Debye-Waller factor in rare gas-solids

The Debye-Waller exponents of solidified krypton at various temperatures have been computed by a phenomenological ‘rigid-atom-model’. The model, which takes rare gas atoms as rigid-hard spheres and derives their central and non-central interactions through Buckingham-Corner and Axilrod-Teller potentials, includes zero-point vibration through potential parameters by a self-consistent method and accounts for the cubic and quartic potential terms as perturbation to the harmonic Hamiltonian. The results show a fairly good agreement with the data obtained from the Mössbauer fractions of ⁸³Kr-9.3 keV transition. Possibilities of further improvement of results have been discussed.     

Surface reconstruction and the Debye-Waller factor

A recently developed soft-mode theory of surface reconstruction¹ is used to calculate the surface Debye-Waller factor (SDWF) as a function of temperature near a supposed transition temperature (T₀) between two reconstruction patterns. The soft surface mode gives rise to a sharp decrease in the SDWF as T₀ is approached, suggesting that an examination the LEED Bragg intensities may help verify the soft mode theory.     

Debye-Waller factor of molecules adsorbed on graphite

The Mössbauer effect in¹¹⁹Sn was employed to study the dynamics of molecules of Sn(CH₃)₄, SnCl₄ and SnI₄ adsorbed on graphite. The mean squared displacement normal to the basal plane was obtained from the temperature dependence of the recoilless fraction. These results were used to estimate the binding energy of the adsorbed molecules.     

On the Debye-Waller factor for atom-surface scattering

The Debye-Waller factor in atom-surface scattering is considered. It is found that the equation suggested by Beeby is valid for soft potentials provided that (1) the attractive portion of the potential is stationary, (2) the repulsive portion of the potentials moves but does not distort, and (3) the effect of trajectories that reflect off the attractive well and multiply scatter from the repulsive wall is negligible.     

On the Debye-Waller factor in molecular beam scattering experiments

In the molecular beam scattering experiments against metal surfaces, one often obtains surface Debye temperatures larger than the bulk ones, in apparent contradiction to the larger thermal atomic motion at the surface. We point out that this is a consequence of that the thermal molecules scatter against the outer part of the electron distribution, and that the metal electrons do not follow the atomic motion rigidly. A self-consistent model calculation shows that this is a large enough effect to explain the experimental results.     

Anharmonic contribution to the Debye-Waller factor for silver

Using high intensity 70 Ci¹⁸³Ta Mössbauer sources, which have a negligible source resonance self absorption (SRSA), we have measured the relative elastic scattering fractionF(g, T) and the relative scattering intensities for the (200), (400), and (600) planes of a 6N pure silver single crystal as a function of temperature from 81 to 1211 K. The values ofF(Q, T) denote the relative fraction of incident recoilless gamma-rays that scatter from the crystal elastically. As determined by line-shape analysis of the Mössbauer spectra, theF(Q, T) values are calculated from the ratio of the measured values of the scattered to incident beam recoilless fractions, and are used to correct for thermaldiffuse scattering. The Debye-Waller factors derived from the Bragg intensities have a significant anharmonic contribution above room temperature.This work was prepared with the support of the US Department of Energy, Grant No. DE-FG02-85ER45199.     

On the calculation of the Debye-Waller factor of alloys

Relations are derived and discussed for the temperature dependence of the diffraction of X-rays on alloys. It is shown that in a certain approximation one can separate the influence of the inter-atomic force constants from that of the masses of the alloy atoms. A reciprocal matrix is found for the linear chain and the equation for the diffraction of X-rays on an alloy is transformed in a certain approximation to an equation for diffraction on a perfect chain.     

Multiphonon resonances in the Debye-Waller factor of atom surface scattering

He atom surface scattering by dispersionless phonons is treated employing coupled channel (CC) calculations. At low energies, they predict a behavior opposite to perturbative Born or "exponentiated" Born approximation: strong resonant phonon stimulated elastic and inhibited inelastic scattering. The corresponding resonances have not been observed in earlier CC results since these have considered only the temperature dependence of the Debye-Waller factor at higher energy or omitted the attractive well. The resonances can be interpreted in terms of bound states in the attractive well with several excited vibrational quanta. They may be observable for, e.g., He scattering by a cold Xe/Cu surface.     

Observation of the Debye-Waller factor in a colloidal single crystal

Bragg diffraction spot intensities of a colloidal single crystal were found to exhibit a DebyeWaller factor-like decay. The Lindemann parameter and the mean-square displacement after infinite times of the colloidal particles could be obtained.     

The influence of the substrate on the Debye-Waller factor of very thin films

The temperature dependence of the Debye-Waller factor of very thin undisturbed films is studied. It is found that if interaction between the substrate and the film does not exist the Debye-Waller factor diverges, with the exception of the case of zero absolute temperature, so that under these conditions resonance recoilless absorption and emission of X- and-rays or neutrons by the nucleus of the film cannot occur. In the presence of interaction between the film and substrate the Debye-Waller factor is finite; its temperature dependence was determined.     

A semiclassical approach to inelastic scattering from solid surfaces and to the Debye-Waller factor

A semiclassical formulation of inelastic atom-surface scattering is presented. This formulation is a mixture of classical S-matrix theory and a classical path model. A Debye-Waller factor enters this theory very naturally as the probability of elastic reflection in the presence of inelastic channels. Because of its importance the Debye-Waller factor is discussed in some detail. Finally, assuming a simplified model of the gas-surface system, the whole scattering problem is solved analytically.     

Debye-Waller factor in atom-surface scattering: Elastic and inelastic channels

Explicit expressions for the Debye-Waller factor for the elastic and one-phonon channels are presented to lowest order in the phonon displacement, using a hard wall model to represent the atom-surface interaction. The periodicity of the crystal is accounted for; thus we explicitly generalize to all elastic channels the reflectance result found by Garcia et al. within the plane-surface model, and we include the contribution of the umklapp processes to the inelastic channels. We show how for high incident energy of the atom all Debye-Waller factors reduce to the standard result.     

The Debye-Waller factor in helium scattering by a crystalline surface

On the dense face of a metallic crystal or the (001) face of lithium fluoride, the distance between nearest neighbours is of the same order of magnitude as the helium diameter. The hypothesis of simultaneous interaction between incident helium atoms of thermal kinetic energy and surface atoms belonging to a surface unit cell is therefore introduced. Then the exponent of the Debye-Waller factor contains mean square and mean correlated displacement between atoms of the cell. This gives for the (00) peak an apparent surface Debye temperature Θ_sa higher than the corresponding value usually deduced for instance from LEED measurements. For the (001) face of copper the calculated value is: 323 < Θ_sa < 354 K according to the value ascribed to bulk Debye temperature. On this face, recent experimental results show that helium atoms are coherently scattered in the specular (00) peak. From its intensity analysis is deduced an attractive well depth of 0.009 ± 0.002 eV and Θ_s = 370 ± 10 K, very close to the calculated result. The simultaneous interaction implies that the one phonon exchange can only take place with phonons of long wavelength. This theoretical expectation seems to be in qualitative agreement with experimental data.     

Pressure effects on EXAFS Debye-Waller factor and melting curve of solid krypton

The pressure effects on atomic mean-square displacement, extended X-ray absorption fine structure (EXAFS) Debye-Waller factor and melting temperature of solid krypton have been investigated in within the statistical moment method scheme in quantum statistical mechanics. By assuming the interaction between atoms can be described by Buckingham potential, we performed the numerical calculations for krypton up to pressure 120?GPa. Our calculations show that the atomic mean-square displacement and EXAFS Debye-Waller factor of krypton crystal depend strongly on pressure. They make the robust reduction of the EXAFS peak height. Our results are in good and reasonable agreements with available experimental data. This approach gives us a relatively simple method for qualitatively calculating high-pressure thermo-physical properties of materials. Moreover, it can be used to verify future high-pressure experimental and theoretical works.     

X-ray determination of the Debye-Waller factors and Debye temperatures of europium monochalcogenides

X-ray powder diffractograms of EuS and EuTe have been recorded. The integrated intensites have been measured and corrected for TDS. From an analysis of the intensity data, Debye-Waller factors and Debye temperatures have been evaluated.     

An effective hard wall model for the calculation of the Debye-Waller factor in atom-surface diffraction

Assuming that: (i) the atom surface repulsive potential is approximated by an effective hard wall which coincides with an isopotential surface, (ii) the repulsive potential is derived from a sum of pairwise interaction (proportional to atomic charge densities) and (iii) the attractive part of the potential is taken into account through the usual Beeby correction, then the Debye-Waller factor may be calculated without any adjustable parameters. The result of a preliminary calculation is found to be in reasonable agreement with recent experimental data.     

A comment on form factor mass singularities in flavor-changing neutral currents

Flavor-changing effective verticesq _l q _h V ⁰, whereV ⁰ represents a neutral gauge boson (,Z ⁰,g), involving a heavy external quark, are discussed within the standard model at one-loop level and second-order approximation in external momenta and masses: the logarithmic singular terms in the form factors at vanishing mass of the internal quark in the loop have to be replaced by pieces coming from next order in external momenta. Implications in thebd+X penguin transitions are commented.