Showing papers in "Physical Review in 1954"
TL;DR: In this paper, a kinetic theory approach to collision processes in ionized and neutral gases is presented, which is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high pressure limit where the aerodynamic equations are valid.
Abstract: A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high-pressure limit where the aerodynamic equations are valid. It is also possible to satisfy the correct microscopic boundary conditions. The method consists in altering the collision terms in the Boltzmann equation. The modified collision terms are constructed so that each collision conserves particle number, momentum, and energy; other characteristics such as persistence of velocities and angular dependence may be included. The present article illustrates the technique for a simple model involving the assumption of a collision time independent of velocity; this model is applied to the study of small amplitude oscillations of one-component ionized and neutral gases. The initial value problem for unbounded space is solved by performing a Fourier transformation on the space variables and a Laplace transformation on the time variable. For uncharged gases there results the correct adiabatic limiting law for sound-wave propagation at high pressures and, in addition, one obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the difference in the nature of the organization in the low-pressure plasma oscillations and in high-pressure sound-type oscillations is studied. Two important cases are distinguished. If the wavelengths of the oscillations are long compared to either the Debye length or the mean free path, a small change in frequency is obtained as the collision frequency varies from zero to infinity. The accompanying absorption is small; it reaches its maximum value when the collision frequency equals the plasma frequency. The second case refers to waves shorter than both the Debye length and the mean free path; these waves are characterized by a very heavy absorption.
6,627 citations
TL;DR: In this article, the authors considered a radiating gas as a single quantum-mechanical system, and the energy levels corresponding to certain correlations between individual molecules were described, where spontaneous emission of radiation in a transition between two such levels leads to the emission of coherent radiation.
Abstract: By considering a radiating gas as a single quantum-mechanical system, energy levels corresponding to certain correlations between individual molecules are described. Spontaneous emission of radiation in a transition between two such levels leads to the emission of coherent radiation. The discussion is limited first to a gas of dimension small compared with a wavelength. Spontaneous radiation rates and natural line breadths are calculated. For a gas of large extent the effect of photon recoil momentum on coherence is calculated. The effect of a radiation pulse in exciting "super-radiant" states is discussed. The angular correlation between successive photons spontaneously emitted by a gas initially in thermal equilibrium is calculated.
5,672 citations
TL;DR: In this paper, the effect of diffusion on free precession in nuclear resonance has been studied, and a new scheme for measuring the transverse relaxation time is described, which largely circumvents the diffusion effect.
Abstract: Nuclear resonance techniques involving free precession are examined, and, in particular, a convenient variation of Hahn's spin-echo method is described. This variation employs a combination of pulses of different intensity or duration ("90-degree" and "180-degree" pulses). Measurements of the transverse relaxation time ${T}_{2}$ in fluids are often severely compromised by molecular diffusion. Hahn's analysis of the effect of diffusion is reformulated and extended, and a new scheme for measuring ${T}_{2}$ is described which, as predicted by the extended theory, largely circumvents the diffusion effect. On the other hand, the free precession technique, applied in a different way, permits a direct measurement of the molecular self-diffusion constant in suitable fluids. A measurement of the self-diffusion constant of water at 25\ifmmode^\circ\else\textdegree\fi{}C is described which yields $D=2.5(\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$ ${\mathrm{cm}}^{2}$/sec, in good agreement with previous determinations. An analysis of the effect of convection on free precession is also given. A null method for measuring the longitudinal relaxation time ${T}_{1}$, based on the unequal-pulse technique, is described.
5,630 citations
TL;DR: In this paper, the shape of the curve of reflected x-ray intensity vs glancing angle in the region of total reflection was analyzed to study certain structural properties of the mirror surface about 10 to several hundred angstroms deep.
Abstract: Analysis of the shape of the curve of reflected x-ray intensity vs glancing angle in the region of total reflection provides a new method of studying certain structural properties of the mirror surface about 10 to several hundred angstroms deep. Dispersion theory, extended to treat any (small) number of stratified homogeneous media, is used as a basis of interpretation.Curves for evaporated copper on glass at room temperature are studied as an example. These curves may be explained by assuming that the copper (exposed to atmospheric air at room temperature) has completely oxidized about 150A deep. If oxidation is less deep, there probably exists some general reduction of density (e.g., porosity) and an electron density minimum just below an internal oxide seal. This seal, about 25A below the nominal surface plane, arrests further oxidation of more deeply-lying loose-packed copper crystallites.All measurements to date have been carried out under laboratory atmospheric conditions which do not allow satisfactory separation or control of the physical and chemical variables involved in the surface peculiarities. The method, under more controlled conditions of preparation and treatment of the surface, promises to be useful.
4,437 citations
TL;DR: In this paper, the LCAO interpolation method was used as an interpolation technique in connection with more accurate calculations made by the cellular or orthogonalized plane-wave methods.
Abstract: The LCAO, or Bloch, or tight binding, approximation for solids is discussed as an interpolation method, to be used in connection with more accurate calculations made by the cellular or orthogonalized plane-wave methods. It is proposed that the various integrals be obtained as disposable constants, so that the tight binding method will agree with accurate calculations at symmetry points in the Brillouin zone for which these calculations have been made, and that the LCAO method then be used for making calculations throughout the Brillouin zone. A general discussion of the method is given, including tables of matrix components of energy for simple cubic, face-centered and body-centered cubic, and diamond structures. Applications are given to the results of Fletcher and Wohlfarth on Ni, and Howarth on Cu, as illustrations of the fcc case. In discussing the bcc case, the splitting of the energy bands in chromium by an antiferromagnetic alternating potential is worked out, as well as a distribution of energy states for the case of no antiferromagnetism. For diamond, comparisons are made with the calculations of Herman, using the orthogonalized plane-wave method. The case of such crystals as InSb is discussed, and it is shown that their properties fit in with the energy band picture.
3,696 citations
3,408 citations
TL;DR: In this paper, a calculation of the indirect exchange type coupling of nuclear magnetic moments in a metal by means of the hyperfine interaction with the conduction electrons was given, which appears to account qualitatively for the broad nuclear spin resonance lines observed in natural metallic silver.
Abstract: A calculation is given of the indirect exchange ${\mathrm{I}}_{i}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{I}}_{j}$ type coupling of nuclear magnetic moments in a metal by means of the hyperfine interaction with the conduction electrons. The interaction appears to account qualitatively for the broad nuclear spin resonance lines observed in natural metallic silver. It is expected that the interaction may sharpen the resonances in pure isotopic specimens. The line shape of the minority isotope in a binary mixture may tend to be Gaussian, while that of the majority isotope may tend to be Lorentzian, if the indirect exchange interaction is dominant.
2,716 citations
TL;DR: In this article, it was pointed out that the usual principle of invariance under isotopic spin rotation is not consistant with the concept of localized fields, and the possibility of having invariance in local isotope spin rotations was explored.
Abstract: It is pointed out that the usual principle of invariance under isotopic spin rotation is not consistant with the concept of localized fields. The possibility is explored of having invariance under local isotopic spin rotations. This leads to formulating a principle of isotopic gauge invariance and the existence of a b field which has the same relation to the isotopic spin that the electromagnetic field has to the electric charge. The b field satisfies nonlinear differential equations. The quanta of the b field are particles with spin unity, isotopic spin unity, and electric charge $\ifmmode\pm\else\textpm\fi{}e$ or zero.
2,635 citations
TL;DR: In this paper, a natural time-dependent generalization for the well-known pair distribution function $g(mathrm{r})$ of systems of interacting particles is given, which gives rise to a very simple and entirely general expression for the angular and energy distribution of Born approximation scattering by the system.
Abstract: A natural time-dependent generalization is given for the well-known pair distribution function $g(\mathrm{r})$ of systems of interacting particles. The pair distribution in space and time thus defined, denoted by $G(\mathrm{r}, t)$, gives rise to a very simple and entirely general expression for the angular and energy distribution of Born approximation scattering by the system. This expression is the natural extension of the familiar Zernike-Prins formula to scattering in which the energy transfers are not negligible compared to the energy of the scattered particle. It is therefore of particular interest for scattering of slow neutrons by general systems of interacting particles: $G$ is then the proper function in terms of which to analyze the scattering data.After defining the $G$ function and expressing the Born approximation scattering formula in terms of it, the paper studies its general properties and indicates its role for neutron scattering. The qualitative behavior of $G$ for liquids and dense gases is then described and the long-range part exhibited by the function near the critical point is calculated. The explicit expression of $G$ for crystals and for ideal quantum gases is briefly derived and discussed.
2,015 citations
TL;DR: In this article, the complete tensor piezoresistance has been determined experimentally for these materials and expressed in terms of the pressure coefficient of resistivity and two simple shear coefficients.
Abstract: Uniaxial tension causes a change of resistivity in silicon and germanium of both $n$ and $p$ types. The complete tensor piezoresistance has been determined experimentally for these materials and expressed in terms of the pressure coefficient of resistivity and two simple shear coefficients. One of the shear coefficients for each of the materials is exceptionally large and cannot be explained in terms of previously known mechanisms. A possible microscopic mechanism proposed by C. Herring which could account for one large shear constant is discussed. This so called electron transfer effect arises in the structure of the energy bands of these semiconductors, and piezoresistance may therefore give important direct experimental information about this structure.
1,779 citations
TL;DR: In this paper, the effect of spin-orbit coupling on the usual band theory of electrons in a lattice is considered, and particular attention is given to the bands in impurity semiconductors with diamond-type structure.
Abstract: The effect of spin-orbit coupling on the usual band theory of electrons in a lattice is considered. Particular attention is given to the bands in impurity semiconductors with diamond-type structure. $g$-values are calculated for electron states typical of various possible cases and it is found that different values are obtained according as to whether the Fermi level is near or distant from a band degeneracy. The spin-lattice relaxation time is calculated so that the effect of spin-orbit coupling on the wave functions is included, and times in fair agreement with those observed in silicon and alkali metals are obtained.
TL;DR: In this article, the problem of solving the Schrodinger equation in a periodic lattice is studied from the point of view of the variation-iteration method, which leads to a very compact scheme if the potential $V(r)$ is spherically symmetrical within the inscribed spheres of the atomic polyhedra and constant in the space between them.
Abstract: The problem of solving the Schr\"odinger equation in a periodic lattice is studied from the point of view of the variation-iteration method. This approach leads to a very compact scheme if the potential $V(r)$ is spherically symmetrical within the inscribed spheres of the atomic polyhedra and constant in the space between them. The band structure of the lattice is then determined by (1) geometrical structure constants, characteristic of the type of lattice and (2) the logarithmic derivatives, at the surface of the inscribed sphere, of the $s, p, d, \dots{}$ functions corresponding to $V(r)$. By far the greater part of the labor is involved in the calculation of (1), which needs to be done only once for each type of lattice; (2) can be obtained by numerical integration or directly from the atomic spectra. Although derived from a different point of view, this scheme turns out to be essentially equivalent to one proposed by Korringa on the basis of the theory of lattice interferences. The present paper also contains an application to the conduction band of metallic lithium.
TL;DR: In this article, it was shown that the interband matrix elements of the applied electric potential energy combine with the spin-orbit perturbation to give a current perpendicular to both the field and the magnetization.
Abstract: Both the unusually large magnitude and strong temperature dependence of the extraordinary Hall effect in ferromagnetic materials can be understood as effects of the spin-orbit interaction of polarized conduction electrons. It is shown that the interband matrix elements of the applied electric potential energy combine with the spin-orbit perturbation to give a current perpendicular to both the field and the magnetization. Since the net effect of the spin-orbit interaction is proportional to the extent to which the electron spins are aligned, this current is proportional to the magnetization. The magnitude of the Hall constant is equal to the square of the ordinary resistivity multiplied by functions that are not very sensitive to temperature and impurity content. The experimental results behave in such a way also.
TL;DR: In this paper, it was shown that the perturbation series to all orders in the coupling constant takes on very simple asymptotic forms and that the series satisfy certain functional equations by virtue of the renormalizability of the theory.
Abstract: The renormalized propagation functions DFC and SFC for photons and electrons, respectively, are investigated for momenta much greater than the mass of the electron. It is found that in this region the individual terms of the perturbation series to all orders in the coupling constant take on very simple asymptotic forms. An attempt to sum the entire series is only partially successful. It is found that the series satisfy certain functional equations by virtue of the renormalizability of the theory. If photon self-energy parts are omitted from the series, so that D_(FC)=D_F, then S_(FC) has the asymptotic form A[p^2m^2]^n[iγ⋅p]^(−1), where A=A(e_1^2) and n=n(e_1^2). When all diagrams are included, less specific results are found. One conclusion is that the shape of the charge distribution surrounding a test charge in the vacuum does not, at small distances, depend on the coupling constant except through a scale factor. The behavior of the propagation functions for large momenta is related to the magnitude of the renormalization constants in the theory. Thus it is shown that the unrenormalized coupling constant e_0^2/4πℏc, which appears in perturbation theory as a power series in the renormalized coupling constant e_1^2/4πℏc with divergent coefficients, may behave in either of two ways: (a) It may really be infinite as perturbation theory indicates; (b) It may be a finite number independent of e_1^2/4πℏc.
TL;DR: In this paper, the nucleation and growth of ferroelectric domains in barium titanate have been studied as a function of applied electric field and temperature, and experiments were made on thin single-crystal plates normal to $c$ the polar direction.
Abstract: The nucleation and growth of ferroelectric domains in barium titanate have been studied as a function of applied electric field and temperature. The optical and electrical measurements were made on thin single-crystal plates normal to $c$, the polar direction. When the electric field applied along this direction is reversed, new domains with opposite polarization are formed. The manner of growth of these domains is very different from that of domain growth in ferromagnetic materials. The sidewise motion of the 180\ifmmode^\circ\else\textdegree\fi{} side walls (walls between domains with antiparallel polarization) which is common in ferromagnetic crystals is almost never found in barium titanate. Instead its polarization is changed by the formation of very many new anti-parallel domains which are extremely thin (${10}^{\ensuremath{-}4}$ cm) and appear to grow only in the forward direction. The explanation of this behavior is found in the weak coupling between the dipoles perpendicular to the dipole direction. The wall thickness is small, of the order of one to a few lattice constants; the wall energy in BaTi${\mathrm{O}}_{3}$ is of the order of 10 erg/${\mathrm{cm}}^{2}$. Electrical pulsing experiments substantiate the optical observations very clearly. Pulsing the samples at different temperatures shows that the nucleation rate of new domains is accelerated at elevated temperatures. Furthermore, the growth of the new domains is faster at higher temperatures. Experimental results are presented showing how the switching current and the switching time depend on applied electrical field, on temperature, and on the size of the sample.
TL;DR: In this paper, a theory of the Auger transition is presented in which the form of the distribution in energy and relative total yield of ejected electrons are derived, and the effects of variation of atomic energy levels near the metal surface and the Heisenberg uncertainty principle makes it possible to account in detail for the experimentally observed energy distributions as well as the variation of these and of ion kinetic energy.
Abstract: Electrons ejected from atomically clean metals by slow ions of the noble gases arise in Auger transitions which involve either the direct neutralization of the ion or the de-excitation of an excited atom. A theory of these processes is presented in which the form of the distribution in energy and relative total yield, ${\ensuremath{\gamma}}_{i}$, of ejected electrons are derived. Matrix elements are not evaluated from first principles, but specific use of experimental results at two points in the theory leads to a determination of the dependence of the matrix element on distance between the atomic particle and the metal surface and the angle between the excited electron's velocity and the surface normal. Inclusion of the effects of variation of atomic energy levels near the metal surface and the Heisenberg uncertainty principle makes it possible to account in some detail for the experimentally observed energy distributions as well as the variation of these and of ${\ensuremath{\gamma}}_{i}$ with ion kinetic energy. The effect upon the resonance ionization and neutralization processes of the variation of atomic energy levels near the metal surface has also been investigated. The theory predicts a critical distance from the metal surface outside which resonance neutralization and inside which resonance ionization are possible. It has also been possible for the specific case of noble gas ions on tungsten, used as an illustrative example, to determine the relative proportion of electrons ejected by each of the possible Auger processes, to estimate ${\ensuremath{\gamma}}_{i}$ values for ions incident upon a metal with thermal energies, and to fix limits on the width of the filled portion of the conduction band in the metal. The role of the state density function in the metal and the effect of possible variation of the matrix element with electron energy in the band are also investigated.
TL;DR: The spectral distribution of photon generation for the photon-radiative recombination of electrons and holes in germanium is determined from known optical properties by application of the principle of detailed balance.
Abstract: The spectral distribution of the rate of photon generation for the photon-radiative recombination of electrons and holes in germanium is determined from known optical properties by application of the principle of detailed balance. Quantities characterizing the process are evaluated: The thermal equilibrium recombination rate at 300\ifmmode^\circ\else\textdegree\fi{}K is 1.57\ifmmode\times\else\texttimes\fi{}${10}^{13}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ ${\mathrm{sec}}^{\ensuremath{-}1}$, which corresponds to a recombination cross section of 2.9\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}21}$ ${\mathrm{cm}}^{2}$ and a decay time for a small disturbance in carrier concentration in intrinsic material of 0.75 sec. The extension to the steady-state case of added current carriers is given, and estimates are included of the dependence of the quantities on temperature.
TL;DR: In this article, the authors measured the electrical conductivity and Hall effect of single-crystal silicon containing arsenic and boron and derived the intrinsic Hall mobility from Hall coefficient and conductivity.
Abstract: Electrical conductivity and Hall effect have been measured from 10\ifmmode^\circ\else\textdegree\fi{} to 1100\ifmmode^\circ\else\textdegree\fi{} Kelvin on single-crystal silicon containing arsenic and boron. Extrinsic carrier concentration is computed from Hall coefficient. Analysis of extrinsic carrier concentration indicates the ionization energy of arsenic donor levels to be 0.049 ev and of boron acceptor levels to be 0.045 ev for low impurity concentrations. Fermi degeneracy is found to occur in the range ${10}^{18}$ to ${10}^{19}$ ${\mathrm{cm}}^{\ensuremath{-}3}$ impurity concentration. Extrinsic Hall mobility is computed from Hall coefficient and conductivity. Curves of Hall mobility against resistivity at 300\ifmmode^\circ\else\textdegree\fi{}K are computed from theory and compared with experiment. The temperature dependence of lattice-scattering mobility is found from conductivity to be ${T}^{\ensuremath{-}2.6}$ for electrons and ${T}^{\ensuremath{-}2.3}$ for holes. From conductivity mobility and intrinsic conductivity, it is found that carrier concentration at any temperature below 700\ifmmode^\circ\else\textdegree\fi{}K is given by the expression: $np=1.5\ifmmode\times\else\texttimes\fi{}{10}^{33}{T}^{3}\mathrm{exp}(\ensuremath{-}\frac{1.21}{\mathrm{kT}})$. The temperature dependence of the ratio Hall mobility/conductivity mobility is determined for holes and electrons.
TL;DR: In this paper, an optical, dielectric, and structural study has been carried out on NaNb${\mathrm{O}}_{3}$ and the solid solution (Na,K)Nb€{O}$, both single crystals and ceramics being used.
Abstract: An optical, dielectric, and structural study has been carried out on NaNb${\mathrm{O}}_{3}$ and the solid solution (Na,K)Nb${\mathrm{O}}_{3}$, both single crystals and ceramics being used. No evidence for ferroelectricity in NaNb${\mathrm{O}}_{3}$ was found, and the crystal seems to be antiferroelectric in accordance with the nonpolar structure reported by Vousden. It is shown that a small addition of KNb${\mathrm{O}}_{3}$ to pure NaNb${\mathrm{O}}_{3}$ produces a new ferroelectric phase, the existence of which suggests a possible explanation of the conflicting dielectric and structural properties previously reported.The phase diagram of NaNb${\mathrm{O}}_{3}$-KNb${\mathrm{O}}_{3}$ is given. This, together with the optical and x-ray studies of pure NaNb${\mathrm{O}}_{3}$, shows that the three phase transitions in NaNb${\mathrm{O}}_{3}$ are quite different in nature from the BaTi${\mathrm{O}}_{3}$-type transitions in KNb${\mathrm{O}}_{3}$.
TL;DR: In this article, it was argued that the wave function representing an excitation in liquid helium should be nearly of the form Σif(ri)φ, where φ is the ground-state wave function, f(r) is some function of position, and the sum is taken over each atom i.i.
Abstract: It is argued that the wave function representing an excitation in liquid helium should be nearly of the form Σif(ri)φ, where φ is the ground-state wave function, f(r) is some function of position, and the sum is taken over each atom i. In the variational principle this trial function minimizes the energy if f(r)=exp(ik·r), the energy value being E(k)=2k2/2mS(k), where S(k) is the structure factor of the liquid for neutron scattering. For small k, E rises linearly (phonons). For larger k, S(k) has a maximum which makes a ring in the diffraction pattern and a minimum in the E(k) vs k curve. Near the minimum, E(k) behaves as Δ+2(k-k0)2/2μ, which form Landau found agrees with the data on specific heat. The theoretical value of Δ is twice too high, however, indicating need of a better trial function.
Excitations near the minimum are shown to behave in all essential ways like the rotons postulated by Landau. The thermodynamic and hydrodynamic equations of the two-fluid model are discussed from this view. The view is not adequate to deal with the details of the λ transition and with problems of critical flow velocity.
In a dilute solution of He3 atoms in He4, the He3 should move essentially as free particles but of higher effective mass. This mass is calculated, in an appendix, to be about six atomic mass units.
TL;DR: A boundary condition at t = ± ∞ (t being the relative time variable) is obtained for the four-dimensional wave function of a two-body system in a bound state as mentioned in this paper.
Abstract: A boundary condition at t = ± ∞ (t being the “relative” time variable) is obtained for the four-dimensional wave function of a two-body system in a bound state. It is shown that this condition implies that the wave function can be continued analytically to complex values of the “relative time” variable; similarly the wave function in momentum space can be continued analytically to complex values of the “relative energy” variable P 0. In particular one is allowed to consider the wave function for purely imaginary values of t, or respectively p 0, i.e., for real values of x 4 = ict and p 4 = ip 0. A wave equation satisfied by this function is obtained by rotation of the integration path in the complex plane of the variable p 0, and it is further shown that the formulation of the eigenvalue problem in terms of this equation presents several advantages in that many of the ordinary mathematical methods become available.
TL;DR: In this article, the two-nucleon problem is considered in terms of an interaction of the form $(\math{p}|V|{\mathrm{p}}^{\ensuremath{'}})=\ensure-math{-}(\frac{\enuremath{\lambda}}{M})g(\mathrm {p})g({\mathrm''p''''sigma''n}\ifmmode\cdot\else\textperiodcentered\fi{}
Abstract: The two-nucleon problem is considered in terms of an interaction of the form $(\mathrm{p}|V|{\mathrm{p}}^{\ensuremath{'}})=\ensuremath{-}(\frac{\ensuremath{\lambda}}{M})g(\mathrm{p})g({\mathrm{p}}^{\ensuremath{'}})$, with $g(\mathrm{p})=C(p)+(\frac{1}{\ensuremath{\surd}8}){3{p}^{\ensuremath{-}2}({\ensuremath{\sigma}}^{p}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{p})({\ensuremath{\sigma}}^{n}\ifmmode\cdot\else\textperiodcentered\fi{}\mathrm{p})\ensuremath{-}({\ensuremath{\sigma}}^{p}\ifmmode\cdot\else\textperiodcentered\fi{}{\ensuremath{\sigma}}^{n}}T(p)$, where the second term gives rise to a mixture of $D$ state with the $S$ state determined by the first term. After deriving general results valid for any form of $g(\mathrm{p})$, we discuss in detail the special case of $C(p)={({p}^{2}+{\ensuremath{\beta}}^{2})}^{\ensuremath{-}1}$ and $T(p)=\ensuremath{-}t{p}^{2}{({\ensuremath{\gamma}}^{2}+{p}^{2})}^{\ensuremath{-}2}$ and compared it with observed data. The photodisintegration of the deuteron is also discussed and it is found that the $D$-wave part of the deuteron plays an important role at high photon energies leading to a larger cross section than given by other calculations to date.
TL;DR: In this paper, an analytical formula for obtaining the mobility from lattice and impurity mobilities is included, and the effect of electron-electron collisions on the mobility is considered in a qualitative manner.
Abstract: Measurements of conductivity and Hall effect from 11\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K on a set of $n$-type germanium samples covering the range from intrinsic to degenerate are reported. The purity and uniformity of the samples and extensiveness of the data permit a more thorough-going comparison with theory than has been possible in previous work.The theory of mobility is reviewed briefly. The treatment of impurity scattering by Brooks and Herring is presented, and their formula for the impurity mobility is used throughout. An analytical formula for obtaining the mobility from lattice and impurity mobilities is included. The effect of electron-electron collisions on the mobility is considered in a qualitative manner.The principal conclusions concerning the mobility are as follows: (1) Over the range 20.4\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K the lattice mobility varies as ${T}^{\ensuremath{-}1.64}$ rather than the theoretically predicted ${T}^{\ensuremath{-}1.5}$. (2) The impurity mobility increases with temperature less rapidly than the theoretical formula predicts, the exponent of $T$ in the numerator being apparently between 1.0 and 1.5. (3) The Erginsoy formula for neutral impurity scattering appears to fit the experimental data reasonably well for a value of effective mass about one-third the free electron mass. (4) Dislocation scattering is negligible, leading to the conclusion that the density of edge-type dislocations is less than ${10}^{6}$/${\mathrm{cm}}^{2}$.In fitting the concentration data, the parameters involved are activation energy, acceptor concentration, and effective mass. An effective mass in the neighborhood of one-quarter the free electron mass gives the best fit for all samples. The values of acceptor concentration obtained in this way for this effective mass agree well with those calculated from the low-temperature mobility values. The activation energy obtained for the purer samples is 0.0125 ev, in agreement with the value calculated from the hydrogen-like model for one-quarter the free electron mass.The variation of activation energy with concentration does not agree with that observed by Pearson and Bardeen for $p$-type silicon. The effects which have been proposed to explain the variation are: residual potential energy of attraction between free electrons and ionized donors, screening of trapping centers by the free electrons, polarization of neutral centers by free electrons. It is concluded that a combination of the three effects will probably be required to explain the results.The ratio of Hall to drift mobility is shown to agree with the theoretically predicted value within about 10 percent in the range 78\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K.
TL;DR: In this article, a physical model leading to the causal interpretation of quantum theory is proposed, where a set of fields which are equivalent in many ways to a conserved fluid act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid.
Abstract: In this paper, we propose a physical model leading to the causal interpretation of the quantum theory. In this model, a set of fields which are equivalent in many ways to a conserved fluid, with density ${|\ensuremath{\psi}|}^{2}$, and local stream velocity, $\frac{d\ensuremath{\xi}}{\mathrm{dt}}=\frac{\ensuremath{
abla}S}{m}$, act on a particle-like inhomogeneity which moves with the local stream velocity of the equivalent fluid. By introducing the hypothesis of a very irregular and effectively random fluctuation in the motions of the fluid, we are able to prove that an arbitrary probability density ultimately decays into ${|\ensuremath{\psi}|}^{2}$. Thus, we answer an important objection to the causal interpretation, made by Pauli and others. This result is extended to the Dirac equation and to the many-particle problem.
TL;DR: In this article, some special problems of interacting fields that contain removable divergences are treated in detail, and a comparison with the power series renormalization procedures is made with respect to their closed forms.
Abstract: Some special problems of interacting fields that contain removable divergences are treated in detail. Comparisons with the power series renormalization procedures are made. Examination of the closed forms of the solutions before and after renormalization shows that, in one special case solved, coupling-constant renormalization cannot be obtained by any limiting processes that involve only real values of the unrenormalized coupling constant.
TL;DR: In this paper, a simple model for the description of the scattering and the compound nucleus formation by nucleons impinging upon complex nuclei is proposed, which is referred to as the "gross-structure" problem.
Abstract: A simple model is proposed for the description of the scattering and the compound nucleus formation by nucleons impinging upon complex nuclei. It is shown that, by making appropriate averages over resonances, an average problem can be defined which is referred to as the "gross-structure" problem. Solution of this problem permits the calculation of the average total cross section, the cross section for the formation of the compound nucleus, and the part of the elastic-scattering cross section which does not involve formation of the compound nucleus. Unambiguous definitions are given for the latter cross sections.The model describing these properties consists in replacing the nucleus by a one-body potential which acts upon the incident nucleon. This potential $V={V}_{0}+i{V}_{1}$ is complex; the real part represents the average potential in the nucleus; the imaginary part causes an absorption which describes the formation of the compound nucleus. As a first approximation a potential is used whose real part ${V}_{0}$ is a rectangular potential well and whose imaginary part is a constant fraction of the real part ${V}_{1}=\ensuremath{\zeta}{V}_{0}$.This model is used to reproduce the total cross sections for neutrons, the angular dependence of the elastic scattering, and the cross section for the formation of the compound nucleus. It is shown that the average properties of neutron resonances, in particular the ratio of the neutron width to the level spacing, are connected with the gross-structure problem and can be predicted by this model.The observed neutron total cross sections can be very well reproduced in the energy region between zero and 3 Mev with a well depth of 42 Mev, a factor $\ensuremath{\zeta}$ of 0.03, and a nuclear radius of $R=1.45\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{A}^{\frac{1}{3}}$ cm. The angular dependence of the scattering cross section at 1 Mev is fairly well reproduced by the same model. The theoretical and experimental values for the ratios of neutron width to level distance at low energies and the reaction cross sections at 1 Mev do not agree too well but they show a qualitative similarity.
TL;DR: In this paper, it was shown that the first two terms in the expansion of the scattering amplitude of light by a system of spin \textonehalf{} in powers of the frequency can be simply expressed in terms of the macroscopic properties of the system.
Abstract: It is shown that the first two terms in the expansion of the scattering amplitude of light by a system of spin \textonehalf{} in powers of the frequency can be simply expressed in terms of the macroscopic properties of the system. The first term is the well known Thomson amplitude, and depends only on the total charge and mass. The second term is found to depend only on the charge, mass, and magnetic moment of the system.
TL;DR: In this paper, a physical observable called the reaction is defined to simplify the formulation of boundary value problems in electromagnetic theory, which is used to obtain formulas for scattering coefficients, transmission coefficients, and aperture impedances.
Abstract: A physical observable called the reaction is defined to simplify the formulation of boundary value problems in electromagnetic theory. To illustrate its value it is used to obtain formulas for scattering coefficients, transmission coefficients, and aperture impedances. An approximate solution to problems of this type is obtained by replacing the correct source (of the scattered field for example) with an approximate source which is adjusted so that its reaction with certain "test" sources is correct. This insures that the approximate source "looks" the same as the correct source according to the physical tests which are inherent in the problem. The formulas so obtained have a stationary character (for the cases considered) and thus the results could also be obtained from a variational approach. However the physical approach has two important advantages. It is general whereas the variational technique has to be worked out for each problem. It is conceptually simple and leads directly to results which might not be uncovered by the variational approach because of the complexity of the mathematical formulation. The problem of scattering by a dielectric body is used to illustrate this latter point.