Christopher T. Chantler, Martin D. de Jonge,
Chanh Q. Tran, Zwi Barnea
School of Physics, University of Melbourne
Victoria 3010, AustraliaWe have applied the xray extended range technique (XERT) to measure mass attenuation coefficients up to two orders of magnitude more accurately than any previously reported in the literature. In this article we describe the application of the XERT to the investigation of systematic effects due to harmonic energy components in the xray beam, scattering and fluorescence from the absorbing sample, the bandwidth of the xray beam, and thickness variations across the absorber. The measurements are used for comparison with different calculations of mass attenuation coeffcients, and to identify particular regions where these calculations fail. Absolutely scaled data with robust error estimates in the region of the xray absorption fine structure (XAFS) can be used to improve the accuracy of XAFS analysis and can provide a rigourous test of the absolute scale of XAFS modelling. Results of the technique can determine atomic and solid state properties, including bond radii, to high accuracy.
I. INTRODUCTIONThe accuracy of xray optical constants can significantly affect the optimisation of an experimental arrangement and the interpretation of experimental results. For example, xray atomic form factors and mass attenuation coeffcients can affect the interpretation of tomographic and crystallographic experiments, and can influence the design of xray refractive lenses and zoneplates. Despite significant differences between various tabulations of mass attenuation coefficients, these tabulations are often used with little discrimination.
Fig. 1 presents the mass attenuation coeffcients of molybdenum appearing in the FFAST [1–3] and XCOM [4, 5] tabulations as raw values (top) and as a difference from the FFAST tabulated values (bottom). Significant differences between the calculated values are observed across a wide range of energies both above and below the absorption edge of molybdenum at about 20 keV, and exceed 15% at some energies. Similar discrepancies between calculated values appear to be present in all tabulations, for most elements, and across most xray energy ranges.
The results of measurements of the mass attenuation coefficients of molybdenum compiled by Hubbell et al. [6, 7] are plotted on Fig. 1.
Figure 1
Top: The mass attenuation coeffcient for molybdenum as given in the FFAST [1–3] and XCOM tabulations [4, 5], and as reported by a variety of previous experimental investigations (experimenter code referred to in the key
is as per Hubbell et al. [6, 7]).
Bottom: Discrepancies between theoretical predictions and experimental measurements presented as a percentage difference from the FFAST tabulation. The difference between the FFAST and XCOM tabulations is greater than about 4% over most of the energy range shown here, but rises to over 15% over several keV above the absorption edge. Differences of 10%–20% between measured values whose typical claimed uncertainties are about 2% indicate the presence of unrecognized systematic errors affecting these measurements.These experimental results generally claim uncertainties of between 1% and 3%. The spread of the measurements demonstrates that the measurement accuracy is much poorer than claimed. These measurements are unable to resolve differences between the FFAST and XCOM or other tabulations. Discrepancies between the results of independent investigations indicate that there are significant and undiagnosed systematic errors which have affected the accuracy of the measurements, particularly addressed in earlier years by Creagh et al.[8, 9]. This article describes our methods for investigating these particular sources of systematic error and their effects on the measured mass attenuation coefficients.
II. METHODOLOGY
Fig. 2 presents a schematic of the experimental setup that we have used to measure mass attenuation coefficients. The exact details of the experimental arrangement vary slightly in response to the operational details of the synchrotron beamline. We have used bending magnet, undulator, and elliptical multipole wiggler sources to produce a spectrum of high-brilliance x-rays. The x-ray beam is monochromated by double reflection from a monochromator, usually silicon, and preferably from planes [such as (111) or (311) ] with a ‘forbidden’ second order reflection. The monochromator is usually detuned to reduce the passage of higher-order harmonics into the beam [16, 17].
We use the XERT and probe a number of dimensions of the measurement parameterspace to determine the influence of a range of systematic effects on the measured values [10]. We make measurements under optimum conditions and continue these measurements well beyond the optimum range as, for example, represented by the Nordfors criterion. Critical examination of the nature of the breakdown of the measurement is used to identify the cause of the measurement failure, and thus to estimate the implications, if any, on measurements made under optimal conditions. This article describes our treatment of systematic errors arising from the presence of harmonic energy components, the effects of secondary radiation (scattering and fluorescence), the influence of the finite xray energy bandwidth, and from absorber thickness variations. These methods have been applied to measurements of the mass attenuation coefficients of copper [11], silicon [12], silver [13], and molybdenum [15]. While such examples can be seen to be ideal test standards, other investigations in progress investigate tin, but also compounds such as ZnSe and to glass (SiO2). The technique is quite general and can be applied to metals, crystals and glasses. Of course, synchrotron beamlines vary dramatically and therefore corresponding experimental techniques must be adapted accordingly. Again, the final accuracy attainable may be limited by specific systematics of the sample, of the
beamline, or of the time allocated by the synchrotron for the particular experiment.
Explicit tests for the effects of a wide range of systematic errors have enabled us to rigorously justify experimental accuracies of between 0.02%–0.7%.
Figure 2
Typical arrangement of the experimental components used to employ the XERT
Counting statistics have limited measurement precision in a number of reported measurements of mass attenuation coefficients [18–21]. We have used high-brilliance synchrotron sources to obtain measurements with high statistical precision. The improved statistical precision of our measurements has made it possible to detect a range of systematic effects which would otherwise not be discernible from the data.
The x-ray beam is collimated to a cross-section of approximately 1×1mm2 by the use of two orthogonal slits. An ‘upstream’ ion chamber is used to monitor the intensity of the incident beam, and a ‘downstream’ ion chamber to record the intensities of the attenuated and unattenuated beams. We use matched ion chambers, and optimize for strong positive correlations between the counts recorded in the upstream and downstream ion chambers [22, 23]. Accordingly, gas is flowed through the ion chambers in a serial configuration. We have generally recorded measurements with correlation coefficient R
0.99 between the upstream and downstream detectors, which enables us to determine the ratio of the measured intensities to high precision. A loss of correlation is a signature of poorly matched or optimised ion chambers, of excessive sample attenuation or electronic noise, or of other undiagnosed errors of the experimental technique. This precision enables us to detect the effects of systematic errors on the measurement with high sensitivity.
A number of specimens of widely differing attenuation (0.1
[ρt]
10) are used to measure the x-ray attenuation at each energy. Conventional literature in the past has investigated XAFS and related phenomena with only a single sample, and has attempted to follow the Nordfors criterion above and below the edge with the same sample. Creagh et al.[8, 9] noted that different thicknesses yield different magnitudes of systematics from particular sources. Hence a range of multiple thicknesses is generally needed for accurate determinations.
The samples are mounted on the sample stage, shown in Fig. 2, which is located mid-way between the upstream and the downstream ion chambers. The stage can be rotated about two axes and translated in two directions orthogonal to the beam. The samples are placed and replaced in the path of the beam to high precision by the use of a computer-controlled motorized driving system.
Daisy-wheels [24] are located between the sample stage and the ion chambers. These daisy-wheels have on their perimeters a series of apertures which are used to admit different amounts of secondary (fluorescent and scattered) photons into the ion chambers. In addition to these apertures, a large number of attenuating foils are mounted on the perimeter of the daisy-wheels and these can, like the apertures, be placed in the path of the beam by suitable rotation of the daisy-wheel. The thicknesses of the daisy-wheel foils are chosen to span an extremely large range of x-ray attenuations, typically with ( 0.01
III. HARMONIC COMPONENTS
When attenuation measurements are made using a monochromatic x-ray beam, the logarithm of the intensity plotted as a function of the absorber thickness t falls on a straight line whose slope is the product of the mass attenuation coefficient
ln
= -
where I and I0 are the attenuated and unattenuated intensities respectively. The productIn practice, ln(I /I0 )can be non-linear with thickness due to the presence of other spectral components in the beam. In particular, harmonic multiples of the fundamental x-ray energy may be present in the beam, especially when their intensities in the source spectrum are significant. While detuning of the monochromator crystal may suppress the propagation of these harmonic components in the beam, the residual effect on the measured attenuation may remain significant.
The relative efficiency of detection of the fundamental and of the harmonic x-rays influences the effect of any harmonic components on an attenuation experiment. For example, the ion-chamber detectors used in our work exhibit a rapid decrease in detection efficiency with increasing x-ray energy. However, the effective harmonic content, i.e., as perceived by the detector, can still be significant, as was the case in our measurement of the mass attenuation coefficient of silicon [12].
For a fraction x of harmonic x-rays (with attenuation coefficient
Fig. 3 shows the measured attenuation of eleven sets of aluminium foils (with thicknesses between 15 µm and 1 mm) in the path of an x-ray beam monochromated by a detuned, double-reflection silicon (111) channelcut monochromator set to select 5 keV x-rays. These foils were placed in the beam by suitable rotation of the daisy wheel. This technique is accurate, reproducible and rapid. This work was performed at the bending magnet beamline 20B of the Photon Factory synchrotron at Tsukuba.
Figure 3
The attenuation ln(I/I0) as a function of the thickness of aluminium absorber in the x-ray beam with a silicon monochromator set to 5 keV.
o = experimental values; solid line = curve of best fit corresponding to an admixture of
(1.09 + 0.02)% third-order harmonic (15 keV) following Equation (2)-
The experimental values follow a straight line until the thickness of aluminium increases to such an extent that the detected radiation consists overwhelmingly of the more energetic 15 keV third-order harmonic. When this occurs, one observes an inflection with the gradient approaching that of
This inflection in the plot provides clear evidence for the presence of a third-order harmonic [the (222) second order reflection for silicon is ‘forbidden’]. The solid curve in Fig. 3 is the calculated thickness dependence of the attenuation of aluminium for 5 keV x-rays with an admixture of (1.09 ± 0.02)% of the 15 keV third-order harmonic, as can be confirmed by extrapolating the second ‘linear’ portion of the graph back to zero thickness.
A minimum of three samples of accurately known thickness is required to simultaneously determine x,
In Fig. 4 we have determined the harmonic content of the beam at several energies by using three well-calibrated thicknesses of silicon. Attenuation measurements
of the foils at the harmonic energy, yielding
Figure 4
A harmonic component measurement with three well calibrated thicknesses provides a constant and reliable indicator of accuracy in attenuation measurements.
Measurements made at :
◊–5.0 keV, o–5.2 keV, □–5.4 keV, and ∆–5.6 keV.
Calculated curves for each energy pass through the measured values.
IV. SECONDARY RADIATION
The mass attenuation coefficient
We have made measurements with apertures of various diameters placed between the absorbing specimen and the ion chambers. These apertures, mounted on the daisy wheels, admit various amounts of the secondary radiation into the detectors. The secondary radiation yields a systematic change in the measured mass attenuation coefficients correlating with the aperture diameter and the sample thickness, and also varying as a function of photon energy.
Fig. 5 shows the percentage discrepancy in the measured mass attenuation coefficients of silver, comparing those obtained with a large (16 mm diameter) and medium (8 mm diameter) aperture. This figure shows that the reading recorded using the large aperture is up to 0.2% less than that measured using the medium aperture. This effect is largest immediately above the silver absorption edge at about 25.2 keV, where the fluorescent yield is greatest and the incident beam is most attenuated.
Figure 5Percentage discrepancy between the mass attenuation coefficients of silver measured using the large and medium diameter apertures. The dashed and dash-dot lines show the prediction for the 10 µm and 100 µm foils used for the measurements.
We have modelled the effect of the dominant fluorescent and Rayleigh scattered x-rays on the measured mass attenuation coefficients [25]. Our model calculates the contribution to the counts recorded in the upstream and the downstream ion chambers resulting from fluorescent radiation emitted by the absorber and from Rayleigh scattering by the absorbing material and the air-path between the ion chambers. Self-absorption corrections are applied to all secondary photons. The prediction of this model is shown on Fig. 5, and is in good agreement with the observed discrepancies. These investigations have been carried out in the course of determining the mass attenuation coefficients of silver accurate to 0.27%– 0.7%.[13].
V. X-RAY BANDWIDTH
Even the most highly monochromatic source produces a spectrum of x-rays of finite bandwidth. Typical bandwidths vary from
10-5 to
Away from absorption edges the mass attenuation coefficient varies sufficiently slowly for the bandwidth effect not to be detected. However, on the absorption edge the mass attenuation coefficient changes rapidly and the effect of the bandwidth is significant.
Fig. 6 presents the values obtained from measurements made along the molybdenum absorption edge. This work was performed on the 1-ID beamline at XOR sector 1 at the APS. The beam was produced by an undulator and was monochromated by reflection from the (3,1,1) planes of a pair of silicon crystals. The values presented in Fig. 6 are in good agreement and could be used to report ‘excellent’ x-ray absorption near-edge structure (XANES).
Figure 6
Values of the mass attenuation coefficient of molybdenum obtained from measurements in the near-edge region. At each energy measurements have been made with three thicknesses of foil, represented by three different symbols:
× — 100 µm; + — 50 µm; ◊— 25 µm.
The consistency of the experimental values is too good for the measurements at each energy to be clearly resolved on this scale. The gradient of the weighted mean of the measurements made at each energy is plotted as a dotted line, on a relative scale.However, the discrepancies between the values obtained using foils of different thickness, which cannot easily be resolved in Fig. 6, are presented in Fig. 7, and in fact lead to beam-line dependencies in reported XANES studies [14].
Figure 7
Percentage difference between the mass attenuation coefficients obtained using the thick (top) and medium (bottom) foils and that obtained using the thin foil. The prominent dip in the discrepancies occurring at about 19.995 keV coincides with the point where the gradient of the mass attenuation coefficient reaches its maximum value (c.f. Fig. 6)
This figure presents the percentage difference between the mass attenuation coefficients determined using the thick (top figure) and medium (bottom figure) foils and that determined using the thinnest foil. There is a gradient correlated discrepancy between the measured values, the magnitude of the discrepancy increasing as the thickness of the foil used to make the measurement increases. The measured mass attenuation coefficient (subscript m) is related to the beam energy profile and the ‘true’ mass attenuation coefficient (subscript t ) by [26]
exp
=
exp
dE (3)
where E0 is the central energy of the beam profile and Ĩ0 is the normalized incident beam energy profile, defined as
Ĩ0 =
(4)
We have inverted Eq. (3) under the assumption of the approximate linearity of the mass attenuation coefficient on the scale of the beam bandwidth, and have determined the bandwidth of our x-ray beam to be 1.57 eV±0.03 eV at 20 keV [26]. We have also used the linearised approach to determine a correction to the mass attenuation coefficients measured on the absorption edge of molybdenum [15].
Fig. 8 presents the correction to the mass attenuation coefficients obtained along the absorption edge, and shows that the finite x-ray bandwidth has affected the measurements by up to 1.4%. The structure shown in Fig. 8 is significant, and would be of particular interest in XAFS and XANES investigations.
Figure 8
Percentage correction to the mass attenuation coefficients measured in the neighborhood of the absorption edge and in the region of the XAFS, evaluated using the linearized approximation.
VI. THE FULL-FOIL MAPPING TECHNIQUE
In a number of recent reports [18–20, 27–30] it has been observed that, for accuracies between 0.5%–2%, a dominant source of error in the measurement of mass attenuation coefficients is the accurate determination of the sample thickness along the actual path traversed by the x-ray beam. We have developed a full-foil mapping technique for determining the mass attenuation coefficient on an absolute scale which overcomes previous limitations due to uncertainties in the sample thickness.
Traditionally the local value of the integrated column density has been determined as the product of the density and the thickness. The local thickness was determined by a variety of techniques using micrometry [11, 12, 21, 30, 31], profilometry [11], optical microscopy [32], step-profilometry [33], and x-ray scanning techniques [11, 12, 21]. Measurements of sample thickness have an advantage in that they probe the variation of the thickness across the surface of the foil. However, each of the techniques mentioned above is subject to a range of fundamental limitations affecting precision and accuracy which are difficult to overcome [11, 12, 34], which represent a major limitation on the precision and accuracy of the determination of the mass attenuation coefficient.
More recent measurements have used the areal density of the absorber, which we term the integrated column density, for the determination of the mass attenuation coefficient [11, 12, 18–20, 27, 29, 35–37]. These measurements have generally been limited to accuracies of 0.5%– 2% due to variation in the thickness, which has limited the determination of the local integrated column density of the absorbing specimen along the column actually traversed by the beam.
The Beer-Lambert equation describes the attenuation of x-rays of a given energy passing through an absorber by
- ln
where I and I0 represent the attenuated and unattenuated beam intensities respectively,
The mass attenuation coefficient of a foil absorber can be determined by measuring the attenuation at (x, y) locations to determine an attenuation profile - ln
=
=
=
(6)
where the mass m of a given area A of the foil is used to determine the average integrated column density
. The mass and area of the foil can be measured to high accuracy using well-established techniques, for example by using an optical comparator to determine area and an accurate microgram balance to measure mass. In contrast with earlier techniques, XERT can determine the mass attenuation coefficient to high accuracy without directly determining the local integrated column density at any point of the absorber.
Figure 9 shows the attenuation profile of a nominally 254-µm-thick molybdenum foil. This attenuation profile has been determined from the attenuation measurements of the sample mounted in a plastic holder. To determine the attenuation profile of the absorbing sample alone we have subtracted the small fitted holder component from the measured attenuation profile.
Using this technique we have recently determined the mass attenuation coefficients of molybdenum to an accuracy of 0.028% [38] and of silver to accuracies in the range 0.27%–0.7% [34]. Measurements of the attenuation profile of the silver foils at different energies have confirmed the reproducibility of the measurement at this high accuracy.
Figure 9
Attenuation profile of a molybdenum foil. The attenuation profile was produced from an x-ray scan of the foil mounted in a plastic holder. The small holder contribution was fitted and subtracted from these measurements. The x-ray beam used to make the measurements was 1×1 mm2 and measurements were taken at 1 mm intervals across the foil.
VII. INFORMING THEORIES OF PHOTOABSORPTION
We have measured the mass attenuation coefficients of copper [11], silicon [12], silver [13], and molybdenum [15] using various synchrotron sources. Following the principles of the XERT, measurements were made over an extended range of the measurement parameter-space, and were investigated for evidence of systematic errors. We have developed a technique to determine an accurate value of the mass attenuation coefficient from raster measurements made across the surface of an absorber. We have also detected and corrected effects resulting from a small fraction of harmonic energy components in the synchrotron beam, from fluorescent radiation produced in an absorbing specimen, and from the finite bandwidth of the x-ray beam. By applying these techniques we have improved measurement accuracies by over one order of magnitude.
Figure 10
Our measured values of the mass attenuation coefficients of molybdenum as a percentage difference from the tabulated FFAST values [1–3]. Measurement uncertainties of 0.02%–0.15% are indicated by the error bars. The spike in the difference of our values from the FFAST tabulated values occurring near the absorption edge is due to the XANES, which is not modelled by either tabulation.
Also shown are the percentage differences between the tabulated XCOM [4, 5] values and the experimental values tabulated by Hubbell et al. [6, 7], compared with FFAST.
Figure 10 presents our measured values for molybdenum compared with the FFAST tabulated values. Also shown are the XCOM calculated values and the experimental values tabulated in Hubbell et al. [6, 7], compared to the FFAST values. The trend of the percentage difference between our values and the FFAST tabulation is generally smooth to within the claimed measurement uncertainty, indicating that the uncertainties are appropriately estimated. By contrast, the point-to-point variations in the trend of individual experimental measurements tabulated by Hubbell et al. is typically no better than 1%–2% and is therefore the limiting possible precision of any of these measurements. The accuracy must necessarily be lower than this point-wise inconsistency. The larger inconsistencies between the different sets of measurements prove the magnitude of present systematic errors in those data sets. Our measurements are free from many such systematic errors because we have explicitly investigated our measurements for their presence and have proven that each has a small or negligible remaining signature in our results.
The XCOM tabulated values exhibit a large oscillation with respect to the FFAST values over the energy range from 20 keV to 30 keV or 40 keV. Oscillatory behavior in the calculated values has been observed elsewhere [1, 2] and may be the result of an incompletely converged calculation. Our measurements clearly show that the XCOM tabulation is in error in this region. Above about 40 keV the XCOM values are in good agreement with our measurements.
The FFAST tabulation estimates uncertainties – arising from calculational convergence precision and the limitations of various approximations –
at about 50% within EK
E
10% -20% within 1.001EK
3% within 1.1EK
1% for E1.2EK (EK is the K-shell absorption-edge energy).
These estimates are in accord with the differences of Fig. 10. The difference between our measurements and the FFAST tabulation is stable at about 0.5%–1% at energies above 25 keV. Below the absorption edge the measurements exhibit a more complex pattern of discrepancy, but fall between the XCOM and FFAST values. Although this supports the accuracies of the FFAST tabulation, the higher accuracy of the experimental measurements also implies a possible systematic error of this magnitude in this atomic regime. The measured values are 1%–3% higher than the FFAST tabulated values within a range of about 5 keV above the absorption edge. Although this is within the FFAST uncertainty, a similar above-edge enhancement observed for copper [11] and silver [13] suggests that the FFAST values are systematically low in this region.
The presence of this discrepancy in measurements of three elements indicates new physics in the above-edge energy region [13, 15, 39, 40]. Further experiments are required to determine whether this discrepancy is present for other elements and above other (e.g.L-shell) absorption edges. Such measurements will provide further clues which will inform future calculations of the mass attenuation coefficients. XAFS structures are solved routinely and hundreds of publications appear per annum. Limitations in theoretical predictions and XAFS analytical frameworks lead to significant uncertainty in results, impairing structural predictions and preventing ab initio determination. Our accurate measurements and robust error estimates of the attenuation of molybdenum in the above-edge region have been used to improve the XAFS determinations by between 5% and 70% [41]. A deeper understanding of the interactions between x-rays and matter requires accurate measurements so that each contributing process may be compared with theoretical models. Relative measurements provide crucial information but absolute attenuation measurements provide additional demanding tests of theory and computation.
ACKNOWLEDGEMENTS
We acknowledge the assistance of the synchrotron staff at each of the beamline facilities involved in this work, being beamline 20B of the Photon Factory in Tsukuba, and beamlines 1BM, 1ID, and 12BM of the APS. This work was supported by the Australian Synchrotron Research Program, which is funded by the Commonwealth of Australia under the Major National Research Facilities Program, and by a number of grants of the Australian Research Council. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research, under Contract No. W31109Eng38.
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