Fluorescence yields and Coster–Kronig probabilities for the atomic L subshells. Part II: The L1 subshell revisited

Atomic Data and Nuclear Data Tables 95 (2009) 115–124

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Fluorescence yields and Coster–Kronig probabilities for the atomic L subshells. Part II: The L1 subshell revisited J.L. Campbell * Guelph-Waterloo Physics Institute, University of Guelph, Guelph, Ont., Canada N1G 2W1

a r t i c l e

i n f o

Article history: Available online 5 October 2008

a b s t r a c t Our recently recommended values for the L1 subshell fluorescence yield x1 and Coster–Kronig probabilities f13 and f12 in the atomic number range 64 6 Z 6 92 are re-assessed in the light of new experimental data. Special attention is paid to the regions of atomic number in which discontinuities arise due to the onset of L1L2N1, L1L3M4, and L1L3M5 transitions. Attention is drawn to large scatter and to systematic differences in the data from different experimental techniques, both of which result in large uncertainties being attached to the recommended values. The urgent need for additional refined measurements is emphasized. Ó 2008 Elsevier Inc. All rights reserved.

* Fax: +1 519 836 9967. E-mail address: [email protected] 0092-640X/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.adt.2008.08.002

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Contents 1. 2. 3. 4. 5. 6. 7. 8.

Introduction: The previous review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review and selection of f13 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review and selection of f12 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review and selection of x1 data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Consistency of recommendations for x1, f13 and f12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explanation of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tables 1. Experimental fluorescence yield and Coster–Kronig probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Recommended fluorescence yield and Coster–Kronig probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction: The previous review In a recent critical review [1], the present author examined all published experimental measurements of the fluorescence and Coster–Kronig probabilities for the atomic L1 subshell; the selection of ‘‘recommended values” relied in addition on the most recent comprehensive set of theoretical calculations of these quantities. The work [2] employed radiative and non-radiative transition rates computed in the Dirac–Hartree–Slater (DHS) model. Some of the published experimental data were re-analyzed with values for assumed quantities brought up-to-date. The recommendations were limited to the atomic number region 64 6 Z 6 96 due to the paucity of experimental data at lower values. Excellent agreement was observed between the DHS predictions of Chen et al. [2] and the overall trend of 42 experimental data points for the L3 fluorescence yield x3. The recommended values for the L2–L3 Coster–Kronig probability f23 lay a few percent below the DHS values, and were in better agreement with graphical results of Chen’s more recent Dirac–Fock approach [3] including exchange (DFEX). Experimental values for x2 were then deduced from 32 values of the measurable quantity t2 = x2 + x3 f23 by substituting the fitted experimental values of x3 and f23 in this expression. Again, the results were so close to the DHS trend that we were able to adopt the latter as our recommended values for x2. This situation for the L2 and L3 fluorescence yields mirrors that for the K fluorescence yield, where the DHS predictions [4] agree well with Bambynek’s semi-empirical fit to measured data, the results of which are presented by Hubbell et al. [5]. In view of this encouraging situation for the L2 and L3 subshells, earlier theoretical compilations, such as the extensive work of McGuire [6] using the non-relativistic Herman–Skillman approach, were not discussed. The x3 values given by that calculation begin to exceed those of Ref. [2] around Z = 65, the difference reaching 10% at Z = 90; the x2 values are in better agreement, the difference being just 5% at Z = 90. For the L1 subshell the situation was much less straightforward, reflecting the smaller quantity of published experimental data, the rejection of some of these data via preset criteria, and the very large scatter inherent in the remaining accepted data. It is known from the general behavior of L1 widths [7] that the DHS predictions for L1 Coster–Kronig rates are not accurate, and there is, as yet, no theoretical work on these rates using the DFEX approach. (Nor do we know of systematic work with a multi-configurational Dirac– Fock approach) The situation was particularly difficult near the atomic numbers 75 and 78 at which the intense L1L3M4 and L1L3M5 Coster–Kronig transitions are predicted to become energetically possible [8]; Ref. [1] provided only a single experimental

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data point (f13) at Z = 75 and none at Z = 76. While the L1–L3 Coster–Kronig probability f13 showed a steep rise above Z = 74, no second discontinuity was discernible above Z = 77. For x1, the scatter of the data entirely masked these two discontinuities. For these reasons, the only one of the expected L1L3M4, L1L3M5 major discontinuities that we felt justified in building into our plots for f13 and x1 was the first one in the case of f13. Additional experimental data have been published since our review. In addition to introducing these new data, we shall also reevaluate here some of the data which were utilized in our previous work. The data concerned are given in Table 1. Our approach will enable us to refine our understanding of the situation and to probe the different outcomes of different experimental methods. We shall then revise our recommendations for these important but poorly known and poorly understood L1 subshell quantities. This attempt at refinement is justified by the important role that is played by the L subshell fluorescence and Coster–Kronig yields in the interpretation of experiments that attempt to measure the inner-shell ionization cross sections of electrons, photons and light ions (H, He) by recording L X-ray spectra. However, a second objective for the present work is to lay out clearly the problems in the available database and to stress the resulting need for new experimental data of high quality. 2. Experimental data Ref. [1] contained a full list of experimental data, a subset of which was used to develop our recommended values. This subset is now supplemented in two ways. First, data published since Ref. [1] are added. Second, a small group of older data points which were not used in reaching recommendations are now used. Most of the new data [9,10] are from a single research group which has previously published data from a photoionization (PI) method that employs the 59-keV c-rays from an intense 241Am radioNuclide source; Ref. [11] is an example. The c-rays from this source are first used to excite the entire L X-ray spectrum of the element of interest. Then they are used to fluoresce a secondary target selected to have characteristic K X-rays that can in turn only photoionize the L3 subshell of the chosen element. The f13 Coster–Kronig probability is determined by comparing the La X-ray intensities in these two energy-dispersed spectra; careful attention is paid by the authors to several experimentally measured corrections. Values have to be assumed for the Coster–Kronig parameters f12 and f23, but the uncertainties in these quantities do not cause significant uncertainty in the resulting value of f13. It is also necessary to assume various ratios between theoretical photoelectric cross sections [12] for the three subshells at the two

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excitation energies. An extension of this method using additional secondary excitation energies permits determination of the three subshell fluorescence yields; here again values of the photoelectric cross sections must be assumed from theory. We include one new radioNuclide measurement [13]. This is the first experiment where spectator-vacancy satellites have been included in the data analysis. In addition, it is only the second experiment in the field that has, in the data analysis, represented X-ray spectra rigorously by using Voigtian lineshapes, which is superior to the customary Gaussian approximation which ignores the natural Lorentzian profile of any X-ray line. We also include here several Coster–Kronig probability measurements that were listed in Ref. [1] but were not used in reaching our recommendations. These are all work of the synchrotron radiation group at the University of Cordoba. These were excluded before because they omit an electron correlation correction that was introduced in other synchrotron photoionization work. The decision to include them was taken as a matter of balance; we believe that further work on the corrections is necessary to clarify which, if either, is the appropriate method. 3. General approach Our approach, as before, will be to deduce best fits to the selected experimental data. We recognize that if considerable scatter, beyond statistical uncertainties, is present within a given data set, this approach will mix good and bad data and therefore may not generate a ‘‘correct” result. But as long as unexplained systematic errors are present we discern no option but this one, and we will compensate for its weakness by taking a conservative approach to uncertainty estimation. This process will be guided in a qualitative sense by the DHS predictions [2] for the L1 subshell. The predicted values for x1, F13 and a1 are shown in Fig. 1. Here F13 is the sum of the total L1 Coster–Kronig probability, i.e.,

F 13 ¼ f12 þ f13

ð1Þ

and the Auger probability is

a1 ¼ 1  ðx1 þ F 13 Þ

ð2Þ

Given the known problems with L1 level widths [7], there is, of course, no expectation or requirement that experimental data and recommended values should conform quantitatively to these predictions. But there is an expectation that overall trends of Z-dependence should be followed. As a first example, the data would be expected to exhibit discontinuities, analogous to those predicted

Fig. 1. Predictions of the DHS approach for the L1 subshell de-excitation probabilities. The small L1L2N1 discontinuity and the two larger L1L3M5 and L1L3M4 discontinuities are marked from left to right.

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at the closure of the L1L2N1 Coster–Kronig channels and at the opening of the L1L3M4 and L1L3M5 channels; the atomic numbers where these ‘‘jumps” are expected from Ref. [2] are marked on Fig. 1; there are other predicted discontinuities (L1L2N2, L1L2N3, L1L2N4, and L1L2N5) but these are very small and unlikely to be observable given the uncertainties of the available data. We shall attempt to incorporate the expected discontinuities in our recommended values, but there will be situations where either the absence or the scatter of data renders this impossible. (In our earlier work [1] the lack of data for f13 made it impossible to delineate the discontinuities in that quantity.) Second, with the exceptions of the discontinuities, the overall L1 Coster–Kronig probability would be expected to decrease with increasing kinetic energy of the ejected electron, i.e., with increasing atomic number. And finally, the a1 values derived from our values for the fluorescence and Coster–Kronig probabilities should follow qualitatively the same overall trend as the DHS predictions for a1. 4. Review and selection of f13 data The measured values that were used in the original review [1] are plotted together with the new data in Fig. 2, which indicates the experimental technique that is the source for each data point. Details of these techniques were given in [1] and possible systematic errors were noted. The data based upon photoionization measurements using synchrotron radiation come from just two research groups (see [1]); data from the first, a German collaboration, are denoted PISR/LX/GC, and the second data set, from the University of Cordoba, is denoted PISR/LX/UC. The new data based upon photoionization using radioNuclide c-ray emitters all come from one research group and are denoted PI(PG + SK) in this discussion; in Ref. [1] these were further subdivided depending on whether one or both of the primary (PIPG) and secondary (PISK) excitation methods discussed above were employed. The data points arising from L X-ray spectroscopy with thin radioNuclide sources and Si(Li) detectors are here collectively denoted ‘‘Nuclide” for simplicity, but were separated into subgroups in [1] according to variants of that methodology. In the region from Z = 63 up to the expected discontinuity at Z = 74, there are four Nuclide data points from quite different approaches, seven PISR points from two groups, and three points from the PI(PG + SK) method. The Nuclide point at Z = 64 is from

Fig. 2. L1 Coster–Kronig probability f13 versus atomic number Z. The marker between Z = 67 and 68 indicates the L1L2N1 discontinuity, which cannot be resolved by this dataset. Refer to Explanation of Tables for the abbreviations.

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one of the most recent such experiments; the treatment of spectrum fitting issues was much more rigorous than is normal in this area and the error estimate was ±5%; the Nuclide point at Z = 70 is from the only reported conversion-electron X-ray coincidence experiment, where there is the rare ability to resolve the L1-vacancy marker signals. There is fair agreement among the Nuclide data and both sets of PISR results, with one clear exception in the UC subset of the latter, which we have therefore classified as anomalous. Some of the PISR/UC results carry very large uncertainty estimates—as high as ±20%, much larger than the ±3% estimates of the PISR/GC data and the typical 5–10% estimates of the Nuclide data. The data are fit well empirically by a rising function in this small region, but such a behavior is qualitatively at variance with the DHS prediction [2], which suggests essentially flat behavior both above and below a small jump between Z = 67 and 68, an effect which presumably reflects the closure of the L1L2N1 channel causing a drop in f12 and a concomitant increase in f13; Coster–Kronig energy calculations by Chen et al. [8] using a Hartree–Fock–Slater code place this closure between Z = 69 and 70. The f13 data cannot distinguish between these two options. The DHS calculation [2] has the f13 value increasing by a factor 1.05 at the discontinuity. If we were simply to adopt the average values of experimental f13 data in the regions 64 6 Z 6 67 and 69 6 Z 6 74, the jump factor would be 1.15. There is too much scatter in the data for us to make an objective overall recommendation that emulates the DHS prediction of flat behavior of f13 above and below Z  69. Our task would be easier if the f12 data (see next section) showed a concomitant, opposite transition between two well-defined flat regions, but this is clearly not the case. We are, therefore, forced to resort to an empirical, rising linear fit as proposed at the outset of this paragraph; this fit must be regarded as a temporary measure until better data that clearly delineate the discontinuity become available. Given the quoted errors and the actual spread of the data, we believe that an uncertainty estimate of 15% is appropriate in this region. Fortunately, we now have a single experimental datum from PI(PG + SK) at Z = 75 (although no data exist at Z = 76), and this clearly shows a sharp rise relative to the data at lower atomic number, an effect which must be attributed to the onset of L1L3M5 transitions. The Nuclide value at Z = 77 is far from accord with the two PI data points there, and we now can reject it as anomalous. The discontinuity that is experimentally established here for f13 confirms that there should be a similar but opposite effect in x1. It is not possible with the given f13 data to observe any discontinuity arising from the closure of the L1L2N2 channel at Z = 76. There are two data points at Z = 77 and two at Z = 78; in each case we simply adopt the mean of these. We then set the values at Z = 75 and 76 equal to that at 77. There is a quite well-defined second jump between Z = 77 and 78, presumably due to the onset of L1L3M4 transitions. Rather than defining the upper edge of the jump as the mean value at Z = 78, we shall delay this decision until after a consideration of the entire cluster of data in the 79 6 Z 6 83 region. We now consider the region 79 6 Z 6 83. The Nuclide point at Z = 81 appears anomalously low, and because the x1 and f23 values from this experiment are also anomalous with respect to the recommended values [1], we believe it can be set aside. The Nuclide point at Z = 82 is one of several measurements made upon lead over many years by the group in Atlanta. These results show a large variation and so we feel that the appropriate choice is their final result, which emerged after significant refinement of their measurements and data analysis; the value shown in Fig. 1 was revised in Ref. [1]. The experimental data overall now visually suggest at first glance a continued, but slow rise, and in Ref. [1] we had accepted this as the recommended behavior. However, the expected trend predicted by the DHS calculations is a slow fall; the only significant deviation from this might be a small discontinuity due to the

closure of the L1L2N3 channel, which would cause a drop in f12 and a concomitant rise in f13. Both Ref. [2] and the Coster–Kronig energy tabulation of Chen et al. [8] suggest that the L1L2N3 channel will close between Z = 80 and 81. The predicted change in the value of f13 at this discontinuity is only about 1%. Across the narrow region 79 6 Z 6 83 the DHS calculation suggests an overall change in f13 of only about 2%; however, the experimental data range from 0.57 to 0.66, and the authors’ estimates of uncertainty range (with one exception) from 1.5% to 7.6%. We decided, therefore, to set a single recommended value at Z = 81 as the median of the nine data points. We built in a very small jump between Z = 81 and 82, quantitatively as per the DHS prediction. It then remains to decide upon the recommended Z-dependence in the broader 79 6 Z 6 92 region. In the region Z = 90 and beyond, there are only two PI(PG + SK) points and a Nuclide point. The Nuclide datum at Z = 96 was derived using assumed values of x2, x3 and f23 that are in disagreement with the recommended values [1]. Moreover, if we take the measured x1 and f13 values for Z = 96, and add our own recommended f12 value, we deduce a value of 0.035 for the L1 Auger probability a1. For comparison with this, an independent value of the Auger probability can be predicted directly as the ratio of the Auger width to the total L1 level width. The former quantity was computed in the DHS approach by Chen et al. [2], and a value for the latter can be taken from recent tables of recommended values [7]; the result is a1 = 0.14. The difference between this expected value and that given by the Nuclide experiment at Z = 96 is large in relative terms, and hence we choose to exclude this Z = 96 datum from further consideration. There remain only the two PI(PG + SK) points, with no evidence from other methods to support or to dispute them. The situation then is difficult. We have a single recommended f13 datum at Z = 81. Theory compels us to define a downward trend for F13 (omitting any L1L2N3 discontinuity) as in Fig. 1. The final behavior of F13 will be discussed in Section 7 below after the quantities x1 and f12 have been reviewed. For now, if we were to demand that the slope of such an F13 curve be equal to that of the DHS prediction, this would force our f13 curve to fall well below the lower extrema of uncertainty of the PI data points at Z = 90 and 92. We have therefore compromised by defining a trend for f13 which intersects the lower extrema of these data. We do not find this outcome particularly satisfactory; but we do not see how the situation can be improved until new measurements clarify the situation around Z = 81 and the situation around Z = 92. Returning to the f13 value at Z = 78, this falls well below the overall trend which we have just established for the region 79 6 Z 6 92. Using multi-configuration Dirac–Fock intermediate coupling calculations for the L2-L3M4,5 transitions, Chen [3] demonstrated for the f23 discontinuities at Z = 90 and 92 that the f23 values there are very strongly influenced by fine structure splitting in the two-hole configuration. This effect might explain why our f13 value at Z = 78 lies lower than the expectation. Taking Z = 79 as defining the L1L3M5 jump, the L1L3M5/L1L3M4 jump ratio is 1.14, as opposed to the DHS prediction of 3.4. Our estimates of uncertainty for this region are in Table A of Appendix A. 5. Review and selection of f12 data The measured values that were used in the review [1] are plotted together with the new data in Fig. 3, which indicates the experimental technique that is the source for each data point. Details of these techniques were given in [1] and possible systematic errors were noted. Typical error bars for PI results are given for selected cases; all error estimates are included for Nuclide results. The markedly different trends of the Nuclide and the PISR/LX/GC data in the region 77 6 Z 6 83 are noteworthy.

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Fig. 3. L1 Coster–Kronig probability f12 versus atomic number Z. The markers above the Z axis indicate the positions of the expected L1L2N1, L1L3M5 and L1L3M4 discontinuities, which cannot be resolved by this dataset. Refer to Explanation of Tables for the abbreviations.

In our previous paper [1], an attempt was made to reflect expected discontinuities in f12 when generating recommended values. The DHS calculation predicts a 17% drop in f12 between Z = 67 and 68 due to closure of the L1L2N1 Coster–Kronig channel, while the calculation of Ref. [8] puts this discontinuity between Z = 69 and 70. This is a much larger relative effect than the concomitant 5% change in f13, and so our initial hope was that we would be able to attempt the incorporation of the L1L2N1 discontinuity in our recommended f12 curve. The large scatter of the experimental data points rendered this impossible and forced us to resort to an empirical downward trend. This trend is consistent with our upward empirical trend for f13 (see previous section), but, as in that case, must be regarded as a temporary measure until data of higher quality become available. Further drops in f12 are expected when the strong L1L3M5 and L1L3M4 channels open at Z = 75 and Z = 78, respectively. However, as was the case with x1, we have no f12 data points at Z = 75 and 76; and then the data above Z = 77 again exhibit a large scatter. Therefore, we made no attempt to fit the region of the discontinuities. The fit to the region below the discontinuities was done by eye; the fit to the region above began with a weighted linear fit which was then adjusted to introduce modest curvature. This approach provides an indication, admittedly somewhat subjective, of the existence of the major discontinuities, while being entirely unable to delineate them. In Section 3 above, we mentioned the DHS prediction of a small jump in the f13 value at Z = 82, due to the closure of the L1L2N3 channel, and so one expects a corresponding drop in the f12 values here; but the scatter of the existing data precludes any attempt to reflect this effect in our recommended values. As regards the remaining discontinuities (L1L2N4 and L1L2N5), the only practical option is the conservative one of plotting the best visual fit to our selected data, and avoiding any temptation to estimate the magnitude of discontinuities. The percentage uncertainty of our recommendations increases with atomic number as shown in Table A of Appendix A. 6. Review and selection of x1 data The measured values that were used in the review [1] are plotted together with new data in Fig. 4, which indicates the experimental technique that is the source for each data point. Details of these techniques were given in [1] and possible systematic

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Fig. 4. L1 subshell fluorescence yield x1 versus atomic number Z. The markers above the Z axis indicate the positions of the expected L1L2N1, L1L3M5, and L1L3M4 discontinuities, none of which can be depicted with confidence in this figure. Refer to Explanation of Tables for the abbreviations.

errors were noted. The data points do not specifically delineate the two L1L3M4,5 discontinuities at Z = 75 (onset of L1L3M5 transitions) and at Z = 78 (onset of L1L3M4 transitions), which must occur, given the previously noted behavior of f13. However, the juxtaposition of a rising trend up to Z = 74 followed by a cluster of lower data points beyond Z = 78 does provide qualitative evidence for one or more reductions in the value of x1 in the region of the two L1L3M4,5 edges. As in the case of f13, there are two independent sets of synchrotron data. The PISR/UC data have typical error estimates of ±15%, while the PISR/GC data have typical error estimates of 5–6%. The two sets are in good agreement, except for one high-lying point at Z = 73. We consider first the region of Z below the predicted discontinuity at Z = 74–75, and compare the combined PISR trend with 5 Nuclide data points. Three of the Nuclide results are in acceptable agreement with the PISR trend; these are from the same experiments that produced the Nuclide points used in the f13 analysis given above. The 1970-era Nuclide datum at Z = 65 lies very high and its small error estimate (±9%) precludes agreement with the overall trend. The Nuclide result at Z = 63 carries a 33% error estimate and so sits at the limit of agreement with the trend that we have selected on the basis of the PISR data point and the three consistent Nuclide data points. Therefore, we have assigned the Nuclide points at Z = 63 and 65 as anomalous. The PISR/GC datum at Z = 65 was in error in Ref. [1] and has been corrected here. The PISR/GC point at Z = 73 is anomalously high with respect to the trend of all other PISR data points. The agreement of the Nuclide datum at Z = 73 with the two PISR/GC points at 72 and 74 provides further justification for classifying the Z = 73 PISR/GC datum as anomalous. The trend below Z = 75 is thus set by nine data points out of the original 12, and our original approach [1] was simply to perform a linear fit to these. However, bearing in mind the discontinuity between Z = 67 and 68 (or between Z = 68 and 69) that is predicted by the DHS calculations, with a 10% increase in value, we finally fitted this region in two separate segments. The resulting small discontinuity fits better between Z = 68 and 69 and its magnitude is equal to the value predicted by the DHS method. We have insufficient experimental information to define the expected discontinuity due to the onset of L1L3M5 transitions; we have an estimate from our fit of the final value of x1 (at Z = 74) on the low atomic number side, but we have no experimental data at

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Z = 75 or 76. In this situation, we must refrain from making any estimates or recommendations at these atomic numbers. Above Z = 77 the second sharp decrease in x1 (due to L1L3M4) should appear. At Z = 77, we have one PISR/GC datum (f13 = 0.145) together with one Nuclide point which has very large associated uncertainty. This particular Nuclide experiment yields an f13 value that is anomalously low, which suggests that its x1 value is likely to be anomalously large. Therefore, we have excluded it from further consideration. This leaves us with only one point at Z = 77 to define the lower edge of the L1L3M4 discontinuity. Above Z = 77, we have several data points, but these fail to provide one consistent trend over all methods. The Nuclide point at Z = 82 is the most recent point measured for lead by the group in Atlanta. At Z = 81 there is very close agreement between a Nuclide point and a new PI(PG) point [10], while an early Atlanta Nuclide point which has been revised by the present author [1] lies very low and is classified here as anomalous; this point was erroneously plotted at Z = 80 in our previous paper. This leaves us with five Nuclide data points. There are three PI(PG + SK) data points, having error estimates of about 68%, which form a consistent trend, but this trend lies much lower than that of the three PISR points. Of the five Nuclide points, two favor the PI(PG + SK) trend and the remainder sit between the two PI trends, leaving the difference in the two PI trends unresolved. The figure shows our best estimate of the overall trend of the data at high atomic numbers. The very large systematic difference between the two PI methods is a matter of significance. Two concerns can be raised regarding the PI(PG + SK) method. First, the authors rely entirely on a single set of theoretical photoelectric cross sections [12] for the L subshells; it is difficult to estimate the uncertainty contributed by this assumption. However, an alternate set of cross sections is available within the Rayleigh Scattering Database (RTAB) [14]. Taking a hypothetical f13 determination as an example, the critical quantity is the ratio between photoelectric cross sections at an energy of 59.5 keV and at an energy about 1 keV above the L3 edge. We observe only a 1% difference in this ratio if we replace the Scofield [12] cross sections by the RTAB cross sections. The second issue is that when the secondary excitation energy is chosen very close to the L3 edge energy, the photoelectric cross section can depart from the independent particle model prediction [12,14] due to electron correlation effects, as demonstrated in Ref. [15]; but while this issue is not addressed in the PI(PG + SK) work, the effect appears to be only of the order of 1–2%. Thus we do not have an explanation for the large discrepancy observed in the results of the two methods in this region of atomic number. Looking at the upper [PISR] and lower [PI(PG + SK)] trend lines of this region, relative to the fit, and at the uncertainties attached to the original data points, we are compelled to suggest that an uncertainty on the order of ±30% should be associated with our best-fit recommendations in the range 78 6 Z 6 85, and ±35% beyond that value.

after publication of Ref. [7] offers us the possibility to replace all the older data and thereby avoid the circularity inherent in any reliance on the widths deduced from the PISR/LX/GC data. The Auger yields thus predicted exist for the seven Z values for which new XES measurements were done and are shown in Table B of Appendix A; in fact additional XES data points were taken [17] but these were in a region of atomic number where non-lifetime broadening effects were present, therefore, they are not useful here. If, for these seven Z values, we combine the deduced a1 values with the recommended f12, f13 and x1 values, we have a consistency test, insofar, as this sum should be equal to unity. Two obvious caveats must be stated. First, the uncertainty limits render this a semiquantitative test at best; and second, agreement with unity could be the spurious effect of one recommended parameter being too high and one too low. Nevertheless, examination of the overall Z-dependence of any agreement or disagreement may prove to be useful. There is good agreement, within the considerable uncertainties, above Z = 78, but below Z = 78 our sum of decay parameters falls below unity. It is important to note that our recommended values represent nothing more than our attempt at a best fit to selected data points within the constraints of the physics involved, and this in some cases involves a compromise between the very different trends of the data from different experimental methods. The test described above suggests that our compromise may be wrong for at least the 70 < Z < 75 region; alternately the XES level widths and the PISR data are not consistent. Finally we show in Fig. 5 our recommended values for x1, F13 and a1. Comparing Figs. 1 and 5, we see that the recommended values qualitatively follow the Z-dependent trends of the DHS predictions, although the absolute values of the quantities are often quite different. 8. Discussion It is clear from the foregoing that both new measurements and new calculations are urgently needed on L1-vacancy de-excitation. More accurate data are required for all three of the quantities studied here in order to delineate the expected L1L2N1 discontinuity. For x1, work at atomic numbers close to the L1L3M5 discontinuity is crucial since we have no acceptable data points there from Ref. [1]. At the L1L3M4 discontinuity new data are needed to better define the jump and to resolve the discrepancy among the various methods at higher Z values. For f13, we do have one data point at the first of these two discontinuities but more are needed; beyond the second discontinuity, the scatter of data needs to be greatly

7. Consistency of recommendations for x1, f13 and f12 We can probe the consistency of our proposed best values (Table 2) by bringing completely independent values of the Auger probability a1 into the discussion. In principle, this quantity can be obtained independently of the present work as the ratio between the L1 Auger width CA(L1) from DHS theory [2] and the recommended total width C(L1) from a critical review [7]. Unfortunately, in the region 64 6 Z 6 92, the recommended values of Ref. [7] are, of necessity, based upon a fit to a very limited set of experimental data. Close to half of these data points are from very early (1944) X-ray emission spectroscopy (XES), and most of the remainder are deduced from the PISR/LX/GC measurements discussed above. However, a modern set of XES measurements [16,17] done mostly

Fig. 5. Recommended values for F13, a1 and x1.

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reduced if the L1L3M4 and the L1L2N3 discontinuities are to be understood. For both f13 and x1, the existing results at Z P 90 need to be rigorously tested in order to improve the ‘‘anchoring” of the curves at the highest values of atomic number; this is a major source of uncertainty here. In the case of f12, the systematic difference around Z = 80 between the PISR and the Nuclide data is of concern, as is our inability to discern any of the expected discontinuities due to the high degree of scatter in a very small data set. The problems discussed here will not be simple to solve. Future experimental work needs to be highly focused, to show significant improvements in accuracy, to provide convincing support for the claimed accuracy, to probe systematic errors in detail, and to analyze measured spectra by the most sophisticated available techniques. The troubling scatter of data observed here suggests methodological problems, even within the carefully selected subset that we extracted from a larger volume in the literature [1]. For example, the PISR/LX/GC data for x1 are mutually consistent except for one quite dramatic anomaly. The same is true of the PISR/LX/UC data for f13. In each case one has to ask why one measurement within a series differed so much from all the others. Papp et al. [18] have raised significant new issues concerning the electronic systems associated with semiconductor detectors, and we suggest that some of the problems may lie here. In future work, dependence of experimental results upon assumed theoretical results needs to be minimized, so that the theory is tested in a completely independent fashion. We strongly recommend that any new work should use Voigtian lineshapes (i.e., Lorentzian–Gaussian convolutes) in fitting L X-ray spectra and extracting peak intensities. It has been shown that use of the Gaussian approximation, which represents only detector resolution, can cause significant error [19]. The fluorescence and Coster–Kronig processes under study here are manifestations of the finite natural width of atomic levels; the energy distribution of these levels is Lorentzian. It is ironic that this fact should have been ignored in the great majority of measurements to date. In addition to all of this, it is desirable that the recent XES work on L1 widths be expanded to cover a much larger set of Z values. On the theoretical side, it appears an urgent matter to derive f12 and f13 using the DFEX approach that was demonstrated by Chen [3] in the case of f23; exchange has a significant effect in these low energy transitions. It is also important to apply the MCDF approach to investigate the behavior of major Coster–Kronig channels close to discontinuities, where fine structure states from the two-vacancy configurations may become favorable or unfavorable. Finally, we suggest that the values which we have recommended here should be used in a conservative fashion. If they are used or quoted, the estimates of uncertainty should be attached; indeed, the reader may prefer to form his or her own estimate of uncertainty from inspection of Figs. 2–4. Our initial goal was to develop the best possible set of recommendations. In the end, the most important outcome is the demonstration that much needs to be done before we can claim to understand radiative and non-radiative de-excitation of L1 vacancies. Acknowledgments The support of the Interdisciplinary Science Awards Committee of the Natural Sciences and Engineering Research Council of Canada is sincerely acknowledged. Discussions with Dr. Tibor Papp were very valuable and deeply appreciated.

Appendix A Table A. Uncertainty estimates (%) for recommended values of x1, f13 and f12 Quantity

Range of Z

Error estimate (%)

x1

64 6 Z 6 74 Z = 75, 76 77 6 Z 6 85 Z > 85

15 No recommendation 30 35

f13

64 6 Z 6 74 75 6 Z 6 77 Z = 78 79 6 Z 6 92

15 15 20 15

f12

64 6 Z 6 74 Z = 75, 76 77 6 Z6 84 Z > 84

20 No recommendation 30–40 50–100

Table B. Test of the sum of L1 Auger yields (deduced from total and Auger widths C and CA) plus the recommended L1 fluorescence yields and Coster–Kronig probabilities Z

C(eV)

CA (eV)

a1 = CA/C

a1 + x1 + f12 + f13

70 74 78 80 83 90 92

5.4 6.41 8.55 10.67 12.5 14.3 16.0

1.864 1.951 2.038 2.086 2.154 2.314 2.359

0.345 0.304 0.238 0.196 0.172 0.162 0.147

0.93 0.90 0.97 1.0 0.99 0.98 0.97

References [1] J.L. Campbell, Atom. Data Nucl. Data Tables 85 (2003) 203. [2] M.H. Chen, B. Crasemann, H. Mark, Phys. Rev. A24 (1981) 177. [3] M.H. Chen, X-ray and Inner-Shell Processes, in: T.A. Carlson, M.O. Krause, S.T. Manson (Eds.), AIP Conf. Proc. 215 (1990) 391. [4] M.H. Chen, B. Crasemann, H. Mark, Phys. Rev. A21 (1980) 436. [5] J.H. Hubbell, P.N. Trehan, N. Singh, B. Chand, D. Mehta, M.L. Garg, R.R. Garg, S. Singh, S. Puri, J. Phys. Chem. Ref. Data 23 (1994) 339. [6] E.J. McGuire, Phys. Rev. A3 (1971) 587. [7] J.L. Campbell, T. Papp, Atom. Data Nucl. Data Tables 77 (2001) 1. [8] M.H. Chen, B. Crasemann, K.-N. Huang, M. Aoyagi, H. Mark, Atom. Data Nucl. Data Tables 19 (1977) 97. [9] M. Sharma, P. Singh, S. Puri, D. Mehta, N. Singh, Phys. Rev. A69 (2004) 032501. [10] M. Sharma, P. Singh, J.S. Shahi, D. Mehta, N. Singh, X-Ray Spectrom. 34 (2005) 35. [11] A. Kumar, S. Puri, D. Mehta, M.L. Garg, N. Singh, X-Ray Spectrom. 31 (2002) 103. [12] J.H. Scofield, Lawrence Livermore National Laboratory Report UCRL-51326, 1973. [13] J.L. Campbell, J. Phys. B At. Mol. Opt. Phys. 36 (2003) 3219. [14] L. Kissel, Radiat. Phys. Chem. 59 (2000) 185. [15] U. Werner, W. Jitschin, Phys. Rev. A38 (1988) 4009. [16] P.-A. Raboud, J.-Cl. Dousse, J. Hoszowska, I. Savoy, Phys. Rev. A61 (2000) 12507. [17] P.-A. Raboud, M. Berset, J.-Cl. Dousse, Y.-P. Maillard, Phys. Rev. A65 (2002) 022512. [18] T. Papp, A.T. Papp, J.A. Maxwell, Anal. Sci. 21 (2005) 737. [19] T. Papp, J.L. Campbell, S. Raman, J. Phys. B At. Mol. Opt. Phys. 26 (1993) 4007. [20] M.O. Krause, J. Phys. Chem Ref. Data 8 (1979) 307.

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Explanation of Tables Table 1.

Experimental fluorescence yield and Coster–Kronig probabilities The table contains two classes of experimental values for L1 subshell fluorescence yield x1, L1–L2 Coster–Kronig probability f12, and L1–L3 Coster–Kronig probability f13. The first class comprises data from measurements reported since publication of Ref. [1]. The second comprises older measurements that were not used in Ref. [1] but are considered here. Italics denote uncertainty estimates. Brackets denote results that were anomalous with respect to the overall adopted trend Z Method

x1 f12 f13

Atomic number and element symbol Method of experimental measurement L1 subshell fluorescence yield or probability L1–L2 Coster–Kronig transition probability L1–L3 Coster–Kronig transition probability

Method of measurement Measured data are from the following techniques: Nuclide Singles and coincidence mode X-ray spectroscopy using energy-dispersive semiconductor detectors PISR/LX Photonization using monochromatic synchrotron radiation scanned over L absorption edges to excite L X-rays GC: work from German collaboration UC: work from University of Cordoba PI(PG + SK) Photoionization using primary c-rays directly from a radioactive source and secondary K X-rays excited by c-rays from the same source (obs) Experimentally observed value without correction for radiative L1L3 transition Rec Recommended value Anom Value is anomalous with respect to overall trend

Table 2.

Recommended fluorescence yield and Coster–Kronig probabilities Recommended values for L1 subshell fluorescence yields and the L1–L2 and L1–L3 Coster–Kronig probabilities, together with predictions of the DHS theory and earlier recommendations of Krause [20] Rec DHS Kr

Recommended value Dirac–Hartree–Slater calculation [2] Semi-empirical recommendation of Krause [20]

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J.L. Campbell / Atomic Data and Nuclear Data Tables 95 (2009) 115–124 Table 1 Experimental fluorescence yield and Coster–Kronig probabilities. See page 122 for Explanation of Tables Z

Method

65 70 70 71 73 74 75 79 79 80 80 81 81 83

PISR/LX/UC PISR/LX/UC PI(PG + SK) PI(PG + SK) PISR/LX/UC PI(PG + SK) PI(PG + SK) PI(PG + SK) PI(PG + SK) PI(PG + SK) PI(PG + SK) PI(PG + SK) PI(PG + SK) Nuclide

x1 (obs)

f12

f13(obs)

Ref.

0.201 14 (0.249 21)

0.281 (0.408 0.348 0.338 0.322 0.354 0.450 0.580

95Sa01 96Sa01 04SH01 04SH01 96Sa01 04SH01 04SH01 04SH01 05SH01 04SH01 05SH01 05SH01 04SH01 03Ca01

(0.168 39)

27 55) 21 21 72 24 27 30

0.097 7 0.582 30 0.102 8 0.110 9 0.131 6

References: 95Sa01: H.J. Sánchez, R.D. Pérez, M. Rubio, G. Castellano, X-Ray Spectrom. 24 (1995) 221. 96Sa01: H.J. Sánchez, R.D. Pérez, M. Rubio, G. Castellano, Radiat. Phys. Chem. 48 (1996) 701. 02Ba01: R.A. Barrea, C.A. Pérez, H.J. Sánchez, J. Phys. B At. Mol. Opt. Phys. 35 (2002) 3167. 03Ca01: J.L. Campbell, J. Phys. B At. Mol. Opt. Phys. 36 (2003) 3219. 04SH01: M. Sharma, P. Singh, S. Puri, D. Mehta, N. Singh, Phys. Rev. A69 (2004) 032501. 05SH01: M. Sharma, P. Singh, J. S. Shahi, D. Mehta, N. Singh, X-ray Spectrom. 34 (2005) 35.

0.069 8

0.568 30 0.66 5

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Table 2 Recommended fluorescence yield and Coster–Kronig probabilities. See page 122 for Explanation of Tables Z

x1 DHS

x1 Rec

x1 Kr

f12 DHS

f12 Rec

f12 Kr

f13 DHS

f13 Rec

f13 Kr

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92

0.083 0.087 0.091 0.095 0.105 0.109 0.114 0.12 0.125 0.131 0.136 0.084 0.088 0.093 0.074 0.078 0.082 0.088 0.093 0.098 0.103 0.109 0.114 0.12 0.126 0.133 0.139 0.147 0.149

0.102 0.107 0.111 0.116 0.121 0.131 0.134 0.138 0.141 0.144 0.148

0.079 0.083 0.089 0.094 0.1 0.106 0.112 0.12 0.128 0.137 0.147 0.144 0.13 0.12 0.114 0.107 0.107 0.107 0.112 0.117 0.122 0.128 0.134 0.139 0.146 0.153 0.161 0.162 0.176

0.216 0.216 0.217 0.219 0.182 0.183 0.184 0.185 0.186 0.186 0.186 0.087 0.088 0.088 0.067 0.068 0.069 0.054 0.064 0.064 0.064 0.064 0.064 0.056 0.056 0.057 0.058 0.053 0.051

0.190 0.182 0.174 0.166 0.158 0.150 0.142 0.134 0.126 0.118 0.110

0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18 0.17 0.16 0.16 0.15 0.14 0.14 0.13 0.13 0.12 0.11 0.11 0.1 0.1 0.1 0.09 0.09 0.09 0.08 0.08

0.334 0.334 0.335 0.338 0.354 0.354 0.354 0.353 0.352 0.351 0.352 0.64 0.636 0.631 0.716 0.711 0.707 0.713 0.708 0.703 0.697 0.691 0.685 0.679 0.672 0.666 0.659 0.655 0.66

0.279 0.285 0.290 0.296 0.301 0.306 0.312 0.317 0.322 0.328 0.333 0.482 0.482 0.482 0.545 0.615 0.615 0.615 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62 0.62

0.3 0.3 0.3 0.3 0.3 0.29 0.29 0.28 0.28 0.28 0.28 0.33 0.39 0.45 0.5 0.53 0.56 0.57 0.58 0.58 0.58 0.59 0.58 0.58 0.58 0.58 0.57 0.58 0.57

0.145 0.114 0.117 0.121 0.124 0.128 0.132 0.135 0.138 0.142 0.146 0.150 0.154 0.159 0.164 0.168

0.076 0.075 0.074 0.072 0.069 0.066 0.063 0.060 0.057 0.053 0.050 0.047 0.044 0.040 0.038 0.035