On the accuracy of the L-subshell ionization cross sections for proton impact I. Spectrum fitting

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Nuclear Instruments

and Methods in Physics Research B 114 (1996) 225-23 1

NOMB

Beam Interactions with Materials&Atoms

ELSEVIER

On the accuracy of the L-subshell ionization cross sections for proton impact I. Spectrum fitting T. Papp a3* , J.L. Campbell b a Institute of Nuclear Research of the Hungarian Academy of Sciences (ATOMKI),P.O.Box 51, H-4001 Debrecen, Hungary b Guelph-Waterloo Program for Graduate Work in Physics, University of Guelph, Guelph, Ontario, Canada NIG 2Wi Received 30 September

1995; revised form received

1.5 December

1995

Abstract The L X-ray spectrum of thorium following 1 MeV proton impact has been measured using a Si(Li) detector_ The spectrum was analysed by various commonly used procedures. It is shown that the separation of the L-y group into X-ray transitions of L 1 and L, subshells is affected by the natural lineshape model. The common practice of neglecting the natural lineshape (Lorentzian or other shapes where there are many-body effects) introduces systematic errors which are larger than the frequently-cited error bars. In the example used here, the effects of neglecting natural lineshape are similar in size to the difference in theoretical L ,/L, and L,/L, subshell ionization cross section ratios predicted for protons by the ECPSSR and coupled channel calculations. Similar systematic errors will arise in other contexts. The optimum accuracy inherent in Si(Li) spectroscopy will only be realised if full account is taken of natural lineshapes in the treatment of the X-ray spectra. PACS: 32.30 Rj; 34.5O.Fa; 32.80.Hd

1. Introduction There are numerous data for the L 1, L, and L, subshell ionization cross sections [I] for proton impact due to both fundamental interest and potential applicability in materials analysis. These data were obtained by analyzing $e L shell X-ray spectra collected by Si(Li) detectors. The ycertainty estimated in any one determination of the gubshell ionization cross sections can be as low as a few Ejercent. Such high accuracy might reasonably be expected, Since X-ray spectroscopy is a mature field with long tiadition. But, on the other hand, the data overall show large scatter, amounting to more than 25% for the L subshells [l-5]. In order to derive L subshell ionization cross sections from the directly measured X-ray production cross sections, a set of atomic parameters, viz. Coster-Kronig bans&ion probabilities, fluorescence yields, total and partial decay widths are used. Although various sets of these atomic parameters are available, the fluorescence yields and Coster-Kronig transition probabilities of Krause [6], &d the X-ray transition rates of Scofield [7] (Hartree-Fock

* Corresponding

author.

instead of the more advanced Dirac-Fock) are most commonly used. Some authors [8,9] work in the opposite direction, assuming a particular set of the subshell ionization cross-sections to be accurate, and deriving fluorescence yields and/or Coster-Kronig transition probabilities. The present situation as regards the most widely studied collision system, i.e. proton impact on a gold target, is very interesting. From the subshell ionization cross sections of this collision it was concluded [lo] that there is clear evidence for the existence of an L subshell coupling effect. In contrast, another report [I I] claimed that the ECPSSR theory provided a completely satisfactory description of this system. There are various ways to approach the problem of widely scattered cross-section values, a typical one being to average the experimental data [l]. Collection of the reported measured data is a very important activity; however, it is unlikely that the large scatter of the L-subshell ionization cross section can be attributed to statistics. If this scatter of the experimental data is caused by sources other than statistics, it would be desirable to find out the particular experimental problems that cause different results, and thereby remove any subjectivity in preselecting the data for averaging.

0168-583X/%/$15.00 Copyright 0 1996 Elsevier Science B.V. All rights reserved PfI SOl68-583X(96)00144-9

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Instr. and Meth. in Phys. Res. B 114 (1996) 225-231

We plan to investigate the problem of large scatter of experimental data by examining the various steps used to arrive at the results. The first problem which we discuss is the physical nature of the line shapes of X-ray transitions and their role in spectrum analysis. Our aim was to study the role of the natural line shape (Lorentzian) of the X-ray transition in the separation of background and the various peaks. This is important in any absolute or relative measurement. We have measured a thorium L X-ray spectrum with a Si(Li) detector following ionization by 1 MeV proton impact. The spectrum was analyzed in various ways.

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2. Line shape of X-ray transitions It is customary to assume in X-ray spectroscopy carried out with Si(Li) detectors that the width of the detector response function (which is approximately Gaussian) is so much larger than the natural width of the X-ray transitions that there is no need for consideration of any natural line shape effect. The natural line shape for the majority of X-ray transitions between inner shells has Lorentzian form. The Lorentzian falls off much more slowly than the Gaussian with increasing energy separation from the peak centroid. Thus, although the Lorentzian’s full width at halfmaximum (FWHM) is much less than the Gaussian FWHM, the Lorentzian broadening will eventually exceed the Gaussian broadening at some energy separation, provided that the background is of low intensity and that the statistical precision of the spectrum is high. The energy resolution of Si(Li) detectors has attained the level where the Lorentzian broadening has visible effects for L and M shell transitions. Even if the applied detector has moderate resolution the Lorentzian wings give important contributions to the background under overlapping peaks. It is common to determine the L 1, L, and L, subshell ionization cross sections from the La, L y , and LY,,,,~ line intensities, following the pioneering work of Datz et al. [ 121. The Lorentzian tail of the Ly, peak protrudes under the LY,,,,~ peaks and it is not clear how much the background subtraction is affected by the smoothly varying overlapping Lorentzian wings. This question will be studied in a later section. The Lorentzian line shape is an approximate form and deviations may be expected when the initial or final states of the transition cannot be treated as a stationary state. Well-documented examples are the 4f elements. The spectrum contains configuration interaction satellites instead of a single Lorentzian. Fig. 1 shows a typical example for the L,N,,, X-rays of rare earth elements measured by high resolution spectroscopy and reported by Ohno and LaVilla 2131; this spectrum is a close approximation to the true energy distribution of the photons. The Ly, peak is expected at an energy of 7.4668 keV and the Ly, at 7.4867 keV, after Bearden [14]. The Ly,,, (L,N,, L,N,) transi-

7462

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(eV)

Fig. 1. The 2s-4p region of X-ray transitions of samarium, measured with a double crystal spectrometer. After Ref. [ 131with permission from the author.

tions strongly deviate from the Lorentzian distribution, as a result of the so called many-electron effect. For example, for Nd (Z = 60), similarly to Sm, only 50-60% of the original strength of the L y, transition corresponds to the relaxed 4p3/2 state. The remaining strength is distributed over a 40-50 eV range of energies. The strength of this broad structure is so large that the Ly, transition can hardly be recognized in the spectrum. This unusual line shape will obviously affect the separation of the Ly group into individual lines. The usual analysis of the Ly group by fitting with a group of Gaussians or Lorentzian-Gaussian convolutes is obviously not appropriate here. In cases such as this it is difficult to derive the relative intensities of L, and L, X-rays in the spectrum from a Si(Li) detector, and of course the matter is complicated further by the fact that the ratio depends upon the energy of the proton responsible for the ionization. The degree of misfit is therefore likely to depend on the projectile energy. General correction factors cannot be established since the obtained results depend on the detector resolution and the detector response function model [15]. The importance of the many-electron effect in the analysis of xenon L shell X-rays measured with a Si(Li) detector is demonstrated in Ref. [ 161.

3. Experiment Our aim was to establish the range of systematic errors of the L-subshell X-ray yields caused by the neglect of the natural line shape. For this case study a target element should be selected where line shape effects other than the Lorentzian broadening are minimal. We have found thorium (Z = 90) to be an excellent choice. The detector response function is best-suited to the energy range of the Th X-rays, because the tailing on the low energy side of

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Instr. and Meth. in Phys. Res. B 114 (1996) 225-231

the peaks due to incomplete charge collection in the detector is minimal. There will be M-shell spectator vacancy satellite effects upon lineshape of L, X-rays beCoster-Kronig process is energeticause the L,L,M,,, cally possible for atomic numbers above Z = 79; however the L,L,M,,, Coster-Kronig channel is not open below Z = 92 [17]. We have used 1 MeV proton impact where the double ionization to single ionization cross section ratio is less than 1% [ 181. The 1 MeV proton beam, supplied by the Van de Graaff accelerator at Guelph, was collimated to 1 mm and had 2.5 X 10m3 md angular divergence. The target was a 1 mg/cm* thick self-supporting metal foil of thorium and it was placed in the scattering chamber reported in Ref. [19]. To reduce the possible pile-up with the intense M-shell X-rays, an 0.5 mm thick Mylar absorber was placed between the target and detector. The Si(Li) detector, manufactured by Link Analytical plc. had 133 eV resolution at 5.9 keV, 32 mm2 active area, and 3 mm nominal thickness. A collimator was placed in front of the detector to restrict the illuminated detector area to 1 mm in diameter. The aim was to use only the central area of the detector to reduce the area of incomplete charge collection. The target to detector distance was 6.5 cm. The counting rate was around 1200 counts per second, and it was stable within 10%.

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4. Data evaluation and results The various energy-dispersive X-ray spectra were fitted using a non-linear least-squares method [20-221, in which the variables to be determined by the fit included the peak heights, the parameters of a background having either linear or quadratic energy-dependence, the two parameters of a linear energy versus channel calibration, and the two parameters governing the relationship between detector resolution width and X-my energy. In such gies

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Table 1 Binding energies of thorium (in keV) [24-261. Only those states are presented, which substantionally differ in Refs. [24] and [25]

R,(6p3/2) p,(6pl/2>

P,(6s l/2) O&d5/2) 0&5d3/2) 0,(5p3/2) 0,(5pl/2> 0,(5sl/2) N&4p3/2)

Bearden and Burr [25]

Bancroft et al. 1261

Sevier

0.043 0.049 0.060 0.088 0.095 0.182 0.229 0.290 0.968

0.0166 0.0245 0.0414 0.0854 0.0925 0.1772 0.234

0.0173 0.0258 0.0414 0.0873 0.0941 0.1808 0.232 0.290 0.967

0.966

Fig.2.AfittotheM,,M,andpartoftheM,X-raygroupofthe M spectrum of thorium. The spectrum was measured with a Si(Li) detector (see text). In the fit the X-ray transitions were modeled with Lorentzians. This Lorentzian line shape was convoluted with the detector response function which included a Gaussian and a tail function at the low energy side of the peak. The energy values of Bearden and Burr [25] were used. The residuals arc the differences between the measured and fitted spectra expressed in units of one standard deviation.

1241

energy values given by Bearden [14] was not successful. There have been suggestions that the energy versus channel calibration is not a linear function, implying that a quadratic term needs to be added [23]. Such an exercise did indeed improve the fit. However, when the M-shell spectrum was also included in the analysis, acceptable results could not be achieved, whichever type of energy calibration was adopted. There is a more up to date compilation of binding energies given by Sevier [24]. In Table 1 we have presented three different sets of binding energy data for thorium [24-261. In Fig. 2 the 3.80-5.6 keV energy range of the M-shell spectrum of Th is shown, with a fit. In this fit the X-ray energy values of Bearden [14], Bearden and Burr [25] were assumed, and the transitions had natural line shapes that were described as Lorentzian; this X-ray spectrum was convoluted with the detector response function. The energy calibration was

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ITISW.and Meth. in Phys. Rex B I14 (1996) 225-231

linear. The detector response function contained a Gaussian, an escape peak, and the detector tailing which was modeled as a Hypermet function [23] with adjustable parameters; this function is simply an exponential tail which decreases in intensity towards lower energies from a maximum at the corresponding Gaussian centroid; it is convoluted with a unit-area Gaussian to reflect the broadening and smoothing effect of the detector response function. Obviously this fit could not be considered acceptable. In Fig. 3 the same fit is shown, with the energy values obtained from [24] instead of [ 14,251. There is a dramatic improvement in the reduced chi-square (from 153 to 3.6), and also in the residuals. The residual is defined as the difference between the data and the fit, normalised to one standard deviation. This result is not surprising since for example the X-ray energy of the Ml-P,,, transition is expected to be 5.136 keV after Bearden and Burr [25] and to be 5.161 keV after Sevier [24]. When we let the energy values be free parameters of the fit, we have obtained 5.163 keV for this transition. This does not mean that the energy values are best determined by Si(Li) detectors, but it proves the importance of the proper data base. It also suggests that in cases where a quadratic energy calibration is needed to provide a good fit, the energy values are

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Fig. 4. L X-ray spectrum of thorium target bombarded by 1 MeV protons and measured with a Si(Li) detector at 57.5” observation angle. The continuous line corresponds to the best fit. The peaks were represented by Lorentzians and convoluted with the detector response function, which included a Gaussian and an exponential tailing component. A quadratic background was assumed and it is shown by the dashed line.

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Fig. 3. ‘Ihe same as Fig. 2, except the transition energies taken from Sevier [24] instead of Bearden and Burr [25].

were

inaccurate; when the correct energy values are available, a linear calibration suffices. Based on this experience we have used the energy values of Sevier [24] for the analysis of the L-shell spectrum, although we were aware of possible many-body effects as reported in Ref. [27]. The best fits are presented in Fig. 4 and 5. For the fit presented in Fig. 4, we have used a background whose energy-dependence was quadratic. The heights of the peaks were allowed to vary, except in the cases of the Lp and Ly groups, where strong overlap appears between numerous peaks. In these cases, the intensities of the weaker lines were locked to those of major lines in the same subshell series, using the tbeoretical intensity ratios [28] corrected for relative detector efficiency and transmission through the Mylar foils. In the X-ray energy regime that is relevant here, these corrections are very small, and do not introduce any significant error into the results. The relative intensities of the La X-rays of the Le group except the L& transition were locked to the L& transition. Any effect of anisotropic angular distribution is not expected [29], since the spectrum was measured

T. Papp, J.L. Campbell/Nucl.

Instr. and Meth. in Phys. Rex B I14 (1996) 225-231

229

Table 2 Lorentziau width values [30] used to tit the thorium L X-ray spectra Transition

Widths (eV)

L3Ml (LI) L3M2 (L t) L3M3 (L s) L3M4 (La 1) L3M.5 (La 2) L2Ml (Lq) L3Nl (Li36) L3N4,5 (Lp15,2) LlM2 (Lp4) LlM3 (Ll33) L2M4 (LPI) LIM4,5 (LJS10,9) L2N4(Lyl) LlN2 (Ly2)

24.2 21.9 15.1 10.8 10.7 24.7 18.5 11.5 29.1 22.3 11.2 18.0 11.9 18.7

l-1

I

I

10

15

20

Fig. 5. The analog of Fig. 4, except all the peaks had unrcalistitally small Lorentziau width values, 0.01 eV. A comparison with Fig. 4 demonstrates the importance of the natural lineshape in the analysis of X-ray spectra.

at 57.5“. The intensity of the L,M, transition was locked to the L,M,. In the Ly group the intensities of the L,-0,, L,-O,, L,-P, lines were kept fixed in the above manner relative to the L,-N, transition, while the L ,-N,, L i-N4,:, L,-0,, Li-Os, L,-P,,s were fixed relative to the principal L,-N, line of that series. The X-my transitions were assumed to have Lorentzian line shape, the width values being taken from an extensive survey of experimental X-my linewidths, which includes a set of recommended values [30]; the widths used here are given in Table 2. The natural line shape. was convoluted with the detector response function. The height and slope parameters of the exponential tail component of the detector response function were obtained from K X-ray L-X-ray coincidence measurements using a 233Pa source as described in [31]. For this detector in the lo-20 keV energy range the tailing due to incomplete charge collection is very small. There are small deviations on each side of the Lp,,, transitions, which we attribute to the inaccuracies of the energy and Lorentzian width values. As is visible in Fig. 4, an excellent fit was achieved. We will refer to this fit as Fit 1. The first row of Table 3 presents the peak intensities from this

fit. The separation of the Ly group into X-rays of the L, and L, subshells is crucial in the derivation of L-subshell ionization cross sections. In Table 3, therefore, we present the sum of the X-ray yields for those transitions which originate from the L, subshell and fall into the Ly range; (L,(Y) = L,-N,, L,-N,, L,-N,,,, L,-O,, L,-0,, L,P2,s) this quantity is denoted by L i(r) in the last column of the table. The same approach is used to obtain the analogous quantity L,(y); (Lz(y) = Lz-N,, L,-N,, L,-0,, L,-o,, L,-Pi). During various analyses of X-ray spectra we have observed that the result is very sensitive to the Lorentzian broadening [32]. A clear demonstration is given in Fig. 5. The continuous curve essentially is the same as in Fig. 4, with all the parameters of the fit except the Lorentzian widths being the same. The Lorentzian widths were set to a negligibly small value, viz. 0.01 eV, instead of the values

Table 3 The results of the various fitting approaches (see text) in integrated peak counts Fit

1 1

2 3 4 5 6 7

Ll 1280432

Lo 20088730

Lrl 147776

1.oo 0.93 0.94 1.oo 0.90 1.oo 0.95

1.00 0.96 0.98 1.00 0.97 1.00 0.97

1.00 0.93 1.05 0.91 0.83 0.88 0.83

,I1

LPS5 402474

LPZH 4709844

L,(Y) 1049980

L Jr) 172018

1.oo

1.oo 0.94 0.94 0.93 0.89 0.98 0.92

1.0 0.94 0.97 0.97 0.93 0.99 0.94

1.0 0.97 1.05 1.02 0.93 1.06 0.99

0.94

1.oo 0.89 0.82 0.94 0.87

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Instr. and Meth. in Phys. Rex B 114 (1996) 225-231

shown in Table 2. The importance of the Lorentzian broadening is obvious. If we permit the fitting code [21] to iterate, it pulls up the quadratic background, and the “missing” Lorentzian wings are accounted for by the background, resulting in smaller peak areas. In order to present a simple and clear comparison of the results of this and other fits with the best fit shown in Fig. 4, the intensities of the various peaks and peak groups have been normalised to unity in the second row of the table. Subsequent rows present the peak areas from subsequent fits, each expressed as a multiple of the unit value that results from tit number 1. The fit with the Lorentzian values fixed at 0.01 eV is designated as fit number 2 in the table; all the peak intensities are significantly smaller than the results of fit number 1. If the physically well-justified Lorentzian line shape is not used, there are various other, empirical, ways to account for the missing Lorentzian wings and improve the fit. The chi square would certainly be improved if a parabola is put under each peak [33]. The detector tailing function may be increased in relative intensity, and in a complex spectrum this can also improve the chi square, although the result does not mean that the data extracted from such a fit are superior. The result in this case may be better if the tail area is added to the Gaussian peak area. We can test this possibility with a fit where the background is fixed at the same size as in Fit 1, the Lorentzian is neglected, and the parameters of the tails are variables of the fit. The results of this third fit show that some of the peak areas are indeed improved but others are obviously over-compensated. There is in general an unsolved problem in the analysis of X-ray spectra, namely the shape of the background. In the present measurement we had a small background. Indeed as can be seen in Fig. 6, the background between the main peaks is smaller than the Lorentzian wing contribution. The results described so far were obtained using a quadratic polynomial background. Another approach is the method of digital filtering [21,22]. We have repeated the analysis using a top-hat digital filter to remove the background, but using the same values as in Fit 1 for the detector tailing and for Lorentzian widths. This fit (number 4) gives values close to those of Fit 1 for the well separated peaks, such as Ll, La, while it underestimates the components and the total areas of the L@ and Ly complexes and the weak Lr) line. When we repeat this fit neglecting Lorentzian lineshape, significantly lower values are generated (fit number 5 in Table 2). The general practice in analyzing L-shell X-ray spectra is to fit only part of the spectrum [34], e.g. the Lp or the L-y group, presumably because this makes it easier to find an empirical background expression that gives a good fit. We have also carried out fits with this approach, separating the spectrum into three ranges. The first range covered the L I- L (Y peaks, incorporating approximately flat areas on both sides. The second and third ranges were the Lp and

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F f 8 103

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Fig. 6. Details pertinent to fitting the Ly region of Th only. The short dashed line is the fitted quadratic background for that fit. The long dashed line shows the result of the full-spectrum fit of Fig. 4 but with the fitted Ly peaks removed. The Lorentzian tails of the Lp transitions are responsible for one third of the background under the Ly peak.

groups respectively. In the first of these fits (number 6) the Lorentzian widths of Table 2 were used, and in the second (fit number 7) the Lorentzian widths were set to 0.01 eV. The small value for the intensity of L, X-rays in the Ly group can be understood on the basis of Fig. 6. The short dashed line corresponds to the quadratic polynomial background generated in the fit to the third region i.e. the Ly region. The long dashed line was obtained by the following manipulation of Fit 1, i.e. of the best full-spectrum fit: the L y peaks were removed from the fitted spectrum; the overall continuum “background” in the Ly region was then taken as the sum of the quadratic polynomial background and all the X-ray transitions with their Lorentzian line shape (except the L-y lines). It is noteworthy that the contribution of the Lorentzians of the Lp group under the Ly group is one third of the overall background. However, this area should be attributed to the LB X-rays and not to the background. When only the Ly range is fitted, the fitted background comes out as a parabola pointing upwards. The parameters of the fitted background are basically determined by the shape of the decaying Lorentzian curves on both sides of the Ly range. This will result in an underestimation of the peak areas. This analysis shows that error is incurred by isolating individual portions of the spectrum and then attributing simple background expressions to these; the most accurate result occurs when the full spectrum is fitted and the Lorentzian broadening is included in full. Ly

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Instr. and Meth. in Phys. Res. B 114 (1996) 225-231

5. Conclusion We have analyzed a thorium L X-ray spectrum in various ways. We have concluded that the natural line shape (Lorentzian, or other shapes in special cases where there are many-body effects> must be included in the analysis if the most accurate results are desired. The errors in L ,, L, and L, subshell ionization cross sections that arise from neglect of Lorentzian lineshape are about 14%, 8% and 4%, respectively. For the derivation of the subshell ionization cross sections from the X-ray production cross sections, the fluorescence yields and Coster-Kronig factors of Chen et al. 1351 were used. For the L,/Ls and LJL, cross section ratios, which are commonly compared with the theoretical calculations, the under-estimates of peak intensities partly cancel out, but errors of 10% and 5% respectively persist in the ratios. It is worthwhile to note that comparison of ECPSSR [36] and coupled channel calculation predictions for the cross section ratios in lead [37] shows a difference of 10% except in the adiabatic region, where it reaches 20-30% at 0.2 MeV proton energy. It follows that conventional fitting approaches used to date are in error to much the same degree as the size of the effect that is of physical interest. The inaccuracies in the fluorescence yields and Caster-Kronig transition probabilities are a source of additional uncertainty. Of course, the involvement of Lorentzian broadening and other line shape elements (such as configuration interaction satellites) introduces additional parameters into the spectrum analysis. The use of tabulated Lorentzian widths might introduce model dependence, since the experimental Lorentzian widths have large scatter and deviate from the theoretical values in some cases [30]. The line shape effect is expected to be even more important in the case of multiply ionized atoms [38]. The line shape effect on the line intensities and consequently on any physical parameters subsequently determined is larger than frequently claimed experimental limits of error. The line shape can also be important in trace element analysis when the fit is based on x2 minimization, since the line shape of the major peaks may dominate the x2. Acknowledgements The current work was supported in part by the Natural Sciences and Engineering Research Council of Canada through an operating grant for ILC. TP acknowledges financial support from the Hungarian Research Fund under contract number T-016636.

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