Use of the Bethe equation for inner-shell ionization by electron impact

We analyzed calculated cross sections for K-, L-, and M-shell ionization by electron impact to determine the energy ranges over which these cross sections are consistent with the Bethe equation for inner-shell ionization. Our analysis was performed with K-shell ionization cross sections for 26 elements, with L-shell ionization cross sections for seven elements, L3-subshell ionization cross sections for Xe, and M-shell ionization cross sections for three elements. The validity (or otherwise) of the Bethe equation could be checked with Fano plots based on a linearized form of the Bethe equation. Our Fano plots, which display theoretical cross sections and available measured cross sections, reveal two linear regions as predicted by de Heer and Inokuti [in Electron Impact Ionization, edited by T. D. M€ark and G. H. Dunn, (Springer-Verlag, Vienna, 1985), Chap. 7, pp. 232–276]. For each region, we made linear fits and determined values of the two element-specific Bethe parameters. We found systematic variations of these parameters with atomic number for both the lowand the high-energy linear regions of the Fano plots. We also determined the energy ranges over which the Bethe equation can be used. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4948700]


I. INTRODUCTION
Cross sections for ionization of inner-shell electrons in neutral atoms by electron impact are needed in many branches of physics, including atomic physics, plasma physics, radiation physics, materials analysis by electronprobe microanalysis (EPMA), surface analysis by Augerelectron spectroscopy (AES), and thin-film analysis by electron energy-loss spectroscopy.These cross sections are often used in Monte Carlo simulations to predict, for example, radiation transport in matter and the generation of X-ray and Auger-electron signal intensities in EPMA and AES.
In 1930, Bethe 1 developed a non-relativistic asymptotic expression for the cross section, per atom, for ionization of shell i in the form where E ¼ mv 2 =2 is the kinetic energy of the incident electrons, v is the electron velocity, E i is the binding energy of electrons in shell i, Z i is the number of electrons in that shell, and e and m are the electronic charge and mass, respectively.The parameters b i and c i in Eq. (1) are given by and where c is the velocity of light.The term M 2 i in Eqs. ( 2) and ( 3) is the square of the total dipole-matrix element for ionizing collisions while the term C i in Eq. ( 3) is given by an integral of the generalized oscillator strength for inner-shell ionization. 2 The parameter b i was estimated by Bethe (using hydrogenic wave functions) to be between 0.2 and 0.6 for inner shells and, for a given shell, to be a function of atomic number, Z.The parameter c i was estimated by Bethe to be about 4. Equation (1) was derived with use of the first Born approximation and is expected to be valid when E ) E i and when the average energy transfer in the ionizing collision is much less than E. 2 A relativistic version of Eq. (1), given as Eq. ( 4) below, was published by Fano. 3 Despite the wide use of Eq. ( 1) and its relativistic counterpart for estimating cross sections for inner-shell ionization, there is limited knowledge of the Bethe parameters b i and c i and of the energy ranges over which these equations can be expected to be valid.Powell [4][5][6] reported values of the Bethe parameters from fits to measured cross sections for inner-shell ionization, but the resulting parameter values were of necessity limited to the elements and energy ranges for which measurements were available.
We report a new analysis to determine the ranges of incident energy over which the Bethe equation is valid and the corresponding values of the Bethe parameters b i and c i .We make use of cross sections for K-, L-, and M-shell ionization that were calculated by Bote et al. 7,8 using a composite algorithm that combines the distorted-wave and plane-wave Born approximations.Llovet et al. 9 recently reviewed these calculated cross sections and the available measured cross sections.They identified elements for which there were at least three (for K shells) or two (for L and M subshells) mutually consistent sets of cross-section measurements and for which the cross sections varied with energy as expected by theory; these selections were referred to as superior values.Llovet et al.  found that the overall average root-mean-square deviation between the calculated cross sections and the selected measured values was 10.9% and the overall average deviation was À2.5%.This degree of agreement was considered to be very satisfactory given the difficulties of making absolute crosssection measurements. 94][5][6]10 As suggested by the non-relativistic Bethe equation [Eq.( 1)], a Fano plot is made by plotting r i EE i (e.g., from a set of measured or calculated cross sections over a particular energy range) versus lnðE=E i Þ; as we will see, a similar type of plot can be made based on the relativistic Bethe equation.5][6] However, de Heer and Inokuti 11 have pointed out that Fano plots should display two linear regions if the energy range for the data set is sufficiently large.At relatively low energies, one linear region is found typically beginning at several times the threshold energy for inner-shell ionization, E i . 4Another linear region is expected at higher energies with a smaller slope that occurs in what is termed the asymptotic Bethe region.
The cross sections for K-, L-, and M-shell ionization that were calculated by Bote et al. 7,8 extend from the threshold energy for ionization to 100 GeV.The large energy range for these cross sections is thus well suited for a more FIG.1. Fano plots for C, N, O, Ne, Al, and Si (solid lines) based on K-shell ionization cross sections calculated by Bote et al. 7,8 in which Y Fano from Eq. ( 7) is plotted against X from Eq. (6).The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) in the asymptotic highenergy region of the Fano plots.The dashed lines have been extrapolated to a lower energy at which the deviation from the solid lines is 5%.The solid circles show values of Y Fano calculated from the preferred measured K-shell ionization cross sections of Llovet et al. (Tables 2 and 7 of Ref. 9). extensive analysis of the Bethe equation based on Fano plots.Llovet et al. 9 identified 26 elements (C, N, O, Ne, Al, Si, Ar, Ca, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ga, Ge, Se, Y, Pd, Ag, Sn, Sb, Au, Pb, and Bi) with measured K-shell ionization cross sections that satisfied their selection criteria.They similarly identified seven elements (Ag, Sn, Sm, Ta, Pb, Bi, and U) with satisfactory measurements of total L-shell ionization cross sections and one element (Xe) with satisfactory measurements of L 3 -subshell ionization cross sections.Finally, Llovet et al. identified three elements (Au, Pb, and Bi) that had satisfactory measurements of total M-shell ionization cross sections.We present Fano plots for each of these elements based on the calculated ionization cross sections.These plots clearly show the existence of two distinct linear regions, and we derive values of the Bethe parameters for each linear region.We then show systematic variations of the derived Bethe parameters with atomic number for the two linear regions and systematic differences in the Bethe parameters for K-, L-, and M-shell ionization.

II. THE FANO PLOT
The relativistic form of the Bethe equation for innershell ionization is 2,3 FIG. 2. Fano plots for Ar, Ca, Ti, Cr, Mn, and Fe; see caption to Fig. 1.
where b is the ratio of the electron velocity to the velocity of light, c.It is convenient to rewrite Eq. (4) as where A Fano plot is prepared by plotting versus X as suggested by Eq. ( 5).For linear regions of the Fano plot, The Bethe parameter b i is the slope of the linear region of the plot, and the parameter c i is determined by the ordinate at X ¼ 0, where Y Fano ¼ b i lnc i .In the nonrelativistic limit, X % lnðE=E i Þ ¼ lnU and mc 2 b 2 % 2E where U ¼ E=E i is the overvoltage and E is the electron kinetic energy.Equation ( 7) then becomes For ionization of L and M shells, we need to determine appropriate values of E i and Z i depending on whether particular or all subshells contribute to the relevant ionization process.For ionization of all L subshells, for example, we have where M 2 Lj is the total dipole-matrix element squared for ionization of L-subshell j and c Lj is the corresponding value of the parameter c i .The total number of electrons in the L shell is and the average L-shell binding energy, E L , is where Z Lj is the number of electrons in L-subshell j and E Lj is the binding energy for these electrons.Similar equations can be written for ionization of all M subshells.

A. K-shell ionization cross sections
Figures 1-5 show Fano plots in which Y Fano from Eq. ( 7) is plotted (solid lines) against X from Eq. ( 6) for C, N, O, Ne, Al, Si, Ar, Ca, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ga, Ge, Se, Y, Pd, Ag, Sn, Sb, Au, Pb, and Bi.In these plots, we have used the K-shell ionization cross sections, r K , calculated by Bote et al. 7,8 and the K-shell binding energies tabulated by Williams. 12The abscissa also shows the corresponding electron kinetic energy, E. The Fano plots extend from the threshold for K-shell ionization (X ¼ 0) to X ¼ 20 where the maximum energy ranges from 600 MeV to 10 GeV.The solid circles in Figs.1-5 show values of Y Fano calculated from Eq. (7) with the preferred measurements of K-shell ionization cross sections of Llovet et al. (Tables 2 and 7 of Ref. 9) versus X from Eq. ( 6).
The Fano plots in Figs.1-5 display two linear regions, as expected from the discussion of de Heer and Inokuti. 11e will now determine values of the Bethe parameters for each linear region.The dashed lines in Figs.1-5 show linear fits with Eq. ( 8) to the Y Fano values from the calculated cross sections in the asymptotic high-energy regions of the Fano plots.The dashed lines have been extrapolated to a minimum energy, E min , at which the deviation from the solid lines becomes 5%.Table I shows values of the Bethe parameters b K and c K from the fits for each element, the values of E min , and the corresponding overvoltage, U min .We note that no uncertainties are given for the values of b K and c K since values of b K were determined from the asymptotic slopes of the Fano plots and then values of c K from the corresponding ordinate values at X ¼ 0 where Y ¼ b K lnc K .We see that the values of b K in Table I are 8) over energy regions, E min to E max , where the deviations with respect to the solid lines are less than 5%.
Table II shows values of the derived Bethe parameters b K and c K and their one-standard-deviation uncertainties from the fits for each element, the values of E min and E max , and the corresponding overvoltage values, U min and U max .We see that b K decreases from 0.806 for C and about 0.84 for N, O, and Ne to about 0.58 for Pb and Bi.These b K values are about 50% larger than those in Table I for the low-Z

TABLE I.
Values of the Bethe parameters, b K and c K , derived from the linear fits with Eq. ( 8) in the high-energy asymptotic regions of the Fano plots with K-shell ionization cross sections in Figs.1-5.We also show the minimum kinetic energy, E min , at which the deviation of the fitted lines in the Fano plots of Figs.1- We note that the fractional uncertainties for c K in Table II are larger than those for b K because the uncertainties in c K are determined from the uncertainties of both b K and b K lnc K in each fit with Eq. ( 8).We see from Table II that the linear regions in Figs.6-10 range from U min % 1:7 to U max % 8 for the high-Z elements while the linear regions for the low-Z elements range from U min % 2:2 to U max % 20.The linear regions for the elements C through Ni occur, as a group, from U min ¼ 2:3 to U max ¼ 15:6.This overvoltage range is comparable with that of the 1976 Powell 4 analysis (4-24) although the lower limit of the overvoltage range found here (2.3) is about a factor of 2 smaller than in the sets of data analyzed by Powell.
Values of U min for the high-energy linear regions in Table I range from about 40 for the low-Z elements to about 400 for the high-Z elements.Values of U max for the lowenergy linear regions in Table II range from about 20 for the low-Z elements to about 8 for the high-Z elements.The Fano plots thus change their slopes (between the two linear regions) over a narrow range of X for the low-Z elements in Fig. 1 while the slope change is much more gradual for the high-Z elements in Figs. 4 and 5.   3 and 9 of Ref. 9).Values of Y Fano from Eq. ( 7) were calculated with the average L-shell binding energies determined from the L-subshell binding energies of Williams 12 for each element that were weighted by the corresponding subshell occupancy numbers, as indicated by Eq. (11b).The abscissas in Figs.11 and 12 also show the electron kinetic energy.As before, the Fano plots extend from the ionization threshold (X ¼ 0) to X ¼ 20 where the maximum energy is between 2 GeV and 5 GeV.

B. L-shell ionization cross sections
The Fano plots in Figs.11 and 12 display two linear regions like those in Figs.1-5 for K-shell ionization.We have again made linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) in the asymptotic highenergy linear regions of each Fano plot.These fits, indicated by the dashed lines, have been extrapolated, as before, to a minimum energy, E min , at which the deviations from the solid lines become 5%.We also show Fano plots in Figs.

184904-10
and U min ¼ 61.2.These values are similar to the corresponding values from the total L-shell ionization cross sections for neighboring elements in Table III.For the linear fit to the low-energy region for Xe in Fig. 13, we found b L3 ¼ 0.741 6 0.001, c L3 ¼ 1.165 6 0.003, E min ¼ 9.3 keV, E max ¼ 89.8 keV, U min ¼ 1.9, and U max ¼ 18.8.These values are also similar to the corresponding values in Table IV derived from total L-shell ionization cross sections.
Finally, we note that the linear regions of the lowenergy Fano plots in Figs. 13 and 14 range from U min ¼ 1.5 to U max % 19 for Ag to U min ¼ 1.9 to U max % 12 for U. We also see that the values of U min in Table III for the highenergy Fano plots in Figs.11 and 12 range from about 60 for Ag to about 162 for U.The Fano plots in Figs.11 and 12 thus change their slopes (between the two linear regions) over a smaller range of X than for the Fano plots in Figs.1-5.7) were calculated with the average M-shell binding energies determined from the M-subshell binding energies of Williams 12 for each element that were weighted by the corresponding subshell occupancy numbers using an equation analogous to Eq. (11b) for L-shell ionization.The abscissas in Figs.11 and 12 also show the electron kinetic energy.As before, the Fano plots extend from the ionization threshold (X ¼ 0) to X ¼ 20 where the maximum energy is between 1 GeV and 2 GeV.

C. M-shell ionization cross sections
Like Figs.1-5, 11, and 12, the Fano plots in Fig. 15 show two linear regions.We have made linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) in the asymptotic high-energy linear regions of each Fano plot.These fits (dashed lines) have been extrapolated to a minimum energy, E min , at which the deviations from the solid lines become 5%.Figure 16 shows expanded views of the low-energy regions, from X ¼ 0 to X ¼ 4, of the Fano plots in Fig. 15.As previously mentioned, the dashed lines in Fig. 16 are linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) over energy regions, E min to E max , where the deviations with respect to the solid line are less than 5%.
Table VI shows values of the resulting Bethe parameters, b M and c M , and their one-standard-deviation uncertainties as TABLE II.Values of the Bethe parameters, b K and c K , and their one-standard-deviation uncertainties derived from fits with Eq. ( 8) in the low-energy linear regions of the Fano plots with K-shell ionization cross sections in Figs.6-10.We also show the minimum and maximum kinetic energies, E min and E max , at which the deviation of the fitted lines in the Fano plots of Figs.6-10   7,8 In these plots, Y Fano from Eq. ( 7) is plotted against X from Eq. ( 6).The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) in the asymptotic high-energy region of the Fano plots.8) in the high-energy asymptotic regions of the Fano plots in Figs.11 and 12 for Ag, Sn, Sm, Ta, Pb, Bi, and U with total L-shell ionization cross sections.We also show the minimum kinetic energy, E min , at which the deviation of the fitted lines in the Fano plots of Figs.11 and  13.Expanded view of the Fano plots for Ag, Sn, Xe, Sm, Ta, and Pb in Fig. 11 (solid lines) from X ¼ 0 to X ¼ 4. The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) over the energy range, E min to E max , where the deviations from the solid lines are less than 5%.The solid circles show values of Y Fano calculated from the measured total Lshell ionization cross sections (or, for Xe, the preferred measured L 3 -subshell ionization cross sections) of Llovet et al. (Tables 3 and 9 of Ref. 9).

IV. DISCUSSION
Figure 17 shows plots of the Bethe parameters b i and c i (i ¼ K, L, or M) as a function of Z from fits with Eq. ( 8) in the high-and low-energy regions of the Fano plots in Figs.1-16.The top panels of Fig. 17   The values of c K from the high-energy fits (top-right panel of Fig. 17) increase monotonically with Z from about 5.9 for C to 116 for Bi.The values of c L , ranging from about 15 for Ag to about 70 for U, are roughly half of the values of c K for medium-Z and high-Z elements.Finally, the values of c M , between 7.8 and 8.9 for Au, Pb, and Bi, are about 13 times smaller than the values of c K for the same elements.
We turn now to the values of b i and c i in the bottom panels of Fig. 17  Overall, the b i values from the low-energy fits range from between 0.806 for C and about 0.84 for N, O, and Ne (for K-shell ionization) to about 0.7 as a rough average of the values of b K , b L , and b M for the high-Z elements.In comparison, the b i values from the high-energy fits range from about 0.52 for C, N, O, and Ne to b K values of about 0.28 for Au, Pb, and Bi, to b L values of about 0.32 for Pb, Bi, and U, and to b M values of about 0.53 for Au, Pb, and Bi.The b K values from the low-energy fits are thus about 60% larger than those from the high-energy fits for the low-Z elements, the b L values from the low-energy fits are roughly double those from the high-energy fits for the medium-and high-Z elements, and the b M values from the low-energy fits are about 50% larger than those from the high-energy fits.
The values of c K from the low-energy fits (bottom-right panel of Fig. 17) increase monotonically with Z from 0.796 for C to 1.043 for Sb but are then about 1.3 for Au, Pb, and Bi.The values of c L range from 1.078 for Ag to 1.297 for U while the values of c M are about 1.26 for Au, Pb, and Bi.These c i values are appreciably less than those obtained from the high-energy fits, which can be larger by between roughly one to two orders of magnitude.
The values of b i from the high-energy fits (top-left panel of Fig. 17 8) in the low-energy linear regions of the Fano plots with total L-shell ionization cross sections in Figs. 13 and 14.We also show the minimum and maximum kinetic energies, E min and E max , at which the deviation of the fitted lines in the Fano plots of Figs. 13 and 14 are less than 5% and the corresponding values of the overvoltage, U min and U max .
E max (keV) U min U max (0.2 to 0.6) with hydrogenic wave functions.However, the values of c i from the high-energy fits (top-right panel of Fig. 17), between 5 and 116, are generally much larger than the value estimated by Bethe (about 4).We now comment briefly on the measured cross sections for K-shell ionization shown as solid circles in Figs.1-5.For some of the measured K-shell ionization cross sections (e.g., Si in Fig. 1; Ca, Cr, and Mn in Fig. 2; Ni, Cu, and Ge in Fig. 3; and Y, Ag, Sn, and Au in Fig. 4), there are sufficient measurements in both the low-energy and high-energy regions of the Fano plots to define adequately the two expected linear regions. 11These two regions have not been separately observed before because cross-section measurements made in a particular laboratory were generally made over a limited range of incident electron energies.Other measured K-shell ionization cross sections (e.g., C, N, O, and Ne in Fig. 1; Ti and Fe in Fig. 2; Ga in Fig. 3; and Sb in Fig. 4) were made with incident energies that occurred only in the low-energy regions of the Fano plots.There are fewer measured cross sections for L-shell ionization in Figs.11 and 12 and for M-shell ionization in Fig. 15 than for K-shell ionization and it is consequently harder to discern, for any one element, the existence of two linear regions in the Fano plots.The calculated ionization cross sections of Bote et al. 7,8 were thus invaluable not only for providing a framework for the evaluation of the measured cross sections 9 but also for clearly indicating, by the Fano plots shown here, the existence of two linear regions in the Fano plots for each element.

V. SUMMARY
2][3] The Bethe equation is a relatively simple analytical expression with two element-specific parameters, b i and c i .The validity (or otherwise) of the Bethe equation for a particular set of calculated or measured ionization cross sections can be checked with a Fano plot 3 based on a linearized form of the Bethe equation.in which Y Fano from Eq. ( 7) is plotted against X from Eq. ( 6).The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. ( 8) in the asymptotic high-energy region of the Fano plots.The dashed lines have been extrapolated to a lower energy at which the deviation from the solid lines is 5%.The solid circles show values of Y Fano calculated from the preferred measured total M-shell ionization cross sections of Llovet et al. (Tables 4 and 11 of Ref. 9).TABLE V. Values of the Bethe parameters, b M and c M , derived from the linear fits with Eq. ( 8) in the high-energy asymptotic regions of the Fano plots in Fig. 15 for Au, Pb, and Bi with total M-shell ionization cross sections.We also show the minimum kinetic energy, E min , at which the deviation of the fitted lines in the Fano plots of Fig. 15  Our analysis was performed with calculated 7,8 K-shell ionization cross sections for 26 elements (C, N, O, Ne, Al, Si, Ar, Ca, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ga, Ge, Se, Y, Pd, Ag, Sn, Sb, Au, Pb, and Bi), for total L-shell ionization cross sections for seven elements (Ag, Sn, Sm, Ta, Pb, Bi, and U), with L 3 -subshell ionization cross sections of Xe, and with total M-shell ionization cross sections of three elements (Au, Pb, and Bi).We have previously shown that the calculated cross sections for these elements and shells, available from the threshold energy for ionization to 1 GeV, agreed satisfactorily with sets of measured cross sections that were mutually consistent with each other and which varied with energy as expected by theory. 9ur Fano plots for each element and shell showed two linear regions as predicted by de Heer and Inokuti. 11For each region, we made linear fits and obtained values of the Bethe parameters, b i and c i .Values of b i for the high-energy regions were determined from the asymptotic high-energy slopes of the Fano plots after which c i values could be calculated.The resulting straight lines were extrapolated to lower energies at which the deviations between the straight lines and each plot became 5%.The linear fits for the low-energy regions were made over energy ranges for which the deviations between the fit and each plot were also less than 5%.Our values of b i and c i for each element and shell are presented in Tables I-VI together with the energy ranges for which they can be used.For the high-energy fits, the minimum overvoltage ranged from about 40 for K-shell ionization of C, N, O, and Ne to around 430 for Pb and Bi while the minimum overvoltage varied from about 60 for L-shell ionization of Ag and Sn to around 160 for U; for M-shell ionization of Au, Pb, and Bi, the minimum overvoltage was around 38.For the low-energy fits, the overvoltage ranged from between around 2 and 20 for K-shell ionization of C, N, O, and Ne to between 1.7 and about 8 for Au, Pb, and Bi, and the corresponding overvoltage ranged from between 1.6 and about 18 for L-shell ionization of Ag and Sn to between 1.9 and about 12 for Pb, Bi, and U; for M-shell ionization, the overvoltage ranged between about 2 and about 30 for Au, Pb, and Bi.
We found systematic variations of b i and c i with Z. Values of b K for the high-energy regions of the Fano plots decreased monotonically from about 0.52 for C, N, O, and  8) in the lowenergy linear regions of the Fano plots with total M-shell ionization cross sections in Fig. 16.We also show the minimum and maximum kinetic energies, E min and E max , at which the deviation of the fitted lines in the Fano plots of Fig. 16  Ne to about 0.28 for Au, Pb, and Bi while values of b L decreased from about 0.41 for Ag and Sn to about 0.32 for Pb, Bi, and U, and b M was about 0.53 for Au, Pb, and Bi.There were much larger variations in c i with Z for the highenergy regions, with c K increasing from about 5.9 for C to 116 for Bi, and c L increasing from about 15 for Ag to about 70 for U.For the low-energy regions of the Fano plots, b K decreased from about 0.84 for N, O, and Ne to 0.581 for Bi, the values of b L decreased from about 0.74 for Ag to about 0.68 for U, and b M was about 0.79 for Au, Pb, and Bi.There are much smaller variations of c i with Z for the low-energy fits than for the high-energy fits, with c K varying between 0.796 for C to about 1.3 for Au, Pb, and Bi, with c L ranging between 1.078 for Ag to 1.297 for U, and with c M about 1.26 for Au, Pb, and Bi.Our values of b i and c i for the elements in Tables I-VI can be used to determine cross sections from the Bethe equation for ionization of K, L, and M shells by electron impact for the designated energy or overvoltage ranges.Similar evaluations can be made for other elements with values of b i and c i determined by interpolation.For many applications, however, it may be more convenient to determine these cross sections, for any element and for any energy from the ionization threshold to 100 GeV, from the empirical equations and parameters published by Bote et al. 8
Figures 1-5 show Fano plots in which Y Fano from Eq. (7) is plotted (solid lines) against X from Eq. (6) for C, N, O, Ne, Al, Si, Ar, Ca, Ti, Cr, Mn, Fe, Ni, Cu, Zn, Ga, Ge, Se, Y, Pd, Ag, Sn, Sb, Au, Pb, and Bi.In these plots, we have used the K-shell ionization cross sections, r K , calculated by Bote et al.7,8 and the K-shell binding energies tabulated by Williams.12The abscissa also shows the corresponding electron kinetic energy, E. The Fano plots extend from the threshold for K-shell ionization (X ¼ 0) to X ¼ 20 where the maximum energy ranges from 600 MeV to 10 GeV.The solid circles in Figs.1-5show values of Y Fano calculated from Eq. (7) with the preferred measurements of K-shell ionization cross sections of Llovet et al. (Tables2 and 7of Ref. 9) versus X from Eq. (6).The Fano plots in Figs.1-5 display two linear regions, as expected from the discussion of de Heer and Inokuti.11We will now determine values of the Bethe parameters for each linear region.The dashed lines in Figs.1-5show linear fits with Eq. (8) to the Y Fano values from the calculated cross sections in the asymptotic high-energy regions of the Fano plots.The dashed lines have been extrapolated to a minimum energy, E min , at which the deviation from the solid lines becomes 5%.TableIshows values of the Bethe parameters b K and c K from the fits for each element, the values of E min , and the corresponding overvoltage, U min .We note that no uncertainties are given for the values of b K and c K since values of b K were determined from the asymptotic slopes of the Fano plots and then values of c K from the corresponding ordinate values at X ¼ 0 where Y ¼ b K lnc K .We see that the values of b K in TableIare about 0.52 for C, N, O, and Ne, and then gradually decrease with increasing Z to 0.281 for Bi.The values of c K in Table I, however, increase from 5.85 for C to 116 for Bi.The solid lines in Figs.6-10 show expanded views of the low-energy regions of the Fano plots in Figs.1-5, from X ¼ 0 to X ¼ 4. The dashed lines in Figs.6-10 indicate linear

FIG. 6 .
FIG. 6. Expanded view of the Fano plots for C, N, O, Ne, Al, and Si in Fig. 1 (solid lines) from X ¼ 0 to X ¼ 4. The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. (8) over the energy range, E min to E max , where the deviations from the solid lines are less than 5%.The solid circles show values of Y Fano calculated from the preferred measured K-shell ionization cross sections of Llovet et al. (Tables 2 and 7 of Ref. 9).

Figures 11 and 12
Figures 11 and 12 show Fano plots based on total cross sections for L-shell ionization, r L , in Ag, Sn, Sm, Ta, Pb, Bi, and U.As for Figs.1-10, we have plotted Y Fano from Eq. (7) versus X from Eq. (6).The solid lines show Y Fano with the calculated cross sections of Bote et al. 7,8 and the solid circles show Y Fano with the preferred measured cross sections of Llovet et al. (Tables3 and 9of Ref. 9).Values of Y Fano from Eq. (7) were calculated with the average L-shell binding energies determined from the L-subshell binding energies of Williams12 for each element that were weighted by the corresponding subshell occupancy numbers, as indicated by Eq. (11b).The abscissas in Figs.11 and 12 also show the electron FIG. view of the Fano plots for Ni, Cu, Zn, Ga, Ge, and Se in Fig.3from X ¼ 0 to X ¼ 4; see caption to Fig.6.

Figure 15 shows
Figure15shows Fano plots based on total crosssections for M-shell ionization, r M , in Au, Pb, and Bi.As for Figs.1-14, we have plotted Y Fano from Eq. (7) versus X from Eq. (6).The solid lines show Y Fano with the calculated cross sections of Bote et al.7,8 and the solid circles show Y Fano with the preferred measured cross sections of Llovet et al. (Tables4 and 11of Ref. 9).Values of Y Fano from Eq. (7) were calculated with the average M-shell binding energies determined from the M-subshell binding energies Fig. 15 are about 38.Unlike the L-shell Fano plots in Figs.11 and 12, the M-shell Fano plots in Fig. 15 change their slopes between the two linear regions over a narrow range of X.
Figure17shows plots of the Bethe parameters b i and c i (i ¼ K, L, or M) as a function of Z from fits with Eq. (8) in the high-and low-energy regions of the Fano plots in Figs.1-16.The top panels of Fig.17show plots of b i and c i from the fits in the high-energy regions of the Fano plots shown in Figs.1-5 for K-shell ionization, Figs.11 and 12 for L-shell ionization, and Fig. 15 for M-shell ionization, as listed in Tables I, III, and V.We see that the values of b K (top-left panel) decrease monotonically from about 0.52 for C, N, O, and Ne to about 0.28 for Au, Pb, and Bi.The values of b L in the top-left panel, decreasing from about 0.41 for Ag and Sn to about 0.32 for Pb, Bi, and U, are about 15% larger than the b K values for medium-Z (Ag and Sn) and high-Z elements (Pb and Bi).In contrast, the values of b M in the top-left panel of Fig. 17(about 0.53 for Au, Pb, and Bi) are about 90% larger than the b K values for the same elements (0.28).The values of c K from the high-energy fits (top-right panel of Fig.17) increase monotonically with Z from about 5.9 for C to 116 for Bi.The values of c L , ranging from about 15 for Ag to about 70 for U, are roughly half of the values of c K for medium-Z and high-Z elements.Finally, the values of c M , between 7.8 and 8.9 for Au, Pb, and Bi, Figure17shows plots of the Bethe parameters b i and c i (i ¼ K, L, or M) as a function of Z from fits with Eq. (8) in the high-and low-energy regions of the Fano plots in Figs.1-16.The top panels of Fig.17show plots of b i and c i from the fits in the high-energy regions of the Fano plots shown in Figs.1-5 for K-shell ionization, Figs.11 and 12 for L-shell ionization, and Fig. 15 for M-shell ionization, as listed in Tables I, III, and V.We see that the values of b K (top-left panel) decrease monotonically from about 0.52 for C, N, O, and Ne to about 0.28 for Au, Pb, and Bi.The values of b L in the top-left panel, decreasing from about 0.41 for Ag and Sn to about 0.32 for Pb, Bi, and U, are about 15% larger than the b K values for medium-Z (Ag and Sn) and high-Z elements (Pb and Bi).In contrast, the values of b M in the top-left panel of Fig. 17(about 0.53 for Au, Pb, and Bi) are about 90% larger than the b K values for the same elements (0.28).The values of c K from the high-energy fits (top-right panel of Fig.17) increase monotonically with Z from about 5.9 for C to 116 for Bi.The values of c L , ranging from about 15 for Ag to about 70 for U, are roughly half of the values of c K for medium-Z and high-Z elements.Finally, the values of c M , between 7.8 and 8.9 for Au, Pb, and Bi, that were obtained from the fits in the lowenergy regions of the Fano plots shown in Figs.6-10 for K-shell ionization, Figs. 13 and 14 for L-shell ionization, and Fig. 16 for M-shell ionization, as listed in Tables II, IV, and VI.The value of b K (bottom-left panel) is 0.806 for C, increases to about 0.84 for N, O, and Ne, and then gradually decreases to 0.581 for Bi.The values of b L for Ag and Sn (about 0.73) are essentially the same as the values of b K for the same elements.However, the values of b L for Pb and Bi (about 0.69) are around 17% larger than the values of b K for the same elements (about 0.59).We see that the values of b M for Au, Pb, and Bi (about 0.79) are about 34% larger than the b K values for the same elements (about 0.59).
FIG. 14. Expanded view of the Fano plots for Bi and U in Fig. 12 from X ¼ 0 to X ¼ 4; see caption to Fig. 13.

FIG. 15 .
FIG.15.Fano plots for Au, Pb, and Bi (solid lines) based on total M-shell ionization cross sections calculated by Bote et al.7,8 in which Y Fano from Eq. (7) is plotted against X from Eq. (6).The abscissa also shows the electron kinetic energy, E. The dashed lines show linear fits to the Y Fano values from the calculated cross sections with Eq. (8) in the asymptotic high-energy region of the Fano plots.The dashed lines have been extrapolated to a lower energy at which the deviation from the solid lines is 5%.The solid circles show values of Y Fano calculated from the preferred measured total M-shell ionization cross sections of Llovet et al. (Tables4 and 11of Ref. 9).

FIG. 16 .
FIG. 16.Expanded of the Fano plots for Au, Pb, and Bi in Fig. 15 (solid lines) from X ¼ 0 to X ¼ 4; see also caption to Fig. 15.
5 becomes 5% and the corresponding value of the overvoltage, U min .elementsandabout100% larger than those in TableIfor the high-Z elements.In contrast to the relatively large increases of c K in Table I with increasing Z for the high-energy regions of the Fano plots (an increase by a factor of about 20), the values of c K in Table II increase from 0.796 6 0.002 for C to 1.333 6 0.007 for Bi, an increase of less than a factor of 2.
Table III shows the Bethe parameters, b L and c L , from each fit, the values of E min , and the corresponding values of U min .We see that the values of b L in Table III decrease from about 0.42 for Ag to about 0.31 for FIG.Expanded view of the Fano plots for Ar, Ca, Ti, Cr, Mn, and Fe in Fig. 2 from X ¼ 0 to X ¼ 4; see caption to Fig. 6.U while the values of c L increase from about 15 for Ag to about 70 for U.The b L values in Table III are between 11% and 17% larger than the values of b K in Table I that decrease from about 0.36 for Ag to about 0.28 for Bi.However, the c L values in Table III are appreciably smaller than the c K values in Table I which range from 34 for Ag to 116 for Bi.Figures13 and 14show expanded views of the lowenergy regions, from X ¼ 0 to X ¼ 4, of the Fano plots in Figs.11 and 12.The dashed lines in Figs. 13 and 14 are, as before, linear fits to the Y Fano values from the calculated cross sections with Eq. (8) over energy regions, E min to E max , where the deviations with respect to the solid lines are less than 5%.TableIVshows values of the derived Bethe parameters b L and c L and their one-standard-deviation uncertainties from the fits for each element, the values of E min and E max , and the corresponding overvoltage values, U min and U max .We see that b L decreases from about 0.74 for Ag to 0.68 for U while c L increases slightly from about 1.08 for Ag to about 1.30 for U.The b L values in TableIVare larger than the corresponding values in Table III (by around a factor of 2 for Ta, Pb, Bi, and U) while the values of c L in Table IV are more than an order of magnitude smaller than those in Table III.We also see that the values of b L and c L in Table IV are roughly comparable with the values of b K and c K in Table II for the elements Ag through Bi.
Table V shows the Bethe parameters, b M and c M , from each fit, the values of E min , and the corresponding values of U min .The values of b M in Table V for Pb and Bi are about 0.53, appreciably larger than the b L values for these elements in Table III (0.33).In contrast, the c M values in Table V for Pb and Bi (about 8.8) are much smaller than the c L values for these elements in Table III (about 51).
are less than 5% and the corresponding values of the overvoltage, U min and U max .
well as the values of E min , E max , U min , and U max .We see that the values of b M in Table VI for Pb are Bi (about 0.790) are larger than the b L values for these elements in Table IV (0.690) while the c M values in Table VI for the same elements (about 1.27) are essentially the same as the corresponding c L values in Table IV (1.24).The linear regions of the low-energy Fano plots in Fig.16extend from U min ¼ 2.1 for Au, Pb, and Bi to values of U max ranging from about 32 for Au to about 29 for Bi.The values of U min for the high-energy Fano plots in

TABLE III .
Values of the Bethe parameters, b L and c L , derived from the linear fits with Eq. ( 12becomes 5% and the corresponding value of the overvoltage, U min .
becomes 5% and the corresponding value of the overvoltage, U min .
are less than 5% and the corresponding values of the overvoltage, U min and U max .