Showing papers in "Physical Review in 1969"

Axial vector vertex in spinor electrodynamics

Abstract: Working within the framework of perturbation theory, we show that the axial-vector vertex in spinor electrodynamics has anomalous properties which disagree with those found by the formal manipulation of field equations. Specifically, because of the presence of closed-loop "triangle diagrams," the divergence of axial-vector current is not the usual expression calculated from the field equations, and the axial-vector current does not satisfy the usual Ward identity. One consequence is that, even after the external-line wave-function renormalizations are made, the axial-vector vertex is still divergent in fourth- (and higher-) order perturbation theory. A corollary is that the radiative corrections to ${\ensuremath{ u}}_{l}l$ elastic scattering in the local current-current theory diverge in fourth (and higher) order. A second consequence is that, in massless electrodynamics, despite the fact that the theory is invariant under ${\ensuremath{\gamma}}_{5}$ tranformations, the axial-vector current is not conserved. In an Appendix we demonstrate the uniqueness of the triangle diagrams, and discuss a possible connection between our results and the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ and $\ensuremath{\eta}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ decays. In particular, we argue that as a result of triangle diagrams, the equations expressing partial conservation of axial-vector current (PCAC) for the neutral members of the axial-vector-current octet must be modified in a well-defined manner, which completely alters the PCAC predictions for the ${\ensuremath{\pi}}^{0}$ and the $\ensuremath{\eta}$ two-photon decays.

Theory of Sputtering. I. Sputtering Yield of Amorphous and Polycrystalline Targets

Abstract: Sputtering of a target by energetic ions or recoil atoms is assumed to result from cascades of atomic collisions. The sputtering yield is calculated under the assumption of random slowing down in an infinite medium. An integrodifferential equation for the yield is developed from the general Boltzmann transport equation. Input quantities are the cross sections for ion-target and target-target collisions, and atomic binding energies. Solutions of the integral equation are given that are asymptotically exact in the limit of high ion energy as compared to atomic binding energies. Two main stages of the collision cascade have to be distinguished: first, the slowing down of the primary ion and all recoiling atoms that have comparable energies---these particles determine the spatial extent of the cascade; second, the creation and slowing down of low-energy recoils that constitute the major part of all atoms set in motion. The separation between the two stages is essentially complete in the limit of high ion energy, as far as the calculation of the sputtering yield is concerned. High-energy collisions are characterized by Thomas-Fermi-type cross sections, while a Born-Mayer-type cross section is applied in the low-energy region. Electronic stopping is included when necessary. The separation of the cascade into two distinct stages has the consequence that two characteristic depths are important for the qualitative understanding of the sputtering process. First, the scattering events that eventually lead to sputtering take place within a certain layer near the surface, the thickness of which depends on ion mass and energy and on ion-target geometry. In the elastic collision region, this thickness is a sizable fraction of the ion range. Second, the majority of sputtered particles originate from a very thin surface layer (\ensuremath{\sim}5 \AA{}), because small energies dominate. The general sputtering-yield formula is applied to specific situations that are of interest for comparison with experiment. These include backsputtering of thick targets by ion beams at perpendicular and oblique incidence and ion energies above \ensuremath{\sim}100 eV, transmission sputtering of thin foils, sputtering by recoil atoms from $\ensuremath{\alpha}$-active atoms distributed homogeneously or inhomogeneously in a thick target, sputtering of fissionable specimens by fission fragments, and sputtering of specimens that are irradiated in the core of a reactor or bombarded with a neutron beam. There is good agreement with experimental results on polycrystalline targets within the estimated accuracy of the data and the input parameters entering the theory. There is no need for adjustable parameters in the usual sense, but specific experimental setups are discussed that allow independent checks or accurate determination of some input quantities.

Non-Lagrangian Models of Current Algebra

Abstract: An alternative is proposed to specific Lagrangian models of current algebra. In this alternative there are no explicit canonical fields, and operator products at the same point [say, ${j}_{\ensuremath{\mu}}(x){j}_{\ensuremath{\mu}}(x)$] have no meaning. Instead, it is assumed that scale invariance is a broken symmetry of strong interactions, as proposed by Kastrup and Mack. Also, a generalization of equal-time commutators is assumed: Operator products at short distances have expansions involving local fields multiplying singular functions. It is assumed that the dominant fields are the $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ currents and the $\mathrm{SU}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(3)$ multiplet containing the pion field. It is assumed that the pion field scales like a field of dimension $\ensuremath{\Delta}$, where $\ensuremath{\Delta}$ is unspecified within the range $1\ensuremath{\le}\ensuremath{\Delta}l4$; the value of $\ensuremath{\Delta}$ is a consequence of renormalization. These hypotheses imply several qualitative predictions: The second Weinberg sum rule does not hold for the difference of the ${K}^{*}$ and axial-${K}^{*}$ propagators, even for exact $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)$; electromagnetic corrections require one subtraction proportional to the $I=1$, ${I}_{z}=0\ensuremath{\sigma}$ field; $\ensuremath{\eta}\ensuremath{\rightarrow}3\ensuremath{\pi}$ and ${\ensuremath{\pi}}_{0}\ensuremath{\rightarrow}2\ensuremath{\gamma}$ are allowed by current algebra. Octet dominance of nonleptonic weak processes can be understood, and a new form of superconvergence relation is deduced as a consequence. A generalization of the Bjorken limit is proposed.

Nucleon-Nucleus Optical-Model Parameters, A > 40 , E 50 MeV

Abstract: Proton-nucleus and neutron-nucleus standard optical-model parameters are given that represent, quite well, much of the elastic scattering data in the range $Ag40$, $El50$ MeV. These parameters were determined by fitting simultaneously a large sample of the available proton data, and independently, a large sample of the available neutron data. Explicit energy- and isospin-dependent terms were included and their coefficients obtained directly from the data analysis. The results are shown to be consistent with the range and strength of the central and isospin components of the two-body interaction.

Power spectrum of light scattered by two-level systems

Abstract: The power spectrum of the light scattered by a two-level atom driven near resonance by a monochromatic classical electric field is evaluated. The atom is assumed to relax to equilibrium with the driving field via radiation damping, which is treated by explicitly coupling the atom to the quantized electromagnetic field modes. The power spectrum of the scattered field is directly obtainable from the two-time atomic dipole moment correlation function, which is evaluated by a method based on a Markoff-type assumption analogous to that used to evaluate the time evolution of single-time atomic expectation values.

Structure of phenomenological lagrangians. ii.

Abstract: The general method for constructing invariant phenomenological Lagrangians is described. The fields are assumed to transform according to (nonlinear) realizations of an internal symmetry group, given in standard form. The construction proceeds through the introduction of covariant derivatives, which are standard forms for the field gradients. The case of gauge fields is also discussed.

Surface Plasmons in Thin Films

Abstract: The dispersion relations for surface plasma oscillations in normal metals are investigated for single- and multiple-film systems taking retardation effects into account. The simple dielectric function $\ensuremath{\epsilon}(\ensuremath{\omega})=1\ensuremath{-}\frac{{{\ensuremath{\omega}}_{p}}^{2}}{{\ensuremath{\omega}}^{2}}$ is found to be adequate for the high-frequency region in which oscillations remain undamped. Two types of possible modes of oscillation are found. One type corresponds to dispersion relations which behave linearly for not-so-high frequency, with a phase velocity always smaller than the velocity of light in the dielectric, but at least ten times larger than the Fermi velocity, while the other type consists of high-frequency modes ($\ensuremath{\omega}\ensuremath{\sim}{\ensuremath{\omega}}_{p}$). The role of these oscillations in the problem of transition radiation is reexamined. In the case of a thin metal film, a new interpretation is proposed for the peak observed in the transition radiation spectrum. Finally, the work is extended to superconducting metals where, in the frequency range $\ensuremath{\hbar}\ensuremath{\omega}l2\ensuremath{\Delta}$ ($2\ensuremath{\Delta}$ is the superconducting energy gap), we have justified the use of a dielectric function of the same functional form as given above but with ${{\ensuremath{\omega}}_{p}}^{2}$ replaced by an almost frequency-independent quantity ${{\ensuremath{\omega}}_{\mathrm{ps}}}^{2}$, where ${\ensuremath{\omega}}_{\mathrm{ps}}=\frac{c}{{\ensuremath{\lambda}}_{\mathrm{ps}}}$ and ${\ensuremath{\lambda}}_{\mathrm{ps}}$ is the actual penetration depth. In this frequency range, the oscillations are essentially undamped and play an important role in the electromagnetic properties of the multiple-film systems, and particularly when the systems exhibit the ac Josephson effect.

Self-Consistent Model of Hydrogen Chemisorption

Abstract: The chemisorption of a hydrogen atom on a transition-metal surface is treated theoretically on the basis of the Anderson Hamiltonian in Hartree-Fock approximation, which includes the interelectronic interaction within the $1s$ orbital. One-electron theory is shown to be inadequate for this problem. The localized states which may occur are discussed. A simple expression for the chemisorption energy $\ensuremath{\Delta}E$ is obtained, and a variational method is given for obtaining its self-consistent value. The metal eigenfunctions enter $\ensuremath{\Delta}E$ only through a function $\ensuremath{\Delta}(\ensuremath{\epsilon})$, and the foregoing results are exemplified and applied when this function is semielliptical. When the band is half-filled, a single analytic formula for the one-electron part of $\ensuremath{\Delta}E$ is obtained, in accord with the Kohn-Majumdar theorem. With some further assumptions, $\ensuremath{\Delta}E$ and the charge on the atom are calculated for adsorption on Ti, Cr, Ni, and Cu. The values of the hopping integral between the $1s$ orbital and a neighboring metal $d$ orbital required to fit the experimental $\ensuremath{\Delta}E$ are found to be similar and are reasonable. The correct prediction that ${|\ensuremath{\Delta}E|}_{\mathrm{Ni}}g{|\ensuremath{\Delta}E|}_{\mathrm{Cu}}$ is believed to be significant. A suggestive correlation is found between observations of catalytic ortho-para hydrogen interconversion on Pd-Au alloys and a rigidband calculation of $\ensuremath{\Delta}E$.

Phase Transitions of the Lennard-Jones System

Abstract: Monte Carlo computations have been performed in order to determine the phase transitions of a system of particles interacting through a Lennard-Jones potential. The fluid-solid transition has been investigated using a method recently introduced by Hoover and Ree. For the liquid-gas transition a method has been devised which forces the system to remain always homogeneous. A comparison is made with experiment in the case of argon. An indirect determination of the phase transition of the hard-sphere gas is made which is essentially in agreement with the results of the more direct calculations.

Quantized Fields and Particle Creation in Expanding Universes. I

Abstract: Spin-0 fields of arbitrary mass and massless fields of arbitrary spin are considered. The equations governing the fields are the covariant generalizations of the special-relativistic free-field equations. The metric, which is not quantized, is that of a universe with an expanding (or contracting) Euclidean 3-space. The spin-0 field of arbitrary mass is quantized in the expanding universe by the canonical procedure. The quantization is consistent with the time development dictated by the equation of motion only when the boson commutation relations are imposed. This consistency requirement provides a new proof of the connection between spin and statistics. We show that the particle number is an adiabatic invariant, but not a strict constant of the motion. We obtain an expression for the average particle density as a function of the time, and show that particle creation occurs in pairs. The canonical creation and annihilation operators corresponding to physical particles during the expansion are specified. Thus, we do not use an $S$-matrix approach. We show that in a universe with flat 3-space containing only massless particles in equilibrium, there will be precisely no creation of massless particles as a result of the expansion, provided the Einstein field equations without the cosmological term are correct. Furthermore, in a dust-filled universe with flat 3-space there will be precisely no creation of massive spin-0 particles in the limit of infinite mass, again provided that the Einstein field equations are correct. Conversely, without assuming any particular equations, such as the Einstein equations, as governing the expansion of the universe, we obtain the familiar Friedmann expansions for the radiation-filled and the dust-filled universes with flat 3-space. We only make a very general and natural hypothesis connecting the particle creation rate with the macroscopic expansion of the universe. In one derivation, we assume that in an expansion of the universe in which a particular type of particle is predominant, the type of expansion approached after a long time will be such as to minimize the average creation rate of that particle. In another derivation, we use the assumption that the reaction of the particle creation back on the gravitational field will modify the expansion in such a way as to reduce, if possible, the creation rate. This connection between the particle creation and the Einstein equations is surprising because the Einstein equations themselves played no part at all in the derivation of the equations governing the particle creation. Finally, on the basis of a so-called infinite-mass approximation, we argue that in the present predominantly dust-filled universe, only massless particles of zero spin might possibly be produced in significant amounts by the present expansion. In this connection, we show that massless particles of arbitrary nonzero spin, such as photons or gravitons, are not created by the expansion, regardless of its form.

Density operators and quasiprobability distributions.

Abstract: The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the $P$ representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the $P$ representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter $s$ assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for $\mathrm{Res}l0$, has bounded eigenvalues for $\mathrm{Res}=0$, and has infinite eigenvalues for $s=1$. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter $s$ is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to $s=+1$, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the $P$ representation is appropriate.

Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State

Abstract: A method of self-consistent fields is used to study the equilibrium configurations of a system of self-gravitating scalar bosons or spin-\textonehalf{} fermions in the ground state without using the traditional perfect-fluid approximation or equation of state. The many-particle system is described by a second-quantized free field, which in the boson case satisfies the Klein-Gordon equation in general relativity, ${\ensuremath{ abla}}_{\ensuremath{\alpha}}{\ensuremath{ abla}}^{\ensuremath{\alpha}}\ensuremath{\varphi}={\ensuremath{\mu}}^{2}\ensuremath{\varphi}$, and in the fermion case the Dirac equation in general relativity ${\ensuremath{\gamma}}^{\ensuremath{\alpha}}{\ensuremath{ abla}}_{\ensuremath{\alpha}}\ensuremath{\psi}=\ensuremath{\mu}\ensuremath{\psi}$ (where $\ensuremath{\mu}=\frac{\mathrm{mc}}{\ensuremath{\hbar}}$). The coefficients of the metric ${g}_{\ensuremath{\alpha}\ensuremath{\beta}}$ are determined by the Einstein equations with a source term given by the mean value $〈\ensuremath{\varphi}|{T}_{\ensuremath{\mu}\ensuremath{ u}}|\ensuremath{\varphi}〉$ of the energy-momentum tensor operator constructed from the scalar or the spinor field. The state vector $〈\ensuremath{\varphi}|$ corresponds to the ground state of the system of many particles. In both cases, for completeness, a nonrelativistic Newtonian approximation is developed, and the corrections due to special and general relativity explicitly are pointed out. For $N$ bosons, both in the region of validity of the Newtonian treatment (density from ${10}^{\ensuremath{-}80}$ to ${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, and number of particles from 10 to ${10}^{40}$) as well as in the relativistic region (density \ensuremath{\sim}${10}^{54}$ g ${\mathrm{cm}}^{\ensuremath{-}3}$, number of particles \ensuremath{\sim}${10}^{40}$), we obtain results completely different from those of a traditional fluid analysis. The energy-momentum tensor is anisotropic. A critical mass is found for a system of $N\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{2}\ensuremath{\sim}{10}^{40}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}25}$ g) self-gravitating bosons in the ground state, above which mass gravitational collapse occurs. For $N$ fermions, the binding energy of typical particles is ${G}^{2}{m}^{5}{N}^{\frac{4}{3}}{\ensuremath{\hbar}}^{\ensuremath{-}2}$ and reaches a value $\ensuremath{\sim}{\mathrm{mc}}^{2}$ for $N\ensuremath{\sim}{N}_{\mathrm{crit}}\ensuremath{\sim}{[\frac{(\mathrm{Planck}\mathrm{mass})}{m}]}^{3}\ensuremath{\sim}{10}^{57}$ (for $m\ensuremath{\sim}{10}^{\ensuremath{-}24}$ g, implying mass \ensuremath{\sim}${10}^{33}$ g, radius \ensuremath{\sim}${10}^{6}$ cm, density \ensuremath{\sim}${10}^{15}$ g/${\mathrm{cm}}^{3}$). For densities of this order of magnitude and greater, we have given the full self-consistent relativistic treatment. It shows that the concept of an equation of state makes sense only up to ${10}^{42}$ g/${\mathrm{cm}}^{3}$, and it confirms the Oppenheimer-Volkoff treatment in extremely good approximation. There exists a gravitational spin-orbit coupling, but its magnitude is generally negligible. The problem of an elementary scalar particle held together only by its gravitational field is meaningless in this context.

Observation of Anderson Localization in an Electron Gas

Abstract: Anderson has shown that there is no diffusion of an electron in certain random lattices, and Mott has pointed out that, for electrons in materials in which there is a potential energy varying in a random way from atom to atom, Anderson's work predicts that there should be a range of energies at the bottom of the conduction band for which an electron can move only by thermally activated hopping from one localized state to another. An energy ${E}_{c}$ will separate the energies where this happens from the nonlocalized range of energies where there is no thermal activation. Cerium sulfide, investigated some years ago by Cutler and Leavy, is a particularly suitable material testing whether this is so because, in the neighborhood of the composition ${\mathrm{Ce}}_{2}$${\mathrm{S}}_{3}$, $\frac{1}{9}$ of the cerium sites are vacancies distributed at random, and the number of free electrons can be varied with only very small changes in the number of vacancies. It is shown that the experimental results find a natural explanation in terms of this model: Conduction is by hopping when the concentration of electrons is low and the Fermi energy ${E}_{F}$ lies below ${E}_{c}$; but when the concentration is higher and ${E}_{F}g{E}_{c}$, conduction is by the usual band mechanism with a short mean free path. The thermoelectric power is examined in both ranges, and the Hall mobility in the hopping region (${E}_{F}l{E}_{c}$) seems in fair agreement with the theory of Holstein and Friedman.

Singularities in the X-Ray Absorption and Emission of Metals. III. One-Body Theory Exact Solution

Abstract: The singularities of x-ray absorption or emission in metals are studied by a new "one-body" method, which describes the scattering of conduction electrons by the transient potential due to the deep hole. Using the linked-cluster theorem, the net transition rate in the time representation is expressed as the product of two factors: a one-electron transient Green's function $L$, and the deep-level Green's function $\mathcal{G}$. These factors obey simple Dyson equations, which can be solved asymptotically by using Muskhelishvili's method. The x-ray transition rate is found to behave as $\frac{1}{{\ensuremath{\epsilon}}^{\ensuremath{\alpha}}}$, where $\ensuremath{\epsilon}$ is the frequency measured from the threshold, and $\ensuremath{\alpha}$ an exponent involving the various phase shifts ${\ensuremath{\delta}}_{l}$ which describe scattering by the deep hole. $\ensuremath{\alpha}$ may be g0 (infinite threshold) or 0 (zero threshold). The experimental implications of these results and their relation to the Friedel sum rule are briefly discussed.

First-Order Raman Effect in Wurtzite-Type Crystals

Abstract: First-order Raman scattering from BeO, ZnO, ZnS, and CdS, all having the wurtzite structure (${C}_{6v}$), has been investigated. A discussion of the effects of the competition between the long-range electrostatic forces and the short-range forces due to anisotropy in the interatomic force constants on the vibrational spectrum has been included. A series of scattering diagrams are presented showing the geometrical arrangements necessary to observe all the $k=0$ phonons for this type of crystal structure. In BeO the ${E}_{2}$ mode was resolved from the transverse modes for the first time. The assignments of the ${E}_{2}$ modes in ZnS differ from previous investigations. From absolute intensity measurements, electro-optic coefficients for BeO, ZnO, and CdS were determined.

Ordered Expansions in Boson Amplitude Operators

Abstract: The expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required $c$-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values $s=+1,0,\ensuremath{-}1$, respectively, of an order parameter $s$. In terms of this convention it is shown that for bounded operators the coefficients are finite when $sg0$, and the series are convergent when $sg\frac{1}{2}$. For each value of the order parameter $s$, a correspondence between operators and $c$-number functions is defined. Each correspondence is one-to-one and has the property that the function $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the $P$ representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter $s$.

Asymptotic Sum Rules at Infinite Momentum

Abstract: By combining the ${q}_{0}\ensuremath{\rightarrow}i\ensuremath{\infty}$ method for asymptotic sum rules with the $P\ensuremath{\rightarrow}\ensuremath{\infty}$ method of Fubini and Furlan, we relate the structure functions ${W}_{2}$ and ${W}_{1}$ in inelastic lepton-nucleon scattering to matrix elements of commutators of currents at almost equal times at infinite momentum. We argue that the infinite-momentum limit for these commutators does not diverge, but may vanish. If the limit is nonvanishing, we predict $\ensuremath{ u}{W}_{2}(\ensuremath{ u}, {q}^{2})\ensuremath{\rightarrow}{f}_{2}(\frac{\ensuremath{ u}}{{q}^{2}})$ and ${W}_{1}(\ensuremath{ u}, {q}^{2})\ensuremath{\rightarrow}{f}_{1}(\frac{\ensuremath{ u}}{{q}^{2}})$ as $\ensuremath{ u}$ and ${q}^{2}$ tend to $\ensuremath{\infty}$. From a similar analysis for neutrino processes, we conclude that at high energies the total neutrino-nucleon cross sections rise linearly with neutrino laboratory energy until nonlocality of the weak current-current coupling sets in. The sum of $\ensuremath{ u}p$ and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{ u}}p$ cross sections is determined by the equal-time commutator of the Cabibbo current with its time derivative, taken between proton states at infinite momentum.

Quantum Dielectric Theory of Electronegativity in Covalent Systems. I. Electronic Dielectric Constant

Abstract: Electronegativity difference is redefined as a scaling parameter, generalizing the concept of valence difference. A procedure for its evaluation is developed in terms of the dielectric constants of diatomic crystals. A simple alternative to the Clausius-Mossotti theory of the electronic dielectric constant is developed in terms of this concept. The effect of $d$-electron states and of hydrostatic pressure are discussed, and procedures for their approximate evaluation are developed. The treatment is extended to 68 crystals of the diamond, zincblende, wurtzite, and rock-salt types; values of the electronegativity parameter are tabulated for these crystals.

Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation

Abstract: We consider two simple field-theoretic models, (a) spinor electrodynamics and (b) the $\ensuremath{\sigma}$ model with the Polkinghorne axial-vector current, and show in each case that the axial-vector current satisfies a simple anomalous divergence equation exactly to all orders of perturbation theory. We check our general argument by an explicit calculation to second order in radiative corrections. The general argument is made tractable by introducing a cutoff, but to check the validity of this artifice, the second-order calculation is carried out entirely in terms of renormalized vertex and propagator functions, in which no cutoff appears.

Anomalous Ward identities in spinor field theories

Abstract: We consider the model of a spinor field with arbitrary internal degrees of freedom having arbitrary nonderivative coupling to external scalar, pseudoscalar, vector, and axial-vector fields. By carefully defining the $S$ matrix in the interaction picture, the vector and axial-vector currents associated with the external vector and axial-vector fields are found to satisfy anomalous Ward identities. If we require that the vector currents satisfy the usual Ward identities, the divergence of the axial-vector current contains well-defined anomalous terms. These terms are explicitly calculated.

Diabatic and adiabatic representations for atomic collision problems.

Abstract: The equations of the general Born-Oppenheimer separation into electronic and heavy-particle coordinates are re-examined, and the coupled equations that result for the heavy-particle motion are expressed in a particularly simple form. This is accomplished by introducing a generalized matrix operator for the effective momentum associated with the heavy particles; the matrix portion of this operator represents a coupling of the nuclear momentum with the electronic motion. The commutator between the momentum and potential matrices is a force matrix, which provides an alternative means of evaluating the momentum matrix. The momentum coupling has both radial and angular parts; the angular momentum coupling agrees with Thorson's expression. In the usual adiabatic molecular representation, the potential energy matrix is diagonalized, and all the coupling is thrown into the radial and angular momentum matrices. For collision problems it is often more important to diagonalize the radial momentum matrix, putting the radial off-diagonal coupling into the potential matrix; this generates a family of diabatic representations, the most important of which dissociates to unique separated atom states. This standard diabatic representations has the properties called for by Lichten, is uniquely defined even with the inclusion of configuration interaction, and leads immediately to the Landau-Zener-Stueckelberg limiting case under appropriate conditions.

Possible Pairing without Superconductivity at Low Carrier Concentrations in Bulk and Thin-Film Superconducting Semiconductors

Abstract: Simultaneous equations for the BCS parameter $\ensuremath{\Delta}$ and for the Fermi energy in the BCS state are studied at low carrier concentrations in bulk and thin-film superconducting semiconductors at $T=0$ for large cutoff energies, and solutions are plotted in the form of universal curves. For electron-electron attraction greater than some critical value in bulk material or for any attractive interaction in very thin films, the pair-binding energy tends to a constant limit for low $n$, and pairing without superconductivity is expected in some temperature range. Solutions of simultaneous equations for ${T}_{c}$ as obtained by formal application of the BCS theory, and for the Fermi energy at ${T}_{c}$, are also obtained in bulk material; but it is shown that at low carrier concentrations the temperature ${T}_{p}$ at which pairing takes place is given by ${T}_{p}=2{T}_{c}$, while it is thought that superconductivity will not set in until a lower temperature, of the order of the Bose-Einstein condensation temperature of the pairs, is reached. By combining the above theory with a previously published model for superconductivity in Zr-doped SrTi${\mathrm{O}}_{3}$, predictions for this material in the region of low carrier concentrations are made.

Exchange Interaction in Alloys with Cerium Impurities

Abstract: Starting with the Anderson model for the $4{f}^{1}$ configuration of cerium, the transformation of Schrieffer and Wolff is performed, taking into account combined spin and orbit exchange scattering. The resultant interaction Hamiltonian differs qualitatively from the conventional $s\ensuremath{-}f$ exchange interaction. The Kondo effect, the spin-disorder resistivity, the Ruderman-Kittel interaction, and the depression of the super-conducting transition temperature with impurity concentration are worked out for alloys containing cerium impurities on the basis of this new interaction.

Flux creep in type-ii superconductors.

Abstract: We have made measurements of the evanescent decay of the irreversible magnetization induced by magnetic cycling of solid superconducting cylinders in order to elucidate the mechanisms of Anderson's thermally activated flux-creep process. A superconducting quantum interferometer device coupled to the creep specimen by a superconducting flux transformer made possible observations of flux changes with a resolution of one part in ${10}^{9}$. The general applicability of Anderson's theory of flux creep was confirmed and the results were analyzed to show that: (1) The total flux in the specimen changed logarithmically in time, i.e., $\ensuremath{\Delta}\ensuremath{\varphi}\ensuremath{\propto}\frac{\mathrm{ln}t}{{t}_{0}}$. (2) The logarithmic creep rate $\frac{d\ensuremath{\varphi}}{d\mathrm{ln}t}$ is proportional to the critical current density ${J}_{c}$ and to the cube of the specimen radius. (3) The logarithmic creep rate appears to be only weakly temperature-dependent because a proportionality to $T$ is nearly compensated by the proportionality to ${J}_{c}$, which decreases as $T$ increases. (4) The creep process is a bulk process that is not surface-limited (in this case). (5) Flux enters and leaves the surface in discrete events containing from about one flux quantum up to at least ${10}^{3}$ flux quanta. (6) On departing from the critical state to a subcritical condition, the creep process tends to remain logarithmic in time, but the rate is decreased exponentially by decreasing $T$ and is decreased extremely rapidly by backing off of the applied field from the critical state. (7) At magnetic fields $Hl{H}_{c1}$ on the initial magnetization curve, no flux creep was observed, but the logarithmic creep rate showed a modest increase above ${H}_{c1}$ and a broad rise as $H$ approached ${H}_{c2}$. The creep process is characterized by a dimension parameter $\mathrm{VX}$ consisting of a flux bundle volume $V$ and pinning length $X$, and by an energy ${U}_{0}$, both of which are supposed to be material-sensitive parameters characteristic of the irreversible processes. These parameters were determined from the experiments. Bundle volumes $V\ensuremath{\approx}{10}^{\ensuremath{-}12}$ ${\mathrm{cm}}^{3}$ and energies ${U}_{0}\ensuremath{\approx}1$ eV were found, indicating that groups of fluxoids must be pinned and must move cooperatively. The results are found compatible with a recent model for flux pinning that includes these cooperative effects.

Scaling Laws for Dynamic Critical Phenomena

Abstract: The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from ${T}_{c}$, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near ${T}_{c}$, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.

Polarization analysis of thermal-neutron scattering.

Abstract: A triple-axis neutron spectrometer with polarization-sensitive crystals on both the first and third axes is described. The calculation of polarized-neutron scattering cross sections is presented in a form particularly suited to apply to this instrument. Experimental results on nuclear incoherent scattering, paramagnetic scattering, Bragg scattering, and spin-wave scattering are presented to illustrate the possible applications of neutron-polarization analysis.

Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice

Abstract: The critical-point anomaly of a plane square $m\ifmmode\times\else\texttimes\fi{}n$ Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ with $\ensuremath{\xi}=\frac{m}{n}$ fixed. Among other results, it is shown that for fixed $\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, the specific heat per spin of a large lattice is given by $\frac{{C}_{\mathrm{mn}}(T)}{{k}_{\mathrm{B}}\mathrm{mn}}={A}_{0}\mathrm{ln}n+B(\ensuremath{\tau}, \ensuremath{\xi})+{B}_{1}(\ensuremath{\tau})\frac{(\mathrm{ln}n)}{n}+\frac{{B}_{2}(\ensuremath{\tau}, \ensuremath{\xi})}{n}+O[\frac{{(\mathrm{ln}n)}^{3}}{{n}^{2}}],$ where explicit expressions can be given for ${A}_{0}$ and for the functions $B$, ${B}_{1}$, and ${B}_{2}$. It follows that the specific-heat peak of the finite lattice is rounded on a scale $\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$, while the maximum in ${C}_{\mathrm{mn}}(T)$ is displaced from ${T}_{c}$ by $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$. For ${\ensuremath{\xi}}_{0}g\ensuremath{\xi}g{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, where ${\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}$, the maximum lies above ${T}_{c}$; but for $\ensuremath{\xi}g{\ensuremath{\xi}}_{0}$ or $\ensuremath{\xi}l{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the maximum is depressed below ${T}_{c}$; when $\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the relative shift in the maximum from ${T}_{c}$ is only of order $\frac{(\mathrm{ln}n)}{{n}^{2}}$. Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.

Hypothesis of Limiting Fragmentation in High-Energy Collisions

Abstract: A hypothesis of limiting fragmentation of the target and of the projectile in a high-energy lepton-hadron or hadron-hadron collision is defined Arguments are given for the hypothesis Comparisons with various models and concepts are made Further speculations are made, including the absence of pionization processes in high-energy collisions and the dependence of multiplicity on the momentum transfer Experiments are suggested

Mössbauer Study of Several Ferrimagnetic Spinels

Abstract: The ferrimagnetic spinels ${\mathrm{Fe}}_{3}$${\mathrm{O}}_{4}$, Ni${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, Co${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, Mn${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, and Mg${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$ have been prepared and studied with the M\"ossbauer-effect technique over a wide temperature range both with and without large applied magnetic fields. The cation distributions have been determined and compared with magnetization measurements. For Co${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$ and Mg${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, this distribution depends on the heat treatment; two extremes---quenched and slowly cooled samples---have been investigated. The hyperfine magnetic fields at ${\mathrm{Fe}}^{57}$ nuclei in $A$ and $B$ sites have been obtained as a function of temperature. A number of hyperfine fields are identified with the $B$ sites of Co${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, Mn${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, and Mg${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$ and attributed to the kind and distributions of cations in the nearest-neighbor $A$ sites. From the data, the ratio of the $\mathrm{Co}(A)\ensuremath{-}\mathrm{Fe}(B)$ and $\mathrm{Mn}(A)\ensuremath{-}\mathrm{Fe}(B)$ to the $\mathrm{Fe}(A)\ensuremath{-}\mathrm{Fe}(B)$ superexchange interactions is found to be 0.68 and 0.66, respectively. Since the M\"ossbauer spectra provides no evidence for ${\mathrm{Fe}}^{2+}$ ions in Mn${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, a canted spin arrangement for the Mn ions in $A$ and $B$ sites is proposed to account for the small observed magnetization.