Debye-Waller temperature factor

Finally, a note on disorder of the membrane stacks and on attempts to correct for it in the analysis of diffraction data. Generally, two kinds of disorder are being discussed in crystal structure Disorder of the first kind refers to displacements of the structural elements (for example the one-dimensional unit cell of a membrane stack) from the ideal positions prescribed by the periodic lattice. The effect on the diffraction pattern is indistinguishable from that of thermal vibrations and may, therefore, be expressed as a Debye-Waller temperature factor so that the structure factor, expressed as a cosine series, includes a Gaussian terra, according to... 

Table 5 also summarized other pertinent crystallographic data the Debye-Waller temperature factor coefficients, B in exp[ —2B(sin0/A) ] at 300 K and 5 K the carbon positional parameters, z in (000, ) (OOz), (z = 0 for the R atoms) the intramolecular C-C distances and the nearest C-R distances. 

Here, S = (S -S o)//l and S =2sin0/A S refers to the scattering vector of X-ray analysis, with S o, and S being the direction of incident and scattered radiation, respectively. Here, X is the radiation wavelength and 0 is the scattering angle. yX ), 7XS), and q, are the atomic structure factor, the Debye Waller temperature factor, and the position, respectively, of atom i. The intensity observed in an X-ray experiment is related to the power... 

G(r, t) - space- and time-dependent pair correlation function = Debye-Waller temperature factor Ge(r) = equilibrium spatial pair cor relation function for atoms Go(r) == instantaneous spatial pair correlation function... 

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). 

Here, is the surface temperature and IF is a Debye-Waller-type factor which is a function of and the effective surface Debye temperature 0 there is currently some discussion as to the most appropriate form for t. 239,240 Atomic- and molecular-beam scattering provides three types of information. 

The analysis of the data for FeQ2, in terms of the SOD contribution, results in the estimate of 8d (SOD) WOO 20 K. This value compares well with the effective Debye temperature derived from the Mdssbauer-Debye-Waller (MDW) factor (Table 1). In contrast, the analysis of the FeCl3 data yields 0d(SOD) 400 25 K, which is nearly twice as large... 

The values of the structure amplitudes were used to determine the atomic scattering factors/ of the gallium and phosphorus ions. The absolute values of/were found by comparison with a standard. The standard was a fine-grained powder of nickel. The comparison line was 311. The Cromer dispersion correction [2] and the Debye-Waller temperature correction were applied to the functions /. 

Millis et al. [21] suggested that the evidence of lattice involvement in charge transport may be observed in the Debye-Waller (DW) factor. Indeed Dai et al. [25] reported an anomalous temperature dependence of the DW factor and concluded that polarons must be formed. However, the evidence provided by the DW factor is qualitative as we discussed earlier. The DW approximation assumes a harmonic (Gaussian) distribution of atomic displacements. If only a small number of atoms deviate significantly from the average positions, the DW approximation greatly underestimates the actual displacements. 

Thermal properties of overlayer atoms. Measurement of the intensity of any diffracted beam with temperature and its angular profile can be interpreted in terms of a surface-atom Debye-Waller factor and phonon scattering. Mean-square vibrational amplitudes of surfece atoms can be extracted. The measurement must be made away from the parameter space at which phase transitions occur. 

An important accessory in many applications of Mossbauer spectroscopy is a cryostat for low temperature and temperature-dependent measurements. This may be necessary to keep samples frozen or to overcome small Debye-Waller factors of the absorbers at room temperature in the case of an isotope with high y-energy. Paramagnetic samples are measured at liquid-helium temperatures to slow down... 

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.

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. 

Neutron-irradiated platinum Debye-Waller factor of Au in Pt after low-temperature neutron irradiation, lattice defect... 

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 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. ... 

The damping factors take into account 1) the mean free path k(k) of the photoelectron the exponential factor selects the contributions due to those photoelectron waves which make the round trip from the central atom to the scatterer and back without energy losses 2) the mean square value of the relative displacements of the central atom and of the scatterer. This is called Debye-Waller like term since it is not referred to the laboratory frame, but it is a relative value, and it is temperature dependent, of course It is important to remember the peculiar way of probing the matter that EXAFS does the source of the probe is the excited atom which sends off a photoelectron spherical wave, the detector of the distribution of the scattering centres in the environment is again the same central atom that receives the back-diffused photoelectron amplitude. This is a unique feature since all other crystallographic probes are totally (source and detector) or partially (source or detector) external probes , i.e. the measured quantities are referred to the laboratory reference system. 

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 ... 

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