Debye-Waller term

The errors in the fitting parameters may be obtained from the covariance matrix of the fit if it is available, but they are more commonly estimated by varying one parameter away from its optimal value while optimizing all other parameters until a defined increase in the statistical function is obtained. However, the statistical error values obtained do not represent the true accuracies of the parameters. In fact, it is difficult to determine coordination numbers to much better than 5%, and 20% is more realistic when the data are collected at room temperature taking into account the strong coupling between the coordination number and Debye Waller terms, the error in the latter may be 30%. 

Cd, and others. For example, nickel EXAFS of carbon monoxide dehydrogenase (CODH) from Clostridium thermoaceticum strain DSM 521 were collected and the Fourier filtered data was best fit to sulfur at 0.216 nm. A reasonable fit to oxygen/nitrogen could only be obtained by setting the zIEq to 40 and the Debye-Waller term to 5xl0 nm. A reasonable Ni-S fit was also obtained by assuming two Ni-S bond lengths at 0.222 and 0.211 nm, respectively. Addition of a Ni-M (M=Fe, Ni, Zn) term at 0.325 nm, also improved the... 

Figure 7 (a) Cd K-edge X-ray absorption spectrum of rat liver Cd5Zn2MT-l. (b) Cd K-edge EXAFS (solid line) and a theoretical simulation (dashed line) with 4 Cd-S distances at 2.53 A and a Debye-Waller term, of... 

As was the case for XANES, here electronic structure calculations including solvent effects are used to draw conclusions on the most probable coordination isomer and from that structure the interatomic bond distances can be directly compared to the computed EXAFS spectra [150-152]. As EXAFS spectroscopy probes the structure of a statistically averaged system, the most appropriate way of comparing theoretical EXAFS data to experimental ones is to use molecular dynamics trajectories to sample the configuration space, select snapshots and finally compute a statistically average spectrum [89,153] with a direct estimate of the mean square relative disorder (MSRD, also called or the EXAFS Debye-Waller term). However, as discussed in Section 11.2.3, the relevance of MD simulations hinges on how accurate intermolecular interactions are, given that these are usually obtained at DFT level (in Car-Parinello MD simulations) or with force-fields (in classical MD simulations). 

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 ... 

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. 

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 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. 

AEq = 0.37 for the LS (1A1) state (relative to natural ion at 298 K). Debye-Waller factors were determined for the two states. The values -lnf(sT2) and -lnfi A,) follow the Debye model between 175 and 250 K with 0D (ST2) = 126 K and between 105 and 225 K with 0D (J Aj) = 150 K, respectively. Deviations encountered outside these regions were considered as evidence for the formation of cooperative domains as suggested by Sorai and Seki34,87T The difference between the Debye temperatures 0d(5T2) and 0d( Ai) may well be understood in terms of more rigidity in the lattice of the 2A state as compared to that of the ST2 state. A study of the magnetic hyper-fine interaction at 4.2K yielded VZZ( A1) < O. VZZ(5T2) > O, however, was concluded from the spin reversal of the texture-induced asymmetry of the Mossbauer line intensities. 

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 exponential term in Eq. (1) is a Debye-Waller factor and includes a sum over all the modes, including the lowest energy modes, which may have an appreciable population at room temperature. To minimize the influence of the Debye-Waller factor, IINS measurements are routinely carried out temperatures below 30 K. 

An alternative approach to the snapshot method is to characterize a working catalyst. Such an experiment would be best done on a chopper instrument, because they provide access to the low-g region of the spectrum, in contrast to FDS, TOSCA, or INlBeF. At room temperature, the Debye-Waller factor (the exponential term in Eq. (1)) is large however, operating at low-g reduces its impact and should allow measurements to be carried out. There are limitations the temperature should be as low as possible, and only steady-state operation could be investigated, because the measurement will be at least several hours in duration. Nonetheless, there are reactions that fit these criteria, and the view inside a working catalytic reactor would be spectacular to behold. 

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). 

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