Measurements of L-shell x-ray production cross sections of W , Pt , and Au by 10 – 30-keV electrons

We present results from measurements of La x-ray production cross sections of the elements W, Pt, and Au by impact of electrons with energies in the range 10–30 keV. The cross sections were obtained by measuring La x-ray intensities emitted from very thin films of the studied elements deposited on thick carbon substrates. The directional and energy spreading of the electron beam within the active film and the x-ray enhancement due to electron backscattering from the substrate were accounted for by means of Monte Carlo simulation. Recorded x-ray intensities were converted to absolute x-ray production cross sections by using two different methods; the first employs measured values of the sample thickness and the number of incident electrons and estimated detector efficiencies; the second is based on a comparison between measured and calculated bremsstrahlung intensities. Experimental data are compared with the results of simple analytical formulas of common use in practical electron probe microanalysis, with calculated cross sections obtained from the distorted-wave Born approximation and with other experimental data available in the literature.


I. INTRODUCTION
Accurate cross sections for inner-shell ionization by electron impact are required for multiple applications, particularly for electron probe microanalysis ͑EPMA͒ and Augerelectron spectroscopy.In spite of this need, a systematic method for calculating accurate ionization cross sections from first-principles remains to be found.The usual practice consists of using semiempirical formulas, which have limited ranges of validity and accuracy; too frequently different formulas lead to significantly different results.Calculations within the plane-wave first Born approximation provide reliable results for high-energy electrons ͓1͔; however, this approach is not satisfactory near the ionization threshold.A more appropriate theoretical tool is provided by the distorted-wave Born approximation ͑DWBA͒, even though the calculations are extremely time consuming and difficult to validate ͓2͔.
In the energy range of interest in EPMA, say 1-50 keV, experimental measurements of inner-shell ionization cross sections deal mostly with K shells; cross-section data for L and M shells are very scarce.Absolute L-shell ionization cross sections have only been reported for a few elements such as Ar ͓3,4͔, Kr ͓5͔, Xe ͓5,6͔, Au ͓7-10͔, and W ͓11͔ ͑see also Refs.͓12-14͔͒.Moreover, available experimental data are affected by large uncertainties and important discrepancies are found when comparing data from different authors ͓15,16͔.As a consequence, it is difficult to assess the reliability of cross sections calculated with the DWBA or other approximation.The situation is even worse for M shells, for which experimental data are extremely rare.New, accurate experimental measurements of L-and M-shell ionization cross sections by keV electron impact are therefore urgently needed.
Inner-shell ionization cross sections can be determined by measuring the intensity of characteristic x-rays emitted from self-supporting thin films of the considered elements, bombarded by an electron beam ͓17͔.The determination of absolute values of the cross section, however, poses numerous difficulties ͑see e.g., Ref. ͓18͔͒.In the case of L and M shells, adding to these difficulties is the fact that vacancies in a given subshell can be produced not only by electron impact but also by nonradiative ͑Coster-Kronig͒ transitions between the subshells.As a consequence, the intensity of a given x-ray line depends on the ionization cross sections of all the subshells, weighted by the corresponding Coster-Kronig coefficients.Consequently, to determine subshell ionization cross sections we have to measure the intensities of a number of x-ray lines, some of which may not be clearly resolved or may have very low intensities.Moreover, Coster-Kronig coefficients are generally affected by large uncertainties, which would propagate to the derived ionization cross sections.Considering these difficulties, it is advisable to report the cross section for x-ray production, usually for the most intense lines, rather than the cross section for inner-shell ionization.By proceeding in this way, Coster-Kronig coefficients, fluorescence yields, and fractional emission rates do not affect the reported experimental data.It should also be noted that for many applications, including EPMA, the goal is to calculate x-ray intensities from irradiated samples, which can be obtained from knowledge of the cross section for x-ray production.
In this work, we report on experimental measurements of L␣ x-ray production cross sections for the elements W (Z ϭ74), Pt (Zϭ78), and Au (Zϭ79), for incident electron energies 10-30 keV.Cross sections were obtained by measuring L␣ x-ray intensities emitted from very thin films of the studied elements, which were deposited on carbon substrates.Measurements were performed with an electron microprobe, by using both a Si͑Li͒ detector and a crystal spectrometer.The effects of spatial and energy spreading of the electron beam within the active film and the x-ray enhancement due to electron backscattering from the substrate were corrected by using Monte Carlo ͑MC͒ simulation results generated by considering the particular geometry of each sample.L␣ x-ray intensities were converted to L␣ x-ray production cross sections by two different methods ͑i͒ by using measured values of the sample thickness and the number of incident electrons and estimated detector efficiencies and ͑ii͒ by using measured and calculated bremsstrahlung intensities.Results are compared with simple analytical formulas of common use in EPMA and with calculated cross sections obtained from the DWBA.To make this comparison possible, theoretical ionization subshell cross sections have been converted to x-ray production cross sections by using relaxation data available from the literature.Our experimental results are also compared with measurements of other authors.

II. EXPERIMENTAL PROCEDURE
The apparatus used for electron bombardment and for the detection of x-ray spectra from the specimen was a CAM-ECA SX-50 electron microprobe ͑Microprobe Laboratory, Universidade Federal do Rio Grande do Sul͒.In this instrument x-ray spectra can be measured simultaneously by a KEVEX Si͑Li͒ detector and by four wavelength-dispersive ͑WD͒ spectrometers.According to the manufacturer's specifications, the Si͑Li͒ detector consists of an active 3-mm-thick diode covered by a 7-m-thick beryllium window and it has an active area of 12.5 mm 2 .Each WD spectrometer consists of a crystal monochromator and a gas proportional counter ͑gas-flow-open-type, with a Be or polypropylene entrance window͒ with an argon-methane ͑90:10͒ mixture as counter gas.Two of the WD spectrometers are separated from the electron column by means of a mylar window, while in the other two, the separation window is made of polypropylene.The Si͑Li͒ detector and the WD spectrometers are located at directions forming angles of 40°with respect to the sample surface.
To determine x-ray production cross sections we have to measure very low x-ray intensities and thus it is preferable to use the WD spectrometer rather than the Si͑Li͒ detector.The reason is that the former has better energy resolution than the latter and therefore peak-to-background ratios obtained from measurements with the WD spectrometer are higher than those recorded with the Si͑Li͒ detector.Therefore, with the WD spectrometer we obtain net peak intensities with a lower uncertainty, especially when the incident electron energy is close to the ionization threshold.However, the absolute efficiency of the WD spectrometer depends on the incoming photon energy in a rather complicated way ͓19͔ and it is difficult to estimate.Conversely, the efficiency of the Si͑Li͒ spectrometer is almost constant in the photon energy of interest in this work ͑say 8 -10 keV͒ and it can be determined to good accuracy ͑see e.g., Ref. ͓18͔͒.Our approach in the present work consists of combining measurements with both spectrometers: the WD is used to obtain relative x-ray intensities, while the Si͑Li͒ detector is employed to obtain the absolute value of the x-ray emission cross section for an electron energy of 20 keV, which is well above the ionization threshold.In the final stage, the x-ray intensities measured with the WD spectrometer are scaled and converted into absolute x-ray production cross sections by matching the crosssection value determined from the measurement using the Si͑Li͒ detector.
X-ray measurements were performed on the WD spectrometer using a LiF diffracting crystal for incident electron energies ranging from 10 keV to 30 keV in 1-2 keV steps.The electron-beam current and beam diameter were 40 nA and 20 m, respectively.The intensity of characteristics x-rays was counted on the wavelength channel corresponding to the maximum of the characteristic peak and the background was subtracted using linear interpolation of the intensities on channels at both sides of the peak.Counting times of about 100 sec were used for each measured channel.For each sample and beam energy, three measurements were performed at different positions on the sample to test for possible inhomogeneities in the thickness of the active film.Therefore, the standard deviation of the measurements accounts for uncertainties not only due to counting statistics, but also to ͑minor͒ inhomogeneities of the active layer.The relative uncertainties of the cross-section values reported below range from ϳ1% to 3.8% ͑at 1 level͒.
The Si͑Li͒ measurements at 20 keV were performed with a beam current of 5 nA, a beam diameter of 20 m and an acquisition time of 1000 sec.A typical example of an x-ray spectrum obtained with this spectrometer from the W target bombarded with electrons of 20 keV is displayed in Fig. 1.Usually two spectra were acquired from each sample.For the studied high-Z elements, the characteristic L peaks are well separated and their intensities N L␣ can be easily obtained either by summing the channel counts directly or by fitting the peak with a Gaussian function, after subtraction of the linearly interpolated background.The statistical uncertainties of the resulting L␣ intensities of interest were less than 1%.The number of incident electrons N e was evaluated by multiplying the probe current by the ''live'' acquisition time.To determine the efficiency of the Si͑Li͒ detector, x-ray spectra were also acquired from a pure graphite substrate and the same instrumental conditions ͑see below͒.
The studied samples were thin W, Pt, and Au films deposited on graphite substrates.Au films were produced by resistive evaporation.W and Pt films were obtained by sputtering.Graphite was selected as the substrate because of its low atomic number and the associated small electron backscattering.
The thickness of the Au films was controlled by a quartz crystal during evaporation.As the accuracy of the quartz crystal reading was not known, the following calibration procedure was adopted.A number of samples with Au overlayers of different thicknesses were produced, corresponding to various crystal frequency variations ⌬.The thicknesses t of the overlayers were determined by Rutherford backscattering spectrometry ͑RBS͒ as follows ͑see e.g., Ref. ͓20͔͒.The samples were irradiated with a 0.9-MeV He ϩϩ beam, at normal incidence, in a Tandetron accelerator from a 3-MeV high-voltage engineering ͓21͔ at the Ion Implantation Laboratory of the Universidade Federal do Rio Grande do Sul.Ion spectra were detected at 165°, with respect to the incident beam, and were analyzed by means of an analytical method ͓22͔ and the RUMP ͑version 2.0͒ simulation code ͓23͔.The two methods gave results that agreed to within 1.5%.This procedure allowed the calibration of the quartz crystal for thicknesses larger than about 20 nm, for which the RBS data were reliable enough ͑the relative uncertainty increases rapidly for decreasing thicknesses͒.The calibration gave an almost perfect linear dependence of ⌬ and t for tϾ20 nm, which was also assumed to hold for thinner films.Once the crystal was calibrated, it was utilized in the production of a thin Au overlayer for the x-ray measurements; its thickness was 7.1 nm, small enough to minimize the effect of electron angular and energy straggling.
W and Pt films were generated by sputtering and their thicknesses were determined from EPMA measurements.This method requires measuring characteristic lines from the overlayer and the substrate at different electron incident energies ͑for a general description of the thickness determination technique by EPMA; see, e.g., Scott et al. ͓45͔͒.As the EPMA measurements of carbon are difficult because of the low energy of C K␣ x rays ͑see e.g., Ref. ͓24͔͒, during the sample fabrication runs, twin films of the studied elements were deposited on ultrapure Si targets, which were used for the thickness determination as follows.We measured the ratio of x-ray intensities ͑the so-called k ratio͒ of W M ␣, Pt M ␣, and Si K␤ peaks from the W/Si and Pt/Si targets to the x-ray intensities of the same peaks from pure W, Pt, and Si targets, from 6 keV to 20 keV electron incident energies, in 2 keV steps.We also measured the Au overlayer on Si ͑Au M ␣), which allowed us to compare the EPMA results with the thickness determined by using the quartz crystal.Measured k ratios were analyzed with the help of the X-FILM ͓25͔ and LAYERF ͓26͔ EPMA analysis codes.These codes estimate the thickness and the elemental concentration of a multilayer target by fitting an analytical model to the experimental k ratios.As an example, Fig. 2 shows the k ratio as a function of the incident electron energy for W M ␣ and Si K␤ lines from a W/Si sample, as well as the best predictions of the LAYERF and X-FILM codes.Thicknesses obtained using the different methods are summarized in Table I.We can see that for the Au sample, the result obtained from the calibrated quartz crystal is in good agreement with the result obtained from the two EPMA methods, as their results are also seen to agree satisfactorily.For each sample, the adopted thickness was set equal to the average of the values obtained from the available methods.

III. DATA CORRECTION
Although the active films are very thin ͑see Table I͒, incident electrons do suffer some scattering and lose energy within the film.Therefore, the ''effective'' path length of electrons is somewhat larger than the film thickness and the effective energy is slightly smaller than the energy of incidence.Furthermore, electrons backscattered from the substrate may also produce ionization within the active film, which causes an enhancement of emitted characteristic x rays, and also bremsstrahlung photons.To account for these effects, we have carried out MC simulations of x-ray spectra emitted from samples with the two-layer structure used in our measurements for incident electron beams with energies between 10 and 30 keV.The simulations were performed by using a dedicated MC subroutine set named PEN-XR ͓27͔, which is based on the latest version of the general-purpose subroutine package PENELOPE ͓28͔.PEN-XR implements the most accurate interaction models available to date.Bremsstrahlung emission is described by the means of differential cross sections ͑DCS͒, differential in the energy and direction of the emitted photon, obtained by combining scaled DCS ͑i.e., photon energy spectra͒ interpolated from Seltzer and Berger's database ͓29͔ and angular distributions obtained from an accurate parametrization of the ''shape functions'' tabulated by Kissel et al. ͓30͔.The ionization of inner shells by electron impact is described by using theoretical total ionization cross sections, calculated by means of a relativistic DWBA code ͓2,31͔.The relaxation of the produced ions is simulated using transition probabilities from the evaluated atomic data library ͑EADL͒ ͓32͔.
From the simulated x-ray spectra, L␣ x-ray intensities I L␣,MC were obtained in absolute units ͑i.e., as the number of photons emitted per unit solid angle per incident bombarding electron͒ and subsequently converted into L␣ x-ray production cross sections by using the equation where N is the number of atoms per unit volume and t is the simulated target thickness.The DWBA ionization cross sections Li used in the MC simulations were in turn converted to L␣ x-ray production cross sections L␣ using the formula ͑see e.g., Ref.
where ⌫ M 4,5 ϪL 3 and ⌫ TotalϪL 3 are the x-ray emission rates for L␣ (M 4,5 -L 3 transition͒ and total (M ,N,O-L 3 transitions͒ x rays, respectively, L 3 is the fluorescence yield for the L 3 shell, and f i j are the Coster-Kronig transition probabilities.These parameters were obtained by combining the corresponding transition probabilities extracted from the EADL. Figure 3͑a͒ compares theoretical L␣ x-ray production cross sections L␣,th , calculated within the DWBA and using relaxation data obtained from the EADL, with the production cross section inferred from the MC simulation, L␣,MC , of emission from a 7.1-nm-thick Au/C sample.The error bars associated with the MC results represent statistical uncertainties ͑three standard deviations͒.Notice that the DWBA calculated cross section is that used in the simulation and, therefore, the two data sets displayed in Fig. 3͑a͒ would coincide in the case of a self-supporting, infinitely thin film.The objective of this figure is to reveal the combined effect of finite film thickness and backscattering from the substrate, which can be expressed by means of a correction factor where L␣,MC and L␣,th are given by Eqs.͑1͒ and ͑2͒, respectively.This correction is expected to be fairly insensitive to the details of the adopted ionization cross section.As seen in Fig. 3͑b͒, and also for other overlayers and substrates used in this work, the largest correction is of the order of 10%.
Correction factors f c calculated in this way have been used to transform measured x-ray intensities into relative x-ray production cross sections, i.e., x-ray intensities that would result from a bare active film with neither electron scattering nor energy loss.As illustrated in Fig. 3͑b͒, the correction factor increases as the electron incident energy E 0 approaches the ionization threshold.In order to minimize the effect of statistical uncertainties, the correction factor has been approximated by the following analytical expression: 3. Theoretical L␣ x-ray production cross sections vs electron incident energy for Au, calculated for a thin self-supporting film by using ionization cross sections calculated from the DWBA and atomic relaxation data from the EADL ͑solid line͒ and derived from the Monte Carlo simulation of emission from a 7.1-nm-thick Au/C sample using identical ionization cross sections and relaxation data ͑symbols͒ ͑a͒ and correction factors f c vs incident electron energy for the same composite sample ͑b͒.The solid line in the right panel is the curve given by Eq. ͑4͒, with fitted parameter values.
with parameters determined by fitting the calculated crosssection ratios.In Fig. 4, we compare our corrected measurements of relative L␣ x-ray production cross sections of Au with the measurements of Llovet et al. ͓33͔, which were obtained by using a 3.9-nm-thick self-supporting film.The agreement between these two independent experiments corroborates the correctness of the correction procedure adopted here.

IV. DATA ANALYSIS
For an homogeneous film of thickness t and normal electron-beam incidence, the absolute x-ray production cross section L␣ can be expressed as ͑see e.g., Ref.
where N L␣ is the intensity of the characteristic L␣ peak, E 0 is the incident electron energy, E ph is the emitted photon energy, N is the number of atoms per unit volume, t is the target thickness, N e is the number of incident electrons, ⑀(E ph ) and ⌬⍀ are the intrinsic efficiency and solid angle subtended by the spectrometer, and f c (E 0 ) is the correction factor defined above, which accounts for electron-transport corrections.Notice that for an ideal, infinitely thin selfsupporting film, we would have f c (E 0 )ϭ1.
As mentioned above, it is difficult to determine the efficiency and the solid angle subtended by a WD spectrometer.To avoid this difficulty, we have derived absolute crosssection values from measurements with the Si͑Li͒ detector, whose efficiency and solid angle can be determined more accurately, by using two alternative methods.The first is based on Eq. ͑5͒ and requires determining all the parameters in this equation.The second method, which is described in the Appendix, relies on calculated and experimental bremsstrahlung cross sections ͓5͔.Evidently, uncertainties in the adopted bremsstrahlung cross sections will spread to the re-sulting x-ray production cross sections.Absolute cross sections obtained from these two scaling methods are consistent with the estimated uncertainties.
Although in the photon energy range of interest, say 8 -10 keV, the absolute efficiency of a Si͑Li͒ spectrometer can be determined by relatively simple means ͑see e.g., Ref. ͓18͔͒, the solid angle of the detector was not known with sufficient accuracy.To overcome this difficulty, we have estimated the detector efficiency with the help of MC simulation as follows.Let us consider the number of photons N exp (E) detected per unit energy interval and unit solid angle per incident electron from a thick solid target, irradiated with an electron beam of energy E 0 .This can be expressed as where N ch (E) is the number of counts in a particular photon energy channel of width ⌬E centered at E. If we replace N exp (E) by the result from a MC simulation expressed in absolute units (N MC ), then ⑀⌬⍀ can be calculated as In a previous work, we have shown that simulated thicktarget bremsstrahlung spectra from pure carbon targets, obtained with the aid of our MC tool PEN-XR, are in very good agreement with absolute x-ray spectra measured with a Si͑Li͒ detector ͓27͔.Therefore, we have obtained the detector efficiency from Eq. ͑7͒ by combining our measured thicktarget bremsstrahlung spectra on graphite targets with MC simulations.As the detector efficiency is nearly constant on a wide energy interval ͑see Fig. 5͒, we have adopted the average value of ⑀ ⌬⍀ in a photon energy region around the energies of interest, i.e., W L␣ (E ph ϭ8.396 keV), Pt L␣ (E ph ϭ9.441 keV), and Au L␣ (E ph ϭ9.712 keV).
Cross-section measurements are affected by relative uncertainties, which arise mainly from counting statistics, background subtraction, sample nonuniformity, and instrumental drift during measurements; they were, on an average, 1.9% for W, 2.5% for Pt, and 2.4% for Au.Relative uncertainties only affect the shape of the cross-section curve.The conver- sion ͑scaling͒ from relative cross sections to absolute cross sections introduces additional uncertainties of a systematic nature, which are the same for all measured energies.These are estimated to be ϳ8% and originate from uncertainties in the determination of film thicknesses ͑4.5% for W, 5.3% for Pt, and 3.7% for Au͒, detection efficiency ͑6%͒, number of incident electrons ͑2%͒, and the statistical uncertainties of peak measurements with the Si͑Li͒ ͑1%͒.With the alternative method described in the Appendix, systematic uncertainties are estimated to be ϳ10.5% and arise from the uncertainty of the adopted ͑calculated͒ bremsstrahlung cross sections ͑10%͒ ͓30͔ and from the statistical uncertainties of measured peak-to-background ratios with the Si͑Li͒ ͑3.2% for W, 1.9% for Pt, and 2.2% for Au͒.
The x-ray production cross-section values reported in this study are calculated as the mean of the cross sections obtained from Eqs. ͑5͒ and ͑A3͒ ͑see the Appendix͒.These have been found to agree to within ϳ7%.This agreement provides a consistency check of the conversion procedure.The absolute uncertainty of the reported cross sections, obtained by combining relative and systematic uncertainties in quadrature, is estimated to be ϳ11%.

V. RESULTS AND DISCUSSION
In what follows, we compare our results with the predictions of various theoretical ionization cross-section models.We consider the classical formula of Gryzinski ͓34͔ and the semiempirical formula of Worthington-Tomlin ͓35͔, which are widely used in EPMA.We also consider results from recent relativistic DWBA calculations by Segui et al. ͓2͔.These calculations go beyond the better known plane-wave Born approximation by accounting for ͑1͒ the distortion of the projectile wave functions by the field of the target atom and ͑2͒ electron exchange effect.
The L-shell ionization cross sections obtained from these models/calculations have been converted into L␣ x-ray production cross sections by using Eq.͑2͒.The x-ray emission rates, fluorescence yields, and Coster-Kronig coefficients have been taken from different available bibliographic sources ͓32,36 -43͔.As already pointed out, these relaxation data are affected by sizable uncertainties and, for a given set of theoretical L-shell ionization cross sections, the L␣ x-ray production cross sections obtained with relaxation data from the various available sources are found to lie within an ''uncertainty band'' whose width is ϳ8% for Pt, ϳ17% for W, and ϳ8% for Au.Notice that if we had tried to infer L-shell ionization cross sections from measured L␣ x-ray intensities, these uncertainties would have added to those arising from the measurement.In order to facilitate the comparisons, we have adopted the theoretical fractional emission rates given by Scofield ͓36͔ and the experimental fluorescence yields, and Coster-Kronig coefficients given by Werner and Jitschin ͓39͔, which are summarized in Table II.
Cross sections for L␣ x-ray production of Au, W, and Pt are listed in Table III.The data are plotted in Fig. 6 together with representative absolute uncertainties that, as mentioned above, are of the order of 11%.Notice, however, that the shape of the cross-section curve is much more accurate (ϳ3%), since it is only affected by relative uncertainties.This is clearly seen from the smoothness of the measured cross section vs energy plots ͑Fig.6͒.Our measurements are also compared with experimental data from other authors in Fig. 6.Some of the experimental results found in the literature were given as subshell ionization cross sections ͓7,10,11͔, therefore these have been converted into L␣ x-ray production cross sections by using the relaxation parameters adopted by the authors to do the reverse transformation.However, in some cases ͓7,10͔, the authors did not provide experimental information on all the required subshells.In these cases, we have assumed subshell cross-section ratios identical to those obtained from DWBA calculations at 20 keV.Namely, for Au, L3 / L1 ϭ0.356 and L3 / L2 ϭ0.246.
In the case of Au ͓Fig.6͑c͔͒, our results agree quite well with the experiments of Shima et al.  ical results, for the three measured elements, the DWBA calculations are in excellent agreement with our experimental data, while results from the Gryzinski and Worthington-Tomlin formulas are systematically lower than our data.
Figure 7 shows our cross sections for the various elements normalized to their corresponding maxima max , as functions of the overvoltage uϭE 0 /E L␣ , where A 1 ϭ3.950,A 2 ϭ0.70,A 3 ϭ0.10,and A 4 ϭ0.184 are parameters, which have been determined from a numerical fit to all the measured data.It is seen that, within the experimental uncertainties, this function describes all the measurements reasonably well for the three considered elements.On the other hand, the small spread of the experimental data about the fitted curve does confirm the relative accuracy of our measurements.
In conclusion, we have reported measurements of L␣ x-ray production cross sections for the elements W, Pt, and Au, from threshold up to 30 keV.The adopted experimental procedure and evaluation methods allowed us to reduce the relative and absolute uncertainties to about 3% and 11%, respectively.The DWBA-based calculations of Segui et al. ͓2͔, combined with x-ray emission rates from Scofield ͓36͔, fluorescence yields, and Coster-Kronig coefficients from Werner and Jitschin ͓39͔, have been found to be in excellent agreement with our measurements.

FIG. 1
FIG. 1. X-ray spectrum from a 7.5-nm-thick W film deposited on C obtained with the Si͑Li͒ detector for an incident electron energy of 20 keV.

FIG. 2 .
FIG.2.K ratios for W M ␣ and Si K␤ lines from a 7.5-nm-thick W film deposited on Si substrate, as functions of the incident electron energy.The k ratio was determined with respect to pure W for the W M ␣ line and relative to pure Si for the Si K␤ line.Open triangles are k ratios for the W M ␣ line; filled triangles are k ratios for the Si K␤ line.Curves are the best predictions of the LAYERF and X-FILM EPMA codes.

FIG. 4 .
FIG. 4. Relative L␣ x-ray production cross sections for Au vs incident electron energy.Full circles are the results of the present measurements using Au/C samples.Open circles represent measurements by Llovet et al. ͓33͔ using self-supporting thin film samples.The continuous line is the result of a fit with the analytical form given by Eq. ͑8͒.

FIG. 6 .
FIG. 6. L␣ x-ray production cross section vs incident electron energy for W ͑a͒, Pt ͑b͒, and Au ͑c͒.The curves have been obtained by means of Eq. ͑2͒ using ionization cross sections calculated from different formulas and approximations.The dotted curves are from the Gryzinski formula ͓34͔; the dashed curves represent results from the Worthington-Tomlin ͓35͔ formula; the continuous curves were obtained from distorted-wave ionization cross sections calculated by Segui et al. ͓2͔.Full circles represent the results from the present measurements.Open symbols are experimental data from by Chang ͓11͔ ͑circles͒, Shima et al. ͓9͔ ͑circles͒, Schneider et al. ͓10͔ ͑squares͒, Davis et al. ͓8͔ ͑triangles͒, and Salem and Moreland ͓7͔ ͑inverted triangles͒.

TABLE I .
Film thicknesses and associated uncertainties obtained by EPMA using the codes LAYERF and X-FILM and by the quartz crystal, calibrated by RBS.
͓9͔, Davis et al. ͓8͔, and  Schneider et al. ͓10͔.The latter measurements, however, are smaller than ours and the difference increases with incident electron energies, whereas the measurements of Davis et al. ͓8͔ at 20 keV are somewhat higher than ours.The measurements of Salem and Moreland ͓7͔ lie systematically ͑about 60%͒ below our results.For W ͓Fig. 6͑a͔͒, the measurements of Chang ͓11͔ are about 40% lower than our experimental results.For Pt ͓Fig.6͑b͔͒, no experimental data were found in the literature.As regards the comparison with the theoret-

TABLE II .
X-ray emission rates, fluorescence yields, and Coster-Kronig transition probabilities used in this work, taken fromScofield ͓36͔ and Werner and Jitschin ͓39͔.