Plane Wave Model Research Articles

On: “Theory of microwave dielectric constant logging using the electromagnetic wave propagation method,” by R. Freedman and J. Vogiatzis (GEOPHYSICS, May, 1979, p. 969–986).

In the interesting paper by Freedman and Vogiatzis, the Schlumberger EPT dielectric well logging method is discussed. The authors pay particular attention to the analysis of the formation traveltime and calculate, for several realistic sources, the corrections to the values of the plane wave model used by Schlumberger.

Explicit formalism for acousto-optic multiple plane-wave scattering

In a previous article, a plane-wave model for strong acousto-optic interaction was developed. In this earlier model, any particular order is contributed to by a multiple-scattering process, involving all other orders. The model involves “direct” scattering paths and “feedback” paths and, in general, provides an implicit solution. The present article describes a variant of this multiple-scattering formalism, in which feedback paths and direct paths are combined so that all orders are explicitly derivable from the incident-light field. We show the usefulness and validity of this approach by rigorously deriving the well-known expressions for Raman-Nath and Bragg diffraction.

Projectile break-up in continuous particle spectra from nuclear reactions induced by 156 MeV6Li

The break-up of 156 MeV6Li ions colliding with12C,60Ni,90Zr,120Sn and208Pb has been experimentally studied. It was found that the break-up cross section decreases rapidly with increasing emission angles of the fragments and increases with increasing target mass. It comprises a fairly large fraction of the total reaction cross section. The observed features suggest that the6Li fragmentation is mainly a peripheral process, and is dominated by the cluster properties of the incident projectile. A simple plane-wave break-up model gives a rather good description of the shapes of the break-up bumps. The angular distributions of the fragments emitted in the6Li+208Pb reaction reveal the transition from peripheral break-up to incomplete fusion processes by which beam-velocity fragments are captured by the target nucleus. In addition to the (α+d) dissociation the (3He+t) break-up has been observed, but the triton spectra appear to be rather complex indicating further components due to more complicated reaction paths.

A plane-wave reflection loss model including sediment rigidity

A method is presented for computation of the complex plane-wave reflection coefficient of an acoustic wave impinging upon a horizontally stratified ocean bottom consisting of a single inhomogeneous (fluid or solid) sediment layer overlying a semi-infinite homogeneous (fluid or solid) substrate. Within the sediment layer the density, the compressional wave speed and attenuation, and the shear wave speed and attenuation are permitted to vary arbitrarily and independently. A matrix formulation is used in which the depth-separated wave equations are replaced by the ’’propagator’’ matrix. The elements of the propagator are calculated by numerical integration of the Helmholtz equations with depth-dependent wavenumbers.

Alpha-particle breakup at incident energies of 20 and 40 MeV/nucleon

The breakup of alpha particles at incident energies of 20 and 40 MeV/nucleon on $^{27}\mathrm{Al}$, $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, and $^{209}\mathrm{Bi}$ has been studied. It was found that the breakup cross section decreases rapidly with increasing angles and increases with increasing target mass and incident energy. The total breakup yield, summed over all charged fragments, is \ensuremath{\sim}15-35% of the alpha-particle total reaction cross section, and has an approximate ${A}^{\frac{1}{3}}$ dependence. The ratios of breakup yields among different fragments are approximately $p:d:t:^{3}\mathrm{He}\ensuremath{\approx}13:3:1:2$, and are roughly independent of the incident energy and the target nucleus. These features suggest that the alpha-particle fragmentation is a peripheral process and is dominated by the properties of the incident projectile. A simple plane-wave alpha-particle breakup model gives a rather good description to the experimental data. In addition to the breakup deuteron peak at half of the beam energy, a second peak at quarter of the beam energy (or the same energy as the breakup proton peak) is observed. This peak might be due to a two-step breakup-pickup process.NUCLEAR REACTIONS $^{27}\mathrm{Al}$, $^{58}\mathrm{Ni}$, $^{90}\mathrm{Zr}$, $^{209}\mathrm{Bi}$($\ensuremath{\alpha}$,$\mathrm{xp}$), ($\ensuremath{\alpha}$,$\mathrm{xd}$), ($\ensuremath{\alpha}$,$\mathrm{xt}$), ($\ensuremath{\alpha}$, $x^{3}\mathrm{He}$,) ${E}_{\ensuremath{\alpha}}=80, 160$ MeV; $\ensuremath{\theta}=6\ifmmode^\circ\else\textdegree\fi{}\ensuremath{-}30\ifmmode^\circ\else\textdegree\fi{}$, measured $\frac{{d}^{2}\ensuremath{\sigma}}{d\ensuremath{\Omega}\mathrm{dE}}$, deduced alpha-particle breakup yield, Comparisons with alpha-particle breakup model.

Ultrasonic standing wave modes in axially stressed rods

Ultrasonic propagation in axially stressed metal resonators is examined from the standpoint of a propagating plane wave model. The theoretical results are compared with experimental data obtained with a new resonant instrument called an Ultrasonic Bolt Tensioning Monitor, capable of resolving changes in stress of less than 104 Pa (1 lb./in.2). The applied load produces a coupling to the acoustic resonant modes via a stress induced strain as well as a stress induced change in sound velocity. The coupling is shown to produce a linear relationship between load and resonant frequency characterized by a continuous wave stress acoustic constant R. Some geometrical considerations may impact R and are considered. For example, non-parallelism is shown to have produced a severe degradation of signal for conventional transducers, and the accompanying stress gradients alter the value of R. Also, the percentage of length actually under load changes R and is shown to produce an effect predicted by the plane wave model. In addition to the linear data obtained with the ultrasonic tension instrument, spectral characteristics of the samples are presented.

Theory of microwave dielectric constant logging using the electromagnetic wave propagation method

The composite dielectric constants of earth formations at microwave frequencies are strongly dependent on formation water saturations and relatively independent of water salinities. Therefore, microwave frequency dielectric constant logging offers an attractive new electromagnetic (EM) method of formation evaluation. The EM wave propagation method of dielectric constant logging attempts to deduce the dielectric properties of earth formations from phase shift and attenuation measurements of EM field, which have been propagated in the formation. A device which utilizes this method of well logging has been proposed by Calvert (1974) and Rau (1976) in two recent U.S. patents. We discuss the basic physics underlying the operation of a device of this type and describe the plane wave procedure discussed by these authors for relating the phase shift and attenuation measurements made by such a device to the formation dielectric properties. This procedure is suspect, since it is based on an unrealistic plane wave model which fails to treat the radiation field correctly and ignores the presence of a layer of mud cake which separates the antenna pad from the formation. To determine the errors likely to be inherent in using this procedure in practice, we consider several simple theoretical models of an EM wave propagation tool. Computer experiments performed on these theoretical models indicate that the apparent formation traveltimes obtained by using this procedure are semiquantitatively accurate with relative errors less than five percent in most cases. For our theoretical models, correction plots or departure curves are demonstrated which enable one to deduce the true formation traveltimes, given the apparent values and a knowledge of the dielectric properties and thickness of the mud cake. The problems which remain if this new method of logging is to attain its full potential (e.g., the accurate determination of formation fluid saturations) are discussed.

(α,He3) breakup reaction on nuclei and the neutron momentum distribution in theαparticle

The ($\ensuremath{\alpha}$,$^{3}\mathrm{He}$) breakup reaction on $^{62}\mathrm{Hi}$ and $^{209}\mathrm{Bi}$ targets at incident $\ensuremath{\alpha}$-particle energies of 172.5 and 140 MeV respectively has been studied by a plane-wave spectator model. The results have been compared with the Serber model calculations as well as with those of the distorted-wave Born approximation. It is found that plane wave methods are not appropriate to describe this reaction correctly. Apart from the distortions in the $\ensuremath{\alpha}$ and $^{3}\mathrm{He}$ channels, the proper consideration of the neutron target interaction is also very important. The validity of some of the assumptions of the Serber model has also been investigated. It is found that these assumptions are not valid for the reactions under investigation and consequently some of the conclusions drawn previously from the Serber model calculations for this reaction are not justified.NUCLEAR REACTIONS $^{209}\mathrm{Bi}$ ($\ensuremath{\alpha}$,$^{3}\mathrm{He}$), $^{62}\mathrm{Ni}$($\ensuremath{\alpha}$,$^{3}\mathrm{He}$) breakup reactions, calculated ($\frac{{d}^{2}\ensuremath{\sigma}}{d\ensuremath{\Omega}\mathrm{dE}}$); ${E}_{\ensuremath{\alpha}}=140\ensuremath{-}172.5$ MeV plane wave spectator model analysis; comparison with DWBA.

The use of the coherence function for the automatic recognition of chorus and hiss observed by GEOS

A technique for identifying the nature of electromagnetic emissions is developed. It consists of testing plane wave models against random wave models, using the coherence function. It has the advantage that it can be performed entirely automatically. It is applied here to make an automatic selection of chorus and hiss observed by GEOS.

Electron refraction and light polarization effects in angle-resolved photoemission spectra of Cu surfaces using 45 eV radiation

Angle-resolved photoemission spectra of the low Miller-index faces of copper were taken with polarized 45 eV light for both surface normal and off-normal photoemission orientations. They reveal that low-energy photoelectrons emitted off-normal to the surface are strongly refracted toward the surface at the solid-vacuum interface; measured refraction angles are significantly (5̃0%) larger than predicted from a simple plane wave model. Furthermore, the spectra are found to be very sensitive to the orientation of the polarization of the exciting radiation with respect to the crystalline axes and the sample surface. Spectral dependence on light polarization is discussed in terms of direct transition selection rules for primary Mahan cones, influence of directionality on matrix elements, and direct transition contributions to the spectra from surface photoemission.

Interpretation of angle-resolved x-ray photoemission spectra

This paper presents several new considerations concerning angle-resolved x-ray photoemission. Phononassisted processes are explicitly considered; it is found that even when such processes dominate direct transitions, agreement between theoretical predictions and experiment is obtained only if the final states are treated as Bloch-like and highly mixed. The use of augmented-plane-wave final states and a proper statistical treatment of the x-ray photon field is outlined for copper and found to yield nearly quantitative agreement with experimental results. The relationship between this and older plane-wave model calculations is discussed in detail, as is the physical significance of the new results.

Shock formation and cavitation as limitations on acoustic source level

The nonlinear phenomena of shock formation and cavitation impose different limitations on acoustic projector source level. It is found, however, that both effects can be conveniently parametrized in terms of the so‐called scaled source level L0* of Mellen and Moffett [J. Acoust. Soc. Am. 61, 325 (1977)]. The well‐known cavitation data of Esche [Akust. Beih. (Acustica) 4, AB208 (1952)] are reformulated as functions of L0*. and directivity index DI for a piston radiator. The shock wave formation distance is calculated in terms of these parameters and the Rayleigh distance R0. A linear model based on weak shock theory is used to combine existing plane and spherical wave models. The comparison between maximum achieveable L0* due to shock formation and cavitation is shown as a function of frequency with DI and water depth as parameter.

Longitudinal magnetoresistance size effect of aluminium single crystals: Effect of the cyclotron orbit nature

The magnetoresistance size effect of aluminium single-crystal foils in the longitudinal and MacDonald geometries has been measured at 4.2K for various crystallographic orientations, and the path integral calculations of the magnetoconductivity tensor have been made with a four orthogonalised plane wave (4-OPW) model of aluminium. It is shown that the origin of the anomaly in the longitudinal magnetoresistance size effect for the magnetic field parallel to the (111) crystallographic orientation reported by Amundsen et al. is in the cyclotron orbit nature, and is not in the anisotropic mean free path of the conduction electrons as suggested by these authors.

Effects of density gradients on bottom reflection loss for a class of marine sediments

The sensitivity of bottom reflection loss to a density gradient is examined within the context of a physically meaningful model consisting of a fluid sediment layer having depth-dependent density and sound speed overlying a substrate having solid properties. The bottom grazing-angle range is divided into two regions, the low-angle and the high-angle regions. In the low-angle region sediment sound-speed gradients refract incident energy upward before it encounters the substrate. In the high-grazing-angle region, energy impinges on the substrate. Through the use of a numerical plane-wave reflection coefficient model it is shown that the effect of a density gradient on low-angle bottom loss is small. An expression indicating this effect is derived and is shown to agree with numerical bottom-loss calculations. The high-angle bottom loss is more influenced by a density gradient because of the impedance changes at the sediment–substrate interface which accompany a density gradient. It is shown that introduction of density gradient of 0.002 g/cm3/m can cause the high-angle bottom loss to change on the average by 12.5%, 7%, and 3% in 100-m clay, silt, and sand layers, respectively.

Multiple plane wave calculation of the resistivity of Pb and dilute Pb alloys

We compare with each other and with experiment results for the ideal resistivity of Pb based on several approximate solutions of the Boltzmann transport equation. The Fermi surface geometry, Fermi velocities, and electronic wave functions in the four plane wave model are calculated from the pseudopotential of Anderson and Gold obtained from de Haas – van Alphen data. The lattice dynamics is taken from the Born–von Karman force constant model of Cowley obtained from neutron scattering data on the dispersion curves. In the electron–phonon matrix element, we use the pseudopotential form factor of Appapillai and Williams. The calculations are extended to the case of Mg in Pb and deviations from Matthiessen's rule obtained.

(e, 2e) spectroscopy of H 2O — separation energy spectra and valence orbital electron momentum distributions

The noneoplanar symmetric (e, 2e) reaction has been applied to H 2O at 1200 eV and 400 eV. By keeping the energies of the two emitted electrons fixed (and equal) and varying the incident energy, separation energy spectra are obtained in the valence region. In addition to the four prominent peaks at 12.7 eV, 18.8 eV and 32.4 eV, corresponding to the ejection of electrons from the 1b 1, 3a 1, 1b 2, and 2a 1 orbitals respectively, other structure extending up to 50 eV is observed. Most of this structure can be assigned to the 2a 1 orbital. Electron momentum profiles are measured for the above four transitions. The momentum distributions are energy independent, in agreement with the plane wave model. They are compared with plane wave calculations using for the overlap functions between the target ground state and final ion states either simple SCF orbitals or overlap amplitudes calculated directly using the propagator formalism. The latter overlap functions include the effects of core relaxation and pair correlations, and they are in much better agreement with the data. The momentum profile for the ground state transition (1b 1) is, however, not well described by either theory. The data show a greater probability of having low momentum electrons, corresponding to an increased electron density far from the nuclei.

Comparison of final state approximations in the calculation of total and differential photoemission cross sections of neon

Differential photoemission cross sections for the 1s, 2s, and 2p shells of neon were calculated by several different approximations for photon energies up to 2000 eV. Specifically, plane wave (PW), orthogonalized plane wave (OPW), and Hartree–Fock functions (with and without consideration of relaxation in the final ionic state) were used to compute transition matrix elements in both velocity and length approximations. Plane wave and orthogonalized plane wave continuum functions were found to have very limited applicability to cross section calculations, with both approximations giving spurious local minima and incorrect angular distributions. The reasons for these failures were analyzed, and limits were set on the n, l, and z values for which the PW model yields qualitatively correct total cross sections. Calculations using Hartree–Fock continuum functions agree very well with experiment, emphasizing the necessity of considering atomic potentials explicitly in photoemission processes. Further, the effects of relaxation in the final bound system were investigated. They were small for valence electrons and only slightly more important for Ne 1s core electrons. Applications of these findings to photoemission from molecules and absorbates are discussed briefly.

Electron ionization spectroscopy of acetylene: Momentum distributions of valence orbitals and correlation effects

The noncoplanar symmetric (e, 2e) reaction has been applied to C 2H 2 at 1200 and 400 eV. In addition to the four prominent peaks in the separation energy spectra at 11.4, 16.8, 18.8 and 23.7 eV corresponding to the ejection of electrons from the 1π u, 3σ g, 2σ u and 2σ g molecular orbitals, respectively, considerable structure extending up to 45 eV is observed. Most of this satellite structure can be assigned to the 2σ g orbital. The state at 23.7 eV has not previously been observed in electron separation energy spectra. Electron momentum distributions are measured for the above four transitions as well as for a transition at a separation energy of 29 eV. The shapes of the momentum distributions are energy independent, in agreement with the plane wave model. The relative magnitudes and shapes of the momentum distributions are compared with plane wave calculations, using either simple SCF orbitals for the overlap functions between the initial and final states or the generalized overlap amplitude calculated directly using the one-particle Green's function method. The spectroscopic factors (pole strengths) obtained from the latter calculation are in good agreement with the data. The shape of the momentum profile for the ground state (1π u) transition is, however, not well described by either theory. The data show a greater probability of low momentum electrons, i.e. a more diffuse wave function.

Relativistic correction to the plane-wave Born-approximation model forK-shell ionization by proton impact

We consider a simple relativistic correction to the plane-wave Born-approximation (PWBA) model, in the refined form of Basbas, Brandt, and Laubert. This correction improves the agreement between the PWBA model and experimental data.

A modified plane wave model for calculating UV photo-ionization cross-sections

Photoionization cross-sections are calculated for a number of molecules, using a plane wave method. Agreement with experimental data is considerably improved with respect to common plane wave results if the energy of the photoelectron is assumed to equal the incident photon energy.