Fichas técnicas de datos y características de cada equipo

With the experimentalχ(E) obtained from (4.17), it is possible to convert to χ(k) by virtue of equation (2.7) and then fit the resulting data to the fine-structure function (2.11).

The primary goal here is to acquire values for all structural parameters that contribute to the fine-structure, and then fix them. This provides a reference point from which to measure the thermally induced atomic perturbations observed in corresponding DiffEX- AFS spectra. For such a reference to be valid,χ(E) must have been extracted from a spectrum taken at close to the same temperature as the DiffEXAFS measurements. This reference is obtained by firstly using FEFF to determine which scattering paths in the sample are significant and to generate their phase and amplitude information. Scat- tering path lengths, sj, and shell coordination numbers, Nj, are fixed from the outset by providing FEFF with a list coordinates for the position of each atom surrounding the absorbing atom. The Debye-Waller factors, σ_j2, may also be calculated by FEFF via the Correlated Debye Model2

, which provides a good initial approximation to each σ_j2. These are not, however, sufficiently good to serve as a reference, and so must be improved by fittingχ(k).

2

FEFF may also calculateσ2

j through either the Equation of Motion or Recursion methods. These

both tend to be more accurate than the Correlated Debye Model, but require knowledge of the sample’s phonon spectrum.

It is important to stress that this fitting process works on a restricted number of param- eters. Only theσ2_j are fitted, along with the experimental shake-off, S₀2, and absolute edge energy, E03. All other components of the fine-structure function are either fixed

or calculated from first principles. This approach can therefore be considered ’pseudo- ab-initio’. It also serves to mitigate any deficiencies in theoretically calculated spectra by absorbing them into the fitted parameters.

The fitting itself is performed with FitChi2, a new code developed for this thesis that is based in part upon the FitChi code [58]. Like DXAS Calibration, it utilises the Levenberg-Marquardt algorithm to determine the optimal parameter values. Parameter errors are also calculated, and are discussed in more detail in section 4.4.2. By default, every significant scattering path returned by FEFF is considered by FitChi2 irrespective of path length or whether it is a single or multiple scattering path. This typically results in many tens of fitting parameters, which can potentially present a problem with conditioning the fit so as to maximise orthogonality between different parameters. To help identify any such problems, FitChi2 outputs the full fit correlation and vari- ance/covariance matrices, and plots containing each individual path contribution to the overall fine structure. In the event of poor fit conditioning, paths of negligible amplitude may be discarded, and then, if conditioning is still poor, the experimental spectra may be Fourier filtered to limit the maximum scattering path length to further reduce the number of parameters. Experience gained from fitting Fe, SrF2, and Ni2MnGa data

has indicated that Fourier filtering spectra to eliminate contributions beyond the fourth or fifth single-scattering path, leaving around ten paths to consider, produces the best results. However, some degree of trial-and-error is necessary to obtain optimal fits. Once a good fit is found. Just one problem remains, which relates to the representation of the Debye-Waller factors. FitChi calculates them asexp(−2σ2_jk2), based uponk, the photoelectron momentum. However, FEFF generates scattering phase and amplitude information in more general terms, based upon a complex local momentum p2 =k2+

k2

F − Σ(E, p)−Σ(EF, pF) (in atomic units), where the F subscripts denote k, E, 3

Although not strictly structural parameters,S2

0 andE0 must be known sinceαand∂σ2j/∂T scale

with signal amplitude, and hence also withS2

0, and since previous work [61] has shown that when varying

scattering path lengths - as is the case in determiningα- parameter correlation withE0 is extremely

andp at the Fermi energy. This includes many-body effects from the ’dressing’ of the electron. The Debye-Waller factor is thereforeexp(ip.r), and there is thus a correction for the inner atomic potential, which is σ2_j dependent [70]. The difference in results between these two treatments is small, particularly in the EXAFS regime. However, since the complex local momentum treatment results in a small phase correction, it is important include this in subsequent analysis in order to obtain the correct thermal expansion coefficient.

It is therefore necessary to ensure self-consistency is achieved between the FitChi results and FEFF. The Debye-Waller factors are taken from FitChi, and inserted back in to FEFF so that it may re-generate the scattering phase and amplitude information with the experimental parameters included. This new information is then passed back to FitChi, and another, identical, fit performed in order to obtain a correction to the FEFF Debye-Waller factors. These corrections are applied to those already in FEFF, another set of phase and amplitude information generated, and so on until self-consistency is reached, and FitChi no longer changes any of the Debye-Waller factors.