Showing papers in "Physical Review in 1956"
TL;DR: The question of parity conservation in β decays and in hyperon and meson decays is examined in this paper, where possible experiments are suggested which might test parity conservation of these interactions.
Abstract: The question of parity conservation in β decays and in hyperon and meson decays is examined. Possible experiments are suggested which might test parity conservation in these interactions.
1,980 citations
TL;DR: In this article, a new type of magnetic anisotropy was discovered which is best described as an exchange anisotropic, which is the result of an interaction between an antiferromagnetic material and a ferromagnetic materials.
Abstract: A new type of magnetic anisotropy has been discovered which is best described as an exchange anisotropy. This anisotropy is the result of an interaction between an antiferromagnetic material and a ferromagnetic material. The material that exhibits this exchange anisotropy is a compact of fine particles of cobalt with a cobaltous oxide shell. The effect occurs only below the N\'eel temperature of the antiferromagnetic material, which is essentially room temperature for the cobaltous oxide. An exchange torque is inferred to exist between the metal and oxide which has a maximum value at 77\ifmmode^\circ\else\textdegree\fi{}K of \ensuremath{\sim}2 dyne-cm/${\mathrm{cm}}^{2}$ of interface.
1,846 citations
1,559 citations
TL;DR: In this paper, the most general form of the Hamiltonian of an electron or hole in a semiconductor such as Si or Ge, in the presence of an external homogeneous magnetic field, is given.
Abstract: The most general form of the Hamiltonian of an electron or hole in a semiconductor such as Si or Ge, in the presence of an external homogeneous magnetic field, is given. Two methods of obtaining the corresponding energy levels are discussed. The first should yield very accurate values for the magnetic field in the (111) direction for either Si or Ge. The second is a perturbation method and is expected to give good results only for Ge.
1,475 citations
TL;DR: In this paper, a general rule is obtained for introducing a new field in a definite way with a definite type of interaction with the original fields by postulating the invariance of these systems under a wider group derived by replacing the parameters of the original group with a set of arbitrary functions.
Abstract: Some systems of fields have been considered which are invariant under a certain group of transformations depending on $n$ parameters. A general rule is obtained for introducing a new field in a definite way with a definite type of interaction with the original fields by postulating the invariance of these systems under a wider group derived by replacing the parameters of the original group with a set of arbitrary functions. The transformation character of this new field under the wider group is determined from the invariance postulate. The possible types of the equations of the new fields can be also derived, giving rise to a certain conservation law owing to the invariance. As examples, the electromagnetic, the gravitational and the Yang-Mills fields are reconsidered following this line of approach.
1,276 citations
TL;DR: In this paper, a general expression for the emitted current as a function of field, temperature, and work function is set up in the form of a definite integral, and each type of emission is associated with a technique for approximating the integral and with a characteristic dependence on the three parameters.
Abstract: Although the theories of thermionic and field emission of electrons from metals are very well understood, the two types of emission have usually been studied separately by first specifying the range of temperature and field and then constructing the appropriate expression for the current. In this paper the emission is treated from a unified point of view in order to establish the ranges of temperature and field for the two types of emission and to investigate the current in the region intermediate between thermionic and field emission. A general expression for the emitted current as a function of field, temperature, and work function is set up in the form of a definite integral. Each type of emission is then associated with a technique for approximating the integral and with a characteristic dependence on the three parameters. An approximation for low fields and high temperatures leads to an extension of the Richardson-Schottky formula for thermionic emission. The values of temperature and field for which it applies are established by considering the validity of the approximation. An analogous treatment of the integral, for high fields and low temperatures, gives an extension of the Fowler-Nordheim formula for field emission, and establishes the region of temperature and field in which it applies. Also another approximate method for evaluating the integral is given which leads to a new type of dependence of the emitted current on temperature and field and which applies in a narrow region of temperature and field intermediate between the field and thermionic emission regions.
1,242 citations
TL;DR: In this paper, the authors generalized Bose-Einstein condensation to a system of interacting particles and showed that B.E. condensation is present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity.
Abstract: The mathematical description of B.E. (Bose-Einstein) condensation is generalized so as to be applicable to a system of interacting particles. B.E. condensation is said to be present whenever the largest eigenvalue of the one-particle reduced density matrix is an extensive rather than an intensive quantity. Some transformations facilitating the practical use of this definition are given.An argument based on first principles is given, indicating that liquid belium II in equilibrium shows B.E. condensation. For absolute zero, the argument is based on properties of the ground-state wave function derived from the assumption that there is no "long-range configurational order." A crude estimate indicates that roughly 8% of the atoms are "condensed" (note that the fraction of condensed particles need not be identified with $\frac{{\ensuremath{\rho}}_{s}}{\ensuremath{\rho}}$). Conversely, it is shown why one would not expect B.E. condensation in a solid. For finite temperatures Feynman's theory of the lambda-transition is applied: Feynman's approximations are shown to imply that our criterion of B.E. condensation is satisfied below the lambda-transition but not above it.
1,224 citations
TL;DR: In this paper, an ideal model of a ferromagnet is studied, consisting of a lattice of identical spins with cubic symmetry and with isotropic exchange coupling between nearest neighbors.
Abstract: An ideal model of a ferromagnet is studied, consisting of a lattice of identical spins with cubic symmetry and with isotropic exchange coupling between nearest neighbors. The aim is to obtain a complete description of the thermodynamic properties of the system at low temperatures, far below the Curie point. In this temperature region the natural description of the states of the system is in terms of Bloch spin waves. The nonorthogonality of spin-wave states raises basic difficulties which are examined and overcome.The following new results are obtained: a practical method for calculating thermodynamic quantities in terms of a nonorthogonal set of basic states; a proof that in 3 dimensions there do not exist states (shown by Bethe to exist in a one-dimensional chain of spins) in which two spins are bound together into a stable complex and travel together through the lattice; a calculation of the scattering cross section of two spin waves, giving a mean free path for spin-spin collisions proportional to ${T}^{\ensuremath{-}\frac{7}{2}}$ at low temperatures; and an exact formula for the free energy of the system, showing explicitly the effects of spin-wave interactions.Quantitative results based on this theory will be published in a second paper.
1,132 citations
Abstract: An ohmic contact between a metal and an insulator facilitates the injection of electrons into the insulator. Subsequent flow of the electrons is space-charge limited. In real insulators the trapping of electrons in localized states in the forbidden gap profoundly influences the current flow. The interesting features of the current density-voltage ($J\ensuremath{-}V$) characteristic are confined within a "triangle" in the $logJ\ensuremath{-}logV$ plane bounded by three limiting curves: Ohm's law, Child's law for solids ($J\ensuremath{\propto}{V}^{2}$) and a traps-filled-limit curve which has a voltage threshold and an enormously steep current rise. Simple inequalities relating the true field at the anode to the ohmic field facilitate qualitative discussion of the $J\ensuremath{-}V$ characteristic. Exact solutions have been obtained for an insulator with a single, discrete trap level in a simplified theory which idealizes the ohmic contact and neglects the diffusive contribution to the current. The discrete trap level produces the same type of nonlinearity discovered by Smith and Rose and attributed by them to traps distributed in energy.
1,079 citations
TL;DR: In this article, the phenomenological Bloch equations in nuclear magnetic resonance are generalized by the addition of terms due to the transfer of magnetization by diffusion, and the revised equations describe phenomena under conditions of inhomogeneity in magnetic field, relaxation rates, or initial magnetization.
Abstract: The phenomenological Bloch equations in nuclear magnetic resonance are generalized by the addition of terms due to the transfer of magnetization by diffusion. The revised equations describe phenomena under conditions of inhomogeneity in magnetic field, relaxation rates, or initial magnetization. As an example the equations are solved in the case of the free precession of magnetic moment in the presence of an inhomogeneous magnetic field following the application of a 90\ifmmode^\circ\else\textdegree\fi{} pulse with subsequent applications of a succession of 180\ifmmode^\circ\else\textdegree\fi{} pulses. The spin-echo amplitudes agree with the results of Carr and Purcell from a random walk theory.
1,026 citations
TL;DR: In this article, a transport theory which allows for anisotropy in the scattering processes is developed for semiconductors with multiple nondegenerate band edge points, and the main effects of scattering on the distribution function over each ellipsoidal constant-energy surface can be described by a set of three relaxation times, one for each principal direction; these are the principal components of an energy-dependent relaxation-time tensor.
Abstract: A transport theory which allows for anisotropy in the scattering processes is developed for semiconductors with multiple nondegenerate band edge points. It is found that the main effects of scattering on the distribution function over each ellipsoidal constant-energy surface can be described by a set of three relaxation times, one for each principal direction; these are the principal components of an energy-dependent relaxation-time tensor. This approximate solution can be used if all scattering processes either conserve energy or randomize velocities. Expressions for mobility, Hall effect, low- and high-field magnetoresistance, piezoresistance, and high-frequency dielectric constant are derived in terms of the relaxation-time tensor. For static-field transport properties the effect of anisotropic scattering is merely to weight each component of the effective-mass tensor, as it appears in the usual theory, with the reciprocal of the corresponding component of the relaxation-time tensor.The deformation-potential method of Bardeen and Shockley is generalized to include scattering by transverse as well as longitudinal acoustic modes. This generalized theory is used to calculate the acoustic contributions to the components of the relaxation-time tensor in terms of the effective masses, elastic constants, and a set of deformation-potential constants. For $n$ silicon and $n$ germanium, one of the two deformation-potential constants can be obtained from piezoresistance data. The other one can at present only be roughly estimated, e.g., from the anisotropy of magnetoresistance. Insertion of these constants into the theory yields a value for the acoustic mobility of $n$ germanium which is in reasonable agreement with observation; a more accurate check of the theory may be possible when better input data are available. For $n$ silicon, available data do not suffice for a check of the theory.
TL;DR: A rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part.
Abstract: "Strict causality" is the assumption that no signal whatsoever can be transmitted over a space-like interval in space-time, or that no signal can travel faster than the velocity of light in vacuo. In this paper a rigorous proof is given of the logical equivalence of strict causality ("no output before the input") and the validity of a dispersion relation, e.g., the relation expressing the real part of a generalized scattering amplitude as an integral involving the imaginary part. This proof applies to a general linear system with a time-independent connection between the output and a freely variable input and has the advantage over previous work that no tacit assumptions are made about the analytic behavior or single-valuedness of the amplitude, but, on the contrary, strict causality is shown to imply that the generalized scattering amplitude is analytic in the upper half of the complex frequency plane. The dispersion relations are given first as a relation between the real and imaginary parts of the generalized scattering amplitude and then in terms of the complex phase shift.
TL;DR: In this paper, the fluctuations of the neutron reduced widths from the resonance region of intermediate and heavy nuclei have been analyzed by a statistical procedure which is based on the method of maximum likelihood.
Abstract: The fluctuations of the neutron reduced widths from the resonance region of intermediate and heavy nuclei have been analyzed by a statistical procedure which is based on the method of maximum likelihood. It is found that a chi-squared distribution with one degree of freedom is quite consistent with the data while a chi-squared distribution with two degrees of freedom (an exponential distribution) is not. The former distribution corresponds to a Gaussian distribution for the reduced-width amplitude, and a plausibility argument is given for it which is based on the consideration of the matrix elements for neutron emission from the compound nucleus and of the central limit theorem of statistics. This argument also suggests that within the framework of the compound-nucleus theory all reduced-width amplitudes have Gaussian distributions, and that many of the distributions for the various channels may be independent. One consequence of the latter suggestion is that the total radiation width for a given spin state which is formed in neutron capture will be essentially constant, in agreement with some observations, because it is the sum of many partial radiation widths. The fluctuations of the provisional fission widths of ${\mathrm{U}}^{235}$ are best described by a chisquared distribution with about 2\textonehalf{} degrees of freedom, indicating that there are effectively only a few independently contributing fission channels.
TL;DR: In this article, the conduction-electron magnetic susceptibility of graphite has been calculated by using the Wallace two-dimensional band structure, taking into account the large (in this case) effects of band-to-band transitions which are not included in the Landau-Peierls treatment.
Abstract: The conduction-electron magnetic susceptibility of graphite has been calculated by using the Wallace two-dimensional band structure. The energy levels induced by the magnetic field are calculated by the method of Luttinger and Kohn, taking into account the large (in this case) effects of band-to-band transitions which are not included in the Landau-Peierls treatment. Agreement with the susceptibility observed at high temperatures is obtained with a choice of 2.6 ev for the resonance-integral parameter ${\ensuremath{\gamma}}_{0}$. The details of the de Haas-van Alphen effect cannot be reproduced, indicating that a more complicated band structure is needed to account for the low-temperature experiments.
TL;DR: In this article, the effect of multiple scattering on bremsstrahlung and pair production was considered and it was shown that the probability of these processes decreases considerably at energies ε(n −1 −1/n −2 −3/n) = 10 −13/n ev.
Abstract: The effect of multiple scattering on bremsstrahlung and pair production is considered. The probability of these processes decreases considerably at energies \ensuremath{\gtrsim}${10}^{13}$ ev.The calculations are carried out with the aid of the density matrix. The formulas thus obtained yield the probability of pair production and bremsstrahlung for arbitrary electron and photon energies.
TL;DR: In this paper, a simple Born-approximation analysis of the elastic data yields values of the effective radii and surface thicknesses of the nuclear charge densities which (if suitably corrected for failure of the Born approximation) are in substantial agreement with the results of Hahn, Ravenhall, and Hofstadter.
Abstract: A survey has been made of the differential scattering cross sections for 187-Mev electrons on the even-even nuclei $_{12}\mathrm{Mg}^{24}$, $_{14}\mathrm{Si}^{28}$, $_{16}\mathrm{S}^{32}$, $_{18}\mathrm{A}^{40}$, and $_{28}\mathrm{Sr}^{88}$. It has been possible to separate the elastic scattering from the inelastic in all cases and to resolve the inelastic groups from specific nuclear levels for at least one level in all cases. A simple Born-approximation analysis of the elastic data yields values of the effective radii and surface thicknesses of the nuclear charge densities which (if suitably corrected for failure of the Born approximation) are in substantial agreement with the results of Hahn, Ravenhall, and Hofstadter; i.e., a radius parameter of $c\ensuremath{\cong}1.08 {A}^{\frac{1}{3}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm (radius to half-maximum of the charge distribution) and a surface thickness of $t\ensuremath{\cong}2.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm (thickness from 10% to 90% of the maximum of the charge distribution). Phenomenological analysis of the inelastic scattering along the lines laid down by Schiff yields some tentative multipolarity assignments, and application of some results of Ravenhall yields estimates of (radiative) partial level widths; for the $E2$ transitions these correspond to lifetimes of \ensuremath{\sim}19\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ sec (Mg 1.37 Mev) to \ensuremath{\sim}1.4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ sec (Sr 1.85 Mev). The observed strengths of the transitions are compared to those predicted by Weisskopf theory.
TL;DR: The octahedral sites in the spinel structure form one of the anomalous lattices in which it is possible to achieve essentially perfect short-range order while maintaining a finite entropy.
Abstract: The octahedral sites in the spinel structure form one of the anomalous lattices in which it is possible to achieve essentially perfect short-range order while maintaining a finite entropy. In such a lattice nearest-neighbor forces alone can never lead to long-range order, while calculations indicate that even the longrange Coulomb forces are only 5% effective in creating long-range order. This is shown to have many possible consequences both for antiferromagnetism in "normal" ferrites and for ordering in "inverse" ferrites.
TL;DR: In this paper, the authors analyzed a model which incorporates the above characteristics of a photoconductive film, except for the space charge effects, and derived numerical values for responsivity, noise, and sensitivity.
Abstract: Photoconductive films of the lead salt family are composed of a system of crystallites separated by intercrystalline barriers. The crystallites are lead salts while the intercrystalline barriers are an oxide of lead or of the lead salt. Space charge regions are present at the surface of the crystallites. The lifetime of hole-electron pairs is determined in part by surface states while the resistivity is strongly affected by intercrystalline barriers.We analyze a model which incorporates the above characteristics of a photoconductive film, except for the space charge effects. It is assumed that the primary photoeffect is absorption of light and production of hole-electron pairs in the crystallites. The change in conductivity results from a change in majority carrier density in the crystallites, and from reduction of intercrystalline barrier potentials.Equations are developed for the response to radiation, for the noise, and for the limit of sensitivity of the detector. These expressions contain familiar semiconductor parameters, and a new parameter which characterizes the relative importance of the change in carrier density as compared to the change in barrier potential. No attempt is made to calculate the parameters, but measurements necessary for their evaluation are briefly discussed. This permits a prediction of numerical values for responsivity, noise, and sensitivity which can be compared with experiment.
TL;DR: In this paper, a new wave function was proposed to represent an excitation (phonon or roton) in liquid helium, inserted into a variational principle for the energy, gave an energy-momentum curve having the qualitative shape suggested by Landau; but the value computed for the minimum energy Δ of a roton was 19.1°K, while thermodynamic data require Δ=9.6°K.
Abstract: A wave function previously used to represent an excitation (phonon or roton) in liquid helium, inserted into a variational principle for the energy, gave an energy-momentum curve having the qualitative shape suggested by Landau; but the value computed for the minimum energy Δ of a roton was 19.1°K, while thermodynamic data require Δ=9.6°K. A new wave function is proposed here. The new value computed for Δ is 11.5°K. Qualitatively, the wave function suggests that the roton is a kind of quantum-mechanical analog of a microscopic vortex ring, of diameter about equal to the atomic spacing. A forward motion of single atoms through the center of the ring is accompanied by a dipole distribution of returning flow far from the ring.
In the computation both the two-atom and three-atom correlation functions appear. The former is known from x-rays, while for the latter the Kirkwood approximation of a product of three two-atom correlation functions is used. A method is developed to estimate and correct for most of the error caused by this approximation, so that the residual uncertainty due to this source is negligible.
TL;DR: In this article, the wave functions for the singlet and triplet states of a two-electron system in a given nuclear framework are investigated as superpositions of configurations and are shown to be transformationally equivalent to quadratic forms having certain ranks and signatures.
Abstract: The wave functions for the singlet and triplet states of a two-electron system in a given nuclear framework are investigated as superpositions of configurations and are shown to be transformationally equivalent to quadratic forms having certain ranks and signatures. By introducing the "natural orbitals" diagonalizing the generalized first-order density matrix, the total wave functions may also be brought to principal form. If the basis contains $M$ one-electron functions, the singlet and triplet wave functions contain respectively $\frac{M(M+1)}{2}$ and $\frac{M(M\ensuremath{-}1)}{2}$ configurations, but the transformation to natural orbitals reduces the number of terms to $M$ and [$\frac{M}{2}$], respectively. The natural expansion having the best convergence is also characterized by another important extremum property. The approximate wave function of rank $r$ having the smallest quadratic deviation from the exact eigenfunction is obtained by interrupting the natural expansion of the eigenfunction after $r$ terms and renormalizing the result. For the singlet state, the wave function of rank two and signature zero has a special importance as giving a simple extension of the visual one-electron picture to include a large part of the correlation effects. The theory is illustrated by some results on helium obtained by using radial configuration interaction.
TL;DR: In this paper, it was deduced that in the interstitial state the solubility of copper is about 10 − 2 + 2 times less and the diffusivity many orders of magnitude greater than in the substitutional state.
Abstract: To explain the rapid diffusivity of copper in germanium and its dependence on structure, it is proposed that the copper be dissolved in two states, interstitial and substitutional. It is deduced that in the interstitial state the solubility of copper is about ${10}^{\ensuremath{-}2}$ times less and the diffusivity many orders of magnitude greater than in the substitutional state. Conversion from the interstitial to the substitutional state is effected by lattice vacancies which are generated at free surfaces and dislocations; this accounts for the structure dependence of the diffusivity observed by Tweet and Gallagher.
TL;DR: In this paper, surface oscillations in nuclei with deformation potentials independent of the shape parameter are discussed, and the regularities in even-even nuclei of the type discussed by Scharff-Goldhaber and Weneser are described.
Abstract: Surface oscillations in nuclei with deformation potentials independent of the shape parameter $\ensuremath{\gamma}$ are discussed, and are found to describe qualitatively the regularities in even-even nuclei of the type discussed by Scharff-Goldhaber and Weneser.
TL;DR: In this article, the free energy of an ideal Heisenberg-model ferromagnet is calculated as a power series in the temperature $T$ using the mathematical machinery developed in an earlier paper.
Abstract: The free energy of an ideal Heisenberg-model ferromagnet is calculated as a power series in the temperature $T$, using the mathematical machinery developed in an earlier paper. The spontaneous magnetization in zero external field is given by $[\frac{M(T)}{M(0)}]=S\ensuremath{-}{a}_{0}{\ensuremath{\theta}}^{\frac{3}{2}}\ensuremath{-}{a}_{1}{\ensuremath{\theta}}^{\frac{5}{2}}\ensuremath{-}{a}_{2}{\ensuremath{\theta}}^{\frac{7}{2}}\ensuremath{-}{a}_{3}{S}^{\ensuremath{-}1}{\ensuremath{\theta}}^{4}+O({\ensuremath{\theta}}^{\frac{9}{2}}).$ Here $\ensuremath{\theta}$ is the temperature in dimensionless units, and ${a}_{0}$, ${a}_{1}$, ${a}_{2}$, ${a}_{3}$ are positive numerical coefficients which are computed for the three types of cubic crystal lattice. The first two terms are the result of the simple Bloch theory in which spin waves are treated as noninteracting Bose particles with constant effective mass. The ${a}_{1}$ and ${a}_{2}$ corrections come from the variation of effective mass with velocity. The ${a}_{3}$ term is the lowest order correction arising from interaction between spin waves. This result is in violent contradiction to earlier published calculations which gave interaction effects proportional to ${T}^{\frac{7}{4}}$ and ${T}^{2}$.The smallness of the thermodynamic effects of spin-wave interactions is discussed in physical terms, and partially explained, in the introduction of this paper. A general proof is given that the thermodynamic effects of the "exclusion principle," which forbids more than ($2S$) spin deviations to occupy the same atom, are of order $\mathrm{exp}(\ensuremath{-}a{\ensuremath{\theta}}^{\ensuremath{-}1})$ and give zero contribution to any finite power of $\ensuremath{\theta}$. The residual dynamical interaction between 2 spin waves gives rise to a second virial coefficient ${{b}_{2}}^{\ensuremath{'}}$ which is calculated and shown to be of order ${T}^{\frac{5}{2}}$. The ${a}_{3}$ term in the magnetization is proportional to ${{b}_{2}}^{\ensuremath{'}}$. Effects of interaction of 3 or more spin waves are estimated and found to be of order ${\ensuremath{\theta}}^{5}$ or higher.
TL;DR: In this article, it was shown that the number of light spots increases with the current rather than individual spots growing brighter, and that all the breakdown current is carried through the junction by these localized light-emitting spots.
Abstract: Visible light is emitted from reverse-biased silicon $p\ensuremath{-}n$ junctions at highly localized regions where avalanche breakdown is taking place. The emission occurs in both grown and diffused junctions. By using junctions diffused to a depth of only 2 microns below the crystal surface, it was established that the light sources are randomly spaced over the whole area of the junction as well as around the periphery where the junction intercepts the surface. The light sources are too small to be resolved under a high-power microscope. Their sites are reproducible with current cycling and their intensity and color are relatively insensitive to the field distribution, to the junction width, and to temperature. The number of light spots increases with the current rather than individual spots growing brighter. It is concluded that all the breakdown current is carried through the junction by these localized light-emitting spots.The spectral distribution of the light is continuous with a long tail extending to photon energies greater than 3.3 ev. It is concluded that recombination between free electrons and free holes within the junction region is responsible for the light at the shorter wavelengths, the carrier energies in excess of the energy gap being supplied by the field. At longer wavelengths there appears to be a considerable contribution to the emission from intraband transitions.A tentative figure for the emission efficiency over the visible spectrum is one photon for every ${10}^{8}$ electrons crossing the junction. The recombination cross section required is reasonable, being about ${10}^{\ensuremath{-}22}$ ${\mathrm{cm}}^{2}$.
TL;DR: In this paper, the vacuum expectation values of products of neutral scalar field operators are discussed and the properties of these distributions arising from Lorentz invariance, the absence of negative energy states and the positive definiteness of the scalar product are determined.
Abstract: Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from Lorentz invariance, the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.
TL;DR: In this paper, the Low scattering equation is studied in the one-meson approximation with both charged and neutral scalar meson theories and the general solution is found for each of these cases.
Abstract: The Low scattering equation is studied in the one-meson approximation with both charged and neutral scalar meson theories. The general solution is found for each of these cases. It has the general character of a Wigner-Eisenbud dispersion formula and contains an infinite number of adjustable parameters. It follows that the Low equation, in this approximation at least, does not determine the scattering, but only expresses a property of the scattering which is independent of the internal structure of the scatterer.
TL;DR: In this article, the second law of thermodynamics was modified to the following: it is impossible to construct an engine that will operate in a closed cycle and provide no effect other than (1) the extraction of heat from a positive temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with corresponding work being done on the engine.
Abstract: The circumstances under which negative absolute temperatures can occur are discussed, and principles of thermodynamics and statistical mechanics at negative temperatures are developed. If the entropy of a thermodynamic system is not a monotonically increasing function of its internal energy, it possesses a negative temperature whenever ${(\frac{\ensuremath{\partial}S}{\ensuremath{\partial}U})}_{X}$ is negative. Negative temperatures are hotter than positive temperatures. When account is taken of the possibility of negative temperatures, various modifications of conventional thermodynamics statements are required. For example, heat can be extracted from a negative-temperature reservoir with no other effect than the performance of an equivalent amount of work. One of the standard formulations of the second law of thermodynamics must be altered to the following: It is impossible to construct an engine that will operate in a closed cycle and provide no effect other than (1) the extraction of heat from a positive-temperature reservoir with the performance of an equivalent amount of work or (2) the rejection of heat into a negative-temperature reservoir with the corresponding work being done on the engine. A thermodynamic system that is in internal thermodynamic equilibrium, that is otherwise essentially isolated, and that has an energetic upper limit to its allowed states can possess a negative temperature. The statistical mechanics of such a system are discussed and the results are applied to nuclear spin systems.
TL;DR: In this article, Brueckner and Levinson developed a self-contained and largely new description for studying the nucleus as a system of strongly interacting particles, which is applicable to a nucleus of finite size and to present the theory in sufficient detail that there are no ambiguities of interpretation and the nature of the approximations required for actual computation.
Abstract: A self-contained and largely new description is given of Brueckner's method for studying the nucleus as a system of strongly interacting particles. The aim is to develop a method which is applicable to a nucleus of finite size and to present the theory in sufficient detail that there are no ambiguities of interpretation and the nature of the approximations required for actual computation is clear.It is shown how to construct a model of the nucleus in which each nucleon moves in a self-consistent potential matrix of the form (${\mathrm{r}}^{\ensuremath{'}}|V|\mathrm{r}$) (Sec. II). The potential is obtained by calculating the reaction matrix for two nucleons in the nucleus from scattering theory. Some complications arise in the definition of the energy levels of excited nucleons (Sec. III). The actual wave function is obtained from the model function by an operator which takes into account multiple scattering of the nucleons by each other (Sec. IV).The method of Brueckner is a vast improvement over the normal Hartree-Fock method since, in calculating the self-consistent potential acting on an individual particle in the model, account is already taken of the correlations between paris of nucleons which arise from the strong internucleon forces (Sec. V). Although the actual wave function is derivable from a wave function which corresponds essentially to the shell model, the probability of finding a large nucleus of mass number $A$ "actually" in its shell model state is small (of order ${e}^{\ensuremath{-}\ensuremath{\alpha}A}$, where $\ensuremath{\alpha}$ is a constant) (Sec. VI). The influence of spin is investigated (Sec. VIII). In the case of an infinite nucleus, an integral equation is obtained for the reaction matrix, just as in the theory of Brueckner and Levinson (Sec. IX).The exclusion principle must be applied in intermediate states in solving the integral equation for the reaction matrix. This makes an enormous difference for the solution. When the exclusion principle is used, the scattering matrix is very nearly given by the Born approximation, for any well-behaved potential (Sec. X). Numerical results are given for the case when nucleons interact only in $S$ states, an assumption which leads to saturation without a repulsive core. The agreement with observation is fair to poor, owing to the poor assumption for the interaction (Sec. XI). Brueckner's result that three-particle clusters give a small contribution to the energy is confirmed, although the numerical value is many times his result; the calculation is then extended to the case of a repulsive core (Sec. XII). The dependence of the binding energy on the mass number $A$ is investigated for saturating and nonsaturating interactions (Sec. XIII). Terms of relative order $\frac{1}{A}$ are calculated, and it is shown that these terms are much smaller than Brueckner and Levinson found, making the method also applicable to relatively small nuclei (Sec. XIV). Some aspects of the problem of the finite nucleus are discussed, including that of degeneracy (Sec. XVI).
TL;DR: In this paper, a quantitative study of π→μ+ν decay using the techniques of dispersion theory is presented, based on a model in which the decay occurs through pion disintegration into a nucleon-antinucleon pair, the latter annihilating via a Fermi interaction to produce the leptons.
Abstract: A quantitative study of π→μ+ν decay is presented using the techniques of dispersion theory. The discussion is based on a model in which the decay occurs through pion disintegration into a nucleon-antinucleon pair, the latter annihilating via a Fermi interaction to produce the leptons. The weak vertex contains effectively both axial vector and pseudoscalar couplings even if one adopts the point of view of a universal axial vector and vector Fermi interaction. The pion-nucleon vertex which enters our model is also calculated using dispersion techniques. Under the assumption that this vertex is damped for large momentum transfers, we obtain a result for the pion lifetime largely independent of the detailed properties of the vertex and one which is in very close agreement with experiment. The precise prediction of our theory depends on the energy dependence of the complex phase shift for nucleon-antinucleon scattering in the 1S0 isotopic triplet state.
TL;DR: In this paper, the authors re-examined the theory of $p$-wave pion-nucleon scattering using the formalism recently proposed by one of the authors (F.E.L.).
Abstract: The theory of $p$-wave pion-nucleon scattering is reexamined using the formalism recently proposed by one of the authors (F.E.L.). On the basis of the cut-off Yukawa theory without nuclear recoil it is found, for not too high values of the coupling constant, that: (a) For each $p$-wave phase shift a certain function of the cotangent should be approximately linear at low energies and should extrapolate to the Born approximation at zero total energy. The value of the renormalized unrationalized coupling constant determined in this way from experiment is ${f}^{2}=0.08$. A special feature of the predicted energy dependence of the phase shifts is that ${\ensuremath{\delta}}_{33}$ is positive and the other $p$ phase shifts are negative. (b) The so-called "crossing theorem" requires a relation between the four $p$ phase shifts, so that in addition to the coupling constant only two further constants are needed to completely specify the low-energy behavior. (c) The direction of the energy variation in the (3,3) state is such that a resonance will occur for a sufficiently large cut-off ${\ensuremath{\omega}}_{max}$. Rough estimates indicate that ${\ensuremath{\omega}}_{max}\ensuremath{\approx}6$ will produce a resonance at the energy required by experiment. It is argued that the results (a) and (b) are very probably also consequences of a relativistic theory but that (c) may not be.