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Showing papers in "Physical Review in 1955"


Journal ArticleDOI
TL;DR: In this article, the theory of double exchange was applied to perovskite-type manganites and detailed qualitative predictions about the magnetic lattice, the crystallographic lattice and the electrical resistivity were made.
Abstract: The theory of semicovalent exchange is reviewed and applied to the perovskite-type manganites $[\mathrm{La}, M(\mathrm{II})]\mathrm{Mn}{\mathrm{O}}_{3}$. With the hypothesis of covalent and semicovalent bonding between the oxygen and manganese ions plus the mechanism of double exchange, detailed qualitative predictions are made about the magnetic lattice, the crystallographic lattice, the electrical resistivity, and the Curie temperature as functions of the fraction of ${\mathrm{Mn}}^{4+}$ present. These predictions are found to be in accord with recent findings from neutron-diffraction and x-ray data as well as with the earlier experiments on this system by Jonker and van Santen.

3,148 citations


Journal ArticleDOI
I. Solomon1
TL;DR: Abragam and Pound's method for the calculation of the longitudinal relaxation time was extended to the transverse relaxation time in this paper, which gave rise to a steady and transient Overhauser effect.
Abstract: Abragam and Pound's method for the calculation of the longitudinal relaxation time ${T}_{1}$ has been extended to the transverse relaxation time ${T}_{2}$. Explicit calculations have been carried out for a pure dipole-dipole interaction, showing that for an interacting pair of like spins, or for nuclei in paramagnetic solution, ${T}_{1}$ is exactly equal to ${T}_{2}$ in the extreme narrow case. For a pair of interacting unlike spins, it is shown that the longitudinal components of the magnetic moments do not have simple exponential decays. This gives rise to a steady and transient Overhauser effect. The transverse components, however, have in all cases, simple exponential decay defined by a single relaxation time ${T}_{2}$. A set of modified Bloch's equations is found, giving the correct equation of motion of the macroscopic magnetic moments of such a system of pairs of unlike spins.The equality of ${T}_{1}$ and ${T}_{2}$ has been verified in paramagnetic solutions, and a nuclear Overhauser effect has been observed in anhydrous hydrofluoric acid. If one assumes that the extreme narrow case corresponds to the actual motion, the experimental results are not consistent with the picture of a pure dipole-dipole interaction between the hydrogen and fluorine nuclei of a molecule without taking into account the effect of the other molecules.

2,852 citations


Journal ArticleDOI
TL;DR: In this article, a character table for the group of the wave vector at certain points of symmetry in the Brillouin zone is given, and a possible reason for the complications which may make a simple effective mass concept invalid for some crystals of this type structure is presented.
Abstract: Character tables for the "group of the wave vector" at certain points of symmetry in the Brillouin zone are given. The additional degeneracies due to time reversal symmetry are indicated. The form of energy vs wave vector at these points of symmetry is derived. A possible reason for the complications which may make a simple effective mass concept invalid for some crystals of this type structure will be presented.

2,833 citations


Journal ArticleDOI
TL;DR: In this article, the authors define a set of generalized density matrices for the Hermitean density matrix of order $k, which is further antisymmetric in each set of these indices.
Abstract: In order to calculate the average value of a physical quantity containing also many-particle interactions in a system of $N$ antisymmetric particles, a set of generalized density matrices are defined. In order to permit the investigation of the same physical situation in two complementary spaces, the Hermitean density matrix of order $k$ has two sets of indices of each $k$ variables, and it is further antisymmetric in each set of these indices.Every normalizable antisymmetric wave function may be expanded in a series of determinants of order $N$ over all ordered configurations formed from a basic complete set of one-particle functions ${\ensuremath{\psi}}_{k}$, which gives a representation of the wave function and its density matrices also in the discrete $k$-space. The coefficients in an expansion of an eigenfunction to a particular operator may be determined by the variation principle, leading to the ordinary secular equation of the method of configurational interaction. It is shown that the first-order density matrix may be brought to diagonal form, which defines the "natural spin-orbitals" associated with the system. The situation is then partly characterized by the corresponding occupation numbers, which are shown to lie between 0 and 1 and to assume the value 1, only if the corresponding spin-orbital occurs in all configurations necessary for describing the situation. If the system has exactly $N$ spin-orbitals which are fully occupied, the total wave function may be reduced to a single Slater determinant. However, due to the mutual interaction between the particles, this limiting case is never physically realized, but the introduction of natural spin-orbitals leads then instead to a configurational expansion of most rapid convergence.In case the basic set is of finite order $M$, the best choice of this set is determined by a form of extended Hartree-Fock equations. It is shown that, in this case, the natural spin-orbitals approximately fulfill some equations previously proposed by Slater.

2,724 citations


Journal ArticleDOI
J. M. Luttinger1, Walter Kohn1
TL;DR: In this paper, a new method of developing an "effective-mass" equation for electrons moving in a perturbed periodic structure is discussed, particularly adapted to such problems as arise in connection with impurity states and cyclotron resonance in semiconductors such as Si and Ge.
Abstract: A new method of developing an "effective-mass" equation for electrons moving in a perturbed periodic structure is discussed. This method is particularly adapted to such problems as arise in connection with impurity states and cyclotron resonance in semiconductors such as Si and Ge. The resulting theory generalizes the usual effective-mass treatment to the case where a band minimum is not at the center of the Brillouin zone, and also to the case where the band is degenerate. The latter is particularly striking, the usual Wannier equation being replaced by a set of coupled differential equations.

2,260 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that while the states of large total spin have both the highest and lowest energies, their average energy is the same as those of low total spin.
Abstract: Zener has suggested a type of interaction between the spins of magnetic ions which he named "double exchange." This occurs indirectly by means of spin coupling to mobile electrons which travel from one ion to the next. We have calculated this interaction for a pair of ions with general spin $S$ and with general transfer integral, $b$, and internal exchange integral $J$.One result is that while the states of large total spin have both the highest and lowest energies, their average energy is the same as for the states of low total spin. This should be applicable in the high-temperature expansion of the susceptibility, and if it is, indicates that the high-temperature Curie-Weiss constant $\ensuremath{\theta}$ should be zero, and $\frac{1}{\ensuremath{\chi}}$ vs $T$ a curved line. This is surprising in view of the fact that the manganites, in which double exchange has been presumed to be the interaction mechanism, obey a fairly good Curie-Weiss law.The results can be approximated rather well by a simple semiclassical model in which the spins of the ion cores are treated classically. This model is capable of rather easy extension to the problem of the whole crystal, but the resulting mathematical problem is not easily solved except in special circumstances, e.g., periodic disturbances (spin waves).

2,086 citations


Journal ArticleDOI
TL;DR: In this article, the infrared spectra of 7 ferrites of the formula $M{\mathrm{Fe}}{2}{\mathrm {O}}_{4}, where $M$ designates a divalent metal, are presented and analyzed.
Abstract: The infrared spectra of 7 ferrites of the formula $M{\mathrm{Fe}}_{2}{\mathrm{O}}_{4}$, where $M$ designates a divalent metal, are presented and analyzed. Electronic absorption was observed in the visible and near-infrared regions. Two absorption bands arising from interatomic vibrations were measured and force constants calculated for the stretching of bonds between octahedral or tetrahedral metal ions and oxide ions. These force constants are in agreement with the elastic and thermodynamic properties of these compounds and are sensitive to distribution of metal ions between the alternate sites. The integrated vibrational band intensities were measured: they are compatible with predominantly ionic bonding for these structures.

1,957 citations


Journal ArticleDOI
TL;DR: In this paper, the magnetic properties of perovskite-type compounds have been investigated using x-ray diffraction measurements of lattice distortions and ferromagnetic saturation data.
Abstract: A study has been made of the magnetic properties of the series of perovskite-type compounds $[(1\ensuremath{-}x)\mathrm{La}, x\mathrm{Ca}]\mathrm{Mn}{\mathrm{O}}_{3}$ The investigations have been made primarily by neutron diffraction methods, but x-ray diffraction measurements of lattice distortions and ferromagnetic saturation data are also included This series of compounds exhibits ferromagnetic and antiferromagnetic properties which depend upon the relative trivalent and tetravalent manganese ion content The samples are purely ferromagnetic over a relatively narrow range of composition ($x\ensuremath{\sim}035$) and show simultaneous occurrence of ferromagnetic and antiferromagnetic phases in the ranges ($0lxl025$) and ($040lxl05$) Several types of antiferromagnetic structures at $x=0$ and $xg05$ have also been determined The growth and mixing of the various phases have been followed over the whole composition range, the ferromagnetic and antiferromagnetic moment contributions to the coherent reflections have been determined, and Curie and N\'eel temperatures have been measured The results have been organized into a scheme of structures and structure transitions which is in remarkable accord with Goodenough's predictions based on a theory of semicovalent exchange

1,656 citations


Journal ArticleDOI
Albert Rose1
TL;DR: The presence of traps not only reduces the magnitude of space-charge-limited currents, but also is likely to distort the shape of the currentvoltage curve from an ideal square law to a much higher power dependence on voltage.
Abstract: Currents, far in excess of ohmic currents, can be drawn through thin, relatively perfect insulating crystals. These currents are the direct analog of space-charge-limited currents in a vacuum diode. In actual crystals, the space-charge-limited currents are less than their theoretical value for an ideal crystal by the ratio of free to trapped carriers. Space-charge-limited currents become, therefore, a simple tool for measuring the imperfections in crystals even in the range of one part in ${10}^{15}$.The presence of traps not only reduces the magnitude of space-charge-limited currents, but also is likely to distort the shape of the current-voltage curve from an ideal square law to a much higher power dependence on voltage. The particular shape can be used to determine the energy distribution of traps.The presence of traps tends to uniformize the charge distribution between electrodes, to introduce a temperature dependence of the current, and to give rise to certain transient effects from which capture cross sections of traps may be computed.Space-charge-limited currents offer another mechanism for electrical breakdown in insulators.

1,526 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the derivative of the scattering phase shift with respect to energy, dn/dE, must exceed a certain limit if the interaction of scattered particle and scatterer vanishes beyond a certain distance.
Abstract: It is shown that the derivative of the scattering phase shift with respect to energy, dn/dE, must exceed a certain limit if the interaction of scattered particle and scatterer vanishes beyond a certain distance. This limitation of dn/dE is, fundamentally, a consequence of the principle of causality; it is derived, however, from a property of the derivative matrix R.

1,505 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present general relations obtaining in relativistic cosmology and show that a simple change over to anisotropy without the introduction of spin does not solve any of the outstanding difficulties of isotropic cosmological models.
Abstract: The paper presents some general relations obtaining in relativistic cosmology. It appears from these that a simple change over to anisotropy without the introduction of spin does not solve any of the outstanding difficulties of isotropic cosmological models.

Journal ArticleDOI
TL;DR: In this article, a method is developed for calculating the effects of a strong oscillating field on two states of a quantum-mechanical system which are connected by a matrix element of the field.
Abstract: A method is developed for calculating the effects of a strong oscillating field on two states of a quantum-mechanical system which are connected by a matrix element of the field. Explicit approximate solutions are obtained for a variety of special cases, and the results of numerical computations are given for others. The effect of an rf field on the $J=2\ensuremath{\rightarrow}1$ $l$-type doublet microwave absorption lines of OCS has been studied in particular both experimentally and theoretically. Each line was observed to split into two components when the frequency of the rf field was near 12.78 Mc or 38.28 Mc, which are the frequencies separating the $J=1$ and $J=2$ pairs of levels, respectively. By measuring the rf frequency, ${\ensuremath{ u}}_{0}$, at which the microwave lines are split into two equally intense components, one may determine the separation between the energy levels. The measured value of ${\ensuremath{ u}}_{0}$ depends upon the intensity of the rf field and the form of this dependence has been calculated and found to be in good agreement with the experimental results.

Journal ArticleDOI
TL;DR: In this paper, a variational principle is developed for the lowest energy of a system described by a path integral, which is applied to the problem of the interaction of an electron with a polarizable lattice, as idealized by Frohlich.
Abstract: A variational principle is developed for the lowest energy of a system described by a path integral. It is applied to the problem of the interaction of an electron with a polarizable lattice, as idealized by Frohlich. The motion of the electron, after the phonons of the lattice field are eliminated, is described as a path integral. The variational method applied to this gives an energy for all values of the coupling constant. It is at least as accurate as previously known results. The effective mass of the electron is also calculated, but the accuracy here is difficult to judge.

Journal ArticleDOI
TL;DR: In this paper, it was shown that if the total wave function is approximated by a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant.
Abstract: In treating a system of $N$ antisymmetric particles, it is shown that, if the total Hamiltonian ${\mathcal{H}}_{\mathrm{op}}$ is degenerate, the eigenstates of the operator used for classifying the corresponding degenerate states may be selected by means of a "projection operator" $\mathcal{O}$. If the total wave function is approximated by such a projection of a single determinant, the description of the system may be reduced to the ordinary Hartree-Fock scheme treating this determinant, if the original Hamiltonian is replaced by a complete Hamiltonian ${\ensuremath{\Omega}}_{\mathrm{op}}={\mathcal{O}}^{\ifmmode\dagger\else\textdagger\fi{}}{\mathcal{H}}_{\mathrm{op}}\mathcal{O}$ containing also many-particle interactions. This approach corresponds to a "fixed" configurational interaction, but the scheme has preserved the physical simplicity and visuality of the Hartree-Fock approximation. The idea of "doubly filled" orbitals is abandoned, and the orbitals associated with different spins will automatically try to arrange themselves in such a way that particles having antiparallel spins will tend to avoid each other due to their mutual repulsion.

Journal ArticleDOI
W. C. Dash1, R. Newman1
TL;DR: In this article, the intrinsic absorption spectra of high-purity single-crystal germanium and silicon have been measured at 77\ifmmode^\circ\else\text degree\fi{}K and 300\ifmode^''circ\decrease\textdegree\fi {}K, respectively.
Abstract: The intrinsic absorption spectra of high-purity single-crystal germanium and silicon have been measured at 77\ifmmode^\circ\else\textdegree\fi{}K and 300\ifmmode^\circ\else\textdegree\fi{}K. The spectral regions studied encompassed a range of absorption coefficient from 0.1 ${\mathrm{cm}}^{\ensuremath{-}1}$ to ${10}^{5}$ ${\mathrm{cm}}^{\ensuremath{-}1}$ for each material. The germanium data may be interpreted as indicating a threshold for direct transitions at 0.81 ev at 300\ifmmode^\circ\else\textdegree\fi{}K and at 0.88 ev at 77\ifmmode^\circ\else\textdegree\fi{}K. The threshold for indirect transitions was placed at 0.62 ev and 0.72 ev for 300\ifmmode^\circ\else\textdegree\fi{}K and 77\ifmmode^\circ\else\textdegree\fi{}K, respectively. For silicon the data were not as readily interpreted However, there is an indication that the threshold for direct transitions should be placed at about 2.5 ev and the threshold for indirect transitions at 1.06 ev and 1.16 ev at 300\ifmmode^\circ\else\textdegree\fi{}K and 77\ifmmode^\circ\else\textdegree\fi{}K, respectively.

Journal ArticleDOI
TL;DR: In this paper, a variational treatment based on wave functions was applied to the problem of the $N$-particle system with strong interactions, and the expectation values, which cannot be factored into singleparticle integrals, were evaluated by a cluster development in powers of the particle density.
Abstract: A trial solution constructed from two-particle functions is applied to the problem of the $N$-particle system with strong interactions. In a variational treatment based on this wave function the expectation values, which cannot be factored into single-particle integrals, are evaluated by a cluster development in powers of the particle density. The procedure is illustrated by a calculation of the pair distribution function and zero-point energy of the hard sphere gas, for Bose and Fermi statistics.

Journal ArticleDOI
TL;DR: In this paper, the results of cyclotron resonance experiments on charge carriers in silicon and germanium single crystals near 4\ifmmode^\circ\else\textdegree\fi{}K were given.
Abstract: An experimental and theoretical discussion is given of the results of cyclotron resonance experiments on charge carriers in silicon and germanium single crystals near 4\ifmmode^\circ\else\textdegree\fi{}K. A description is given of the light-modulation technique which gives good signal-to-noise ratios. Experiments with circularly polarized microwave radiation are described. A complete study of anisotropy effects is reported. The electron energy surfaces in germanium near the band edge are prolate spheroids oriented along $〈111〉$ axes with longitudinal mass parameter ${m}_{l}=(1.58\ifmmode\pm\else\textpm\fi{}0.04)m$ and transverse mass parameter ${m}_{t}=(0.082\ifmmode\pm\else\textpm\fi{}0.001)m$. The electron energy surfaces in silicon are prolate spheroids oriented along $〈100〉$ axes with ${m}_{l}=(0.97\ifmmode\pm\else\textpm\fi{}0.02)m$; ${m}_{t}=(0.19\ifmmode\pm\else\textpm\fi{}0.01)m$. The energy surfaces for holes in both germanium and silicon have the form $E(k)=A{k}^{2}\ifmmode\pm\else\textpm\fi{}{[{B}^{2}{k}^{4}+{C}^{2}({{k}_{x}}^{2}{{k}_{y}}^{2}+{{k}_{y}}^{2}{{k}_{z}}^{2}+{{k}_{z}}^{2}{{k}_{x}}^{2})]}^{\frac{1}{2}}.$ We find, for germanium, $A=\ensuremath{-}(13.0\ifmmode\pm\else\textpm\fi{}0.2)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$, $|B|=(8.9\ifmmode\pm\else\textpm\fi{}0.1)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$, $|C|=(10.3\ifmmode\pm\else\textpm\fi{}0.2)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$; and for silicon, $A=\ensuremath{-}(4.1\ifmmode\pm\else\textpm\fi{}0.2)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$, $|B|=(1.6\ifmmode\pm\else\textpm\fi{}0.2)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$, $|C|=(3.3\ifmmode\pm\else\textpm\fi{}0.5)(\frac{{\ensuremath{\hbar}}^{2}}{2m})$. A discussion of possible systematic errors in these constants is given in the paper.

Journal ArticleDOI
TL;DR: In this paper, the theory of paramagnetic resonance absorption by conduction electrons in a metal is worked out, taking into account the diffusion of the electrons in and out of the thin skin into which the radio-frequency field penetrates.
Abstract: The theory of paramagnetic resonance absorption by the conduction electrons in a metal is worked out, taking into account the diffusion of the electrons in and out of the thin skin into which the radio-frequency field penetrates. Calculations are carried through in detail for the case of a flat metal plate. It is found that the diffusion has no marked effect on the width of the resonance absorption line, but has a radical effect on the shape of the line. In particular, for a piece of metal thick compared to the skin depth and with a relaxation time long compared to the diffusion time, the line is antisymmetrical about its center and has an unusual characteristic shape.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the theory of Bloembergen, Purcell, and Pound is applicable only to solids at rf magnetic field intensities well above the saturation level and which is in reasonable agreement with the experimental observations.
Abstract: Nuclear spin-lattice relaxation times of ${\mathrm{Al}}^{27}$ in pure Al and ${\mathrm{Cu}}^{63}$ in annealed pure Cu have been measured with a nuclear induction spectrometer, by the method of saturation. The experimental values of ${T}_{1}$ are 4.1\ifmmode\pm\else\textpm\fi{}0.8 milliseconds for ${\mathrm{Al}}^{27}$ and 3.0\ifmmode\pm\else\textpm\fi{}0.6 milliseconds for ${\mathrm{Cu}}^{63}$, in reasonable agreement with theory.The dispersion mode of the nuclear resonance was also observed, and it was found that ${\ensuremath{\chi}}^{\ensuremath{'}}$ (the real part of the rf susceptibility) does not saturate at the same level as the absorption, ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$, but remains roughly constant out to a radio-frequency field intensity of about 2 gauss. Both ${\ensuremath{\chi}}^{\ensuremath{'}}$ and ${\ensuremath{\chi}}^{\ensuremath{'}\ensuremath{'}}$ become narrower and nearly Lorentzian in shape above saturation. When the dc magnetic field modulation is increased from 14 to 41 cps the phase of the dispersion signal lags behind the modulation, presumably because the modulation period is then comparable to ${T}_{1}$. Large dispersion signals above saturation have also been observed for the ${\mathrm{Na}}^{23}$ resonance in NaCl.This behavior of the dispersion mode is in conflict with the predictions of Bloembergen, Purcell, and Pound and of the Bloch equations. The validity of these theories is re-examined, and it is concluded that although they are applicable to nuclear resonance in liquids and gases, and to solids at small rf intensities, they contain incorrect assumptions as applied to solids at high rf power levels. The theory of Bloembergen, Purcell, and Pound is based on an assumption equivalent to that of a spin temperature. It is shown that the spin state cannot be strictly described by a spin temperature because the phases of the spin quantum states are not incoherent, as required by the temperature concept. The transverse decay of the nuclear magnetization predicted by the Bloch equations is shown to be partially forbidden by energy and entropy considerations if a large rf field at the resonance frequency is continuously applied to the solid.A theory is developed which is applicable only to solids at rf magnetic field intensities well above the saturation level and which is in reasonable agreement with the experimental observations. The Hamiltonian is transformed to a coordinate system rotating at the frequency of the rf field. The resulting time-dependent parts of the spin-spin interaction are nonsecular perturbations on the time-independent part, and can therefore be ignored. Statistical mechanics is applied to the remaining stationary spin Hamiltonian; specifically it is assumed that the spin system is in its most probable macrostate (a canonical distribution of quantum states) with respect to the transformed spin Hamiltonian. This assumption is justified because the transformed spin Hamiltonian is effectively time independent and the spin-lattice interaction is small, and it is analogous to assumptions basic to classical acoustics and fluid mechanics. The spin-lattice interaction merely determines the expectation value of the transformed spin Hamiltonian, which can be readily calculated under the assumption that the expectation value of the spin angular momentum of each spin is relaxed independently to its thermal equilibrium value by the lattice in time ${T}_{1}$. Both fast and slow modulation of the dc magnetic field can be treated."Rotary saturation" is observed by applying an audio-frequency magnetic field to the sample in the dc field direction while observing the dispersion derivative at resonance with a large rf field ${H}_{1}$. When the audio-frequency approaches $\ensuremath{\gamma}{H}_{1}$ the dispersion signal decreases and goes through a minimum. The effect is easily treated theoretically in solids, liquids and gases by using a rotating coordinate system, and is a rotary analogue of ordinary saturation. It is a convenient method for calibrating rf magnetic fields and appears potentially capable of providing useful information on the solid state. Experimental data on rotary saturation are presented and discussed.

Journal ArticleDOI
TL;DR: In this paper, the electron spin resonance absorption in metals was investigated at 300 and 9000 Mc/sec, in the temperature range from 4\ifmmode^\circ\else\textdegree\fi{}K to 296\ifmode^''circ\circ''else''textdegree''fi{ }K, with too poor a signal-to-noise ratio to make accurate measurements.
Abstract: The electron spin resonance absorption in metals was investigated at 300 and 9000 Mc/sec, in the temperature range from 4\ifmmode^\circ\else\textdegree\fi{}K to 296\ifmmode^\circ\else\textdegree\fi{}K. The metals investigated were: Li, Na, K, Be, Mg, Al, Pd, W. Resonances were observed in Li, Na, Be, for which accurate $g$ values and relaxation times were obtained and compared with existing theories. K gave a signal only at 4\ifmmode^\circ\else\textdegree\fi{}K and 300 Mc/sec, with too poor a signal-to-noise ratio to make accurate measurements. No resonances were observed in Mg, Al, Pd, W. The line shapes predicted by Dyson's theory for the resonance absorption in metals were checked experimentally for different diffusion times and relaxation times. A satisfactory agreement between the theory and experiment is obtained for both the normal and the anomalous skin effect region.

Journal ArticleDOI
TL;DR: In this paper, the authors used the recently measured effective masses for $n$-type Si, and obtained approximate solutions of the resulting effective mass Schroedinger equation for the case of the ground state where the error is attributed to failure of the effective mass theory near the donor nucleus.
Abstract: By using the recently measured effective masses for $n$-type Si, ${m}_{1}=0.98$ $m$ and ${m}_{2}=0.19$ m, approximate solutions of the resulting effective mass Schroedinger equation are obtained. The accuracy of the solutions was tested in the limiting cases where $\frac{{m}_{2}}{{m}_{1}}=1 \mathrm{and} 0$ respectively. The nature of the resulting states and their degeneracy is discussed in some detail, taking into account the fact that the conduction band of Si has six equivalent minima. Experimentally measured ionization energies show that the effective mass theory is seriously in error in the case of the ground state. This error is attributed to failure of the effective mass theory near the donor nucleus, and allowance for this failure is made in the case of higher states. This leads finally to a theoretical spectrum for the electrons bound by P, As, or Sb donors.

Journal ArticleDOI
TL;DR: In this article, a type of device is described which can be used as a microwave amplifier, spectrometer, or oscillator, and experimental results are given for the expected sensitivity, stability and purity of the oscillation.
Abstract: A type of device is described which can be used as a microwave amplifier, spectrometer, or oscillator. Experimental results are given. When operated as a spectrometer, the device has good sensitivity, and, by eliminating the usual Doppler broadening, a resolution of 7 kc/sec has been achieved. Operated as an oscillator, the device produced a frequency stable to at least 4 parts in ${10}^{12}$ in times of the order of a second, and stable over periods of an hour or more to at least a part in ${10}^{10}$. The device is examined theoretically, and results are given for the expected sensitivity of the spectrometer, the stability and purity of the oscillation, and the noise figure of the amplifier. Under certain conditions a noise figure approaching the theoretical limit of unity, along with reasonably high gain, should be attainable.

Journal ArticleDOI
TL;DR: In this article, a new magnetic exchange mechanism was proposed to explain the magnetic properties of spinels, which is consistent with the covalent model of the spinels' magnetic properties.
Abstract: Elastic- and electrostatic-energy considerations are insufficient for an explanation of cation ordering between tetrahedral and octahedral sites in spinels. Tetrahedral, octahedral, and square covalent bonds are also important when cations are to be accommodated in these sites. The square bonds in octahedral sites can cause the tetragonal distortion observed in ${\mathrm{Mn}}_{3}$${\mathrm{O}}_{4}$ $\ensuremath{\gamma}$-${\mathrm{Mn}}_{2}$${\mathrm{O}}_{3}$, Zn${\mathrm{Mn}}_{2}$${\mathrm{O}}_{4}$, Cu${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$, Cu${\mathrm{Cr}}_{2}$${\mathrm{O}}_{4}$, Ca${\mathrm{In}}_{2}$${\mathrm{O}}_{4}$, Cd${\mathrm{In}}_{2}$${\mathrm{O}}_{4}$, and metallic indium. A new magnetic exchange mechanism, "semicovalent exchange," which is consistent with the covalent model, is used to explain the magnetic properties of spinels.

Journal ArticleDOI
TL;DR: In this paper, the coercive force predicted by theory for single-domain particles with shape anisotropy frequently far exceeds the observed value, and several approximate models are suggested which may exist in certain experimental situations.
Abstract: The coercive force predicted by theory for single-domain particles with shape anisotropy frequently far exceeds the observed value. From an examination of the results of Paine, Mendelsohn, and Luborsky on elongated iron particles, several approximate models are suggested which may exist in certain experimental situations. Detailed calculations are presented for a "chain-of-spheres" model remagnetizing by several mechanisms. Comparison with experiment on certain single-domain particles of known elongation favors the chain-of-spheres model as a suitable description of their magnetic behavior. A successful calculation is made of the coercive force of material prepared by several earlier workers. A comparison of the new models with the older ones indicates a direction for experimental advance.

Journal ArticleDOI
TL;DR: In this paper, it is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles.
Abstract: An approximation method developed previously to deal with many particles in strong interaction is examined in further detail. It is shown that the series giving the interaction energy is a development in a sequence of linked or irreducible cluster terms each of which gives a contribution to the energy proportional to the total number of particles. Consequently the convergence of the expansion is independent of the total number of particles. The origin of this simple feature is illustrated by showing that a similar situation exists in the expansion of standard perturbation theory. The numerical convergence of the expansion is quantitatively discussed for the nuclear problem where it is shown that the correction arising from the first cluster term involving three particles is less than the leading term by a factor of about ${10}^{\ensuremath{-}4}$. The smallness of the correction is largely a result of the action of the exclusion principle.

Journal ArticleDOI
TL;DR: The lattice constants of a graphite single crystal, natural graphite powder, and Artificial Graphite powder have been measured at 297, 4.2, and 4.0005 A. as discussed by the authors.
Abstract: The lattice constants of a graphite single crystal, natural graphite powder, and artificial graphite powder have been measured at 297\ifmmode^\circ\else\textdegree\fi{}K, 78\ifmmode^\circ\else\textdegree\fi{}K, and 4.2\ifmmode^\circ\else\textdegree\fi{}K. The $a$-spacing does not change in this temperature range within the error of \ifmmode\pm\else\textpm\fi{}0.0005 A. The material with the greatest $c$-spacing showed between 297\ifmmode^\circ\else\textdegree\fi{} and 78\ifmmode^\circ\else\textdegree\fi{} the greatest change of $c$-spacing and the smallest change below 78\ifmmode^\circ\else\textdegree\fi{}. The $c$-spacing of the artificial powder did not return to its original value at 297\ifmmode^\circ\else\textdegree\fi{} after a thermal cycle, demonstrating irreversible deformation.

Journal ArticleDOI
TL;DR: In this paper, a photographic technique for the nearly simultaneous determination of shock and free surface velocities is presented, and measurements for aluminum, copper, and zinc are given, respectively.
Abstract: Shock wave pressure magnitudes from about 150 to 500 kilobars have been attained for metals by using high explosives. A photographic technique for the nearly simultaneous determination of shock and free surface velocities is presented, and measurements for aluminum, copper, and zinc are given.


Journal ArticleDOI
TL;DR: Magnetic crystal anisotropy and magnetostriction have been measured in various single crystals of ferrites having compositions represented approximately by $M{\mathrm{Fe}}_{2}{\mathrm{\O}}_{4}$, where $M$ stands for Mn, Fe, Co, Ni, and Zn in various proportions.
Abstract: Magnetic crystal anisotropy and magnetostriction have been measured in various single crystals of ferrites having compositions represented approximately by $M{\mathrm{Fe}}_{2}{\mathrm{O}}_{4}$, where $M$ stands for Mn, Fe, Co, Ni, and Zn in various proportions. Special attention is given to heat-treatment in a magnetic field.The magnetic anisotropy of cobalt ferrite at room temperature is as high as 4\ifmmode\times\else\texttimes\fi{}${10}^{6}$ ergs/${\mathrm{cm}}^{2}$. Magnetostriction is as high as 800\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$. Magnetic anneal is effective at temperatures as low as 150\ifmmode^\circ\else\textdegree\fi{}C, and causes the hysteresis loop to become square. In polycrystalline material the response to magnetic anneal is a maximum at compositions intermediate between Co${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$ and ${\mathrm{Fe}}_{3}$${\mathrm{O}}_{4}$.The constants for the various specimens are tabulated. Values of the anisotropy constants of Mn${\mathrm{Fe}}_{2}$${\mathrm{O}}_{4}$ at 20\ifmmode^\circ\else\textdegree\fi{}C and -196\ifmmode^\circ\else\textdegree\fi{}C are the same as those determined from ferromagnetic resonance experiments. At -196\ifmmode^\circ\else\textdegree\fi{}C the constant for ${\mathrm{Ni}}_{0.75}$${\mathrm{Fe}}_{2.25}$${\mathrm{O}}_{4}$ differs markedly from that determined by ferromagnetic resonance; this is to be expected from the relaxation phenomena observed by Galt, Yager, and Merritt.

Journal ArticleDOI
TL;DR: In this paper, the velocities of 10-Mc/sec acoustic waves in copper single crystals in the range from 4.2\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmode^''circ\circ''else\ text degree\fi {}K were determined.
Abstract: The ultrasonic pulse technique has been used in conjunction with a specially devised cryogenic technique to measure the velocities of 10-Mc/sec acoustic waves in copper single crystals in the range from 4.2\ifmmode^\circ\else\textdegree\fi{}K to 300\ifmmode^\circ\else\textdegree\fi{}K. The values and the temperature variations of the elastic constants have been determined. The room temperature elastic constants were found to agree well with those of other experimental works. Fuchs' theoretical ${c}_{44}$ at 0\ifmmode^\circ\else\textdegree\fi{}K is 10 percent larger than our observed value but his theoretical ${c}_{11}$, ${c}_{12}$, $K$ and (${c}_{11}\ensuremath{-}{c}_{12}$) agree well with the observations. The isotropy, $\frac{({c}_{11}\ensuremath{-}{c}_{12})}{2{c}_{44}}$, was observed to remain practically constant from 4.2\ifmmode^\circ\else\textdegree\fi{}K to 180\ifmmode^\circ\else\textdegree\fi{}K, then to diminish gradually at higher temperatures. Some general features of the temperature variations of elastic constants are discussed.