Wave vector, photoelectron

The photoelectron wave-vector k is evaluated using = 2m(E — E ) where E is the energy of the X-ray photon, , a reference energy and m, the mass of the electron. x(k) is multiplied by k"(n = 2 or 3 usually) to magnify the faint EXAFS at large k (Lytle et al, 1975) /c"x(k) is Fourier transformed to yield the RSF, < (R). In the model compound, the first peak at a distance Rj represents the distance to the nearest-neighbour shell and may be compared to R[, the known distance. We can then define a as (R — Rj), which represents the experimentally determined phase correction. In principle, 2a should be equal to the theoretically estimated k-dependent part of /k), viz. if the identity of the scatterer environment has been correctly assumed. It must be emphasized that wherever scatterer identities are obscure (e.g. in several covalently bonded and disordered systems) use of a (and not j) is advisable. Further, the k-dependence of < /k) introduces an intrinsic limitation to its quantitative accuracy. 

X-ray diffraction data were obtained using Rigaku RAD-C with a copper X-ray tube in air atmosphere. X-ray absorption measurements of Cu K-edge were performed with laboratory EXAFS equipment (Technos EXAC-820). The X-ray source with a rotating Mo target and a LaBe filament was operated at 17 kV, 100 mA (EXAFS) and 20 kV 150 mA (XANES). The samples were pressed into wafers with methyl cellulose as a binder. The measurements were carried out in air atmosphere at room temperature. EXAFS Fourier transformations were carried out over the ranges of photoelectron wave vector, k, of 2.5 -10.0 A 1. 

If the independent variable, E, is replaced by the photoelectron wave-vector, k = 2ir/A = V2me(E — E0)/h2, where me is the mass of the electron, and E0 is the threshold energy for the excitation of a core electron, the EXAFS is given by ... 

Although EXAFS and XANES are recorded as a function of energy, it is conventional to plot the data as a function of k, the photoelectron wave vector [Eq. (3)] where E is the energy of the X-ray photon, Eq is... 

The photoelectron wave vector is k, the magnitude of the amplitude for backscattering from the th atom is f k) and A is the electron mean free path. The phase shift 5j is the sum of contributions from the p-wave shift in the absorbing-atom s potential and the shift in the backscattering amplitude from the th atom. The final exponential term is a Debye-Waller-type factor representing the probability distribution caused by lattice vibrations and disorder, Oj being the root mean square deviation from Rj (also known as the thermal correlation factor ). 

In Eq. (1), k is the photoelectron wave vector relative to Eq (k = 0) N is the the number of neighboring atoms of the same kind at a distance r., of is the mean-square relative displacement (MSRD) of the absorber-scatterer atom pair from their equilibrium inter-atomic distance or in molecular spectroscopy terminology, the mean-square amplitude of vibration other terms have their usual meaning Using standard Fourier transform and curve fitting procedures, we can derive the coordination number, bond length and local dynamics (MSRD) from EXAFS. 

The EXAFS modulations are better expressed as a function of the photoelectron wave-vector k, which is related to the de Broglie wavelength described above as follows ... 

The EXAFS intensity oscillations are described by Eq. 10.19, which gives the relative modulation, X, of the absorption coefficient, fi, of the atom as a function of the variable k (the photoelectron wave vector). 

Here, n is the diffraction order, V s( s) is the scattering phase shift, and k is the magnitude of the photoelectron wave vector. In the calculated pattern of... 

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