Debye-Waller equation

X-Ray results provide important information regarding molecular and atomic motions, through determination of the thermal factor (B), which gives a measure of the mean square (harmonic) displacement (u ) of an atom or group from its equilibrium position. The two are related by the Debye—Waller equation B = A highly mobile pro-... 

From equation 5, it is apparent that each shell of scatterers will contribute a different frequency of oscillation to the overall EXAFS spectrum. A common method used to visualize these contributions is to calculate the Fourier transform (FT) of the EXAFS spectrum. The FT is a pseudoradial-distribution function of electron density around the absorber. Because of the phase shift [< ( )], all of the peaks in the FT are shifted, typically by ca. —0.4 A, from their true distances. The back-scattering amplitude, Debye-Waller factor, and mean free-path terms make it impossible to correlate the FT amplitude directly with coordination number. Finally, the limited k range of the data gives rise to so-called truncation ripples, which are spurious peaks appearing on the wings of the true peaks. For these reasons, FTs are never used for quantitative analysis of EXAFS spectra. They are useful, however, for visualizing the major components of an EXAFS spectrum. 

The 0 term is a Boltzmann temperature, T, factor the <1) term is a Debye-Waller (DW) factor, well known from diffraction work and finally, the I are Bessel functions of the first kind. This single, simple, harmonic system is quite unrepresentative of any realistic molecular solid, however, it is instructive to express the equation with appropriate experimental parameters. The translational optic modes of water (see below) appear at about 35 meV (260 cm ) and typical experimental temperatures are about 20 K (= 2 meV). 

The last two terms in equation 1 are, in effect, linearly related to the frequency distribution, g(Ho). All of the remaining factors are known or are established by the experimental conditions, except the Debye-Waller factor, exp (—2 IF). It has been customary to set this factor equal to unity, because 2 IF is small under the conditions where the one-phonon approximation is valid (14). 

In terms of the Debye Waller factor, B, which is used in some crystallographic computing codes. Equation (21) is ... 

The observed Bragg rod intensity / ( z) is actually a sum over those (h,k) reflections whose Bragg rods coincide at a particular horizontal angle or position. In the upper equation, the most important variation is due to the molecular structure factor amplitude The Debye-Waller factor = exp[-( jj[4 -i-... 

At this point, the question about the structure of the dimers arises. I he authors tested a letialie-dral mid octahedral geometry and calculated the corresponding I 1 and 1 C distances front the experimentally determined Mg 1 distance and a Mg - C distance, which is taken from other data, preferably from crystal structure data. As the eoor dination number is known from the peak area and the values of the Debye Waller factors of these distances were set, the contribution of the distances 1-1 and 1—C (methyl) to the distinct pan can he calculated according to equation (10.20). 

In these equations, hco is the neutron final energy, Or and aj. are, respectively, the incoherent and coherent scattering lengths for the rth nucleus, [co/(q)] is the Bose population factor, K is the momentum transferred by the neutron, 2H is the Debye-Waller factor for the rth nucleus, and 2nris a reciprocal lattice vector the remaining terms have been previously defined. 

The Debye-Waller factor takes account of the mean-square amplitude of displacements, (u ). The equation describes how the Bragg peaks are still sharp, but their intensities are depressed at high angles. 

The EXAFS amplitude falls off as 1 /R. This reflects the decrease in photoelectron amplitude per unit area as one moves further from the photoelectron source (i.e., from the absorbing atom). The main consequence of this damping is that the EXAFS information is limited to atoms in the near vicinity of the absorber. There are three additional damping terms in Equation (2). The 5 q term is introduced to allow for inelastic loss processes and is typically not refined in EXAFS analyses. The first exponential term is a damping factor that arises from the mean free path of the photoelectron (A(k)). This serves to limit further the distance range that can be sampled by EXAFS. The second exponential term is the so-called Debye-Waller factor. This damping reflects the fact that if there is more than one absorber-scatterer distance, each distance will contribute EXAFS oscillations of a... 

However, due to the cheap computer power available, evaluation of single-crystal neutron diffraction data (at the level of atomic coordinates in the unit cell and the Debye-Waller factors associated with them) is a routine task nowadays. The trick is to apply direct methods of phase determination - methods that depend on the many mutual relationships (inequalities and probabilities) between structure factors and their phases. The simplest such constraint is that scattering power in solids is concentrated on atoms, which are well-defined regions by their size and shape. (For pioneering applications, see the works published by Karle and Karle (1966), Bernal and Watkins (1972) and Jonsson and Hamilton (1972).) Similar conditions cannot be cast into exact mathematical equations, but computer algorithms can easily cope with them. 

In Eq. (3-51), and in a few equations which will follow the vibrational frequencies are not expressed as v in wavenumbers, but as co in Hz. In Eq. (3-51), A is the scattering cross-section of the scattering nucleus (since the H atom has a very large cross section, namely Ah = 82.5 barn, Ac = 5.5 barn, molecules containing hydrogen are very suitable for neutron experiments), kf and ko are the moduli of the wave-vectors of the scattered and incident neutron respectively, e " the Debye-Waller factor or temperature factor, K = kf - ko, M the mass of the unit cell, g(vibrational states, and P = h/2kBT (where ks and T are the Boltzman factor and the temperature respectively). 

In this equation the first term is the Debye Waller Factor, the second one is the detailed balance term which interrelates the intensity of... 

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