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Stable nuclei only occur in a very narrow band in the ZN plane close to the line Z ¼ N (see Figure 2.7). All other nuclei are unstable and decay spontaneously in various ways. Isobars with a large surplus of neutrons gain energy by converting a
Figure 2.7 The distribution of stable nucle... | {
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"Header 2": "2.3 Nuclear Instability",
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Before looking in more detail at different classes of instability, we will consider the general formalism describing the rate of radioactive decay. The probability per unit time that a given nucleus will decay is called its decay constant and is related to the activity A by
$$\mathscr{A} = -\mathrm{d}N/\mathrm{d}t = ... | {
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"Header 2": "2.4 Radioactive Decay",
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Apart from the lightest elements and a few special isolated very stable nuclei, the binding energy data of Figure 2.2 can be fitted by a simple formula containing just a few free parameters. This is the semi-empirical mass formula (SEMF), first written down in 1935 by Weizsa¨cker. It is a semi-empirical formula, becaus... | {
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"Header 2": "2.5 Semi-Empirical Mass Formula: The Liquid Drop Model",
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Using the numerical values of Equation (2.54), the relative sizes of each of the terms in the SEMF may be calculated and for the case of odd-A are shown in Figure 2.11. In this diagram, the volume term is shown as positive and the other terms are subtracted from it to give the final SEMF curve.
![](_page_65_Figure_... | {
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By rearranging terms, the SEMF (2.46) may be written
$$M(Z, A) = \alpha A - \beta Z + \gamma Z^2 + \frac{\delta}{A^{\frac{1}{2}}},$$
(2.57)
where
$$\alpha = M_n - a_v + \frac{a_s}{A^{\frac{1}{3}}} + \frac{a_a}{4}$$
$$\beta = a_a + (M_n - M_p - m_e)$$
$$\gamma = \frac{a_a}{A} + \frac{a_c}{A^{\frac{1}{3}}}$$
... | {
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"Header 2": "2.6 b-Decay Phenomenology",
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Lifetimes of $\beta$ emitters vary enormously from milliseconds to $10^{16}$ years. They
FISSION 59
depend very sensitively on the Q-value for the decay and on the properties of the nuclei involved, e.g. their spins.
#### 2.7 Fission
Spontaneous fission has been defined as the process whereby a parent nucle... | {
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When a heavy nucleus disintegrates by either - or -decay, or by fission, the daughter nucleus is often left in an excited state. If this state is below the excitation energy for fission, it will de-excite, usually by emitting a high-energy photon. The energy of these photons is determined by the average energy level sp... | {
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"Header 2": "2.8 c-Decays",
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In Chapter 1 and earlier sections of the present chapter we discussed various aspects of reactions. In particle physics, because the projectiles and targets have relatively simple structures, this is all that is required in classifying reactions. In nuclear physics, however, because the target has a rich structure it i... | {
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"Header 2": "2.9 Nuclear Reactions",
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However, for production by very slow (thermal) neutrons with energies of the order of 0.02 eV, the available decay kinetic energy will reflect the initial energy of the projectile, which is very small. Therefore, in these cases, photon emission is often preferred. We shall see in Chapter 8 that the fact that radiative ... | {
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"Header 2": "2.9 Nuclear Reactions",
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We have seen that the spin- $\frac{1}{2}$ leptons are one of the three classes of elementary particles in the standard model and we shall start with a discussion of their basic properties. Then we shall look in more detail at the neutral leptons, the neutrinos and, amongst other things, examine an interesting property... | {
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"Header 2": "3.1 Leptons",
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Figure 3.1 Single-photon exchange in the reaction e<sup>þ</sup>e ! <sup>þ</sup>
Figure 3.2 Dominant Feynman diagram for the decay ! ee
In weak interactions more general possibilities are allowed, which still conserve lepton numbers. For example, in th... | {
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Then, in order to preserve the orthonormality of the states, we can write
$$\nu_{\mu} = \nu_1 \cos \alpha + \nu_2 \sin \alpha \tag{3.16}$$
and
$$\nu_x = -\nu_1 \sin \alpha + \nu_2 \cos \alpha. \tag{3.17}$$
Here is a mixing angle which must be determined from experiment. If 6¼ 0 then some interesting predictions... | {
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This publication contains a wealth of useful data about elementary particles and their interactions and we will refer to it in future simply as PDG04
<sup>&</sup>lt;sup>6</sup>Cosmic neutrinos were first detected (independently) by Raymond Davis Jr. and Masatoshi Koshiba, for which they were jointly awarded the 2002 ... | {
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A simultaneous analysis of the data from this experiment and the solar neutrino data yields the result:
$$7.6 \times 10^{-5} \le \Delta(m^2c^4) \le 8.8 \times 10^{-5} (\text{eV})^2, \quad 0.32 \le \tan^2(\alpha) \le 0.48. \tag{3.36}$$
The existence of neutrino oscillations (flavour changing), and by implication non... | {
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These also interact by the weak and electromagnetic interactions, although such effects can often be neglected compared with the strong interactions. To this extent we are entering the realm of 'strong interaction physics'.
#### 3.2.1 Evidence for quarks
Several hundred hadrons (not including nuclei) have been obse... | {
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Except for the top quark, these masses are inferred indirectly from the observed masses of their hadron
**Table 3.2** Properties of quarks: all have spin $\frac{1}{2}$ and masses are given units of GeV/ $c^2$ ; the antiparticles (not shown) have the same masses as their associated particles, but the electric charge... | {
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In principle, the properties of atoms and nuclei can be explained in terms of their proton, neutron and electron constituents, although in practice many details are too complicated to be accurately calculated. However, the properties of these constituents can be determined without reference to atoms and nuclei by study... | {
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"Header 2": "3.3 Hadrons",
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HADRONS 95
Secondly, by constructing all the $\pi N$ isospin states by analogy with Equations (3.50) and (3.51) we can show that
$$\left|\pi^{-}p\right\rangle = \frac{1}{\sqrt{3}}\left|\pi N; \frac{3}{2}, -\frac{1}{2}\right\rangle - \sqrt{\frac{2}{3}}\left|\pi N; \frac{1}{2}, -\frac{1}{2}\right\rangle$$
(3.53a)... | {
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| Table 3.3 | Some examples | of baryons | and | mesons, | with | their | major | decay |
|------------|--------------------|------------|-----|---------|------|-------|-------|-------|
| modes; mas | sses are in MeV/c² | | | | | | | |
| Particle ... | {
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From experiments such as electron scattering we know that hadrons have typical radii r of the order of 1 fm and hence associated time scales r/c of the order of 10<sup>23</sup> s. The vast majority are highly unstable resonances, corresponding to excited HADRONS 101
states of the various quark systems, and decay to... | {
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The 'spin-up' baryon state is given by
$$\begin{vmatrix} B; S = \frac{1}{2}, S_z = \frac{1}{2} \end{vmatrix} = \sqrt{\frac{2}{3}} b; S = \frac{1}{2}, S_z = -\frac{1}{2} \begin{vmatrix} aa; S = 1, S_z = 1 \end{vmatrix}
- \sqrt{\frac{1}{3}} b; S = \frac{1}{2}, S_z = \frac{1}{2} \begin{vmatrix} aa; S = 1, S_z = 0 \end{v... | {
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As we cannot calculate the equivalent quark–quark wavefunction, for the purposes of a phenomenological analysis we will write the contribution to the hadron mass as
$$\Delta M \propto \frac{\mathbf{S}_1 \cdot \mathbf{S}_2}{m_1 m_2}.\tag{3.78}$$
This of course assumes that $|\psi(0)|^2$ is the same for all states,... | {
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Then,
$$M_{\Lambda} = m_u + m_d + m_s + b \left[ \frac{\mathbf{S}_u \cdot \mathbf{S}_d}{m_u m_d} + \frac{\mathbf{S}_u \cdot \mathbf{S}_s}{m_u m_s} + \frac{\mathbf{S}_d \cdot \mathbf{S}_s}{m_d m_s} \right]. \tag{3.90}$$
| The original values | | ... | {
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Assuming maximal mixing and a mean neutrino energy of 3 MeV, use this result to estimate upper and lower bounds on the squared mass of the e.
- 3.12 Comment on the feasibility of the following reactions:
- (a) p þ p ! <sup>þ</sup> þ ;
- (b) p ! e<sup>þ</sup> þ ;
- (c) <sup>0</sup> ! þ ;
- (d) p þ p ! <sup>þ</sup> þ n þ... | {
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To explore the structure of nuclei (nuclear physics) or hadrons (particle physics) requires projectiles whose wavelengths are at least as small as the effective radii of the nuclei or hadrons. This determines the minimum value of the momentum p ¼ h= and hence the energy required. The majority of experiments are conduct... | {
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All accelerators use electromagnetic forces to boost the energy of stable charged particles. These are injected into the machine from a device that provides a highintensity source of low-energy particles, for example an electron gun (a hot filament), or a proton ion source. The accelerators used for nuclear structure s... | {
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"Header 2": "4.2 Accelerators and Beams",
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Accelerators using radio frequency (r.f.) electric fields may conveniently be divided into linear and cyclic varieties.
#### Linear accelerators
In a linear accelerator (or linac) for acclerating ions, particles pass through a series of metal pipes called drift tubes, that are located in a vacuum vessel and connect... | {
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"Header 2": "4.2.2 AC accelerators",
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The use of the terms storage rings and colliders as synonymous is not strictly correct, because we will see that the former can also describe a machine that stores a single beam for use on both internal and external fixed targets.
new particles. Almost all new machines for particle physics are therefore colliders, ... | {
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In order to be detected, a particle must undergo an interaction with the material of a detector. In this section we discuss these interactions, but only in sufficient detail to be able to understand the detectors themselves.
The first possibility is that the particle interacts with an atomic nucleus. For example, thi... | {
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In the 'minimum ionization' region where $\beta\gamma\approx 3$ –4, the minimum value of $-\mathrm{d}E/\mathrm{d}x$ can be calculated from Equation (4.11) and for a particle with unit charge is given approximately by
$$\left(-\frac{\mathrm{d}E}{\mathrm{d}x}\right)_{\mathrm{min}} \approx 3.5 \frac{Z}{A} \mathrm{MeV... | {
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When a charged particle traverses matter it can also lose energy by radiative collisions, especially with nuclei. The electric field of a nucleus will accelerate and decelerate the particles as they pass, causing them to radiate photons, and hence lose energy. This process is called bremsstrahlung (literally 'braking r... | {
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"Header 2": "4.3.3 Radiation energy losses",
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The detection of a particle means more than simply its localization. To be useful this must be done with a resolution sufficient to enable particles to be separated in both space and time in order to determine which are associated with a particular event. We also need to be able to identify each particle and measure it... | {
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"Header 2": "4.4 Particle Detectors",
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Most gas detectors detect the ionization produced by the passage of a charged particle through a gas, typically an inert one such as argon, either by collecting the ionization products or induced charges onto electrodes, or (historically) by making the ionization track visible in some form. The average energy needed to... | {
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2 m long and 1 m in diameter, surrounding the beam pipe of a collider. At each end of the chamber is a segmented layer of proportional counters. The electric drift field E, due to a negative high-voltage electrode plane at the centre of the chamber, and a strong magnetic field B are aligned parallel and anti-parallel... | {
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For charged particles we have seen that energy losses occur due to excitation and ionization of atomic electrons in the medium of the detector. In suitable materials, called scintillators, a small fraction of the excitation energy re-emerges as visible light (or sometimes in the UV region) during de-excitation. In a sc... | {
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"Header 2": "4.4.2 Scintillation counters",
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This method is used, for example, in nuclear physics experiments using very low-energy neutron beams.
However, since all high-energy particles have velocities close to the speed of light, the method ceases to be useful for even quite moderate momenta. This can been seen by taking the relativistic limit of Equation (4... | {
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We assume:
• each electron with $E > E_{\rm C}$ travels one radiation length and then gives up half of its energy to a bremsstrahlung photon;
- each photon with E > E<sup>C</sup> travels one radiation length and then creates an electron–positron pair with each particle having half the energy of the photon;
- elec... | {
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As stated earlier, in particle physics it is necessary to combine several detectors in a single experiment to extract the maximum amount of information from it. Typically, working out from the interaction region, there will be a series of wire chambers, followed further out by calorimeters and at the outermost limits, ... | {
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"Header 2": "**4.5 Layered Detectors**",
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- 4.1 At a collider, a 20 GeV electron beam collides with a 300 GeV proton beam at a crossing angle of 10. Evaluate the total centre-of-mass energy and calculate what beam energy would be required in a fixed-target electron machine to achieve the same total centre-of-mass energy.
- 4.2 What is the length L of the longe... | {
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In Chapter 3 we described the basic properties of quarks and in particular their static properties and how these are used to construct the quark model of hadrons. We now look in more detail at how quarks interact and the role of gluons in the strong interactions. Thus we will be considering dynamical properties and the... | {
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The theory that describes strong interactions in the standard model is called *quantum chromodynamics*, or QCD for short (chromos means colour in Greek). Although QCD is not tested to the same extent or precision as quantum electrodynamics (QED), the quantum theory of electromagnetic interactions, it is nevertheless in... | {
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Some of the features of QCD discussed above are illustrated by considering the static potential between a heavy quark and an antiquark. Such systems give rise to bound states and because the quarks are so heavy they move slowly enough within the resulting hadrons to be treated non-relativistically to a first approximat... | {
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**Table 5.2** Predicted $c\bar{c}$ and $b\bar{b}$ states with $L \leq 2$ and masses up to and just above the charm and bottom thresholds $(3.74\,\text{GeV/c}^2)$ and $10.56\,\text{GeV/c}^2$ , respectively), compared with experimentally observed states (masses are given in MeV/c<sup>2</sup>)
| $n^{2S+1}L_J$... | {
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The strong interaction derives its name from the force that, among other things, binds quarks into hadrons. However, some remarkable phenomena depend on the fact that the interaction gets weaker at short distances; that is, on asymptotic
<sup>10</sup>The equivalent coupling in QED also varies with distance, but the v... | {
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A striking feature of many high-energy particle collisions is the occurrence of jets of hadrons in the final state. We have already mentioned these in Section 3.2.1 when we discussed the experimental evidence for quarks and again when we discussed basic properties of quarks and gluons interactions earlier in this chapt... | {
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"Header 2": "5.5 Jets and Gluons",
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What evidence is there that quarks exist in just three colour states? This question can be settled by using data from electron–positron annihilation. The cross-sections for electron-positron annihilation to hadrons and for electron-positron annihilation to muons<sup>12</sup> both decrease rapidly with energy, but their... | {
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"Header 2": "5.6 Colour Counting",
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In Chapter 2 we discussed the scattering of electrons from nuclei to determine their radial charge distributions. This was done by assuming a form for the charge distribution, calculating the resulting form factor (i.e. the Fourier transform of the charge distribution) and using it to fit experimental cross-sections. I... | {
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Finally, using this mass in Equation(5.36) yields the Callan–Gross relation. Figure 5.16 shows some results for the ratio $2xF_1/F_2$ . It is clear that spin- $\frac{1}{2}$ is strongly favoured.
To deduce the parton charges is more complicated. We will assume that the constituent partons are quarks and show that th... | {
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We have seen that the average binding energy of nucleons in heavy nuclei is of the order of 7–8 MeV per nucleon. As this energy is much smaller than those used in deep inelastic scattering experiments, it might be thought safe to ignore nuclear effects (except those due to the internal motion of the nucleons – the Ferm... | {
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Like the strong and electromagnetic interactions, the weak interaction is also associated with elementary spin-1 bosons, which act as 'force carriers' between quarks and/or leptons. Until 1973 all observed weak interactions were consistent with the hypothesis that they were mediated by the exchange of the charged boson... | {
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"Header 2": "**6.1** Charged and Neutral Currents",
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In this section we will discuss the parity (P) and charge conjugation (C) operators, which were introduced in Chapter 1. These are conserved in the strong and
electromagnetic interactions. The first indication that parity might be violated in weak interactions came from observations on the pionic decays of K-mesons, ... | {
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"Header 2": "6.2 Symmetries of the Weak Interaction",
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Specifically, if we apply the *CP* operator to muon decays, the parity operator changes $\theta$ to $\pi - \theta$ as before, while the *C* operator changes particles to antiparticles. Hence *CP*-invariance alone implies that the condition obtained from *P*-invariance is replaced by the weaker condition
$$\Gamma_... | {
"Header 1": "6",
"Header 2": "6.2 Symmetries of the Weak Interaction",
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We turn now to the spin structure of the weak interactions, which is closely related to the symmetry properties discussed above. As this spin structure takes its simplest form for zero-mass particles, we will discuss the case of neutrinos and antineutrinos first, assuming that they have zero mass for the purpose of thi... | {
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"Header 2": "6.3 Spin Structure of the Weak Interactions",
"token_count": 2006,
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**Figure 6.7** Helicities of the charged leptons in pion decays: the short arrows denote spin vectors and the longer arrows denote momentum vectors
For the case of a positive muon this is unimportant, since it is easy to check that it recoils non-relativistically and so both chiralit... | {
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"Header 2": "6.3 Spin Structure of the Weak Interactions",
"token_count": 1318,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The three intermediate vector bosons mediating weak interactions, the two charged bosons $W^+$ and $W^-$ and the neutral $Z^0$ , were all discovered at CERN in 1983 in the reactions
$$\bar{p} + p \to W^+ + X^-, \quad \bar{p} + p \to W^- + X^+, \quad \text{and} \quad \bar{p} + p \to Z^0 + X^0, \quad (6.20)$$
wh... | {
"Header 1": "6",
"Header 2": "6.4 $W^{\\pm}$ and $Z^{0}$ Bosons",
"token_count": 1223,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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A typical semileptonic decay (i.e. one that involves both hadrons and leptons) is that of the neutron, which at the quark level is
$$d \to u + e^- + \bar{\nu}_e, \tag{6.26}$$
**Figure 6.10** Quark diagram for the decay $n \rightarrow pe^-\bar{\nu}_e$
as illustrated in Figure 6.10, ... | {
"Header 1": "6",
"Header 2": "6.5.1 Semileptonic decays",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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#### 6.5.2 Neutrino scattering
Consider the elastic scattering process $\nu_e + e^- \rightarrow \nu_e + e^-$ at high energies, proceeding via the exchange of a W-meson, i.e. a charged current weak interaction. We know the W-meson couples only to left-handed fermions and from the discussion of Section 6.3.1 that n... | {
"Header 1": "6",
"Header 2": "6.5.1 Semileptonic decays",
"token_count": 1961,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Neutral mesons are of particular interest not only because they enable very sensitive tests of CP-conservation to be made, but also because the application of basic quantum mechanics leads to surprising effects that, for example, allow the symmetry between particles and antiparticles to be tested with extraordinary pre... | {
"Header 1": "6",
"Header 2": "6.6 Neutral Meson Decays",
"token_count": 2046,
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we may write
$$|K_{\mathcal{S}}^{0},\mathbf{0}\rangle = \frac{1}{(1+|\varepsilon|^{2})^{1/2}} \left[ |K_{1}^{0},\mathbf{0}\rangle - \varepsilon|K_{2}^{0},\mathbf{0}\rangle \right]$$
(6.56a)
and
$$|K_{L}^{0}, \mathbf{0}\rangle = \frac{1}{(1+|\varepsilon|^{2})^{1/2}} \left[\varepsilon |K_{1}^{0}, \mathbf{0}\rangle ... | {
"Header 1": "6",
"Header 2": "6.6 Neutral Meson Decays",
"token_count": 1985,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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For example, the neutral kaon produced in the strong interaction
$$\pi^{-} + p \to K^{0} + \Lambda^{0}$$
$$S = 0 \qquad 0 \qquad 1 \qquad -1$$
(6.60)
must necessarily be a $K^0$ state with S=1, in order to conserve strangeness. However, if the produced particle is allowed to travel through free space and its st... | {
"Header 1": "6",
"Header 2": "6.6 Neutral Meson Decays",
"token_count": 1702,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Neutral current reactions are those that involve the emission, absorption or exchange of $Z^0$ bosons. The unified electroweak theory predicted the existence
**Figure 6.18** Higher order contribution to the reaction $e^+\mu^- \to e^+\mu^-$ from the exchange of two W-bosons
of suc... | {
"Header 1": "6",
"Header 2": "6.7 Neutral Currents and the Unified Theory",
"token_count": 2029,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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At still higher energies, the cross-section is dominated by a very large peak at an energy corresponding to the Z<sup>0</sup> mass, as illustrated in Figure 6.20. At this energy the low-energy approximation is irrelevant and Figure 6.20 corresponds to the formation of physical Z<sup>0</sup> bosons in the process e<sup>... | {
"Header 1": "6",
"Header 2": "6.7 Neutral Currents and the Unified Theory",
"token_count": 1906,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Ignore final states that are Cabibbo-suppressed relative to the lepton modes.
- **6.6** The couplings of the $Z^0$ to right-handed (R) and left-handed (L) fermions are given by
$$g_{\rm R}(f) = -q_f \sin^2 \theta_{\rm W}, \quad g_{\rm L}(f) = \pm 1/2 - q_f \sin^2 \theta_{\rm W},$$
where $q_f$ is the electric ch... | {
"Header 1": "6",
"Header 2": "6.7 Neutral Currents and the Unified Theory",
"token_count": 1899,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The existence of stable nuclei implies that overall the net nucleon–nucleon force must be attractive and much stronger than the Coulomb force, although it cannot be attractive for all separations, or otherwise nuclei would collapse in on themselves. So at very short ranges there must be a repulsive core. However, the r... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.1 The Nucleon -- Nucleon Potential",
"token_count": 1805,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In this model, the protons and neutrons that make up the nucleus are assumed to comprise two independent systems of nucleons, each freely moving inside the nuclear volume subject to the constraints of the Pauli principle. The potential felt by every nucleon is the superposition of the potentials due to all the other nu... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.2 Fermi Gas Model",
"token_count": 1716,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The nuclear shell model is based on the analogous model for the orbital structure of atomic electrons in atoms. In some areas it gives more detailed predictions than the Fermi gas model and it can also address questions that the latter model cannot. Firstly, we recap the main features of the atomic case.
#### 7.3.1 S... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.3 Shell Model",
"token_count": 1946,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
} |
$$\mathbf{L} \cdot \mathbf{S} = \frac{1}{2} (\mathbf{J}^2 - \mathbf{L}^2 - \mathbf{S}^2) \tag{7.23}$$
and hence the expectation value of $\mathbf{L} \cdot \mathbf{S}$ , which we write as $\langle \ell s \rangle$ , is
$$\langle \ell s \rangle = \frac{\hbar^2}{2} [j(j+1) - \ell(\ell+1) - s(s+1)] = \begin{cases} \... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.3 Shell Model",
"token_count": 1947,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Using a notation similar to that used in atomic physics, we can write the nuclear magnetic moment as
$$\mu = g_i \, j\mu_{\rm N},\tag{7.28}$$
where $\mu_N$ is the *nuclear magneton* that was used in the discussion of hadron magnetic moments in Section 3.3.3, $g_j$ is the *Landé g-factor* and j is the nuclear sp... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.3 Shell Model",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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So far we have discussed only spherical nuclei, but with non-sphericity new phenomena are allowed, including additional modes of excitation and the possibility of an electric quadrupole moment.
#### 7.4.1 Electric quadrupole moments
The charge distribution in a nucleus is described in terms of electric multipole mo... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.4 Non-Spherical Nuclei",
"token_count": 1847,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The Rainwater model is equivalent to assuming an aspherical liquid drop and Aage Bohr (the son of Neils Bohr) and Mottelson showed that many properties of heavy nuclei could be ascribed to the surface motion of such a drop. However, the singleparticle shell model cannot be abandoned because it explains many general fea... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.4.2 Collective model",
"token_count": 385,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The shell model is based upon the idea that the constituent parts of a nucleus move independently. The liquid-drop model implies just the opposite, since in a drop of incompressible liquid, the motion of any constituent part is correlated with the motion of all the neighbouring pairs. This emphasizes that models in phy... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.5 Summary of Nuclear Structure Models",
"token_count": 268,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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This model assumes that all nuclei have similar mass densities, with binding energies approximately proportional to their masses, just as in a classical charged liquid drop. The model leads to the SEMF, which gives a good description of the average masses and binding energies. It is largely classical, with some quantum... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "Liquid-drop model",
"token_count": 1998,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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\tag{7.52}$$
Finally, since $E_{\alpha}$ is typically 5 MeV and the height of the barrier is typically 40 MeV, $r_{\rm C} \gg R$ and from (7.52), $G \approx 4\pi\alpha Z/\beta$ , where $\beta = v_{\alpha}/c$ and $v_{\alpha}$ is the velocity of the alpha particle within the nucleus.
The probability per unit... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "Liquid-drop model",
"token_count": 823,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In Chapter 2 we discussed in some detail the phenomenology of $\beta$ -decay using the SEMF. In this section we return to these decays and examine their theoretical interpretation.
#### 7.7.1 Fermi theory
The first successful theory of nuclear $\beta$ -decay was proposed in the 1930s by Fermi, long before the W a... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "**7.7** β-Decay",
"token_count": 2032,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Using these in Equation (7.57) and setting $M = G_{\rm F}/V$ , gives
$$\frac{\mathrm{d}\omega}{\mathrm{d}E_e} = \frac{G_\mathrm{F}^2}{2\pi^3 \hbar^7 c^4} p_e E_e p_\nu E_\nu \tag{7.67}$$
where in general
$$p_{\nu}c = \sqrt{E_{\nu}^2 - m_{\nu}^2 c^4} = \sqrt{(E - E_e)^2 - m_{\nu}^2 c^4}.$$
(7.68)
Finally, it is... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "**7.7** β-Decay",
"token_count": 1860,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In Chapter 2 we mentioned that excited states of nuclei frequently decay to lower states (often the ground state) by the emission of photons in the energy range appropriate to -rays and that in addition it is possible for the nucleus to de-excite by ejecting an electron from a low-lying atomic orbit. We shall discuss t... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.8 c-Emission and Internal Conversion",
"token_count": 1975,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Finally, from the work in Chapter 2 on nuclear sizes, we can substitute R ¼ R0A1<sup>=</sup>3, with R<sup>0</sup> ¼ 1:21 fm, to give the final results:
$$B^{E}(L) = \frac{e^{2}}{4\pi} \left(\frac{3}{L+3}\right)^{2} (R_{0})^{2L} A^{2L/3}$$
(7.79a)
and
$$B^{\rm M}(L) = \frac{10}{\pi} \left(\frac{e\hbar}{2m_{\rm p}c... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.8 c-Emission and Internal Conversion",
"token_count": 1997,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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To see the nature of the problem, estimate the fraction of electrons with energies within 10 eV of the end-point.
- 7.11 The electron energy spectra of $\beta$ -decays with very low-energy end-points $E_0$ may be approximated by $d\omega/dE = E^{1/2}(E_0 E)^2$ . Show that in this case the mean energy is $\frac{1}{... | {
"Header 1": "Models and Theories of Nuclear Physics",
"Header 2": "7.8 c-Emission and Internal Conversion",
"token_count": 295,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In Chapter 2 we saw that for a nucleus with A 240, the Coulomb barrier, which inhibits spontaneous fission, is between 5 and 6 MeV. If a neutron with zero kinetic energy enters a nucleus to form a compound nucleus, the latter will have an excitation energy above its ground state equal to the neutron's binding energy in... | {
"Header 1": "8",
"Header 2": "8.1.1 Induced fission -- fissile materials",
"token_count": 1339,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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We have seen in Chapter 2 that in each fission reaction a large amount of energy is produced, which of course is what is needed for power production. However, just as important is the fact that the fission decay products contain other neutrons. For example, we have said that in the case of fission of 235U, on average n... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Because of this, a neutron is much more likely to interact with a nucleus of <sup>238</sup>U. However, a 2 MeV neutron from the primary fission has very little chance of inducing fission in a nucleus of <sup>238</sup>U. Instead it is much more likely to scatter inelastically, leaving the nucleus in an excited state and... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Consider, for example, a reactor containing 100 tonnes of natural uranium, generating a neutron flux of 10<sup>13</sup> cm<sup>2</sup> s<sup>1</sup> and with a fission cross-section for 235U of 580 b at the appropriate energy (see Figure 8.1). Since the fraction of 235U in natural uranium is 0.072 per cent, the number ... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 1986,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The temperature necessary may be estimated from the relation E = kT, where $k_{\rm B}$ is Boltzmann's constant, given by $k_{\rm B} = 8.6 \times 10^{-5}\,{\rm eV}\,{\rm K}^{-1}$ . For an energy of 4.8 MeV, this implies a temperature of $5.6 \times 10^{10}\,{\rm K}$ . This is well above the typical temperature of $... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 2025,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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We saw in Chapter 3 that this is only possible if neutrinos have mass, so a definitive measurement of neutrino masses would be an important piece of evidence to finally resolve the solar neutrino problem. Such measurements should be available in a few years.
The process whereby heavier elements (including the 12C req... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 2030,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The solid curve is the combined effect and is proportional to the overall probability of fusion with a peak at E<sup>0</sup> and a width of E<sup>0</sup>
Figure 8.5 The exponential part of the integrand in Equation (8.39) for the case of pp fusion at a temperature of 2 107 K
FUSION 27... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 2020,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In case (a) the particle traverses a circular orbit of fixed radius (compare the principle of the cyclotron discussed in Chapter 4) and in case (b) the path is a helix of fixed pitch along the direction of the field (compare the motion of electrons in a time projection
chamber, also discussed in Chapter 4). Two techn... | {
"Header 1": "8",
"Header 2": "8.1.2 Fission chain reactions",
"token_count": 941,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Radiation therapy is a long-standing treatment for cancer, often combined with chemotherapy and/or surgery. By damaging DNA, the ability of the cell to reproduce is inhibited and so tumour tissue can, in principle, be destroyed. However, the same of course applies to healthy tissue so, when using radiation in a medical... | {
"Header 1": "8",
"Header 2": "8.3.1 Biological effects of radiation: radiation therapy",
"token_count": 1943,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Alternatively, the radiation may interact directly with the molecule RH again releasing a free radical R:
$$RH \xrightarrow{\text{radiation}} RH^+ + e^-; \quad RH^+ \to R^{\bullet} + H^-.$$
(8.53)
Finally, if the irradiated material is rich in oxygen, yet another set of reactions is possible:
$$R^{\bullet} + O_2 ... | {
"Header 1": "8",
"Header 2": "8.3.1 Biological effects of radiation: radiation therapy",
"token_count": 1989,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Ott, Royal Marsden Hospital, London, UK)
#### Computed tomography
A radiographic image is a two-dimensional display of a three-dimensional structure and although the overlapping images give a useful three-dimensional effect, details are always partially obscured by the superposition of information from underlying a... | {
"Header 1": "8",
"Header 2": "8.3.1 Biological effects of radiation: radiation therapy",
"token_count": 2019,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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In the absence of an external magnetic field, the two states corresponding to the two values of the magnetic quantum number $m_s = \pm \frac{1}{2}$ are equally populated and the net magnetization $\mathbf{M}$ (i.e. the average magnetic moment per unit volume) is zero. In the presence of a static magnetic field $\m... | {
"Header 1": "8",
"Header 2": "8.3.1 Biological effects of radiation: radiation therapy",
"token_count": 1976,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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This means that patients with heart pacemakers, or metal implants cannot in general be scanned and care has to be taken to screen out people who have had an occupational exposure to microscopic fragments of steel (such as welders) as these may well have lodged in critical organs such as the eyes and the latter could be... | {
"Header 1": "8",
"Header 2": "8.3.1 Biological effects of radiation: radiation therapy",
"token_count": 1751,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The Higgs boson is an electrically neutral spin-0 boson whose existence is predicted by the unified electroweak theory, but which has not yet been observed. It is required because of a fundamental symmetry associated with theories in which the force carriers are spin-1 bosons. This symmetry is called gauge invariance a... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.1 The Higgs boson",
"token_count": 1789,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Whether or not the Higgs boson exists is the most pressing unanswered question of the standard model but, even if it is found with its predicted properties, this is not the end of the story, because one of the goals of particle theory is to have a single universal theory that explains all the phenomena of the subject. ... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.2 Grand unification",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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#### 9.1.3 Supersymmetry
One of the problems with GUTs is that if there are new particles associated with the unification energy scale, then they would have to be included as additional contributions in the higher-order calculations in the electroweak theory, for example for the mass of the *W*-boson. These contrib... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.2 Grand unification",
"token_count": 2022,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
} |
#### 9.1.4 Particle astrophysics
Particle physics and astrophysics interact in an increasing number of areas and the resulting field of particle astrophysics is a rapidly expanding one. The interactions are particularly important in the field of cosmology where, for example, the detection of neutrinos can provide u... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.2 Grand unification",
"token_count": 2038,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The ice then refreezes around them. In the first phase of the experiment in 1993/94 (AMANDA-A) four detector strings were located at depths of between 800 and 1000 m. The ice at
Figure 9.8 A schematic diagram of the AMANDA neutrino detector
these depths is filled with air bubbles and ... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.2 Grand unification",
"token_count": 1930,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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The absence of antimatter is completely unexpected because, in the original big bang, it would be natural to assume a total baryon number B ¼ 0.7 Then during the period when kT was large compared with hadron energies, baryons and antibaryons would be in equilibrium with photons via reversible reactions such as
$$p + ... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.1 Particle Physics",
"Header 3": "9.1.2 Grand unification",
"token_count": 631,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Despite more than a century of research, nuclear physics is by no means a 'closed' subject. Even the basic strong nucleon–nucleon force is not fully understood at a phenomenological level, let alone in terms of the fundamental quark–gluon strong interaction. Indeed one of the outstanding problems of nuclear physics is ... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.2 Nuclear Physics",
"token_count": 2042,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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Very often in science new insights are achieved by pushing experiments to their limits. Nuclear physics is no exception. One such limit is the quest for super-heavy elements. Discovery of elements beyond those currently known could explore questions about possible limits on nuclear charges and masses. According to nu... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.2 Nuclear Physics",
"token_count": 2016,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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However, we have also seen in Section 9.1.4 above that the mechanism of violation that can explain meson decays is unable to explain the observed matter/antimatter asymmetry in the universe. Thus it is likely there exists another CP-violating mechanism and hence another source of T violation.
There are several ways i... | {
"Header 1": "Outstanding Questions and Future Prospects",
"Header 2": "9.2 Nuclear Physics",
"token_count": 2035,
"source_pdf": "datasets/websources/Physics_v1/Physics/Martin - Nuclear and Particle Physics - An Introduction.pdf"
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