Post-print, re-edited for inclusion in T. Tabata, edited with commentary,The Collected Works of Tatsuo Tabata Volume 11, IDEA-TR 15 (2018), of the paperpublished in Nuclear Instruments and Methods in Physics Research B, Volume 31,

Issue 3, 1 May 1988, Pages 375–381 (doi:10.1016/0168-583X(88)90334-5)Copyright © 1988 by Elsevier B.V.

Analytic Cross Sections for Charge Transfer of HydrogenAtoms and Ions Colliding with Metal Vapors∗

Tatsuo Tabata and Rinsuke ItoRadiation Center of Osaka Prefecture, Sakai, Osaka 593, Japan

Yohta Nakai, Toshizo Shirai and Masao SatakaTokai Research Establishment, Japan Atomic Energy Research Institute, Naka-gun,

Ibaraki 319-11, Japan

Toshio Sugiura

Integrated Arts and Sciences, University of Osaka Prefecture, Sakai, Osaka 591, Japan

(Received 11 November 1987 and in revised form 12 January 1988)

Analytic formulas are given for the total cross sections, σ, for the charge transferin collisions of H+, H− and H with metal vapors. The cross sections considered areσ10, σ0−1, σ1−1, σ01, σ−10, and σ−11, where the subscripts represent the initial andthe final charge state of the projectile. The functional form of the formulas is amodification of the semiempirical formula proposed by Green and McNeal for σ10 ofH+ in gaseous atoms and molecules. Values of adjustable parameters in the formulashave been determined by least-squares fits to a compiled set of experimental data.The root-mean-square deviation of the data from the formula is from 7 to 34% foreach type of cross section. The main cause of larger deviations is the discrepancyamong the data.

1. IntroductionCharge transfer in collisions between atomic particles is one of the important processes

in plasma physics, radiation physics, astrophysics and other areas. In particular, knowl-edge of the cross sections for charge transfer of hydrogen ions and atoms is necessaryfor thermonuclear fusion research [1]. In this research the charge transfer of hydrogen inmetal vapors has relevance to (1) neutralization of negative ions for neutral-beam heating,(2) interaction with impurities at the plasma edge and (3) use of ion beams of metals forplasma diagnostics [2]. The latest compilation of the total cross section for charge transferof hydrogen in metal vapors has been given by Morgan et al. [3].

∗Part of this work was performed at the Radiation Center of Osaka Prefecture (RCOP) under acontract with Japan Atomic Energy Research Institute. A preliminary account of the work has beengiven in RCOP Technical Report No. 6 (1987).

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Several theoretical approaches have been developed to calculate the charge-transfercross sections (see, for example, ref. [3]). These approaches, however, require time-consuming computation, and it is impracticable to incorporate them in computationsof the problems that call for the cross sections as input data. For rapid evaluation of thecross sections, therefore, it is desirable to have analytic formulas fitted to experimentalor computed data. In a previous paper [4], we have given analytic formulas to expressthe experimental total cross sections for the charge transfer of hydrogen in gaseous atoms

Fig. 1. Cross section as a function of pro-jectile energy. σ10 of H+ in Li vapor. Thesolid line: the analytic formula fitted tothe data; the dashed line: the values rec-ommended by Morgan et al. ref. [3]; var-ious symbols: experimental data. In thelegend, the letter D in parentheses afterauthor’s name indicate that the resultswere deduced from experiment with deu-terium projectiles.

Fig. 2. Cross section as a function of pro-jectile energy. σ0−1 of H+ in Cs vapor.The solid line: the analytic formula fittedto the data; the dashed line: the valuesrecommended by Morgan et al. [3]; vari-ous symbols: experimental data. In thelegend, the letter D in parentheses afterauthor’s name indicate that the resultswere deduced from experiment with deu-terium projectiles.

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and molecules. The present paper gives similar formulas for the targets of metal vapors.The charge-transfer processes considered are:

(1) σ10 for single-electron capture of H+,(2) σ0−1 for single-electron capture of H,(3) σ1−1 for double-electron capture of H+,(4) σ01 for single-electron loss of H,(5) σ−10 for single-electron loss of H−,

Fig. 3. Cross section as a function of pro-jectile energy. σ1−1 of H+ in Mg vapor.The solid line: the analytic formula fittedto the data; the dashed line: the valuesrecommended by Morgan et al. [3]; vari-ous symbols: experimental data.

Fig. 4. Cross section as a function of pro-jectile energy. σ01 of H+ in Sr vapor. Thesolid line: the analytic formula fitted tothe data; the dashed line: the values rec-ommended by Morgan et al. [3]; varioussymbols: experimental data.

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(6) σ−11 for double-electron loss of H−.Here the subscripts to σ represent the initial and the final charge state of the projectile.For each of the six processes, the experimental database included targets of Li, Na, K,Rb and Cs vapors as well as all or some of the following targets: Mg, Ca, Sr, Cd, Ba andPb vapors. The parameters in the formulas have been determined by least-squares fits tothe experimental data compiled.

Fig. 5. Cross section as a function of pro-jectile energy. σ−10 of H+ in Ca vapor.The solid line: the analytic formula fittedto the data; various symbols: experimen-tal data.

Fig. 6. Cross section as a function of pro-jectile energy. σ−11 of H+ in Na vapor.The solid line: the analytic formula fittedto the data; the dashed line: the valuesrecommended by Morgan et al. [3]; vari-ous symbols: experimental data.

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Table 1Parameters in the analytic formula for the single-electron-capture cross section σ10of H+. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 8) = adjustableparameters; a3, a5 and a8 in keV. Values in parentheses are those assumed in thefit

Target Et a1 a2 a3 a4Li −8.21E−03 2.88E+04 (2.0E+00) 6.3 E−01 −7.1 E−01Na −8.46E−03 8.16E+04 (2.0E+00) (6.0 E−01) −5.0 E−01Mg −5.95E−03 4.69E+01 (3.0E−01) (1.5 E+00) −8.5 E−01K −9.26E−03 2.89E+05 (2.0E+00) (6.0 E−01) 4.0 E−01Ca −7.48E−03 1.55E+04 (2.0E+00) (6.0 E−01) −7.53E−01Rb −9.42E−03 4.71E+05 (2.0E+00) (4.5 E−01) (4.0 E−01)Sr −7.90E−03 1.68E+04 (2.0E+00) (6.0 E−01) −7.5 E−01Cs −9.70E−03 3.99E+04 (1.0E+00) 1.67E−01 5.00E−01Ba −7.79E−03 2.02E+04 (2.0E+00) (6.0 E−01) −6.4 E−01Pb −6.18E−03 1.75E+03 (2.0E+00) (1.4 E+00) −(6.4 E−01)

Target a5 a6 a7 a8Li 3.72E+00 2.96E+00 2.8 E−03 1.56E+01Na 3.21E+00 2.86E+00 5.4 E−03 1.75E+01Mg 1.28E+01 3.34E+00 4.5 E−03 (1.0 E+01)K 3.06E+00 3.66E+00 3.00E−02 1.10E+01Ca 4.55E+00 3.07E+00 (2.0 E−02) (1.0 E+01)Rb (2.3 E+00) (3.7 E+00) (3.0 E−02) (1.0 E+01)Sr 3.67E+00 2.54E+00 3.65E−02 (1.0 E+01)Cs 2.53E+00 2.21E+00 (3.0 E−02) (5.0 E+01)Ba 2.88E+00 2.20E+00 3.35E−02 (1.0 E+01)Pb 6.48E+00 1.95E+00 (3.0 E−02) (1.0 E+01)

2. Data set usedThe database used were prepared by Nakai et al. [5]. References up to the middle of

1984 were covered [6–27]. No limitation has been placed on the projectile energy in thecompilation. The data collected are stored in the Atomic and Molecular Data Storage andRetrieval System (AMSTOR) of Japan Atomic Energy Research Institute; magnetic-tapecopy of the data is available upon request.

3. FormulationThe curve of the total cross-section for the charge-transfer processes between unlike

atomic species versus projectile energy, when plotted in logarithmic scales, shows generallya linear increase at the lowest energies, a broad peak,† and a linear decrease at the highestenergies. A structure in the form of a subpeak or a shoulder often appears (see figs. 1–6and also figures in ref. [4]).

First we consider an analytic function to express the basic feature of the cross-sectioncurve without the structure. For this purpose. we use the semiempirical function of Greenand McNeal [28]. They used this function to fit the data on σ10 of H+ in gaseous atoms

†The maximum of the cross section for the charge-transfer processes of hydrogen atoms and ionscolliding with other atoms and molecules often lies around the energy ER defined later.

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and molecules. We rewrite it as follows:

f(E1) = a1(E1/ER)a2/[1 + (E1/a3)a2+a4 + (E1/a5)a2+a6 ], (1)

where f is the quantity that gives the cross section when multiplied by a convenient crosssection unit σ0 (here 10−16 cm2), E1 is given by

E1 = E0 − Et, (2)

ER (= 25.00 keV) is the Rydberg energy multiplied by the ratio of the atomic-hydrogenmass to the electron mass, E0 is the projectile energy, and Et is the threshold energy ofthe process.

The physical bases of eq. (1) are as follows [28]:(1) The factor Ea2

1 , introduced to provide for the threshold behavior of the crosssection, agrees with the low-energy form of the approximate expression for σ10 derivedtheoretically by Rapp and Francis [29], if a2 is made equal to 2.

(2) In the intermediate-energy region, eq. (1) is nearly proportional to E−a4 , which isa good approximation to the form of the Rapp–Francis expression for this region.

(3) At high energies, the asymptotic form of eq. (1) simulates the results of Bohr’ssemiclassical argument and the Born approximation, if a6 takes on a value from 3 to 6.

The approximate theories cited above are not always applicable to the various charge-transfer processes we consider. From the mathematical point of view, however, eq. (1)

Table 2Parameters in the analytic formula for the single-electron-capture cross section σ0−1of H. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 8) = adjustableparameters; a3, a5 and a8 in keV. Values in parentheses are those assumed in thefit

Target Et a1 a2 a3 a4Li 4.64E−03 5.60E+04 (3.5 E+00) (1.0E+00) 1.5 E−01Na 4.38E−03 6.88E+05 (4.0 E+00) 9.7E−01 −7.7 E−02Mg 6.89E−03 1.52E+03 (3.0 E+00) (2.4E+00) (1.5 E−01)K 3.58E−03 6.26E+02 1.48E+00 1.0E+00 1.36E+00Ca 5.36E−03 5.41E+02 (1.5 E+00) (1.0E+00) 6.40E−01Rb 3.42E−03 4.37E+02 (1.0 E+00) 8.4E−01 5.6 E−01Sr 4.94E−03 2.3 E+02 (1.0 E+00) 5.8E−01 7.3 E−01Cs 3.14E−03 7.4 E+02 8.9 E−01 4.3E−01 1.05E+00Pb 6.66E−03 4.24E+01 (1.0 E+00) (1.7E+00) (7.3 E−01)

Target a5 a6 a7 a8Li 3.08E+00 (3.5 E+00) (8.0 E−03) (2.0 E+01)Na 2.54E+00 3.4 E+00 8.1 E−03 2.1 E+01Mg 5.22E+00 2.97E+00 5.37E−03 (2.0 E+01)K 2.76E+00 4.29E+00 2.01E−02 2.71E+01Ca 6.46E+00 5.44E+00 (2.0 E−02) (3.0 E+01)Rb 1.28E+00 (2.0 E+00) 4.93E−03 (4.0 E+01)Sr 4.7 E+00 4.5 E+00 2.19E−02 6.3 E+00Cs 2.60E+00 5.56E+00 1.57E−02 3.01E+01Pb 1.32E+01 (4.5 E+00) (2.0 E−02) (6.0 E+00)

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has a possibility of reproducing the basic features common to these processes, if we makethe most of its flexibility. In our extended use of eq. (1), therefore, we regard all theparameters from a1 to a6 adjustable (in fact, Green and McNeal have also adjusted theseparameters except a2), and allow a4 to take on either positive or negative values.

Next we consider how to reproduce the structure in the cross-section curve. Thestructure is caused by different contributing processes. These processes are, for example,the capture of different orbital electrons of the target atom in the case of σ10, and theformation of different final products, B2+ and B3+ +e, in the case of σ1−1, where B denotesthe target atom [30]. Therefore, using the same functional form as eq. (1) for each of thetwo contributing processes, we write a generalized expression for the cross sections as

σqq′ = σ0[f(E1) + a7f(E1/a8)], (3)

where qq′ represents any one of the six combinations considered of the initial and the finalcharge-state of the projectile, and a7 and a8 are adjustable parameters. The differencebetween the threshold energies for the two contributing processes is neglected, and thethreshold energy of the total process is used commonly in the two terms of eq. (3).

An additional minor structure seen in some sets of data is considered uncertain, andthe formula was not designed to reproduce it. Thus we have eight adjustable parameters

Table 3Parameters in the analytic formula for the double-electron-capture cross section σ1−1of H+. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 8) = adjustableparameters; a3, a5 and a8 in keV. Values in parentheses are those assumed in thefit. When the values of a3 and a4 are absent, the second term in the denominatorof eq. (1) should be omitted. When the values of a7 and a8 are absent, the crosssection is expressed by the first term only of eq. (3)

Target Et a1 a2 a3 a4Li 6.67E−02 1.68E+00 8.3 E−01 . . . . . .Na 3.81E−02 8.2 E+01 1.45E+00 . . . . . .Mg 8.32E−03 9.6 E+00 1.11E+00 . . . . . .K 2.18E−02 2.81E+00 6.3 E−01 . . . . . .Ca 3.63E−03 1.63E+00 (1.0 E+00) . . . . . .Sr 2.73E−03 2.97E+00 (1.0 E+00) (1.0E+00) 2.3E−01Cs 1.46E−02 3.14E+01 (1.0 E+00) (8.0E−01) 9.6E−01Ba 1.46E−03 1.27E+00 (1.0 E+00) 3.2E+00 7.8E−01Pb 8.10E−03 3.15E+01 (1.0 E+00) . . . . . .

Target a5 a6 a7 a8Li 8.4 E+00 3.97E+00 . . . . . .Na 4.47E+00 1.94E+00 . . . . . .Mg 1.26E+00 4.09E+00 1.79E+00 8.3 E+00K 3.3 E+00 1.76E+00 . . . . . .Ca 2.22E+00 3.11E+00 8.45E−01 4.72E+00Sr 4.57E+00 (4.0 E+00) 1.69E−01 1.08E+01Cs 2.94E+00 (3.2 E+00) . . . . . .Ba 5.8 E+00 4.1 E+00 1.95E−01 7.60E+00Pb 2.44E+00 4.58E+00 (2.0 E−01) (8.0 E+00)

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at most to fit eq. (3) to the compiled data. When a subpeak or a shoulder is not presentin the cross-section curve, the second term in eq. (3) is omitted. When the cross sectionshows a rather narrow peak, we can omit the middle term in the denominator of eq. (1).

Uncertainties of the data are not given in all the sources of the compiled data. There-fore, we have given equal significance to all the data in the sense that the relative standarddeviations are assumed to be the same. This assumption has been taken into account byusing the two-step, least-squares method [31]. This method was developed to fit a func-tion to those data whose dependent variable changes by several orders of magnitude. Inthe first step of the method, the logarithm of the function is fitted to the logarithm of thedata with the uniform weight. In the logarithmic fit, however, the values of the functionobtained tend to be lower than the arithmetic mean of fluctuating data. To remove thisdefect, the second step is used. In the second step, the function is fitted to the data withthe weights inversely proportional to the values of the function obtained in the first step.

When some parameters could not be determined by the fitting procedure because ofinsufficient data, these parameters were fixed to an assumed value, and the remainingparameters were adjusted.

The following sets of data were excluded in the fit because of appreciable deviationfrom the plural sets of data reported by other authors:

Table 4Parameters in the analytic formula for the single-electron-loss cross section σ01 ofH. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 8) = adjustableparameters; a3, a5 and a8 in keV. Values in parentheses are those assumed in thefit. When the values of a7 and a8 are absent, the cross section is expressed by thefirst term only of eq. (3)

Target Et a1 a2 a3 a4Li 1.36E−02 5.92E+00 (1.5 E+00) (2.0 E+01) 2.95E−01Na 1.36E−02 4.40E+00 1.79E+00 2.52E+01 (3.0 E−01)Mg 1.36E−02 4.75E+00 2.32E+00 2.32E+01 (3.0 E−01)K 1.36E−02 2.20E+01 (2.3 E+00) (1.6 E+01) 2.06E−01Ca 1.36E−02 7.58E+01 3.50E+00 1.16E+01 5.1 E−01Rb 1.36E−02 1.71E+01 (3.1 E+00) 1.80E+01 1.25E−01Sr 1.36E−02 3.49E+01 3.14E+00 1.33E+01 2.73E−01Cd 1.36E−02 1.45E+00 1.56E+00 (1.1 E+02) (2.7 E−01)Cs 1.36E−02 6.27E+00 2.48E+00 3.06E+01 5.27E−01Pb 1.36E−02 1.39E+00 1.29E+00 (1.2 E+02) (5.0 E−01)

Target a5 a6 a7 a8Li (6.0E+01) (1.0E+00) . . . . . .Na (2.0E+02) (1.0E+00) 1.8 E−02 (3.3 E−02)Mg (2.0E+02) (1.0E+00) 7.0 E−02 (1.0 E−01)K (1.4E+02) (1.0E+00) (7.0 E−02) (1.0 E−01)Ca (6.0E+01) (1.0E+00) 2.39E−01 2.35E−01Rb (1.6E+02) (1.0E+00) (2.4 E−01) (2.0 E−01)Sr (8.0E+01) (1.0E+00) 1.66E−01 1.86E−01Cd (6.4E+02) (1.0E+00) (1.7 E−01) (1.9 E−01)Cs (2.0E+02) (1.0E+00) 7.9 E−02 1.44E−01Pb (8.0E+02) (1.0E+00) (8.0 E−02) (1.5 E−01)

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(1) The data by Nagata [21] on σ10 in Na and K vapors.(2) The data by Gruebler et al. [11] on σ10 in Cs vapor.

Values of Et were calculated from ionization energies and electron affinities. Whenthese were unavailable, assumed values of Et were used.

4. Results and discussionValues of parameters in the analytic formula are presented in tables 1–6. The assumed

values of Et and of adjustable parameters are shown in parentheses in these tables. Thecurves given by the analytic formula are compared with the experimental data for rep-resentative cases in figs. 1–6. The curves are shown, when possible, over the range ofprojectile energy from Emin/10 to 10Emax, where Emin and Emax are the minimum andthe maximum energy of available experimental data.

In favorable cases, experimental data on the cross sections considered are available forprojectile energies down to about 100 eV and up to about 200 keV. However, the data donot always cover the energy region where the process is significant. Such deficiency canbe remedied to a considerable extent by extrapolation of the analytic formulas.

The cross sections σ10 and σ0−1 show a shoulder in the high energy region (figs. 1 and2); σ1−1 has a subpeak for some of the targets (fig. 3). The cross section σ01 shows ashoulder in the low energy region for some of the targets (fig. 4); σ−10 exhibits a ratherbroad peak (fig. 5). This wide variety of behavior can be fitted well by eq. (3).

In figs. 1–4 and 6, the dashed line connects the recommended values tabulated byMorgan et al. [3]. These values are generally close to the present best-fit values. When

Table 5Parameters in the analytic formula for the single-electron-loss cross section σ−10 ofH−. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 6) = adjustableparameters; a3 and a5 in keV. Values in parentheses are those assumed in the fit.Use the first term only of eq. (3)

Target Et a1 a2 a3 a4Li 1.7 E−04 8.86E+03 (1.0 E+00) (2.3 E−01) 3.75E−01Na 3.5 E−04 4.56E+04 (1.4 E+00) (2.3 E−01) 2.98E−01Mg 1.35E−03 8.97E+03 (1.0 E+00) (2.3 E−01) 3.43E−01K 1.5 E−04 1.30E+04 (1.0 E+00) (2.3 E−01) 3.27E−01Ca (0.0 E+00) 2.32E+03 1.36E+00 2.34E+00 5.5 E−01Rb (0.0 E+00) 1.22E+04 (1.0 E+00) (2.3 E−01) 2.94E−01Sr (0.0 E+00) 3.73E+03 1.50E+00 1.89E+00 5.58E−01Cs 5.5 E−04 5.53E+03 (1.0 E+00) 2.24E−01 1.78E−01

Target a5 a6Li (1.0E+02) (1.0E+00)Na (1.5E+02) (1.0E+00)Mg (1.0E+02) (1.0E+00)K (1.0E+02) (1.0E+00)Ca (1.5E+01) (1.0E+00)Rb (1.0E+02) (1.0E+00)Sr (2.0E+01) (1.0E+00)Cs (1.0E+02) (1.0E+00)

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Table 6Parameters in the analytic formula for the single-electron-loss cross section σ−11 ofH−. Et = threshold energy of process in keV; ai (i = 1, 2, . . . , 6) = adjustableparameters; a3 and a5 in keV. Values in parentheses are those assumed in the fit.Use the first term only of eq. (3)

Target Et a1 a2 a3 a4Li 1.4 E−02 1.43E+00 (2.0E+00) (2.3 E+01) 6.38E−01Na 1.4 E−02 1.07E+00 (2.0E+00) 2.30E+01 1.54E−01K 1.4 E−02 3.95E+00 (2.0E+00) 2.88E+01 6.27E−01Rb 1.4 E−02 1.04E+01 (2.0E+00) 1.62E+01 3.71E−01Cs 1.4 E−02 3.11E+01 (2.0E+00) 7.1 E−01 −1.28E+00

Target a5 a6Li (2.0E+02) (1.0 E+00)K (2.0E+02) (1.0 E+00)Na (2.0E+02) (1.0 E+00)Rb (2.0E+02) (1.0 E+00)Cs 1.5E+01 1.05E+00

Table 7Number of data points n used in fit and percentage rms deviation δ of the datapoints from the analytic formula

σ10 σ0−1 σ1−1 σ01 σ−10 σ−11Target n δ n δ n δ n δ n δ n δ

Li 51 41 6 4 25 34 8 1 8 3 8 6Na 57 41 38 50 36 39 21 26 16 5 13 7Mg 35 11 12 4 23 15 15 14 3 13 – –K 44 31 29 5 25 28 8 2 8 2 8 3Ca 13 11 13 8 12 8 12 8 10 9 – –Rb 22 11 26 4 – – 8 1 8 1 8 3Sr 16 7 16 7 16 11 15 3 15 10 – –Cd – – – – – – 3 8 – – – –Cs 93 22 48 16 16 82 13 6 29 23 16 10Ba 30 7 – – 30 7 – – – – – –Pb 6 10 7 2 6 15 7 2 – – – –

Overall 367 28 195 24 189 34 110 13 97 14 53 7

data sets of different authors show large discrepancy, Morgan et al. have selected the setwith the highest stated-accuracy. This treatment, however, is not free from problems.Furthermore, polynomial extrapolation of tabulated values is known often to give anoscillatory curve and unreasonable values; on the other hand, the present analytic formulasbehave well also in the extrapolation regions.

In the following cross sections, appreciable differences exist between the present best-fit values and the recommended values of Morgan et a1.:(1) σ0−1 in K; σ1−1 in Na and K; σ01 in Na, K and Rb.(2) σ0−1 in Cs (fig. 2).(3) σ−10 and σ−11 in Cs.

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The causes of these differences are as follows: For (1) and (2), Morgan et al. used the datanot yet included in the database AMSTOR used in the present work: Ebel’s unpublisheddata [32] for (1) and the data of Miethe et al. [33] for (2). For (3), Morgan et al. used thedata of Leslie et al. [15] multiplied by two (the reason not given).

In table 7, the numbers n of the experimental data points used in the fit and the relativeroot-mean-square (rms) deviations δ of the data from the corresponding analytic formulaare shown. The value of δ for all the data on each type of cross section, shown in the lastrow of table 7, is from 7 to 34%. In general, two factors contribute to δ: (1) uncertainty inexperimental data and (2) systematic error of the fitting function. Graphical comparison‡

of the analytic formula with the data indicates that the main cause of larger deviationis the discrepancy among the data sets of different authors. From this, we can concludethat factor (1) is dominant over factor (2) in most of the present values of δ. Therefore,the values of δ in table 7 serve as a measure of factor (1).

Rudd et al. [34] fitted a modified Green–McNeal formula, similar to eq. (3), to theirown experimental data on σ10 of hydrogen in gaseous atoms and molecules. They appliedeq. (1) to each electronic subshell in the target atoms or molecules using the numberof electrons in and the binding energy of the subshell, and took summation over thesubshells. This treatment is instructive for further refinement of the present formulas.

References[1] H.W. Drawin and K. Katsonis, Phys. Scripta 23 (1981) 69.[2] D.E. Post, Princeton Plasma Physics Institute Rep. PPPL-1877 (1982).[3] T.J. Morgan, R.E. Olson, A.S. Schlachter and J.W. Gallagher, J. Phys. Chem. Ref. Data

14 (1985) 971.[4] Y. Nakai, T. Shirai, T. Tabata and R. Ito, At. Data Nucl. Data Tables 37 (1987) 69.[5] Y. Nakai, T. Shirai, M. Sataka and T. Sugiura, Jpn. At. Energ. Res. Inst. Rep. JAERI-M

84-169 (1984).[6] C.J. Anderson, R.J. Girnius, A.M. Howald and L.W. Anderson, Phys. Rev. A22 (1980) 822.[7] C.J. Anderson, A.M. Howald and L.W. Anderson, Nucl. Instr. and Meth. 165 (1979) 583.[8] R.A. Baragiola and E.R. Salvatelli, Nucl. Instr. and Meth. 110 (1973) 503.[9] K.H. Berkner, R.V. Pyle and J.W. Stearns, Phys. Rev. 178 (1969) 248.

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‡Figures for all the available combinations of the processes and targets are given in the report citedin the footnote on the first page of this paper.

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(File made April 27, 2019)

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