit.
Several genetic diseases may be the result of splice site mutations. For example, mutations that cause the incorrect splicing of β-globin mRNA are responsible of some cases of β-thalassemia . Another Example is TTP (thrombotic thrombocytopenic purpura). TTP is caused by deficiency of ADAMTS-13 . A splice site mutation of ADAMTS-13 gene can therefore cause TTP. It is estimated that 15% of all point mutations causing human genetic diseases occur within a splice site. [ 13 ]
When a splice site mutation occurs in intron 2 of the gene that produces the parathyroid hormone , a parathyroid deficiency can prevail. In one particular study, a G to C substitution in the splice site of intron 2 produces a skipping effect in the messenger RNA transcript. The exon that is skipped possesses the initiation start codon to produce parathyroid hormone. [ 14 ] Such failure in initiation causes the deficiency.
Using the model organism Drosophila melanogaster , data has been compiled regarding the genomic information and sequencing of this organism. A prediction model exists in which a researcher can upload his or her genomic information and use a splice site prediction database to gather information about where the splice sites could be located. The Berkeley Drosophila Project can be used to incorporate this research, as well as annotate high quality euchromatic data. The splice site predictor can be a great tool for researchers studying human disease in this model organism .
Splice site mutations can be analyzed using information theory . [ 15 ] | https://en.wikipedia.org/wiki/Splice_site_mutation |
A spliceosome is a large ribonucleoprotein (RNP) complex found primarily within the nucleus of eukaryotic cells . The spliceosome is assembled from small nuclear RNAs ( snRNA ) and numerous proteins. Small nuclear RNA (snRNA) molecules bind to specific proteins to form a small nuclear ribonucleoprotein complex (snRNP, pronounced "snurps"), which in turn combines with other snRNPs to form a large ribonucleoprotein complex called a spliceosome. The spliceosome removes introns from a transcribed pre-mRNA, a type of primary transcript . This process is generally referred to as splicing . [ 1 ] An analogy is a film editor, who selectively cuts out irrelevant or incorrect material (equivalent to the introns ) from the initial film and sends the cleaned-up version to the director for the final cut. [ citation needed ]
However, sometimes the RNA within the intron acts as a ribozyme, splicing itself without the use of a spliceosome or protein enzymes. [ citation needed ]
In 1977, work by the Sharp and Roberts labs revealed that genes of higher organisms are "split" or present in several distinct segments along the DNA molecule. [ 2 ] [ 3 ] The coding regions of the gene are separated by non-coding DNA that is not involved in protein expression. The split gene structure was found when adenoviral mRNAs were hybridized to endonuclease cleavage fragments of single stranded viral DNA. [ 2 ] It was observed that the mRNAs of the mRNA-DNA hybrids contained 5' and 3' tails of non-hydrogen bonded regions. When larger fragments of viral DNAs were used, forked structures of looped out DNA were observed when hybridized to the viral mRNAs. It was realised that the looped out regions, the introns , are excised from the precursor mRNAs in a process Sharp named "splicing". The split gene structure was subsequently found to be common to most eukaryotic genes. Phillip Sharp and Richard J. Roberts were awarded the Nobel Prize in Medicine 1993 for the discovery of introns and the splicing process.
Each spliceosome is composed of five small nuclear RNAs (snRNA) and a range of associated protein factors. When these small RNAs are combined with the protein factors, they make RNA-protein complexes called snRNPs ( s mall n uclear r ibo n ucleo p roteins, pronounced "snurps").
The snRNAs that make up the major spliceosome are named U1 , U2 , U4 , U5 , and U6 , so-called because they are rich in uridine , and participate in several RNA-RNA and RNA-protein interactions. [ 1 ]
The assembly of the spliceosome occurs on each pre-mRNA (also known as heterogeneous nuclear RNA, hn-RNA) at each exon:intron junction. The pre-mRNA introns contains specific sequence elements that are recognized and utilized during spliceosome assembly. These include the 5' end splice site, the branch point sequence, the polypyrimidine tract, and the 3' end splice site. The spliceosome catalyzes the removal of introns, and the ligation of the flanking exons. [ citation needed ]
Introns typically have a GU nucleotide sequence at the 5' end splice site, and an AG at the 3' end splice site. The 3' splice site can be further defined by a variable length of polypyrimidines, called the polypyrimidine tract (PPT), which serves the dual function of recruiting factors to the 3' splice site and possibly recruiting factors to the branch point sequence (BPS). The BPS contains the conserved adenosine required for the first step of splicing. [ citation needed ]
Many proteins exhibit a zinc-binding motif, which underscores the importance of zinc in the splicing mechanism. [ 4 ] [ 5 ] [ 6 ] The first molecular-resolution reconstruction of U4/U6.U5 triple small nuclear ribonucleoprotein (tri-snRNP) complex was reported in 2016. [ 7 ]
Cryo-EM has been applied extensively by Shi et al. to elucidate the near-/atomic structure of spliceosome in both yeast [ 9 ] and humans. [ 10 ] The molecular framework of spliceosome at near-atomic-resolution demonstrates Spp42 component of U5 snRNP forms a central scaffold and anchors the catalytic center in yeast. The atomic structure of the human spliceosome illustrates the step II component Slu7 adopts an extended structure, poised for selection of the 3'-splice site. All five metals (assigned as Mg2+) in the yeast complex are preserved in the human complex. [ citation needed ]
Alternative splicing (the re-combination of different exons ) is a major source of genetic diversity in eukaryotes. Splice variants have been used to account for the relatively small number of protein coding genes in the human genome , currently estimated at around 20,000. One particular Drosophila gene, Dscam , has been speculated to be alternatively spliced into 38,000 different mRNAs , assuming all of its exons can splice independently of each other. [ 11 ]
Pre-mRNA splicing factors were originally found to be concentrated in nuclear bodies known as nuclear speckles. [ 12 ] It was originally postulated that nuclear speckles are either sites of mRNA splicing or storage sites of mRNA splicing factors. It is now understood that nuclear speckles help concentrate splicing factors near genes that are physically located close to them. Genes located farther from speckles can still be transcribed and spliced, but their splicing is less efficient compared to those closer to speckles. [ 13 ] RNA splicing is a biochemical reaction, and like all biochemical reactions, its rate depends on the concentration of enzymes and substrates. In this case, the enzymes are the spliceosomes, and the substrates are the pre-mRNAs. By varying the concentration of spliceosomes and pre-mRNAs based on their proximity to nuclear speckles, cells could potentially regulate the efficiency of splicing. [ 13 ]
The model for formation of the spliceosome active site involves an ordered, stepwise assembly of discrete snRNP particles on the pre-mRNA substrate. The first recognition of pre-mRNAs involves U1 snRNP binding to the 5' end splice site of the pre-mRNA and other non-snRNP associated factors to form the commitment complex, or early (E) complex in mammals. [ 14 ] [ 15 ] The commitment complex is an ATP-independent complex that commits the pre-mRNA to the splicing pathway. [ 16 ] U2 snRNP is recruited to the branch region through interactions with the E complex component U2AF (U2 snRNP auxiliary factor) and possibly U1 snRNP. In an ATP-dependent reaction, U2 snRNP becomes tightly associated with the branch point sequence (BPS) to form complex A. A duplex formed between U2 snRNP and the pre-mRNA branch region bulges out the branch adenosine specifying it as the nucleophile for the first transesterification. [ 17 ]
The presence of a pseudouridine residue in U2 snRNA, nearly opposite of the branch site, results in an altered conformation of the RNA-RNA duplex upon the U2 snRNP binding. Specifically, the altered structure of the duplex induced by the pseudouridine places the 2' OH of the bulged adenosine in a favorable position for the first step of splicing. [ 18 ] The U4/U5/U6 tri-snRNP (see Figure 1) is recruited to the assembling spliceosome to form complex B, and following several rearrangements, complex C is activated for catalysis. [ 19 ] [ 20 ] It is unclear how the tri-snRNP is recruited to complex A, but this process may be mediated through protein-protein interactions and/or base pairing interactions between U2 snRNA and U6 snRNA. [ citation needed ]
The U5 snRNP interacts with sequences at the 5' and 3' splice sites via the invariant loop of U5 snRNA [ 21 ] and U5 protein components interact with the 3' splice site region. [ 22 ]
Upon recruitment of the tri-snRNP, several RNA-RNA rearrangements precede the first catalytic step and further rearrangements occur in the catalytically active spliceosome. Several of the RNA-RNA interactions are mutually exclusive; however, it is not known what triggers these interactions, nor the order of these rearrangements. The first rearrangement is probably the displacement of U1 snRNP from the 5' splice site and formation of a U6 snRNA interaction. It is known that U1 snRNP is only weakly associated with fully formed spliceosomes, [ 23 ] and U1 snRNP is inhibitory to the formation of a U6-5' splice site interaction on a model of substrate oligonucleotide containing a short 5' exon and 5' splice site. [ 24 ] Binding of U2 snRNP to the branch point sequence (BPS) is one example of an RNA-RNA interaction displacing a protein-RNA interaction. Upon recruitment of U2 snRNP, the branch binding protein SF1 in the commitment complex is displaced since the binding site of U2 snRNA and SF1 are mutually exclusive events. [ citation needed ]
Within the U2 snRNA, there are other mutually exclusive rearrangements that occur between competing conformations. For example, in the active form, stem loop IIa is favored; in the inactive form a mutually exclusive interaction between the loop and a downstream sequence predominates. [ 20 ] It is unclear how U4 is displaced from U6 snRNA, although RNA has been implicated in spliceosome assembly, and may function to unwind U4/U6 and promote the formation of a U2/U6 snRNA interaction. The interactions of U4/U6 stem loops I and II dissociate and the freed stem loop II region of U6 folds on itself to form an intramolecular stem loop and U4 is no longer required in further spliceosome assembly. The freed stem loop I region of U6 base pairs with U2 snRNA forming the U2/U6 helix I. However, the helix I structure is mutually exclusive with the 3' half of an internal 5' stem loop region of U2 snRNA. [ citation needed ]
Some eukaryotes have a second spliceosome, the so-called minor spliceosome . [ 25 ] A group of less abundant snRNAs, U11 , U12 , U4atac , and U6atac , together with U5, are subunits of the minor spliceosome that splices a rare class of pre-mRNA introns, denoted U12-type. The minor spliceosome is located in the nucleus like its major counterpart, [ 26 ] though there are exceptions in some specialised cells including anucleate platelets [ 27 ] and the dendroplasm ( dendrite cytoplasm) of neuronal cells. [ 28 ] | https://en.wikipedia.org/wiki/Spliceosome |
A splicing factor is a protein involved in the removal of introns from strings of messenger RNA , so that the exons can bind together; the process takes place in particles known as spliceosomes . Splicing factors regulate the binding of the snRNPs U1 and U2 to the 3' and 5' ends of the intron during splicing and can either be splicing promoters or splicing repressors. [ 1 ]
In a research paper, splicing factors were found to be produced upon application of resveratrol analogues, which induced senescent cells to rejuvenate. [ 2 ] [ 3 ] The expression of splicing factors may be altered during aging. [ 4 ]
Splicing factor 3b is a protein complex consisting of the following proteins: PHF5A , SF3B1 , SF3B2 , SF3B3 , SF3B4 , SF3B5 , SF3B6 . [ citation needed ]
This protein -related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Splicing_factor |
The split-Hopkinson pressure bar ( SHPB ), named after Bertram Hopkinson , sometimes also called a Kolsky bar , is an apparatus for testing the dynamic stress – strain response of materials.
The Hopkinson pressure bar was first suggested by Bertram Hopkinson in 1914 [ 1 ] as a way to measure stress pulse propagation in a metal bar. Later, in 1949 Herbert Kolsky [ 2 ] refined Hopkinson's technique by using two Hopkinson bars in series, now known as the split-Hopkinson bar, to measure stress and strain, incorporating advancements in the cathode ray oscilloscope in conjunction with electrical condenser units to record the pressure wave propagation in the pressure bars as pioneered by Rhisiart Morgan Davies a year earlier in 1948. [ 3 ]
Later modifications have allowed for tensile, compression, and torsion testing.
Although there are various setups and techniques currently in use for the split-Hopkinson pressure bar, the underlying principles for the test and measurement are the same. The specimen is placed between the ends of two straight bars, called the incident bar and the transmitted bar . [ 4 ] At the end of the incident bar (some distance away from the specimen, typically at the far end), a stress wave is created which propagates through the bar toward the specimen. This wave is referred to as the incident wave , and upon reaching the specimen, splits into two smaller waves. One of which, the transmitted wave , travels through the specimen and into the transmitted bar, causing plastic deformation in the specimen. The other wave, called the reflected wave , is reflected away from the specimen and travels back down the incident bar. [ 5 ]
Most modern setups use strain gauges on the bars to measure strains caused by the waves. Assuming deformation in the specimen is uniform, the stress and strain can be calculated from the amplitudes of the incident, transmitted, and reflected waves. [ 6 ]
For compression testing, two symmetrical bars are situated in series, with the sample in between. The incident bar is struck by a striker bar during testing. The striker bar is fired from a gas gun. The transmitted bar collides with a momentum trap (typically a block of soft metal). Strain gauges are mounted on both the incident and transmitted bars. [ 6 ]
Tension testing in a Split Hopkinson pressure bar (SHPB) is more complex due to a variation of loading methods and specimen attachment to the incident and transmission bar. [ 7 ] The first tension bar was designed and tested by Harding et al. in 1960; the design involved using a hollow weight bar that was connected to a yoke and threaded specimen inside of the weight bar. A tensile wave was created by impacting the weight bar with a ram and having the initial compression wave reflect as a tensile wave off the free end [ 8 ] Another breakthrough in the SHPB design was done by Nichols who used a typical compression setup and threaded metallic specimens on both the incident and transmission ends, while placing a composite collar over the specimen. The specimen had a snug fit on the incident and transmission side in order to bypass an initial compression wave. Nichols' setup would create an initial compression wave by an impact in the incident end with a striker, but when the compression wave reached the specimen, the threads would not be loaded. The compression wave would ideally pass through the composite collar and then reflect off the free end in tension. The tensile wave would then pull on the specimen. [ 7 ] The next loading method was revolutionized by Ogawa in 1984. A hollow striker was used to impact a flange that is threaded to end on an incident bar. This striker was propelled by using either a gas gun or a rotating disk. The specimen was once again attached to the incident and transmission bar via threading. [ 9 ]
As with tension testing, there exist a variety of methods for specimen attachment and loading when subjecting materials to torsion on a SHPB.
One way of applying loading called the stored-torque method involves clamping the midsection of the incident bar while a torque is applied to the free end. The incident wave is created by suddenly releasing the clamp, which sends a torsion wave toward the specimen. [ 5 ]
Another loading technique known as explosive-loading uses explosive charges on the free end of the incident bar to create the incident wave. This method is particularly sensitive to error because each charge must apply an equal impulse to the incident bar (to create pure torsion without bending) and must both detonate simultaneously. Explosive-loading is also unlikely to produce clean incident waves, which may cause uneven strain rates throughout the test. This method however has the advantage of having a very small rise time as compared to the stored-torque method. [ 10 ] | https://en.wikipedia.org/wiki/Split-Hopkinson_pressure_bar |
In algebra , a split-complex number (or hyperbolic number , also perplex number , double number ) is based on a hyperbolic unit j satisfying j 2 = 1 {\displaystyle j^{2}=1} , where j ≠ ± 1 {\displaystyle j\neq \pm 1} . A split-complex number has two real number components x and y , and is written z = x + y j . {\displaystyle z=x+yj.} The conjugate of z is z ∗ = x − y j . {\displaystyle z^{*}=x-yj.} Since j 2 = 1 , {\displaystyle j^{2}=1,} the product of a number z with its conjugate is N ( z ) := z z ∗ = x 2 − y 2 , {\displaystyle N(z):=zz^{*}=x^{2}-y^{2},} an isotropic quadratic form .
The collection D of all split-complex numbers z = x + y j {\displaystyle z=x+yj} for x , y ∈ R {\displaystyle x,y\in \mathbb {R} } forms an algebra over the field of real numbers . Two split-complex numbers w and z have a product wz that satisfies N ( w z ) = N ( w ) N ( z ) . {\displaystyle N(wz)=N(w)N(z).} This composition of N over the algebra product makes ( D , +, ×, *) a composition algebra .
A similar algebra based on R 2 {\displaystyle \mathbb {R} ^{2}} and component-wise operations of addition and multiplication, ( R 2 , + , × , x y ) , {\displaystyle (\mathbb {R} ^{2},+,\times ,xy),} where xy is the quadratic form on R 2 , {\displaystyle \mathbb {R} ^{2},} also forms a quadratic space . The ring isomorphism D → R 2 x + y j ↦ ( x − y , x + y ) {\displaystyle {\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}} is an isometry of quadratic spaces.
Split-complex numbers have many other names; see § Synonyms below. See the article Motor variable for functions of a split-complex number.
A split-complex number is an ordered pair of real numbers, written in the form
z = x + j y {\displaystyle z=x+jy}
where x and y are real numbers and the hyperbolic unit [ 1 ] j satisfies
j 2 = + 1 {\displaystyle j^{2}=+1}
In the field of complex numbers the imaginary unit i satisfies i 2 = − 1. {\displaystyle i^{2}=-1.} The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit j is not a real number but an independent quantity.
The collection of all such z is called the split-complex plane . Addition and multiplication of split-complex numbers are defined by
( x + j y ) + ( u + j v ) = ( x + u ) + j ( y + v ) ( x + j y ) ( u + j v ) = ( x u + y v ) + j ( x v + y u ) . {\displaystyle {\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}}
This multiplication is commutative , associative and distributes over addition.
Just as for complex numbers, one can define the notion of a split-complex conjugate . If
z = x + j y , {\displaystyle z=x+jy~,}
then the conjugate of z is defined as
z ∗ = x − j y . {\displaystyle z^{*}=x-jy~.}
The conjugate is an involution which satisfies similar properties to the complex conjugate . Namely,
( z + w ) ∗ = z ∗ + w ∗ ( z w ) ∗ = z ∗ w ∗ ( z ∗ ) ∗ = z . {\displaystyle {\begin{aligned}(z+w)^{*}&=z^{*}+w^{*}\\(zw)^{*}&=z^{*}w^{*}\\\left(z^{*}\right)^{*}&=z.\end{aligned}}}
The squared modulus of a split-complex number z = x + j y {\displaystyle z=x+jy} is given by the isotropic quadratic form
‖ z ‖ 2 = z z ∗ = z ∗ z = x 2 − y 2 . {\displaystyle \lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.}
It has the composition algebra property:
‖ z w ‖ = ‖ z ‖ ‖ w ‖ . {\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~.}
However, this quadratic form is not positive-definite but rather has signature (1, −1) , so the modulus is not a norm .
The associated bilinear form is given by
⟨ z , w ⟩ = R e ( z w ∗ ) = R e ( z ∗ w ) = x u − y v , {\displaystyle \langle z,w\rangle =\operatorname {\mathrm {Re} } \left(zw^{*}\right)=\operatorname {\mathrm {Re} } \left(z^{*}w\right)=xu-yv~,}
where z = x + j y {\displaystyle z=x+jy} and w = u + j v . {\displaystyle w=u+jv.} Here, the real part is defined by R e ( z ) = 1 2 ( z + z ∗ ) = x {\displaystyle \operatorname {\mathrm {Re} } (z)={\tfrac {1}{2}}(z+z^{*})=x} . Another expression for the squared modulus is then
‖ z ‖ 2 = ⟨ z , z ⟩ . {\displaystyle \lVert z\rVert ^{2}=\langle z,z\rangle ~.}
Since it is not positive-definite, this bilinear form is not an inner product ; nevertheless the bilinear form is frequently referred to as an indefinite inner product . A similar abuse of language refers to the modulus as a norm.
A split-complex number is invertible if and only if its modulus is nonzero ( ‖ z ‖ ≠ 0 {\displaystyle \lVert z\rVert \neq 0} ), thus numbers of the form x ± j x have no inverse. The multiplicative inverse of an invertible element is given by
z − 1 = z ∗ ‖ z ‖ 2 . {\displaystyle z^{-1}={\frac {z^{*}}{{\lVert z\rVert }^{2}}}~.}
Split-complex numbers which are not invertible are called null vectors . These are all of the form ( a ± j a ) for some real number a .
There are two nontrivial idempotent elements given by e = 1 2 ( 1 − j ) {\displaystyle e={\tfrac {1}{2}}(1-j)} and e ∗ = 1 2 ( 1 + j ) . {\displaystyle e^{*}={\tfrac {1}{2}}(1+j).} Idempotency means that e e = e {\displaystyle ee=e} and e ∗ e ∗ = e ∗ . {\displaystyle e^{*}e^{*}=e^{*}.} Both of these elements are null:
‖ e ‖ = ‖ e ∗ ‖ = e ∗ e = 0 . {\displaystyle \lVert e\rVert =\lVert e^{*}\rVert =e^{*}e=0~.}
It is often convenient to use e and e ∗ as an alternate basis for the split-complex plane. This basis is called the diagonal basis or null basis . The split-complex number z can be written in the null basis as
z = x + j y = ( x − y ) e + ( x + y ) e ∗ . {\displaystyle z=x+jy=(x-y)e+(x+y)e^{*}~.}
If we denote the number z = a e + b e ∗ {\displaystyle z=ae+be^{*}} for real numbers a and b by ( a , b ) , then split-complex multiplication is given by
( a 1 , b 1 ) ( a 2 , b 2 ) = ( a 1 a 2 , b 1 b 2 ) . {\displaystyle \left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)~.}
The split-complex conjugate in the diagonal basis is given by ( a , b ) ∗ = ( b , a ) {\displaystyle (a,b)^{*}=(b,a)} and the squared modulus by
‖ ( a , b ) ‖ 2 = a b . {\displaystyle \lVert (a,b)\rVert ^{2}=ab.}
On the basis {e, e*} it becomes clear that the split-complex numbers are ring-isomorphic to the direct sum R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } with addition and multiplication defined pairwise.
The diagonal basis for the split-complex number plane can be invoked by using an ordered pair ( x , y ) for z = x + j y {\displaystyle z=x+jy} and making the mapping
( u , v ) = ( x , y ) ( 1 1 1 − 1 ) = ( x , y ) S . {\displaystyle (u,v)=(x,y){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=(x,y)S~.}
Now the quadratic form is u v = ( x + y ) ( x − y ) = x 2 − y 2 . {\displaystyle uv=(x+y)(x-y)=x^{2}-y^{2}~.} Furthermore,
( cosh a , sinh a ) ( 1 1 1 − 1 ) = ( e a , e − a ) {\displaystyle (\cosh a,\sinh a){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=\left(e^{a},e^{-a}\right)}
so the two parametrized hyperbolas are brought into correspondence with S .
The action of hyperbolic versor e b j {\displaystyle e^{bj}\!} then corresponds under this linear transformation to a squeeze mapping
σ : ( u , v ) ↦ ( r u , v r ) , r = e b . {\displaystyle \sigma :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.}
Though lying in the same isomorphism class in the category of rings , the split-complex plane and the direct sum of two real lines differ in their layout in the Cartesian plane . The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a dilation by √ 2 . The dilation in particular has sometimes caused confusion in connection with areas of a hyperbolic sector . Indeed, hyperbolic angle corresponds to area of a sector in the R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } plane with its "unit circle" given by { ( a , b ) ∈ R ⊕ R : a b = 1 } . {\displaystyle \{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.} The contracted unit hyperbola { cosh a + j sinh a : a ∈ R } {\displaystyle \{\cosh a+j\sinh a:a\in \mathbb {R} \}} of the split-complex plane has only half the area in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of R ⊕ R {\displaystyle \mathbb {R} \oplus \mathbb {R} } .
A two-dimensional real vector space with the Minkowski inner product is called (1 + 1) -dimensional Minkowski space , often denoted R 1 , 1 . {\displaystyle \mathbb {R} ^{1,1}.} Just as much of the geometry of the Euclidean plane R 2 {\displaystyle \mathbb {R} ^{2}} can be described with complex numbers, the geometry of the Minkowski plane R 1 , 1 {\displaystyle \mathbb {R} ^{1,1}} can be described with split-complex numbers.
The set of points
{ z : ‖ z ‖ 2 = a 2 } {\displaystyle \left\{z:\lVert z\rVert ^{2}=a^{2}\right\}}
is a hyperbola for every nonzero a in R . {\displaystyle \mathbb {R} .} The hyperbola consists of a right and left branch passing through ( a , 0) and (− a , 0) . The case a = 1 is called the unit hyperbola . The conjugate hyperbola is given by
{ z : ‖ z ‖ 2 = − a 2 } {\displaystyle \left\{z:\lVert z\rVert ^{2}=-a^{2}\right\}}
with an upper and lower branch passing through (0, a ) and (0, − a ) . The hyperbola and conjugate hyperbola are separated by two diagonal asymptotes which form the set of null elements:
{ z : ‖ z ‖ = 0 } . {\displaystyle \left\{z:\lVert z\rVert =0\right\}.}
These two lines (sometimes called the null cone ) are perpendicular in R 2 {\displaystyle \mathbb {R} ^{2}} and have slopes ±1.
Split-complex numbers z and w are said to be hyperbolic-orthogonal if ⟨ z , w ⟩ = 0 . While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the simultaneous hyperplane concept in spacetime.
The analogue of Euler's formula for the split-complex numbers is
exp ( j θ ) = cosh ( θ ) + j sinh ( θ ) . {\displaystyle \exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).}
This formula can be derived from a power series expansion using the fact that cosh has only even powers while that for sinh has odd powers. [ 2 ] For all real values of the hyperbolic angle θ the split-complex number λ = exp( jθ ) has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as λ have been called hyperbolic versors .
Since λ has modulus 1, multiplying any split-complex number z by λ preserves the modulus of z and represents a hyperbolic rotation (also called a Lorentz boost or a squeeze mapping ). Multiplying by λ preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a group called the generalized orthogonal group O(1, 1) . This group consists of the hyperbolic rotations, which form a subgroup denoted SO + (1, 1) , combined with four discrete reflections given by
z ↦ ± z {\displaystyle z\mapsto \pm z} and z ↦ ± z ∗ . {\displaystyle z\mapsto \pm z^{*}.}
The exponential map
exp : ( R , + ) → S O + ( 1 , 1 ) {\displaystyle \exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)}
sending θ to rotation by exp( jθ ) is a group isomorphism since the usual exponential formula applies:
e j ( θ + ϕ ) = e j θ e j ϕ . {\displaystyle e^{j(\theta +\phi )}=e^{j\theta }e^{j\phi }.}
If a split-complex number z does not lie on one of the diagonals, then z has a polar decomposition .
In abstract algebra terms, the split-complex numbers can be described as the quotient of the polynomial ring R [ x ] {\displaystyle \mathbb {R} [x]} by the ideal generated by the polynomial x 2 − 1 , {\displaystyle x^{2}-1,}
R [ x ] / ( x 2 − 1 ) . {\displaystyle \mathbb {R} [x]/(x^{2}-1).}
The image of x in the quotient is the "imaginary" unit j . With this description, it is clear that the split-complex numbers form a commutative algebra over the real numbers. The algebra is not a field since the null elements are not invertible. All of the nonzero null elements are zero divisors .
Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a topological ring .
The algebra of split-complex numbers forms a composition algebra since
‖ z w ‖ = ‖ z ‖ ‖ w ‖ {\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~}
for any numbers z and w .
From the definition it is apparent that the ring of split-complex numbers is isomorphic to the group ring R [ C 2 ] {\displaystyle \mathbb {R} [C_{2}]} of the cyclic group C 2 over the real numbers R . {\displaystyle \mathbb {R} .}
Elements of the identity component in the group of units in D have four square roots.: say p = exp ( q ) , q ∈ D . then ± exp ( q 2 ) {\displaystyle p=\exp(q),\ \ q\in D.{\text{then}}\pm \exp({\frac {q}{2}})} are square roots of p . Further, ± j exp ( q 2 ) {\displaystyle \pm j\exp({\frac {q}{2}})} are also square roots of p .
The idempotents 1 ± j 2 {\displaystyle {\frac {1\pm j}{2}}} are their own square roots, and the square root of s 1 ± j 2 , s > 0 , is s 1 ± j 2 {\displaystyle s{\frac {1\pm j}{2}},\ \ s>0,\ {\text{is}}\ {\sqrt {s}}{\frac {1\pm j}{2}}}
One can easily represent split-complex numbers by matrices . The split-complex number z = x + j y {\displaystyle z=x+jy} can be represented by the matrix z ↦ ( x y y x ) . {\displaystyle z\mapsto {\begin{pmatrix}x&y\\y&x\end{pmatrix}}.}
Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of z is given by the determinant of the corresponding matrix.
In fact there are many representations of the split-complex plane in the four-dimensional ring of 2x2 real matrices. The real multiples of the identity matrix form a real line in the matrix ring M(2,R). Any hyperbolic unit m provides a basis element with which to extend the real line to the split-complex plane. The matrices
m = ( a c b − a ) {\displaystyle m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}}
which square to the identity matrix satisfy a 2 + b c = 1. {\displaystyle a^{2}+bc=1.} For example, when a = 0, then ( b,c ) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a subring of M(2,R). [ 3 ] [ better source needed ]
The number z = x + j y {\displaystyle z=x+jy} can be represented by the matrix x I + y m . {\displaystyle x\ I+y\ m.}
The use of split-complex numbers dates back to 1848 when James Cockle revealed his tessarines . [ 4 ] William Kingdon Clifford used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called split-biquaternions . He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the circle group . Extending the analogy, functions of a motor variable contrast to functions of an ordinary complex variable .
Since the late twentieth century, the split-complex multiplication has commonly been seen as a Lorentz boost of a spacetime plane. [ 5 ] [ 6 ] [ 7 ] [ 8 ] [ 9 ] [ 10 ] In that model, the number z = x + y j represents an event in a spatio-temporal plane, where x is measured in seconds and y in light-seconds . The future corresponds to the quadrant of events { z : | y | < x } , which has the split-complex polar decomposition z = ρ e a j {\displaystyle z=\rho e^{aj}\!} . The model says that z can be reached from the origin by entering a frame of reference of rapidity a and waiting ρ nanoseconds. The split-complex equation
e a j e b j = e ( a + b ) j {\displaystyle e^{aj}\ e^{bj}=e^{(a+b)j}}
expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity a ;
{ z = σ j e a j : σ ∈ R } {\displaystyle \{z=\sigma je^{aj}:\sigma \in \mathbb {R} \}}
is the line of events simultaneous with the origin in the frame of reference with rapidity a .
Two events z and w are hyperbolic-orthogonal when z ∗ w + z w ∗ = 0. {\displaystyle z^{*}w+zw^{*}=0.} Canonical events exp( aj ) and j exp( aj ) are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to j exp( aj ) .
In 1933 Max Zorn was using the split-octonions and noted the composition algebra property. He realized that the Cayley–Dickson construction , used to generate division algebras, could be modified (with a factor gamma, γ ) to construct other composition algebras including the split-octonions. His innovation was perpetuated by Adrian Albert , Richard D. Schafer, and others. [ 11 ] The gamma factor, with R as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for Mathematical Reviews , N. H. McCoy wrote that there was an "introduction of some new algebras of order 2 e over F generalizing Cayley–Dickson algebras." [ 12 ] Taking F = R and e = 1 corresponds to the algebra of this article.
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in Contribución a las Ciencias Físicas y Matemáticas , National University of La Plata , República Argentina (in Spanish). These expository and pedagogical essays presented the subject for broad appreciation. [ 13 ]
In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the nine-point hyperbola of a triangle inscribed in zz ∗ = 1 . [ 14 ]
In 1956 Mieczyslaw Warmus published "Calculus of Approximations" in Bulletin de l’Académie polonaise des sciences (see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra. [ 15 ] D. H. Lehmer reviewed the article in Mathematical Reviews and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.
In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.
Different authors have used a great variety of names for the split-complex numbers. Some of these include: | https://en.wikipedia.org/wiki/Split-complex_number |
Split-intein circular ligation of peptides and proteins ( SICLOPPS ) is a biotechnology technique that permits the creation of cyclic peptides . These peptides are produced by ribosomal protein synthesis , followed by an intein -like event that splices the protein into a loop. By contrast with the nonribosomal peptide synthetases that produces some cyclic peptides like gramicidin S , SICLOPPS offers the advantage that the peptides' structure can be encoded by DNA in a simple manner according to the genetic code , but for this reason it imposes limitations on the types of amino acids incorporated that are comparable to those that apply to ordinary proteins . As implemented there is also some constraint on the peptide sequence of the cyclic sequence; for example, libraries may use the sequence S G XX..XX P L to increase the efficiency of circularization of the peptide. [ 1 ] [ 2 ] SICLOPPS is frequently used with a library of randomized DNA sequence that permits the simultaneous production and screening of large numbers of constructs at once, followed by the recovery of the DNA sequences responsible for the activity of the clone of interest.
A number of natural antimicrobial peptides are cyclic, and the products of SICLOPPS are "increasingly viewed as ideal backbones for modulation of protein-protein interactions." [ 3 ] Circular peptides tend to be resistant to protease activity, and may be suitable for use as orally administered drugs. Once a cyclic peptide is identified with a biological activity of interest, it may also be possible to identify the target of the peptide (a gene that encodes a protein with which it interacts) by functional complementation , facilitating a better understanding of its mechanism of action . [ 1 ] | https://en.wikipedia.org/wiki/Split-intein_circular_ligation_of_peptides_and_proteins |
A split-ring resonator ( SRR ) is an artificially produced structure common to metamaterials . Its purpose is to produce the desired magnetic susceptibility (magnetic response) in various types of metamaterials up to 200 terahertz .
Split ring resonators (SRRs) consist of a pair of concentric metallic rings, etched on a dielectric substrate, with slits etched on opposite sides. SRRs can produce the effect of being electrically smaller when responding to an oscillating electromagnetic field . These resonators have been used for the synthesis of left-handed and negative refractive index media, where the necessary value of the negative effective permeability is due to the presence of the SRRs. When an array of electrically small SRRs is excited by means of a time-varying magnetic field , the structure behaves as an effective medium with negative effective permeability in a narrow band above SRR resonance . SRRs have also been coupled to planar transmission lines for the synthesis of metamaterials transmission line . [ 4 ] [ 5 ] [ 6 ] [ 7 ] These media create the necessary strong magnetic coupling to an applied electromagnetic field not otherwise available in conventional materials. For example, an effect such as negative permeability is produced with a periodic array of split ring resonators. [ 8 ]
A single-cell SRR has a pair of enclosed loops with splits in them at opposite ends. The loops are made of nonmagnetic metal like copper and have a small gap between them. The loops can be concentric or square, and gapped as needed. A magnetic flux penetrating the metal rings will induce rotating currents in the rings, which produce their own flux to enhance or oppose the incident field (depending on the SRR resonant properties). This field pattern is dipolar . The small gaps between the rings produces large capacitance values, which lowers the resonating frequency . Hence the dimensions of the structure are small compared to the resonant wavelength. This results in low radiative losses and very high quality factors . [ 8 ] [ 9 ] [ 10 ]
The split ring resonator is a microstructure design featured in the paper by Pendry et al in 1999 called, "Magnetism from Conductors and Enhanced Nonlinear Phenomena". [ 11 ] It proposed that the split ring resonator design, built out of nonmagnetic material, could enhance the magnetic activity unseen in natural materials. In the simple microstructure design, it is shown that in an array of conducting cylinders , with an applied external H 0 {\displaystyle H_{0}} field parallel to the cylinders, the effective permeability can be written as the following. (This model is very limited and the effective permeability cannot be less than zero or greater than one.) [ 5 ]
Where σ {\displaystyle \sigma } is the resistance of the cylinder surface per unit area, a is the spacing of the cylinders, ω {\displaystyle \omega } is the angular frequency, μ 0 {\displaystyle \mu _{0}} is the permeability of free space and r is the radius. Moreover, when gaps are introduced to a double cylinder design similar to the image above, we see that the gaps produce a capacitance. This capacitor and inductor microstructure design introduces a resonance that amplifies the magnetic effect. The new form of the effective permeability resembles a familiar response [ 12 ] known in plasmonic materials.
Where d is the spacing of the concentric conducting sheets [ clarification needed ] . The final design replaces the double concentric cylinders with a pair of flat concentric c-shaped sheets, placed on each side of a unit cell. The unit cells are stacked on top of each other by a length, l. The final result of the effective permeability can be seen below.
where c is the thickness of the c-shaped sheet and σ {\displaystyle \sigma } is the resistance of unit length of the sheets measured around the circumference. [ 5 ]
The split ring resonator and the metamaterial itself are composite materials. Each SRR has an individual tailored response to the electromagnetic field. However, the periodic construction of many SRR cells is such that the electromagnetic wave interacts as if these were homogeneous materials . This is similar to how light actually interacts with everyday materials; materials such as glass or lenses are made of atoms, an averaging or macroscopic effect is produced.
The SRR is designed to mimic the magnetic response of atoms [ clarification needed ] , only on a much larger scale. Also, as part of periodic composite structure, the SRR is designed to have a stronger magnetic coupling than is found in nature. The larger scale allows for more control over the magnetic response, while each unit is smaller than the radiated electromagnetic wave .
SRRs are much more active [ clarification needed ] than ferromagnetic materials found in nature. The pronounced magnetic response in such lightweight materials [ clarification needed ] demonstrates an advantage over heavier, naturally occurring materials. Each unit can be designed to have its own magnetic response. The response can be enhanced or lessened as desired. In addition, the overall effect reduces power requirements. [ 8 ] [ 13 ]
There are a variety of split-ring resonators and periodic structures: rod-split-rings, nested split-rings, single split rings, deformed split-rings, spiral split-rings, and extended S-structures. The variations of split ring resonators have achieved different results, including smaller and higher frequency structures. The research which involves some of these types are discussed throughout the article. [ 14 ]
To date (December 2009) the capability for desired results in the visible spectrum has not been achieved. However, in 2005 it was noted that, physically, a nested circular split-ring resonator must have an inner radius of 30 to 40 nanometers for success in the mid-range of the visible spectrum. [ 14 ] Microfabrication and nanofabrication techniques may utilize direct laser beam writing or electron beam lithography depending on the desired resolution. [ 14 ]
Split-ring resonators (SRR) are one of the most common elements used to fabricate metamaterials . [ 16 ] Split-ring resonators are non-magnetic materials, which initially were fabricated from circuit board material to create metamaterials. [ 17 ]
Looking at the image directly to the right, it can be seen that at first a single SRR looks like an object with two square perimeters, with each perimeter having a small section removed. This results in square "C" shapes on fiberglass printed circuit board material. [ 16 ] [ 17 ] In this type of configuration it is actually two concentric bands of non-magnetic conductor material. [ 16 ] There is one gap in each band placed 180° relative to each other. [ 16 ] The gap in each band gives it the distinctive "C" shape, rather than a totally circular or square shape. [ 18 ] [ 16 ] [ 17 ] Then multiple cells of this double band configuration are fabricated onto circuit board material by an etching technique and lined with copper wire strip arrays. [ 17 ] After processing, the boards are cut and assembled into an interlocking unit. [ 17 ] It is constructed into a periodic array with a large number of SRRs. [ 17 ]
There are now a number of different configurations that use the SRR nomenclature.
A periodic array of SRRs was used for the first demonstration of a negative index of refraction . [ 17 ] For this demonstration, square shaped SRRs , with the lined wire configurations, were fabricated into a periodic, arrayed, cell structure. [ 17 ] This is the substance of the metamaterial. [ 17 ] Then a metamaterial prism was cut from this material. [ 17 ] The prism experiment demonstrated a negative index of refraction for the first time in the year 2000; the paper about the demonstration was submitted to the journal Science on January 8, 2001, accepted on February 22, 2001 and published on April 6, 2001. [ 17 ]
Just before this prism experiment, Pendry et al. was able to demonstrate that a three-dimensional array of intersecting thin wires could be used to create negative values of ε. In a later demonstration, a periodic array of copper split-ring resonators could produce an effective negative μ. In 2000 Smith et al. were the first to successfully combine the two arrays and produce a so-called left-handed material , which has negative values of ε and μ for a band of frequencies in the GHz range. [ 17 ]
SRRs were first used to fabricate left-handed metamaterials for the microwave range, [ 17 ] and several years later for the terahertz range. [ 19 ] By 2007, experimental demonstration of this structure at microwave frequencies has been achieved by many groups. [ 20 ] In addition, SRRs have been used for research in acoustic metamaterials. [ 21 ] The arrayed SRRs and wires of the first left-handed metamaterial were melded into alternating layers. [ 22 ] This concept and methodology was then applied to (dielectric) materials with optical resonances producing negative effective permittivity for certain frequency intervals resulting in " photonic bandgap frequencies". [ 21 ] Another analysis showed left-handed materials to be fabricated from inhomogeneous constituents, which yet results in a macroscopically homogeneous material. [ 21 ] SRRs had been used to focus a signal from a point source, increasing the transmission distance for near field waves . [ 21 ] Furthermore, another analysis showed SRRs with a negative index of refraction capable of high-frequency magnetic response , which created an artificial magnetic device composed of non-magnetic materials (dielectric circuit board). [ 17 ] [ 21 ] [ 22 ]
The resonance phenomena that occurs in this system is essential to achieving the desired effects. [ 20 ]
SRRs also exhibit resonant electric response in addition to their resonant magnetic response. [ 22 ] The response, when combined with an array of identical wires, is averaged over the whole composite structure which results in effective values, including the refractive index. [ 23 ] The original logic behind SRRs specifically, and metamaterials generally was to create a structure, which imitates an arrayed atomic structure only on a much larger scale.
In research based in metamaterials, and specifically negative refractive index , there are different types of split-ring resonators. Of the examples mentioned below, most of them have a gap in each ring. In other words, with a double ring structure, each ring has a gap. [ 24 ]
There is the 1-D Split-Ring Structure with two square rings, one inside the other. One set of cited " unit cell " dimensions would be an outer square of 2.62 mm and an inner square of 0.25 mm. 1-D structures such as this are easier to fabricate compared with constructing a rigid 2-D structure. [ 24 ]
The Symmetrical -Ring Structure is another classic example. Described by the nomenclature these are two rectangular square D type configurations, exactly the same size, lying flat, side by side, in the unit cell . Also these are not concentric . One set of cited dimensions are 2 mm on the shorter side, and 3.12 mm on the longer side. The gaps in each ring face each other, in the unit cell. [ 24 ]
The Omega Structure , as the nomenclature describes, has an Ω-shaped ring structure. [ 25 ] There are two of these, standing vertical, side by side, instead of lying flat, in the unit cell. In 2005 these were considered to be a new type of metamaterial. One set of cited dimensions are annular parameters of R=1.4 mm and r=1 mm, and the straight edge is 3.33 mm. [ 24 ]
Another new metamaterial in 2005 was a coupled S-shaped structure. There are two vertical S-shaped structures, side by side, in a unit cell. There is no gap as in the ring structure; however, there is a space between the top and middle parts of the S and space between the middle part and bottom part of the S. Furthermore, it still has the properties of having an electric plasma frequency and a magnetic resonant frequency. [ 24 ] [ 26 ]
On May 1, 2000, research was published about an experiment which involved conducting wires placed symmetrically within each cell of a periodic split-ring resonator array . This effectively achieved negative permeability and permittivity for electromagnetic waves in the microwave regime. The concept was and still is used to build interacting elements smaller than the applied electromagnetic radiation. In addition, the spacing between the resonators is much smaller than the wavelength of the applied radiation. [ 27 ]
Additionally, the splits in the ring allow the SRR unit to achieve resonance at wavelengths much larger than the diameter of the ring. The unit is designed to generate a large capacitance, lower the resonant frequency, and concentrate the electric field. Combining units creates a design as a periodic medium. Furthermore, the multiple unit structure has strong magnetic coupling with low radiative losses. [ 27 ] Research has also covered variations in magnetic resonances for different SRR configurations. [ 28 ] [ 29 ] [ 30 ] Research has continued into terahertz radiations with SRRs [ 31 ] Other related work fashioned metamaterial configurations with fractals [ 25 ] and non-SRR structures. These can be constructed with materials such as periodic metallic crosses, or an ever-widening concentric ring structures known as Swiss rolls. [ 32 ] [ 33 ] [ 34 ] [ 35 ] Permeability for only the red wavelength at 780 nm has been analyzed and along with other related work. [ 36 ] [ 37 ] [ 38 ] | https://en.wikipedia.org/wiki/Split-ring_resonator |
A split in phylogenetics is a bipartition of a set of taxa , and the smallest unit of information in unrooted phylogenetic trees : each edge of an unrooted phylogenetic tree represents one split, and the tree can be efficiently reconstructed from its set of splits. Moreover, when given several trees, the splits occurring in more than half of these trees give rise to a consensus tree, and the splits occurring in a smaller fraction of the trees generally give rise to a consensus split network . Pairs of splits are compatible if any of the subsets defined by each split do not overlap. [ 2 ]
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Split_(phylogenetics) |
The split TEV technique [ 1 ] is a molecular method to monitor protein-protein interactions in living cells. It is based on the functional reconstitution of two previously inactive fragments derived from the NIa protease of the tobacco etch virus ( TEV protease ). These fragments, either an N-terminal (NTEV) or C-terminal part (CTEV), are fused to protein interaction partners of choice. Upon interaction of the two candidate proteins, the NTEV and CTEV fragments get into close proximity, regain proteolytic activity, and activate specific TEV reporters which indicate an occurred protein-protein interaction. | https://en.wikipedia.org/wiki/Split_TEV |
The split and pool (split-mix) synthesis is a method in combinatorial chemistry that can be used to prepare combinatorial compound libraries. It is a stepwise , highly efficient process realized in repeated cycles. The procedure makes it possible to prepare millions or even trillions of compounds as mixtures that can be used in drug research .
According to traditional methods, most organic compounds are synthesized one by one from building blocks coupling them together one after the other in a stepwise manner. Before 1982 nobody was even dreaming about making hundreds or thousands of compounds in a single process. Not speaking about millions or even trillions. So the productivity of the split and pool method invented by Prof. Á. Furka (Eötvös Loránd University Budapest Hungary), in 1982 seemed incredible at first sight. The method had been described it in a document notarized in the same year. The document is written in Hungarian and translated to English [ 1 ] Motivations that led to the invention are found in a 2002 paper [ 2 ] and the method was first published in international congresses in 1988 [ 3 ] then in print in 1991. [ 4 ]
The split and pool synthesis (S&P synthesis) differs from traditional synthetic methods. The important novelty is the use of compound mixtures in the process. This is the reason of its unprecedentedly high productivity. Using the method one single chemist can make more compounds in a week than all chemists produced in the whole history of chemistry.
The S&P synthesis is applied in a stepwise manner by repeating three operations in each step of the process:
The original method is based on the solid-phase synthesis of Merrifield [ 5 ] The procedure is illustrated in the figure by the flowing diagram showing of a two-cycle synthesis using the same three BBs in both cycles.
Choosing the solid phase method in the S&P synthesis is reasonable since otherwise removal of the by-products from the mixture of compounds would be very difficult.
The high efficiency is the most important feature of the method. In a multi step (n) synthesis using equal number of BBs (k) in every step the number of components in a forming combinatorial library (N) is:
N=k n
This means that the number of components increases exponentially with the number steps (cycles) while the number of the required couplings increases only linearly. If a different number of building BBs are used in the cycles (k1, k2, k3....kn) the number of the formed components is:
N=k 1 .k 2 .k 3 ...k n .
This feature of the procedure offers the possibility to synthesize a practically unlimited number of compounds. For example, if 1000 BBs are used in four cycles 1 trillion compounds are expected to form. The number of needed couplings is only 4000!
The explanation of the extraordinary efficiency is the use of mixtures in the synthetic steps. If in a traditional reaction one compound is coupled with one reactant and one new compound is formed. If a mixture of compounds containing n components is coupled with a single reactant the number of new compounds formed in the single coupling is n.
The difference between the traditional and the split and pool synthesis is convincingly shown by the number of coupling steps in the traditional and the split and pool synthesis of 3,2 million pentapeptides.
Conventional synthesis: 3,200,000x5=16,000,000 coupling steps cca 40,000 years
S&P synthesis: 20x5=100 coupling steps cca 5 days
It is possible to conduct the conventional synthesis rational way as is shown in the figure. In this case, the number of coupling cycles is:
20+400+8,000+160,000+3,200,000=3,368,420 cca 9,200 years
As often mentioned the split and pool method makes it possible to synthesize an unlimited number of compounds. In fact, the theoretical maximum number of components depends on the quantity of the library expressed in moles. If for example, 1 mol library is synthesized the maximum number of components is equal to the Avogadro number:
6,02214076·10 23
In such a library each component would be represented by a single molecule.
As far as the chemistry of the couplings makes it possible the components of the libraries form in nearly equal molar quantity. This is made possible by dividing of the mixtures into equal samples and by homogenization of the pooled samples by thoroughly mixing them. The equal molar quantity of components of the library is very important considering their applicability. The presence of compounds in unequal quantities may lead to difficulties in evaluation of the results in screening. The solid phase method makes it possible to use the reagents in excess to drive the reactions close to completion since the surplus can easily be removed by filtration.
In principle, the use of two mixtures in the S&P synthesis can lead to the same combinatorial library that forms in the usual S&P method. The differences in the reactivity of BBs however, bring about large differences in the concentrations of components, and the differences are expected to increase after each step. Although a considerable amount of labor could be saved by using the two mixtures approach when a high number of BBs are coupled in each position, it is advisable to stick to the normally used S&P procedure.
Formation of all structural variants that can be deduced from the BBs is an important feature of the S&P synthesis. Only the S&P method can achieve this in a single process. On the other hand, the presence of all possible structural varieties in a library assures that the library is a combinatorial one and is prepared by combinatorial synthesis.
The consequence of using a single BB in couplings is the formation of a single compound in each bead. The formation of OBOC libraries is an inherent property of the S&P synthesis. The reason is explained in the figure. The structure of the compound formed in a bead depends on the reaction vessels in which the bead happens to occur in the synthetic route.
It depends on the decision of the chemist to use the library in the tethered (OBOC) form or cleave down the compounds from the beads and use it as a solution.
The split and pool synthesis was first applied to prepare peptide libraries on solid support. The synthesis was realized in a home-made manual device shown in the figure. The device has a tube with 20 holes to which reaction vessels could be attached. One end of the tube is linked to a waste container and a water pump. Left shows loading and filtering, right coupling-shaking position.
In the early years of combinatorial chemistry, an automatic machine was constructed and commercialized at AdvancedChemTech (Louisville KY USA). All operations of the S&P synthesis are carried automatically under computer control. At present, the Titan 357 automatic synthesizer is available at aapptec (Louisville KY, USA). [ 6 ]
Although in the S&P synthesis a single compound forms on each bead its structure is not known. For this reason, encoding methods had been introduced to help to determine the identity of the compound contained in a selected bead. Encoding molecules are coupled to the beads in parallel with the coupling of the BBs. The structure of the encoding molecule has to be easier determined than that of the library member on the bead.
Ohlmeyer et al. published a binary encoding method. [ 7 ] They used mixtures of 18 tagging molecules that after cleaving them from the beads could be identified by Electron Capture Gas Chromatography.
Nikolajev et al. applied peptide sequences for encoding [ 8 ] Sarkar et al. described chiral oligomers of pentenoic amides (COPAs) that can be used to construct mass encoded OBOC libraries. [ 9 ] Kerr et al. introduced an innovative kind of encoding. [ 10 ] An orthogonally protected removable bifunctional linker was attached to the beads. One end of the linker was used to attach the non-natural BBs of the library while to the other end the encoding amino acid triplets were linked.
One of the earliest and very successful encoding methods was introduced by Brenner and Lerner [ 11 ] in 1992. They proposed to attach DNA oligomers to the beads for encoding their content.
The method was implemented by Nielsen, Brenner, and Janda [ 12 ] using the bifunctional linker of Kerr et al. to attach the encoding DNA oligomers. This made it possible to cleave down the compound with the DNA encoding oligomer attached to it.
Han et al. described a method that made it possible to keep the advantages of both the high efficiency of S&P synthesis and that of a homogeneous media in the chemical reactions. [ 13 ] In their method polyethyleneglycol (PEG) was used as soluble support in S&P synthesis of peptide libraries.
MeO-CH 2 -CH 2 -O-(CH 2 -CH 2 -O)n-CH 2 -CH 2 -OH
PEG proved suitable for this purpose since it is soluble in a wide variety of aqueous and organic solvents and its solubility provides homogeneous reaction conditions even when the attached molecule itself is insoluble in the reaction medium. Separation from the solution of the polymer and the synthesized compounds bound to it can be achieved by precipitation and filtration. The precipitation requires concentrating the reaction solutions then diluting with diethyl ether or tert-butyl methyl ether. Under carefully controlled precipitation conditions the polymer with the bound products precipitates in crystalline form and the unwanted reagents remain in solution.
In the solid phase, S&P synthesis a single compound forms on each bead, and as a consequence, the number of compounds can't exceed the number of beads. So, the theoretical maximum number of compounds depends on the quantity of the solid support and the size of the beads. On 1 g polystyrene resin, for example, a maximum of 2 million compounds can be synthesized if the diameter of the resin beads is 90 μm, and 2 billion can be made if the bead size is 10 μm. In practice, the solid support is used in excess (often tenfold) to be sure that all expected components are formed.
The above limitation is completely removed if the solid support is omitted and the synthesis is carried out in solution. In this case, there is no upper limit concerning the number of components of the library. Both the number of components and the quantity of the library can be freely decided based only on practical considerations.
An important modification was introduced in the synthesis of DNA encoded combinatorial libraries by Harbury and Halpin. [ 14 ] The solid support in their case is replaced by the encoding DNA oligomers. This makes it possible to synthesize libraries containing even trillions of components and screen them using affinity binding methods.
A different way of carrying out solution-phase S&P synthesis is applying scavenger resins to remove the byproducts. Scavenger resins are polymers having functional groups that make it possible to react with and bind components of the excess of reagents then filtered them out from the reaction mixture [ 15 ] Two examples: a resin containing primary amino groups can remove the excess of acyl chlorides from reaction mixtures while an acyl chloride resin removes amines.
A fluorous technology was described by Curran [ 16 ] The fluorous synthesis employs functionalized perfluoroalkyl (Rf) groups like 4,4,5,5,6,6,7,7,8,8,9,9,9-Tridecafluorononyl {CF 3 (CF 2 ) 4 CF 2 CH 2 CH 2 -} group attached to substrates or reagents. The Rf groups make it possible to remove either the product or the reagents from the reaction mixture. At the end of the procedure, the Rf groups attached to the substrate can be removed from the product. By attaching Rf groups to the substrate the synthesis can be carried out in solution and the product can be separated from the reaction mixture by liquid extraction using a fluorous solvent like perfluoromethylcyclohexane or perfluorohexane. It can be seen that the function of the Rf groups in the synthesis is similar to that of the solid or soluble support. If the Rf tag is attached to reagent its excess can be removed from the reaction mixture by extraction.
Polymer supported reagents are also used in S&P synthesis. [ 17 ]
One of the best examples of the special features caused by DNA encoding is the synthesis of the self-assembling library introduced by Mlecco et al. [ 18 ] First, two sublibraries are synthesized. In one of the sublibraries BBs are attached to the 5’ end of an oligonucleotide containing a dimerization domain followed by the codes of the BBs. In the other sublibrary the BBs are attached to the 3’ end of the oligonucleotides also containing a dimerization domain and the codes of another set of BBs. The two sublibraries are mixed in equimolar quantities, heated to 70 °C then allowed to cool to room temperature, heterodimerize and form the self-assembling combinatorial library. One member of such two pharmacophore library is shown in the figure. In affinity screening, the two BBs of the pharmacophore may interact with the two adjacent binding sites of the target protein.
In the synthesis of DNA templated combinatorial libraries, the ability of the DNA double helix to direct region-specific chemical reactions is harnessed by Gartner et al. [ 19 ] [ 20 ] The DNA- linked reagents are kept in close proximity. This is equivalent to the virtual increase of local concentration that is nearly constant within a distance of 30 nucleotides. The proximity effect helps reactions to proceed.
Two libraries are synthesized. A template library containing at one end one of the BBs and its code followed by two annealing regions for the codes of the BBs of the two reagent libraries. Each of the two reagent libraries contains a coding oligonucleotide linked with cleavable bonds to the reagent (BB) capable of forming a bond with the already linked BB taking advantage of the proximity effect. The synthesis is realized in two steps as shown in the figure. Each step has three operations: mixing, annealing, coupling-cleaving.
The yoctoreactor method introduced by Hansen et al. [ 21 ] is based on the geometry and stability of a three-dimensional DNA structure that creates a yoctoliter (10 −24 L) size chemical reactor in which proximity of BBs brings about reactions among them. The DNA oligomers comprise the DNA-barcode for the attached BBs and form the structural elements of the reactor. One kind of yoctoreactor format is shown in the figure.
Harbury and Halpin developed DNA template libraries that direct like genes the synthesis of DNA encoded organic libraries. [ 22 ] [ 23 ] The members of the template combinatorial library contain the codes of all BBs and their order of couplings. The figure shows one member of a simple ssDNA template library (A) containing the codes of three BBs (2, 4, 6) that planned to be successively attached. The coding regions are separated by the same non-coding regions (1, 3, 5, 7) in all members.
The sequence directed procedure uses a series of columns of resin beads each coated with the anticodon of one of the BBs (B). When the template library is transferred to an anticodon column the proper template member is captured by hybridization then is coupled with the appropriate BB. After finished with all anticodon columns of a coupling position (CP) the libraries are eluted from the beads of the anticodon columns mixed and the mentioned operations are repeated with the series of anticodon columns of the next CP. In figure, C shows one member of the template library captured by the “yellow” second CP anticodon library. The template contains the “red” BB already coupled in CP1 and the “yellow” BB attached after its capture. The final library contains all of the synthesized organic compounds attached to their encoding DNA oligomers.
One of the most forward-looking method commonly used for DNA-encoding is applied in the synthesis of single-pharmacophore libraries. [ 24 ] As the figure shows the library is built repeating the usual cycles of S&P synthesis, The second operation of the cycle is modified: in addition to coupling with the BBs the encoding DNA oligomer is elongated by attaching the code of the BB by ligation.
Modifications had been developed enabling the split and pool synthesis to produce known compounds in larger quantities than the content of a bead of solid support and retain the high efficiency of the original method.
As published by Moran et al. [ 25 ] and Nicolau et al. [ 26 ] the resin normally used in the solid phase synthesis was enclosed into permeable capsules including a radiofrequency label recording the BBs in order of their coupling. Both manual and automatic machine was constructed to sort the capsules into the appropriate reaction vessels.
A different kind of labeled macroscopic solid support unit was introduced by Xiao et al. [ 27 ] The supports are 1x1 cm polystyrene grafted square plates. The medium carrying the code is a 3x3 mm ceramic plate in the center of the synthesis support The code is etched into the ceramic support by a CO 2 laser in the form of a two-dimensional bar code that can be read by a special scanner.
The String Synthesis introduced by Furka et al. [ 28 ] uses stringed macroscopic solid support units (crowns) and the units are identified by their position occupied on the string. One string is assigned for every building block in the synthesis. In the coupling stage, the string is in the proper reaction vessel. The content of the strings coming out from a synthetic step must be redistributed into the strings of the next step. The units are not pooled. The redistribution demonstrated in the figure follows the combinatorial distribution rule: all products formed in a synthetic step are equally divided among all reaction vessels of the next synthetic step. Different distribution formats can be followed that allows the identification the content of each crown depending on the position on the new string and the destination reaction vessel of the string. [ 29 ]
The stringed crowns and the trays used in manual sorting are shown in the figure. The destination tray is moved step by step in the direction of the arrow. The crowns are transferred in groups from the slots of the source tray into the all opposite slots of the destination tray. The transfers are directed by computer and the products are identified by the positions of the crowns occupied on the final strings.
A fast automatic sorter machine had also been described. [ 30 ] The sorter is outlined in the figure. It has two sets of aligned tubes. The lower ones are step by step moving in the direction showed by the arrow and the coin-like units are dropped from the upper source tubes into the lower destination ones. The tubes may serve as reaction vessels too.
A software had also been developed that can direct sorting if not a full combinatorial library is synthesized only a set of its components are prepared that are picked out from the full library. [ 31 ] | https://en.wikipedia.org/wiki/Split_and_pool_synthesis |
The term split exact sequence is used in two different ways by different people. Some people mean a short exact sequence that right-splits (thus corresponding to a semidirect product) and some people mean a short exact sequence that left-splits (which implies it right-splits, and corresponds to a direct product). This article takes the latter approach, but both are in common use. When reading a book or paper, it is important to note precisely which of the two meanings is in use.
In mathematics , a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way.
A short exact sequence of abelian groups or of modules over a fixed ring , or more generally of objects in an abelian category
is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones:
The requirement that the sequence is isomorphic means that there is an isomorphism f : B → A ⊕ C {\displaystyle f:B\to A\oplus C} such that the composite f ∘ a {\displaystyle f\circ a} is the natural inclusion i : A → A ⊕ C {\displaystyle i:A\to A\oplus C} and such that the composite p ∘ f {\displaystyle p\circ f} equals b . This can be summarized by a commutative diagram as:
The splitting lemma provides further equivalent characterizations of split exact sequences.
A trivial example of a split short exact sequence is
where M 1 , M 2 {\displaystyle M_{1},M_{2}} are R -modules, q {\displaystyle q} is the canonical injection and p {\displaystyle p} is the canonical projection.
Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis .
The exact sequence 0 → Z → 2 Z → Z / 2 Z → 0 {\displaystyle 0\to \mathbf {Z} \mathrel {\stackrel {2}{\to }} \mathbf {Z} \to \mathbf {Z} /2\mathbf {Z} \to 0} (where the first map is multiplication by 2) is not split exact.
Pure exact sequences can be characterized as the filtered colimits of split exact sequences. [ 1 ] | https://en.wikipedia.org/wiki/Split_exact_sequence |
The split gene theory is a theory of the origin of introns , long non-coding sequences in eukaryotic genes between the exons . [ 1 ] [ 2 ] [ 3 ] The theory holds that the randomness of primordial DNA sequences would only permit small (< 600bp) open reading frames (ORFs), and that important intron structures and regulatory sequences are derived from stop codons . In this introns-first framework, the spliceosomal machinery and the nucleus evolved due to the necessity to join these ORFs (now "exons") into larger proteins, and that intronless bacterial genes are less ancestral than the split eukaryotic genes. The theory originated with Periannan Senapathy .
The theory provides solutions to key questions concerning the split gene architecture, including split eukaryotic genes, exons, introns, splice junctions, and branch points, based on the origin of split genes from random genetic sequences. It also provides possible solutions to the origin of the spliceosomal machinery, the nuclear boundary and the eukaryotic cell.
This theory led to the Shapiro–Senapathy algorithm , which provides the methodology for detecting the splice sites, exons and split genes in eukaryotic DNA, and which is the main method for detecting splice site mutations in genes that cause hundreds of diseases.
Split gene theory requires a separate origin of all eukaryotic species. It also requires that the simpler prokaryotes evolved from eukaryotes. This completely contradicts the scientific consensus about the formation of eukaryotic cells by endosymbiosis of bacteria. In 1994, Senapathy wrote a book about this aspect of his theory - The Independent Birth of Organisms. It proposed that all eukaryotic genomes were formed separately in a primordial pool. Dutch biologist Gert Korthoff criticized the theory by posing various problems that cannot be explained by a theory of independent origins. He pointed out that various eukaryotes need nurturing and called this the 'boot problem', in that even the initial eukaryote needed parental care. Korthoff notes that a large fraction of eukaryotes are parasites. Senapathy's theory would require a coincidence to explain their existence. [ 4 ] [ 5 ] Senapathy's theory cannot explain the strong evidence for common descent ( homology , universal genetic code, embryology , fossil record .) [ 6 ]
Genes of all organisms, except bacteria, consist of short protein-coding regions ( exons ) interrupted by long sequences ( introns ). [ 1 ] [ 2 ] When a gene is expressed, its DNA sequence is copied into a “primary RNA” sequence by the enzyme RNA polymerase . Then the “spliceosome” machinery physically removes the introns from the RNA copy of the gene by the process of splicing, leaving only a contiguously connected series of exons, which becomes messenger RNA (mRNA). This mRNA is now read by the ribosome , which produces the encoded protein. Thus, although introns are not physically removed from a gene, a gene's sequence is read as if introns were not present.
Exons are usually short, with an average length of about 120 bases (e.g. in human genes). Intron lengths vary widely from 10 to 500,000, but exon lengths have an upper bound of about 600 bases in most eukaryotes. Because exons code for protein sequences, they are important for the cell, yet constitute only ~2% of the sequences. Introns, in contrast, constitute 98% of the sequences but seem to have few crucial functions, except for enhancer sequences and developmental regulators in rare instances. [ 7 ] [ 8 ]
Until Philip Sharp [ 9 ] [ 10 ] and Richard Roberts [ 11 ] discovered introns [ 12 ] within eukaryotic genes in 1977, it was believed that the coding sequence of all genes was always in one single stretch, bounded by a single long ORF. The discovery of introns was a profound surprise, which instantly brought up the questions of how, why and when the introns came into the eukaryotic genes.
It soon became apparent that a typical eukaryotic gene was interrupted at many locations by introns, dividing the coding sequence into many short exons. Also surprising was that the introns were long, as long as hundreds of thousands of bases. These findings prompted the questions of why many introns occur within a gene (for example, ~312 introns occur in the human gene TTN), why they are long, and why exons are short.
introns in the gene
It was also discovered that the spliceosome machinery was large and complex with ~300 proteins and several SnRNA molecules. The questions extended to the origin of the spliceosome. Soon after the discovery of introns, it became apparent that the junctions between exons and introns on either side exhibited specific sequences that directed the spliceosome machinery to the exact base position for splicing. How and why these splice junction signals came into being was another important question.
The discovery of introns and the split gene architecture of the eukaryotic genes started a new era of eukaryotic biology. The question of why eukaryotic genes had fragmented genes prompted speculation and discussion almost immediately.
Ford Doolittle published a paper in 1978 in which he stated that most molecular biologists assumed that the eukaryotic genome arose from a ‘simpler’ and more ‘primitive’ prokaryotic genome rather like that of Escherichia coli . [ 13 ] However, this type of evolution would require that introns be introduced into the coding sequences of bacterial genes. Regarding this requirement, Doolittle said, “It is extraordinarily difficult to imagine how informationally irrelevant sequences could be introduced into pre-existing structural genes without deleterious effects.” He stated “I would like to argue that the eukaryotic genome, at least in that aspect of its structure manifested as ‘genes in pieces’ is in fact the primitive original form.”
James Darnell expressed similar views in 1978. He stated, “The differences in the biochemistry of messenger RNA formation in eukaryotes compared to prokaryotes are so profound as to suggest that sequential prokaryotic to eukaryotic cell evolution seems unlikely. The recently discovered non-contiguous sequences in eukaryotic DNA that encode messenger RNA may reflect an ancient, rather than a new, distribution of information in DNA and that eukaryotes evolved independently of prokaryotes.” [ 14 ]
However, in an apparent attempt to reconcile with the idea that RNA preceded DNA in evolution, and with the concept of the three evolutionary lineages of archea , bacteria and eukarya, both Doolittle and Darnell deviated from their original speculation in a joint paper in 1985. [ 15 ] They suggested that the ancestor of all three groups of organisms, the ‘ progenote ,’ had a genes-in-pieces structure, from which all three lineages evolved. They speculated that the precellular stage had primitive RNA genes which had introns, which were reverse transcribed into DNA and formed the progenote. Bacteria and archea evolved from the progenote by losing introns, and ‘urkaryote’ evolved from it by retaining introns. Later, the eukaryote evolved from the urkaryote by evolving a nucleus and absorbing mitochondria from bacteria. Multicellular organisms then evolved from the eukaryote.
These authors predicted that the distinctions between the prokaryote and the eukaryote were so profound that the prokaryote to eukaryote evolution was not tenable, and had different origins. However, other than the speculations that the precellular RNA genes must have had introns, they did not address the key questions of intron origin. No explanations described why exons were short and introns were long, how the splice junctions originated, what the structure and sequence of the splice junctions meant, and why eukaryote genomes were large.
Around the same time that Doolittle and Darnell suggested that introns in eukaryotic genes could be ancient, Colin Blake [ 16 ] and Walter Gilbert [ 17 ] [ 18 ] published their views on intron origins independently. In their view, introns originated as spacer sequences that enabled convenient recombination and shuffling of exons that encoded distinct functional domains in order to evolve new genes. Thus, new genes were assembled from exon modules that coded for functional domains, folding regions, or structural elements from preexisting genes in the genome of an ancestral organism, thereby evolving genes with new functions. They did not specify how exons or introns originated. In addition, even after many years, extensive analysis of thousands of proteins and genes showed that only extremely rarely do genes exhibit the supposed exon shuffling phenomenon. [ 19 ] [ 20 ] Furthermore, molecular biologists questioned the exon shuffling proposal, from a purely evolutionary view for both methodological and conceptual reasons, and, in the long run, this theory did not survive.
Around the time introns were discovered, Senapathy was asking how genes themselves could have originated. He surmised that for any gene to come into being, genetic sequences (RNA or DNA) must have been present in the prebiotic environment. A basic question he asked was how protein-coding sequences could have originated from primordial DNA sequences at the origin of the first cells.
To answer this, he made two basic assumptions:
He also surmised that codons must have been established prior to the origin of the first genes. If primordial DNA did contain random nucleotide sequences, he asked: Was there an upper limit in coding-sequence lengths, and, if so, did this limit play a crucial role in the formation of the structural features of genes at the origin of genes?
His logic was the following. The average length of proteins in living organisms, including the eukaryotic and bacterial organisms, was ~400 amino acids. However, much longer proteins existed, even longer than 10,000-30,000 amino acids in both eukaryotes and bacteria. [ 21 ] Thus, the coding sequence of thousands of bases existed in a single stretch in bacterial genes. In contrast, the coding sequence of eukaryotes existed only in short segments of exons of ~120 bases regardless of the length of the protein. If the coding sequence ORF lengths in random DNA sequences were as long as those in bacterial organisms, then long, contiguous coding genes were possible in random DNA. This was not known, as the distribution of ORF lengths in a random DNA sequence had never been studied.
As random DNA sequences could be generated in the computer, Senapathy thought that he could ask these questions and conduct his experiments in silico . Furthermore, when he began studying this question, sufficient DNA and protein sequence information existed in the National Biomedical Research Foundation (NBRF) database in the early 1980s.
Senapathy analyzed the distribution of the ORF lengths in computer-generated random DNA sequences first. Surprisingly, this study revealed that about 200 codons (600 bases) was the upper limit in ORF lengths. The shortest ORF (zero base in length) was the most frequent. At increasing lengths of ORFs, their frequency decreased logarithmically, approaching zero at about 600 bases. When the probability of ORF lengths in a random sequence was plotted, it revealed that the probability of increasing lengths of ORFs decreased exponentially and tailed off at a maximum of about 600 bases. From this “negative exponential” distribution of ORF lengths, it was found that most of ORFs were far shorter than the maximum.
This finding was surprising because the coding sequence for the average protein length of 400 AAs (with ~1,200 bases of coding sequence) and longer proteins of thousands of AAs (requiring >10,000 bases of coding sequence) would not occur at a stretch in a random sequence. If this was true, a typical gene with a contiguous coding sequence could not originate in a random sequence. Thus, the only possible way that any gene could originate from a random sequence was to split the coding sequence into shorter segments and select these segments from short ORFs available in the random sequence, rather than to increase the ORF length by eliminating consecutive stop codons. This process of choosing short segments of coding sequences from the available ORFs to make a long ORF would lead to a split structure.
If this hypothesis was true, eukaryotic DNA sequences should reflect it. When Senapathy plotted the distribution of ORF lengths in eukaryotic DNA sequences, the plot was remarkably similar to that from random DNA sequences. This plot was also a negative exponential distribution that tailed off at a maximum of about 600 bases, as with eukaryotic genes, [ 1 ] [ 22 ] [ 3 ] which coincided exactly with the maximum length of ORFs observed in both random DNA and eukaryotic DNA sequences.
The split genes thus originated from random DNA sequences by choosing the best of the short coding segments (exons) and splicing them. The intervening intron sequences were left-over vestiges of the random sequences, and thus were earmarked to be removed by the spliceosome. These findings indicated that split genes could have originated from random DNA sequences with exons and introns as they appear in today's eukaryotic organisms. Nobel Laureate Marshall Nirenberg , who deciphered the codons, stated that these findings strongly showed that the split gene theory for the origin of introns and the split structure of genes must be valid. [ 1 ] [ 23 ]
Blake proposed the Gilbert-Blake hypothesis in 1979 for the origin of introns and stated that Senapathy's split gene theory comprehensively explained the origin of the split gene structure. In addition, he stated that it explained several key questions including the origin of the splicing mechanism: [ 16 ]
Recent work by Senapathy, when applied to RNA, comprehensively explains the origin of the segregated form of RNA into coding and non-coding regions. It also suggests why a splicing mechanism was developed at the start of primordial evolution. He found that the distribution of reading frame lengths in a random nucleotide sequence corresponded exactly to that for the observed distribution of eukaryotic exon sizes. These were delimited by regions containing stop signals, the messages to terminate construction of the polypeptide chain, and were thus non-coding regions or introns. The presence of a random sequence was therefore sufficient to create in the primordial ancestor the segregated form of RNA observed in the eukaryotic gene structure. Moreover, the random distribution also displays a cutoff at 600 nucleotides, which suggests that the maximum size for an early polypeptide was 200 residues, again as observed in the maximum size of the eukaryotic exon. Thus, in response to evolutionary pressures to create larger and more complex genes, the RNA fragments were joined together by a splicing mechanism that removed the introns. Hence, the early existence of both introns and RNA splicing in eukaryotes appears to be very likely from a simple statistical basis. These results also agree with the linear relationship found between the number of exons in the gene for a particular protein and the length of the polypeptide chain.”
Under the split gene theory, an exon is defined by an ORF. It requires a mechanism to recognize an ORF to have originated. As an ORF is defined by a contiguous coding sequence bounded by stop codons, these stop codon ends had to be recognized by the exon-intron gene recognition system. This system could have defined the exons by the presence of a stop codon at the ends of ORFs, which should be included within the ends of the introns and eliminated by the splicing process. Thus, the introns should contain a stop codon at their ends, which would be part of the splice junction sequences.
If this hypothesis was true, the split genes of today's living organisms should contain stop codons exactly at the ends of introns. When Senapathy tested this hypothesis in the splice junctions of eukaryotic genes, he found that the vast majority of splice junctions did contain a stop codon at the end of each intron, outside of the exons. In fact, these stop codons were found to form the “canonical” GT:AG splicing sequence, with the three stop codons occurring as part of the strong consensus signals. Thus, the basic split gene theory for the origin of introns and the split gene structure led to the understanding that the splice junctions originated from the stop codons. [ 2 ]
Sequence data for only about 1,000 exon-intron junctions were available when Senapathy thought about this question. He took the data for 1,030 splice junction sequences (donors and acceptors) and counted the codons occurring at
each of the 7- base positions in the donor signal sequence [CAG:GTGAGT] and each of the possible 2-base positions in the acceptor signal [CAG:G] from the GenBank database. He found that the stop codons occurred at high frequency only at the 5th base position in the donor signal and the first base position in the acceptor signal. These positions are the* start of the intron (in fact, one base after the start) and at the end of the intron, as Senapathy had predicted. The codon counts at only these positions are shown. Even when the codons at these positions were not stop
codons, 70% of them began with the first two bases of the stop codons TA and TG [TAT = 75; TAC = 59; TGT = 70].
All three stop codons (TGA, TAA and TAG) were found after one base (G) at the start of introns. These stop codons are shown in the consensus canonical donor splice junction as AG:GT(A/G)GGT, wherein the TAA and TGA are the stop codons, and the additional TAG is also present at this position. Besides the codon CAG, only TAG, which is a stop codon, was found at the ends of introns. The canonical acceptor splice junction is shown as (C/T)AG:GT, in which TAG is the stop codon. These consensus sequences clearly show the presence of the stop codons at the ends of introns bordering the exons in all eukaryotic genes, thus providing a strong corroboration for the split gene theory. Nirenberg again stated that these observations fully supported the split gene theory for the origin of splice junction sequences from stop codons. [ 2 ] [ 24 ]
Soon after the discovery of introns by Philip Sharp and Richard Roberts , it became known that mutations within splice junctions could lead to diseases. Senapathy showed that mutations in the stop codon bases (canonical bases) caused more diseases than the mutations in non-canonical bases. [ 1 ]
An intermediate stage in the process of eukaryotic RNA splicing is the formation of a lariat structure. It is anchored at an adenosine residue in intron between 10 and 50 nucleotides upstream of the 3' splice site. A short conserved sequence (the branch point sequence) functions as the recognition signal for the site of lariat formation. During the splicing process, this conserved sequence towards the end of the intron forms a lariat structure with the beginning of the intron. [ 25 ] The final step of the splicing process occurs when the two exons are joined and the intron is released as a lariat RNA. [ 26 ]
Several investigators found the branch point sequences in different organisms [ 25 ] including yeast, human, fruit fly, rat, and plants. Senapathy found that, in all of these sequences, the codon ending at the branch point adenosine is consistently a stop codon. What is interesting is that two of the three stop codons (TAA and TGA) occur almost all of the time at this position.
C TAA T
C TGA T
C TAA C
CTCAC
Lariat (branch point) sequences have been identified from many different
organisms. These sequences consistently show that the codon ending in
the branching adenosine is a stop codon, either TAA or TGA, which are shown in red.
These findings led Senapathy to propose that the branch point signal originated from stop codons. The finding that two different stop codons (TAA and TGA) occur within the lariat signal with the branching point as the third base of the stop codons corroborates this proposal. As the branching point of the lariat occurs at the last adenine of the stop codon, it is possible that the spliceosome machinery that originated for the elimination of the stop codons from the primary RNA sequence created an auxiliary stop-codon sequence signal as the lariat sequence to aid its splicing function. [ 2 ]
The small nuclear U2 RNA found in splicing complexes is thought to aid splicing by interacting with the lariat sequence. [ 27 ] Complementary sequences for both the lariat sequence and the acceptor signal are present in a segment of only 15 nucleotides in U2 RNA. Further, the U1 RNA has been proposed to function as a guide in splicing to identify the precise donor splice junction by complementary base-pairing. The conserved regions of the U1 RNA thus include sequences complementary to the stop codons. These observations enabled Senapathy to predict that stop codons had operated in the origin of not only the splice-junction signals and the lariat signal, but also some small nuclear RNAs.
Senapathy proposed that the gene-expression regulatory sequences (promoter and poly-A addition site sequences) also could have originated from stop codons. A conserved sequence, AATAAA, exists in almost every gene a short distance downstream from the end of the protein-coding message and serves as a signal for the addition of poly(A) in the mRNA copy of the gene. [ 28 ] This poly(A) sequence signal contains a stop codon, TAA. A sequence shortly downstream from this signal, thought to be part of the complete poly(A) signal, also contains the TAG and TGA stop codons.
Eukaryotic RNA-polymerase-II-dependent promoters can contain a TATA box (consensus sequence TATAAA), which contains the stop codon TAA. Bacterial promoter elements at ~10 bases exhibits a TATA box with a consensus of TATAAT (which contains the stop codon TAA), and at -35 bases exhibits a consensus of TTGACA (containing the stop codon TGA). Thus, the evolution of the whole RNA processing mechanism seems to have been influenced by the too-frequent occurrence of stop codons, thus making the stop codons the focal points for RNA processing.
CAG:G TGA GT
C TAA C
Senapathy discovered that stop codons occur as key parts in every genetic element in eukaryotic genes. The table and figure show that the key parts of the core promoter elements, the lariat signal, the donor and acceptor splice signals, and the poly-A addition signal consist of one or more stop codons. This finding corroborates the split gene theory's claim that the underlying reason for the complete split gene paradigm is the origin of split genes from random DNA sequences, wherein random distribution of an extremely high frequency of stop codons were used by nature to define these genetic elements.
Research based on the split gene theory sheds light on other basic questions of exons and introns. The exons of eukaryotes are generally short (human exons average ~120 bases, and can be as short as 10 bases) and introns are usually long (average of ~3,000 bases, and can be several hundred thousands bases long), for example genes RBFOX1, CNTNAP2, PTPRD and DLG2. Senapathy provided a plausible answer to these questions, the only explanation to date. If eukaryotic genes originated from random DNA sequences, they have to match the lengths of ORFs from random sequences, and possibly should be around 100 bases (close to the median length of ORFs in random sequence). The genome sequences of living organisms exhibit exactly the same average lengths of 120 bases for exons, and the longest exons of 600 bases (with few exceptions), which is the same length as that of the longest random ORFs. [ 1 ] [ 2 ] [ 3 ] [ 22 ]
If split genes originated in random DNA sequences, then introns would be long for several reasons. The stop codons occur in clusters leading to numerous consecutive short ORFs: longer ORFs that could be defined as exons would be rarer. Furthermore, the best of the coding sequence parameters for functional proteins would be chosen from the long ORFs in random sequence, which may occur rarely. In addition, the combination of donor and acceptor splice junction sequences within short lengths of coding sequence segments that would define exon boundaries would occur rarely in a random sequence. These combined reasons would make introns long compared to exons.
This work also explains why genomes such as the human genome have billions of bases, and why only a small fraction (~2%) codes for proteins and other regulatory elements. [ 29 ] [ 30 ] If split genes originated from random primordial DNA sequences, they would contain a significant amount of DNA that represented by introns. Furthermore, a genome assembled from random DNA containing split genes would also include intergenic random DNA. Thus, genomes that originated from random DNA sequences had to be large, regardless of the complexity of the organism.
The observation that several organisms such as the onion (~16 billion bases [ 31 ] ) and salamander (~32 billion bases [ 32 ] ) have much larger genomes than humans (~3 billion bases [ 33 ] [ 34 ] ) while the organisms are no more complex than humans comports with the theory. Furthermore, the fact that several organisms with smaller genomes have a similar number of genes as human, such as C. elegans (genome size ~100 million bases, ~19,000 genes) [ 35 ] and Arabidopsis thaliana (genome size ~125 million bases, ~25,000 genes), [ 36 ] supports the theory. The theory predicts that the introns in the split genes in these genomes could be the “reduced” (or deleted) form compared to larger genes with long introns, thus leading to reduced genomes. [ 1 ] [ 22 ] In fact, researchers have recently proposed that these smaller genomes are actually reduced genomes. [ 37 ]
Senapathy addressed the origin of the spliceosomal machinery that edits out the introns from RNA transcripts. If the split genes had originated from random DNA, then the introns would have become an unnecessary but integral part of eukaryotic genes along with the splice junctions. The spliceosomal machinery would be required to remove them and to enable the short exons to be linearly spliced together as a contiguously coding mRNA that can be translated into a complete protein. Thus, the split gene theory argues that spliceosomal machinery exists to remove the unnecessary introns. [ 1 ] [ 2 ]
Blake states, “Work by Senapathy, when applied to RNA, comprehensively explains the origin of the segregated form of RNA into coding and noncoding regions. It also suggests why a splicing mechanism was developed at the start of primordial evolution.” [ 16 ]
Senapathy proposed a plausible mechanistic and functional rationale why the eukaryotic nucleus originated, a major question in biology. [ 1 ] [ 2 ] If the transcripts of the split genes and the spliced mRNAs were present in a cell without a nucleus, the ribosomes would try to bind to both the un-spliced primary RNA transcript and the spliced mRNA, which would result in chaos. A boundary that separates the RNA splicing process from the mRNA translation avoids this problem. The nuclear boundary provides a clear separation of the primary RNA splicing and the mRNA translation.
These investigations thus led to the possibility that primordial DNA with essentially random sequence gave rise to the complex structure of the split genes with exons, introns and splice junctions. Cells that harbored split genes had to be complex with a nuclear cytoplasmic boundary, and must have a spliceosomal machinery. Thus, it was possible that the earliest cell was complex and eukaryotic. [ 1 ] [ 2 ] [ 3 ] [ 22 ] Surprisingly, findings from extensive comparative genomics research from several organisms since 2007 overwhelmingly show that the earliest organisms could have been highly complex and eukaryotic, and could have contained complex proteins, [ 38 ] [ 39 ] [ 40 ] [ 41 ] [ 42 ] [ 43 ] [ 44 ] as predicted by Senapathy's theory.
The spliceosome is a highly complex mechanism, containing ~200 proteins and several SnRNPs . Collins and Penny stated, “We begin with the hypothesis that ... the spliceosome has increased in complexity throughout eukaryotic evolution. However, examination of the distribution of spliceosomal components indicates that not only was a spliceosome present in the eukaryotic ancestor but it also contained most of the key components found in today's eukaryotes. ... the last common ancestor of extant eukaryotes appears to show much of the molecular complexity seen today.” This suggests that the earliest eukaryotic organisms were complex and contained sophisticated genes and proteins. [ 45 ]
Genes with uninterrupted coding sequences that are thousands of bases long - up to 90,000 bases - that occur in many bacterial organisms [ 21 ] were practically impossible to have occurred. However, the bacterial genes could have originated from split genes by losing introns, the only proposed way to arrive at long coding sequences. It is also a better [ clarification needed ] way than by increasing the lengths of ORFs from short random ORFs to long ORFs by specifically removing the stop codons by mutation. [ 1 ] [ 2 ] [ 3 ]
According to the split gene theory, this process of intron loss could have happened from prebiotic random DNA. These contiguously coding genes could be tightly organized in the bacterial genomes without any introns and be more streamlined. According to Senapathy, the nuclear boundary that was required for a cell containing split genes would not be required for a cell containing only uninterrupted genes. Thus, the bacterial cells did not develop a nucleus. Based on split gene theory, the eukaryotic genomes and bacterial genomes could have independently originated from the split genes in primordial random DNA sequences.
Senapathy developed algorithms to detect donor and acceptor splice sites, exons and a complete split gene in a genomic sequence. He developed the position weight matrix (PWM) method based on the frequency of the four bases at the consensus sequences of the donor and acceptor in different organisms to identify the splice sites in a given sequence. Furthermore, he formulated the first algorithm to find the exons based on the requirement of exons to contain a donor sequence (at the 5’ end) and an acceptor sequence (at the 3’ end), and an ORF in which the exon should occur, and another algorithm to find a complete split gene. These algorithms are collectively known as the Shapiro-Senapathy algorithm (S&S). [ 46 ] [ 47 ]
This algorithm aids in the identification of splicing mutations that cause disease and adverse drug reactions. [ 46 ] [ 47 ] Scientists used the algorithm to identify mutations and genes that cause cancers, inherited disorders, immune deficiency diseases and neurological disorders. It is increasingly used in clinical practice and research to find mutations in known disease-causing genes in patients and to discover novel genes that are causal of different diseases. Furthermore, it is used in defining the cryptic splice sites and deducing the mechanisms by which mutations can affect normal splicing and lead to different diseases. It is also employed in basic research.
Findings based on S&S have impacted major questions in eukaryotic biology and in human medicine. [ 48 ]
The split gene theory implies that structural features of split genes predicted from computer-simulated random sequences occur in eukaryotic split genes. This is borne out in most known split genes. The sequences exhibit a nearly perfect negative exponential distribution of ORF lengths. [ 1 ] [ 2 ] [ 22 ] [ 3 ] With rare exceptions, eukaryotic gene exons fall within the predicted 600 base maximum.
The theory correctly predicts that exons are delimited by stop codons, especially at the 3’ ends of exons. Actually they are precisely delimited more strongly at the 3’ ends of exons and less strongly at the 5’ ends in most known genes, as predicted. [ 1 ] [ 2 ] [ 22 ] [ 3 ] These stop codons are the most important functional parts of both splice junctions. The theory thus provides an explanation for the “conserved” splice junctions at the ends of exons and for the loss of these stop codons along with introns when they are spliced out. The theory correctly predicts that splice junctions are randomly distributed in eukaryotic DNA sequences. [ 3 ] [ 25 ] [ 46 ] [ 47 ] The theory correctly predicts that splice junctions present in transfer RNA genes and ribosomal RNA genes, do not contain stop codons. The lariat signal, another sequence involved in the splicing process, also contains stop codons. [ 1 ] [ 2 ] [ 3 ] [ 22 ] [ 25 ] [ 46 ] [ 47 ]
The theory correctly predicts that introns are non-coding and that they are mostly non-functional. Except for some intron sequences including the donor and acceptor splice signal sequences and branch point sequences, and possibly the intron splice enhancers that occur at the ends of introns, which aid in the removal of introns, the vast majority of introns are devoid of any functions. The theory does not exclude rare sequences within introns that could be used by the genome and the cell, especially because introns are so long.
Thus, the theory's predictions are precisely corroborated by the major elements in modern eukaryotic genomes.
Comparative analysis of the modern genome data from several living organisms found that the characteristics of split genes trace back to the earliest organisms. These organisms could have contained the split genes and complex proteins that occur in today's living organisms. [ 49 ] [ 50 ] [ 51 ] [ 52 ] [ 53 ] [ 54 ] [ 55 ] [ 56 ] [ 57 ]
Studies employing maximum likelihood analysis found that the earliest eukaryotic organisms contained the same genes as modern organisms with yet a higher intron density. [ 58 ] Comparative genomics of many organisms including basal eukaryotes [ 59 ] (considered to be primitive eukaryotic organisms such as Amoeboflagellata, Diplomonadida , and Parabasalia ) showed that intron-rich split genes accompanied and spliceosome from modern organisms were present in their earliest forebears, and that the earliest organisms came with all the eukaryotic cellular components. [ 60 ] [ 49 ] [ 61 ] [ 62 ] [ 63 ] [ 58 ] | https://en.wikipedia.org/wiki/Split_gene_theory |
For a given set of taxa , and a set of splits S on the taxa, usually together with a non-negative weighting, which may represent character changes distance, or may also have a more abstract interpretation, if the set of splits S is compatible, then it can be represented by an unrooted phylogenetic tree and each edge in the tree corresponds to exactly one of the splits. More generally, S can always be represented by a split network , [ 1 ] which is an unrooted phylogenetic network with the property that every split in S is represented by an array of parallel edges in the network.
This genetics article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Split_networks |
In particle physics , split supersymmetry is a proposal for physics beyond the Standard Model .
It was proposed separately in three papers. The first by James Wells in June 2003 in a more modest form that mildly relaxed the assumption about naturalness in the Higgs potential . In May 2004 Nima Arkani-Hamed and Savas Dimopoulos argued that naturalness in the Higgs sector may not be an accurate guide to propose new physics beyond the Standard Model and argued that supersymmetry may be realized in a different fashion that preserved gauge coupling unification and has a dark matter candidate. In June 2004 Gian Giudice and Andrea Romanino argued from a general point of view that if one wants gauge coupling unification and a dark matter candidate, that split supersymmetry is one amongst a few theories that exists.
The new light (~ TeV ) particles in Split Supersymmetry (beyond the Standard Models particles) are
The Lagrangian for Split Supersymmetry is constrained from the existence of high energy supersymmetry. There are five couplings in Split Supersymmetry: the Higgs quartic coupling and four Yukawa couplings between the Higgsinos, Higgs and gauginos. The couplings are set by one parameter, tan β {\displaystyle \tan \beta } , at the scale where the supersymmetric scalars decouple. Beneath the supersymmetry breaking scale, these five couplings evolve through the renormalization group equation down to the TeV scale. At a future Linear collider , these couplings could be measured at the 1% level and then renormalization group evolved up to high energies to show that the theory is supersymmetric at an exceedingly high scale.
The striking feature of split supersymmetry is that the gluino becomes a quasi-stable particle with a lifetime that could be up to 100 seconds long. A gluino that lived longer than this would disrupt Big Bang nucleosynthesis or would have been observed as an additional source of cosmic gamma rays . The gluino is long lived because it can only decay into a squark and a quark and because the squarks are so heavy and these decays are highly suppressed. Thus, the decay rate of the gluino can roughly be estimated, in natural units , as m g 5 m s q 4 {\displaystyle {{m_{g}}^{5} \over {m_{sq}}^{4}}} where m g {\displaystyle m_{g}} is the gluino rest mass and m s q {\displaystyle m_{sq}} the squark rest mass. For gluino mass of the order of 1 TeV , the cosmological bound mentioned above sets an upper bound of about 10 9 {\displaystyle 10^{9}} GeV on squarks masses.
The potentially long lifetime of the gluino leads to different collider signatures at the Tevatron and the Large Hadron Collider . There are three ways to see these particles:
Split supersymmetry allows gauge coupling unification as supersymmetry does, because the particles which have masses way beyond the TeV scale play no major role in the unification. These particles are the gravitino - which has a small coupling (of order of the gravitational interaction) to the other particles, and the scalar partners to the standard model fermions - namely, squarks and sleptons . The latter move the beta functions of all gauge couplings together, and do not influence their unification, because in the grand unification theory they form a full SU(5) multiplet , just like a complete generation of particles.
Split supersymmetry also solves the gravitino cosmological problem , because the gravitino mass is much higher than TeV .
The upper bounds on proton decay rate can also be satisfied because the squarks are very heavy as well.
On the other hand, unlike conventional supersymmetry , split supersymmetry does not solve the hierarchy problem which has been a primary motivation for proposals for new physics beyond the Standard Model since 1979. One proposal is that the hierarchy problem is "solved" by assuming fine-tuning due to anthropic reasons .
The initial attitude of some of the high energy physics community towards split supersymmetry was illustrated by a parody called supersplit supersymmetry . Often when a new notion in physics is proposed there is a knee-jerk backlash. When naturalness in the Higgs sector was initially proposed as a motivation for new physics, the notion was not taken seriously. After the supersymmetric Standard Model was proposed, Sheldon Glashow quipped that 'half of the particles have already been discovered.' After 25 years, the notion of naturalness had become so ingrained in the community that proposing a theory that did not use naturalness as the primary motivation was ridiculed. Split supersymmetry makes predictions that are distinct from both the Standard Model and the Minimal Supersymmetric Standard Model and the ultimate nature of the naturalness in the Higgs sector will hopefully be determined at future colliders.
Many of the original proponents of naturalness no longer believe that it should be an exclusive constraint on new physics. Kenneth Wilson originally advocated for it, but has recently called it one of his biggest mistakes during his career. [ citation needed ] Steven Weinberg relaxed the notion of naturalness in the cosmological constant and argued for an environmental explanation for it in 1987. Leonard Susskind , who initially proposed technicolor , is a firm advocate of the notion of a landscape and non-naturalness. Savas Dimopoulos , who initially proposed the supersymmetric Standard Model, proposed split supersymmetry. | https://en.wikipedia.org/wiki/Split_supersymmetry |
In mathematics , the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles . In the theory of vector bundles, one often wishes to simplify computations, for example of Chern classes . Often computations are well understood for line bundles and for direct sums of line bundles. Then the splitting principle can be quite useful.
One version of the splitting principle is captured in the following theorem. This theorem holds for complex vector bundles and cohomology with integer coefficients. It also holds for real vector bundles and cohomology with Z 2 {\displaystyle \mathbb {Z} _{2}} coefficients.
Theorem — Let ξ : E → X {\displaystyle \xi \colon E\rightarrow X} be a vector bundle of rank n {\displaystyle n} over a paracompact space X {\displaystyle X} . There exists a space Y = F l ( E ) {\displaystyle Y=Fl(E)} , called the flag bundle associated to E {\displaystyle E} , and a map p : Y → X {\displaystyle p\colon Y\rightarrow X} such that
In the complex case, the line bundles L i {\displaystyle L_{i}} or their first characteristic classes are called Chern roots.
Another version of the splitting principle concerns real vector bundles and their complexifications : [ 1 ]
Theorem — Let ξ : E → X {\displaystyle \xi \colon E\rightarrow X} be a real vector bundle of rank 2 n {\displaystyle 2n} over a paracompact space X {\displaystyle X} . There exists a space Y {\displaystyle Y} and a map p : Y → X {\displaystyle p\colon Y\rightarrow X} such that
The fact that p ∗ : H ∗ ( X ) → H ∗ ( Y ) {\displaystyle p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)} is injective means that any equation which holds in H ∗ ( Y ) {\displaystyle H^{*}(Y)} — for example, among various Chern classes — also holds in H ∗ ( X ) {\displaystyle H^{*}(X)} . Often these equations are easier to understand for direct sums of line bundles than for arbitrary vector bundles. So equations should be understood in Y {\displaystyle Y} and then pushed forward to X {\displaystyle X} .
Since vector bundles on X {\displaystyle X} are used to define the K-theory group K ( X ) {\displaystyle K(X)} , it is important to note that p ∗ : K ( X ) → K ( Y ) {\displaystyle p^{*}\colon K(X)\rightarrow K(Y)} is also injective for the map p {\displaystyle p} in the first theorem above. [ 2 ]
Under the splitting principle, characteristic classes for complex vector bundles correspond to symmetric polynomials in the first Chern classes of complex line bundles; these are the Chern classes . | https://en.wikipedia.org/wiki/Splitting_principle |
Spodium , ( Latin for ashes or soot ) refers to burned bone (usually used for medical purposes), or the act of divination with ash.
Spodium may also refer to other types of ash, such as the scrapings from the inside of a furnace.
Spodium has a long history of medical usage, mentioned by Hippocrates and, for example, in the Medical Poem of Salerno "...Who knows the cause why Spodium stancheth bleeding?..." (in this case spodium referring to oxen bone ashes).
This ancient Greece –related article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spodium |
A spoil tip (also called a boney pile , [ 1 ] culm bank , gob pile , waste tip [ 2 ] or bing ) [ 3 ] is a pile built of accumulated spoil – waste material removed during mining . [ 4 ] Spoil tips are not formed of slag , but in some areas, such as England and Wales, they are referred to as slag heaps . In Scotland the word bing is used. In North American English the term is mine dump [ 5 ] or mine waste dump . [ 6 ]
The term "spoil" is also used to refer to material removed when digging a foundation, tunnel, or other large excavation. Such material may be ordinary soil and rocks (after separation of coal from waste ), or may be heavily contaminated with chemical waste, determining how it may be disposed of. Clean spoil may be used for land reclamation .
Spoil is distinct from tailings , which is the processed material that remains after the valuable components have been extracted from ore.
The phrase originates from the French word espoilelier , a verb conveying the meaning: to seize by violence, to plunder, to take by force. [ 7 ]
Spoil tips may be conical in shape, and can appear as conspicuous features of the landscape , or they may be much flatter and eroded, especially if vegetation has established itself. In Loos-en-Gohelle , in the former mining area of Pas-de-Calais , France, are a series of five very perfect cones, of which two rise 100 metres (330 ft) from the plain.
Most commonly the term is used for the piles of waste earth materials removed during an excavation process.
Spoil banks can also refer to refuse heaps formed from removal of excess surface materials. For example, alongside livestock lots, spoil banks are formed of manure and other slurry periodically removed from the surface of the livestock lot areas.
Spoil tips sometimes increased to millions of tons, and, having been abandoned, remain as huge piles today. They trap solar heat, making it difficult (although not impossible) for vegetation to take root; this encourages erosion and creates dangerous, unstable slopes. Existing techniques for regreening spoil tips include the use of geotextiles to control erosion as the site is resoiled and simple vegetation such as grass is seeded on the slope.
The piles also create acid rock drainage , which pollutes streams and rivers. Environmental problems have included surface runoff of silt , and leaching of noxious chemical compounds from spoil banks exposed to weathering . These cause contamination of ground water , and other problems. [ 8 ] [ 9 ]
In the United States, current state and federal coal mining regulations require that the earth materials from excavations be removed in such a fashion that they can be replaced after the mining operations cease in a process called mine reclamation , with oversight of mining corporations. This requires adequate reserves of monetary bonds to guarantee a completion of the reclamation process when mining becomes unprofitable or stops. (See for example, the Surface Mining Control and Reclamation Act of 1977 .)
In some spoil tips, the waste resulting from industries such as coal or oil shale production can contain a relatively high proportion of hydrocarbons or coal dust. Spontaneous subterranean combustion may result, which can be followed by surface fires. In some coal mining districts, such fires were considered normal and no attempt was made to extinguish them. [ 10 ]
Such fires can follow slow combustion of residual hydrocarbons. Their extinction can require complete encasement, which can prove impossible for technical and financial reasons. Sprinkling is generally ineffective and injecting water under pressure counter-productive, because it carries oxygen, bringing the risk of explosion.
The perceived weak environmental and public health effects of these fires leads generally to waiting for their natural extinction, which can take a number of decades.
The problem of landslides in spoil tips was first brought to public attention in October 1966 in the English speaking world when a spoil tip at Aberfan in Glamorgan , Wales, gave way, killing 144 people, 116 of them children. The tip was built over a spring, increasing its instability, and its height exceeded guidelines. Water from heavy rainfall had built up inside the tip, weakening the structure, until it suddenly collapsed onto a school below. [ 11 ]
The wider issue of stability had been known about prior to the Aberfan disaster ; for example, it was discussed in a paper by Professor George Knox in 1927, [ 12 ] but received little serious consideration by professional engineers and geologists — even to those directly concerned with mining. [ 13 ] Also the Aberfan disaster was not the first landslide with casualties: for example, in 1955 two successive landslides killed 73 people in Sasebo, Nagasaki in Japan. [ 14 ] [ja]
In February 2013, a spoil tip landslip caused the temporary closure of the Scunthorpe to Doncaster railway line in England. [ 15 ]
Landslides are rare in spoil tips after settling and vegetation growth act to stabilise the spoil. However, when heavy rain falls on spoil tips that are undergoing combustion, infiltrated water changes to steam ; increasing pressure that may lead to a landslide. [ 16 ] In Herstal , Belgium, a landslide on the Petite Bacnure spoil tip in April 1999 closed off a street for many years. [ 17 ]
Several techniques of re-utilising the spoil tips exist, usually including either geotechnics or recycling . Most commonly, old spoil tips are partially revegetated to provide valuable green spaces since they are inappropriate for building purposes. At Nœux-les-Mines , an artificial ski slope has been constructed on the tip. If spoil tips are considered to contain sufficient amounts of residual material, various methods are employed to remove the spoil from the site for subsequent processing.
The oldest coal-based spoil tips may still contain enough coal to begin spontaneous slow combustion . This results in a form of vitrification of the shale, which then acquires sufficient mechanical strength to be of use in road construction. [ 18 ] Some can therefore have a new life in being thus exploited; for example, the flattened pile of residue from the 11/19 site of Loos-en-Gohelle. Conversely, others are painstakingly preserved on account of their ecological wealth. With the passage of time, they become colonised with a variety of flora and fauna , sometimes foreign to the region. This diversity follows the mining exploitation. In South Wales some spoil tips are protected as Sites of Special Scientific Interest because they provide a unique habitat for 57 species of Lichen , several of which are at risk due to their limited environment being developed and by vegetation development. [ 19 ]
For example, because the miners threw their apple or pear cores into the wagons, the spoil tips became colonised with fruit trees. One can even observe the proliferation of buckler-leaved sorrel (French sorrel – Rumex scutatus ), the seeds of which have been carried within the cracks in the pine timber used in the mines. Furthermore, on account of its dark colour, the south face of the spoil tip is significantly warmer than its surroundings, which contributes to the diverse ecology of the area. In this way, the spoil tip of Pinchonvalles, at Avion , hosts 522 different varieties of higher plants. Some sixty species of birds nest there. [ 20 ]
Some are used to cultivate vines, as in the case of Spoil Tip No. 7 of the coal-mining region of Mariemont-Bascoup near Chapelle-lez-Herlaimont ( province of Hainaut ). It produces some 3,000 litres of wine each year from a vineyard on its slopes.
Some spoil tips are used for various sporting activities. The slopes of the spoil tips of 11/19 at Loos-en-Gohelle , or again, at Nœux-les-Mines , are used for winter sports , for example ski and luge . A piste was built on the flank of the heap. In Belgium , a long distance footpath along the spoil tips (GR-412, Sentier des terrils ) was opened in 2005. It leads from Bernissart in western Hainaut to Blegny in the province of Liège .
In the United States, coal mining companies have not been allowed to leave behind abandoned piles since the Surface Mining Control and Reclamation Act was passed in 1977. The Virginia City Hybrid Energy Center uses coal gob as a fuel source for energy production.
One of the highest, at least in Western Europe , is in Loos-en-Gohelle in the former mining area of Pas-de-Calais , France. It comprises a range of five cones, of which two reach 180 metres (590 ft), surpassing the highest peak in Flanders , Mont Cassel . One of the regions of Europe most "littered" with (mountainous) spoil heaps is the Donbas , in Ukraine, especially around the city of Donetsk , which alone boasts about 130 of them. [ 21 ] In Ukrainian , they are called terykony ( терикони ; singular терикон , terykon , 'soil cone') because of their shape.
In Heringen , Hesse , Germany, is the locally called " Monte Kali ", made of spoil from potash mining and rising some 200 meters above the surrounding terrain. [ 22 ] "La Muntanya de Sal" (The Salt Mountain), another potash mine spoil heap, lies in Cardona , Catalonia , at about 120 meters in height. [ 23 ] [ 24 ] A larger and higher pile is that of "El runam del Cogulló" (The Spoil Heap of El Cogulló), also known as "El runam de la democràcia" (The Slag Heap of Democracy) or "Montsalat" (Salty Mountain), in Sallent , which has already grown higher than the small mountain it was named after (El Cogulló, 474 meters above sea level). [ 25 ] [ 26 ]
Richard Llewellyn 's novel How Green Was My Valley (1939) describes the social and environmental effects of coal mining in Wales at the turn of the 20th century. The local mine's spoil tip, which he calls a slag heap, is the central figure of devastation. Eventually the pile overtakes the entire valley and crushes Huw Morgan's house:
The slagheap is moving again. I can hear it whispering to itself, and as it whispers, the walls of this brave little house are girding themselves to withstand the assault. For months, more than I ever thought it would have the courage to withstand, that great mound has borne down upon these walls, this roof. And for those months the great bully has been beaten, for in my father’s day men built well for they were craftsmen. Stout beams, honest blocks, good work, and love for the job, all that is in this house. But the slag heap moves, pressing on, down and down, over and all round this house which was my father’s and my mother’s and now is mine. Soon, perhaps in an hour, the house will be buried, and the slag heap will stretch from the top of the mountain right down to the river in the Valley. Poor river, how beautiful you were, how gay your song, how clear your green waters, how you enjoyed your play among the sleepy rocks (102). [ 27 ] | https://en.wikipedia.org/wiki/Spoil_tip |
A sponge bomb is a specialized device designed to seal the end of a tunnel. Small enough that it can be set by a single person, it is a non-explosive, chemical bomb that releases a burst of expanding foam that quickly hardens. [ 1 ] [ 2 ]
The sponge bomb was developed by the Israel Defense Forces (IDF) to address the use of tunnels by Hamas in Gaza . [ 3 ] [ 4 ]
Housed in a plastic container, the bomb has a metal partition that separates two liquid reagents. Once the partition is removed, the liquids mix and react, causing them to rapidly expand and then solidify, creating a physical barrier blocking the tunnel. The device is either set at its target by an individual or thrown. [ 1 ]
In 2021, testing of sponge bombs was reportedly conducted by IDF in simulated tunnels. [ 5 ]
During initial testing of these bombs, the liquid emulsion was found to be hazardous to work with when mishandled – some Israeli soldiers lost their eyesight. [ 1 ]
This is not the first time that sticky foam has been used by a military force. Reportedly, the U.S. Marine Corps and the U.S. Army have used streams of foam as non-lethal tools for crowd control or restraint of hostile combatants. [ 4 ]
This article relating to bombs is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Sponge_bomb |
Sponge grounds , also known as sponge aggregations , are intertidal to deep-sea habitats formed by large accumulations of sponges ( glass sponges and/or demosponges ), often dominated by a few massive species. Sponge grounds were already reported more than 150 years ago, [ 1 ] but the habitat was first fully recognized, studied and described in detail around the Faroe Islands during the inter-Nordic BIOFAR 1 programme 1987–90. [ 2 ] These were called Ostur (meaning "cheese" and referring to the appearance of the sponges) by the local fishermen and this name has to some extent entered the scientific literature. [ 1 ] Sponge grounds were later found elsewhere in the Northeast Atlantic [ 3 ] and in the Northwest Atlantic, [ 4 ] as well as near Antarctica. [ 1 ] They are now known from many other places worldwide and recognized as key marine habitats. [ 5 ]
Sponge grounds are important habitats supporting diverse ecosystems . During a study of outer shelf and upper slope sponge grounds at the Faroe Islands, 242 invertebrate species were found in the vicinity and 115 were associated with the sponges. [ 6 ] In general, fish fauna associated with sponge grounds are poorly known, but include rockfish and gadiforms . [ 1 ] Sponge grounds are threatened, especially by bottom trawling and other fishing gear , dredging , oil and gas exploration and undersea cables , but potentially also by deep sea mining , carbon dioxide sequestration , pollution and climate change . [ 1 ]
By studying spicules in sediments cores taken from sponge grounds on the slopes of the Flemish Cap and Grand Bank (off Newfoundland, Canada), scientists managed to detect the presence of sponges in the past. [ 7 ] The oldest record for Geodiidae sponges in this region was found in a long core collected in the slope of the Grand Bank, where typical sterraster spicules were found in the top of a submarine landslide deposit older than 25 000 BP. Continuous presence of sponges was recorded on the southeastern region of the Flemish Cap as far as 130 000 BP. It seems the distribution range of the Geodiidae in this area significantly expended after the deglaciation. | https://en.wikipedia.org/wiki/Sponge_ground |
The sponge iron reaction (SIR) is a chemical process based on redox cycling of an iron-based contact mass, the first cycle is a conversion step between iron metal (Fe) and wuestite (FeO), the second cycle is a conversion step between wuestite (FeO) and magnetite (Fe 3 O 4 ). [ 1 ] In application, the SIT is used in the reformer sponge iron cycle (RESC) in combination with a steam reforming unit.
The process has two modes, a reduction mode and an oxidation mode.
FeO + H 2 ↔ Fe + H 2 O
Fe 3 O 4 + H 2 ↔ 3 FeO + H 2 O
The reformer sponge iron cycle is a two step cycle to produce hydrogen from hydrocarbon fuels based SIR and steam. [ 2 ]
In the first step the hydrocarbon fuel is reformed to syngas in the reformer which is then used to reduce the iron oxide ( magnetite —Fe 3 O 4 ) to iron ( wüstite —FeO), in the second step steam is utilized to re-oxidise the iron into magnetite and hydrogen. The iron oxide pellets are placed in a pelletbed and have a service life of several thousand redox cycles. [ 3 ] | https://en.wikipedia.org/wiki/Sponge_iron_reaction |
Lacking an immune system , protective shell, or mobility, sponges have developed an ability to synthesize a variety of unusual compounds for survival. C- nucleosides isolated from Caribbean Cryptotethya crypta , were the basis for the synthesis of zidovudine ( AZT ), aciclovir ( Cyclovir ), cytarabine ( Depocyt ), and cytarabine derivative gemcitabine ( Gemzar ).
Semisynthetic analogs of the sponge isolate jasplakinolide, were submitted to National Cancer Institute ’s Biological Evaluation Committee in 2011.
Trabectedin , aplidine , didemnin , were isolated from sea squirts . Monomethyl auristatin E is a derivative of a dolastatin 10, a compound made by Dolabella auricularia . Bryostatins were first isolated from Bryozoa .
Salinosporamides are derived from Salinispora tropica . Ziconotide is derived from the sea snail Conus magus . | https://en.wikipedia.org/wiki/Sponge_isolates |
Spongin , a modified type of collagen protein , forms the fibrous skeleton of most organisms among the phylum Porifera , the sponges. It is secreted by sponge cells known as spongocytes. [ 1 ]
Spongin gives a sponge its flexibility. True spongin is found only in members of the class Demospongiae . [ 2 ] Its molecular structure remains incompletely characterized, however it shares similarities with both collagen and keratin. [ 3 ] [ 4 ] [ 5 ]
Researchers have found spongin to be useful in the photocatalytic degradation and removal of bisphenols (such as BPA ) in wastewater. A heterogeneous catalyst consisting of a spongin scaffold for iron phthalocyanine (SFe) in conjunction with peroxide and UV radiation has been shown to remove phenolic wastes more quickly and efficiently than conventional methods. [ 6 ] Other research using spongin scaffolds for the immobilization of Trametes versicolor Laccase has shown similar results in phenol degradation. [ 7 ]
Structure of spongin remains incompletely understood due to limitations in protein analytical methods. Although its chemical composition shares some features with collagen and keratin, spongin is a distinct biopolymer characterized by halogenated amino acids, primary bromine, with smaller amounts of iodine and chlorine. Additionally, the presence of xylose and significant mineralization with calcium carbonates and silica further differentiates spongin from collagen and keratin. [ 8 ] [ 9 ]
Spongin and collagen exhibit comparable filament structure, both displaying a hierarchical organization of nanofibrils, microfibrils, and fibers, as well as a triple-helical based structure. Spongin microfibrilis measure approximately 10nm in diameter and exhibit a periodic banding pattern every 60nm, comparing to collagen's 67nm periodicity. [ 10 ] Despite the structural similarities, amino acid analysis reveals that spongin contains significantly higher levels of tyrosine residues, approximately 90% of which are mono- or di-brominated derivatives. This abundance of tyrosine is related to the oxidation of phenylalanine residues, which are prevalent in collagen but nearly absent in spongin. The brominated tyrosine derivatives are hypothesized to play a crucial role in stabilizing spongin's triple-helical structure through cross-linking. [ 9 ]
Spongin also exhibits compositional similarities to keratin, particularly in its sulfur content and thermal stability. It withstands temperatures up to 300ºC, which is more characteristic for keratin than collagen. However, due to spongin's distinct biochemical features, its full molecular classification remains unknown. [ 8 ]
This poriferan - (or sponge-) related article is a stub . You can help Wikipedia by expanding it .
This biochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spongin |
Spongy degeneration of the central nervous system , also known as Canavan's disease , Van Bogaert-Bertrand type or Aspartoacylase (AspA) deficiency , is a rare autosomal recessive neurodegenerative disorder . [ 1 ] It belongs to a group of genetic disorders known as leukodystrophies , [ 1 ] where the growth and maintenance of myelin sheath in the central nervous system (CNS) are impaired. [ 2 ] There are three types of spongy degeneration: infantile, congenital and juvenile, with juvenile being the most severe type. [ 3 ] Common symptoms in infants include lack of motor skills , weak muscle tone , and macrocephaly . [ 4 ] It may also be accompanied by difficulties in feeding and swallowing, seizures and sleep disturbances . [ 4 ] Affected children typically die before the age of 10, but life expectancy can vary. [ 5 ]
The cause of spongy degeneration of the CNS is the mutation in a gene coding for aspartoacylase (AspA) , an enzyme that hydrolyzes N-acetyl aspartic acid (NAA) . [ 6 ] In the absence of AspA, NAA accumulates and results in spongy degeneration. [ 7 ] The exact pathophysiological causes of the disease are currently unclear, but there are developing theories. [ 8 ] Spongy degeneration can be diagnosed with neuroimaging techniques and urine examination. [ 9 ] There is no current treatment for spongy degeneration, but research utilising gene therapy to treat the disease is underway. [ 8 ] Spongy degeneration is found to be more prevalent among Ashkenazi Jews , with an incidence of 1/6000 amongst this ethnic group. [ 10 ]
Spongy Degeneration of the CNS is classified into three types: infantile, juvenile and congenital; based on the age of onset and severity of symptoms.
The infantile type is the most common type of spongy degeneration of the CNS. [ 11 ] Usually, affected infants appear normal for the first few months of life. [ 12 ] The age of onset is around 6 months, where infants begin to develop noticeable psychomotor defects . [ 12 ] Various motor skills such as turning over and stabilising head movements are affected. [ 11 ] Hypotonia and macrocephaly are also observed in the first few months. [ 13 ]
During the latter part of the first year, most children's eyes fail to respond to visual stimuli , with episodic saccadic eye movements observed, rendering most children blind in the second year. [ 5 ]
The symptoms in the terminal stage of disease development are sweating, emesis , hyperthermia , seizures, and hypotension , which usually results in the death of the child. [ 13 ] Life expectancies of affected infants vary, but most infants do not live past the age of ten. [ 5 ]
The age of onset is typically a few days after birth in the congenital type. Pregnancy and delivery are not affected and the child is born with a normal appearance and no health issues. [ 12 ] However, affected infants may become lethargic in the following days and find movements such as sucking and swallowing difficult. [ 14 ] As the disease progresses, patients may have decreased muscle tone and inactivation of Moro reflex , also known as startle reflex. [ 12 ] This may lead to the development of Cheyne Stokes respiration after a few weeks or even days after delivery, which may be fatal. [ 12 ]
The age of onset of the juvenile type is around five years of age. Most patients with the juvenile type survive until late adolescence . [ 15 ] Affected toddlers typically develop progressive cerebellar syndrome and mental deterioration, which is followed by vision loss, optic atrophy , and generalised spasticity . [ 16 ] Unlike the infantile form, there is no macrocephaly exhibited. [ 12 ]
Although the pathophysiological causes of CD symptoms are still unclear, there are developing theories on the causes of myelination issues, gelatinous cortical white matter and seizures. [ 8 ]
Increased cerebrospinal fluid (CSF) pressure and intramyelinic edema in CD patients suggest the existence of an efficient MWP in the brain. [ 17 ] [ 8 ] The MWP is a membrane protein responsible for pumping water molecules, along with dissolved NAA molecules, from the intraneuronal space to the interstitial space . [ 10 ] In healthy individuals, NAA is first transported down the concentration gradient through the MWP from neurons to the interstitial space and subsequently hydrolyzed by AspA in neighbouring oligodendrocytes . [ 10 ]
In patients with CD, it is theorized that AspA deficiency causes accumulation of NAA in the interstitial space, inducing an osmolyte imbalance and accumulation of water in the interstitial space. [ 8 ] This increases hydrostatic pressure between interlamellar spaces and extracellular periaxonal and parenchymatous space, loosening the tight junctions between them, thus causing intramyelinic edema. [ 18 ] Subsequent demyelination possibly contributes to vacant spaces in the white matter or spongy degeneration. [ 8 ]
NAA-derived acetates are involved in the synthesis of fatty acids , which are subsequently incorporated into myelin lipids . [ 3 ] [ 8 ] It is hypothesized that in CD patients, AspA deficiency reduces NAA-derived acetates, and consequently decreases the synthesis of myelin-associated lipids. [ 6 ] This leads to dysmyelination , which promotes the formation vacuoles in interstitial space and spongy degeneration. [ 8 ] However, it has been shown that spongy degeneration is not directly caused by the disrupted synthesis of myelin. [ 19 ] Animal models show that myelination may still occur in AspA lacking species, possibly due to parallel pathways for myelination during the initial stages of myelinogenesis . [ 19 ]
Deficiency of AspA lowers acetyl coenzyme A (CoA) expression in cells, which may be responsible for stabilization and correct folding of proteins . [ 20 ] This leads to protein degradation, with a particularly large effect in oligodendrocytes. [ 20 ] In animal studies of AspA deficient species, protein degradation in oligodendrocytes has been shown to cause severe loss in myelin proteins. [ 21 ]
The deficiency in AspA, which is vital in oligodendrocytes to produce NAA derived acetate, leads to a lack of regulation in the genetic structure and expression in these cells. [ 1 ] This results in the death of oligodendrocytes, hence induces neuronal injury and the formation of vacuoles in the subcortical matter . [ 22 ] These vacuoles contribute to the formation of gelatinous-textured subcortical white matter found in many CD patients. [ 22 ]
The pathophysiological causes of seizures and neurodegeneration in CD patients are likely due to oxidative stress generated by NAA accumulation. [ 23 ] It is postulated that NAA promotes oxidative stress through promoting reactive oxygen species , as well as reducing non-enzymatic antioxidant defenses. [ 23 ] NAA also affects multiple antioxidant enzymes, such as catalase and glutathione peroxidase , impairing the detoxification of hydrogen peroxide . [ 24 ] Recent animal studies have shown the chronic oxidative stress may cause dysfunction in mitochondria , rendering the brain more susceptible to epileptic seizures . [ 24 ] [ 25 ]
Canavan's disease is initially recognized by the appearance of symptoms, yet further examinations are needed for definitive diagnosis. Neuroimaging techniques such as Computed Tomography (CT) scan or Magnetic Resonance imaging (MRI) are typically used to detect the presence of degenerative subcortical white matter . [ 26 ] Microscopy of the cerebrospinal fluid can also be used for diagnosis, where swollen astrocytes with distorted and elongated mitochondria can be seen in patients. [ 5 ]
Urine examinations are used to differentiate CD patients from other neurodegenerative disorders with similar morphology , such as Alexander diseases and Tay-Sachs diseases (which similarly exhibit macrocephaly ), as patients with CD uniquely display increased excretion of NAA. [ 5 ] [ 13 ]
DNA analysis is generally used to determine if parents are carriers of the mutant gene. [ 27 ] Prenatal diagnosis through either DNA analysis or determination of NAA in amniotic fluid (which would be increased in an affected pregnancy) can also be used when DNA analysis cannot be performed on parents. [ 28 ] It has been observed that there is an abnormally high carrier rate in the Ashkenazi Jewish population. [ 4 ] The risk of their offspring having spongy degeneration is one in four if both parents are carriers of the mutant gene. [ 28 ]
There are currently no specific forms of treatment known for spongy degeneration of the CNS. [ 29 ] Certain treatment modules are under experimental trials and current patients are supported by palliative measures, all of which are described below.
Current patients are supported by the care guidelines for other paediatric neurodegenerative diseases. [ 30 ] For patients with respiratory issues, suction machines are used to clear mucous from the upper airway of the lungs. [ 8 ] Oxygen concentrators are also administered for airway clearance and continuous supply of air to aid breathing. [ 8 ] As for infants with hypotonia, it is addressed by the provision of positioning equipment like specialized strollers, bath chairs and feeder seats. [ 31 ]
Lipoic acid (which can cross the blood brain barrier), has recently been trialed in preclinical studies , where it has been injected into tremor rats intraperitoneally . [ 32 ] Tremor rats are deemed as the naturally occurring model for spongy degeneration of the CNS as NAA induces oxidative stress. [ 33 ] Positive results have emerged from these studies, suggesting that lipoic acid may be a possible approach for symptomatic treatments . [ 32 ]
A possible treatment is to employ neuroprotective techniques to offset the neurological damage in the CNS caused by the accumulation of NAA. [ 8 ] One potential treatment that has been identified is lithium , which has been observed to induce neuroprotective effects in dementia patients. [ 34 ] Administration of intraperitoneal lithium has been tested in both tremor and wild-type rats, causing a decrease in NAA levels in both species. [ 35 ] In human trials, NAA levels in patient's brain and urine was found to drop after one year of treatment. [ 29 ] This is coupled with the elevation of alertness and visual tracking. [ 29 ] However. CD symptoms including axial hypotonia and spastic diplegia remained. [ 8 ]
Since CD arises from a monogenic defect and is localized in the CNS, gene replacement therapy is a potential treatment. [ 8 ] This therapy involves replacing the mutant gene of the disease with a fully functional gene using a vector, which transports therapeutic DNA into cells, allowing cells to produce AspA. [ 36 ] Adeno-associated Viruses (AAVs) are widely used as vectors for gene therapy. [ 8 ] They are adopted as they do not replicate themselves and are almost non-toxic. [ 8 ] There are two serotypes used for the treatment: AAV2 and AAV9. [ 37 ] The difference of the stereotypes is that AAV2 is limited by blood-brain-barrier (BBB), whilst AAV9 can cross the BBB, allowing for treatment even at the later stages of the disease. [ 38 ] However, current research shows that AAVs may trigger unwanted immune responses in infants and have limited gene encapsulating capacity. [ 39 ]
Spongy degeneration of the CNS is pan-ethnic , due to its prevalence among Ashkenazi Jews . There are two common mutations found among them: missense mutation (Glu285AIa) and nonsense mutation (Tyr231X). [ 40 ] In the missense mutation, there is a substitution of glutamic acid to alanine . [ 41 ] As for the nonsense mutation, the tyrosine codon is replaced by a termination codon . [ 41 ] Genetic screening reveals that around 1 in 40 healthy Jews are carriers and the incidence of this disease in this population is as high as 1 in 6000. [ 10 ]
The first case of spongy degeneration of the CNS was reported in 1928 by Globus and Strauss, [ 42 ] who designated the case as Schilder's disease, a term for diffuse myelinoclastic sclerosis . [ 43 ] [ 44 ] [ 45 ] In 1931, Canavan reported a case where the megalencephaly of brain degeneration is different from that caused by a tumour . [ 46 ] However, she failed to recognize the spongy alterations that suggest a unique pathological cause that distinguishes her case from Schilder's disease. [ 47 ] Later in 1937, Eislebergl reported six cases from Jewish families and discovered the familial characteristics of spongy degeneration, but she classified these cases as Krabbe's sclerosis . [ 47 ] [ 48 ] It was not until 1949 when Van Bogaert and Bertrand reported five cases from Jewish families, whereupon further pathological analysis confirmed that spongy degeneration is the nosologic entity. [ 47 ] | https://en.wikipedia.org/wiki/Spongy_degeneration_of_the_central_nervous_system |
Spontaneous absolute asymmetric synthesis is a chemical phenomenon that stochastically generates chirality based on autocatalysis and small fluctuations in the ratio of enantiomers present in a racemic mixture . [ 1 ] In certain reactions which initially do not contain chiral information , stochastically distributed enantiomeric excess can be observed. The phenomenon is different from chiral amplification, where enantiomeric excess is present from the beginning and not stochastically distributed. Hence, when the experiment is repeated many times, the average enantiomeric excess approaches 0%. [ 2 ] The phenomenon has important implications concerning the origin of homochirality in nature. [ 3 ]
This stereochemistry article is a stub . You can help Wikipedia by expanding it . | https://en.wikipedia.org/wiki/Spontaneous_absolute_asymmetric_synthesis |
Spontaneous combustion or spontaneous ignition is a type of combustion which occurs by self-heating (increase in temperature due to exothermic internal reactions ), followed by thermal runaway (self heating which rapidly accelerates to high temperatures) and finally, autoignition . [ 1 ] It is distinct from (but has similar practical effects to) pyrophoricity , in which a compound needs no self-heat to ignite. The correct storage of spontaneously combustible materials is extremely important, as improper storage is the main cause of spontaneous combustion. Materials such as coal, cotton, hay, and oils should be stored at proper temperatures and moisture levels to prevent spontaneous combustion.
Reports of spontaneous human combustion are not considered truly spontaneous, but due to external ignition. [ 2 ]
Spontaneous combustion can occur when a substance with a relatively low ignition temperature such as hay, straw, peat, etc., begins to release heat. This may occur in several ways, either by oxidation in the presence of moisture and air, or bacterial fermentation , which generates heat. These materials are thermal insulators that prevent the escape of heat causing the temperatures of the material to rise above its ignition point. Combustion will begin when a sufficient oxidizer , such as oxygen, and fuel are present to maintain the reaction into thermal runaway.
Thermal runaway can occur when the amount of heat produced is greater than the rate at which the heat is lost. Materials that produce a lot of heat may combust in relatively small volumes, while materials that produce very little heat may only become dangerous when well insulated or stored in large volumes. Most oxidation reactions accelerate at higher temperatures, so a pile of material that would have been safe at a low ambient temperature may spontaneously combust during hotter weather.
Hay [ 3 ] and compost piles [ 4 ] may self-ignite because of heat produced by bacterial fermentation , which then can cause pyrolysis and oxidation that leads to thermal runaway reactions that reach autoignition temperature. Rags soaked with drying oils or varnish can oxidize rapidly due to the large surface area, and even a small pile can produce enough heat to ignite under the right conditions. [ 5 ] [ 6 ] Coal can ignite spontaneously when exposed to oxygen, which causes it to react and heat up when there is insufficient ventilation for cooling. [ 7 ] Pyrite oxidation is often the cause of coal's spontaneous ignition in old mine tailings . Pistachio nuts are highly flammable when stored in large quantities, and are prone to self-heating and spontaneous combustion. [ 8 ] Large manure piles can spontaneously combust during conditions of extreme heat. Cotton and linen can ignite when they come into contact with polyunsaturated vegetable oils (linseed, massage oils); bacteria will slowly decompose the materials, producing heat. If these materials are stored in a way so the heat cannot escape, the heat buildup increases the rate of decomposition and thus the rate of heat buildup increases. Once ignition temperature is reached, combustion occurs with oxidizers present (oxygen). Nitrate film , when improperly stored, can deteriorate into an extremely flammable condition and combust. The 1937 Fox vault fire was caused by spontaneously combusting nitrate film.
Hay is one of the most widely studied materials in spontaneous combustion. It is very difficult to establish a unified theory of what occurs in hay self-heating because of the variation in the types of grass used in hay preparation, and the different locations where it is grown. It is anticipated that dangerous heating will occur in hay that contains more than 25% moisture. The largest number of fires occur within two to six weeks of storage, with the majority occurring in the fourth or fifth week.
The process may begin with microbiological activity (bacteria or mold) which ferments the hay, creating ethanol. Ethanol has a flash point of 14 °C (57 °F). So with an ignition source such as static electricity, e.g. from a mouse running through the hay, combustion may occur. The temperature then increases, igniting the hay itself.
Microbiological activity reduces the amount of oxygen available in the hay. At 100 °C, wet hay absorbed twice the amount of oxygen of dry hay. There has been conjecture that the complex carbohydrates present in hay break down to simpler sugars, which are more readily fermented to ethanol. [ 9 ]
Charcoal, when freshly prepared, can self-heat and catch fire. This is separate from hot spots which may have developed from the preparation of charcoal. Charcoal that has been exposed to air for a period of eight days is not considered to be hazardous. There are many factors involved, among them the type of wood and the temperature at which the charcoal was prepared. [ 10 ]
Extensive studies have been completed on the self-heating of coal. Improper storage of coal is a main cause of spontaneous combustion, as there can be a continuous oxygen supply and the oxidization of coal produces heat that doesn't dissipate. Over time, these conditions can cause self-heating. [ 11 ] The tendency to self-heat decreases with the increasing rank of the coal. Lignite coals are more active than bituminous coals , which are more active than anthracite coals. Freshly mined coal consumes oxygen more rapidly than weathered coal, and freshly mined coal self-heats to a greater extent than weathered coal. The presence of water vapor may also be important, as the rate of heat generation accompanying the absorption of water in dry coal from saturated air can be an order of magnitude or more than the same amount of dry air. [ 12 ]
Cotton too can be at great risk of spontaneous combustion. [ 13 ] In an experimental study on the spontaneous combustion of cotton, three different types of cotton were tested at different heating rates and pressures. Different cotton varieties can have different self-heating oxidation temperature and larger reactions. Understanding what type of cotton is being stored will help reduce the risk of spontaneous combustion. [ 14 ] A striking example of a cargo igniting spontaneously occurred on the ship Earl of Eldon in the Indian Ocean on 24 August 1834.
Oil seeds and residue from oil extraction will self-heat if too moist. Typically, storage at 9–14% moisture is satisfactory, but limits are established for each individual variety of oil seed. In the presence of excess moisture that is just below the level required for germinating seed, the activity of mold fungi is a likely candidate for generating heat. This was established for flax and sunflower seeds, and soy beans. Many of the oil seeds generate oils that are self-heating. Palm kernels, rapeseed, and cotton seed have also been studied. [ 15 ] Rags soaked in linseed oil can spontaneously ignite if improperly stored or discarded. [ 16 ]
Copra , the dried, white flesh of the coconut from which coconut oil is extracted, [ 17 ] has been classed with dangerous goods due to its spontaneously combustive nature. [ 18 ] It is identified as a Division 4.2 substance.
There have been unconfirmed anecdotal reports of people spontaneously combusting. This alleged phenomenon is not considered true spontaneous combustion, as supposed cases have been largely attributed to the wick effect , whereby an external source of fire ignites nearby flammable materials and human fat or other sources. [ 19 ]
There are many factors that can help predict spontaneous combustion and prevent it. The longer a material sits, the higher the risk of spontaneous combustion. Preventing spontaneous combustion can be as simple as not leaving materials stored for extended periods of time, controlling air flow, moisture, methane, and pressure balances. There are also many materials that prevent spontaneous combustion. For example, spontaneous coal combustion can be prevented by physical based materials such as chlorine salts, ammonium salts, alkalis, inert gases, colloids, polymers, aerosols, and LDHs, as well as chemical-based materials like antioxidants, ionic liquids, and composite materials. [ 20 ] | https://en.wikipedia.org/wiki/Spontaneous_combustion |
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