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Showing papers in "Physical Review in 1950"


Journal ArticleDOI
TL;DR: In this article, the authors used the method of effective mass, extended to apply to gradual shifts in energy bands resulting from deformations of the crystal lattice, to estimate the interaction between electrons of thermal energy and the acoustical modes of vibration.
Abstract: The method of effective mass, extended to apply to gradual shifts in energy bands resulting from deformations of the crystal lattice, is used to estimate the interaction between electrons of thermal energy and the acoustical modes of vibration. The mobilities of electrons and holes are thus related to the shifts of the conduction and valence-bond (filled) bands, respectively, associated with dilations of longitudinal waves. The theory is checked by comparison of the sum of the shifts of the conduction and valence-bond bands, as derived from the mobilities, with the shift of the energy gap with dilation. The latter is obtained independently for silicon, germanium and tellurium from one or more of the following: (1) the change in intrinsic conductivity with pressure, (2) the change in resistance of an $n\ensuremath{-}p$ junction with pressure, and (3) the variation of intrinsic concentration with temperature and the thermal expansion coefficient. Higher mobilities of electrons and holes in germanium as compared with silicon are correlated with a smaller shift of energy gap with dilation.

2,530 citations


Journal ArticleDOI
W. T. Read1, William Shockley1
TL;DR: In this article, the authors investigated the energies and motions of grain boundaries between two crystallites using the dislocation model of grain boundary and provided a quantitative expression for energy per unit area for small angles.
Abstract: The energies and motions of grain boundaries between two crystallites are investigated theoretically using the dislocation model of grain boundaries. Quantitative predictions made for simple boundaries for cases in which the plane of the boundary contains the axis of relative rotation of the grains appear to agree with available experimental data. The quantitative expression for energy per unit area for small angles is approximately $[\frac{\mathrm{Ga}}{4\ensuremath{\pi}(1\ensuremath{-}\ensuremath{\sigma})}]\ensuremath{\theta}[A\ensuremath{-}\mathrm{ln}\ensuremath{\theta}]$ where $G$ is the rigidity modulus, $a$ the lattice constant, $\ensuremath{\sigma}$ Poisson's ratio, $\ensuremath{\theta}$ the relative rotation and $A$ approximately 0.23. Grain boundaries of the form considered may permit intercrystalline slip and may act as stress raisers for the generation of dislocations.

1,767 citations


Journal ArticleDOI
TL;DR: In this paper, a canonical transformation on the Dirac Hamiltonian for a free particle is obtained in which positive and negative energy states are separately represented by two-component wave functions.
Abstract: By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.

1,715 citations


Journal ArticleDOI
Peter J. Anderson1
TL;DR: In this article, the general formalism of Kramers indicating the existence of superexchange interaction has been reduced, under simplifying assumptions, to the point where actual formulas for the interaction can be written down directly in terms of spin operators, with certain exchange and transition integrals as parameters.
Abstract: In this paper the general formalism of Kramers indicating the existence of superexchange interaction has been reduced, under simplifying assumptions, to the point where actual formulas for the interaction can be written down directly in terms of spin operators, with certain exchange and transition integrals as parameters. Two results of physical interest are the following: (a) superexchange must be expected to show the directional properties (as far as directional relations of interacting magnetic ions are concerned) of the orbitals in the outer shell of the non-magnetic connecting ions; and (b) the sign of the effective exchange integral depends upon the sign of the internal exchange coupling of an added electron on the magnetic ion.

1,304 citations


Journal ArticleDOI
TL;DR: In this paper, Dyson's systematic approach to the reduction of the Heisenberg $S$-Matrix into a sum of "graph" terms can be simplified, and a notation is introduced and an algebraic theorem is proved.
Abstract: Dyson's systematic approach to the reduction of the Heisenberg $S$-Matrix into a sum of "graph" terms can be simplified. A notation is introduced and an algebraic theorem is proved, which allow one to handle the reduction problem quite easily and in the same manner for any type of field.

1,048 citations


Journal ArticleDOI
TL;DR: In this article, an expression for the magnetic field at a nucleus resulting from the application of an external magnetic field to a polyatomic molecule which has no resultant electron orbital or spin angular momenta in the absence of the external field is developed.
Abstract: An expression is developed for the magnetic field at a nucleus resulting from the application of an external magnetic field to a polyatomic molecule which has no resultant electron orbital or spin angular momenta in the absence of the external field. The field at the nucleus is not the same as the externally applied field because of the field arising from the motion of the electrons in the molecule. The expression for the electron contribution to the magnetic field is shown to consist of two parts. The first is a simple term that is similar to the diamagnetic correction developed by Lamb for atoms. The second is a complicated one arising from second-order paramagnetism and is analogous to the term dependent on the high frequency matrix elements in the theory of molecular diamagnetism. Under certain circumstances the second-order paramagnetic term can become quite large. Since both of these terms are altered when the same nucleus is in different molecules, they at least partially and perhaps completely explain the chemical effect that has been reported by various observers in measurements of nuclear moments. For linear molecules, the second-order paramagnetic term is shown to be directly related to the experimentally measurable spin-rotational magnetic interaction constant of the molecule. This relation is particularly valuable in the important case of molecular hydrogen where it is shown that the correction for second-order paramagnetism is -0.56\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$. When this is added to the Lamb-type term as calculated by Anderson, the total magnetic shielding constant for molecular ${\mathrm{H}}_{2}$ becomes 2.68\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$.

1,021 citations


Journal ArticleDOI
TL;DR: In this article, the long-and short-range order parameters for alloys of arbitrary composition in such binary systems as that of the copper-gold alloys were derived from simple thermodynamic reasoning, involving a certain degree of approximation.
Abstract: Short-range order parameters ${\ensuremath{\alpha}}_{i}$ are defined to express the interaction of a given atom in an alloy with the atoms of the ith shell of atoms surrounding it. From simple thermodynamic reasoning, involving a certain degree of approximation, equations relating the ${\ensuremath{\alpha}}_{i}$ with energy terms and the temperature are derived. Equations for the long-range order parameter, $S$, are obtained by considering the limiting case of $i$ very large. The values of the long- and short-range order parameters obtained by solving these equations are in good agreement with experimental values recently found by x-ray diffraction methods. The theory is extended to the calculation of configurational energy and specific heat, and to the determination of order parameters for alloys of arbitrary composition in such binary systems as that of the copper-gold alloys. In these cases qualitative agreement with experimental observations is obtained.

902 citations


Journal ArticleDOI
Gregory H. Wannier1
TL;DR: In this article, the statistical mechanics of a two-dimensionalally infinite set of Ising spins are worked out for the case in which they form either a triangular or a honeycomb arrangement.
Abstract: In this paper the statistical mechanics of a two-dimensionally infinite set of Ising spins is worked out for the case in which they form either a triangular or a honeycomb arrangement. Results for the honeycomb and the ferromagnetic triangular net differ little from the published ones for the square net (Curie point with logarithmically infinite specific heat). The triangular net with antiferromagnetic interaction is a sample case of antiferromagnetism in a non-fitting lattice. The binding energy comes out to be only one-third of what it is in the ferromagnetic case. The entropy at absolute zero is finite; it equals $S(0)=R\frac{2}{\ensuremath{\pi}}\ensuremath{\int}{0}^{\frac{\ensuremath{\pi}}{3}}\mathrm{ln}(2cos\ensuremath{\omega})d\ensuremath{\omega}=0.3383R$The system is disordered at all temperatures and possesses no Curie point.

890 citations


Journal ArticleDOI
TL;DR: In this paper, a new molecular beam resonance method using separated oscillating fields at the incident and emergent ends of the homogeneous field region is theoretically investigated and an expression is obtained for the quantum mechanical transition probability of a system between two states when the system is subjected to such separated oscillators.
Abstract: A new molecular beam resonance method using separated oscillating fields at the incident and emergent ends of the homogeneous field region is theoretically investigated in this paper. An expression is obtained for the quantum mechanical transition probability of a system between two states when the system is subjected to such separated oscillating fields. This is numerically averaged over the molecular velocity distribution and provides the theoretical shape of the resonance curves. It is found that resonances with such a technique have a theoretical half-width only 0.6 as great as those by conventional molecular beam resonance methods. In addition to producing sharper resonance minima, the new method has its resonances much less broadened by inhomogeneities of the fixed field, it makes possible resonance experiments in regions into which an oscillating field cannot be introduced, and it is more convenient and effective with short wave-length radiation.

884 citations



Journal ArticleDOI
TL;DR: In this article, a systematic treatment of the application of variational principles to the quantum theory of scattering is presented, starting from the time-dependent theory, a pair of Variational principles are provided for the approximate calculation of the unitary (collision) operator that describes the connection between the initial and final states of the system.
Abstract: A systematic treatment is presented of the application of variational principles to the quantum theory of scattering.Starting from the time-dependent theory, a pair of variational principles is provided for the approximate calculation of the unitary (collision) operator that describes the connection between the initial and final states of the system. An equivalent formulation of the theory is obtained by expressing the collision operator in terms of an Hermitian (reaction) operator; variational principles for the reaction operator follow. The time-independent theory, including variational principles for the operators now used to describe transitions, emerges from the time-dependent theory by restricting the discusson to stationary states. Specialization to the case of scattering by a central force field establishes the connection with the conventional phase shift analysis and results in a variational principle for the phase shift.As an illustration, the results of Fermi and Breit on the scattering of slow neutrons by bound protons are deduced by variational methods.

Journal ArticleDOI
TL;DR: In this article, a relation between the amplitude for a given process in an arbitrary unquantized potential and in a quantum electrodynamical field is established, which permits a simple general statement of the laws of quantum mechanics.
Abstract: The validity of the rules given in previous papers for the solution of problems in quantum electrodynamics is established. Starting with Fermi's formulation of the field as a set of harmonic oscillators, the effect of the oscillators is integrated out in the Lagrangian form of quantum mechanics. There results an expression for the effect of all virtual photons valid to all orders in e2/hc. It is shown that evaluation of this expression as a power series in e2/hc gives just the terms expected by the aforementioned rules. In addition, a relation is established between the amplitude for a given process in an arbitrary unquantized potential and in a quantum electrodynamical field. This relation permits a simple general statement of the laws of quantum electrodynamics. A description, in Lagrangian quantum-mechanical form, of particles satisfying the Klein-Gordon equation is given in an Appendix. It involves the use of an extra parameter analogous to proper time to describe the trajectory of the particle in four dimensions. A second Appendix discusses, in the special case of photons, the problem of finding what real processes are implied by the formula for virtual processes. Problems of the divergences of electrodynamics are not discussed.


Journal ArticleDOI
TL;DR: In this article, selection rules for the disintegration of a meson into two photons were derived from the general principle of invariance under rotation and inversion, and the polarization state of the photons was completely fixed by the selection rules.
Abstract: Selection rules governing the disintegration of a particle into two photons are derived from the general principle of invariance under rotation and inversion. The polarization state of the photons is completely fixed by the selection rules for initial particles with spin less than 2. These results which are independent of any specific assumption about the interactions may possibly offer a method of deciding the symmetry nature of mesons which decay into two photons.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the simple theory of lattice scattering alone cannot explain the temperature dependence of the resistivity of germanium semiconductors and that another probable source of resistance is scattering by ionized impurity centers.
Abstract: Experiments by Lark-Horovitz and collaborators on the Hall effect and resistivity of germanium semiconductors have shown that the simple theory of lattice scattering alone cannot explain the temperature dependence of the resistivity. Another probable source of resistance is scattering by ionized impurity centers. This resistance can be calculated by using the Rutherford scattering formula. Evaluation of the collision terms in the Lorentz-Boltzmann equation of state is made by assuming that scattering of an electron by one ion is approximately independent of all other ions. This results in a resistivity given by (in ohm cm): $\ensuremath{\rho}=2.11\ifmmode\times\else\texttimes\fi{}{10}^{2}{\ensuremath{\kappa}}^{\ensuremath{-}2}{T}^{\ensuremath{-}\frac{3}{2}}\mathrm{ln}{1+36{\ensuremath{\kappa}}^{2}{d}^{2}{(\mathrm{kT})}^{2}{e}^{\ensuremath{-}4}}$ where $d$ is half the average distance between impurity ions and $\ensuremath{\kappa}$ the dielectric constant of the semiconductor.

Journal ArticleDOI
TL;DR: In this paper, it was investigated whether suitable generalizations of the field equations of current field theories to equations of higher order may be of help in eliminating the divergent features of the present theory.
Abstract: It is investigated whether suitable generalizations of the field equations of current field theories to equations of higher order may be of help in eliminating the divergent features of the present theory. It turns out to be difficult, if feasible, to reconcile in this way the requirements of convergence, of positive definiteness of the free field energy, and of a strictly causal behavior of the state vector of a physical system. Progress may perhaps be made by relinquishing the condition of unlimited localizability of any space time event.




Journal ArticleDOI
TL;DR: In this article, a double probe system is used to measure the plasma potential of a pair of probes joined by a variable potential source in varying or decaying plasmas, where the double probes are joined by the same potential source.
Abstract: Measurements with a Langmuir probe in varying or decaying plasmas are defeated by the fact that the plasma potential in these cases follows that of the most positive electrode it can contact. The difficulties can be obviated by the use of a pair of probes joined by a variable potential source. The double probe system "floats" with respect to the discharge system.From the measurements of probe current vs. differential probe voltage, electron temperature and plasma densities can be determined. The method is also applicable to "going" discharges where it has the advantage over the single probe of exerting a negligible influence on the discharge.




Journal ArticleDOI
TL;DR: In this paper, the integro-differential diffusion equation of the multiple scattering problem in an infinite, homogeneous, medium is studied without the usual small-angle approximation, and an expansion in spherical harmonics is carried out which is rapidly convergent in the case of large-angle scattering, whose coefficients can be exactly determined, and leads to expressions for the various moments of the spatial and angular distributions.
Abstract: The integro-differential diffusion equation of the multiple scattering problem in an infinite, homogeneous, medium, is studied without the usual small-angle approximation. An expansion in spherical harmonics is carried out which is rapidly convergent in the case of large-angle scattering, whose coefficients can be exactly determined, and which leads to expressions for the various moments of the spatial and angular distributions. The latter alone has previously been obtained by Goudsmit and Saunderson, and, in the small-angle approximation, by Snyder and Scott. Our results are shown to include these.

Journal ArticleDOI
TL;DR: In this paper, the Schroedinger equation for the perturbed core wave functions for Li, Al, and Cl was used to calculate the correction to nuclear quadrupole moments on account of the quadrupoles induced in the electron shells.
Abstract: The correction to nuclear quadrupole moments on account of the quadrupole moment induced in the electron shells has been obtained by solving the Schroedinger equation for the perturbed core wave functions for Li, Al, and Cl. The correction factor by which the average $〈\frac{1}{{r}^{3}}〉$ over the valence electron function in the equation for the quadrupole coupling should be multiplied to take account of the induced effect is 1.11, 0.83, and 0.68, respectively. The previously described Thomas-Fermi calculation of this effect has been carried out for 13 additional elements.

Journal ArticleDOI
TL;DR: In this article, it was shown that the postulate that the quantum mechanical operators obey the classical equations of motion uniquely determines the commutation relations of the Hamiltonian with the operator corresponding to any physical quantity gives the operator which corresponds to the time derivative of that quantity.
Abstract: The commutator of the Hamiltonian with the operator corresponding to any physical quantity gives the operator which corresponds to the time derivative of that quantity. One can ask, hence, whether the postulate, that the quantum mechanical operators obey the classical equations of motion, uniquely determines the commutation relations. The answer is found to depend on the form of the Hamiltonian and is in the negative for a free particle and for the harmonic oscillator.


Journal ArticleDOI
TL;DR: In this paper, the velocity of propagation of a single domain boundary in a crystal of silicon iron with a simple domain structure is given by a relation of the form $v=G(H\ensuremath{-}{H}_{0})$, where $G$ is a constant and H is the starting field.
Abstract: Experimental results are given on the velocity of propagation of a single domain boundary in a crystal of silicon iron with a simple domain structure. In weak applied magnetic fields (\ensuremath{\sim}0.003 oersted) the velocity is given by a relation of the form $v=G(H\ensuremath{-}{H}_{0})$, where $G$ is a constant \ensuremath{\sim}4 cm/sec./oersted in this crystal, and ${H}_{0}\ensuremath{\cong}0.003$ oersted is the starting field. Calculation of the eddy current losses accompanying the motion of a plane boundary gives a theoretical expression for $G$ in good agreement with experimental values; the predicted linear dependence on the resistivity was approximately verified by measurements at 78\ifmmode^\circ\else\textdegree\fi{}, 194\ifmmode^\circ\else\textdegree\fi{}, and 293\ifmmode^\circ\else\textdegree\fi{}K. In stronger fields (g5 oersteds) there is evidence that the wall closes on itself, and the experimental velocity of collapse of the wall as deduced from flux changes agrees with the theoretical result based on a model of eddy current losses accompanying a collapsing cylindrical boundary. The results have a bearing on the well-known eddy current anomaly, namely, the fact that the total loss in a ferromagnetic material undergoing a.c. magnetization is often two or three times larger than the eddy-current and hysteresis losses calculated in the usual way assuming a spatially uniform and isotropic classical permeability.

Journal ArticleDOI
R. V. Pound1
TL;DR: In this article, the effects of the presence of nuclear electric quadrupole moments on the nuclear magnetic resonance absorption in solids were studied and a method was described to determine the frequency of a large NE splitting, by observation of the dependence of the frequency for the magnetic splitting of the levels of an odd half-integral spin.
Abstract: A study is made of the effects of the presence of nuclear electric quadrupole moments on the nuclear magnetic resonance absorption in solids. Necessary theoretical background is followed by descriptions of experiments in which the spectra obtained with an r-f spectrograph, described elsewhere, are found to show marked splitting resulting from the quadrupole interaction with the electric field gradient. Both powder patterns and spectra of single crystals are reported, for ${\mathrm{Li}}^{7}$, ${\mathrm{Na}}^{23}$, and ${\mathrm{Al}}^{27}$, with splittings varying from about 20 kc to 718 kc, depending on the crystal and the nucleus. Unambiguous indication of the spin values is obtained. The possibility of the detection of a resonance resulting from the electric quadrupole splittings in the absence of a static magnetic field is discussed and an unsuccessful attempt to find such a resonance in ICl is described and the failure tentatively explained. Available signal-to-noise ratios are calculated for such resonances. A method is described of determining the frequency of a large electric quadrupole splitting, by observation of the dependence of the frequency of the magnetic splitting of the levels $m=\ifmmode\pm\else\textpm\fi{}\frac{1}{2}$, for an odd half-integral spin, on the angle between the direction of a magnetic field and the symmetry axis of the crystalline field. The electric quadrupole moment is shown probably to account for relaxation times of nuclei of spin greater than \textonehalf{} even in cubic crystals and, as evidence, resonances of ${\mathrm{Na}}^{23}$, ${\mathrm{Li}}^{7}$, ${\mathrm{I}}^{127}$, and ${\mathrm{Br}}^{81}$ are reported. Assuming partial saturation and electric quadrupolar relaxation, calculations are made to account for a departure of the intensity ratios of the lines, in the ${\mathrm{Na}}^{23}$ spectrum in NaN${\mathrm{O}}^{3}$, from the ratios of the transition probabilities for magnetic dipole radiation. The possibility of determining nuclear electric quadrupole moments from splittings or relaxation times in ionic crystals is discussed.

Journal ArticleDOI
TL;DR: The theory of the elastic scattering of electrons on protons at very high energies is discussed in detail in this article, where a formula for the cross section of the cross-section is given for the virtual photon and meson fields.
Abstract: The theory of the elastic scattering of electrons on protons at very high energies is discussed in detail. A formula is given for the cross section. This formula contains certain parameters which depend on the action of the virtual photon and meson fields. In particular, curves have been calculated on the assumption of scalar and pseudoscalar meson theory. While these perturbation theory calculations are not very trustworthy, and the results depend on the choice of coupling constants, it is felt that qualitative features can be checked with experiment. It is concluded that at low relativistic energies ($El50$ Mev) the experiment provides a valuable check on quantum electrodynamics. At higher energies it should yield data on the nature of the meson cloud of the proton.