One-phonon Cross Section Research Articles

Incoherent approximation for neutron up-scattering cross sections and its corrections for slow neutrons and low crystal temperatures

The incoherent approximation (IA) is often used for calculating the one-phonon inelastic neutron scattering cross section for arbitrary solids. It is valid for thermal neutrons but for slow neutrons it requires a correction, which is significant for isotopes that are strong coherent scatterers. In this article, we present the extension of the Placzek--Van Hove corrections for slow neutrons in the limit of low temperatures using the example of solid \emph{ortho}-deuterium (sD$_2$). Our approach yields realistic one-phonon up-scattering cross sections for sD$_2$ and shows the IA to be a factor of 2 to 5 too high for ultracold neutron (UCN) up-scattering in sD$_2$. Our calculations are compared with previously published Monte Carlo calculations of the one-phonon cross section based on the dynamic structure function $S(q,\omega)$ of polycrystalline \emph{ortho}-deuterium and are found to be consistent with them. Furthermore, we provide the means for easily replicable calculations of the one-phonon up-scattering cross sections of solid \emph{ortho}-deuterium for slow neutrons. These should from now on be used in calculations and simulations of UCN scattering in sD$_2$.

Multiphonon effects in the one-phonon cross section of Al

Multiphonon effects in the one-phonon cross section of Al

Resonant and kinematical enhancement of He scattering from LiF(001) surface and pseudosurface vibrational normal modes.

One-phonon cross sections calculated from sagittally polarized vibrational normal modes account for most salient inelastic-scattering intensities seen in He-LiF(001) 〈100〉 and 〈110〉 measurements published by Brusdeylins, Doak, and Toennies. We have found that most inelastic intensities which cannot be attributed to potential resonances can be explained as kinematically enhanced scattering from both surface and pseudosurface bulk modes.

One-phonon scattering of He from the LiF(001)〈100〉 Rayleigh mode

The one-phonon cross-section from the Rayleigh mode calculated with a point-ion lattice dynamical theory accounts for all salient inelastic scattering intensities seen in He-LiF(001)〈100〉 measurements published by Brusdeylins, Doak and Toennies [J. Chem. Phys. 75 (1981) 1784]. Scattering cross-sections are found by the Rayleigh method, wherein the repulsive part of the scattering potential is approximated as a hard-wall. The solution is in the form of a perturbation expansion in the elastic corrugation and phonon motion of the surface. For each phonon order the solution is infinite in the elastic corrugation. Resonant intensity enhancement occurs either through selective desorption or selective adsorption, both of which have amplitudes which explicitly appear in the perturbation expansion and can be separately calculated. Intensity enhancement also occurs by kinematical focusing. We find a new type of kinematical focusing in addition to that discussed by Benedek [Phys. Rev. Letters 35 (1975) 234].

Collective Excitations in Ethane Monolayers Adsorbed on Graphite (0001) Surfaces

Inelastic incoherent neutron spectra below 4.1 THz have been obtained for the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}4$ herringbone and $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ on-end monolayer phases of ethane adsorbed on graphite basal planes. One-phonon cross sections calculated independently with empirical atom-atom potentials make possible identification of the collective vibrational modes observed and reproduce qualitatively the rather different spectra of the two phases.

Calculations of rheed scattering intensities

A study of the factors giving rise to the observed effects in reflection high energy electron diffraction is presented. A model calculation of the temperature dependence of the scattering intensity in the one-phonon cross-section is made and the possible experimental consequences are discussed.

Thermal diffuse x-ray scattering in simple metals

Calculations are reported for the ionic structure factor and x-ray scattering cross section of sodium (at $T=0 \mathrm{and} 90\ifmmode^\circ\else\textdegree\fi{}$K) and lithium (both isotopes at $T=0$\ifmmode^\circ\else\textdegree\fi{}K) within the harmonic approximation. An evaluation of the appropriate displacement-displacement correlation function by the special-point method circumvents the need for a multiphonon expansion. In the case of sodium, the structure in the one-phonon scattering is straightforwardly accounted for and an approximate expansion is obtained for all multiphonon scattering. By treating core and conduction electrons on an equal footing it is shown that information on the conduction-electron system is present in the forward-scattering component. In lithium the one-phonon cross section at small angles aids in the determination of the effective electron-ion interaction.

Theory of Inelastic Scattering of Slow Electrons by Long-Wavelength Surface Optical Phonons

The purpose of this paper is to present a quantum-mechanical theory of the inelastic scattering of slow electrons by long-wavelength surface optical phonons for simple models of an ionic crystal and for a nonionic crystal such as silicon. It is argued that a quantum-mechanical approach is necessary for this problem. However, the expression we obtain for the one-phonon cross section is found to be identical to the one that follows from the earlier classical theory of Lucas and co-workers, provided one replaces their parameter ${P}_{0}$ by the quantum-mechanical reflection coefficient for specular reflection. The angular distribution of the scattered electrons and the energy dependence of the one-phonon cross section are discussed for the case of ZnO and silicon, where the surface optical modes have a very different character. For the surface mode in silicon, we define a dipole-moment effective charge, which is nonzero by virtue of the absence of inversion symmetry in the surface region. A quantitative estimate of the magnitude of this parameter is extracted from the data of Ibach.

Approximate Dispersion Relations and the Total Coherent Inelastic Neutron Scattering Cross Section for Graphite

The total one-phonon coherent neutron scattering cross section for graphite is evaluated using an approximate model of the dispersion relations. A relatively simple phonon frequency distribution is derived and multiphonon terms are calculated in the incoherent approximation. The results are compared with experimental and other theoretical calculations based upon the incoherent approximation. The latter have been consistently lower than the experimental results by a factor of two or more. It is found that the out-of-plane or z-mode contributes ∼ 75% of the one-phonon cross section. The value of [υ Σ01 (υ)]min is found to be ∼70% greater than the same quantity obtained with the same model in the incoherent approximation. Moreover, the λ-law or constant collision frequency domain only begins with neutron wavelengths >∼20 Å (the Bragg cutoff is 6.7 Å).

Lattice Dynamics and Ionic Deformation in Some Alkali Halides

Equations of motion are derived for a shell model in which the shells are allowed to deform. Dispersion curves are then calculated for NaF, NaCl, NaBr, and NaI from measured dielectric, infrared, and elastic constants. A comparison with published phonon frequencies shows that a simple model with radial deformation gives some improvement over a rigid-shell model. A rather more general treatment with increased radial deformation produces even better curves. However, the LO (0,0,1) zone-boundary phonons are not well represented by either model. The eigenvectors from the rigid-shell model and from the best deformable-shell models are then used to work out the ionic form-factor changes in NaCl when ($0,0,\ensuremath{\eta}$) phonon states are excited, and the corresponding changes in the x-ray one-phonon cross sections along the (0,0,1) axis are obtained. It is found that the rigid-shell model gives small variations in the cross sections resulting from the displacements of the shells, and that the radial deformations give additional changes. The models predict the occurrence of asymmetries in x-ray scattering, but the intensity differences are an order of magnitude less than the effects which have been observed. Further calculations with the inclusion of deformation of the inner electrons are required.

Session H: Phase transitions and critical scattering

The one-phonon cross-section from the Rayleigh mode calculated with a point-ion lattice dynamical theory accounts for all salient inelastic scattering intensities seen in He-LiF(001)〈100〉 measurements published by Brusdeylins, Doak and Toennies [J. Chem. Phys. 75 (1981) 1784]. Scattering cross-sections are found by the Rayleigh method, wherein the repulsive part of the scattering potential is approximated as a hard-wall. The solution is in the form of a perturbation expansion in the elastic corrugation and phonon motion of the surface. For each phonon order the solution is infinite in the elastic corrugation. Resonant intensity enhancement occurs either through selective desorption or selective adsorption, both of which have amplitudes which explicitly appear in the perturbation expansion and can be separately calculated. Intensity enhancement also occurs by kinematical focusing. We find a new type of kinematical focusing in addition to that discussed by Benedek [Phys. Rev. Letters 35 (1975) 234].

The Mössbauer effect for an impurity nucleus: I. Theoretical results for bravais crystals

Formal expressions are derived for the no-phonon and one-phonon cross sections for the absorption of a gamma ray photon by an impurity nucleus in an arbitrary Bravais crystal. An expression is also derived for the second order Doppler shift in the energy of the gamma ray. Most of the analysis is carried out on the assumption that the impurity nucleus is a mass defect, but a brief discussion is also given for the case that the force constants associated with the impurity differ from those for a perfect host crystal.