Chapter 3.47. Projected roughness in X-ray absorption spectroscopy

The mass attenuation coefficient reflects the transition probability of a simple atomic system interacting with an incident X-ray. In the 0.1–100 keV range, the energy dependence of the mass attenuation coefficient is strongly influenced by the absorption edge from the detailed configuration of the atomic orbitals and the local chemical environment. Accurate measurement is used to provide tests of atomic theory and calculation methods, and data ranging from absolutely scaled to relative can probe the orbital density of states via X-ray absorption near-edge structure (XANES) and the local chemical environment via extended X-ray absorption fine structure (EXAFS).

We will show that the projected roughness affects local oscillations, amplitudes and X-ray absorption fine structure (XAFS). As such, it affects both relative accuracy and precision and is relevant for any high-accuracy measurement. Since the projected roughness can impact local (XAFS) structure and amplitudes by 2% and more (Glover et al., 2009), the effect is important if these amplitudes are used for principal component analysis or related techniques or for XAFS fitting of paths within some Hanning window. The effect of projected roughness is substantial for the pre-edge and the first few XAFS oscillations; however, if one excludes these from the Hanning window then the effect of projected roughness may sometimes not affect the fitting parameters.

In this chapter, we concentrate on the effect of the variation in local projected thickness (i.e. the projected roughness) in X-ray transmission across the sample at a lateral length scale that is below the size of the probe beam and show that this can have a detrimental impact on both absolute and relatively scaled measurements, with particular caution to measurements in the tender X-ray regime.

In order to advance these arguments, we generalize the concept of roughness as a measure of surface variations into the projected specimen domain to include variations in the integrated column density. Projected roughness can be divided into surface roughness, as perhaps measured by a profilometer or scanning electron microscope, destructively or nondestructively, and internal `roughness', such as at an interfacial layer, in a multilayer, at domain or phase boundaries, or density variations resulting from internal vacuoles or deposition packing mechanics. Projected roughness hence includes such complexity of origin in both measurement and theory. X-ray transmission techniques enable us to measure and investigate both surface and internal roughness. X-ray absorption is affected by any change in composition or density and hence can be used to observe and measure any variation in integrated column density.

Several articles in this volume relate to thickness effects and hence to this general area of inquiry, and we point to these chapters for a variety of important details and comments (Bunker, 2024a,b; Bridges, 2024; De Panfilis & Bardelli, 2024). An excellent text (Bunker, 2010) discusses thickness effects (Chapter 3.6.5, pp. 87–88), including the existence of roughness and sample non-uniformity. In this chapter, we are not interested so much in how to mount powders and solids (Bunker, 2024a), which might yield larger or smaller uniformity, heterogeneity and loss of stress, nor in a general discussion of samples for XAS (Bunker, 2024a), nor in the complex variety of thickness effects (Bridges, 2024), nor directly in the loss of spectral information and amplitudes from bandwidth and divergence (Bunker, 2024b), all of which are important, but in the diagnosis and mapping of the consequences of sample heterogeneity across the X-ray beam. That is, this chapter is concerned with observing, diagnosing and correcting the impact on the near-edge structure resulting from the projected roughness of the sample.

This chapter presents a definition of the projected roughness, investigations of the effect of the projected roughness on measurements of X-ray mass attenuation coefficients, accurate physical models of this effect, illustration through two case studies and a discussion of particular scenarios in which the effect will be significant. We illustrate through measurements mass attenuation coefficients [μ/ρ] which show unique signatures due to projected roughness. In addition to correcting a systematic error in the determination of the mass attenuation coefficient, the study defines and provides nondestructive techniques for determining the magnitude of the projected roughness.

The term roughness is generally understood as relating to the root-mean-square (r.m.s.) variations in the surface height. The roughness distribution can likewise be understood in this context as relating to the distinct form of the surface variations: triangular, Gaussian, top hat etc. Thus, when a projected measurement is undertaken, the usual statistical mathematics are applied to the roughness that comes from the front and rear surface. However, in a transmission measurement, in the absence of significant refraction or diffraction through the specimen, the integrated column density [ρt] is the relevant parameter (Chantler et al., 2004; Bunker, 2010): Here, is the path of a single ray of an X-ray beam for normal incidence, through a point at (x, y) on the specimen, with z parallel to the beam axis. ρ(r) is the density of the sample at the point r = (x, y, z). The square brackets in equation (1) are used to signify that the integrated column density [ρt] is a single object, as distinct from the product of density ρ and thickness t (de Jonge et al., 2004a). We treat the mass attenuation coefficient [μ/ρ] similarly. Whilst ρ(r) is far from experimentally accessible, and t is normally accessible in a macroscopic sense, [ρt] is far more experimentally accessible and is easily determined for foils using , where m is the mass and A is the perpendicular area. We have shown elsewhere (de Jonge et al., 2004b) that [ρt](x, y) can be determined by combining with X-ray absorption measurements made across the entire surface of a foil. With this definition, we will drop the positional reference for the integrated column density, so that [ρt] ≡ [ρt](x, y).

When the integrated column density is used, the concept of roughness needs to be generalized from a surface-only parameter to the projected bulk. Interestingly, we have found no established quantity for this, and have considered terms such as `bulk roughness' and `the r.m.s. of the projected density', but find none particularly satisfying. On the other hand, the term `roughness' is often used with an exclusive intent as `surface roughness' and too easily hides the scope and importance of the new parameter. Accordingly, we here adopt the term `projected roughness' (symbol σ_[ρt]) to distinguish it from the surface roughness σ_t, and define it using the usual convention of r.m.s. deviations, where is a finite region of the specimen with area A.

According to equation (4), the apparent integrated column density obtained by using attenuation measurements with a known [μ/ρ] is smaller than the linearly averaged integrated column density within the beam footprint by the amount . Equivalently, when the integrated column density [ρt] within the beam footprint is determined by linear averaging (for example, by using ), measurements of the attenuation coefficients are overestimated by this factor. Hence, the linearly averaged column density should not be used in the Beer–Lambert relation to determine mass attenuation coefficients.

Equation (5) tells us that the fractional impact of the projected roughness scales with [μ/ρ] and so is not constant across an X-ray spectrum. The fractional impact increases with the mass attenuation coefficient. As a result, the impact cannot be removed by scaling the spectrum data, and so roughness impacts both relative and absolute measurements alike. Specifically, for principal component analysis (PCA), linear combination fitting (LCF) and fingerprinting studies, samples measured with different roughnesses and with different attenuations cannot be compared beyond the level of the impact of the projected roughness. Spectral distortion due to projected roughness will be falsely accommodated by other components that are not actually present in a specimen. For EXAFS methods, spectral distortion will result in peak suppression that will also distort estimates of the Debye–Waller factor, bond length and coordination number, also compromising these methods.

The impact of the disturbance on XAS methods may depend further on the method employed. XANES measurements, by their nature, probe a substantial range of [μ/ρ] and so will be strongly affected. EXAFS measurements, as well as being critically sensitive to above-edge oscillations, may also rely on below-edge and edge-step normalizations, which can still result in significant distortion of data from different specimens.

Absolute data will always be impacted by projected roughness effects, in accordance with equations (4) and (5). However, use of the X-ray extended-range technique (Chantler et al., 2001) to provide measurements over a wide range of [μ/ρ] can enable the clear identification and correction of the effect. When measurements are made at a single energy (i.e. with a single value of [μ/ρ]) it is impossible to resolve between an incorrect value of [ρt] (equation 6) and the effect of projected roughness (equation 5). As a result, the absolute measurement will suffer an unidentifiable, and hence uncorrectable, systematic error.

What is the typical magnitude of the effect? This is difficult to answer in the abstract. However, we consider the case of a relatively scaled measurement across an absorption edge. We consider a specimen selected with negative log attenuation of 2, which is the Nordfors recommended ideal for a single XAFS measurement (Nordfors, 1960). Then, −ln(I/I₀) = [μ/ρ][ρt] ≃ 2 and equation (5) becomesIn this situation, at a single energy, the fractional effect is proportional to the square of the fractional projected roughness, and a 10% projected roughness results in a 1% effect on the mass attenuation coefficient, independent of all other parameters.

At present, we are aware of three nominally equivalent methods for determining σ_[ρt] and a correction to the mass attenuation coefficients:

(i) Track the systematic discrepancy in [μ/ρ] between thin and thick specimens as a function of the X-ray energy E (as in de Jonge et al., 2005, 2007; Glover et al., 2009, 2010); model the impact along with [ρt] errors and apply corrections to minimize the systematic discrepancies.

(ii) Plot Δ_[μ/ρ] versus [μ/ρ]; fit the quadratic and linear terms to determine σ_[ρt] and [ρt] simultaneously (Ekanayake et al., 2021b).

(iii) Plot Δ_[μ/ρ]/[μ/ρ] versus [μ/ρ]; fit the linear and offset terms to determine σ_[ρt] and [ρt] simultaneously (not yet demonstrated).

Early precise studies showed micro-structure from profilometry for copper in the range 200–600 nm for samples of thickness from 10 to 100 µm, with structure at lateral length scales of 10–100 µm. No strong roughness effect was explicitly diagnosed in the measurements of copper (8.85–20 keV; Chantler et al., 2001) or silicon (5–20 keV; Tran et al., 2003) despite these strong profilometry signatures, but it is likely that the signature of the effect was overwhelmed by other measurement uncertainties.

Follow-up investigations of projected roughness for molybdenum (13.5–41.5 keV; de Jonge et al., 2005) and tin (29–60 keV; de Jonge et al., 2007) found no observable roughness. In hindsight, the absence of the effect was due to the measurement with large thicknesses (25–250 µm for molybdenum; 25–500 µm for tin) and relatively smooth surfaces. Nonetheless, rolling defects and structural thickness changes of order 100 nm were observed using full-foil mapping with steps of 1 mm across the surface (de Jonge et al., 2004b). The anticipated projected roughness signatures for these data sets were well below the measurement precision.

Projected roughness was first observed and quantified by Glover et al. (2009) and corrections were applied in Glover et al. (2010) and Ekanayake et al. (2021a). In all cases, observation was only possible due to use of the XERT (Chantler et al., 2001) by probing an extended range of attenuation and energy space throughout the measurement. Here we outline the approach and findings of Glover et al. (2009) and Ekanayake et al. (2021a).

In Glover et al. (2009), projected roughness was measured for thin gold foils of nominal thickness 5 µm. Equation (5) relates σ_[ρt], [μ/ρ] and [ρt] to the change in measured mass attenuation coefficient (Δ_[μ/ρ]) for the thinner specimen. We approximate Δ_[μ/ρ] by comparing it with measurement from a thicker specimen, noting that the impact of the projected roughness is reduced in proportion to the thickness, as discussed in Section 4.

The magnitude of the integrated column density and the projected roughness in the sample were fitted as [ρt] = 10.287 ± 0.007 mg cm⁻² and σ_[ρt] = 1.276 ± 0.033 mg cm⁻², respectively. The best-fit model of the roughness discrepancy is compared with the experimental discrepancy in Fig 1. The agreement between the two is excellent. Note that equation (6) shows that an error in [ρt] would result in a constant fractional change to [μ/ρ] that is independent of the value of [μ/ρ], and so would only impact the vertical location of this curve if a correction for the projected roughness was not included.

Figure 1 A comparison of the output of the roughness model (orange line) with the experimental discrepancy (diamonds with error bars), here defined as the fractional difference in the determined value of [μ/ρ] between the nominal 5 µm sample and the thicker samples. The roughness is thus determined to be σ[ρt] = 1.276 mg cm−2, around 12.4% of [ρt].

The fractional projected roughness, i.e. the ratio of the projected roughness to the integrated column density across the area of the X-ray beam, was σ_[ρt]/[ρt] = 0.124, or 12.4%. If we assume that the gold foil has a uniform density of 19.30 g cm⁻³, this corresponds to a thickness equivalent of t = [ρt]/ρ_nominal = 5.330 ± 0.004 µm and a thickness equivalent roughness of σ_t = σ_[ρt]/ρ_nominal = 661 ± 17 nm. The impact on the determined [μ/ρ] was up to 1.4%, as shown in Fig. 1; this impact can be applied as a correction to the measured values to improve their precision and overall accuracy.

It is interesting to also consider the approach of Ekanayake et al. (2021a), in which the projected roughness was characterized and used to correct the measurement of mass attenuation coefficients for zinc between 8.51 and 11.59 keV, spanning the K-shell absorption edge. Fig. 2 shows the difference between the measured value for a nominal 10 µm foil and that of a thicker 50 µm foil. The 50 µm foil can be used as a zero reference for roughness in this case as the impact of the roughness is one fifth that of the 10 µm foil, presuming similar levels of σ_[ρt] in both foils.

Figure 2 Difference between the measured values of the mass attenuation coefficient Δ[μ/ρ] for the 10 µm foil and the 50 µm foil as a function of mass attenuation coefficient. Projected roughness is shown here as a quadratic variation with [μ/ρ]; incorrect estimation of integrated column density [ρt] would show as a linear variation. It is clear that the two effects are strongly correlated, and one must be careful to probe a suitable range of [μ/ρ]. This fit determines the projected roughness to be σ[ρt] = 0.99 ± 0.004 mg cm−2, corresponding to a surprising σt = 1.392 ± 0.006 µm for this nominal 10 µm thick foil. The red linear trend is discussed in the text.

Fig. 2 includes a red line that indicates the approximate result of a fit with roughness fixed at σ_[ρt] = 0. This demonstrates the measurement space that needs to be probed. The linear fit might seem quite reasonable, except that (i) it misses the clear second-order curvature term and (ii) it determines an incorrect value of [ρt], and hence an incorrect value of [μ/ρ], thereby resulting in a substantial offset in the determined values. Indeed, the [ρt] error that results from ignoring the effect of projected roughness can be estimated by equating the two contributions (equations 5 and 6), Here, the fractional error in the integrated column density also provides the fractional error in [μ/ρ], as per equation (6). The impact of the correlation depends on the range of [μ/ρ] that is probed, and so we estimate the impact by using parameters from Fig. 2 (namely [μ/ρ] ≃ 150 cm² g⁻¹, which is the overall range of [μ/ρ] probed by the data, and σ_[ρt] = 0.99 mg cm⁻²) to give an error in the determined absolute value of [μ/ρ] of the order of 1%. To be clear: this is the error in the determined value of [μ/ρ] when the effect of the projected roughness is ignored for this nominally 10 µm thick specimen. Relatively scaled data probing a similar range of [μ/ρ] as might appear across an absorption edge would also encounter spectral distortions at this level.

This analysis shows how easy it is to overlook the effect of projected roughness, despite the fact that the effect is substantial: here over 2.5% in the high-attenuation region across the K-shell absorption edge for this zinc foil. At this amplitude, the effect will suppress peak amplitudes affecting the fitted XAS parameters and the amplitude-reduction factor .

Projected roughness is present in any solid sample and in the form (shape) of the mount containing liquid samples, and can affect absorption measurements by several percent. Techniques have been developed to measure and investigate these and are recommended for accurate measurement. This approach has the ability to probe surface roughness and is the first technique that can investigate `internal' roughness. It is nondestructive and works equally well with large and small measurement areas. It can be particularly important for the determination of edge and near-edge and pre-edge features, where roughness has the greatest impact. When fingerprinting using LCF or PCA methods, the change in shape and structure can misidentify components, local structures, oxidation states or geometries.

However, the proposed technique is not `standard'. It requires multiple samples (Chantler, 2024c) and preferably a reference standard (i.e. a typically thicker, low-roughness foil reference; Chantler, 2024d). It requires the careful measurement of statistics and dark current and blank measurements (Chantler, 2024e). It requires measurement across significant ranges of mass attenuation coefficient, so is most effective either across an absorption edge or XANES region, or at lower X-ray energies subject to the X-ray penetrating through the sample. A beamline could be optimized for such measurement down to a small level of roughness and this could be at an XAS, X-ray fluorescence microscopy or soft or tender X-ray beamline. Indeed, it could be used as a dedicated resource supplemental to adjacent beamlines. Current results show that the projected roughness is observable, measurable and significant at third-generation XAS beamlines. Combined with normal profilometry or other surface measurements, and combined with X-ray mapping of samples to observe rolling structure and structure across the sample, this technique is able to observe and measure the effect of projected roughness inside the X-ray beam footprint on a sample. X-ray absorption projected roughness measurements show great potential.