4 Absorption Calculation and Experimental Corrections

Proper sample preparation for an XAFS experiment requires knowledge of the absorption depth and edge step size of the material of interest. The statistics of data collection can be optimized by choosing the correct sample thickness. It is also necessary to avoid distortions to the data due to thickness and large particle size effects.

atoms calculates the total cross section of the material above the edge energy of the central atom and divides by the unit cell volume. The number obtained, mu_total, has units of cmˆ-1. Thus, if x is the thickness of the sample in cm, the x-ray beam passing through the sample will be attenuated by exp(-mu_total * x).

atoms also calculates the change in cross section of the central atom below and above the absorption edge and divides by the unit cell volume. This number, delta_mu, multiplied by the sample thickness in cm gives the approximate edge step in a transmission experiment.

The density of the material is also reported. This number assumes that the bulk material will have the same density as the unit cell. It is included as an aid to sample preparation.

Typically, XAFS data is normalized to a single number representing the size of the edge step. While there are compelling reasons to use this simple normalization, it can introduce an important distortion to the amplitude of the chi(k) extracted from the absorption data. This distortion comes from energy response of the bare atom absorption of the central atom. This is poorly approximated away from the edge by a single number. Because this affects the amplitude of chi(k) and not the phase, it can be corrected by including a Debye-Waller factor and a fourth cumulant in the analysis of the data. These two ``McMaster corrections'' are intended to be additive corrections to any thermal or structural disorder included in the analysis of the XAFS.

atoms uses data from McMaster to construct the bare atom absorption for the central atom. atoms then regresses a quadratic polynomial in energy to the natural logarithm of the constructed central atom absorption. Because energy and photo-electron wave number are simply related, E is proportional to kˆ2, the coefficients of this regression can be related to the XAFS Debye-Waller factor and fourth cumulant. The coefficient of the term linear in energy equals 2*sigma_MMˆ2 and the coefficient of the quadratic term equals 43 * sigma_MMˆ4/. The values of sigma_MMˆ2 and sigma_MMˆ2 are written at the top of the output file.

The response of the I0 chamber varies with energy during an XAFS experiment. In a fluorescence experiment, the absorption signal is obtained by normalizing the IF signal by the I0 signal. There is no energy response in the IF signal since all atoms fluoresce at set energies. The energy response of I0 is ignored by this normalization. At low energies this can be a significant effect. Like the McMaster correction, this effect attenuates the amplitude of chi(k) and is is well approximated by an additional Debye-Waller factor and fourth cumulant.

atoms uses the values of the nitrogen, argon and krypton keywords in atoms.inp to determine the content of the I0 chamber by pressure. It assumes that the remainder of the chamber is filled with helium. It then uses McMaster's data to construct the energy response of the chamber and regresses a polynomial to it in the manner described above. sigma_I0ˆ2 and sigma_I0ˆ4 are also written at the top of the output file and intended as additive corrections in the analysis.

If the thickness of a sample is large compared to absorption length of the sample and the absorbing atom is sufficiently concentrated in the sample, then the amplitude of the chi(k) extracted from the data taken on it in fluorescence will be distorted by self-absorption effects in a way that is easily estimated. The absorption depth of the material might vary significantly through the absorption edge and the XAFS wiggles. The correction for this effect is well approximated as

1 + mu_abs / (mu_background+mu_fluor)

where mu_background is the absorption of the non-resonant atoms in the material and mu_fluor is the total absorption of the material at the fluorescent energy of the absorbing atom. atoms constructs this function using the McMaster tables then regresses a polynomial to it in the manner described above. sigma_selfˆ2 and sigma_selfˆ4 are written at the top of the output file and intended as additive corrections in the analysis. Because the size of the edge step is affected by self-absorption, the amplitude of chi(k) is attenuated when normalized to the edge step. Since the amplitude is a measure of S0ˆ2, this is an important effect. The number reported in feff.inp as the amplitude factor is intended to be a multiplicative correction to the data or to the measured S0ˆ2.