The Debye-Waller factor

The phonon excitation can be probed by scattering with neutrons or He-atoms which have De Broglie wavelengths comparable to those of the phonons. For the probing of the softer surface phonons, scattering with low-energy He atom beams is ideal. The elastic scattering probabilities, i.e., the diffraction peaks, contain two factors. One depends on the structure of the solid while the other involves the temperature-dependent effects. Thus the fact that the experiments are 

By introducing the D-W factor, the experimental data are interpreted approximately such that 

As we have seen above, it is possible to formulate a dynamical approach which in a simple manner includes the surface temperature effect. However, schemes which introduce the Debye-Waller factor as a simple multiplicative factor as illustrated by eq. (8.129) are, of course, even simpler to use. 

For surface scattering the derivation of a D-W factor has been considered by a number of researchers (see, e.g., refs. [67, 68, 69]). Previously the theory had been primarily developed for including the effect of inelastic motion in the interpretation of X-ray and neutron scattering data [70]. 

Thus the probability of interest is that of no energy loss to phonons. In the semiclassical model (considered in section 8.3.1) this probability is given as 

For an arbitrary state of the system, we are not interested in individual phonon scattering processes but rather in the thermal average over such events. We are also typically interested in the absolute value squared of the scattering matrix element, which enters in the expression for the cross-section, the latter being the experimentally measurable quantity. When we average over the sums that appear in the absolute value squared of the structure factor, the contribution that survives is 

Taking advantage again of the fact that the magnitude of fj/ vectors is small, we can approximate the factor in square brackets by 

The quantity W is called the Debye-Waller factor. Note that in the expression for W the index n of is irrelevant since it only appears in the complex exponential exp(—iq R ) which is eliminated by the absolute value thus, W contains no dependence on n. Substituting the expression of fj,6 from Eq. (6.90) we find 

We will assume next that the amplitudes are independent of the phonon 

As we discussed earlier, phonons can be viewed as harmonic oscillators, for which the average kinetic and potential energies are equal and each is one-half of the total 

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In the procedure of X-ray refinement, the positions of the atoms and their fluctuations appear as parameters in the structure factor. These parameters are varied to match the experimentally determined strucmre factor. The term pertaining to the fluctuations is the Debye-Waller factor in which the atomic fluctuations are represented by the atomic distribution tensor ... 

Here Pyj is the structure factor for the (hkl) diffiaction peak and is related to the atomic arrangements in the material. Specifically, Fjjj is the Fourier transform of the positions of the atoms in one unit cell. Each atom is weighted by its form factor, which is equal to its atomic number Z for small 26, but which decreases as 2d increases. Thus, XRD is more sensitive to high-Z materials, and for low-Z materials, neutron or electron diffraction may be more suitable. The faaor e (called the Debye-Waller factor) accounts for the reduction in intensity due to the disorder in the crystal, and the diffracting volume V depends on p and on the film thickness. For epitaxial thin films and films with preferred orientations, the integrated intensity depends on the orientation of the specimen. 

Where, /(k) is the sum over N back-scattering atoms i, where fi is the scattering amplitude term characteristic of the atom, cT is the Debye-Waller factor associated with the vibration of the atoms, r is the distance from the absorbing atom, X is the mean free path of the photoelectron, and is the phase shift of the spherical wave as it scatters from the back-scattering atoms. By talcing the Fourier transform of the amplitude of the fine structure (that is, X( )> real-space radial distribution function of the back-scattering atoms around the absorbing atom is produced. 

The used S5mbols are K, scale factor n, number of Bragg peaks A, correction factor for absorption P, polarization factor Jk, multiplicity factor Lk, Lorentz factor Ok, preferred orientation correction Fk squared structure factor for the kth reflection, including the Debye-Waller factor profile function describing the profile of the k h reflection. 

The Bragg scattering of X-rays by a periodic lattice in contrast to a Mossbauer transition is a collective event which is short in time as compared to the typical lattice vibration frequencies. Therefore, the mean-square displacement (x ) in the Debye-Waller factor is obtained from the average over the ensemble, whereas (r4) in the Lamb-Mossbauer factor describes a time average. The results are equivalent. 

Mossbauer nuclei per square centimetre, the Debye-Waller factor/a of the absorber material, and the resonance cross-section Gq of the Mossbauer isotope. For a multiline spectrum, the result must be split into separate values for each line, which are obtained by weighting t with the relative transition probability of each line. 

Conroy and Perlow [235] have measured the Debye-Waller factor for W in the sodium tungsten bronze Nao.gWOs. They derived a value of/= 0.18 0.01 which corresponds to a zero-point vibrational amplitude of R = 0.044 A. This amplitude is small as compared to that of beryllium atoms in metallic beryllium (0.098 A) or to that of carbon atoms in diamond (0.064 A). The authors conclude that atoms substituting tungsten in bronze may well be expected to have a high recoilless fraction. 

Mossbauer spectroscopy with started only in 1965, when Harris et al. [322] measured the Mossbauer absorption spectra of the 99 keV transition of Pt in platinum metal as a function of temperature (between 20 and 100 K) and of absorber thickness and derived the temperature dependence of the Debye-Waller factor. 

Distortions of the First Kind and Thermal Disorder. In crystallography the best-known example for a lattice distortion of the first kind is the reduction of peak intensity from random temperature movement of the atoms. In materials science a frozen-in thermal disorder of nanostructures25 is observed as well. The result of this kind of disorder is a multiplicative26 attenuation of the scattering intensity by the Debye-Waller factor... 

The reason for multiplying with a k weighting factor is to compensate for the decrease of the EXAFS amplitudes at high k values due to the Debye-Waller factor, the backscattering amplitude, and the k 1 dependence of the EXAFS (see, e.g., Ref. (21)). 

The recoilless fraction, /, has been calculated (13) for monotomic lattices using the Debye approximation. When the specific heat Debye temperatures of the alkali iodides are inserted in the Debye-Waller factor, a large variation of f follows (from 0.79 in Lil to 0.15/xCsI). It is not... 

A small problem arises when the crystal thickness and temperature factors are refined simultaneously, because these parameters are highly correlated. Raising both the thickness and the temperature factors results in almost the same least-squares sum. This is not an artifact of the calculation method but lies in the behavior of nature. Increasing the Debye-Waller factor of an atom means a less peaked scattering potential, which in turn results in a less sharply peaked interaction with the ncident electron wave. It can be shown that a thickness of 5 nm anc B=2 will give about the same results as a thickness of 10 nm and B=6 A. ... 

Fig. 3.16 Scaling presentation of the dynamic structure factor from a M =36,000 PE melt at 509 K as a function of the Rouse scaling variable. The solid lines are a fit with the reptation model (Eq. 3.39). The Q-values are from above Q=0.05,0.077,0.115,0.145 A The horizontal dashed lines display the prediction of the Debye-Waller factor estimate for the confinement size (see text)...

The Debye-Waller factor, Eq. 4, describes how the uncertainty in real space (u) determines the range of S(Q) in Q space. Now the exact converse happens with respect to the resolution of the measurement in Q space. If the Q resolution of the instrument is AQ, the PDF will have an envelope exp(- r ( AQ) ), and the oscillations in the PDF decay. Therefore in order to determine the PDF up to large distances it is important to use an instrument with high Q resolution. Since the PDF method was initially applied to glasses and liquids in which atomic correlation decays quickly with distance, this point was not... 

Where N is the effective number of bonds along the normal and in plane directions, as tabulated in Table 3. From this expression, the variation of the Debye-Waller factor between 300 K and 77 K is derived by the ratio method ... 

S is the scattering vector, Mj is the atomic displacement parameter in this simplified notation assumed to be isotropic, 6 is the scattering angle, and 1 the wavelength of the incident radiation. The atomic displacement depends on the temperature, and hence so does the Debye-Waller factor. If an atom is modeled by a classical oscillator, then the atomic displacement would change linearly with temperature ... 

Thermal diffuse scattering (TDS) is ascribed to low-frequency lattice vibrations. The atoms in a perfect crystal are not fixed to their sites and oscillate about their positions. The Bragg intensities are reduced by the Debye-Waller factor, proportional to... 

Taking the phonon source out of equilibrium at a certain frequency range may lead to enhancement in Ipc. On a speculative level, one may visualize shining the electrons with a high intensity beam of non-equilibrium phonons with a narrow frequency range around, say, w0. Icc, resulting from resonant transitions, will be significantly affected only when u>o is close to the differences e — Cj or c( — C(. The effect on the Debye-Waller factor will be small for a narrow-band beam. In this way, Icc will initially increase with the intensity of this radiation, until decoherence effects will take over and Ipc will disappear. 

For both heavy and Tight polaron a dependence of y on the nonadiabaticity parameter fl/T appears. It implies the dependence of the Debye-Waller factor and consequently of the polaron mass on the phonon frequency O. This can be thought of as an analogy of the isotope effect at zero temperature. 

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