Showing papers in "Physical Review in 1959"
TL;DR: In this article, it was shown that there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish.
Abstract: In this paper, we discuss some interesting properties of the electromagnetic potentials in the quantum domain. We shall show that, contrary to the conclusions of classical mechanics, there exist effects of potentials on charged particles, even in the region where all the fields (and therefore the forces on the particles) vanish. We shall then discuss possible experiments to test these conclusions; and, finally, we shall suggest further possible developments in the interpretation of the potentials.
5,553 citations
TL;DR: In this article, a phenomenological model is developed to facilitate calculation of lattice thermal conductivities at low temperatures, where the phonon scattering processes can be represented by frequency-dependent relaxation times.
Abstract: A phenomenological model is developed to facilitate calculation of lattice thermal conductivities at low temperatures. It is assumed that the phonon scattering processes can be represented by frequency-dependent relaxation times. Isotropy and absence of dispersion in the crystal vibration spectrum are assumed. No distinction is made between longitudinal and transverse phonons. The assumed scattering mechanisms are (1) point impurities (isotopes), (2) normal three-phonon processes, (3) umklapp processes, and (4) boundary scattering. A special investigation is made of the role of the normal processes which conserve the total crystal momentum and a formula is derived from the Boltzmann equation which gives their contribution to the conductivity. The relaxation time for the normal three-phonon processes is taken to be that calculated by Herring for longitudinal modes in cubic materials. The model predicts for germanium a thermal conductivity roughly proportional to ${T}^{\ensuremath{-}\frac{3}{2}}$ in normal material, but proportional to ${T}^{\ensuremath{-}2}$ in single-isotope material in the temperature range 50\ifmmode^\circ\else\textdegree\fi{}-100\ifmmode^\circ\else\textdegree\fi{}K. Magnitudes of the relaxation times are estimated from the experimental data. The thermal conductivity of germanium is calculated by numerical integration for the temperature range 2-100\ifmmode^\circ\else\textdegree\fi{}K. The results are in reasonably good agreement with the experimental results for normal and for single-isotope material.
2,390 citations
TL;DR: In this article, the theory of indirect exchange in poor conductors is examined from a new viewpoint in which the $d$ (or $f$) shell electrons are placed in wave functions assumed to be exact solutions of the problem of a single $d-electron in the presence of the full diamagnetic lattice.
Abstract: The theory of indirect exchange in poor conductors is examined from a new viewpoint in which the $d$ (or $f$) shell electrons are placed in wave functions assumed to be exact solutions of the problem of a single $d$-electron in the presence of the full diamagnetic lattice. Inclusion of $d$-electron interactions leads to three spin-dependent effects which, in the usual order of their sizes, we call: superexchange per se, which is always antiferromagnetic; direct exchange, always ferromagnetic; and an indirect polarization effect analogous to nuclear indirect exchange. Superexchange itself is shown to be closely related to the poor conductivity, in agreement with experiment. By means of crystal field theory the parameters determining superexchange can be estimated, and in favorable cases (NiO, LaFe${\mathrm{O}}_{3}$) the exchange integrals can be evaluated with accuracy of several tens of percent. Qualitative understanding of the whole picture of exchange in iron group oxides and fluorides follows from these ideas.
1,876 citations
TL;DR: In this paper, a series of papers dealing with many-particle systems from a unified, nonperturbative point of view is presented, which includes derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers.
Abstract: This is the first of a series of papers dealing with many-particle systems from a unified, nonperturbative point of view. It contains derivations and discussions of various field-theoretical techniques which will be applied in subsequent papers. In a short introduction the general method of approach is summarized, and its relationship to other field-theoretic problems indicated. In the second section the macroscopic properties of the spectra of many-particle systems are described. Asymptotic evaluations are performed which characterize these macroscopic features in terms of intensive parameters, and the relationship of these parameters to thermodynamics is discussed. The special characteristics of the ground state are shown to follow as a limiting case of the asymptotic evaluations. The third section is devoted to the time-dependent field correlation functions, or Green's functions, which describe the microscopic behavior of a multiparticle system. These functions are defined, and related to intensive macroscopic variables when the energy and number of particles are large. Spectral representations and other properties of various one-particle Green's functions are derived. In the fourth section the treatment of non-equilibrium processes is considered. As a particular example, the electromagnetic properties of a system are expressed in terms of the special two-particle Green's function which describes current correlation. The discussion yields specifically a fluctuation-dissipation theorem, a sum rule for conductivity, and certain dispersion relations. The fifth section deals with the differential equations which determine the Green's functions. The boundary conditions that characterize the Green's function equations are exhibited without reference to adiabatic decoupling. A method for solving the equations approximately, by treating the correlations among successively larger numbers of particles, is considered. The first approximation in this sequence is shown to yield a generalized Hartree-like equation. A related, but rigorous, identity for the single-particle Green's function is then derived. A second approximation, which takes certain two-particle correlations into account, is shown to produce various additional effects: The interaction between particles is altered in a manner characterized by the intensive macroscopic parameters, and the modification and spread of the energy-momentum relation come into play. In the final section compact formal expressions for the Green's functions and other physical quantities are derived. Alternative equations and systematic approximations for the Green's functions are obtained.
1,283 citations
TL;DR: In this article, a singularity-free elementary algebraic solution of the field equations is presented and exact values obtained from it compared with the limits prescribed by some of the inequalities.
Abstract: In Part I of this paper certain well known results concerning the Schwarzschild interior solution are generalized to more general static fluid spheres in the form of inequalities comparing the boundary value of ${g}_{44}$ with certain expressions involving only the mass concentration and the ratio of the central energy density to the central pressure. A minimal theorem appropriate to the relativistic domain is derived for the central pressure, corresponding to a well-known classical result. Inequalities involving the proper energy and the potential energy are also considered, as is the introduction of the physical radius in place of the coordinate radius. A singularity-free elementary algebraic solution of the field equations is presented and exact values obtained from it compared with the limits prescribed by some of the inequalities. In Part II an answer is given to the question whether the total amount of radiation emitted during the symmetrical gravitational contraction of an amount of matter whose initial energy, at complete dispersion, is ${W}_{0}$ can ever exceed ${W}_{0}$.
1,161 citations
TL;DR: In this paper, it was shown that the equation of motion for the pair creation operators is the same as that for the one-particle density matrix in the self-consistent field framework.
Abstract: The self-consistent field method in which a many-electron system is described by a time-dependent interaction of a single electron with a self-consistent electromagnetic field is shown to be equivalent for many purposes to the treatment given by Sawada and Brout. Starting with the correct many-electron Hamiltonian, it is found, when the approximations characteristic of the Sawada-Brout scheme are made, that the equation of motion for the pair creation operators is the same as that for the one-particle density matrix in the self-consistent field framework. These approximations are seen to correspond to (1) factorization of the two-particle density matrix, and (2) linearization with respect to off-diagonal components of the one-particle density matrix. The complex, frequency-dependent dielectric constant is obtained straight-forwardly from the self-consistent field approach both for a free-electron gas and a real solid. It is found to be the same as that obtained by Nozi\'eres and Pines in the random phase approximation. The resulting plasma dispersion relation for the solid in the limit of long wavelengths is discussed.
976 citations
TL;DR: In this paper, the Morse parameters were calculated using experimental values for the energy of vaporization, the lattice constant, and the compressibility for both face-centered and body-centered cubic metals.
Abstract: The Morse parameters were calculated using experimental values for the energy of vaporization, the lattice constant, and the compressibility. The equation of state and the elastic constants which were computed using the Morse parameters, agreed with experiment for both face-centered and body-centered cubic metals. All stability conditions were also satisfied for both the face-centered and the body-centered metals. This shows that the Morse function can be applied validly to problems involving any type of deformation of the cubic metals.
928 citations
TL;DR: In this article, it is shown that advantage of crystal symmetry can be taken to construct wave functions which are best described as the smooth part of symmetrized Bloch functions.
Abstract: For metals and semiconductors the calculation of crystal wave functions is simplest in a plane wave representation. However, in order to obtain rapid convergence it is necessary that the valence electron wave functions be made orthogonal to the core wave functions. Herring satisfied this requirement by choosing as basis functions "orthogonalized plane waves." It is here shown that advantage can be taken of crystal symmetry to construct wave functions ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ which are best described as the smooth part of symmetrized Bloch functions. The wave equation satisfied by ${\ensuremath{\phi}}_{\ensuremath{\alpha}}$ contains an additional term of simple character which corresponds to the usual complicated orthogonalization terms and has a simple physical interpretation as an effective repulsive potential. Qualitative estimates of this potential in analytic form are presented. Several examples are worked out which display the cancellation between attractive and repulsive potentials in the core region which is responsible for rapid convergence of orthogonalized plane wave calculations for $s$ states; the slower convergence of $p$ states is also explained. The formalism developed here can also be regarded as a rigorous formulation of the "empirical potential" approach within the one-electron framework; the present results are compared with previous approaches. The method can be applied equally well to the calculation of wave functions in molecules.
921 citations
TL;DR: In this article, the one-dimensional Schr\"odinger equation with a periodic and symmetric potential is considered, under the assumption that the energy bands do not intersect, and it is shown that for each band there exists one and only one Wannier function which is real, symmetric or antisymmetric under an appropriate reflection.
Abstract: The one-dimensional Schr\"odinger equation with a periodic and symmetric potential is considered, under the assumption that the energy bands do not intersect. The Bloch waves, ${\ensuremath{\phi}}_{n,k}$, and energy bands, ${E}_{n,k}$, are studied as functions of the complex variable, $k$. In the complex plane, they are branches of multivalued analytic and periodic functions, ${\ensuremath{\phi}}_{k}$, and ${E}_{k}$, with branch points, ${k}^{\ensuremath{'}}$, off the real axis. A simple procedure is described for locating the branch points. Application is made to the power series and Fourier series developments of these functions. The analyticity and periodicity of ${\ensuremath{\phi}}_{n,k}$ has some consequences for the form of the Wannier functions. In particular, it is shown that for each band there exists one and only one Wannier function which is real, symmetric or antisymmetric under an appropriate reflection, and falling off exponentially with distance. The rate of falloff is determined by the distance of the branch points ${k}^{\ensuremath{'}}$ from the real axis.
873 citations
TL;DR: The theory of photoconduction through the reverse-biased $p\ensuremath{-n$ junction in semiconductors is developed without the customary assumption that carrier generation in the junction depletion layer is negligible as discussed by the authors.
Abstract: The theory of photoconduction through the reverse-biased $p\ensuremath{-}n$ junction in semiconductors is developed without the customary assumption that carrier generation in the junction depletion layer is negligible. Different from previous theories, the more general treatment leads to a voltage dependence of the photocurrent and its spectral distribution. When the incident light beam is modulated at frequencies comparable to the transit time through the depletion layer, a phase shift between the photon flux and photocurrent is noticed and transit-time rectification occurs.
822 citations
TL;DR: In this paper, a set of covariant conservation laws is constructed in the general theory of relativity, and their relationship to the generators of infinitesimal coordinate transformations is indicated.
Abstract: A set of covariant conservation laws is constructed in the general theory of relativity. Their relationship to the generators of infinitesimal coordinate transformations is indicated. In a given coordinate system certain of these quantities may be naturally identified as energy and momentum. We can continue to recognize these conserved quantities in all coordinate systems due to the covariant character of the expressions.
TL;DR: In this article, the Schwinger action principle is used to define the dynamical structure and definition of energy for the classical general theory of relativity, which is a special case of the problem in particle mechanics.
Abstract: The problem of the dynamical structure and definition of energy for the classical general theory of relativity is considered on a formal level. As in a previous paper, the technique used is the Schwinger action principle. Starting with the full Einstein Lagrangian in first order Palatini form, an action integral is derived in which the algebraic constraint variables have been eliminated. This action possesses a "Hamiltonian" density which, however, vanishes due to the differential constraints. If the differential constraints are then substituted into the action, the true, nonvanishing Hamiltonian of the theory emerges. From an analysis of the equations of motion and the constraint equations, the two pairs of dynamical variables which represent the two independent degrees of freedom of the gravitational field are explicitly exhibited. Four other variables remain in theory; these may be arbitrarily specified, any such specification representing a choice of coordinate frame. It is shown that it is possible to obtain truly canonical pairs of variables in terms of the dynamical and arbitrary variables. Thus a statement of the dynamics is meaningful only after a set of coordinate conditions have been chosen. In general, the true Hamiltonian will be time dependent even for an isolated gravitational field. There thus arises the notion of a preferred coordinate frame, i.e., that frame in which the Hamiltonian is conserved. In this special frame, on physical grounds, the Hamiltonian may be taken to define the energy of the field. In these respects the situation in general relativity is analogous to the parametric form of Hamilton's principle in particle mechanics.
TL;DR: In this article, the fastest stable drift was calculated (drift energy $0.9kT$) and the energy of a faster drift was found to be dissipated into instabilities within, typically, 30 plasma periods.
Abstract: The destruction of electron drifts by instabilities is analyzed. The fastest stable drift is calculated (drift energy $0.9kT$) and the energy of a faster drift is found to be dissipated into instabilities within, typically, 30 plasma periods. The growth of a local disturbance in this process is shown to take place without effective propagation. The "turbulent" flow pattern created, eventually, under nonlinear conditions is calculated numerically, demonstrating the tendency towards randomization of the initial drift energy. The effect stops "runaway" in about 100 plasma periods after which there is "heating" by "collective collisions" instead.
TL;DR: The ground-state wave function of the antimony, phosphorus, and arsenic impurities in silicon has been investigated by means of the electron nuclear double resonance (ENDOR) method as discussed by the authors.
Abstract: The ground-state wave function of the antimony, phosphorus, and arsenic impurities in silicon has been investigated by means of the electron nuclear double resonance (ENDOR) method By this method the hyperfine interactions of the donor electron with the ${\mathrm{Si}}^{29}$ nuclei situated at different lattice sites were obtained The isotropic part of the hyperfine interaction agreed with the theory of Kohn and Luttinger to better than 50% From a comparison of the experimental results with their theory a value for the conduction band minimum in silicon of $\frac{{k}_{0}}{{k}_{max}}=085\ifmmode\pm\else\textpm\fi{}003$ was obtained So far no satisfactory theory exists to account quantitatively for the observed anisotropic part of the hyperfine interactionThe observed line shape agreed with the shape predicted by summing up the individual hyperfine interactions which are the cause of the broadening The behavior of an inhomogeneously broadened line observed under adiabatic fast passage conditions is discussed in an appendix The electronic $g$-values were measured with respect to the free carriers in a degenerate $n$-type silicon sample The $g$-value of the free carriers was found to be 199875\ifmmode\pm\else\textpm\fi{}000010 The deviations of the donor $g$-values from the above value is several parts in ${10}^{4}$ and increases monotonically with increasing ionization energy of the donorsBesides the shallow donors Sb, P, and As, several other centers were investigated, but in considerably less detail They include the chemical impurities Bi, Li, Fe, centers associated with the surface of the sample and with the heat treatment of silicon The influence of substitutional germanium atoms on the resonance line in phosphorus-doped silicon has also been investigated
TL;DR: In this article, it was shown that a fully ionized gas cannot exhibit the steady-state behavior characterized by time independent drift velocities which has previously been accredited to it by other authors.
Abstract: Hydrodynamic equations are used to describe the flow of the electrons and ions of a fully ionized gas under the action of an electric field, E, of arbitrary magnitude. The dynamical friction force exerted by the electrons and ions upon each other through the agency of two-body Coulomb encounters is evaluated. In this connection the electrons and ions have been assigned Maxwellian velocity distributions which are displaced from each other by their relative drift velocity. This treatment yields a dynamical friction force which maximizes when the relative drift velocity is equal to the sum of the most probable random electron and ion speeds. For relative drift velocities in excess of this value the friction force decreases rapidly. As a consequence, it is found that a fully ionized gas cannot exhibit the steady-state behavior characterized by time independent drift velocities which has previously been accredited to it by other authors. Instead, it is shown that the electron and ion currents flowing parallel to the existing magnetic fields increase steadily in time (i.e., runaway) as long as a component of the electric field persists along the magnetic field. Drift velocities which greatly exceed the random speeds of the plasma particles can be created in this manner.The theory yields a critical electric field parameter, ${E}_{c}$, which is proportional to the plasma density and inversely proportional to its temperature. It is a measure of the electric field which is required if the velocities are to increase and exceed the most probable random speeds in the gas in one mean free collision time. For electric fields in excess of ${E}_{c}$ runaway proceeds even faster. In smaller fields runaway occurs when Joule heating has depressed ${E}_{c}$ sufficiently. Several interpretations of ${E}_{c}$ are given in terms of the collisional phenomenon involved.Within the framework of the hydrodynamic equations it is shown that the well-known ${(\mathrm{temperature})}^{\frac{3}{2}}$ electrical conductivity law can be recovered, provided $E\ensuremath{\ll}{E}_{c}$ and the electron temperature is held constant.Numerical solutions giving electron temperature and drift velocity as a function of time are presented for a range of the ratio $\frac{E}{{E}_{c}}$. The assumption of the displaced Maxwellian distribution is justified on the basis of a comparison between the rate of Joule heating and the rate of equipartition of random speeds. Moreover, it is found that the use of an anisotropic velocity distribution does not affect the runaway phenomenon in any important way.The possibility of runaway induced across magnetic fields by steep pressure gradients and its relation to diffusion across magnetic fields is examined and discussed in detail.
TL;DR: In this paper, the infrared absorption of a series of plane parallel plates having electrical resistivities ranging from 3 to 0.01 ohm-m has been examined and it is postulated that the electrical conductivity arises from the ionization of either one or two trapped electrons from each oxygen vacancy.
Abstract: Rutile Ti${\mathrm{O}}_{2}$ single crystal plates have been reduced in hydrogen at about 700\ifmmode^\circ\else\textdegree\fi{}C for several minutes to make them semiconducting. The concentration of oxygen vacancies was controlled by variations of time and temperature. The infrared absorption of a series of plane parallel plates having electrical resistivities ranging from 3 to 0.01 ohm-m has been examined. It is postulated that the electrical conductivity arises from the ionization of either one or two trapped electrons from each oxygen vacancy.In samples with electrical resistivity (\ensuremath{\perp} to the $c$ axis) greater than 0.04 ohm-m, the optical absorption at room temperature peaks at about 0.75 ev. For samples with electrical resistivity less than 0.03 ohm-m, the optical absorption shows a new maximum at 1.18 ev. The decrease of thermal activation energy with increasing oxygen vacancy concentration is expected to explain the "optical transition" from 0.75 to 1.18 ev. The ionization energies agree reasonably well with those calculated for a helium atom model of a doubly ionizable donor immersed in a dielectric medium [${K}_{e}={{n}_{o}}^{2}={(2.40)}^{2}$], namely 0.73 ev and 1.64 ev. A modification of this theory is also indicated which predicts the second ionization energy as 1.41 ev in better agreement with the experimental value of 1.18 ev.
TL;DR: In this paper, it was shown that for a repulsive interaction the energy of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.
Abstract: In this paper properties of a boson gas at zero temperature are investigated by means of field-theoretic methods. Difficulties arising from the depletion of the ground state are resolved in a simple way by the elimination of the zero-momentum state. The result of this procedure when applied to the calculation of the Green's functions of the system is identical to that of Beliaev. It is then shown generally that for a repulsive interaction the energy $E(\mathrm{k})$ of a phonon of momentum k, which is found as the pole of a one-particle Green's function, approaches zero for zero momentum, which means that the phonon spectrum does not exhibit an energy gap.The Green's function method is applied to the calculation of the properties of a low-density boson gas. The next order term beyond that calculated by Lee and Yang, and Beliaev for the ground-state energy is obtained and the general form of the series expansion is found to be $(\frac{{E}_{0}}{\ensuremath{\Omega}})=\frac{1}{2}{n}^{2}{f}_{0}[1+a{(n{{f}_{0}}^{3})}^{\frac{1}{2}}+b(n{{f}_{0}}^{3})\mathrm{ln}n{{f}_{0}}^{3}+c(n{{f}_{0}}^{3})+d{(n{{f}_{0}}^{3})}^{\frac{3}{2}}\mathrm{ln}(n{{f}_{0}}^{3})+\ensuremath{\cdots}],$ where $n$ is the density and ${f}_{0}$ is the scattering length for the assumed two-body interaction between the bosons. The coefficients $a$ and $b$ are independent of the shape of the interaction, and are the only terms thus far calculated. The coefficient $b$ is in agreement with the hard-sphere gas calculations of Wu and of Sawada.A discussion is given of the intermediate-density calculation of Brueckner and Sawada, and certain possible improvements in the method of summing a selected set of higher-order terms are proposed.
TL;DR: In this article, it was shown that the time required for mixing to start is inversely proportional to the square of the amplitude of the oscillations, which indicates that multistream flow will usually set in on the first oscillation.
Abstract: Investigations of nonlinear electron oscillations in a cold plasma where the thermal motions may be neglected indicate that except for the simplest one-dimensional situation such oscillations will destroy themselves through the development of multistream flow. It is found possible to give an exact analysis of oscillations with plane, cylindrical, and spherical symmetry. Plane oscillations in a uniform plasma are found to be stable below a critical amplitude. For larger amplitudes it is found that multistream flow or fine-scale mixing sets in on the first oscillation. Oscillations with spherical or cylindrical symmetry develop multistream flow almost always, independent of the amplitude. The time required for mixing to start is inversely proportional to the square of the amplitude. Plane oscillations in a nonuniform plasma are also found to exhibit this type of behavior. Some considerations are also given to more general oscillations and a calculation is presented which indicates that multistream flow will usually set in.
TL;DR: In this article, the effect of magnetic field on optical absorption in semiconductors is developed on the basis of the effective mass approximation, and a detailed treatment of the direct transition in germanium is given in which account is taken of the change in curvature of the bands away from k=0.
Abstract: The theory of the effect of a magnetic field on the optical absorption in semiconductors is developed on the basis of the effective-mass approximation. For simple parabolic conduction and valence bands and a direct transition which is allowed at k=0, absorption peaks occur at energies above the zero-field gap. Since the selection rule for the transition is $\ensuremath{\Delta}n=0$ where $n$ is the magnetic quantum number, the spacing between the peaks is the sum of the cyclotron frequencies for the two bands. For degenerate band edges, the spectrum is more complicated. A detailed treatment of the direct transition in germanium is given in which account is taken of the change in curvature of the bands away from k=0 and the results are in good agreement with the experimental measurements of Zwerdling, Lax, Roth, and Button. The k=0 conduction band mass is found to agree with predictions based on cyclotron resonance in the valence band. In addition, a gyromagnetic ratio for conduction electrons of -2.6 resulted from the calculations. The deviation from $g=2.0$ is due to spin-orbit interaction. In InSb the effect is much greater, the result being $g=\ensuremath{-}50$. These are consistent with experimental results. For bands in which the transition probability vanishes at k=0, absorption peaks will also occur corresponding to $\ensuremath{\Delta}n=\ifmmode\pm\else\textpm\fi{}1$ but absorption edges occur for $\ensuremath{\Delta}n=0$. In the case of indirect transitions, the absorption does not exhibit oscillations but consists of a series of "steps" as has been observed in Ge by Zwerdling, Lax, Roth, and Button.
TL;DR: In this article, the mass-polarization and relativistic corrections derived from the new wave functions were applied to the wave equation of two-electron atoms to obtain a value for the ionization energy of 198 312.
Abstract: The method described previously for the solution of the wave equation of two-electron atoms has been applied to the $1^{1}S$ and $2^{3}S$ states of helium, with the purpose of attaining an accuracy of 0.001 ${\mathrm{cm}}^{\ensuremath{-}1}$ in the nonrelativistic energy values. For the $1^{1}S$ state we have extended our previous calculations by solving determinants of orders 252, 444, 715, and 1078, the last yielding an energy value of -2.903724375 atomic units, with an estimated error of the order of 1 in the last figure. Applying the mass-polarization and relativistic corrections derived from the new wave functions, we obtain a value for the ionization energy of 198 312.0258 ${\mathrm{cm}}^{\ensuremath{-}1}$, as against the value of 198 312.011 ${\mathrm{cm}}^{\ensuremath{-}1}$ derived previously from the solution of a determinant of order 210. With a Lamb shift correction of -1.339, due to Kabir, Salpeter, and Sucher, this leads to a theoretical value for the ionization energy of 198 310.687 ${\mathrm{cm}}^{\ensuremath{-}1}$, compared with Herzberg's experimental value of 198 ${310.8}_{2}$\ifmmode\pm\else\textpm\fi{}0.15 ${\mathrm{cm}}^{\ensuremath{-}1}$.For the $2^{3}S$ state we have solved determinants of orders 125, 252, 444, and 715, the last giving an energy value of -2.17522937822 a.u., with an estimated error of the order of 1 in the last figure. This corresponds to a nonrelativistic ionization energy of 38 453.1292 ${\mathrm{cm}}^{\ensuremath{-}1}$. The mass-polarization and relativistic corrections bring it up to 38 454.8273 ${\mathrm{cm}}^{\ensuremath{-}1}$. Using the value of 74.9 ry obtained by Dalgarno and Kingston for the Lamb-shift excitation energy ${K}_{0}$, we get a Lamb-shift correction to the ionization energy of the $2^{3}S$ state of -0.16 ${\mathrm{cm}}^{\ensuremath{-}1}$. The resulting theoretical value of 38 454.66 ${\mathrm{cm}}^{\ensuremath{-}1}$ for the ionization potential is to be compared with the experimental value, which Herzberg estimates to be 38 454.73\ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. The electron density at the nucleus $D(0)$ comes out 33.18416, as against a value of 33.18388\ifmmode\pm\else\textpm\fi{}0.00023 which Novick and Commins deduced from the hyperfine splitting. We have also determined expectation values of several positive and negative powers of the three mutual distances, which enter in the expressions for the polarizability and for various sum rules.
TL;DR: In this article, a cosmological solution of the Einstein-Maxwell's field equations, corresponding to the case of a uniform (that is, covariant constant) electromagnetic field, is derived by means of simple geometrical arguments; the Riemannian manifold it corresponds to is the product of two ordinary surfaces of constant curvature.
Abstract: A cosmological solution of the Einstein-Maxwell's field equations, corresponding to the case of a uniform (that is, covariant constant) electromagnetic field, is derived by means of simple geometrical arguments; the Riemannian manifold it corresponds to is the product of two ordinary surfaces of constant curvature, whose type and radius depend on the values of the cosmological constant and the invariants of the electromagnetic field. The world-lines of charged test particles have also a very simple geometrical meaning.
IBM1
TL;DR: In this article, a model for the donor wave function is proposed which puts the donor electron principally into an antibonding orbital located on a nitrogen atom and on one of its nearest-neighbor carbon atoms.
Abstract: Electron-spin resonance of bound substitutional nitrogen donors in diamond is observed and discussed. The $g$ factor is isotropic at 2.0024\ifmmode\pm\else\textpm\fi{}0.0005. For a given donor, one of the C-N bond directions is a hyperfine axis with constants $A=40.8$ oersteds, $B=29.2$ oersteds. There are thus four types of donors, equally abundant. A model for the donor wave function is proposed which puts the donor electron principally into an antibonding orbital located on a nitrogen atom and on one of its nearest-neighbor carbon atoms. A C-N bond distortion results which can be regarded as a manifestation of the Jahn-Teller effect. A careful search reveals the presence of an additional weak spectrum due to donors on ${\mathrm{N}}^{14}$-${\mathrm{C}}^{13}$ pairs. (The isotope ${\mathrm{C}}^{13}$ which has a nuclear spin of \textonehalf{} has a natural abundance of 1.1%.) The hyperfine constants measured for a ${\mathrm{C}}^{13}$ atom of an N-C pair are ${A}^{\ensuremath{'}}=60.8$ oersteds, ${B}^{\ensuremath{'}}=25.3$ oersteds. The $s$ and $p$ contributions to all 4 measured hyperfine constants are separated to give the values ${O}_{\mathrm{N}}=(\frac{8\ensuremath{\pi}}{3}){{|\ensuremath{\psi}(0)|}^{2}}_{\mathrm{N}}=2.41 \mathrm{atomic}\mathrm{units},$ ${P}_{\mathrm{N}}={〈\frac{[{z}^{2}\ensuremath{-}\frac{1}{2}({x}^{2}+{y}^{2})]}{{r}^{5}}〉}_{\mathrm{N}}=0.28 \mathrm{atomic}\mathrm{unit},$ ${O}_{\mathrm{C}}=(\frac{8\ensuremath{\pi}}{3}){{|\ensuremath{\psi}(0)|}^{2}}_{\mathrm{C}}=0.78 \mathrm{atomic}\mathrm{unit},$ ${P}_{\mathrm{C}}={〈\frac{[{z}^{2}\ensuremath{-}\frac{1}{2}({x}^{2}+{y}^{2})]}{{r}^{5}}〉}_{\mathrm{C}}=0.25 \mathrm{atomic}\mathrm{unit}.$ These are compared with theoretical values obtained by assuming a simple antibonding wave function composed of nitrogen and carbon tetrahedral orbitals. An increase of several percent in the N-C separation along the hyperfine axis is strongly implied by the comparison.
TL;DR: In this article, the authors measured the nuclear spin-lattice relaxation times in normal and superconducting Al from 0.94 to 4.2 K, and as a function of static field in the normal state.
Abstract: Nuclear spin-lattice relaxation times have been measured in normal and superconducting Al from 0.94\ifmmode^\circ\else\textdegree\fi{}K to 4.2\ifmmode^\circ\else\textdegree\fi{}K, and as a function of static field in the normal state. In the normal state the relaxation rate is proportional to temperature as predicted by Redfield and others. The field dependence is somewhat greater than predicted. Relaxation in the superconductor was studied by a field cycling method which allowed the measurements to be made in the normal state but relaxation to occur in the superconductor. The results disagree with a simple two-fluid model, but are explained by the theory of Bardeen, Cooper, and Schrieffer. The contrast between the temperature dependence of nuclear relaxation and ultrasonic absorption confirms the central feature of the Bardeen-Cooper-Schrieffer theory that electrons of opposite spin and momentum are correlated.
TL;DR: The absorption spectrum of NiO has been measured in the 0.1 to 6 ev range and its reflectivity spectrum from 0.025 to 10 ev as mentioned in this paper, showing a series of lines from 1 to 3.5 ev arising from internal transitions of the Ni ion.
Abstract: The absorption spectrum of single crystal NiO has been measured in the 0.1 to 6 ev range and its reflectivity spectrum from 0.025 to 10 ev. The absorption spectrum shows a series of lines from 1 to 3.5 ev arising from internal transitions of the Ni ion. A continuous background absorption occurs in the range from 0.1 to 3.5 ev whose magnitude increases with impurity concentration. The absorption coefficient rises steeply above 3.5 ev and reaches a value of ${10}^{6}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ at and above 4 ev. An absorption line at 0.24 ev is found to be temperature sensitive in both intensity and frequency in the range above 300\ifmmode^\circ\else\textdegree\fi{}K. Its behavior suggests that it is connected with the antiferromagnetic ordering. The reststrahlen spectrum was observed with the following parameters: high- and low-frequency dielectric constants 5.4 and 12, respectively; energies of longitudinal and transverse optical mode vibrations are 0.076 and 0.044 ev, respectively.
TL;DR: In this paper, the infrared reflectivity measurements on several samples of different carrier concentrations were used to deduce the free-carrier contribution to the electric susceptibility and the electron effective mass.
Abstract: The infrared absorption between 0.85 and 25 microns has been measured as a function of carrier concentration for $n$-type single-crystal gallium arsenide. The absorption in the 1- to 5-micron region is compatible with a model in which there are minima \ensuremath{\sim}0.25 ev above the bottom of the conduction band. Infrared reflectivity measurements on several samples of different carrier concentrations were used to deduce the free-carrier contribution to the electric susceptibility and the electron effective mass. The results indicate a value for the mass of $(0.078\ifmmode\pm\else\textpm\fi{}0.004)m$ with an indication of an increase for the sample of highest carrier concentration. This value is substantially larger than previously reported values.
TL;DR: In this article, the residual ray bands have been observed for the ordinary and extraordinary rays for green alpha (hexagonal) SiC and the resonance frequencies are 2.380 and 2.356, respectively.
Abstract: Infrared transmission and reflectivity measurements from 1 to 25 \ensuremath{\mu} (microns) have been made on several samples of green alpha (hexagonal) SiC. The residual ray bands have been observed for the ordinary and extraordinary rays. The resonance frequencies are 2.380\ifmmode\times\else\texttimes\fi{}${10}^{13}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ (12.60 \ensuremath{\mu}) and 2.356\ifmmode\times\else\texttimes\fi{}${10}^{13}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ (12.73 \ensuremath{\mu}), respectively. From the reflectivity the high-frequency dielectric constant is found to be 6.7. A careful analysis shows that the residual ray bands can be fitted within experimental error by the classical dispersion theory within the correct choice of the dispersion parameters. From the parameters the value 10.0 is obtained for the low-frequency dielectric constant. The effective charge is $0.94e$. Complete description of the residual ray band for the extraordinary ray required, in addition to the main resonance, a weak resonance at 2.647\ifmmode\times\else\texttimes\fi{}${10}^{13}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ (11.33 \ensuremath{\mu}). A study on the effects of several different surface treatments shows the reflectivities reported here are an intrinsic property of the crystal. The room-temperature absorption coefficient as a function of wavelength in the range 1 to 10 \ensuremath{\mu} has been determined from transmission measurements. A number of weak lattice bands are observed between 5 and 10 \ensuremath{\mu}.
TL;DR: In this article, a theory of the thermal conductivity of superconductors is presented, based on the theory of super conductivity due to Bardeen, Cooper, and Schrieffer, which is treated as quasi-particles, allowing a Boltzmann equation to be set up.
Abstract: A theory of the thermal conductivity of superconductors is presented, based on the theory of superconductivity due to Bardeen, Cooper, and Schrieffer. The excited states of the system are treated as quasi-particles, allowing a Boltzmann equation to be set up. The electronic contribution to the thermal conductivity when the dominant scatterers are impurities has been calculated exactly. The result is very close to that of the Heisenberg-Koppe theory which is in fair agreement with experiment. The variational principle of Wilson has been used to find the electronic conductivity when the dominant scatterers are lattice waves. It is concluded that the theory fails to predict the sharp drop in the ratio $\frac{{\ensuremath{\kappa}}_{\mathrm{es}}}{{\ensuremath{\kappa}}_{\mathrm{en}}}$ as the temperature is lowered below ${T}_{c}$, a feature which is characteristic of the experimental results. The effect of the electrons on the lattice conductivity has also been calculated. The theoretical values may be too large.
TL;DR: Monte Carlo calculations of nuclear reactions in the low-energy (E < 50 Mev) region are described in this paper, where the calculations are based on the nuclear evaporation model of Weisskopf.
Abstract: Monte Carlo calculations of nuclear reactions in the low-energy (E < 50 Mev) region are described. The calculations are based on the nuclear evaporation model of Weisskopf. Continuum theory was used for the calculation of inverse reaction cross sections. In the calculation of the level densities of excited nuclei, pairing and shell energy corrections were used in terms of charactertstic level displacements. The accurate equation rather than the approximate Maxwell distribution was used for the selection of the kinetic energy of the evaporated particle. Experimentally determined Q-values for the various reactions were used. The calculations are compared with experimental measurements for about 60 excitation functions of nuclear reactions in the mass range Cr/sup 50/-Se/sup 74/. Cameron's values for pairing energies were used at the outset; but a new set of pairing and shell energy correction values, which leads to substuntially improved agreement with the experimental curves, Is presented. The procedure which was used to arrive at this set is described and several features of the set are discussed. The need for a further downward correction of the level density of symmetrical (A = 2Z) nuclei is indicated. Computed excitation functions are shown for all the reactions studied as well asmore » for several reactions for which experimental data are not yet available. Further experiments on reaction cross sections are suggested which would allow a unique deterntination of the pairing and shell energy corrections of level densities for any value of Z and N in the region under discussion. The existence of a unique set of these correction terms would provide strong evidence for the validity of evaporation theory for the reactions considered. (auth)« less
TL;DR: In this paper, the energy transfer between adjacent resonances in nuclear and electron spin systems is analyzed in terms of the overlap of line-shape functions, which is an enlargement on the original proposal of Kronig and Bouwkamp, and consists of taking partial account of off-diagonal elements in the spin-spin interaction.
Abstract: The energy transfer between adjacent resonances in nuclear and electron spin systems is analyzed in terms of the overlap of line-shape functions. The procedure is an enlargement on the original proposal of Kronig and Bouwkamp, and consists of taking partial account of off-diagonal elements in the spin-spin interaction, which are omitted in Van Vleck's truncated Hamiltonian. If the frequency of these off-diagonal elements is sufficiently small, they give rise to an additional kind of spin-spin relaxation, observed by Gorter and co-workers. They are also responsible for cross-saturation effects in paramagnetic salts of the type observed by Townes and co-workers. A crucial experiment is described which can be explained by spin-spin interactions, but not by the assumption of a hot-phonon region. Implications of the cross-relaxation for the operation of solid state masers are discussed. Special consideration is given to magnetically dilute substances and inhomogeneously broadened lines. Paradoxically, the latter will usually still undergo a homogeneous steady-state saturation.
TL;DR: In this article, it is shown that the strongest absorption bands in the infrared at 78, 83, 91, and 208 \ensuremath{\mu} are considered to be C-N molecular vibrations.
Abstract: Common type I diamonds (as classified by Robertson et al) have additional absorption in the infrared and ultraviolet It is shown that the strongest absorption band in the infrared at 78 \ensuremath{\mu} and the ultraviolet absorption at 3065 A are proportional to the nitrogen concentration of the crystal A corresponding increase in lattice constant is found to be as high as 001% for a nitrogen content of 02% Concentration, X-ray, and density data suggest that nitrogen occupies a substitutional position in the diamond lattice The infrared absorption bands at 78, 83, 91, and 208 \ensuremath{\mu} are considered to be C-N molecular vibrations Several optical, electrical, and thermal properties of diamond are discussed in view of our findings