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.     

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.     

Quantum theory of atom-surface scattering: Debye-Waller factor

The Debye-Waller factor, introduced historically for X-rays, was used later for electrons, neutrons, and atoms as well. In this process of extension, however, the assumptions on which the Debye-Waller theory rested became more and more questionable until in the case of atoms (whose scattering from surfaces is both strong and slow) serious modifications are necessary. In the present article four models are discussed in order. In Model 1 a fast atom impinges on a surface whose atoms all vibrate deviating from their equilibrium positions by the same vector displacement ?. In Model 2 again the impinging atom is fast, but the atoms in the surface vibrate incoherently rather than coherently. It is shown that both Models 1 and 2 yield the conventional Debye-Waller result in the infinite crystal atom mass limit (for Model 2 Einstein oscillators have also to be assumed) and it is also shown how corrections to this result can be built. Turning then to slow impinging atoms, in Model 3 a slow atom impinges on a hard crystal surface, interacting with the rapidly varying potential of the vibrating solid. Model 3 is discussed in detail and it is shown that the Debye-Waller exponent can be written in terms of a time integral of the product of two correlations: the force correlation and the displacement correlation. The result is a dramatic increase of diffraction of relatively heavy atoms (with respect to the conventional theory). Finally, in Model 4 the impinging atom is again slow but the crystal is soft rather than hard. This case is more difficult to treat but a preliminary analysis again indicates a dramatic increase of diffraction since the soft solid adjusts itself to the instantaneous atom position leading to elastic scattering. The experimental implications of the present theory, especially for neon scattering from surfaces, are discussed.     

Debye-Waller effects in atom-surface scattering

The temperature sensitivity of helium atom reflection from NaF ranges from slight in the quantum regime (long wavelength, near-glancing incidence) to considerable in the classical regime (short wavelength, near-normal incidence). The latter is describable by the Debye-Waller theory only over a limited range of angles. In this range, no Beeby well-depth correction is required. The surface Debye temperature of the NaF is found to be 425 ± 20 K.     

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.     

Multiphonon resonant Raman scattering in GaSe

Multiphonon resonant Raman scattering up to 4 phonons in GaSe has been measured. The results are interpreted by a simple cascade theory. It is shown that the dispersion of RRS here is dominated by resonances with the exciton states.     

Transition from resonances to surface waves in elastic scattering

In this article, we study resonances and surface waves in π⁺–p scattering. We focus on the sequence whose spin-parity values are given by . A widely-held belief takes for granted that this sequence can be connected by a moving pole in the complex angular momentum (CAM)-plane, which gives rise to a linear trajectory of the form , which is the standard expression of the Regge pole trajectory. But the phenomenology shows that only the first few resonances lie on a trajectory of this type. For higher J^p this rule is violated and is substituted by the relation JkR, where k is the pion-nucleon c.m.s. momentum, and R1 fm. In this article we prove: (a) Starting from a non-relativistic model of the proton, regarded as composed by three quarks confined by harmonic potentials, we prove that the first three members of this π⁺–p resonance sequence can be associated with a vibrational spectrum of the proton generated by an algebra . Accordingly, these first three members of the sequence can be described by Regge poles and lie on a standard linear trajectory. (b) At higher energies the amplitudes are dominated by diffractive scattering, and the creeping waves play a dominant role. They can be described by a second class of poles, which can be called Sommerfeld’s poles, and lie on a line nearly parallel to the imaginary axis of the CAM-plane. (c) The Sommerfeld’s pole which is closest to the real axis of the CAM-plane is dominant at large angles, and describes in a proper way the backward diffractive peak in both the following cases: at fixed k, as a function of the scattering angle, and at fixed scattering angle θ=π, as a function of k. (d) The evolution of this pole, as a function of k, is given in first approximation by JkR.     

Debye-Waller factor of gold

The temperature dependence of the exponent of Debye-Waller factor of gold for the temperature range extending from 50 to 900°K has been studied using a theoretical model where the interactions are treated in realistic manner. The results are found to be in reasonable agreement with the experimental ones.     

The Debye-Waller factor of graphite

The Debye-Waller factor of graphite is calculated, using the model proposed by Komatsu.     

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.     

The anomalous Debye-Waller factor of hydrogen in niobium as determined by quasielastic neutron scattering

The integrated intensity of quasielastic neutron scattering by protons in polycrystalline NbH_0.16 and in a single crystal of NbH_0.045 was investigated as a function of the scattering vector Q. Strong deviations from a harmonic Debye-Waller factor behavior were observed at elevated temperatures. The results show a temperature dependent delocalization of the proton extending as far as the neighboring sites of the interstitial lattice. Experiments on the single crystal indicate a directional dependent mean-square amplitude of the proton even at room temperature.     

Debye-Waller factor through the glass transition temperature in a-selenium,by incoherent inelastic neutron scattering

We report, in this work, Incoherent Inelastic Neutron Scattering (IINS) measurements in amorphous bulk selenium. Taking into account the low frequency Vibrational Density-of-States(VDOS), we study the modification introduced in the Debye-Waller factor by increasing temperature through the glass transition temperature (T_g). The occurrence, in the vibrational density-of-states, of a ω² dependence in the acoustic region, allowed us to apply the Debye theory from which the variations of the Deybe-Waller factor are calculated in addition. It is shown that the main contribution to it is given by the acoustic region of the vibrational density of states and has a faster increase for temperatures above T_g.     

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.     

The spectroscopy of surface vibrations by atom scattering

The recent progress in the production of highly monochromatic atomic beams is opening new perspectives in surface physics, having paved the way for a full determination of the surface vibrational structure. After a discussion on the possible determination of Rayleigh wave dispersion curves from angular distributions exploiting the kinematical focussing effect, a short review is presented on the direct measurement of surface phonon dispersion curves, first achieved by Brusdeylins, Doak and Toennies in alkali halides, from time-of-flight (TOF) spectra of scattered He atoms. A comparison is made with the existing theories of surface phonons in ionic crystals. The state of the art in the theory of inelastic processes is briefly illustrated in order to discuss the theoretical interpretation of TOF spectra. The one-phonon energy loss spectra of He scattering from LiF(001) calculated for a hard corrugated surface model are found to be in general good agreement with the experimental TOF spectra. From such a comparison evidence is obtained that: i) one-phonon processes are predominant, and ii) in addition to Rayleigh waves important contributions to the inelastic scattering come from the surface-projected density of bulk phonons. Important effects due to inelastic resonances with surface bound states are put in evidence and explained by simple kinematical arguments. The possible observation of surface optical modes in NaF(001) is finally 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 lead nitrate

The X-ray Debye-Waller factors and Debye temperatures of lead nitrate single crystals taken in the powder form have been determined by measuring integrated intensities of selected Bragg reflections at different temperatures. The characteristic specific Debye temperature has been compared with the value obtained from elastic constant data.     

Multiphonon Raman scattering and gap states in cis-polyacetylene

The different features of the Raman scattering in resonance with electronic absorption band in cis and trans-polyacetylene are discussed. The strong multiphonon scattering observed in cis-polyacetylene is accounted for in terms of a localized gap state, which dominates the scattering process near the absorption edge.     

On the interpretation of rainbow scattering in gas atom/surface collisions

A model is proposed to explain the scattering of thermal energy atoms from surfaces. The model allows the behaviour of the scattering trajectories to be assessed. It is shown that when a collimated beam of Neon atoms is scattered from the LiF(100) surface, the peaks observed in the inplane distribution of scattered atoms versus angle of reflection arise from trajectories with one and two repulsive collisions with the same surface atom.