Showing papers in "Physical Review in 1961"

Effects of Configuration Interaction on Intensities and Phase Shifts

Abstract: The interference of a discrete autoionized state with a continuum gives rise to characteristically asymmetric peaks in excitation spectra. The earlier qualitative interpretation of this phenomenon is extended and revised. A theoretical formula is fitted to the shape of the $2s2p^{1}P$ resonance of He observed in the inelastic scattering of electrons. The fitting determines the parameters of the $2s2p^{1}P$ resonance as follows: $E=60.1$ ev, $\ensuremath{\Gamma}\ensuremath{\sim}0.04$ ev, $f\ensuremath{\sim}2 \mathrm{to} 4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. The theory is extended to the interaction of one discrete level with two or more continua and of a set of discrete levels with one continuum. The theory can also give the position and intensity shifts produced in a Rydberg series of discrete levels by interaction with a level of another configuration. The connection with the nuclear theory of resonance scattering is indicated.

Mach's principle and a relativistic theory of gravitation

Abstract: The role of Mach's principle in physics is discussed in relation to the equivalence principle. The difficulties encountered in attempting to incorporate Mach's principle into general relativity are discussed. A modified relativistic theory of gravitation, apparently compatible with Mach's principle, is developed.

Localized Magnetic States in Metals

Abstract: The conditions necessary in metals for the presence or absence of localized moments on solute ions containing inner shell electrons are analyzed. A self-consistent Hartree-Fock treatment shows that there is a sharp transition between the magnetic state and the nonmagnetic state, depending on the density of states of free electrons, the $s\ensuremath{-}d$ admixture matrix elements, and the Coulomb correlation integral in the $d$ shell; that in the magnetic state the $d$ polarization can be reduced rather severely to nonintegral values, without appreciable free electron polarization because of a compensation effect; and that in the nonmagnetic state the virtual localized $d$ level tends to lie near the Fermi surface. It is emphasized that the condition for the magnetic state depends on the Coulomb (i.e., exchange self-energy) integral, and that the usual type of exchange alone is not large enough in $d$-shell ions to allow magnetic moments to be present. We show that the susceptibility and specific heat due to the inner shell electrons show strongly contrasting behavior even in the nonmagnetic state. A calculation including degenerate $d$ orbitals and $d\ensuremath{-}d$ exchange shows that the orbital angular momentum can be quenched, even when localized spin moments exist, and even on an isolated magnetic atom, by kinetic energy effects.

Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity. II

Abstract: Continuing the program developed in a previous paper, a "superconductive" solution describing the proton-neutron doublet is obtained from a nonlinear spinor field Lagrangian. We find the pions of finite mass as nucleon-antinucleon bound states by introducing a small bare mass into the Lagrangian which otherwise possesses a certain type of the ${\ensuremath{\gamma}}_{5}$ invariance. In addition, heavier mesons and two-nucleon bound states are obtained in the same approximation. On the basis of numerical mass relations, it is suggested that the bare nucleon field is similar to the electron-neutrino field, and further speculations are made concerning the complete description of the baryons and leptons.

Dynamical model of elementary particles based on an analogy with superconductivity. part ii

Abstract: Continuing the program developed in a previous paper, a "superconductive" solution describing the protonneutron doublet is obtained from a nonlinear spinor field Lagrangian. The pions of finite mass are found as nucleonantinucleon bound states by introducing a small bare mass into the Lagrangian which otherwise possesses a certain type of the gamma /sub 5/ invariance. In addition, heavier mesons and two-nucleon bound states are obtained in the same approximation. On the basis of numerical mass relations, it is suggested that the bare nucleon field is similar to the electron-neutrino field, and further speculations are made concerning the complete description of the baryons and leptons. (auth)

Conservation laws and correlation functions

Abstract: ln describing transport phenomena, it is vital to build the conservation laws of number, energy, momentum, and angular momentum into the structure of the approximation used to determine the thermodynamic many-particle Green's functions. A method for generating conserving approximations was developed. This method is based on a consideration, at finite temperature, of the equations of motion obeyed by the one-particle propagator G, defined in the presence of a nonlocal external scalar field U. Approximations for G(U) are obtained by replacing the G/ sub 2/(U) which appears in these equations by various functionals of G(U). lf the approximation for G/sub 2/(U) satisfies certain simple symmetry conditions, then the G(U) thus defined obeys all the conservation laws. Furthermore, the two- particle correlation function, generated as ( delta G/ delta U) - /sub U=O/ plus or minus L, in terms of which all linear transport can be described, will obey all the conservation laws as well as several essential sum rules, such as the longitudinal f-sum rule. Examples of conserving approximations are described. The Hartree approximation, G/sub 2/(U) = G(U)G(U), generates the random-phase approximation for L. The Hartree-Fock approximation for G(U) leads to a natural generalization of the random-phase approximation in which hole-particlemore » ladder diagrams are summed. Another conserving approximation for G(U) is obtained by expanding the self-energy to first order in the many-particle scattenring matrix T(U). This T is obtained by summing ladder diagrams in which the sides of the ladder are composed of G(U)'s. The resulting L equation, which involves coefficients proportional to T/sup 2, is analogous to the linearized version of the usual Boltzmann equation. Finally, in order to obtain a description of collisions in a plasma, the selfenergy is expanded to first order in a dynamically shielded potential, V/sub 3/(U). This potential is obtained by summing bubbles composed of two G(U)'s. The resulting L equation is similar in structure to a Boltzmann equation in which the collision cross section is proportional to V/sub 3/sup 2/.(auth)« less

ENERGY DISSIPATION BY IONS IN THE kev REGION

Abstract: At low energies ionic collisions with atoms are largely elastic. Simple theoretical approximations to scattering cross sections, ranges and straggling are derived for power potentials, showing that the scattering is peaked in the forward direction rather than isotropic. Using an approximate universal potential of Thomas-Fermi type a natural measure of range, $\ensuremath{\rho}$, and of energy, $\ensuremath{\epsilon}$, is obtained for all ions in all substances. The corresponding range-energy curve is computed.At higher ion energies the electronic excitation becomes increasingly important. An approximate formula is given for the electronic stopping contribution, increasing proportional to ion velocity at low and moderate velocities. These results are applied in the interpretation of a few isotope effects, observed in range measurements.

Asymptotic Behavior and Subtractions in the Mandelstam Representation

Abstract: It is proved that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form $\mathrm{Cs}{\mathrm{ln}}^{2}s$ at the forward and backward angles, and $C{s}^{\frac{3}{4}}{\mathrm{ln}}^{\frac{3}{2}}s$ at any other fixed angle in the physical region, $C$ being a constant, $s$ being the total squared c.m. energy. This corresponds to cross sections increasing at most like ${\mathrm{ln}}^{2}s$. These restrictions limit the freedom of choice of the subtraction terms to six arbitrary single spectral functions and one subtraction constant.

Memory Effects in Irreversible Thermodynamics

Abstract: A new generalization of Onsager's theory of irreversible processes is presented. The main purpose is to allow for memory effects or causal time behavior, so that the response to a thermodynamic force comes later than the application of the force. This is accomplished by a statistical mechanical derivation of an exact non-Markoffian kinetic equation for the probability distribution in the space of macroscopic state variables. The memory effect in the resulting transport equations is represented by a time convolution of the thermodynamic forces with memory functions. The latter are time-correlation functions in the rates of change of the phase functions corresponding to macroscopic quantities. The resulting transport equations are not restricted to small deviations from thermal equilibrium. Onsager's theory is shown to be the low-frequency limit of our causal theory.

Low-Frequency Conductivity Due to Hopping Processes in Silicon

Abstract: The complex conductivity has been measured in $n$-type silicon with various kinds of impurities at frequencies between ${10}^{2}$ and ${10}^{5}$ cps and temperatures between 1 and 20\ifmmode^\circ\else\textdegree\fi{}K. In most cases it is orders of magnitude larger than the measured dc conductivity and is attributed to polarization caused by hopping processes. The observed frequency dependence in the measured range can be expressed as $A{\ensuremath{\omega}}^{0.8}$, where $A$ is a complex constant. At the low-temperature end the conductivity is roughly proportional to minority impurity concentration and is almost independent of the majority impurity concentration and At higher temperatures the conductivity becomes approximately proportional to the product of both concentrations. A simple theory, based on the currently accepted model of impurity conduction, is given for the higher temperature range. It accounts well for the observed frequency and concentration dependences. However, only order-of-magnitude absolute agreement is obtained.

Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas

Abstract: An electron gas with short-range interactions is considered in the presence of a uniform magnetic field. It is shown that (1) the cyclotron resonance frequency is independent of the interaction; (2) for a two-dimensional gas, the de Haas-van Alphen period is independent of the interaction. The low-lying excited states are briefly discussed.

Simultaneous Effect of Doppler and Foreign Gas Broadening on Spectral Lines

Abstract: By using the classical Fourier integral theory, an expression is given for the shape of a spectral line, broadened by phase changes due to collisions and by the actual changes in velocity of the emitting particles resulting from collisions. The result is not a simple Voigt-type folding of an exponential into a dispersion distribution; it exhibits the contraction noted by Dicke and leads to the usual formulas when the time interval between path-deflecting or phase-disturbing collisions becomes very great.

Coordinate invariance and energy expressions in general relativity

Abstract: The invariance of various definitions proposed for the energy and momentum of the gravitational field is examined. We use the boundary conditions that ${g}_{\ensuremath{\mu}\ensuremath{ u}}$ approaches the Lorentz metric as $\frac{1}{r}$, but allow ${g}_{\ensuremath{\mu}\ensuremath{ u},\ensuremath{\alpha}}$ to vanish as slowly as $\frac{1}{r}$. This permits "coordinate waves." It is found that none of the expressions giving the energy as a two-dimensional surface integral are invariant within this class of frames. In a frame containing coordinate waves they are ambiguous, since their value depends on the location of the surface at infinity (unlike the case where ${g}_{\ensuremath{\mu}\ensuremath{ u},\ensuremath{\alpha}}$ vanishes faster than $\frac{1}{r}$). If one introduces the prescription of space-time averaging of the integrals, one finds that the definitions of Landau-Lifshitz and Papapetrou-Gupta yield (equal) coordinate-invariant results. However, the definitions of Einstein, M\o{}ller, and Dirac become unambiguous, but not invariant.The averaged Landau-Lifshitz and Papapetrou-Gupta expressions are then shown to give the correct physical energy-momentum as determined by the canonical formulations Hamiltonian involving only the two degrees of freedom of the field. It is shown that this latter definition yields that inertial energy for a gravitational system which would be measured by a nongravitational apparatus interacting with it. The canonical formalism's definition also agrees with measurements of gravitational mass by Newtonian means at spacial infinity. It is further shown that the energy-momentum may be invariantly calculated from the asymptotic form of the metric field at a fixed time.

Quantum Fluctuations and Noise in Parametric Processes. I.

Abstract: A quantum mechanical model for parametric interactions is used to evaluate the effect of the measuring (amplifying) process on the statistical properties of radiation. Parametric amplification is shown to be ideal in the sense that it allows a simultaneous determination of the phase and number of quanta of an electromagnetic wave with an accuracy which is limited only by the uncertainty principle. Frequency conversion via parametric processes is shown to be free of zero-point fluctuations.

Relationship Between Crystal Symmetry and Magnetic Properties of Ionic Compounds Containing Mn 3

Abstract: An investigation has been made of the relationship between crystallographic symmetry and ${\mathrm{Mn}}^{3+}$ - ${\mathrm{O}}^{2\ensuremath{-}}$ - ${\mathrm{Mn}}^{3+}$ 180\ifmmode^\circ\else\textdegree\fi{} superexchange interactions in several perovskite systems. In particular, crystallographic and magnetic measurements have been made on a number of samples in the systems $\mathrm{La}({\mathrm{Mn}}_{1\ensuremath{-}x}{M}_{x}){\mathrm{O}}_{3+\ensuremath{\delta}}$, where $M=\mathrm{G}\mathrm{a},\phantom{\rule{0ex}{0ex}}\mathrm{C}\mathrm{o},\phantom{\rule{0ex}{0ex}}\mathrm{N}\mathrm{i}$. In all three systems, the ${\mathrm{Mn}}^{3+}$ - ${\mathrm{O}}^{2\ensuremath{-}}$ - ${\mathrm{Mn}}^{3+}$ interactions are found to be ferromagnetic for $O$-orthorhombic samples having $al\frac{c}{\sqrt{2}}lb$. For $xl0.5$ in the system $M=\mathrm{Ga}$, there is ${O}^{\ensuremath{'}}$-orthorhombic symmetry ($\frac{c}{\sqrt{2}}lalb$) and ferrimagnetism that is suggestive of anisotropic ${\mathrm{Mn}}^{3+}$ - ${\mathrm{O}}^{2\ensuremath{-}}$ - ${\mathrm{Mn}}^{3+}$ interactions, similar to those found in LaMn${\mathrm{O}}_{3}$, and preferential ordering of the ${\mathrm{Ga}}^{3+}$ into one magnetic sublattice. Measurements of Curie temperature vs composition in this system support ordering of the gallium in the compositional range $x\ensuremath{\le}0.4$, partial ordering in the range $0.4lx\ensuremath{\le}0.6$. These observations are consistent with the magnetic measurements of various other workers on the systems $(\mathrm{La}, {M}^{\ensuremath{'}2+})\mathrm{Mn}{\mathrm{O}}_{3+\ensuremath{\delta}}$, La(Mn,Cr)${\mathrm{O}}_{3}$ (La,Ba) (Mn,Ti)${\mathrm{O}}_{3}$.The ferromagnetic ${\mathrm{Mn}}^{3+}$ - anion - ${\mathrm{Mn}}^{3+}$ interactions that occur in the perovskites with $O$-orthorhombic or rhombohedral symmetry and in the NiAs-type compounds cannot be accounted for by present superexchange theory if the electron configuration about a ${\mathrm{Mn}}^{3+}$ ion is assumed fixed with one electron arithmetically averaged over the two ${e}_{g}$ orbitals, or if static, local distortions are randomly distributed through the structure. It is pointed out that Jahn-Teller electronic ordering is fast relative to the atomic vibrations so that there is strong coupling of the vibrational modes and the ${e}_{g}$-electron configuration. This means that the electron configuration that is used in the superexchange calculation must be correlated with the vibrational modes. If this is done, a ferromagnetic ${\mathrm{Mn}}^{3+}$ - anion - ${\mathrm{Mn}}^{3+}$ interaction follows from the superexchange theory.

Statistical Mechanics of Dimers on a Plane Lattice

Abstract: This paper considers the statistical mechanics of hard rigid dimers distributed on a lattice (each dimer occupying two nearest neighbor lattice sites). The problem is solved in exact closed form for a finite $m\ifmmode\times\else\texttimes\fi{}n$ plane square lattice with edges which is completely filled with $\frac{1}{2}\mathrm{mn}$ dimers (close-packed limit). In terms of the activities $x$ and $y$ of horizontal and vertical dimers, the configurational partition function ${Z}_{\mathrm{mn}}(x, y)$ is given in the limit of a large lattice by $limit of\text{}{(\mathrm{mn})}^{\ensuremath{-}1}\mathrm{ln}{Z}_{\mathrm{mn}}(x, y)\text{as}m,n\ensuremath{\rightarrow}\ensuremath{\infty}=\frac{1}{2}\mathrm{ln}y+(\frac{1}{\ensuremath{\pi}})\ensuremath{\int}{0}^{\frac{x}{y}}(\frac{1}{v})invtanv\mathrm{dv}.$ It follows that the free energy and entropy of the system are smooth continuous functions of the densities of horizontal and vertical dimers. The number of ways of filling the lattice with dimers is calculated exactly for $m=n=8$ and is given asymptotically by ${[\mathrm{exp}(\frac{2G}{\ensuremath{\pi}})]}^{\frac{1}{2}\mathrm{mn}}={(1.791623)}^{\frac{1}{2}\mathrm{mn}}$. The results are derived with the aid of operator techniques which reduce the partition function to a Pfaffian and hence to a determinant. Some results are also presented for the more general case with monomers present.

Infrared Lattice Bands of Quartz

Abstract: The infrared lattice bands of $\ensuremath{\alpha}$ quartz have been investigated at 297\ifmmode^\circ\else\textdegree\fi{}K from 5 to 37\ensuremath{\mu} in reflection and transmission with polarized light. Previously published measurements of the optical constants do not agree in this spectral range. It is shown that dispersion theory can fit the data within experimental error throughout the range, and accurate values of the dispersion parameters and the optical constants are obtained. This is the first accurate dispersion analysis of a complex spectrum. A study was made of the accuracy of the Kramers-Kronig method of analysis on this spectrum. The strength, width, and frequency of 14 optically active lattice vibrations are given, 4 of which have not previously been established. From a consideration of published Raman data, 10 of the resonances are assigned according to symmetry type as fundamental vibrations.

Defects in Irradiated Silicon. I. Electron Spin Resonance of the Si-A Center

Abstract: The Si-$A$ center is a major, radiation-damage defect produced in "pulled" silicon by a room temperature irradiation. As a result of studies described in this paper (I), and the following one (II), it is concluded that this center is a lattice vacancy with an oxygen atom impurity bridging two of the four broken bonds associated with the vacancy. Spin resonance and electrical activity arise from an electron trapped in the other two bonds. In this paper (I), the spin resonance studies are described. A molecular orbital treatment of the trapped electron wave-function satisfactorily accounts for the observed $g$ tensor, as well as the hyperfine interaction observed with neighboring 4.7% abundant ${\mathrm{Si}}^{29}$ nuclei. Study of the changes in the spectrum of a sample subjected to uniaxial stress are also described. Under stress, the amplitudes of the individual resonance components (which correspond to different orientations of the defect in the crystal) are observed to change. This results from (1) electronic redistribution of the trapped electrons among the defects, and (2) thermally activated reorientation of the defects themselves under the applied stress. These two effects are separated and a quantitative study of their magnitudes and signs, as well as their rates, is given. The results confirm many of the important microscopic features of the model.

Time in the Quantum Theory and the Uncertainty Relation for Time and Energy

Abstract: Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)g~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.

Dielectric and X-Ray Studies of Ca x Ba 1 − x Ti O 3 and Ca x Sr 1 − x Ti O 3

Abstract: Ceramics of ${\mathrm{Ca}}_{x}{\mathrm{Ba}}_{1\ensuremath{-}x}\mathrm{Ti}{\mathrm{O}}_{3}$ and ${\mathrm{Ca}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}\mathrm{Ti}{\mathrm{O}}_{3}$ have been prepared and their dielectric and structural properties investigated. Firing conditions were adjusted to obtain sharp x-ray back reflections. The Curie point of ${\mathrm{Ca}}_{x}{\mathrm{Ba}}_{1\ensuremath{-}x}\mathrm{Ti}{\mathrm{O}}_{3}$ increases with Ca concentration up to 136\ifmmode^\circ\else\textdegree\fi{}C for $x=0.08$, and then decreases. Both the tetragonal-orthorhombic and the orthorhombic-rhombohedral transition points of ${\mathrm{Ca}}_{x}{\mathrm{Ba}}_{1\ensuremath{-}x}\mathrm{Ti}{\mathrm{O}}_{3}$ decrease monotonically with increasing Ca concentration. ${\mathrm{Ca}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}\mathrm{Ti}{\mathrm{O}}_{3}$ solid solutions with $0.01l~xl~0.10$ are ferroelectric at very low temperatures. SrTi${\mathrm{O}}_{3}$ assumes a tetragonal structure below about 80\ifmmode^\circ\else\textdegree\fi{}K.

Exponential Temperature Dependence of Young's Modulus for Several Oxides

Abstract: Young's modulus was measured over the temperature range 77\ifmmode^\circ\else\textdegree\fi{}-850\ifmmode^\circ\else\textdegree\fi{}K by an accurate resonance technique. Data are presented for single crystals of aluminum oxide with various orientations of the crystallographic axes and for polycrystalline aluminum oxide, thorium oxide, and magnesium oxide. The results show that the range of validity of a ${T}^{4}$ temperature dependence predicted by theory must be quite small. The temperature dependence is very well described over the whole temperature range by $T\mathrm{exp}(\ensuremath{-}\frac{{T}_{0}}{T})$, where ${T}_{0}$ is an empirical parameter.

Origin of effective fields in magnetic materials

Abstract: The origin of the effective magnetic fields at the nuclei of magnetic materials which have been determined by M\"ossbauer, nuclear magnetic resonance, electron paramagnetic resonance, specific heat, and nuclear polarization methods is investigated theoretically by means of the exchange polarization mechanism. Exchange-polarized iron series Hartree-Fock calculations were carried out for (a) free ions and neutral atoms, (b) ions in a (crude) crystalline field (as in a salt), and (c) spin densities and configurations which conform with energy band and neutron magnetic scattering observations for the ferromagnetic metals. The effective field data for metals, ferrites, rare-earth garnets, and salts are then discussed and it is shown that the dominant contribution to the effective field (in almost every case) arises from the (exchange) polarization of the core electrons by the spin density of the unpaired outer electrons. For the transition metals, the role of the conduction electrons is analyzed including some new contributions not previously considered. The data for ions like ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Mn}}^{++}$ may be understood mainly on the basis of the core polarization term but such factors as covalent bonding, charge transfer, crystal field effects (such as distortions from cubic symmetry) must also be included. For ions like ${\mathrm{Fe}}^{++}$ and ${\mathrm{Co}}^{++}$ the (large) field due to unquenched orbital angular momentum must also be considered and several cases in which the orbital field dominates are discussed. The exchange polarization method and the accuracy of the analytic spin-polarized Hartree-Fock functions are discussed with regard to the sensitivity of the internal field to orbital descriptions, the effect of crystalline environments, and to expansion and contraction of the spin density. Each factor is investigated in detail by means of accurate exchange-polarized calculations. In conjunction with these studies a restricted Hartree-Fock calculation for ${\mathrm{Mn}}^{++}$ was carried out (and is reported as an Appendix) which is more accurate than existing calculations and indicates the accuracy of earlier analytic Hartree-Fock calculations.

Core excitations in nondeformed, odd-a nuclei

Abstract: The possibility of describing some excited states of odd-A nuclei in terms of excitations of the even-even core is investigated. No assumption is made on the nature of the core excitation, but certain relations involving electromagnetic transitions and moments are deduced. These seem to fit well some data available on Ag/sup 1//sup 0//sup 7/,Ag/sup 1//sup 0//sup 9/, Au /sup 1//sup 9//sup 7/,Hg/sup 1//sup 9//sup 9/, Tl/sup 2//sup 0//sup 3/, and Tl/sup 2//sup 0// sup 5/. More experimental data are required to test the validity of this model in other cases. (auth)

Generalized Bardeen-Cooper-Schrieffer States and the Proposed Low-Temperature Phase of Liquid He 3

Abstract: Particle interactions in a Fermi gas may be such as to attract pairs near the Fermi surface more strongly in $l=1, 2, 3$ or higher states than in the simple spherically symmetrical $s$ state. In that case the Bardeen-Cooper-Schrieffer condensed state must be generalized, and the resulting state is an anisotropic superfluid. We have studied the properties of this type of state in considerable detail, especially for $l=1 \mathrm{and} 2$. We have derived expressions for the energy, the moment of inertia, the magnetic susceptibility and the specific heat. We also derive the density correlation function and the density-current density correlation; in some cases the latter implies that the liquid has net surface currents and a net orbital angular momentum. The ground state for $l=2$ is different from those previously considered, and has cubic symmetry and no net angular momentum. A general method for replacing the possibly rather complicated potential by a simple scattering matrix is given. A brief discussion of possible collective effects is included. We apply our results to liquid ${\mathrm{He}}^{3}$; after correction for scattering by a method due to Suhl, it is found that the predicted transition should take place below 0.02\ifmmode^\circ\else\textdegree\fi{}K. Other possible applications are suggested.

Fine Structure and Magneto-Optic Effects in the Exciton Spectrum of Cadmium Sulfide

Abstract: The valence band of cadmium sulfide is split by spin-orbit and crystal field effects into three nearly degenerate bands at k=0. The magneto-optic absorption spectrum of direct excitons formed from the top valence band and the conduction band has been studied in detail. Most of the experiments reported have been performed in light polarized parallel to the hexagonal axis. In this geometry, the exciton series consists of weak lines amenable to magneto-optic experiments. When the magnetic field and the wave vector of the light are perpendicular to each other and to the hexagonal axis, the reversal of the magnetic field produces large changes in the absorption spectrum. This effect can be quantitatively understood as an interference effect between allowed exciton transitions (optical matrix elements independent of the wave vector of the light) and forbidden exciton transitions (optical matrix elements proportional to the wave vector of the light). It is shown that in CdS the forbidden processes having a principal quantum number 2 are somewhat stronger than allowed processes of the same quantum number. By using group theory and the effective-mass approximation, the electron and hole anisotropic $g$ values and masses are determined from an analysis of the exciton spectrum. The electron mass, ${0.20}_{5}$ $m$ (almost isotropic), determined in this analysis is compatible with the assumption that the k=0 conduction band valley is the lowest conduction band valley. The hole masses for the top valence band are ${m}_{h\ensuremath{\perp}}=0.7 m$ and ${m}_{h\mathrm{II}}\ensuremath{\approx}5m$. An experimental upper limit on the slope of the conduction band at k=0 is obtained.

Stochastic Dynamics of Quantum-Mechanical Systems

Abstract: The most general dynamical law for a quantum mechanical system with a finite number of levels is formulated. A fundamental role is played by the so-called "dynamical matrix" whose properties are stated in a sequence of theorems. A necessary and sufficient criterion for distinguishing dynamical matrices corresponding to a Hamiltonian time-dependence is formulated. The non-Hamiltonian case is discussed in detail and the application to paramagnetic relaxation is outlined.

Phenomenological Discussion of Magnetic Ordering in the Heavy Rare-Earth Metals

Abstract: The rare-earth metals Gd-Tm have similar crystal structures and their magnetic properties have been partially evaluated by a number of techniques. The magnetic order is complicated, showing several phases in some cases and differing considerably in the various elements. These various orderings can be explained on a molecular field (Bragg-Williams) model if a long-range oscillatory exchange interaction whose minimum Fourier component $J(\mathrm{q})$ is at $q\ensuremath{ e}0$, small quadrupole-quadrupole interaction, and anisotropy are included. A crystal field calculation gives axial and hexagonal anisotropies which vary along the series in a way which accounts for the observed structures. In Tb, Dy, and Ho the moment is forced into the basal plane and the order is a spiral at high $T$, becoming ferromagnetic at low $T$ because of the hexagonal anisotropy. The quadrupole-quadrupole interaction determines the change of pitch with $T$. In Er and Tm the moment is forced along the $c$ axis and the observed order, with sinusoidal variation of this moment, is found to have lowest free energy at high $T$. As $T$ is lowered, transitions to an anti-phase domain structure and then to ferromagnetism are predicted.

Elastic Constants of Iron from 4.2 to 300°K

Abstract: The zero-field elastic constants of iron have been measured from 4.2 to 300\ifmmode^\circ\else\textdegree\fi{}K using the ultrasonic pulse technique. Extrapolation of the data to absolute zero gives ${c}_{11}=2.431\ifmmode\pm\else\textpm\fi{}0.008$, ${c}_{12}=1.381\ifmmode\pm\else\textpm\fi{}0.004$, and ${c}_{44}=1.219\ifmmode\pm\else\textpm\fi{}0.004$, all expressed in units of ${10}^{12}$ dyne ${\mathrm{cm}}^{\ensuremath{-}2}$. The corresponding limiting value of the Debye temperature is ${\ensuremath{\theta}}_{0}=(477\ifmmode\pm\else\textpm\fi{}2)\ifmmode^\circ\else\textdegree\fi{}$K. Using this figure, the low-temperature heat capacity data for iron have been reanalyzed assuming the presence of a spin-wave contribution to the specific heat, i.e., the heat capacity is assumed to follow the relation $C=\ensuremath{\gamma}T+\ensuremath{\beta}{T}^{3}+\ensuremath{\alpha}{T}^{\frac{3}{2}}$. A least squares fit of $\frac{(C\ensuremath{-}\ensuremath{\beta}{T}^{3})}{T}$ versus ${T}^{\frac{1}{2}}$ gives $\ensuremath{\gamma}=(11.7\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ cal ${\mathrm{mole}}^{\ensuremath{-}1}$ ${\mathrm{deg}}^{\ensuremath{-}2}$, $\ensuremath{\alpha}=(2\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5} \mathrm{cal} {\mathrm{mole}}^{\ensuremath{-}} {\mathrm{deg}}^{\ensuremath{-}\frac{5}{2}}$. There is agreement, within experimental error, between the latter figure and the theoretical estimate of $\ensuremath{\alpha}=0.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5} \mathrm{cal} {\mathrm{mole}}^{\ensuremath{-}1} {\mathrm{deg}}^{\ensuremath{-}\frac{5}{2}}$ obtained from the low-temperature magnetization data of Fallot. From the room temperature elastic constants, the compressibility of iron is found to be $K=(5.95\ifmmode\pm\else\textpm\fi{}0.02)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ ${\mathrm{cm}}^{2}$ ${\mathrm{dyne}}^{\ensuremath{-}1}$, which agrees exactly with the static value obtained by Bridgman.

Cancellation of kinetic and potential energy in atoms, molecules, and solids

Abstract: In the energy levels of valence electrons in atoms, molecules, solids, and liquids, there is a contribution from the large negative potential energy inside the core of the atom and the large positive kinetic energy which the electron has there. The kinetic energy can be represented by a repulsive pseudopotential that cancels most of the potential energy inside the core. The explicit representation of the pseudopotential is now developed further to demonstrate more clearly the extent of the cancellation. The formalism justifies the simple models for valence electrons. It is also used to relate similar atoms from different rows of the periodic table, and in particular to discuss the systematic trends in the energy levels of the alkali and noble metal atoms. (auth)