Experimental Corrections

ATOMS can perform several useful calculations using tables of x-ray absorption cross sections, including estimations of absorption length and various experimental corrections. There are several tables x-ray absorption cross sections which the user can choose from:

1.

The 1999 Elam tables (the default)

2.

The 1969 McMaster Tables

3.

The 1993 Henke Tables

4.

The 1995 Chantler Tables

5.

The Brennan-Cowen implementation of the Cromer-Liberman Tables^11.1.

See the references in Chapter  and the technical documentation in Appendix .

   
The Density and the Absorption Lengths

In an EXAFS experiment in transmission mode, proper sample preparation is essential for collection of high quality data. In order to make appropriate choices for the composition and form of the sample it is necessary to know both the absorption length of the sample and its edge step absorption length. ATOMS approximates both of these quantities.

The absorption length is defined as the thickness x of the sample such that the intensity of the x-rays incident upon the sample at an energy 50 eV above the absorption edge is attenuated e-fold. That is, $e^{-x\mu}=1$ where $\mu$ is the total absorption of the sample at that energy. The sample length for unit edge step, the reciprocal of which is called $\Delta\mu$, is the thickness x such that there is an e-fold change in absorption between 50 eV below and above the edge, i.e. for $e^{-x\,\Delta\mu}=1$.

ATOMS computes the free atom cross-sections for each atom specified in the input data at 50 eV above the absorption edge and for the resonant atom at 50 eV below the edge. Using the crystallographic information, it knows the size of the unit cell and how many of each species are in the unit cell and so can correctly calculate the total and unit edge step absorption lengths for the crystal. Using the unit cell size and tabulated masses of the elements, the density of the crystal is also computed. With the absorption lengths and density of the material, proper choices for sample preparation can be made.

Here is an example of a sample of lead titanate, PbTiO₃, prepared using these calculations from ATOMS. Using this structural data

    title PbTiO3 25C
    title Glazer and Mabud, Acta Cryst. B34, 1065-1070 (1978)
    core=ti    Space  P 4 m m
    a=3.905    nitrogen=1   rmax=3.6   c=4.156
    atom
    ! At.type  x        y       z      tag
       Pb     0.0      0.0     0.0
       Ti     0.5      0.5     0.539
       O      0.5      0.5     0.1138  axial
       O      0.0      0.5     0.6169  planar

yields the results shown in Table .

 

Table: Absorption lengths and experimental corrections for lead titanate calculated using the Elam tables.

calculation	value
total $\mu$	4850.42 cm-1
$\Delta\mu$	747.42 cm-1
specific gravity	7.942
Normalization correction	0.00046 Å2
I0 correction	0.00111 Å2
self absorption correction	0.00003 Å2
self absorption amplitude factor	1.093
net correction	0.00160 Å2

 

In this example, the absorption length is 1/4850 cm or about 2.1 microns and the edge step absorption length is 1/747 cm or about 13.4 microns. For my experiment on the titanium K edge of lead titanate, I chose to make a sample which was 5 microns thick, thus with total absorption of about 2.5 and an edge step of about 0.37. Knowing the appropriate thickness for the sample, the dimensions of the die used to press the sample, and the density of lead titanate, I was able to make an sample appropriate to these conditions. An example spectrum obtained with one of these samples is shown in Fig. , which has an edge step calculated by the AUTOBK program of 0.413, only 10% larger than the estimation made by ATOMS.

Unnormalized absorption spectrum for the titanium K edge of lead titanate at 300 K. The edge step for this scan is 0.413, which differs by only 10% from the predicted edge step of 0.37 for a 5 micron sample.

 

   
The Normalization Correction

To avoid introducing systematic errors into the amplitude of the measured $\chi(k)$, an edge-step normalization is typically used. Since the true atomic background absorption, $\mu_0$, has energy dependence, normalization by the edge-step introduces an energy-dependent attenuation to the amplitude of $\chi(k)$. This attenuation is small for heavy elements, but can be of the same order as thermal effects for light elements. ATOMS calculates an approximation to this attenuation called the normalization correction^11.2.

Using the tables of free atoms x-ray cross-sections, ATOMS evaluates the cross-section for the free central atom in a range from 50 to 300 eV above the absorption edge. It then regresses a second order polynomial in the natural log of the energy relative to the edge to the natural log of the free atom cross-section^11.3. The linear term is an approximation to the degree of attenuation introduced by the edge-step normalization. This is intended as additive corrections to the measured $\sigma^2$ of a fit. Neglecting this correction will make the temperature dependence of $\sigma^2$ deviate from an Einstein behavior by a constant offset.

   
The I₀ Correction

In a fluorescence experiment the absorption cross section is obtained from the detected intensities on the I₀ and I_F chambers by measuring their ratio as a function of energy. This introduces an energy dependent error, usually an attenuation, into the amplitude of the measured $\chi(k)$. The secondary photon measured in the fluorescence experiment is always of the same energy. Thus there is no energy dependent part of the signal measured on I_F. There is, though, an energy response to I₀ that is neglected when I_F is normalized by the signal on I₀. ATOMS calculates an approximation to this attenuation called the I₀ correction.

To enable this calculation when using ATOMS, it is necessary to specify the contents of the I₀ chamber. In ATOMS and TkATOMS this is enabled by specifying non-zero values for one or more of the fill gasses. From the supplied values of fill gases, ATOMS approximates the energy response of the I₀ chamber using the tables of free-atom x-ray cross sections.

ATOMS evaluates the cross-section for the gases in the I₀chamber in a range from 50 to 300 eV above the absorption edge. It then regresses a second order polynomial in the natural log of the energy relative to the edge to the natural log of the gas cross-section. The linear term is an approximation to the degree of attenuation introduced by normalizing I_F by I₀. This is intended as an additive correction to the measured $\sigma^2$ of a fit. Neglecting this correction will make the temperature dependence of $\sigma^2$ deviate from an Einstein behavior by a constant offset.

   
The Self-Absorption Correction

The second correction required for fluorescence measurements is called the self-absorption correction. This calculation is also enabled when fill gases are specified. Assuming^11.4 equal entry and exit angles, the formula for the fluorescence signal is given by

$$\frac{I_F}{I_0} \approx \frac{\mue}{\muf + \mub + \mue}$$ (11.1)

$\mue$ is the absorption of the resonant atom, $\mub$ is the rest of the absorption in the material, and $\muf$ is the absorption of the material at the fluorescence energy of the resonant atom. The self-absorption correction is due to the $\mue$ term in the denominator of this equation. In the limit that $\mue\gg\big(\muf+\mub\big)$, the self-absorption correction is enormous, canceling the oscillatory structure of the XAFS spectrum. ATOMS approximates the effect of $\mue$ term in the denominator of Eq. () given the assumption of infinite sample thickness, which, in practice, means that the sample is very thick compared to the absorption length.

The correction presented in this section is a correction to the measured $\chi$. A similar correct for the XANES portion of the $\mu(E)$ spectrum can be calculated, but that is not the purpose of this one. The signal $\chi$ is the normalized variation in I_F/I₀, the oscillatory part of the absorption of the resonant atom. I will now derive expressions for the variation in I_F/I₀ and for the normalization term using the notation of Eq. ().

The variation in the signal, $\delta\big(I_F/I_0\big)$ is due to the variation in $\mue$, $\delta\mue$
 $$\begin{align}\delta\Big(\frac{I_F}{I_0}\Big) =&\> \delta\Big( \frac{\mue}{\muB ... ...\notag \\ =&\> \frac{\delta\mue\cdot\muB}{\big(\muB + \mue\big)^2} \end{align}$$
In this equation $\muB=\big(\muf+\mub\big)$.

The normalization, $\Delta$, is the difference in I_F/I₀ below and above the edge. The symbols $\big\vert _a$ and $\big\vert _b$ denote that the quantity is evaluated above or below the absorption edge.

$$\Delta = \frac{\mue\big\vert _a}{\mue\big\vert _a+\muB} - \frac{\mue\big\vert _b}{\mue\big\vert _b+\muB}$$ (11.2)

For a fluorescence measurement where the self-absorption correction is a significant effect, $\mue\big\vert _a$ is a large term in Eq. (), but $\mue\big\vert _b$ is assumed to be small. I need one more bit of notation,

$$\mue\big\vert _a = \mue\big\vert _b + \Delta\mu_c.$$ (11.3)

Using the assumption that $\mue\big\vert _b$ is small compared to $\muB$,
 $$\begin{align}\Delta =&\> \frac{\mue\big\vert _b + \Delta\mu_c}{\mue\big\vert _a... ...b}{\muB} \notag \\ =&\> \frac{\Delta\mu_c}{\mue\big\vert _a+\muB} \end{align}$$

Finally I write an expression for the normalized variation in I_F/I₀ using Eqs. () and (). Since this derivation is for a correction to $\chi$, I am only concerned with normalized variation above the absorption edge.

 $$\begin{align}\frac{\delta\big({I_F}/{I_0}\big)}{\Delta} =&\> \frac{\delta\mue\... ...delta\mue\cdot\muB}{\Delta\mu_c\big(\mue+\muB\big)} \>\Bigg\vert _a \end{align}$$

The ideal measurement would be undistorted and simply expressed as $\delta\mue / \Delta\mu_c$. The correction factor $\mathcal{C}_{\mathit{self}}$ is the factor by which the measured signal must be multiplied to obtain the ideal signal.
 $$\begin{gather}\mathcal{C}_{\mathit{self}} \cdot \frac{\delta\mue\cdot\muB}{\Del... ...ue+\muB}{\muB} \notag \\ =&\> 1 + \frac{\mue}{\muB} \end{align}\end{gather}$$

ATOMS uses the free atom cross-sections to evaluate Eq. () in the energy range from 50 to 300 eV above the absorption edge^11.5. A second order polynomial in the natural log of the energy relative to the edge is regressed to the natural log of Eq. ().

The amplitude factor is the exponent of the constant term in the regression. This is intended as a multiplicative correction to the amplitude of the data. If this term is neglected in a fit, the measured S₀² will be too small by that factor. If the density of the absorbing atom is large, this factor can be huge. Indeed for pure transition metal K edges, the oscillatopry structure is damped out almost completely.

The other correction is the linear term of the regression and is expressed as a second order cumulants. This is intended as an additive corrections to the measured $\sigma^2$ of a fit. Neglecting this correction will make the temperature dependence of $\sigma^2$deviate from an Einstein behavior by a constant offset.