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Showing papers in "Physical Review in 1951"


Journal ArticleDOI
TL;DR: In this paper, it was shown that the extraction of gauge invariants from a formally gauge invariant theory is ensured if one employs methods of solution that involve only gauge covariant quantities.
Abstract: This paper is based on the elementary remark that the extraction of gauge invariant results from a formally gauge invariant theory is ensured if one employs methods of solution that involve only gauge covariant quantities. We illustrate this statement in connection with the problem of vacuum polarization by a prescribed electromagnetic field. The vacuum current of a charged Dirac field, which can be expressed in terms of the Green's function of that field, implies an addition to the action integral of the electromagnetic field. Now these quantities can be related to the dynamical properties of a "particle" with space-time coordinates that depend upon a proper-time parameter. The proper-time equations of motion involve only electromagnetic field strengths, and provide a suitable gauge invariant basis for treating problems. Rigorous solutions of the equations of motion can be obtained for a constant field, and for a plane wave field. A renormalization of field strength and charge, applied to the modified lagrange function for constant fields, yields a finite, gauge invariant result which implies nonlinear properties for the electromagnetic field in the vacuum. The contribution of a zero spin charged field is also stated. After the same field strength renormalization, the modified physical quantities describing a plane wave in the vacuum reduce to just those of the maxwell field; there are no nonlinear phenomena for a single plane wave, of arbitrary strength and spectral composition. The results obtained for constant (that is, slowly varying fields), are then applied to treat the two-photon disintegration of a spin zero neutral meson arising from the polarization of the proton vacuum. We obtain approximate, gauge invariant expressions for the effective interaction between the meson and the electromagnetic field, in which the nuclear coupling may be scalar, pseudoscalar, or pseudovector in nature. The direct verification of equivalence between the pseudoscalar and pseudovector interactions only requires a proper statement of the limiting processes involved. For arbitrarily varying fields, perturbation methods can be applied to the equations of motion, as discussed in Appendix A, or one can employ an expansion in powers of the potential vector. The latter automatically yields gauge invariant results, provided only that the proper-time integration is reserved to the last. This indicates that the significant aspect of the proper-time method is its isolation of divergences in integrals with respect to the proper-time parameter, which is independent of the coordinate system and of the gauge. The connection between the proper-time method and the technique of "invariant regularization" is discussed. Incidentally, the probability of actual pair creation is obtained from the imaginary part of the electromagnetic field action integral. Finally, as an application of the Green's function for a constant field, we construct the mass operator of an electron in a weak, homogeneous external field, and derive the additional spin magnetic moment of $\frac{\ensuremath{\alpha}}{2\ensuremath{\pi}}$ magnetons by means of a perturbation calculation in which proper-mass plays the customary role of energy.

5,579 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that both electrical conduction and ferromagnetic coupling in these compounds arise from a double exchange process, and a quantitative relation was developed between electrical conductivity and the Ferromagnetic Curie temperature.
Abstract: Recently, Jonker and Van Santen have found an empirical correlation between electrical conduction and ferromagnetism in certain compounds of manganese with perovskite structure. This observed correlation is herein interpreted in terms of those principles governing the interaction of the $d$-shells of the transition metals which were enunciated in the first paper of this series. Both electrical conduction and ferromagnetic coupling in these compounds are found to arise from a double exchange process, and a quantitative relation is developed between electrical conductivity and the ferromagnetic Curie temperature.

5,097 citations


Journal ArticleDOI
TL;DR: In this article, the Hartree-Fock equations can be regarded as ordinary Schrodinger equations for the motion of electrons, each electron moving in a slightly different potential field, which is computed by electrostatics from all the charges of the system, positive and negative, corrected by the removal of an exchange charge, equal in magnitude to one electron, surrounding the electron whose motion is being investigated.
Abstract: It is shown that the Hartree-Fock equations can be regarded as ordinary Schr\"odinger equations for the motion of electrons, each electron moving in a slightly different potential field, which is computed by electrostatics from all the charges of the system, positive and negative, corrected by the removal of an exchange charge, equal in magnitude to one electron, surrounding the electron whose motion is being investigated By forming a weighted mean of the exchange charges, weighted and averaged over the various electronic wave functions at a given point of space, we set up an average potential field in which we can consider all of the electrons to move, thus leading to a great simplification of the Hartree-Fock method, and bringing it into agreement with the usual band picture of solids, in which all electron are assumed to move in the same field We can further replace the average exchange charge by the corresponding value which we should have in a free-electron gas whose local density is equal to the density of actual charge at the position in question; this results in a very simple expression for the average potential field, which still behaves qualitatively like that of the Hartree-Fock method This simplified field is being applied to problems in atomic structure, with satisfactory results, and is adapted as well to problems of molecules and solids

3,390 citations


Journal ArticleDOI
C. G. Koops1
TL;DR: In this paper, the ac resistivity and the apparent dielectric constant of the material show a dispersion which can be explained satisfactorily with the help of a simple model of the solid: there should be wellconducting grains separated by layers of lower conductivity.
Abstract: Semiconducting ${\mathrm{Ni}}_{0.4}$${\mathrm{Zn}}_{0.6}$${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, prepared in different ways, has been investigated. It appeared that the ac resistivity and the apparent dielectric constant of the material show a dispersion which can be explained satisfactorily with the help of a simple model of the solid: there should be well-conducting grains separated by layers of lower conductivity. Dispersion formulas are given. There is good agreement between experiment and theory.

2,915 citations


Journal ArticleDOI
TL;DR: In this article, a relation between the generalized resistance and the generalized forces in linear dissipative systems is obtained, which forms the extension of the Nyquist relation for the voltage fluctuations in electrical impedances.
Abstract: A relation is obtained between the generalized resistance and the fluctuations of the generalized forces in linear dissipative systems. This relation forms the extension of the Nyquist relation for the voltage fluctuations in electrical impedances. The general formalism is illustrated by applications to several particular types of systems, including Brownian motion, electric field fluctuations in the vacuum, and pressure fluctuations in a gas.

2,457 citations


Journal ArticleDOI
TL;DR: In this article, the relativistic $S$-matrix formalism of Feynman is applied to the bound-state problem for two interacting Fermi-Dirac particles.
Abstract: The relativistic $S$-matrix formalism of Feynman is applied to the bound-state problem for two interacting Fermi-Dirac particles. The bound state is described by a wave function depending on separate times for each of the two particles. Two alternative integral equations for this wave function are derived with kernels in the form of an expansion in powers of ${g}^{2}$, the dimensionless coupling constant for the interaction. Each term in these expansions gives Lorentz-invariant equations. The validity and physical significance of these equations is discussed. In extreme nonrelativistic approximation and to lowest order in ${g}^{2}$ they reduce to the appropriate Schr\"odinger equation.One of these integral equations is applied to the deuteron ground state using scalar mesons of mass $\ensuremath{\mu}$ with scalar coupling. For neutral mesons the Lorentz-invariant interaction is transformed into the sum of the instantaneous Yukawa interaction and a retarded correction term. The value obtained for ${g}^{2}$ differs only by a fraction proportional to ${(\frac{\ensuremath{\mu}}{M})}^{2}$ from that obtained by using a phenomenological Yukawa potential. For a purely charged meson theory a correction term is obtained by a direct solution of the relativistic integral equation using only the first term in the expansion of the kernel. This correction is due to the fact that a nucleon can emit, or absorb, positive and negative mesons only alternately. The constant ${g}^{2}$ is increased by a fraction of $1.1(\frac{\ensuremath{\mu}}{M})$ or 15 percent.

1,962 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown that the spin coupling between the incomplete $d$ shells and the conduction electrons leads to a tendency for a ferromagnetic alignment of $d $ spins.
Abstract: It is assumed (1) that the interaction between the incomplete $d$ shells of the transition elements is insufficient to disrupt the coupling between the $d$ electrons in the same shells, and (2) that the exchange interaction between adjacent $d$ shells always has the same sign irrespective of distance of separation. The direct interaction between adjacent $d$ shells then invariably leads to a tendency for an antiferromagnetic alignment of $d$ spins. The body-centered cubic structure of the transition metals V, Cr, Cb, Mo, Ta, and W is thereby interpreted, as well as more complex lattices of certain alloys. It is demonstrated that the spin coupling between the incomplete $d$ shells and the conduction electrons leads to a tendency for a ferromagnetic alignment of $d$ spins. The occurrence of ferromagnetism is thereby interpreted in a much more straightforward manner than through the ad hoc assumption of a reversal in sign of the exchange integral. The occurrence of antiferromagnetism and of ferromagnetism in various systems is readily understood, and certain simple rules are deduced for deciding which type of magnetism will occur in particular alloys.

1,660 citations


Journal ArticleDOI
TL;DR: In this article, a new method of approximation for order-disorder phenomena is developed for the one-dimensional Ising lattice and an improved treatment for the three-dimensional simple cubic Ising model is given.
Abstract: A new method of approximation for order-disorder phenomena is developed. In Sec. A, the method is explained for the one-dimensional Ising lattice. Sections B and C cover the approximations already known, such as those of Bethe (Sec. B) and of Kramers-Wannier (Sec. C), which are shown to be derived as special cases of the method with suitable choices of variables. In Sec. D, an improved treatment is explained for the three-dimensional simple cubic Ising lattice. This approximation is found to agree with the rigorous expansion of the partition function up to the fourth moment by Kirkwood's moment method, so far as the disordered state is concerned. In Sec. E the general formula for the entropy is given. In Sec. H an improved treatment of the face-centered lattice (Ising model) is given.

1,529 citations


Journal ArticleDOI
Conyers Herring1
TL;DR: In this article, the Wulff construction for the equilibrium shape of a small particle and the relations between atomic models of crystal surfaces and the surface free energy function used in this construction are discussed.
Abstract: Although the interpretation of experiments in such fields as the shapes of small particles and the thermal etching of surfaces usually involves problems of kinetics rather than mere equilibrium considerations, it is suggested that a knowledge of the relative free energies of different shapes or surface configurations may provide a useful perspective. This paper presents some theorems on these relative free energies which follow from the Wulff construction for the equilibrium shape of a small particle, and some relations between atomic models of crystal surfaces and the surface free energy function used in this construction. Equilibrium shapes of crystals and of noncrystalline anisotropic media are classified, and it is pointed out that the possibilities for crystals include smoothly rounded as well as sharp-cornered forms. The condition is formulated for thermodynamic stability of a flat crystal face with respect to formation of a hill-and-valley structure. A discussion is presented of the limitations on the applicability of the results imposed by the dependence of surface free energy on curvature; and it is concluded that these limitations are not likely to be serious for most real substances, though they are serious for certain idealized theoretical models.

1,208 citations


Journal ArticleDOI
TL;DR: In this article, an ordering subscript is introduced to indicate the order of operation of noncommuting quantities, such that AsBs′ means AB or BA depending on whether s exceeds s′ or vice versa.
Abstract: An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscript is introduced so that AsBs′ means AB or BA depending on whether s exceeds s′ or vice versa. Then As can be handled as though it were an ordinary numerical function of s. An increase in ease of manipulating some operator expressions results. Connection to the theory of functionals is discussed in an appendix. Illustrative applications to quantum mechanics are made. In quantum electrodynamics it permits a simple formal understanding of the interrelation of the various present day theoretical formulations. The operator expression of the Dirac equation is related to the author's previous description of positrons. An attempt is made to interpret the operator ordering parameter in this case as a fifth coordinate variable in an extended Dirac equation. Fock's parametrization, discussed in an appendix, seems to be easier to interpret. In the last section a summary of the numerical constants appearing in formulas for transition probabilities is given.

986 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that the Lagrange function of half-integral spin fields behaves like a pseudoscalar with respect to time reflection transformations, and that the transformation function is invariant under time reflection if the order of all operators in the lagrange function leaves an integral spin contribution unaltered.
Abstract: The conventional correspondence basis for quantum dynamics is here replaced by a self-contained quantum dynamical principle from which the equations of motion and the commutation relations can be deduced. The theory is developed in terms of the model supplied by localizable fields. A short review is first presented of the general quantum-mechanical scheme of operators and eigenvectors, in which emphasis is placed on the differential characterization of representatives and transformation functions by means of infinitesimal unitary transformations. The fundamental dynamical principle is stated as a variational equation for the transformation function connecting eigenvectors associated with different spacelike surfaces, which describes the temporal development of the system. The generator of the infinitesimal transformation is the variation of the action integral operator, the spacetime volume integral of the invariant lagrange function operator. The invariance of the lagrange function preserves the form of the dynamical principle under coordinate transformations, with the exception of those transformations which include a reversal in the positive sense of time, where a separate discussion is necessary. It will be shown in Sec. III that the requirement of invariance under time reflection imposes a restriction upon the operator properties of fields, which is simply the connection between the spin and statistics of particles. For a given dynamical system, changes in the transformation function arise only from alterations of the eigenvectors associated with the two surfaces, as generated by operators constructed from field variables attached to those surfaces. This yields the operator principle of stationary action, from which the equations of motion are obtained. Commutation relations are derived from the generating operator associated with a given surface. In particular, canonical commutation relations are obtained for those field components that are not restricted by equations of constraint. The surface generating operator also leads to generalized Schr\"odinger equations for the representative of an arbitrary state. Action integral variations which correspond to changing the dynamical system are discussed briefly. A method for constructing the transformation function is described, in a form appropriate to an integral spin field, which involves solving Hamilton-Jacobi equations for ordered operators. In Sec. III, the exceptional nature of time reflection is indicated by the remark that the charge and the energy-momentum vector behave as a pseudoscalar and pseudovector, respectively, for time reflection transformations. This shows, incidentally, that positive and negative charge must occur symmetrically in a completely covariant theory. The contrast between the pseudo energy-momentum vector and the proper displacement vector then indicates that time reflection cannot be described within the unitary transformation framework. This appears most fundamentally in the basic dynamical principle. It is important to recognize here that the contributions to the lagrange function of half-integral spin fields behave like pseudoscalars with respect to time reflection. The non-unitary transformation required to represent time reflection is found to be the replacement of a state vector by its dual, or complex conjugate vector, together with the transposition of all operators. The fundamental dynamical principle is then invariant under time reflection if inverting the order of all operators in the lagrange function leaves an integral spin contribution unaltered, and reverses the sign of a half-integral spin contribution. This implies the essential commutativity, or anti-commutativity, of integral and half-integral field components, respectively, which is the connection between spin and statistics.

Journal ArticleDOI
TL;DR: In this paper, a series of paramagnetic and antiferromagnetic substances are studied and the magnetic form factor of the electrons in the $3d$-shell of the ion has been determined.
Abstract: Neutron scattering and diffraction studies on a series of paramagnetic and antiferromagnetic substances are reported in the present paper. The paramagnetic diffuse scattering predicted by Halpern and Johnson has been studied, resulting in the determination of the magnetic form factor for ${\mathrm{Mn}}^{++}$ ions. From the form factor, the radial distribution of the electrons in the $3d$-shell of ${\mathrm{Mn}}^{++}$ has been determined, and this is compared with a theoretical distribution of Dancoff. Antiferromagnetic substances are shown to produce strong, coherent scattering effects in the diffraction pattern. The antiferromagnetic reflections have been used to determine the magnetic structure of the material below the antiferromagnetic Curie temperature. For some substances the magnetic unit cell is found to be larger than the chemical unit cell. The temperature dependence of the antiferromagnetic intensities has been studied, and the directional effects which characterize neutron scattering by aligned atomic moments have been used to determine the moment alignment with respect to crystallographic axes. From studies with magnetic ions possessing both orbital and spin moments, it is found that the antiferromagnetic intensities contain partial orbital moment components along with the spin moment component. The degree of orbital moment contribution agrees satisfactorily with that predicted by models of lattice quenching.

Journal ArticleDOI
TL;DR: The relativistic two-body equation of Bethe and Salpeter is derived from field theory as discussed by the authors, and it is shown that the Feynman kernel may be written as a sum of wave functions over the states of the system.
Abstract: The relativistic two-body equation of Bethe and Salpeter is derived from field theory. It is shown that the Feynman two-body kernel may be written as a sum of wave functions over the states of the system. These wave functions depend exponentially on the energies of the states to which they correspond and therefore provide a means of calculating energy levels of bound states.

Journal ArticleDOI
TL;DR: The Hartree-Fock equations state that each electron in an atom or molecular system should move in a different potential as mentioned in this paper, which leads to a periodic perturbation of potential, with periodicity twice the atomic periodicity, and leads to splitting of each energy band in half, with a gap in the middle.
Abstract: The Hartree-Fock equations state that each electron in an atom or molecular system should move in a different potential. In some cases, particularly magnetic cases, this leads to important consequences, since electrons with opposite spins move in different potentials. In particular, in an antiferromagnetic substance, electrons of + and - spin have different potentials; and for an electron of + spin, for instance, the potential energy is lower in those atoms whose spins are pointed in the + direction than in those with the opposite spin. This results in a periodic perturbation of potential, with periodicity twice the atomic periodicity, and leads to a splitting of each energy band in half, with a gap in the middle. In a case where the energy band was half full, resulting in a conductor: when we disregard this effect, the resulting half-band will be just filled when we consider it; this may explain the insulating nature of some antiferromagnetics. A similar argument applied to a diatomic molecule like ${\mathrm{H}}_{2}$ can result in two alternative types of solutions of the Hartree-Fock equations: one leading to atomic orbitals, the other to molecular orbitals. The solution with atomic orbitals shows an analogy to the antiferromagnetic problem; that with ordinary molecular orbitals shows an analogy to the band theory of a non-magnetic conductor.

Journal ArticleDOI
David Pines1
TL;DR: In this article, a new approach to the treatment of the interactions in a collection of electrons is developed, which is called the collective description, based on the organized behavior produced by the interactions of an electron gas of high density; this organized behavior results in oscillations of the system as a whole.
Abstract: A new approach to the treatment of the interactions in a collection of electrons is developed, which we call the collective description. The collective description is based on the organized behavior produced by the interactions in an electron gas of high density; this organized behavior results in oscillations of the system as a whole, the so-called "plasma oscillations." The collective description, in contrast to the usual individual particle description, describes in a natural way the long-range correlations in electron positions brought about by their mutual interaction. In this paper we confine our attention to the magnetic interactions between the electrons; the coulomb interactions will be discussed in a subsequent paper.The transition from the usual single-particle description to the collective description of the electron motion in terms of organized oscillations is obtained by a suitable canonical transformation. The complete hamiltonian for a collection of charges interacting with the transverse electromagnetic field is re-expressed as a sum of three terms. One involves the collective field coordinates and expresses the degree of excitation of organized oscillations. The others represent the kinetic energy of the electrons and the residual particle interaction, which is not describable in terms of the organized oscillations, and corresponds to a screened interparticle force of short range.Both a classical and a quantum-mechanical treatment are given, and the criteria for the validity of the collective description are discussed.

Journal ArticleDOI
TL;DR: In this paper, the authors derived an exact solution for the first passage time probability of a stationary one-dimensional Markoffian random function from an integral equation, and derived a recursion formula for the moments.
Abstract: We have derived an exact solution for the first passage time probability of a stationary one-dimensional Markoffian random function from an integral equation. A recursion formula for the moments is given for the case that the conditional probability density describing the random function satisfies a Fokker-Planck equation. Various known solutions for special applications (noise, Brownian motion) are shown to be special cases of our solution. The Wiener-Rice series for the recurrence time probability density is derived from a generalization of Schr\"odinger's integral equation, for the case of a two-dimensional Markoffian random function.

Journal ArticleDOI
H. Y. Fan1
TL;DR: The effect of lattice vibrations on the energy gap has been treated previously on the basis of broadening rather than shifting of the energy levels as mentioned in this paper, and the effect was found to be negligible for nonpolar crystals, whereas according to their treatment it should be much larger.
Abstract: The problem treated is the effect of lattice vibrations in producing a shift of the energy levels which results in a temperature dependent variation of the energy gap in semiconductors. Calculations for silicon and germanium give results of the same order of magnitude as the observed temperature dependent shift of the absorption band edge. The effect of lattice vibrations on the energy gap has been treated previously on the basis of broadening rather than shifting of the energy levels. The effect was found to be negligible for non-polar crystals, whereas according to our treatment it should be much larger. For polar crystals our result turns out to be essentially the same as was given by the previous treatment.

Journal ArticleDOI
TL;DR: In this article, the dielectric constant above and below the antiferroelectric curie point is investigated for both first and second-order transitions, and it is shown that if the transition is second order, the temperature anomaly near the Curie point will be of the same nature and magnitude as in ferroelectrics.
Abstract: An antiferroelectric state is defined as one in which lines of ions in the crystal are spontaneously polarized, but with neighboring lines polarized in antiparallel directions. In simple cubic lattices the antiferroelectric state is likely to be more stable than the ferroelectric state. The dielectric constant above and below the antiferroelectric curie point is investigated for both first- and second-order transitions. In either case the dielectric constant need not be very high; but if the transition is second order, $\ensuremath{\epsilon}$ is continuous across the Curie point. The antiferroelectric state will not be piezoelectric. The thermal anomaly near the Curie point will be of the same nature and magnitude as in ferroelectrics. A susceptibility variation of the form $\frac{C}{(T+\ensuremath{\theta})}$ as found in strontium titanate is not indicative of antiferroelectricity, unlike the corresponding situation in antiferromagnetism.

Journal ArticleDOI
W. H. Furry1
TL;DR: The use of bound-state wave functions in calculations in positron theory is justified by the introduction of a new representation, in a certain sense intermediate between the Heisenberg and interaction representations as mentioned in this paper.
Abstract: The use of bound-state wave functions in calculations in positron theory is justified by the introduction of a new representation, in a certain sense intermediate between the Heisenberg and interaction representations. In the bound-state representation the definition of a stable vacuum state is possible only for a restricted class of external fields. Some attention is given to the problem of vacuum polarization, and it is shown that a very simple procedure accomplishes the charge renormalization with sufficient accuracy to be of use in certain scattering problems. The application to the scattering of radiation is discussed in some detail, in order to show the relation between the different points of view that may be adopted in problems of the coherent scattering by a bound electron and the "Delbr\"uck scattering" by virtual electron pairs.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the distribution of molecular velocities in a strong shock wave in a gas is bimodal and that the assumed distribution changes relatively slowly with time and so is an approximate stationary solution of the Boltzmann equation for strong shocks.
Abstract: It is pointed out that the distribution of molecular velocities in a strong shock wave in a gas is bimodal. Assuming the distribution function to consist of a sum of two maxwellian terms with temperatures and mean velocities corresponding to the subsonic and supersonic streams, it is found that the space distribution, as determined by the solution of a transport equation, is appropriate to describe a shock wave. Comparison of the solutions of two different transport equations shows that the assumed distribution changes relatively slowly with time and so is an approximate stationary solution of the Boltzmann equation for strong shocks. The shock thickness found is considerably greater than that given by previous theories. The nominal thermal conduction coefficient is negative in the after part of the shock.

Journal ArticleDOI
L. I. Schiff1
TL;DR: In this article, the differential bremsstrahlung cross section of Bethe and Heitler is integrated over scattered electron angles to obtain an expression for the distribution in energy and angle of the radiation from fast electrons in very thing targets.
Abstract: The differential bremsstrahlung cross section of Bethe and Heitler is integrated over scattered electron angles to obtain an expression for the distribution in energy and angle of the radiation from fast electrons in very thing targets. Screening is taken into account through the assumption of an atomic potential $(\frac{\mathrm{Ze}}{r})\mathrm{exp}(\ensuremath{-}\frac{r}{a})$, and the calculation is restricted to high energies and to small to moderate angles. The result is the same as that of Sommerfeld for no screening, except that the argument of the logarithm now depends on angle as well as on energy. Integration of this expression over gamma-ray angle gives an analytic formula for the total intensity that is nowhere more than a few percent higher than the Bethe-Heither result calculated numerically on the basis of the Thomas-Fermi potential.

Journal ArticleDOI
TL;DR: In this article, a theorem due to Wannier for treating the motion of electrons in a perturbed periodic field is generalized to include the effect of a slowly varying magnetic field.
Abstract: A theorem due to Wannier for treating the motion of electrons in a perturbed periodic field is generalized to include the effect of a slowly varying magnetic field. It is shown that the problem reduces to that of solving an effective Schr\"odinger equation, which is known as soon as we have solved the problem without perturbing fields.


Journal ArticleDOI
TL;DR: In this paper, the spin resonance condition for a disk-shaped single-domain uniaxial or cubic antiferromagnetic crystal at 0, √ √ K with √ H √ 1/2 parallel to the domain axis is extended by classical calculations to cover finite temperature, ellipsoidal shape, orthorhombic symmetry, generalized two-lattice anisotropy, and arbitrary static field direction.
Abstract: The spin resonance condition $\frac{\ensuremath{\omega}}{\ensuremath{\gamma}}={H}_{0}\ifmmode\pm\else\textpm\fi{}{[{H}_{A}(2{H}_{E}+{H}_{A})]}^{\frac{1}{2}}$ previously given by Kittel for a disk-shaped single-domain uniaxial or cubic antiferromagnetic crystal at 0\ifmmode^\circ\else\textdegree\fi{}K with ${H}_{0}$ parallel to the domain axis is extended by classical calculations to cover finite temperature, ellipsoidal shape, orthorhombic symmetry, generalized two-lattice anisotropy, and arbitrary static field direction. The normal precessional modes are discussed. A quantum-mechanical derivation of the resonance equations is carried out by the method developed by Van Vleck for ferromagnetic resonance; no new features are introduced by the quantum-mechanical calculation. Several factors contributing to the line width are considered. Existing experimental data on antiferromagnetic resonance are reviewed; the data are scanty and taken in circumstances not closely related to the situation envisaged by the theory.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of fitting a covariant field theory into a canonical form and showed that only a finite number of such constraints exist and that they form a function group.
Abstract: In this paper we have considered certain problems which arise when one attempts to cast a covariant field theory into a canonical form. Because of the invariance properties of the theory, certain identities exist between the canonical field variables. To insure that the canonical theory is equivalent to the underlying lagrangian formalism one must require that these identities, once satisfied, will remain satisfied through the course of time. In general, this will be true only if additional constraints are set between the canonical variables. We have shown that only a finite number of such constraints exist and that they form a function group. Our proof rests essentially on the possibility of constructing a generating function for an infinitesimal canonical transformation that is equivalent to an invariant infinitesimal transformation on the lagrangian formalism.Once a hamiltonian is obtained by one of the procedures outlined in previous papers of this series, and the constraints have all been found, the consistent, invariant canonical formulation of the theory is completed. The main results of the paper have been formulated in such a manner as to make them applicable to a fairly general type of invariance. In the last sections we have applied these results to the cases of gauge and coordinate invariance. In the latter case a hamiltonian, corresponding to a quadratic lagrangian, has been constructed in a parameter-free form; and in both cases the constraints, together with the poisson bracket relations between them, have been obtained explicitly. As was to be expected, two constraints were found for a gauge-invariant theory and eight for a coordinate-invariant theory.

Journal ArticleDOI
TL;DR: In this article, the dielectric response of PbZr${\mathrm{O}}_{3}$ ceramic to biasing dc field has been examined, and it was shown that an effect is scarcely detected below the Curie point of 233\ifmmode^\circ\else\textdegree\fi{}C even up to a field strength of 20 kv/cm.
Abstract: In order to distinguish whether it is really a ferroelectric or not, the dielectric response of PbZr${\mathrm{O}}_{3}$ ceramic to biasing dc field has been examined. In contrast to ordinary ferroelectrics, an effect is scarcely detected below the Curie point of 233\ifmmode^\circ\else\textdegree\fi{}C even up to a field strength of 20 kv/cm, except for a slight increase of permittivity at the vicinity of the transition point. In addition, the Curie temperature decreases with increasing biasing field attaining the value $\ensuremath{\Delta}{T}_{c}=\ensuremath{-}1.7$\ifmmode^\circ\else\textdegree\fi{}C for 20 kv/cm. The hysteresis loops of this ceramic have been studied. So long as the amplitude of the applied ac field is less than a critical field of about 23 kv/cm, the $D\ensuremath{-}E$ curves are almost always linear, except that a slight upward curvature becomes perceptible just at the temperature range wherein permittivity increases, and never show any hysteresis characteristics even just below the Curie point. But if the amplitude of the applied field is greater than this threshold value, anomalous hysteresis loops of very impressive form are observed in a narrow temperature range just below the Curie point. These experimental facts all seem to suggest that PbZr${\mathrm{O}}_{3}$ may be, not a ferroelectric, but an antiferroelectric. A preliminary pyroelectric test also seems to show that no net polarization grows spontaneously when the sample is cooled through its Curie point.


Journal ArticleDOI
TL;DR: In this paper, the nuclear photo effect for photons of energy greater than 150 Mev is calculated assuming the two-nucleon model used by Heidmann, and the main features of the nuclear effect are then quite similar to those of the deuteron photo effect.
Abstract: The nuclear photoeffect for photons of energy greater than 150 Mev is calculated assuming the two-nucleon model used by Heidmann. The main features of the nuclear photoeffect are then quite similar to those of the deuteron photoeffect. The cross section for nuclear absorption of a high energy photon is about 1.6A times the cross section for the deuteron photoeffect. The deuteron photoeffect gives a very strongly forward angular distribution for protons of a given energy, observed in the laboratory system. The angular distribution for protons from the nuclear photoeffect is almost as strongly forward: for 90-Mev protons in the laboratory system the ratio of the differential cross section at 60\ifmmode^\circ\else\textdegree\fi{} to that at 90\ifmmode^\circ\else\textdegree\fi{} is about 3. The proton energy spectrum decreases rapidly with proton energy, and becomes steeper for observations at large angles. The calculated angular distributions and proton energy spectra are in fair agreement with measurements by Walker. The absolute value of the differential cross section for 90-Mev protons from carbon at 30\ifmmode^\circ\else\textdegree\fi{} (laboratory system) is about 0.2 \ensuremath{\mu}barn/Mev steradian per $Q$, or about 20 \ensuremath{\mu}barn/steradian per photon. This absolute value is about one-third the absolute value measured by Walker, and is somewhat larger than the absolute value measured by Levinthal and Silverman.

Journal ArticleDOI
TL;DR: In this paper, the effects of diffusion of electrons through a thin $p$-type layer of germanium have been studied in specimens consisting of two $n-type regions with the $p-type region interposed.
Abstract: The effects of diffusion of electrons through a thin $p$-type layer of germanium have been studied in specimens consisting of two $n$-type regions with the $p$-type region interposed. It is found that potentials applied to one $n$-type region are transmitted by diffusing electrons through the $p$-type layer although the latter is grounded through an ohmic contact. When one of the $p\ensuremath{-}n$ junctions is biased to saturation, power gain can be obtained through the device. Used as "$n\ensuremath{-}p\ensuremath{-}n$ transistors" these units will operate on currents as low as 10 microamperes and voltages as low as 0.1 volt, have power gains of 50 db, and noise figures of about 10 db at 1000 cps. Their current-voltage characteristics are in good agreement with the diffusion theory.

Journal ArticleDOI
TL;DR: In this paper, the authors measured transit times between emitter and collector in single crystal rods and determined the mobilities of holes injected into n$-type germanium and of electrons injected into $p$ -type germium.
Abstract: The mobilities of holes injected into $n$-type germanium and of electrons injected into $p$-type germanium have been determined by measuring transit times between emitter and collector in single crystal rods. Strong electric fields in addition to those due to injected current were employed so that spreading effects due to diffusion were reduced. The mobilities at 300\ifmmode^\circ\else\textdegree\fi{}K are 1700 ${\mathrm{cm}}^{2}$/volt-sec for holes and 3600 ${\mathrm{cm}}^{2}$/volt-sec for electrons with an error of probably less than five percent. The value for electrons is about 20 percent higher than the best estimates obtained from the conventional interpretation of the Hall effect and the difference may be due to curved energy band surfaces in the Brillouin zone. Studies of rates of decay indicate that recombination of holes and electrons takes place largely on the surface of small samples with constants varying from ${10}^{2}$ to g ${10}^{4}$ cm/sec for special treatments.