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Ionization yield in xenon due to electron impact
Physica 113C (1982) 237-243 North-Holland Publishing C o m p a n y
IONIZATION YIELD IN XENON DUE T O E L E C T R O N IMPACT DAYASHANKAR Division of Radiological Protection, Bhabha Atomic Research Centre, Bombay 400 085, India Received 10 July 1981 Revised 6 October 1981
T h e ionization yield in xenon for complete energy degradation of electrons with initial energy up to I keV has been calculated by solving the generalized Fowler equation. T h e expression for the energy spectrum of secondary electrons from the O shell was obtained by using the empirical scaling functions to weight the Williams-Weizs/icker cross section for glancing collisions and the Mott cross section for knock-on collisions. T h e total ionization and excitation cross sections were taken from the recent evaluation reported by D e H e e r et al. Contributions from the inner-shell ionization and the A u g e r process were explicitly taken into account. T h e results are expressed in terms of the quantity W, the m e a n energy required to produce an ion pair. T h e W value is found to decrease with increasing incident energy, finally approaching a constant value of 23.1 eV for electrons with an energy above 200 eV. T h e results are generally in good agreement with the available experimental work.
1. Introduction
The total ionization produced in a gas under electron irradiation, for complete energy dissipation of the incident electron (including the successively generated electrons), is the simplest and most accessible measure of an aggregate effect of the incident radiation. Since the yield of ionization (i.e., the number of ion pairs produced) has statistical fluctuations, one usually considers the mean ionization yield for practical purposes. This yield is customarily expressed in terms of the quantity W, the mean energy required to produce an ion pair. Knowledge of the ionization yield or W is useful in many fields such as radiation dosimetry, charged particle spectroscopy, radfation chemistry and upper atmospheric physics. Whereas the W value in xenon for highenergy electrons is well-known experimentally [1], little work has been done for electrons below 1 keV. The work reported below 1 keV consists of only two measurements: one by Samson and Haddad [2] for electrons up to about 50 eV and another by Combecher [3] for electrons up to
I keV. It may be emphasized that no calculations have been reported on W in xenon for electrons below 1 keV. The only calculations reported in the literature are two very approximate ones by Sato and co-workers [4, 5] for electrons with an energy as high as 100 keV. In view of this situation it is obviously worthwhile to calculate the ionization yield in xenon for electrons below 1 keV. The approach in this work is the same as that used recently by the author in the case of krypton [6].
2. Inelastic collision cross sections
The relevant cross sections for calculation of the ionization yield include the total ionization cross section, excitation cross sections for various discrete excited states and the differential ionization cross section (i.e., the energy spectrum of secondary electrons). Since xenon is a sufficiently heavy atomic target, the inner-shell ionization cross sections are also required to account for the contributions of the inner-shell ionization and the Auger process explicitly.
0378-4363/82/0000-0000/$02.75 O 1982 North-Holland
238
D a y a s h a n k a r / Ionization yield in xenon due to electron impact
2.1. Total ionization a n d excitation cross sections For total ionization and excitation cross sections we depend on the analysis and evaluation of electron impact cross sections for noble gases carried out by D e H e e r et al. [7]. For the total ionization cross section, o'i, we have used the counting ionization cross section which is obtained when one measures the n u m b e r of ions irrespective of their charge. D u e to lack of sufficient experimental or theoretical data on excitation cross sections for xenon, D e H e e r et al. have used a scaling scheme to obtain the total excitation cross section. T h e scheme m a k e s use of the semiempirical excitation cross sections of argon and of the fact that o-,E2, is a universal function of T / E , where o', is the excitation cross section for the state n with excitation energy En and T is the kinetic energy of the incident electrons. With this scheme, the total excitation cross section of xenon, O'ex, is obtained by treating it as corresponding to a single level having an excitation energy equal to the average excitation energy of the resonance levels. Since the m a j o r contribution to the total excitation cross section comes from the resonance levels, this assumption will not introduce a serious error in our calculation. The resonance lines of xenon occur at 1296 ,& and 1470,& [8] and correspond to an average excitation energy of 9 e V . The total inelastic cross section, trine~, is the sum of the total ionization cross section and the total excitation cross section, i.e., O'inel = Ori "q- Orex In the energy region covered in this study, the inner-shell ionization from the N shell only need be considered. T h e main contribution to the N-shell ionization comes from ten electrons in the 4d subshell, followed by a small contribution from six electrons in the 4p subsheli and a still smaller contribution from two electrons in the 4s subshell. W e consider the three subshells separately and for their total ionization cross section we use the formula derived from the symmetrical model of the binary-encounter col-
lision theory reported in detail by Vriens [9]. According to this formula, the total ionization cross section, o'i, for a (sub)shell j with ionization t h r e s h o l d / / i s given by
cri(T) -
47raER 2 Sj T+Ij+Ej X
[( ~ 1
1
2E~ (1_ 1 ) _ In (T//j)1 (1)
where a0 is the first Bohr radius, R the Rydberg energy, T the kinetic energy of the incident electron, and Sj the n u m b e r of electrons in the jth (sub)shell having average kinetic energy Ej. T h e values of //. for the 4d, 4p and 4s subshells are assumed to be 67.5 eV, 147 eV and 213 eV, respectively [10] and those of Ej as 232.7eV, 151.3eV and 83.11eV, respectively [4]. It may be mentioned that although the cross sections given by this formula are not quite accurate, they would serve the purpose fairly well in our calculation because the overall contribution of the inner-shell ionization is relatively small. 2.2. Differential ionization cross section The available information on the differential ionization cross section or the energy spectrum of secondary electrons for xenon is rather scarce. The only m e a s u r e m e n t s have been reported by Opal et al. [11] for 500 eV electrons. The O shell in the xenon atom m a k e s a predominant contribution to the ionization of the whole atom. Therefore, the differential ionization cross section for the O shell should be obtained in a reliable way. For this purpose we follow the approach of utilizing the qualitative features of two nearly distinct domains, namely that of glancing collisions (involving small m o m e n t u m transfer) and that of knock-on collisions (involving large m o m e n t u m transfer). These features, discussed in detail by Kim [12], are: (a) for slow secondary electrons, the glancing collisions dominate and the secondary elec-
Dayashankar / Ionization yield in xenon due to electron impact tron spectrum is largely .determined by the differential optical oscillator strength; and (b) for fast secondary electrons, the spectrum should follow the knock-on collision cross section. Combining the contributions of the two kinds of collisions with suitable weight functions, Gerhart [13], Soong [14], Eggarter [15] and Dayashankar [6] have obtained the differential ionization cross sections for H2, Ne, A r and Kr respectively. Using the same approach, the differential cross section, [dtr°(T, E)/dE]th, for ionization of the O shell of xenon, for energy transfer E by an incident electron with kinetic energy T can be written as
d t r °"T.
( , E)]t h =
4zra2R [R d r . 4 T R b ( E ) T m E
1
E ( T + Io - E
÷ ( r + t o - E)
)/]
'
The functions b(E) and ~b(E) were determined by fitting eq. (2) to the available experimental data of Opal et al. [11] on the secondary electron spectrum for 500 eV electrons. These data correspond to an assumed total ionization cross section of 11.07 a02 while the value adopted in the present work is 9.029a02. Therefore, we renormalized the experimental data by a factor Of 0.816. We also made an allowance for the minor contribution of the N shell in order to get the O shell contribution to be used for the fitting. T o estimate the contribution of the three subshells belonging to the N shell, we used the binaryencounter formula [9] corresponding to eq. (1) which gives the differential ionization cross section, do~(T, E)/dE, for the jth (sub)shell as
dtrJ (T, E ) = 4rra~R2 [ 1 4Ei dE E + I j + E j Sj E-'~+ 3E 3
1
+ 4,(E)RSo
239
(2)
1 ~_ 4Ei + ( T + I j - E ) 2 3(T + / j - E ) 3 1
where dr/dE is the differential optical oscillator strength for ionization of the O shell with a threshold energy I0 (12.1 eV) and So is the number of electrons in the O shell. The first term in eq. (2) is essentially the Williams-Weizsiicker cross section representing the contribution of glancing collisions; the coefficient b(E), to be determined empirically, reflects the cutoff of glancing collisions at a certain m o m e n t u m transfer. The second term in eq. (2) is basically the Mott scattering cross section representing the contribution of knock-on collisions. The coefficient 4~(E) is an empirical function which reduces the second term at very low energy transfers and approaches unity as the energy transfer increases. For the differential optical oscillator strength, dr/dE, in eq. (2) we used a smooth representation of the experimental data of Samson [8] for E -< 59 eV. In the region of E > 59 eV, where experimental data are not available, we have made use of the fact that for large energy transfers dr/dE decreases as E -35 [16].
E(T + Ij- E
'
(3)
where the symbols have the same meaning as in eq. (1). It may be mentioned that eq. (3) does not describe the glancing collisions and hence is not valid for the low-energy part of the secondaryelectron spectrum. Ignoring the electron correlation effects in subshell ionization would also lead to a poor estimate of the cross section from eq. (3). However, the contribution of the innershell ionization being relatively minor, the use of approximate cross sections given by eq. (3) will not introduce a serious error in our calculation. It should be noted that for the O shell, which is the major contributor to the ionization, eq. (2) is preferable to eq. (3) because the former describes adequately the low-energy part of the electron spectrum as well as the high-energy part. T o fit eq. (2) to the renormalized experimental data at T = 5 0 0 e V corrected for the N-shell contribution, we proceeded by setting ~b(E)= 0 for E < 18.47 eV and adjusting b(E) only. The
240
Dayashankar
Ionization yield in xenon due to electron impact
i
1.0 0.8 w 0.6 -e-
)
~" O.4
n 0.2 0
I
12
14
~ 16
18
20
22
24
E (eV) Fig. 1. Empirical parameters b(E) and ~b(E) for eq. (2).
resulting b(E), shown in fig. 1, is a smooth function that approaches a value of 0.89 at 18.47 eV. For E < 16.1 eV, experimental data on the secondary-electron spectrum not being available, b(E) was obtained by extrapolation in such a way that the integrated cross section would agree with the a d o p t e d total ionization cross section for the O shell at T = 500 eV. For E > 18.47 eV, adopting b ( E ) = 0.89, eq. (2) was fitted to the data by adjusting the coefficient &(E) only. This coefficient turned out to increase rapidly from zero at E -- 18.47 eV to about unity at E = 2 0 . 5 e V . For E > 2 0 . 5 e V , &(E) was found to be nearly unity at most of the points; at some points, however, especially near the Auger peaks, the value turned out to be different from unity. W e ignored such points and took &(E) = 1 for all energy transfers above 20.5eV. This is justified because the contribution of Auger electrons is considered separately in this work. The function &(E) used in the calculation is also shown in fig. 1. With the coefficients b(E) and &(E) as obtained above, we integrated eq. (2) for various values of the incident energy T between the limits I0 and (T+Io)/2 to check whether the integrated cross section agreed with the adopted total ionization cross section for the O shell. T h e agreement was found to be good (within 8%) for T--> 300 eV. In this region the ratio C ( T ) of the
adopted cross section to the integrated cross section is a slowly varying function of T, ranging from 0.929 at 300 eV to 1.04 at 1 keV, and being exactly unity at 500 eV. The values of C ( T ) are given in table I. It is thus clear that eq. (2) is approximately valid for T _> 300 eV. Since we can rely more on the adopted total ionization cross section, eq. (2) can be improved by normalizing it exactly with respect to the total ionization cross section. Therefore, we multiply the right-handside of eq. (2) by the ratio C(T). This procedure is acceptable only for T -->300 eV where C ( T ) does not change significantly. Thus, for T - > 300 eV, we take the differential ionization cross section of the O shell to be do-°(T, E ) / d E = C(T)[do-°(T, E)/dE],h . For T < 300 eV, the region which is not described well by eq. (2), we resort to the cruder procedure of Miller [17] and G e r h a r t [13] and simply put do'°(T, E ) / d E = K ( T ) E -2 , where K ( T ) is adjusted to fit the total ionization cross section. It may be pointed out that, while considering ionization from the O shell, we have ignored the difference in 5s and 5p electrons and have treated the two electrons in the s subshell as having the same ionization threshold as the six electrons in the p subshell. However, the error
Table I Values of the normalizing parameter C(T)
T
T
(eV)
C(T)
(eV)
C(T)
300 400 500 600
0.929 0.991 1.000 1.014
700 800 900 1000
1.029 1.003 1.027 1.040
Dayashankar / Ionization yield in xenon due to electron impact
involved in this approximation is offset to a large extent by the fitting procedure used. 3. Degradation calculation With the set of inelastic collision cross sections described above, we consider the complete energy degradation of an incident electron of kinetic energy T and all the successively generated electrons in xenon gas having low and uniform density. The gaseous medium is assumed to be so voluminous that all excited and ionized species produced as well as all electrons generated remain within the medium. Under this premise, the mean total number of ion pairs produced in the entire degradation process, Ni(T), is given by the generalized Fowler equation [15, 18]: Ni(T) = pi(T) + ~'~ pn(T)Ni(T - En) n
(T+/])/2
+ ~. 1
[ 1j
p{(T,E)
x [Ni(T - E ) + N~(E - / j ) ] dE
+
~_,
j # outer shell
pA,(T)N,(TA,).
0)
In this integral equation, pi(T)= (ri(T)/orinel(T) is the probability of ionization per inelastic collision and pn(T)= ~r.(T)/~i,el(T) is the analogous probability for excitation to a state n with excitation energy En. Similarly, p~(T, E) = (doJ(T, E)/dE)/tri,et(T) is the probability per unit energy for ionization of the jth shell with an energy loss E and production of a secondary electron with a kinetic energy E - / j . The last term in eq. (4) represents the contribution of Auger electrons with the kinetic energy TAj produced as a result of ionization of the flh shell, pAj(T) being the probability of the Auger emission per inelastic collision. Since in our set of cross sections, the total excitation cross section, o'ex, is treated as cor-
241
responding to a single level, with excitation energy Eav equal to the average excitation energy of the resonance levels, the summation in the second term of eq. (4) is reduced to pex(T)Ni(T Eav), where pex(T)= tr=x(T)/o'ioel(T). In the energy region of interest in our study (i.e., for T-< 1 keV), the third term of eq. (4) includes contributions from the O shell and from the three subshells of the N shell which are considered separately. In the same way, the last term includes Auger electrons only from the N shell. Accordingly, the last term reduces to pA(T)Ni(TA) where pA(T)= rltrr~(T)/tri,e~(T), rt being the Auger yield which we have taken to be unity and TA being the average energy of Auger electrons from the N shell, which we have taken to be 26.6 eV on the basis of the line energies and relative intensities reported in the literature [191. Eq. (4) was solved numerically by starting at T = I0, the first ionization threshold, and ascending in T stepwise. We chose a step size of 1 eV for T -< 50 eV and of 5 eV for higher energies. It is necessary to keep the step size smaller than the excitation energy so that the inputs for solving the equation can be obtained by interpolating the previously calculated Ni values. The Lagrange three-point interpolation formula was used for interpolation of data. A comment may be in order on the necessity of using the generalized Fowler equation in the present work instead of the simple Fowler equation. The generalization of the Fowler equation in our work consists in allowing for inner-shell ionization and for its secondary Auger process. Since xenon is a sufficiently heavy atomic target, inner-shell ionization is quite appreciable. Although we have not evaluated the effect of the generalization of the Fowler equation in the present case, it has been demonstrated in our work on krypton [6] that for sufficiently heavy atoms that are expected to have appreciable cross sections for inner-shell ionization, the simple Fowler equation (based on the assumption of a single ionization threshold) is not adequate.
Dayashankar ~ Ionization yield in xenon due to electron impact
242
4. Results and discussion
Fig. 2 shows the results of the ionization yield calculations in terms of the quantity W, the mean energy required to produce an ion pair, as a function of the incident electron energy. It is found that the value of W initially decreases rapidly with increasing incident energy and then gradually approaches a constant value of 23.1 eV (within ---1%) for electrons with energy higher than 200 eV. This asymptotic value is in good agreement ( - 4 % ) with the experimental value of 22.2 eV for a tritium source of about 4 k e V beta energy as reported by Jesse and Sadauskis [20]. T h e calculated curve also reveals a kink around 21 eV. This kink corresponds to the onset of the excitation process. For comparing our results with the experimental work, we note that only two m e a s u r e m e n t s have been reported in the energy region covered in this study: one by Samson and H a d d a d [2] for electrons up to about 50 e V and another very recently by Combecher [3] for electrons up to l keV. T h e experimental data of these authors are also plotted in fig. 2. T h e calculated values are found to
100
'
'
'
'
' " "
'
'
'
' ' " 7
9O
have only occasional agreement with the experimental data of Samson and Haddad. The data of C o m b e c h e r are generally higher than the calculated values, although the agreement is pretty good for an incident energy above 100 eV. It may be mentioned that his m e a s u r e m e n t s for argon and krypton show a similar tendency with respect to the calculations of Eggarter [15] and D a y a s h a n k a r [6], respectively. It may also be noted that near 2 1 e V these observations only produce a slight shoulder in the curve instead of a kink. It is not possible to give an overall error in the calculated values of W because it depends on the errors in all the input cross sections, most of which are not known exactly. The errors involved in the experimental values of W are not known either. In any case, the large difference between experiment and theory with respect to W at low energies is not quite understandable. The crude formula for differential ionization cross sections used in the calculation for an incident energy lower than 300 eV may be responsible for the large errors in W at these energies. While comparing the results of our calculations with the experimental data it is relevant to mention that at the gas pressures used in the experiments, some of the excited atoms are likely to be converted into diatomic ions by the H o r n b e c k - M o l n a r process [21]:
a0 <
•
Z
Q
Xe* + Xe
PRESENT WORK
- -
70
COMBECHER(19B0) SAMSON AND HADDAD (1976)
60
a_ 50
A
Z
z
30
~
20
< I.U
L
101
J
i
i
~ i
>Xe~ + e - .
ill
t
10 2
i
,
i
i
i+
10 3
INCIDENT ELECTRON ENERGY(eV) Fig. 2. T h e m e a n energy r e q u i r e d to produce an ion pair, W, as a function of the incident electron energy.
In our calculation, however, the contribution of this collateral ionization has not been taken into account because of insufficient collision data. Guided by the estimate of this contribution for argon [15] one would expect this effect to lower the W value obtained in this work by an amount of the order of l e V . In view of this, for the asymptotic value of W, our calculation favours the measured value of Jesse and Sadauskis with respect to that of Combecher. Finally, we conclude that the agreement with the experimental work is, generally, quite encouraging, consider-
Dayashankar / Ionization yield in xenon due to electron impact
ing the assumptions made in constructing the cross sections.
Acknowledgements The author wishes to thank M.A. Prasad and Dr. U. Madhvanath for helpful discussions.
References [1] Average Energy Required to Produce an Ion Pair, Report 31, International Commission on Radiation Units and Measurements, Washington, D.C. (1979). [2] J.A.R. Samson and G.N. Haddad, Radiat. Res. 66 (1976) 1. [3] D. Combecher, Radiat. Res. 84 (1980) 189. [4] S. Sato, K. Okazaki and S. Ohno, Bull. Chem. Soc. Jap. 47 (1974) 2174. [5] K. Okazaki, S. Sato and S. Ohno, Bull. Chem. Soc. Jap. 48 (1975) 1411. [6] Dayashankar, Physica 111C (1981) 134.
243
[7] F.J. de Heer, R.H.J. Jansen and W. van der Kaay, J. Phys. B (Atom. Molec. Phys.) 12 (1979) 979. [8] J.A.R. Samson, Advances in Atomic and Molecular Physics, Vol. 2, D.R. Bates and I. Estermann, eds. (Academic Press, New York, 1966) p. 177. [9] L. Vriens, Case Studies in Atomic Collision Physics I, E.W. McDaniel and M.R.C. McDowell, eds. (NorthHolland, Amsterdam, 1969) p. 335. [10] W. Lotz, J. Opt. Soc. Am. 60 (1970) 206. [11] C.B. Opal, E.C. Beaty and W.K. Peterson, At. Data 4 (1972) 209. [12] Y.K. Kim, Radiat. Res. 61 (1975) 21. [13] D.E. Gerhart, J. Chem. Phys. 62 (1975) 821. [14] S.C. Soong, Radiat. Res. 67 (1976) 187. [15] E. Eggarter, J. Chem. Phys. 62 (1975) 833. [16] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [17] W.F. Miller, Ph.D. Thesis, Purdue University (1956). [18] M. Inokuti, Radiat. Res. 64 (1975) 6. [19] ESCA Applied to Free Molecules, K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heden, K. Hamrin, U. Gelius, T. Bergmark, L.O. Werme, R. Manne and Y. Baer, eds. (North-Holland, Amsterdam, 1969) p. 156. [20] W.P. Jesse and J. Sadauskis, Phys. Rev. 107 (1957) 766. [21] J.A. Hornbeck and J.P. Molnar, Phys. Rev. 84 (1951) 621.
IONIZATION YIELD IN XENON DUE T O E L E C T R O N IMPACT DAYASHANKAR Division of Radiological Protection, Bhabha Atomic Research Centre, Bombay 400 085, India Received 10 July 1981 Revised 6 October 1981
T h e ionization yield in xenon for complete energy degradation of electrons with initial energy up to I keV has been calculated by solving the generalized Fowler equation. T h e expression for the energy spectrum of secondary electrons from the O shell was obtained by using the empirical scaling functions to weight the Williams-Weizs/icker cross section for glancing collisions and the Mott cross section for knock-on collisions. T h e total ionization and excitation cross sections were taken from the recent evaluation reported by D e H e e r et al. Contributions from the inner-shell ionization and the A u g e r process were explicitly taken into account. T h e results are expressed in terms of the quantity W, the m e a n energy required to produce an ion pair. T h e W value is found to decrease with increasing incident energy, finally approaching a constant value of 23.1 eV for electrons with an energy above 200 eV. T h e results are generally in good agreement with the available experimental work.
1. Introduction
The total ionization produced in a gas under electron irradiation, for complete energy dissipation of the incident electron (including the successively generated electrons), is the simplest and most accessible measure of an aggregate effect of the incident radiation. Since the yield of ionization (i.e., the number of ion pairs produced) has statistical fluctuations, one usually considers the mean ionization yield for practical purposes. This yield is customarily expressed in terms of the quantity W, the mean energy required to produce an ion pair. Knowledge of the ionization yield or W is useful in many fields such as radiation dosimetry, charged particle spectroscopy, radfation chemistry and upper atmospheric physics. Whereas the W value in xenon for highenergy electrons is well-known experimentally [1], little work has been done for electrons below 1 keV. The work reported below 1 keV consists of only two measurements: one by Samson and Haddad [2] for electrons up to about 50 eV and another by Combecher [3] for electrons up to
I keV. It may be emphasized that no calculations have been reported on W in xenon for electrons below 1 keV. The only calculations reported in the literature are two very approximate ones by Sato and co-workers [4, 5] for electrons with an energy as high as 100 keV. In view of this situation it is obviously worthwhile to calculate the ionization yield in xenon for electrons below 1 keV. The approach in this work is the same as that used recently by the author in the case of krypton [6].
2. Inelastic collision cross sections
The relevant cross sections for calculation of the ionization yield include the total ionization cross section, excitation cross sections for various discrete excited states and the differential ionization cross section (i.e., the energy spectrum of secondary electrons). Since xenon is a sufficiently heavy atomic target, the inner-shell ionization cross sections are also required to account for the contributions of the inner-shell ionization and the Auger process explicitly.
0378-4363/82/0000-0000/$02.75 O 1982 North-Holland
238
D a y a s h a n k a r / Ionization yield in xenon due to electron impact
2.1. Total ionization a n d excitation cross sections For total ionization and excitation cross sections we depend on the analysis and evaluation of electron impact cross sections for noble gases carried out by D e H e e r et al. [7]. For the total ionization cross section, o'i, we have used the counting ionization cross section which is obtained when one measures the n u m b e r of ions irrespective of their charge. D u e to lack of sufficient experimental or theoretical data on excitation cross sections for xenon, D e H e e r et al. have used a scaling scheme to obtain the total excitation cross section. T h e scheme m a k e s use of the semiempirical excitation cross sections of argon and of the fact that o-,E2, is a universal function of T / E , where o', is the excitation cross section for the state n with excitation energy En and T is the kinetic energy of the incident electrons. With this scheme, the total excitation cross section of xenon, O'ex, is obtained by treating it as corresponding to a single level having an excitation energy equal to the average excitation energy of the resonance levels. Since the m a j o r contribution to the total excitation cross section comes from the resonance levels, this assumption will not introduce a serious error in our calculation. The resonance lines of xenon occur at 1296 ,& and 1470,& [8] and correspond to an average excitation energy of 9 e V . The total inelastic cross section, trine~, is the sum of the total ionization cross section and the total excitation cross section, i.e., O'inel = Ori "q- Orex In the energy region covered in this study, the inner-shell ionization from the N shell only need be considered. T h e main contribution to the N-shell ionization comes from ten electrons in the 4d subshell, followed by a small contribution from six electrons in the 4p subsheli and a still smaller contribution from two electrons in the 4s subshell. W e consider the three subshells separately and for their total ionization cross section we use the formula derived from the symmetrical model of the binary-encounter col-
lision theory reported in detail by Vriens [9]. According to this formula, the total ionization cross section, o'i, for a (sub)shell j with ionization t h r e s h o l d / / i s given by
cri(T) -
47raER 2 Sj T+Ij+Ej X
[( ~ 1
1
2E~ (1_ 1 ) _ In (T//j)1 (1)
where a0 is the first Bohr radius, R the Rydberg energy, T the kinetic energy of the incident electron, and Sj the n u m b e r of electrons in the jth (sub)shell having average kinetic energy Ej. T h e values of //. for the 4d, 4p and 4s subshells are assumed to be 67.5 eV, 147 eV and 213 eV, respectively [10] and those of Ej as 232.7eV, 151.3eV and 83.11eV, respectively [4]. It may be mentioned that although the cross sections given by this formula are not quite accurate, they would serve the purpose fairly well in our calculation because the overall contribution of the inner-shell ionization is relatively small. 2.2. Differential ionization cross section The available information on the differential ionization cross section or the energy spectrum of secondary electrons for xenon is rather scarce. The only m e a s u r e m e n t s have been reported by Opal et al. [11] for 500 eV electrons. The O shell in the xenon atom m a k e s a predominant contribution to the ionization of the whole atom. Therefore, the differential ionization cross section for the O shell should be obtained in a reliable way. For this purpose we follow the approach of utilizing the qualitative features of two nearly distinct domains, namely that of glancing collisions (involving small m o m e n t u m transfer) and that of knock-on collisions (involving large m o m e n t u m transfer). These features, discussed in detail by Kim [12], are: (a) for slow secondary electrons, the glancing collisions dominate and the secondary elec-
Dayashankar / Ionization yield in xenon due to electron impact tron spectrum is largely .determined by the differential optical oscillator strength; and (b) for fast secondary electrons, the spectrum should follow the knock-on collision cross section. Combining the contributions of the two kinds of collisions with suitable weight functions, Gerhart [13], Soong [14], Eggarter [15] and Dayashankar [6] have obtained the differential ionization cross sections for H2, Ne, A r and Kr respectively. Using the same approach, the differential cross section, [dtr°(T, E)/dE]th, for ionization of the O shell of xenon, for energy transfer E by an incident electron with kinetic energy T can be written as
d t r °"T.
( , E)]t h =
4zra2R [R d r . 4 T R b ( E ) T m E
1
E ( T + Io - E
÷ ( r + t o - E)
)/]
'
The functions b(E) and ~b(E) were determined by fitting eq. (2) to the available experimental data of Opal et al. [11] on the secondary electron spectrum for 500 eV electrons. These data correspond to an assumed total ionization cross section of 11.07 a02 while the value adopted in the present work is 9.029a02. Therefore, we renormalized the experimental data by a factor Of 0.816. We also made an allowance for the minor contribution of the N shell in order to get the O shell contribution to be used for the fitting. T o estimate the contribution of the three subshells belonging to the N shell, we used the binaryencounter formula [9] corresponding to eq. (1) which gives the differential ionization cross section, do~(T, E)/dE, for the jth (sub)shell as
dtrJ (T, E ) = 4rra~R2 [ 1 4Ei dE E + I j + E j Sj E-'~+ 3E 3
1
+ 4,(E)RSo
239
(2)
1 ~_ 4Ei + ( T + I j - E ) 2 3(T + / j - E ) 3 1
where dr/dE is the differential optical oscillator strength for ionization of the O shell with a threshold energy I0 (12.1 eV) and So is the number of electrons in the O shell. The first term in eq. (2) is essentially the Williams-Weizsiicker cross section representing the contribution of glancing collisions; the coefficient b(E), to be determined empirically, reflects the cutoff of glancing collisions at a certain m o m e n t u m transfer. The second term in eq. (2) is basically the Mott scattering cross section representing the contribution of knock-on collisions. The coefficient 4~(E) is an empirical function which reduces the second term at very low energy transfers and approaches unity as the energy transfer increases. For the differential optical oscillator strength, dr/dE, in eq. (2) we used a smooth representation of the experimental data of Samson [8] for E -< 59 eV. In the region of E > 59 eV, where experimental data are not available, we have made use of the fact that for large energy transfers dr/dE decreases as E -35 [16].
E(T + Ij- E
'
(3)
where the symbols have the same meaning as in eq. (1). It may be mentioned that eq. (3) does not describe the glancing collisions and hence is not valid for the low-energy part of the secondaryelectron spectrum. Ignoring the electron correlation effects in subshell ionization would also lead to a poor estimate of the cross section from eq. (3). However, the contribution of the innershell ionization being relatively minor, the use of approximate cross sections given by eq. (3) will not introduce a serious error in our calculation. It should be noted that for the O shell, which is the major contributor to the ionization, eq. (2) is preferable to eq. (3) because the former describes adequately the low-energy part of the electron spectrum as well as the high-energy part. T o fit eq. (2) to the renormalized experimental data at T = 5 0 0 e V corrected for the N-shell contribution, we proceeded by setting ~b(E)= 0 for E < 18.47 eV and adjusting b(E) only. The
240
Dayashankar
Ionization yield in xenon due to electron impact
i
1.0 0.8 w 0.6 -e-
)
~" O.4
n 0.2 0
I
12
14
~ 16
18
20
22
24
E (eV) Fig. 1. Empirical parameters b(E) and ~b(E) for eq. (2).
resulting b(E), shown in fig. 1, is a smooth function that approaches a value of 0.89 at 18.47 eV. For E < 16.1 eV, experimental data on the secondary-electron spectrum not being available, b(E) was obtained by extrapolation in such a way that the integrated cross section would agree with the a d o p t e d total ionization cross section for the O shell at T = 500 eV. For E > 18.47 eV, adopting b ( E ) = 0.89, eq. (2) was fitted to the data by adjusting the coefficient &(E) only. This coefficient turned out to increase rapidly from zero at E -- 18.47 eV to about unity at E = 2 0 . 5 e V . For E > 2 0 . 5 e V , &(E) was found to be nearly unity at most of the points; at some points, however, especially near the Auger peaks, the value turned out to be different from unity. W e ignored such points and took &(E) = 1 for all energy transfers above 20.5eV. This is justified because the contribution of Auger electrons is considered separately in this work. The function &(E) used in the calculation is also shown in fig. 1. With the coefficients b(E) and &(E) as obtained above, we integrated eq. (2) for various values of the incident energy T between the limits I0 and (T+Io)/2 to check whether the integrated cross section agreed with the adopted total ionization cross section for the O shell. T h e agreement was found to be good (within 8%) for T--> 300 eV. In this region the ratio C ( T ) of the
adopted cross section to the integrated cross section is a slowly varying function of T, ranging from 0.929 at 300 eV to 1.04 at 1 keV, and being exactly unity at 500 eV. The values of C ( T ) are given in table I. It is thus clear that eq. (2) is approximately valid for T _> 300 eV. Since we can rely more on the adopted total ionization cross section, eq. (2) can be improved by normalizing it exactly with respect to the total ionization cross section. Therefore, we multiply the right-handside of eq. (2) by the ratio C(T). This procedure is acceptable only for T -->300 eV where C ( T ) does not change significantly. Thus, for T - > 300 eV, we take the differential ionization cross section of the O shell to be do-°(T, E ) / d E = C(T)[do-°(T, E)/dE],h . For T < 300 eV, the region which is not described well by eq. (2), we resort to the cruder procedure of Miller [17] and G e r h a r t [13] and simply put do'°(T, E ) / d E = K ( T ) E -2 , where K ( T ) is adjusted to fit the total ionization cross section. It may be pointed out that, while considering ionization from the O shell, we have ignored the difference in 5s and 5p electrons and have treated the two electrons in the s subshell as having the same ionization threshold as the six electrons in the p subshell. However, the error
Table I Values of the normalizing parameter C(T)
T
T
(eV)
C(T)
(eV)
C(T)
300 400 500 600
0.929 0.991 1.000 1.014
700 800 900 1000
1.029 1.003 1.027 1.040
Dayashankar / Ionization yield in xenon due to electron impact
involved in this approximation is offset to a large extent by the fitting procedure used. 3. Degradation calculation With the set of inelastic collision cross sections described above, we consider the complete energy degradation of an incident electron of kinetic energy T and all the successively generated electrons in xenon gas having low and uniform density. The gaseous medium is assumed to be so voluminous that all excited and ionized species produced as well as all electrons generated remain within the medium. Under this premise, the mean total number of ion pairs produced in the entire degradation process, Ni(T), is given by the generalized Fowler equation [15, 18]: Ni(T) = pi(T) + ~'~ pn(T)Ni(T - En) n
(T+/])/2
+ ~. 1
[ 1j
p{(T,E)
x [Ni(T - E ) + N~(E - / j ) ] dE
+
~_,
j # outer shell
pA,(T)N,(TA,).
0)
In this integral equation, pi(T)= (ri(T)/orinel(T) is the probability of ionization per inelastic collision and pn(T)= ~r.(T)/~i,el(T) is the analogous probability for excitation to a state n with excitation energy En. Similarly, p~(T, E) = (doJ(T, E)/dE)/tri,et(T) is the probability per unit energy for ionization of the jth shell with an energy loss E and production of a secondary electron with a kinetic energy E - / j . The last term in eq. (4) represents the contribution of Auger electrons with the kinetic energy TAj produced as a result of ionization of the flh shell, pAj(T) being the probability of the Auger emission per inelastic collision. Since in our set of cross sections, the total excitation cross section, o'ex, is treated as cor-
241
responding to a single level, with excitation energy Eav equal to the average excitation energy of the resonance levels, the summation in the second term of eq. (4) is reduced to pex(T)Ni(T Eav), where pex(T)= tr=x(T)/o'ioel(T). In the energy region of interest in our study (i.e., for T-< 1 keV), the third term of eq. (4) includes contributions from the O shell and from the three subshells of the N shell which are considered separately. In the same way, the last term includes Auger electrons only from the N shell. Accordingly, the last term reduces to pA(T)Ni(TA) where pA(T)= rltrr~(T)/tri,e~(T), rt being the Auger yield which we have taken to be unity and TA being the average energy of Auger electrons from the N shell, which we have taken to be 26.6 eV on the basis of the line energies and relative intensities reported in the literature [191. Eq. (4) was solved numerically by starting at T = I0, the first ionization threshold, and ascending in T stepwise. We chose a step size of 1 eV for T -< 50 eV and of 5 eV for higher energies. It is necessary to keep the step size smaller than the excitation energy so that the inputs for solving the equation can be obtained by interpolating the previously calculated Ni values. The Lagrange three-point interpolation formula was used for interpolation of data. A comment may be in order on the necessity of using the generalized Fowler equation in the present work instead of the simple Fowler equation. The generalization of the Fowler equation in our work consists in allowing for inner-shell ionization and for its secondary Auger process. Since xenon is a sufficiently heavy atomic target, inner-shell ionization is quite appreciable. Although we have not evaluated the effect of the generalization of the Fowler equation in the present case, it has been demonstrated in our work on krypton [6] that for sufficiently heavy atoms that are expected to have appreciable cross sections for inner-shell ionization, the simple Fowler equation (based on the assumption of a single ionization threshold) is not adequate.
Dayashankar ~ Ionization yield in xenon due to electron impact
242
4. Results and discussion
Fig. 2 shows the results of the ionization yield calculations in terms of the quantity W, the mean energy required to produce an ion pair, as a function of the incident electron energy. It is found that the value of W initially decreases rapidly with increasing incident energy and then gradually approaches a constant value of 23.1 eV (within ---1%) for electrons with energy higher than 200 eV. This asymptotic value is in good agreement ( - 4 % ) with the experimental value of 22.2 eV for a tritium source of about 4 k e V beta energy as reported by Jesse and Sadauskis [20]. T h e calculated curve also reveals a kink around 21 eV. This kink corresponds to the onset of the excitation process. For comparing our results with the experimental work, we note that only two m e a s u r e m e n t s have been reported in the energy region covered in this study: one by Samson and H a d d a d [2] for electrons up to about 50 e V and another very recently by Combecher [3] for electrons up to l keV. T h e experimental data of these authors are also plotted in fig. 2. T h e calculated values are found to
100
'
'
'
'
' " "
'
'
'
' ' " 7
9O
have only occasional agreement with the experimental data of Samson and Haddad. The data of C o m b e c h e r are generally higher than the calculated values, although the agreement is pretty good for an incident energy above 100 eV. It may be mentioned that his m e a s u r e m e n t s for argon and krypton show a similar tendency with respect to the calculations of Eggarter [15] and D a y a s h a n k a r [6], respectively. It may also be noted that near 2 1 e V these observations only produce a slight shoulder in the curve instead of a kink. It is not possible to give an overall error in the calculated values of W because it depends on the errors in all the input cross sections, most of which are not known exactly. The errors involved in the experimental values of W are not known either. In any case, the large difference between experiment and theory with respect to W at low energies is not quite understandable. The crude formula for differential ionization cross sections used in the calculation for an incident energy lower than 300 eV may be responsible for the large errors in W at these energies. While comparing the results of our calculations with the experimental data it is relevant to mention that at the gas pressures used in the experiments, some of the excited atoms are likely to be converted into diatomic ions by the H o r n b e c k - M o l n a r process [21]:
a0 <
•
Z
Q
Xe* + Xe
PRESENT WORK
- -
70
COMBECHER(19B0) SAMSON AND HADDAD (1976)
60
a_ 50
A
Z
z
30
~
20
< I.U
L
101
J
i
i
~ i
>Xe~ + e - .
ill
t
10 2
i
,
i
i
i+
10 3
INCIDENT ELECTRON ENERGY(eV) Fig. 2. T h e m e a n energy r e q u i r e d to produce an ion pair, W, as a function of the incident electron energy.
In our calculation, however, the contribution of this collateral ionization has not been taken into account because of insufficient collision data. Guided by the estimate of this contribution for argon [15] one would expect this effect to lower the W value obtained in this work by an amount of the order of l e V . In view of this, for the asymptotic value of W, our calculation favours the measured value of Jesse and Sadauskis with respect to that of Combecher. Finally, we conclude that the agreement with the experimental work is, generally, quite encouraging, consider-
Dayashankar / Ionization yield in xenon due to electron impact
ing the assumptions made in constructing the cross sections.
Acknowledgements The author wishes to thank M.A. Prasad and Dr. U. Madhvanath for helpful discussions.
References [1] Average Energy Required to Produce an Ion Pair, Report 31, International Commission on Radiation Units and Measurements, Washington, D.C. (1979). [2] J.A.R. Samson and G.N. Haddad, Radiat. Res. 66 (1976) 1. [3] D. Combecher, Radiat. Res. 84 (1980) 189. [4] S. Sato, K. Okazaki and S. Ohno, Bull. Chem. Soc. Jap. 47 (1974) 2174. [5] K. Okazaki, S. Sato and S. Ohno, Bull. Chem. Soc. Jap. 48 (1975) 1411. [6] Dayashankar, Physica 111C (1981) 134.
243
[7] F.J. de Heer, R.H.J. Jansen and W. van der Kaay, J. Phys. B (Atom. Molec. Phys.) 12 (1979) 979. [8] J.A.R. Samson, Advances in Atomic and Molecular Physics, Vol. 2, D.R. Bates and I. Estermann, eds. (Academic Press, New York, 1966) p. 177. [9] L. Vriens, Case Studies in Atomic Collision Physics I, E.W. McDaniel and M.R.C. McDowell, eds. (NorthHolland, Amsterdam, 1969) p. 335. [10] W. Lotz, J. Opt. Soc. Am. 60 (1970) 206. [11] C.B. Opal, E.C. Beaty and W.K. Peterson, At. Data 4 (1972) 209. [12] Y.K. Kim, Radiat. Res. 61 (1975) 21. [13] D.E. Gerhart, J. Chem. Phys. 62 (1975) 821. [14] S.C. Soong, Radiat. Res. 67 (1976) 187. [15] E. Eggarter, J. Chem. Phys. 62 (1975) 833. [16] M. Inokuti, Rev. Mod. Phys. 43 (1971) 297. [17] W.F. Miller, Ph.D. Thesis, Purdue University (1956). [18] M. Inokuti, Radiat. Res. 64 (1975) 6. [19] ESCA Applied to Free Molecules, K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heden, K. Hamrin, U. Gelius, T. Bergmark, L.O. Werme, R. Manne and Y. Baer, eds. (North-Holland, Amsterdam, 1969) p. 156. [20] W.P. Jesse and J. Sadauskis, Phys. Rev. 107 (1957) 766. [21] J.A. Hornbeck and J.P. Molnar, Phys. Rev. 84 (1951) 621.