Debye-Waller effect

Informations on the vibrational and electron mean free path properties. Such analysis is possible only if the interface phase is very well defined, and if temperature dependent measurements are done and compared. Debye Waller effects can be tangled with ordering transformation of the interface phase as a function of temperature and so on. If a single phase interface with order at least to the second nearest neighbour is recognised, then a temperature dependent Debye Waller, and mean free path analysis can be attempted. 

Small oscillatory torsions of molecular units within the segment produce a de-localization of the electronic density according to the Debye-Waller effect . In a very distorted structure as in the melt its smooth intensity distribution is not affected to a measurable extent. Rotational isomers, on the other hand, induce remarkable changes of the ru involved. Moreover, it is easily shown that the Boltzmann weighting factor of the molecular conformation ensemble also has a substantia] influence on the ru. It is strictly not possible to calculate the... 

This calculation assumes that long range order exists in the system and that the errors are simple thermal deviations from the positions defined by this order. This diminution of intensity with increasing Q was first discussed by Debye [31,32J and Waller [33J and is hence known as the Debye-Waller effect. 

Analysis of low temperature data using parameters derived from the study of model complexes at room temperature may result in high N values due to Debye-Waller effects (see Ref. 116) Fe-0(dioxygen)= 1.77 A N(0 )= 1.3 ... 

A theoretical calculation of the scattering intensity from 3D-distorted paracrystals was carried out by Matsuoka et al.(9l Three parameters were demonstrated to be crucia I paracrysta 1 1 ine distortion, Debye-Waller effect and crystal size. The theory was compared with experimental data on ionic micelles, silica particles, and latex particles and fairly large degrees of distortion were obtained. 

D. R. Frankl, Debye-Waller effects in atom-surface scattering. Surface Sci. 77 493 (1978). 

It has long been known that thermal displacements of the surface ions lead to temperature-dependent attenuation of the intensity of elastically scattered helium atoms. This Debye-Waller effect has been extensively used to determine the surface Debye temperature of alkali halide single-crystal surfaces, especially... 

Table 7.8 Summary of results obtained for the four Os Mossbauer transitions studied. The absorber thickness d refers to the amount of the resonant isotope per unit area. The estimates of the effective absorber thickness t are based on Debye-Waller factors / for an assumed Debye temperature of 0 = 400 K. For comparison with the full experimental line widths at half maximum, Texp, we give the minimum observable width = 2 S/t as calculated from lifetime data.

Photoionization and therefore EXAFS takes place on a time scale that is much shorter than that of atomic motions so the experiment samples an average configuration of the neighbors around the absorber. Therefore, we need to consider the effects of thermal vibration and static disorder, both of which will have the effect of reducing the EXAFS amplitude. These effects are considered in the so-called Debye-Waller factor which is included as... 

By Fourier transforming the EXAFS oscillations, a radial structure function is obtained (2U). The peaks in the Fourier transform correspond to the different coordination shells and the position of these peaks gives the absorber-scatterer distances, but shifted to lower values due to the effect of the phase shift. The height of the peaks is related to the coordination number and to thermal (Debye-Waller smearing), as well as static disorder, and for systems, which contain only one kind of atoms at a given distance, the Fourier transform method may give reliable information on the local environment. However, for more accurate determinations of the coordination number N and the bond distance R, a more sophisticated curve-fitting analysis is required. 

Single crystals of Te are highly anisotropic and occur as either right-hand or left-hand crystals. It would be interesting to use the Te Mossbauer effect to measure the anisotropic Debye-Waller factor in such crystals. 

According to the model, a perturbation at one site is transmitted to all the other sites, but the key point is that the propagation occurs via all the other molecules as a collective process as if all the molecules were connected by a network of springs. It can be seen that the model stresses the concept, already discussed above, that chemical processes at high pressure cannot be simply considered mono- or bimolecular processes. The response function X representing the collective excitations of molecules in the lattice may be viewed as an effective mechanical susceptibility of a reaction cavity subjected to the mechanical perturbation produced by a chemical reaction. It can be related to measurable properties such as elastic constants, phonon frequencies, and Debye-Waller factors and therefore can in principle be obtained from the knowledge of the crystal structure of the system of interest. A perturbation of chemical nature introduced at one site in the crystal (product molecules of a reactive process, ionized or excited host molecules, etc.) acts on all the surrounding molecules with a distribution of forces in the reaction cavity that can be described as a chemical pressure. 

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 ... 

The Debye Waller analysis of the S—B bonds gives A05 g2(HO) = 2.9 x 10 A. This value is lower than the pure Co value (3.6 x 10 A ). Due to the low density of the (110) face, one mi t have expected a large mean-square relative displacement. The measured small value reveals a stiffening of the force constant of the Co—Cu bond. This is consistent with the large eontraction of the Co—Cu interlayer distance (sell % see above). The stiffening in strongly relaxed surfaces has been observed before and overcompensate the effect of the reduced surface coordination in the perpendicular direction. Reversed surfaee anisotropy of the mean square relative atomic displacements has also been found on an other low-density surfaee C2 x 2 Cl/Cu(l 10) i.e. one half density of Cl vs. Cu(l 10) in plane density where the Cl atoms moves with amplitudes parallel to the surface eomparable with those of the Cu subtrate, but with a much reduced amplitude in the perpendicular direction... 

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. 

When the electronic system is open the energy levels acquire a finite lifetime, and the the transitions no longer conserve the sum of electronic and radiation energies. Nonetheless, the radiation introduces again a unique effect, which goes beyond that of the Debye-Waller exponent. [11] Imagine an... 

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. 

Further Mossbauer effect studies (304 — 12 K) and magnetic susceptibility measurements (301 - 1K) on the neutral complex [Fe(papt)2 ] have been performed recently 1S9 The magnetic data are shown in Fig. 34. The values of - In f(s T2) and - In f(J Aj) have been found to follow the high-temperature approximation of the Debye model above 105 K and 140 K, respectively, if anharmonic corrections have been introduced. No simple model is available at present, which would be capable to account for the complete temperature dependence of the Debye-Waller factors in this crossover system. 

Two of the more direct techniques used in the study of lattice dynamics of crystals have been the scattering of neutrons and of x-rays from crystals. In addition, the phonon vibrational spectrum can be inferred from careful analysis of measurements of specific heat and elastic constants. In studies of Bragg reflection of x-rays (which involves no loss of energy to the lattice), it was found that temperature has a strong influence on the intensity of the reflected lines. The intensity of the scattered x-rays as a function of temperature can be expressed by I (T) = IQ e"2Tr(r) where 2W(T) is called the Debye-Waller factor. Similarly in the Mossbauer effect, gamma rays are emitted or absorbed without loss of energy and without change in the quantum state of the lattice by... 

---