# Showing papers in "Physical Review in 1953"

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3,910 citations

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TL;DR: In this paper, the effect of mutual electron encounters is considered as a problem of diffusion in velocity space, taking into account a term which previously had been neglected, and the appropriate integro-differential equations are then solved numerically.

Abstract: The coefficients of electrical and thermal conductivity have been computed for completely ionized gases with a wide variety of mean ionic charges. The effect of mutual electron encounters is considered as a problem of diffusion in velocity space, taking into account a term which previously had been neglected. The appropriate integro-differential equations are then solved numerically. The resultant conductivities are very close to the less extensive results obtained with the higher approximations on the Chapman-Cowling method, provided the Debye shielding distance is used as the cutoff in summing the effects of two-body encounters.

1,720 citations

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TL;DR: In this paper, the probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.

Abstract: The probability of a given succession of (nonequilibrium) states of a spontaneously fluctuating thermodynamic system is calculated, on the assumption that the macroscopic variables defining a state are Gaussian random variables whose average behavior is given by the laws governing irreversible processes.This probability can be expressed in terms of the dissipation function; the resulting relation, which is an extension of Boltzmann's principle, shows the statistical significance of the dissipation function. From the form of the relation, the principle of least dissipation of energy becomes evident by inspection.

1,411 citations

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1,362 citations

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TL;DR: In this article, the behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations, and the results are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

Abstract: The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

1,290 citations

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TL;DR: In this article, it was shown that if the electron spin resonance of the conduction electrons is saturated, the nuclei will be polarized to the same degree they would be if their gyromagnetic ratio were that of electron spin.

Abstract: A new method for polarizing nuclei, applicable only to metals, is proposed. It is shown that if the electron spin resonance of the conduction electrons is saturated, the nuclei will be polarized to the same degree they would be if their gyromagnetic ratio were that of the electron spin. This action results from the paramagnetic relaxation processes that occur by means of the hyperfine structure interaction between electron and nuclear spins. A shift of the electron spin resonance due to the same interaction will occur for large amounts of polarization and should provide a direct indication of the degree of polarization.

1,271 citations

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TL;DR: In this article, the Doppler effect results from the recoil momentum changing the translational energy of the radiating atom, and it is shown that the assumption that recoil momentum is given to the radii is incorrect if collisions are taking place.

Abstract: Quantum mechanically the Doppler effect results from the recoil momentum changing the translational energy of the radiating atom. The assumption that the recoil momentum is given to the radiating atom is shown to be incorrect if collisions are taking place. If the collisions do not cause broadening by affecting the internal state of the radiator, they result in a substantial narrowing of the Doppler broadened line.

1,199 citations

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TL;DR: In this paper, a set of electronic polarizabilities has been obtained from a least-squares fit of experimental refraction data using simple additivity and a Lorentz factor of $\frac{4\ensuremath{\pi}}{3}$.

Abstract: A set of electronic polarizabilities has been obtained from a least-squares fit of experimental refraction data using simple additivity and a Lorentz factor of $\frac{4\ensuremath{\pi}}{3}$. Except for the fluorides, the electronic polarizability values of the alkali-halide crystals calculated from this set agree with the experimental data within 3 percent. Similar least-squares fits were attempted with various values of the Lorentz factor, the best fit being obtained for $\frac{4\ensuremath{\pi}}{3}$. On the basis of $\frac{4\ensuremath{\pi}}{3}$, the additivity assumption and the alkali-halide set, polarizabilities have been obtained for other ions. The best values for the sodium $D$ line in ${\mathrm{A}}^{3}$ are ${\mathrm{Li}}^{+}$ 0.03, ${\mathrm{Na}}^{+}$ 0.41, ${\mathrm{K}}^{+}$ 1.33, ${\mathrm{Rb}}^{+}$ 1.98, ${\mathrm{Cs}}^{+}$ 3.34, ${\mathrm{F}}^{\ensuremath{-}}$ 0.64, ${\mathrm{Cl}}^{\ensuremath{-}}$ 2.96, ${\mathrm{Br}}^{\ensuremath{-}}$ 4.16, ${\mathrm{I}}^{\ensuremath{-}}$ 6.43, ${\mathrm{Ca}}^{++}$ 1.1, ${\mathrm{Sr}}^{++}$ 1.6, ${\mathrm{Ba}}^{++}$ 2.5, ${\mathrm{O}}^{\ensuremath{-}\ensuremath{-}}$ 0.5-3.2, ${\mathrm{S}}^{\ensuremath{-}\ensuremath{-}}$ 4.8-5.9, ${\mathrm{Se}}^{\ensuremath{-}\ensuremath{-}}$ 6.0-7.5, ${\mathrm{Te}}^{\ensuremath{-}\ensuremath{-}}$ 8.3-10.2, ${\mathrm{Ag}}^{+}$ 2.4, ${\mathrm{Cu}}^{+}$ 1.6, ${\mathrm{Cu}}^{++}$ 0.2, ${\mathrm{Zn}}^{++}$ 0.8, ${\mathrm{Cd}}^{++}$ 1.8, ${\mathrm{Ge}}^{4+}$ 1, ${\mathrm{Sn}}^{4+}$ 3.4, ${\mathrm{Pb}}^{++}$ 4.9. Values represented by a spread indicate ions that cannot be treated additively.

1,051 citations

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TL;DR: In this paper, it was shown that for a crystal, under the assumption of harmonicity for the interatomic forces and as a consequence of the periodic structure, the frequency distribution function of elastic vibrations has analytic singularities.

Abstract: It is shown that for a crystal, under the assumption of harmonicity for the interatomic forces and as a consequence of the periodic structure, the frequency distribution function of elastic vibrations has analytic singularities. In the general case, the nature of the singularities depends only on the number of dimensions of the crystal. For a two-dimensional crystal, the distribution function has logarithmically infinite peaks. In the three-dimensional case, the distribution function itself is continuous whereas its first derivative exhibits infinite discontinuities. These results are elementary consequences of a theorem of Morse on the existence of saddle points for functions defined on a torus.

950 citations

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Bell Labs

^{1}TL;DR: In this paper, the dependence of the yield on the energy just above a threshold is derived and the derivation is not rigorous because it circumvents some of the difficulties of the three-body problem by applying ergodicity, albeit in a weakened form.

Abstract: When an electron hits an atom or ion, it may knock off an electron This process is fundamental in almost all types of gas discharge The reaction is endothermic; hence there is a threshold value in the electron energy below which it does not occur In this paper, the dependence of the yield on the energy just above this threshold is derived The derivation is not rigorous because it circumvents some of the difficulties of the three-body problem by applying ergodicity, albeit in a weakened form The result is that, for atoms, the yield rises as the 1127th power of the energy excess For ions the exponent lies between this number and unity

814 citations

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TL;DR: In this article, the velocities of elastic waves in stressed solids are derived using Murnaghan's theory of finite deformations and third-order terms in the energy.

Abstract: Expressions for the velocities of elastic waves in stressed solids are derived using Murnaghan's theory of finite deformations and third-order terms in the energy. For isotropic materials, in addition to the Lam\'e constants $\ensuremath{\lambda}$ and $\ensuremath{\mu}$, three additional constants, $l$, $m$, and $n$, are required to describe the material.By measuring the transmission time of elastic pulses through the material, the velocities of longitudinal and shear waves are determined as a function of applied stress. By subjecting the material to hydrostatic pressure as well as simple compression, it is found that seven functions of the three constants $l$, $m$, and $n$ can be measured and thus numerical values calculated. Results are given for polystyrene, iron, and Pyrex glass.

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TL;DR: In addition to the direct magnetic interaction between two nuclear spins in a molecule, the nuclei can have an effective mutual interaction as a result of the magnetic interactions between each nuclei and the electrons of the molecule as discussed by the authors.

Abstract: In addition to the direct magnetic interaction between two nuclear spins in a molecule, the nuclei can have an effective mutual interaction as a result of the magnetic interactions between each of the nuclei and the electrons of the molecule. Although these indirect interactions are in general small, they are important in many cases, particularly as a result of the fact that the frequent collisions which characterize most nuclear paramagnetic resonance experiments cause the direct interaction to average to zero while this is not so for the indirect. In the present paper, the indirect interactions are evaluated. Contributions arise from a simple diamagnetic term, a closely related second-order orbital paramagnetism term, and terms due to the magnetic interaction of the nuclei with the electron spins. Expressions for each of these terms are given both for the general case and for the spherically symmetric case which arises when there are many collisions. Specific numerical calculations of the various effects are given for hydrogen deuteride. It is shown for HD that the diamagnetic terms are of the order of a few tenths of a cycle per second. On the other hand, if the suitable mean energy of the molecular excited states is given the reasonable value of 1.4 Rydbergs, the electron spin terms are 43 cps in agreement with experiment.

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TL;DR: In this article, it was shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate, and that the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated.

Abstract: A method of field quantization is investigated which is more general than the usual methods of quantization in accordance with Bose of Fermi statistics, though these are included in the scheme. The commutation properties and matrix representations of the quantized field amplitudes are determined, and the energy levels of the field are derived in the usual way. It is shown that spin-half fields can be quantized in such a way that an arbitrary finite number of particles can exist in each eigenstate. With the generalized statistics, the interchange of two particles of the same kind may or may not be physically significant, according to the type of interaction by means of which they are created or annihilated. Physical consequences of the assumption that there are particles which obey the generalized statistics are briefly examined.

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TL;DR: In this article, Moli et al. derived Moli's theory of multiple scattering of electrons and other charged particles in a mathematically simpler way and showed that the angular distribution depends only on a single parameter, the screening angle.

Abstract: Mole\`ere's theory of multiple scattering of electrons and other charged particles is here derived in a mathematically simpler way. The differential scattering law enters the theory only through a single parameter, the screening angle ${{\ensuremath{\chi}}_{a}}^{\ensuremath{'}}$, Eq. (21). The angular distribution, except for the absolute scale of angles, depends again only on a single parameter $b$, Eq. (22). It is shown that $b$ depends essentially only on the thickness of the scattering foil in g/${\mathrm{cm}}^{2}$, and is nearly independent of $Z$.The transition to single scattering is re-investigated. An asymptotic formula is obtained which agrees essentially with that of Moli\`ere, Snyder, and Scott, but which remains accurate down to smaller angles, Eq. (38).The theory of Goudsmit and Saunderson has a close quantitative relation to that of Moli\`ere, and a good approximation to their distribution function can be obtained by multiplying Moli\`ere's function by ($\frac{\ensuremath{\theta}}{sin\ensuremath{\theta}}$). This relation holds until the scattering angles become so large that only very few terms in the series of Goudsmit and Saunderson need to be taken into account.

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TL;DR: In this article, a system of linear differential equations of the first order is derived for the distribution matrix, analogous to the classical Boltzmann equation, which allows, upon integration, to determine the macroscopic average value of any spin function in its dependence upon time.

Abstract: Starting from the microscopic viewpoint, the dynamics of nuclear induction is derived by means of statistical methods. The only essential lack of generality lies in the assumption that the nuclei in the sample are independent of each other, so that the treatment does not account for features arising from spin-spin interaction. By considering the simultaneous action of an arbitrary external field and of the molecular surroundings upon a representative nucleus a system of linear differential equations of the first order is derived for the "distribution matrix." It is analogous to the classical Boltzmann equation for the distribution function and allows, upon integration, to determine the macroscopic average value of any spin function in its dependence upon time. This general result is particularly applied to the time dependence of the macroscopic nuclear polarization, and the conditions are investigated under which it satisfies the phenomenological differential equation originally proposed by one of the authors (F.B.). Besides the fact that this equation does not describe line structures caused by the interaction of neighboring spins its validity is found to be seriously restricted only for nuclei having a spin larger than unity and in cases where, in addition, quadrupole relaxation is essential. It demands in these cases that the molecular surroundings are isotropic, e.g., as in gaseous and liquid samples, and further, that their characteristic frequencies of interaction with the nuclei are large compared to the Larmor frequency so that there exists equality between the longitudinal and the transverse relaxation time.

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TL;DR: The general theory of Bloembergen, Purcell, and Pound of nuclear spin relaxation has been extended to a more quantitative study of relaxation by translational diffusion as mentioned in this paper, and it has been found necessary to treat the problem by the theory of random walk.

Abstract: The general theory of Bloembergen, Purcell, and Pound of nuclear spin relaxation has been extended to a more quantitative study of relaxation by translational diffusion. It has been found necessary to treat the problem by the theory of random walk. In the case of isotropic diffusion two cases have been studied: one in which the flight distance has a probability distribution, and the other in which it is constant. The problem of random walk to nearest neighbor sites in a lattice is also treated and quantitative results are obtained for a face-centered cubic lattice.

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TL;DR: In this paper, the distribution function of the frequencies of normal modes of vibration of a disordered chain of one-dimensional harmonic oscillators is calculated analytically, in the limit when the chain becomes infinitely long.

Abstract: By a disordered chain we mean a chain of one-dimensional harmonic oscillators, each coupled to its nearest neighbors by harmonic forces, the inertia of each oscillator and the strength of each coupling being a random variable with a known statistical distribution law. A method is presented for calculating exactly the distribution-function of the frequencies of normal modes of vibration of such a chain, in the limit when the chain becomes infinitely long. For some special examples, in which the distribution law of the oscillator parameters is assumed to be of exponential form, the frequency spectra are calculated analytically. The theory applies equally well to a chain of masses connected by elastic springs and making mechanical vibrations, or to an electrical transmission line composed of alternating inductances and capacitances with random characteristics.

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TL;DR: The theory of the influence on angular correlations of perturbing interactions in the intermediate state is reformulated to allow the description of the effects of time-dependent as well as of static perturbations.

Abstract: The theory of the influence on angular correlations of perturbing interactions in the intermediate state is reformulated to allow the description of the effects of time-dependent as well as of static perturbations. For static interactions of the nuclear electric quadrupole moment with crystalline fields of axial symmetry in polycrystalline sources, attenuation factors are calculated for the coefficients of the various terms in the expansion of the correlation function in Legendre polynomials. No matter how strong the quadrupole interaction, some anisotropy must remain for polycrystalline sources but, for the same interaction in simple single crystals, the anisotropy can be either undisturbed or completely destroyed, depending on the orientation of the crystal. Fields of lower symmetry are shown also to leave, for polycrystalline sources, some anisotropy. Expressions for the influence of randomly fluctuating interactions, such as must exist in liquid sources, are calculated and these predict arbitrarily complete destruction of the correlation under certain conditions, but explain the more nearly unperturbed results usually found with such sources. For electronic shells having magnetic moments, the influences of electronic paramagnetic relaxation and of anisotropy of the hyperfine structure interaction are examined. An applied static magnetic field in the presence of static quadrupole interactions in polycrystalline sources is shown to have differing effects depending on the relative strengths of the two interactions. Application of a magnetic field directed toward a counter cannot reduce the disturbance of the intermediate state in liquid sources, except under special circumstances. The influences of an applied field in the presence of time dependent anisotropic hyperfine structure interactions are discussed. Finally, the feasibility of resonance experiments, for the precise determination of nuclear moments in the intermediate state, is explored.

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TL;DR: In this article, the theory of scattering is developed from first principles with strict attention to the question of the preparation of the state vector of the system appropriate to a description of scattering, and the connection between the present formulation and the more conventional interaction representation and S matrix presentations is traced.

Abstract: The theory of scattering is developed from first principles with strict attention to the question of the preparation of the state vector of the system appropriate to a description of scattering. The connection between the present formulation and the more conventional interaction representation and S matrix presentations is traced. The wave matrix of Moller is introduced and the existence of bound states is discussed in connection with it. A number of applications to rather involved processes are discussed. Finally, the problem of self-energies in field-theoretic scattering calculations is treated.

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Bell Labs

^{1}TL;DR: In this paper, it is shown that the shape of the hysteresis loop at temperatures slightly above the Curie point corresponds to the paraelectric state of BaTi.

Abstract: It is known that the Curie point $\ensuremath{\theta}$ of the ferroelectric BaTi${\mathrm{O}}_{3}$ shifts to higher temperatures when a dc bias field is applied. If the crystal shows a sharp transition, we expect by applying an ac field at the Curie temperature that the crystal would become alternately ferroelectric and nonferroelectric in the cycle of the ac field. This can be seen in the shape of the hysteresis loop at temperatures slightly above $\ensuremath{\theta}$. In the center of the polarization $P$ versus field $E$ plot, we observe a linear behavior corresponding to the paraelectric state of BaTi${\mathrm{O}}_{3}$ above $\ensuremath{\theta}$. At both high voltage ends, however, we observe a hysteresis loop corresponding to the ferroelectric state. A change in temperature causes a change in size and shape of the double hysteresis loops, ranging from a line with curves at the ends (higher temperature) to two overlapping loops (lower temperature). The results obtained allow us to calculate the different constants in the free-energy expression of Devonshire and Slater. One of the results shows that the transition is of the first order since the ${P}^{4}$ term turns out to be negative. The properties of the hysteresis loops are discussed, especially the large spontaneous electrical polarization and the low coercive field strength.

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TL;DR: In this article, a linear theory of the ac behavior of solid or liquid materials containing charge carriers which can move freely within the material but cannot leave it through the electrodes is developed for any degree of dissociation of neutral centers and recombination of positive and negative charge carriers.

Abstract: A linear theory is developed of the ac behavior of solid or liquid materials containing charge carriers which can move freely within the material but cannot leave it through the electrodes. The theory applies for any degree of dissociation of neutral centers and recombination of positive and negative charge carriers, but these carriers are assumed to have been produced by dissociation from only one species of neutral center. The mobile carriers may be electrons, positive holes, positive ions, negative ions, positive ion vacancies, or negative ion vacancies. The general solution for the admittance of the material is obtained for an arbitrary ratio between the mobilities of positive and negative carriers, but, because of the complexity of the result, it is only discussed in detail in the present paper for the following special cases: (a) charge carriers of only one sign mobile, arbitrary recombination time; (b) charge carriers of both signs mobile with the same mobility, arbitrary recombination time; and (c) charge carriers of both signs mobile with unequal mobilities and very short recombination time. In case (a), two dispersion regions may appear, with that at lower frequencies arising from recombination and the other from the finite mobility of the carriers. Both regions follow Debye dispersion curves accurately over a wide frequency range, making it possible to represent the electrical behavior of the material for any recombination time by means of a simple equivalent circuit containing only frequency-independent elements. In cases (b) and (c), only the motional dispersion region appears, and it again follows Debye curves. Finally, the results of the present theory are compared with those of other theories of ac space-charge effects in semiconductors and electrolytes.

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TL;DR: The electrical conductivity of black phosphorus has been measured as a function of temperature and pressure up to 350 and 8000 kg/${\mathrm{cm}}^{2}$/volt sec, respectively.

Abstract: The electrical conductivity of black phosphorus has been measured as a function of temperature and pressure up to 350\ifmmode^\circ\else\textdegree\fi{}C and 8000 kg/${\mathrm{cm}}^{2}$. The Hall constant of the same material has been measured as a function of temperature at atmospheric pressure. At low temperatures $p$-type impurity conduction is observed; at high temperatures the phosphorus is an intrinsic semiconductor with a gap width of 0.33 ev. The mobilities at 27\ifmmode^\circ\else\textdegree\fi{}C are 350 ${\mathrm{cm}}^{2}$/volt sec and 220 ${\mathrm{cm}}^{2}$/volt sec for the holes and electrons, respectively. Application of hydrostatic pressure decreases the gap at a rate $\frac{\mathrm{VdW}}{\mathrm{dV}}=8.3$ ev. The results are also interpreted in terms of a two-dimensional semiconductor model.

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TL;DR: In this paper, it was shown that the unusual observed saturation behavior of the microwave electron spin resonance associated with $F$ centers in KCl, NaCl, and KBr crystals can be accounted for if the overall width is ascribed to interaction between the $F $-center electrons and the nuclear magnetic moments of the ions adjacent to the centers.

Abstract: It is shown that the unusual observed saturation behavior of the microwave electron spin resonance associated with $F$ centers in KCl, NaCl, and KBr crystals can be accounted for if the overall width is ascribed to interaction between the $F$-center electrons and the nuclear magnetic moments of the ions adjacent to the $F$ centers. The measured saturation factor gives for $F$ centers in KCl a spin-lattice relaxation time of 2.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ sec at room temperature. The observed saturation behavior in which only the absorption saturates is in marked disagreement with the Kramers-Kronig relations. However it is shown, that the Kramers-Kronig relations are not applicable to saturated systems. Expressions which avoid the use of these relations are presented for saturable systems.

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TL;DR: In this article, the dependence on volume of the velocity of plane waves in one-, two-, and three-dimensional lattices is obtained, and hence the corresponding Gruneisen parameter is predicted for a body with a purely harmonic atomic potential.

Abstract: Under appropriate conditions, the thermal expansion of a solid is closely related to the Gr\"uneisen parameter $\ensuremath{\gamma}$. This in turn may be derived from the variation of the characteristic frequency of the lattice with volume. If, however, this variation is calculated from the usual expressions for the velocity of sound in solids at zero pressure, the $\ensuremath{\gamma}$ does not agree with that predicted from lattice theory, and an anomalous thermal expansion is predicted for a solid with a purely harmonic atomic potential.General expressions for the dependence on volume of the velocity of plane waves in one-, two-, and three-dimensional lattices are obtained, and hence the corresponding Gr\"uneisen $\ensuremath{\gamma}'\mathrm{s}$. The three-dimensional $\ensuremath{\gamma}$ differs by a numerical constant from that used by Slater. All three expressions are now consistent when applied to a body with a purely harmonic atomic potential and predict no thermal expansion.

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TL;DR: In this article, a number of problems arising in the analysis of angular correlation and angular distribution data are considered, including corrections for finite angular resolution, and a brief discussion of the corrections for a decaying source.

Abstract: Consideration of a number of problems which arise in the analysis of angular correlation and angular distribution data is given. These include corrections for finite angular resolution. For a source of constant strength the questions of determination of most probable counting rates, their associated errors and the determination of the most probable coefficients in the Legendre representation of the data, as well as the errors in the determination of these coefficients, are examined. A brief discussion of the corrections for a decaying source is also presented.

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TL;DR: In this paper, the results of the previous paper are extended to second-order systems, i.e., systems with inertia using a generalized definition of the thermodynamic forces, reciprocal relations for the dissipative coefficients in the equations describing irreversible processes are derived.

Abstract: The results of the previous paper are extended to second-order systems, ie, systems with inertia Using a generalized definition of the thermodynamic forces, reciprocal relations for the dissipative coefficients in the equations describing irreversible processes are derived A dissipation function can again be defined, and it can again be used to express the probability functional for fluctuations

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TL;DR: In this article, the effects of the Coulomb interaction between free electrons in an electron gas are considered for a variety of phenomena, and the analysis is based on the collective description, which describes the long-range correlations in electronic positions in terms of the collective oscillations of the system as a whole.

Abstract: The effects of the Coulomb interaction between free electrons in an electron gas are considered for a variety of phenomena. The analysis is based on the collective description, which describes the long-range correlations in electronic positions (due to the Coulomb force) in terms of the collective oscillations of the system as a whole. It is shown that an independent electron model should provide a good description of the electrons in a metal in many cases of interest. The ground state energy of the free electron gas is determined, and an estimate of the correlation energy is obtained, with results in good agreement with those of Wigner. The exchange energy is shown to be greatly reduced by the long-range correlations, so that its effect on the level density and the specific heat is comparatively slight, leading to an electronic specific heat for Na which is approximately 80 percent of the free-electron value. The possible ferromagnetism of a free-electron gas is investigated, and it is found that the long-range Coulomb correlations are such that a free-electron gas will never become ferromagnetic (no matter how low the density). The excitation of the collective oscillations by a fast charged particle is studied, and the semiclassical results obtained by Bohm and Pines are verified by a quantum-mechanical calculation. The results are applied to the experiments of Ruthermann and Lang on the scattering of electrons by thin metallic films and to experiments on the stopping power of light metals for fast charged particles, with resulting good agreement between theory and experiment.