Chapter 3.13. Sample-thickness effects

The homogeneity, uniformity and thickness of the sample are crucial for collecting high-quality extended X-ray absorption fine-structure (EXAFS) and X-ray absorption near-edge structure (XANES) data. When samples are not prepared appropriately the EXAFS oscillations can be too small in amplitude and structure in the XANES can easily be distorted, and it may not be obvious from looking at the data that a problem exists. These issues are more important for transmission data because of the log function used to calculate the absorption coefficient, i.e. μ(E)t = ln(I_o/I₁), where μ(E) is the absorption coefficient at energy E, t is the sample thickness, I_o is the incident flux and I₁ is the transmitted flux. When the sample thickness varies and the incident flux is not uniform these variations can couple, which is particularly important at monochromator-induced glitches (see Section 5). For high-energy edges these errors can easily be avoided by careful preparation, but at lower energies (below roughly 5–6 keV) which require thinner samples it becomes increasingly more difficult to minimize these effects. Uniform EXAFS samples are usually made by (i) rubbing a fine powder onto Scotch tape until it is uniformly covered and then stacking tape layers to give the desired absorption or (ii) by mixing a fine sample powder (enough power to give the desired absorption) with a fine boron nitride powder until the mixture is homogeneous (which may take several minutes) and then pressing into a disk of uniform thickness.

A simple way to understand how sample non-uniformity changes both the EXAFS and the XANES data is to consider a sample with a single pinhole; a small fraction f of the initial beam is then not absorbed and passes through the sample, and the measured transmitted flux I₁ is given by When the last term f begins to be a significant fraction of the first term [(1 − f)exp(−μt)] the edge becomes distorted. Consider a XANES scan; as μ increases through an edge [and exp(−μt) decreases], the resulting decrease in I₁ is proportionally too small because of the flux through the pinhole. The larger μ becomes, the more the fractional change in I₁ becomes compressed, with a larger compression at the top of the edge but relatively little compression near the bottom. For EXAFS the oscillation amplitude will also be too small, as the top of the edge (including the EXAFS range) is the most compressed region, while the measured edge step, used for normalization, is relatively too large. More formally, defining (μ*t) = ln(I_o/I₁) as the measured value, then one can expand equation (1) for f[exp(μt) − 1] < 1 to give the ratio R = (μ*t)/(μt) asFor f = 0.01, this is a good approximation up to μt ≃ 3. As μt increases, R decreases from 1.0 at first linearly, but then more rapidly. This compresses the data at high μt, as discussed above.

Note that the thicker the sample, the faster the term fexp(μt) increases with μ and the larger the distortion. Stern & Kim (1981) have explored this effect in some detail but focused more on the measured EXAFS function (determined by small variations in ), and also the measured step height , when the background absorption below the edge is small. See also the discussion by Heald (1988).

The ratio of the measured value to the actual value of χ, i.e. , approaches 1.0 as the actual edge step height μ_et decreases to 0. Stern and Kim plot the ratios and for a range of values of f and μ_et, but do not consider the effects of a significant background from other atoms in the sample. Since many materials currently being investigated often have a significant absorption background from other atoms in the unit cell, these calculations are expanded below. The effects of this background can be large and may require that the step height be much less than 1.0 when pinholes are present.

First set μ = μ_b + μ_e in equation (1), where μ_b is the background absorption coefficient from other edges or atoms and μ_et is the absorption step at the edge of interest. The measured step height is obtained by subtracting from , and the ratio of the measured step height to the actual step height is = . In Fig. 1(a), is plotted as a function of the actual step height (μ_et) for four values of f (0.005, 0.01, 0.02 and 0.04) and two values of μ_bt (1.0 and 2.0). This ratio decreases with increasing μ_bt and with increasing f; it is clearly very dependent on the value of μ_bt, even for μ_bt = 1.0. Thus, pinholes have an even larger effect when the sample is relatively thick, and the background absorption is comparable to or larger than μ_et. It is therefore important to determine the absorption from the other atoms in the sample. Also note that the pinhole fraction is rarely reported; Lu & Stern (1983) report a fraction f of 1% for a very uniform tape sample made with a fine powder (1.2 µm).

Figure 1 The ratios (a) and (b) as a function of the actual step height (μet) for different values of the pinhole fraction f from 0.005 to 0.04 and for μbt = 1.0 (solid lines) and 2.0 (dashed lines). f increases downwards; numbers near the centre are for μbt = 1.0, while numbers to the right are for μbt = 2.0.

To obtain the experimental EXAFS oscillation function χ, Stern & Kim (1981) considered a small change, , of above the edge and defined χ as Δμ_e/μ_e and as . Then, from equation (3),

In Fig. 1(b), is plotted as a function of the actual step height for several values of f and μ_bt. Even when the step height is only 1.0 there is a significant suppression of when the pinhole leakage is a few percent, and if the background absorption μ_bt is as large as 2.0 it can be a 20% effect. It is important to note that the decrease in partially compensates the decrease in , so that the net decrease in is smaller. Stern & Kim (1981) studied the reduction of and from pinholes in their early paper but did not include the effect of the background absorption μ_b from other atoms. If μ_b is set equal to zero in equations (3) and (4) then the resulting plots agree with Fig. 1 of Stern & Kim (1981); note that Stern and Kim use an effective thickness t′ instead of an effective value for the absorption coefficient , and plot t′/t instead of . With μ_b = 0, the reduction in is much smaller; for example at μ_et = 1, is about 0.96 for f = 0.04 and μ_b = 0, while it is about 0.83 for f = 0.04 and μ_b = 2.

Since partial pinhole effects (i.e. thin sections of samples, cracks etc.) cannot be completely eliminated for powder samples, it is best not to use thick samples. Goulon et al. (1982) have also noted the important effect of leakage through the sample in their analysis of errors in the collection of EXAFS data.

Plots in Stern & Kim (1981) and in Fig. 1 provide guidance in choosing the appropriate sample thickness so that the error in is small: a few percent. When μ_bt is very small then the edge step height (μ_et) can be as large as 1 for typical pinhole fractions of a few percent; this sets a maximum thickness. [For samples with very low pinhole fractions and very uniform thickness, as in uniform metal foils, even μ_bt = 1.5 might be reasonable.] However, as μ_bt increases, the error increases for pinhole fractions of a few percent and for μ_bt = 2.0 the error in for f = 0.04 is about 15% when μ_et = 1. In this case reducing the sample thickness by a factor of two reduces both μ_bt and μ_et by 2, and the error in is less than 5%. For a larger background absorption μ_bt, μ_et must be even smaller than 1/2.

For moderately low concentrations (3–10%) there is often the temptation to make the sample thicker to increase the step height for the dilute element, but this can cause problems. In such cases, it is useful to check two different sample thicknesses to verify that the step height for the atom of interest is proportional to the sample thickness. If the samples are fairly uniform, an easy way to do this is to collect XANES data with the sample rotated at different angles relative to the X-ray beam; for example, perpendicular to the beam and at 45°. For these angles the step height should be 1.4 times larger at 45°. If it is significantly smaller then there are pinhole/uniformity problems; either a thinner sample should be used or the data should be collected using fluorescence mode.

A more general equation for I₁ when both the incident flux and sample thickness are not uniform is given bywhere I_o(x, y) and t(x, y) are the incident flux and sample thickness at point (x, y) within the X-ray beam cross section. One important example of using this equation is usually referred to as the particle-size effect (Lu & Stern, 1983). Consider a single layer of close-packed particles of diameter D. The centre of each particle will be highly absorbing if D is large, while the edges transmit significantly more flux, much like a partial pinhole. Lu & Stern (1983) calculated the transmission through such a particle and treated leakage flux between particles as discussed above for pinholes. They showed that if the particle size is too large there is a serious reduction in the EXAFS amplitude (and also distortion of the XANES, although this was not discussed). How thick is too thick depends on the `unit step height length' at the edge of interest, i.e. the thickness to produce a step height of 1.0, which increases rapidly with X-ray energy. The diameter of the particles should be much less than this length. Using nearly identical layers of FeSi₂ particles (unit step height length ∼12 µm at the Fe K edge) formed by rubbing fine particles onto Scotch tape, Lu & Stern (1983) argue that the error is very small if the absorption edge step per layer is 0.1 or less, i.e. the particle size is of the order of 0.1 of the unit step height length. This is a strong constraint on the particle size, particularly for energies below ∼5–6 keV, as decreases as the X-ray energy decreases for a fixed value of D. Note also that using small particles reduces pinhole effects between particles and using many layers further reduces pinhole effects. Thus, it is best to use very fine powders in making samples, usually <5 µm and, for low energies, down to 1 µm, although this may not be easily attainable. This is important for a large range of materials with edges below 10 keV; for example, for the lanthanide L₃ edges in compounds such as LaF₃ the unit step height length is about 6.5 µm and at the Mn K edge in LaMnO₃ the unit step height length is about 14 µm; but here the background from lanthanum would be large, roughly μ_bt = 3.0, and a thinner sample is needed. Transition metals have even shorter unit step height lengths; for example for chromium metal (K edge ≃ 6 keV) the unit step height length is about 2.9 µm at the Cr K edge and even 1 µm particles would have some particle-size effects using the criteria of Lu & Stern (1983). For such materials, uniform thin films are desirable.

However, Lu & Stern (1983) find only a small difference in the EXAFS amplitude between FeSi₂ particles of 1.2 and 4 µm, and a more reasonable constraint would be that the particle size be less than 1/3 the unit step height length, and perhaps even up to 0.5 of the unit step height length if some decrease in is tolerated (see below). Others have used similar criteria: Cao et al. (2001) used four layers of tape for the Mn K edge, with a total step height of 0.5, while Jiang et al. (2007) also used four layers of tape with a similar step height and noted that all particles on the tape had sizes of <5 µm.

Finally, since the EXAFS oscillations are small and the compression does not change much over an oscillation, the measured EXAFS is to a reasonable approximation given by = χ/α, where α is a constant greater than 1 (that accounts for particle size and small pinholes). This is effectively a reduction of the parameter by α, and likely contributes in part to variations of the reported values of in the literature for a given element.

A second example using equation (5) is for tapered samples when the X-ray flux also varies across the beam; here, the equations developed for glitches (Bridges et al., 1991, 1992) are used. Assume that there is a linear taper of thickness along the x direction and that the X-ray flux also varies in this direction; for simplicity let it be a linear variation and assume no variation along y. Then, where I_o and t_o are the average values of the incident flux and the sample thickness, β and γ are small constants describing the linear variations and the range of integration for x is (−0.5 mm, 0.5 mm), the size of the entrance slit defining the X-ray beam. Carrying out the integration under the assumption that μt_oγx is small, then I₁(E) and are approximately given byThus can be slightly smaller or larger than the value μt_o (calculated using the average thickness), depending on the signs of β and γ; if the integration range is smaller (i.e. smaller slits) then the effects are smaller. If the highest flux passes through the thickest part of the sample then the absorption is larger, whereas if the highest flux passes through the thinnest section of the sample it will be lower. The effects for are similar but the equations are somewhat more complex, particularly if the background value of μt below the edge is significant. The important point to note is that if there are large spatial variations in the flux and the sample is tapered, or more generally non-uniform, the extracted can differ from the real value. Since X-ray beams at synchrotrons can have significant variations, it is important to minimize variations in thickness. Note that for side stations there is a large horizontal variation in flux, and for focused beams there is a large intensity variation across the beam in both the x and y directions unless small slits are used and positioned appropriately.

Similar issues can occur if μ is also a function of x and y, for example as found in soil or powdered rock samples. The distribution of distinct compounds in the samples can then vary dramatically across the samples, and if the grain sizes of various compounds are significantly different this will introduce errors (Tannazi & Bunker, 2005). If the X-ray beam is also not uniform, modelling the XANES as a sum of reference edges may not give a good estimate of the relative fraction of each compound.

More complicated variations of sample thickness need to be addressed on a case-by-case basis. For example, Ottaviano et al. (1994) considered the more complex case of a distribution of metallic particles of different diameters embedded in a matrix. They estimated the particle-size distribution from microscope pictures and then modelled the transmission of this composite to extract a measure of μ. For further details, see Ottaviano et al. (1994).

The sample constraints for fluorescence measurements are much less restrictive because the log function is not involved. Uniform samples are always best, but other considerations might dictate the form of the sample. If the concentration of the element of interest is low then thick samples can be used, as the rest of the sample determines the penetration depth. If pinholes are present then the incident flux that produces fluorescence is reduced: I_o(1 − f). Also, if the incident flux were uniform across the sample then the net fluorescence would only be reduced by 1 − f and the XANES and EXAFS would not be changed. On the other hand, if the beam is not uniform and the spatial intensity distribution across the sample varies with time, this can lead to fluctuations that contribute to noise or to longer term drifts, as if the gain were changing. Usually this is not very important for unfocused beams, but could be important if a focused beam is used, the slits are not small enough to use just the uniform part of the beam and the focused beam moves slightly during a scan.

If the element of interest is relatively concentrated then self-absorption effects must be addressed and corrected (see Bridges & Booth, 2024).

For concentrated samples it is often useful to use a very thin sample (μ_et ≪ 1; see Bridges & Booth, 2024).

However, the issues discussed above when the sample is non-uniform, the beam intensity is non-uniform and/or some beam motion is present still apply.

Variations in sample thickness also play a role in the magnitude of glitches induced by additional Bragg reflections within the monochromator, usually called monochromator glitches (Bridges et al., 1991, 1992; Li et al., 1994). Here, one assumes that harmonics (at two or three times the energy of the main beam) have been sufficiently removed and only considers unfocused beams; the behaviour of focused beams depends on the focusing conditions, but the principles are the same.

At a few particular angles of the monochromator crystals during an EXAFS scan, three or more Bragg reflections can be simultaneously possible at the same X-ray energy, over a small energy range, usually a few electronvolts. (For the extra reflections the d-spacing and Bragg angles are different.) Then, when an EXAFS scan passes through such an energy (angle), some of the desired X-rays go into another reflection and are lost, reducing the incident flux in I_o over a small energy interval. In addition, because of the slight divergence of the X-ray beam from a synchrotron as it moves towards the monochromator, the energy varies slightly in the vertical direction: the variation is about 1.4 eV mm⁻¹ at 10 keV for Si(220) crystals at 20 m from the electron beam (Bridges et al., 1992) and is about 2.4 eV mm⁻¹ for Si(111) crystals; see Fig. 2 in Li et al. (1994). Consequently, if the beam profile is scanned vertically over a range of 4 mm, there will be a dip in the profile at the glitch energy (roughly 1–2 mm wide) that moves across the beam profile. An example is shown in Fig. 2; see experimental measurements on two beamlines at SSRL in Fig. 1 of Bridges et al. (1992) and Li et al. (1994).

Figure 2 Simulation of a monochromator glitch crossing a beam profile; h is the vertical direction and the beam extends from 2.5 to 6.5 mm; this simulation is based on results reported in Bridges et al. (1992) and Li et al. (1994). The glitch is a narrow dip on the profile for a vertical scan (using tiny slits of 0.05 mm) at a fixed X-ray energy. At h = 5 mm, the slope of the beam within a vertical slit of say 1 mm will change from zero, to negative, to zero (the middle of the glitch), to positive and back to zero as the energy is increased through the glitch.

To see how this produces the glitch in an EXAFS scan consider equation (6) but with x now the vertical direction, h. As this dip moves across the slits (that define the beam), the parameter β changes with increasing energy, from nearly zero to negative to positive and back to nearly zero (or vice versa depending on the beamline configuration). The correction term in equation (7), βγ/12, then fluctuates, producing the glitch. If the sample is very uniform (i.e. γ, which describes any taper, is tiny) this effect becomes small; in addition, the glitch size can also be significantly reduced by using narrow slits (Bridges et al., 1992). More complex situations can be modelled in a similar way. For focused beams the position and shape of the glitch on the profile need to be determined.

Several of the issues discussed above (pinholes, thickness effects and glitches), together with beam inhomogeneity and/or beam motion, can contribute significantly to apparent noise in an EXAFS scan. All of these effects are minimized by using very homogeneous/uniform samples, so it is worth the extra effort to prepare such samples.

Assume that the samples are not uniform in some way: pinholes are present, the sample is tapered or more generally the sample varies in thickness across the X-ray beam. Also recognize that all X-ray beams are not uniform to some extent and that intensity variations across the beam can vary slowly with energy. Li et al. (1994) observed small variations with energy in glitch studies (see Figs. 1–3 in Li et al., 1994) but did not comment further; these features are likely to arise from variations in the reflectivity of the monochromator crystals (or mirrors) as the beam moves along the crystal in an X-ray scan. In energy scans, these variations will couple with sample non-uniformity to produce (repeatable) noise-like variations in the transmitted intensity. Beam motion can also produce variations in transmitted flux; if, for example, the highest intensity of the beam shifts towards (away from) a pinhole from one data point to the next, the I₁ intensity will increase (decrease) and this will appear as noise in the scan. A similar variation occurs if the highest beam intensity moves on and off a larger powder particle.

The formation of large glitches, when both the beam and sample are not uniform, was discussed in Section 5; however, there are many more tiny glitches that usually are not important. See, for example, Fig. 2 in Li et al. (1994); there are five tiny glitches between 9850 and 9910 eV that would contribute to the apparent noise.

Because the EXAFS amplitude in k-space decreases at high k, both from the decrease of the backscattering amplitude with k as well as from thermal vibrations, these effects become pronounced at high k and it is then critical to have a very uniform sample to obtain good EXAFS data, typically above k = 12–15 Å⁻¹.