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those with $\mathcal{J}_z = -\frac{1}{2}\hbar$ . Since the magnet M3 has an equal and opposite effect on the two trajectories, the atoms in B1 and B2 will be brought together at the slit S3 and detected at D. Now let us see what happens if the magnet M2 is switched on, which produces a large uniform static field in ... | {
"Header 1": "14.1 MAGNETIC RESONANCE AND THE MEASUREMENT OF THE GYROMAGNETIC RATIOS",
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It is also possible to detect resonance phenomena in bulk samples of materials. Let us consider the case of a material composed of atoms of total angular momentum one-half. In the absence of a magnetic field, the two states of each atom with $\mathcal{J}_z = m\hbar$ , $m = \pm \frac{1}{2}$ have the same energy, and ... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
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Similarly, the rate of change of the energy density because of stimulated emission is
$$\frac{\mathrm{d}\rho_s}{\mathrm{d}t} = N_2(\hbar\omega)W_{12} \tag{14.34}$$
where $N_2$ is the number of atoms in the upper energy level per unit volume, and $W_{12}$ is the transition rate per atom for stimulated emission. ... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
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Level 4 is chosen so that it has a fast decay to level 3, and pumping between levels 1 and 4 immediately produces a population inversion between levels 3 and 2. As level 2 begins to fill up by stimulated emission at the frequency ( $E_3 - E_2$ )/h, the population inversion will decrease. To minimise this, level 2 is ch... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
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Having obtained a population entirely in the state of higher energy $E_2$ , maser action is obtained by stimulated emission of the transition from the level 2 to the level 1 [2], which is reinforced by passing the beam through a cavity tuned to the
14.8 The ammonia maser. Molecules ... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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#### Lasers and spectroscopy
The decades since the discovery of the laser have witnessed a revolution in spectroscopy. Unlike other sources of light, laser light is coherent and very nearly monochromatic. The line width which can be achieved is often smaller than the line widths of the atomic or molecular system to... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
"token_count": 2032,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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This can be achieved by absorbing the neutrons in a lithium blanket, with the advantage of breeding tritium through the exothermic reaction
$$n + {}^{6}Li \rightarrow T(2.7 \text{ MeV}) + {}^{4}He(2.1 \text{ MeV})$$
[14.67]
The isotope <sup>6</sup>Li constitutes about 7.5 per cent of natural lithium, which is readi... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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#### Energy balance and impurities
In a fusion reactor the fundamental energy producing reaction rates are characterised by cross-sections which are of the order of $10^{-30}$ m<sup>2</sup>. In contrast, the cross-sections for atomic collision processes, such as those discussed in Chapters 12 and 13, are of the o... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
"token_count": 2034,
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To produce the injected beam, an ion source capable of yielding pulses of $D^+$ ions at 100 keV is first constructed. The $D^+$ beam is neutralised by passing it through a gas target containing molecular deuterium $(D_2)$ where charge exchange takes place (see Fig. 14.14)
$$D^+ + D_2 \rightarrow D + D_2^+$$ ... | {
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"Header 3": "Paramagnetic resonance and nuclear magnetic resonance in bulk samples",
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hundred of these lines, now called Fraunhofer lines, in the solar spectrum. The origin of the dark lines remained unexplained until G. R. Kirchhoff and R. G. Bunsen discovered that heated gas vapours emit spectra composed of bright lines characteristic of the elements from which the spectrum is emitted. Furthermore, Ki... | {
"Header 1": "The first spectroscopic study of a star was made in 1802 by W. H. Wollaston, who observed that the sun emitted a continuous spectrum interrupted by dark lines. In 1811, J. von Fraunhofer, using a diffraction grating, counted about six hundred of these lines, now called Fraunhofer lines, in the solar sp... |
Indeed, by comparing the stellar spectral lines of a given element with those of a reference spectrum taken in the laboratory, the Doppler shift can be determined, which in turn provides the value of the radial velocity. If a star is moving towards the observer lines are
shifted towards the shorter wavelengths (blue ... | {
"Header 1": "The first spectroscopic study of a star was made in 1802 by W. H. Wollaston, who observed that the sun emitted a continuous spectrum interrupted by dark lines. In 1811, J. von Fraunhofer, using a diffraction grating, counted about six hundred of these lines, now called Fraunhofer lines, in the solar sp... |
In this appendix we show how particles are scattered from a central potential V(r) using classical Newtonian mechanics, and we obtain the Rutherford scattering formula [1.58] for the scattering of a beam of particles by a repulsive Coulomb potential.
The path of a particle in the field of a central potential is confi... | {
"Header 1": "I Classical scattering by a central potential",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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That is
$$\sigma_{\text{tot}} = \int \frac{d\sigma}{d\Omega} (\theta, \phi) d\Omega$$
$$= \int_{0}^{2\pi} d\phi \int_{0}^{\pi} d\theta \sin \theta \frac{d\sigma}{d\Omega} (\theta, \phi)$$
[A1.17]
It is worth noting that in defining the above cross-sections we have considered the simple case of *elastic* collision... | {
"Header 1": "I Classical scattering by a central potential",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Thus, using [A1.19] and [A1.30], we find that
$$\frac{d\sigma_{c}}{d\Omega} = \left(\frac{A}{2}\right)^{2} \frac{1}{\sin^{4}(\theta/2)}$$
[A1.32]
or (see [A1.25])
$$\frac{d\sigma_{\rm c}}{d\Omega} = \left(\frac{q_{\rm A}q_{\rm B}}{4\pi\varepsilon_0}\right)^2 \frac{1}{16 E^2 \sin^4(\theta/2)}$$
[A1.33]
where the... | {
"Header 1": "I Classical scattering by a central potential",
"token_count": 550,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Let us consider a non-relativistic collision between a 'beam' particle A of mass $m_A$ and a 'target' particle B of mass $m_B$ . The *laboratory system* (L) is the framework in which the target particle B is at rest before the collision. In what follows we shall use the subscript L to denote quantities in the labora... | {
"Header 1": "2 The laboratory and centre of mass systems",
"token_count": 1821,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We first define the *relative momentum* $\mathbf{p}$ of two particles 1 and 2 having masses $m_1$ and $m_2$ and momenta $\mathbf{p}_1$ and $\mathbf{p}_2$ by the relation
$$\mathbf{p} = \frac{m_2 \mathbf{p}_1 - m_1 \mathbf{p}_2}{m_1 + m_2}$$
[A2.12]
Using this definition, and evaluating the momenta of the ... | {
"Header 1": "2 The laboratory and centre of mass systems",
"token_count": 2046,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We may also write
$$\cos \theta_{\rm L} = \frac{\cos \theta + \tau_{\rm A}}{(1 + 2\tau_{\rm A}\cos \theta + \tau_{\rm A}^2)^{1/2}}$$
[A2.29]
so that the elastic laboratory and CM differential cross-sections are related by
$$\frac{d\sigma}{d\Omega_{\rm I}}(\theta_{\rm L},\,\phi_{\rm L}) = \frac{(1+\tau_{\rm A}^2+2... | {
"Header 1": "2 The laboratory and centre of mass systems",
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In this appendix we shall show how generating functions may be used to evaluate various integrals involving harmonic oscillator or hydrogenic wave functions.
#### Harmonic oscillator
We have seen in Section 2.4 that the Hermite polynomials $H_n(\xi)$ can be expressed in terms of a generating function $G(\xi, s)$... | {
"Header 1": "Evaluation of integrals by using generating functions",
"token_count": 313,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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For this purpose, we consider the two generating functions $G(\xi, s)$ and
$$G(\xi, t) = e^{-t^2 + 2t\xi} = \sum_{m=0}^{\infty} \frac{H_m(\xi)}{m!} t^m$$
[A3.4]
and use [A3.1] and [A3.4] to write
$$\int_{-\infty}^{+\infty} e^{-\xi^2} G(\xi, s) G(\xi, t) d\xi = \sum_{n=0}^{\infty} \sum_{m=0}^{\infty} \frac{s^n t... | {
"Header 1": "Evaluation of integrals by using generating functions",
"token_count": 2040,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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As a simple illustration of the method, we shall normalise the radial hydrogenic eigenfunctions. We see from [3.51] that for this purpose we need the integral
$$I_{pq,pq}^{p+1} = \int_0^\infty e^{-\rho} \rho^{p+1} [L_q^p(\rho)]^2 d\rho$$
[A3.22]
with p = 2l + 1 and q = n + l. Using the generating functions $U_p(... | {
"Header 1": "Evaluation of integrals by using generating functions",
"token_count": 754,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In this appendix, we collect, for the most part without proof, useful relations and results concerned with angular momentum. A good elementary treatment of angular momentum can be found in the texts by Powell and Crasemann (1962) and Merzbacher (1970) while a more advanced and complete treatment can be found in the mon... | {
"Header 1": "Angular momentum – useful formulae and results",
"token_count": 1916,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
Since both $\psi_{jm}$ and $\psi_{jm+1}$ are normalised to unity, we have from [A4.3]
$$N^{2} = \langle \mathcal{J}_{+} \psi_{jm} | \mathcal{J}_{+} \psi_{jm} \rangle = \langle \psi_{jm} | \mathcal{J}_{-} \mathcal{J}_{+} | \psi_{jm} \rangle$$
= $\hbar^{2} (j(j+1) - m(m+1))$ [A4.13]
where we have used [A4.8].... | {
"Header 1": "Angular momentum – useful formulae and results",
"token_count": 1690,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Consider a system described by two angular momenta $J_1$ and $J_2$ , such that the components of $J_1$ commute with the components of $J_2$ . For example, $J_1$ and $J_2$ could be the angular momenta of different particles, or the orbital and spin angular momenta of a single particle. The normalised simultane... | {
"Header 1": "Angular momentum – useful formulae and results",
"Header 3": "Addition of angular momenta. The Clebsch-Gordan coefficients",
"token_count": 1307,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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#### Useful notations
When adding two orbital angular momenta $L_1$ and $L_2$ , we shall write [A4.31] in the explicit position representation as
$$\mathfrak{Y}_{l_1 l_2}^{lm}(\theta_1 \phi_1; \ \theta_2 \phi_2) = \sum_{m_1 m_2} \langle l_1 l_2 m_1 m_2 | lm \rangle \ Y_{l_1 m_1}(\theta_1, \ \phi_1) Y_{l_2 m_2}... | {
"Header 1": "Angular momentum – useful formulae and results",
"Header 3": "Addition of angular momenta. The Clebsch-Gordan coefficients",
"token_count": 1881,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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It can be shown (Powell and Crasemann, 1962; Merzbacher, 1970) that for an operator to be scalar it must commute with all components of the total angular momentum operator **J**:
$$[\mathcal{G}, \mathbf{I}] = 0$$
[A4.41]
from which it follows that if $\psi_{jm}$ is a simultaneous eigenfunction of $\mathbf{J}^2$ ... | {
"Header 1": "Angular momentum – useful formulae and results",
"Header 3": "Addition of angular momenta. The Clebsch-Gordan coefficients",
"token_count": 1663,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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For the case j' = j, $\alpha' = \alpha$ , C can be found by writing
$$\langle \alpha j m | \mathbf{V} \cdot \mathbf{J} | \alpha j m \rangle = \sum_{m'} \langle \alpha j m | \mathbf{V} | \alpha j m' \rangle \cdot \langle \alpha j m' | \mathbf{J} | \alpha j m \rangle$$
[A4.52]
where we have used the closure property... | {
"Header 1": "Angular momentum – useful formulae and results",
"Header 3": "Addition of angular momenta. The Clebsch-Gordan coefficients",
"token_count": 432,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We have seen in Section 2.1 that the wave function in momentum space, $\phi(\mathbf{p})$ , is defined as the Fourier transform of the ordinary wave function $\psi(\mathbf{r})$ in position space. That is,
$$\phi(\mathbf{p}) = (2\pi\hbar)^{-3/2} \int e^{-i\mathbf{p}\cdot\mathbf{r}/\hbar} \psi(\mathbf{r}) d\mathbf{r}... | {
"Header 1": "5 Hydrogenic wave functions in momentum space",
"token_count": 2045,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Hence $\tilde{V}(\mathbf{p}-\mathbf{p}')$ reduces to
$$\tilde{V}(|\mathbf{p} - \mathbf{p}'|) = (2\pi^2)^{-1} \int_0^\infty dr r^2 j_0(qr) V(r)$$
[A5.18]
which agrees with [A5.14] since $j_0(qr) = \sin(qr)/qr$ .
For a central potential V(r) the Schrödinger equation [A5.10] admits solutions of the form $\psi(\m... | {
"Header 1": "5 Hydrogenic wave functions in momentum space",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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By analogy with the radial hydrogenic wave functions $R_{nl}(r)$ , we shall denote the 'radial' hydrogenic momentum space wave functions by $F_{nl}(p)$ instead of $F_{E,l}(p)$ .
The 'radial' functions $F_{nl}(p)$ , corresponding to the hydrogen atom, obtained either by performing directly the Fourier transformat... | {
"Header 1": "5 Hydrogenic wave functions in momentum space",
"token_count": 1837,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We start from Lagrange's equations of motion (Goldstein, 1962)
$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 \quad i = 1, 2, \dots$$
[A6.1]
where $q_i$ are generalised coordinates and L is the Lagrangian function. For a conservative system... | {
"Header 1": "The Hamiltonian for a charged particle in an electromagnetic field",
"token_count": 779,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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In our discussion of atomic structure and of the interaction of atoms with external electromagnetic fields, we have introduced various interactions which are approximations to a complete relativistic theory and which could not be derived, without additional assumptions, from the non-relativistic Schrödinger equation fo... | {
"Header 1": "The Dirac equation and relativistic corrections to the Schrödinger equation",
"token_count": 2036,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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or
$$i\hbar \frac{\partial}{\partial t} \Psi = -i\hbar c\boldsymbol{\alpha} \cdot \nabla \Psi + \beta mc^2 \Psi$$
[A7.14]
More explicitly, we may write [A7.14] as
$$i\hbar \frac{\partial}{\partial t} \Psi_i = -i\hbar c \sum_{j=1}^N \sum_{k=1}^3 \alpha_{ij}^k \frac{\partial}{\partial x_k} \Psi_j + \sum_{j=1}^N \... | {
"Header 1": "The Dirac equation and relativistic corrections to the Schrödinger equation",
"token_count": 1647,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We have seen above that the wave function $\Psi$ may be considered as a column matrix of the form [A7.11], with four components $\Psi_i(i=1,\ldots,4)$ . We can define $\Psi^{\dagger}$ to be a row matrix with components $\Psi_i^{\star}$ , namely
$$\Psi^{\dagger} = (\Psi_2^{\star} \quad \Psi_2^{\star} \quad \Psi_... | {
"Header 1": "Adjoint equation. Continuity equation. Probability and current densities",
"token_count": 1998,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Thus, in Dirac's theory, simultaneous eigenstates of the Hamiltonian [A7.36] and of the operators $J^2$ and $\mathcal{J}_z$ can be found, with eigenvalues given respectively by E, $j(j+1)\hbar^2$ and $m_j\hbar$ .
#### The non-relativistic limit
Let us return to the system of equations [A7.35] for stationary ... | {
"Header 1": "Adjoint equation. Continuity equation. Probability and current densities",
"token_count": 1778,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Substituting in [A7.44a], we find
$$E'\psi(\mathbf{r}) = c^{2}(i\hbar\boldsymbol{\sigma}\cdot\boldsymbol{\nabla})\frac{1}{E'+2mc^{2}-V(r)}\left(i\hbar\boldsymbol{\sigma}\cdot\boldsymbol{\nabla}\right)\psi(\mathbf{r})+V(r)\psi(\mathbf{r})$$
[A7.53]
Expanding $[E' + 2mc^2 - V(r)]^{-1}$ in powers of $[E' - V(r)]/... | {
"Header 1": "Adjoint equation. Continuity equation. Probability and current densities",
"token_count": 1904,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
Let us consider an atom or ion containing a nucleus of mass M and charge Ze and N electrons of mass m and charge (-e). We denote by $\mathbf{R}_0$ the coordinates of the nucleus with respect to a fixed origin O, and by $\mathbf{R}_1$ , $\mathbf{R}_2$ , . . . $\mathbf{R}_N$ those of the electrons. In the absence o... | {
"Header 1": "Separation of the centre of mass coordinates for an *N*-electron atom",
"token_count": 2049,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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The integrals required in the discussion of the hydrogen molecular ion can all be obtained from the basic integral
$$\mathcal{J} = \int \frac{e^{-pr_A}e^{-qr_B}}{r_A r_B} dr$$
[A9.1]
where r, r<sub>A</sub> and r<sub>B</sub> are defined in Fig. 9.8.
This integral is most easily evaluated by introducing confocal el... | {
"Header 1": "**9** Evaluation of two-centre integrals",
"token_count": 890,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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#### **CHAPTER 1**
- 1.1 (a) $v = 4.55 \times 10^7 \text{ m s}^{-1}$ (b) $e/m = 1.65 \times 10^{11} \text{ C kg}^{-1}$ .
- 1.3 The value of $\lambda_{\text{max}}$ is obtained by solving the equation $d\rho(\lambda)/d\lambda = 0$ , where $\rho(\lambda)$ is given by [1.30]. Setting $x = hc/\lambda kT$ , it is f... | {
"Header 1": "IO Solutions to selected problems",
"token_count": 1799,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We take each of the Cartesian coordinates in turn, thus
$$i\hbar \frac{\mathrm{d}}{\mathrm{d}t} \langle x \rangle = \langle [x, H] \rangle$$
Now,
$$[x, H] = x \left\{ -\frac{\hbar^2}{2m} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \right) + V(x, y, z)... | {
"Header 1": "IO Solutions to selected problems",
"token_count": 1974,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
} |
Thus
$$P_{10}(t = +\infty) = \frac{q^2 \mathcal{E}_0^2}{2m\hbar\omega} \frac{\tau^2}{(\tau\omega)^2 + 1}$$
#### **CHAPTER 3**
- 3.2 (a) $r \ge 2a_0$
- (b) Probability is $13e^{-4} = 0.238$ .
- 3.3 (a) Yes, since $\psi_{100}$ , $\psi_{200}$ and $\psi_{322}$ have even parity.
- (b) $P_{100} = 2/7$ , $P_{20... | {
"Header 1": "IO Solutions to selected problems",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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We now have to integrate over all directions of emission, which is equivalent to keeping the direction of emission fixed and integrating over all orientations of $\mathbf{r}_{ba}$ . We have
$$\int d\Omega \sin^2 \theta = \int_{-1}^{+1} d(\cos \theta)(1 - \cos^2 \theta) \int_{0}^{2\pi} d\phi = 8\pi/3$$
from which w... | {
"Header 1": "IO Solutions to selected problems",
"token_count": 1923,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Taking the case $m_s = 1/2$ the possible transitions from the states of the $np_{3/2}$ level to the $n's_{1/2}$ level and the corresponding transition rates are given (apart from an overall constant) in the following table
Initial states Transition rate
$$\begin{array}{ccccccccccccccccccccccccccccccccccc$$
wh... | {
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"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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7.2
$$\Phi_{2^{3}S}(M_{S}=1) = \frac{1}{\sqrt{2}} \begin{vmatrix} u_{1s\uparrow}(1) & u_{2s\uparrow}(1) \\ u_{1s\uparrow}(2) & u_{2s\uparrow}(2) \end{vmatrix}$$
$$\Phi_{2^{3}S}(M_{S}=0) = \frac{1}{\sqrt{2}} \left\{ \frac{1}{\sqrt{2}} \begin{vmatrix} u_{1s\uparrow}(1) & u_{2s\downarrow}(1) \\ u_{1s\uparrow}(2) & u_{... | {
"Header 1": "IO Solutions to selected problems",
"token_count": 1777,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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For the configuration np nd the possible values of 3 and terms $(j_1j_2)_{\tau}$ are then given by
$$j_1j_2$$
$\mathcal{F}$ Terms $(j_1j_2)_{\mathcal{F}}$
$1/2\ 3/2$ $1, 2$ $(1/2\ 3/2)_1, (1/2\ 3/2)_2$
$1/2\ 5/2$ $2, 3$ $(1/2\ 5/2)_2, (1/2\ 5/2)_3$
$3/2\ 3/2$ $0, 1, 2, 3$ $(3/2\ 3/2)_0, (3/2\ 3/2)_1, ... | {
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Assuming the upper levels are equally populated, the relative intensities of the lines are given by [8.11], through the factors $\sin^2\Theta |\langle \mathcal{J}1M_{\mathcal{J}}0|\mathcal{J}'M_{\mathcal{J}'}\rangle|^2$ for the $\pi$ lines and $\frac{1}{2}(1+\cos^2\Theta)$ $|\langle \mathcal{J}1M_{\mathcal{J}}\p... | {
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For v = 1 we have $R = \exp(-\hbar\omega_0/kT)$ .
- 10.3 $\tilde{B} = 1.93 \text{ cm}^{-1}$ ; $R_0 = 1.12 \text{ Å}$ ; $k = 1.9 \times 10^3 \text{ N/m}$ .
- 10.4 (a) The Deslandres table is constructed from [10.33]
- (b) The wave number of the v'=0 to v=0 transition is $\tilde{\nu}=\tilde{\nu}_{s's}-801.54\,\mathr... | {
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11.9 (a)
$$\tan \delta_0 = -\frac{1}{k} \int_0^\infty \sin^2(kr) U(r) dr$$
$$= -\frac{U_0}{k} \int_0^\infty \sin^2(kr) \frac{e^{-\alpha r}}{r} dr = -\frac{U_0}{4k} \log\left(1 + \frac{4k^2}{\alpha^2}\right)$$
(b) $\tan \delta_0 = -\frac{U_0}{k} \int_0^\infty \sin^2(kr) \frac{1}{(r^2 + d^2)^2} dr$
$$= -\frac{\pi ... | {
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The sum can now be extended to cover the ground state term (j = 0), since, as in [13.13], the ground state matrix element $\langle \psi_0 | V | \psi_0 \rangle$ vanishes. This allows us to use the closure relation $\sum_i |\psi_i \rangle \langle \psi_i| = 1$ , giving
$$E^{(2)}(R) = \frac{1}{E_0} \langle \psi_0 | V^... | {
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"token_count": 1997,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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The physical constants listed in Table A11.1 are mainly taken from 'The 1973 Least-Squares Adjustment of the Fundamental Constants' by E. R. Cohen and B. N. Taylor, in J. Phys. Chem. Ref. Data 2, 663 (1973) and from 'The 1973 Table of the Fundamental Physical Constants', by E. R. Cohen, in Atomic Data and Nuclear Data ... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"token_count": 1399,
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|
| N | $M_{ m n}$ |... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"token_count": 1838,
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= \frac{1}{12}M_{12_{\text{C}}} = 1.66057 \times 10^{-27} \text{kg} = 931.502 \text{ MeV}/c^2
1 \text{ J} = 10^7 \text{ erg} = 0.239 \text{ cal} = 6.24146 \times 10^{18} \text{ eV}
1 \text{ cal} = 4.184 \text{ J} = 2.611 \times 10^{19} \text{ eV}
1 \text{ eV} = 1.60219 \times 10^{-19} \text{ J} = 1.60219 \times 10^{-12... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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(1957) Elementary Theory of Angular Momentum. Wiley, New York.
Sakurai, J. J. (1967) Advanced Quantum Mechanics. Addison-Wesley, Reading, Massachusetts.
Schiff, L. I. (1968) Quantum Mechanics 3rd Edn. McGraw-Hill, New York.
Sobelman, I. I. (1979) Atomic Spectra and Radiative Transitions. Springer-Verlag, Berlin. ... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"token_count": 223,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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| A10 molecule, 439 absorption cross section, 495 absorption in scattering, 494–6, 513 absorption of radiation, 155, 161–3 absorption spectra, 27, 362, 434–5, 584 cross-section for, 162 in the dipole approximation, 167–8 by many electron atoms, 355–8 transition rate for, 161–3 alkali metals alkali halides, 420 and bond... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"Header 3": "Index",
"token_count": 14263,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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292, 320 |
| integral equation for, 484–8<br>and the optical theorem, 468, 496 | Slater determinant, 296-8, 320-1, 322-39 |
| partial wave expansion for, 468-75 | passim, 414 |
| partial wa... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"Header 3": "Index",
"token_count": 1489,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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299–300 | 363-4, 366-8, 448-52, 641 |
| closed shell, 300 | spinors, 93 |
| notation for, 300 ... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"Header 3": "Index",
"token_count": 1358,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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J., 4, 5, 6, 15, 18, 26 | 94 |
| Thomson scattering, 18, 19 | vector operators, 94, 216–17, 357, 618–20 |
| thresholds, 463-4 | vibrational motion of a molecule, 384–6, |
| anomalous... | {
"Header 1": "Fundamental constants, atomic units and conversion factors",
"Header 3": "Index",
"token_count": 1180,
"source_pdf": "datasets/websources/Physics_v1/Physics/Physics_of_atoms_and_molecules_Bransden_Joachain.pdf"
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Troy Lectures Second Edition
🔊 📙 Acoustics for Engineers
Springer
Troy Lectures
Troy Lectures
Second Edition
Prof. Jens Blauert, Dr.-Ing., Dr. Tech. h.c. Institute of Communication Acoustics Ruhr-University Bochum 44780 Bochum Germany E-mail: jens.blauert@rub.de
Prof. Ning ... | {
"Header 1": "Acoustics for Engineers",
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This book provides the material for an introductory course in engineering acoustics for students with basic knowledge in mathematics. It is based on extensive teaching experience at the university level.
Under the guidance of an academic teacher it is sufficient as the sole textbook for the subject. Each chapter deal... | {
"Header 1": "Preface",
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| 1 | | Introduction<br> | 1 |
|---|-----|-------------------------------------------------------------|----|
| | 1.1 | Definition of Three Basic Terms<br> | 1 |
| | 1.2 | Specialized Areas within Acoustics<br> | 3 |
| ... | {
"Header 1": "Contents",
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Human beings are usually considered to predominantly perceive their environment through the visual sense – in other words, humans are conceived as visual beings. However, this is certainly not true for inter-individual communication.
In fact, it is audition and not vision that is the most relevant social sense of hum... | {
"Header 1": "Introduction",
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"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
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When you work your way into acoustics, you will usually start with the phenomenon of hearing. Actually, the term acoustics is derived from the Greek verb ακoυ²ιν ´ [ak´uIn], which means to hear. We thus start with the following definition.
Auditory event ... An auditory event is something that exists as heard. It bec... | {
"Header 1": "1.1 Definition of Three Basic Terms",
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In Fig. 1.1 (a) we present a schematic of a transmission system as it is often used in communication technology. A source renders information that is fed into a sender in coded form and transmitted over a channel. At the receiving end, a receiver picks up the transmitted signals, decodes them, and delivers the informat... | {
"Header 1": "1.2 Specialized Areas within Acoustics",
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"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
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Acoustics is a very old science. Pythagoras already knew, around 500 BC, of the quantitative relationship between the length of a string and the pitch of its accompanying auditory event. In 1643, Torricelli demonstrated the vacuum experimentally and showed that there is no sound propagation in it. At the end of the 19t... | {
"Header 1": "1.3 About the History of Acoustics",
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The following quantities are of particular relevance in acoustics.
- Displacement, elongation
- $\overrightarrow{\xi}$ , in [m] ... displacement of an oscillating particle from its resting position
- Particle velocity
- $\overrightarrow{v}$ , in [m/s] ... alternating velocity of an oscillating particle
- Sound pressu... | {
"Header 1": "1.3 About the History of Acoustics",
"Header 2": "1.4 Relevant Quantities in Acoustics",
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In order to derive some illustrative numerical examples, we consider a plane wave in air³. A plane wave is a wave where all quantities are invariant across areas perpendicular to the direction of wave propagation. The field impedance in a plane wave is a quantity that is specific to the medium and is called the charact... | {
"Header 1": "1.5 Some Numerical Examples",
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As shown above, the range of sound pressures that must be handled in acoustics is at least 1 : 10,000,000, which is 1 : 10<sup>7</sup> . This leads to unhandy numbers when describing sound pressures and sound-pressure ratios. For this and other reasons, a logarithmic measure called the level is frequently used. The oth... | {
"Header 1": "1.6 Levels and Logarithmic Frequency Intervals",
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octave-center frequencies
| Octave-center frequency [Hz] | 16 | 32 | 63 | 125 | 250 | 500 | 1k | 2k | 4k | 8k | 16k |
|------------------------------|----|----|----|-----|------|------|------|------|------|------|------|
| Wave length in air [m] | 20 | 10 | 5 | 2.5 | 1.25 | 0.63 | 0.32 | 0.16 | 0.08... | {
"Header 1": "1.6 Levels and Logarithmic Frequency Intervals",
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By plotting levels over logarithmic frequency intervals, we obtain a doublelogarithmic graphic representation of the original quantities. This way of plotting has some advantages over linear representations and is quite popular in acoustics<sup>5</sup> . Figure 1.2 (a) presents an example of a linear representation, an... | {
"Header 1": "1.7 Double-Logarithmic Plots",
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When physical or other quantities vary in a specific way as a function of time, we say that they oscillate. A common, very broad definition of oscillation is as follows.
Oscillation ... An oscillation is a process with attributes that are repeated regularly in time
Oscillating processes are widespread in our world,... | {
"Header 1": "Mechanic and Acoustic Oscillations",
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Three elements are required to form a simple mechanic oscillator, and they include a mass, a spring and a fluidic damper (dashpot) – shown in Fig. 2.1.
Fig. 2.1. Basic elements of linear time-invariant mechanic oscillation systems, (a) mass, (b) spring, (c) fluidic damper (dashpot)
Fo... | {
"Header 1": "2.1 Basic Elements of Linear, Oscillating, Mechanic Systems",
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We now consider an arrangement where a mass, a spring and a dashpot are connected in parallel by idealized, that is, rigid and massless rods – see Fig. 2.2.
Fig. 2.2. Mechanic parallel oscillator, exited by an alternating force. The second port is grounded here for simplicity
The arr... | {
"Header 1": "2.2 Parallel Mechanic Oscillators",
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In this section we deal with the special case in which the oscillator is in a position away from its resting position, and the introduced force is set to zero, that is F(t) = 0 for t > 0. The differential equation (2.13) then converts into a homogenous differential equation as follows,
$$m\frac{\mathrm{d}^2\xi}{\math... | {
"Header 1": "2.3 Free Oscillations of Parallel Mechanic Oscillators",
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The exciting force was zero for free oscillations, but we will now consider the case where the oscillator is driven by an ongoing sinusoidal force, F(t) = Fˆ cos(ωt + φ), with frequency f = ω/2π. The oscillation of the system at this point is stationary<sup>4</sup> . We call this mode of operation force-driven or force... | {
"Header 1": "2.4 Forced Oscillation of Parallel Mechanic Oscillators",
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To derive the energies and losses in the elements from which the oscillator is built, (2.13) is at first multiplied with v(t) to arrive at what is called instantaneous power, namely
$$P(t) = F(t) \ v(t) = m \frac{dv}{dt} v(t) + r v^{2}(t) + \frac{1}{n} v(t) \int \underbrace{v(t) dt}_{dt} . \tag{2.31}$$
Integration ... | {
"Header 1": "2.5 Energies and Dissipation Losses",
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In addition to the mechanic elements, there is a further class of elements for oscillators that are traditionally called acoustic elements. Please note that the terms mechanic and acoustic are historic in this case. Since sound is mechanic, the oscillators built from both classes of elements are, to be sure, mechanic a... | {
"Header 1": "2.6 Basic Elements of Linear, Oscillating, Acoustic Systems",
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The Helmholtz resonator is the best known example of an oscillator with an acoustic element. A Helmholz resonator is commonly demonstrated by blowing over the open end of a bottle to produce a musical tone. This is an auditory event with a distinct pitch that can be varied by filling the bottle with some water.
What ... | {
"Header 1": "2.7 The Helmholtz Resonator",
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During the discussion of simple mechanic and acoustic oscillators in Chapter 2, readers with some electrical engineering experience may have realized that many mathematical formulae are similar to those that appear when dealing with electric oscillators. There is a general isomorphism of the equations in mechanic, acou... | {
"Header 1": "Electromechanic and Electroacoustic Analogies",
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There are two kinds of analogies possible with mechanic networks. Analogy # 1, usually called impedance analogy<sup>1</sup> , is expressed as
$$\underline{F} = \underline{u} \text{ and } \underline{v} = \underline{i},$$
(3.4)
and analogy # 2, also known as mobile analogy or dynamic analogy, is expressed as
$$\und... | {
"Header 1": "3.1 The Electromechanic Analogies",
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Besides m, n, r and L, C, R, respectively, there is an additional mechanic linear element that is frequently found in practical networks, namely, the me-
<sup>2</sup> An additional type of electroacoustic analogy that allows for waves will be introduced later in Section 8.5
Fig. 3.2. E... | {
"Header 1": "3.3 Levers and Transformers",
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When deriving the analogous electric circuit of a mechanic or acoustic circuit, the mechanic or acoustic one-, two- or triple-port elements must be replaced by analogous electric elements. When connecting those elements, the following rules apply.
For electromechanic analogies – refer to Fig. 3.5,
Chains (cascades)... | {
"Header 1": "3.4 Rules for Deriving Analogous Electric Circuits",
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By looking at the electromechanic analogies given in Fig. 3.7, it becomes apparent that the circuits derived by analogy # 2, namely, with F ˆ= i and v ˆ= u, show the same topology as their mechanic counterpart. This behavior is called circuit fidelity or topological fidelity. Please note that in this case impedances tr... | {
"Header 1": "3.6 Circuit Fidelity, Impedance Fidelity and Duality",
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Two examples of simple oscillators and their electric analogies are described below. The first is a mechanic oscillator with two finite masses. This kind of oscillator can be found in many practical applications, including engines dynamically based on concrete plates, ultrasound-source transducers, and vibrating engine... | {
"Header 1": "3.7 Examples of Mechanic and Acoustic Oscillators",
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In the preceding chapter, we dealt with simple linear, time-invariant mechanic and acoustic networks and their electric analogies. In general, these networks can be quite complicated and may assume any number of degrees of freedom. Yet, regardless of how sophisticated the networks are, the energy and power transported ... | {
"Header 1": "Electromechanic and Electroacoustic Transduction",
"token_count": 327,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
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A coupling element between mechanic and electric domains is generally represented as a three-port representation – shown in Fig. 4.1 (a). In a housing of mass, m, rigidly connected to the terminal 2, there is a movable component which can be operated by means of a rigid, massless rod. This rod penetrates the housing fr... | {
"Header 1": "4.1 Electromechanic Couplers as Two- or Three-Port Elements",
"token_count": 658,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
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An important class of electromechanic couplers consists of couplers where the signal-representing quantities in one network, mechanic or electric, accomplish the coupling by controlling elements of the other network.
An illustrative historic example is the carbon microphone, which was an important part of telephone t... | {
"Header 1": "4.1 Electromechanic Couplers as Two- or Three-Port Elements",
"Header 2": "4.2 The Carbon Microphone – A Controlled Coupler",
"token_count": 535,
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From an application standpoint, the most important electromechanic couplers are those where electric power is directly transformed into mechanic power, or vice versa. This class of coupler is called transducers. The term transducer is usually reserved for those couplers that can work bi-directionally and are intrinsica... | {
"Header 1": "4.3 Fundamental Equations of Electroacoustic Transducers",
"token_count": 1689,
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In mechanics as well as in electric networks, the principle of reciprocity may apply. In mechanics, for example, we can observe it in experiments like the one shown in Fig. 4.8.
Fig. 4.8. Mechanic reciprocity experiment with a bending beam, (a) Force applied at position x2, deflection a... | {
"Header 1": "4.4 Reversibility",
"token_count": 750,
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In order for electromechanic transducers to work, they must be coupled to the sound field in such a way that they can either act as $receivers^3$ by withdrawing power from the field or as $emitters^4$ by delivering power to it. Fig. 4.9 illustrates these roles. In this way, we have moved from electromechanic to ele... | {
"Header 1": "4.5 Coupling of Electroacoustic Transducers to the Sound Field",
"token_count": 933,
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Figure 4.11 (a) schematically illustrates the construction of pressure receivers. There is a closed volume with a membrane of effective area, A, covering part of the surface. The complete arrangement is small compared to the wavelength of the sound, λ.
Fig. 4.11. Pressure receiver, (a) ... | {
"Header 1": "4.6 Pressure and Pressure-Gradient Receivers",
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When linearly superimposing a pressure and pressure-gradient receiver, one arrives at a directional characteristic called cardioid. Figure 4.15 (a) shows such a characteristic. The mathematical expression for it is
$$\Gamma = \frac{1}{2} \left( 1 + \cos \delta \right), \tag{4.27}$$
where the reference direction for... | {
"Header 1": "4.7 Further Directional Characteristics",
"token_count": 770,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
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At the supporting transducer we have, due to reversibility as stated in (4.16),
$$\left|\frac{\underline{u}}{\underline{F}}\right|_{\underline{i}=0} = \left|\frac{\underline{v}}{\underline{i}}\right|_{\underline{F}=0} \text{ and, thus, } \left|\underline{T}_{up}\right|_{X} = \left|\frac{\underline{u}}{\underline{p}}\... | {
"Header 1": "4.7 Further Directional Characteristics",
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\tag{4.31}$$
With the acoustic impedance, $|\underline{p}/\underline{q}| = |\underline{Z}_a|$ , which can be computed for the spherical sound field with the distance known, we arrive at
$$|\underline{u}|_{\mathcal{M}_2} = |\underline{p}| |\underline{T}_{\mathrm{up}}|_{\mathcal{M}} = |\underline{q}| |\underline{Z}_... | {
"Header 1": "4.7 Further Directional Characteristics",
"token_count": 558,
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While dealing with magnetic-field transducers in this chapter and electric-field transducers in the next, we will demonstrate that the force-law relationship between the mechanic force, F, and the coupled electric quantity, u or I, is either linear or quadratic. A linear force law, characterized by F ∼ u or F ∼ i, is o... | {
"Header 1": "Magnetic-Field Transducers",
"token_count": 984,
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#### The Inner Transducer
Consider a rod-shaped conductor exposed to a stationary magnetic field, −→<sup>B</sup> – drawn in Fig. 5.2. If an electric current, i, passes through this conductor, a force, −→<sup>F</sup> , will act on it according to the expression
$$\overrightarrow{F} = i \left( \overrightarrow{l} \tim... | {
"Header 1": "5.1 The Magnetodynamic Transduction Principle",
"token_count": 751,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
#### Dynamic Loudspeakers
The dynamic loudspeaker, illustrated in Fig. 5.4, is arguably the most important magnetodynamic sound emitter.
Fig. 5.4. Section view of a dynamic cone loudspeaker
A membrane is elastically supported and driven by a coil carrying alternating current in a st... | {
"Header 1": "5.2 Magnetodynamic Sound Emitters and Receivers",
"token_count": 2047,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Ribbon microphones can also be built to form spherical or cardioid receivers by positioning an acoustic sink to one side of the ribbon. Ribbon microphones are rare today but sometimes irreplaceable. Trumpet sounds in studios, for instance, are often picked up with these microphones since they are difficult to overloa... | {
"Header 1": "5.2 Magnetodynamic Sound Emitters and Receivers",
"token_count": 625,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
#### The Inner Transducer
Figure 5.12 illustrates the fundamental arrangement for the inner-transducer principle.
Fig. 5.12. Electromagnet with a movable armature
To derive the transducer equation, we must compute the force on the movable armature. This is done by imagining a small ... | {
"Header 1": "5.3 The Electromagnetic Transduction Principle",
"token_count": 1071,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Electromagnetic transducers can be built to be very small and very efficient, but, because of their intrinsically quadratic force law, nonlinear distortions are harder to manage than with magnetodynamic transducers. Examples of traditional applications include telephone-receiver capsules, miniature microphones for hear... | {
"Header 1": "5.3 The Electromagnetic Transduction Principle",
"Header 2": "5.4 Electromagnetic Sound Emitters and Receivers",
"token_count": 482,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Rods made of ferromagnetic material experience a variation of their lengths when exposed to magnetic fields. The effect can be conceptualized by considering the distances between the molecules as forming a fictive air gap. This model leads to the same transducer equations as derived for the electromagnetic transducer. ... | {
"Header 1": "5.5 The Magnetostrictive Transduction Principle",
"token_count": 323,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
Magnetostrictive transducers have a very high mechanic impedance, which makes them well-suited to underwater and/or ultrasound applications. This type of transducer may achieve efficiencies of more than 90 % when used in water at ultrasound frequencies. Sometimes pre-magnetization is not applied when this principle is ... | {
"Header 1": "5.6 Magnetostrictive Sound Transmitters and Receivers",
"token_count": 244,
"source_pdf": "datasets/websources/Physics_v1/Physics/acoustics-for-engineers-2nd_troy-lecture.pdf"
} |
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