Plane Wave Model Research Articles

The Band Structure and Fermi Surface of Cadmium

Calculation of the band structure and the Fermi surface of cadmium are carried out semiempirically on the basis of a k -independent pseudopotential scheme. Three Fourier coefficients of the pseudopotential are used as adjustable parameters. They are determined so as to make the calculated Fermi surface compatible with experimental results of magnetoacoustic and de Haas-van Alphen effects. The Fermi surface calculated with these parameters is slightly deformed from that calculated on a single orthogonalized plane wave model. The `monster' has collapsed as the result of a pinching off of the horizontal arms, and possesses diagonal arms as the result that the `monster' does not contact along the whole Brillouin zone edge, K H . The values of the effective masses and cyclotron masses of electrons on the `pillow' are also calculated, and they agree fairly well with the cyclotron masses determined by Galt.

The magneto-resistance of aluminium

Low -temperature measurements of transverse magneto-resistance were made on 24 single crystals of pure aluminium in magnetic fields up to 20 kG. In contrast with the theory, neither complete saturation of magneto-resistance nor quadratic variation with field were observed in any orientation. The anisotropy of themagneto-resistance was small, but showed a pattern consistent with symmetry requirements. These results appear inexplicable on a basis of Harrison’s four orthogonal plane wave model of the aluminium Fermi surface, but can be explained qualitatively in terms of a slightly different Fermi surface which touches the corners of the second Brillouin zone. This model is not inconsistent with other experimental results.

Wave-Number-Dependent Dielectric Function of Semiconductors

Expressions for the wave-number-dependent dielectric function are derived for various models of a semiconductor. The calculation is carried out for the diagonal part of the dielectric function at zero frequency. It is found that calculations based on plane wave models (such as the free electron model) give poor results for small values of the wave number due to neglect of both Bragg reflections and Umklapp processes. We use instead an isotropic version of the nearly free electron model in which dielectric function depends on only one parameter ${E}_{g}$ representing an average energy gap that can be determined from optical data. It is noted that for small wave numbers Umklapp processes give the major contribution to the dielectricfunction, where-as for large wave numbers normal processes dominate. The dielectric function is evaluated numerically for a value of ${E}_{g}$ appropriate to Si.

Low-Field de Haas-van Alphen Effect in Zinc

A study of the low-field de Haas-van Alphen effect in zinc, employing automatic recording of the variation of the torque as a function of the reciprocal of the applied magnetic field, has revealed six sets of periods. These periods are assigned in a consistent manner to portions of the free-electron (single orthogonalized planewave) model of the Fermi surface. Cyclotron masses associated with the three larger sets of observed periods are found to agree well with the theoretical estimates. Evidence is obtained for the presence of a spin-orbit energy gap across the hexagonal face of the first Brillouin zone near the corners. Thus, the validity of the single Brillouin-zone construction is demonstrated for zinc.

A simple direct reaction model of proton inelastic scattering

The model of McCarthy and Pursey for angular distributions in direct nuclear reactions is extended to the case of proton inelastic scattering, a reaction whose angular distributions at medium energies are unintelligible from the point of view of the plane-wave Born approximation. The model is closely analogous to the plane-wave model in that it does not require a partial wave expansion of the optical model wave functions. The distortion is included by simple but realistic generalizations of the optical model wave functions. The present work includes a parameter representing the focus in the wave functions which is seen to account for the shapes of the angular distributions for the (p, p′) reaction on carbon 12 at medium energies. It is shown how intuition gained from the present simple theory can lead to useful knowledge about nuclei.

Cyclotron Resonance in Aluminum

Preliminary results of cyclotron resonance experiments at 35 kMc/sec in aluminum having a relatively long relaxation time at 4.2\ifmmode^\circ\else\textdegree\fi{}K ($\ensuremath{\omega}\ensuremath{\tau}\ensuremath{\sim}20$) are presented. Discrepancies and errors in earlier cyclotron resonance experiments are resolved. A tentative orbit identification scheme is proposed which indicates (a) that the second-zone Fermi surface is consistent with the single orthogonalized plane wave (OPW) model, and (b) that the third-zone Fermi surface is connected and that the connecting region is thicker than in the single OPW model. The effective masses obtained are compared with the values predicted by the single OPW model and with other experimental values, where pertinent. Departures of the effective mass value of orbits in the second zone from values given by the single OPW model are noted and may be attributable to electron-electron or electron-phonon effects.

Self-Consistent Nuclear Model

A Hamiltonian proposed by Duerr is applied to carry out self-consistent calculations for atomic nuclei. Average exchange forces, calculated from a plane-wave model, are included in the Hartree potential. In order to obtain agreement with the empirical data, two coupling constants and one constant representing the range of the interaction have to be adjusted. The range turns out to be 3.23\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}14}$ cm. Because this range is very short the calculations may not be justified and must be considered as a formal procedure. Having adjusted the three parameters, reasonable neutron and proton binding energies and nucleon densities are obtained for the nuclei ${\mathrm{O}}^{16}$, ${\mathrm{Ca}}^{40}$, ${\mathrm{Ce}}^{140}$, and ${\mathrm{Pb}}^{208}$.

Leaking modes in the crustal waveguide: 1. The oceanicPLwave

The problem of the seismic signal associated with the earliest P wave is treated by application of normal mode theory, in which the signal is regarded as a quasi-surface wave, coupled both to the motion of the earth's layered surface and to body waves propagating in the underlying media. Predictions made for the particular model assumed are relevant to explosion and earth-quake sources. The oscillations following the initial P motion are explained. The transient solution obtained by Rosenbaum for leaking mode propagation in an acoustic waveguide has been generalized to describe propagation in an elastic halfspace overlain by a liquid layer. The early-arriving PL modes known from earthquake studies have been computed for several theoretical models to test the effect of the elastic constants on their dispersion and attenuation. Physical reasoning, based on harmonic plane wave models, appears inadequate for predicting many features of the exact dispersion and attenuation. The analogy between PL waves and normal modes in the case treated by Pekeris is exploited, and it is also believed that PL waves are related to an attenuated pseudo-surface wave of a free solid halfspace. Late-arriving quasi-standing waves are treated briefly and their relevance to certain seismic phenomena is mentioned.

Transfer of Transient Electromagnetic Surface Waves into a Lossy Medium

The propagation of a transient surface wave in the form of an impulse into a lossy medium is examined via a simple two-dimensional plane wave model. The solution found exhibits characteristics of both wave propagation and diffusion. This impulse response is used as a basis for calculating the response to an arbitrary excitation at a point on the surface. For observation times longer than the time duration of the surface disturbance, the form of this solution is equivalent to a moment expansion. The limits imposed on the signalling band width by the losses in the medium are discussed.