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Analytic electron impact inelastic cross sections for water vapor and a study of energy deposition
R&at. Phys. Gem. Vol. 40, No. 6, pp. 523-528,1992 Int. .I. Radtat.Appl. Instrwn.,port c Printedin GreatBritain. All rights reacrved
OM-5724/92 $5.00+ 0.00 copyright 0 1992Pergamon PrcS Ltd
ANALYTIC ELECTRON IMPACT INELASTIC CROSS SECTIONS FOR WATER VAPOR AND A STUDY OF ENERGY DEPOSITION DAYASHANKAR’
and A. E. S. GREEN*
‘Division of Radiological Protection, Bhabha Atomic Research Ccntre, Bombay 400 085, India and *Department of Nuclear Engineering Sciences, University of Florida, Gainesville, FL 32611, U.S.A. (Received I5 January 1992; accepted 6 April 1992) Abstrac-Electron impact cross sections for various electronic excited states and for ionization of water vapor are represented by analytic forms that cover non-relativistic as well as relativistic energy region. This inelastic cross section set may serve as a convenient input for electron transport calculations relevant in several fields. With this cross section input an electron energy deposition study has been carried out using a Monte Carlo simulation giving excitation and ionization yields, the Fano factor and the subexcitation spectra for initial energies up to 1 MeV.
EXCITATION
INTRODUCTION
gas phase cross section inputs used by various groups (Green et al., 1985; Berger, 1984;.Tumer et al., 1982; Paretxke et al., 1986; Zaider et al., 1983; Kaplan et al., 1986) in electron energy deposition studies for the modeling of radiation effects in water are very similar, resembling for the most part the detailed atomic cross section set first assembled in 1971 by the Florida group (Green et al., 1971; Oliver0 et al., 1972). There are, however, at this time, individual group variations and updating and, in this connection, there appears to be a need to establish a base set of water vapor cross section inputs. Then, based upon the same inputs, it would be possible to genuinely compare outputs of different calculating machinery used for energy deposition problems that are relevant in several fields such as microdosimetry, radiation chemistry and planetary aeronomy. In the present work we avail ourselves of the relativistic cross section set used by Berger (1984) in conjunction with his OGRE code and the earlier non-relativistic cross section set assembled by the Florida group (Green et al., 1971; Oliver0 et al., 1972). To avoid any ambiguities in input cross section which might be. reflected in the output of a transport calculation, we have attempted to seek an analytic expression which covers the entire energy range, non-relativistic as well as relativistic. Successful functions of this nature were proposed earlier by Porter, Jackman and Green (PJG) (Porter et al., 1976) who demonstrated the possibility of achieving this objective for N, and 02. Representation of cross sections in analytic forms is very convenient for efficient storage of information as well as for interpolation and extrapolation that are generally necessary in the computational studies.
The
RPC 40/c-”
CROSS SECTIONS
For excitation cross sections, Green et al. (1971) have introduced the approximate non-relativistic characterization
where q. = 4xuiR* = 6.514 x 10-‘4cm2 eV*, a,, being the Bohr radius and R the Rydberg energy. Herefis the optical oscillator strength and W, is the excitation energy, E represents the incident energy and Cl and /I’ are fitting parameters. This form was chosen over twenty years ago to facilitate analytic integrations and other conveniences. These purposes are now largely superceded by recent trends towards strict numerical computations. One of the principal devices for these earlier purposes was the approximation ln(E/ W,A x c(E/ We&’ where c is a constant and q is a small power between 0.25 and 0.33. Since this approximation does not serve a useful purpose in transport calculations carried out by strict numerical methods we abandon equation (1) in developing broad range analytic cross sections for the purpose of unambiguous input. Berger (1984) in his relativistic water vapor energy degradation studies has chosen a relativistic form somewhat similar to that of PJG: u ex
(E)=% R-&w+4
where /I = v/c and H=ln-
fi2 l-82
-82
Berger has evaluated the constants a and b for many excitations by the Fano plot graphical procedure. 523
DAYASHANKAR and A. E. S. GREEN
524
Since his work reflects a recent analysis of experimental data we consider only analytic forms which extrapolate to equations (2) and (3) while going at low energies to reasonable non-relativistic forms. After exploring many such possibilities, including variations of those proposed by PJG, we have settled on the expression %X(E) = (&If/ KX) (2 we, lmc ‘B*)[I? +
w
(4)
where
d=b_ln$ a
(6) ex
and Z is a sigmoid function which in the nonrelativistic region goes to 0 as E + W,, and in the high energy region goes to 1. This form has the property of going over exactly to Berger’s relativistic form if we identify f = aWex/R. The form of j? in the non-relativistic region goes to ln(E/ WA, neglecting the term b*. This is satisfactory in that it almost vanishes as E --, W,. Among the family of Z’s the best one which we have found to have reasonable low energy behavior is the function Z=
(E - w,J K W:,
with K = lpd 1, where p, d and v are fitting parameters. Table 1 contains the best fit parameters to the excitation cross sections based upon equations (4) and (7). In the case of direct excitations we used for fitting the cross section “data” set generated by equation (1) together with the relativistic cross
State Rydtxrg (A + B), n = 3 Rydberg (A+B), n ,4 Rydberg (C+D), n =3 Rydberg (C+D),n,4 Diffuse bands Dissociative continuum 1 Dissociative continuum 2 Triplet It Triplet 2t
Lyman D: Balmer OL Balmer jj Balmer y Balmer 6 018441 01 7114 OH 3064 OH 2800 Wquation
1. Analytic
Threshold W, (ev)
IONIZATION
CROSS SECTION
The total ionization cross section as well as the partial ionization cross section for various ionization states, namely 161, 3a1, 162, 2al and lal, from thresholds to relativistic energies can be fitted very well by using a slight modification in the formula that we have used for discrete excitations. In particular we propose for the ionization cross section
a,=%a 2 -Z(A ’
(7)
+ (E - W,,)
Table
sections generated by equation (2) using Berger’s a and 6 parameters. In the case of dissociative excitations we fitted the experimental data of Beenaker ef al. (1974) in the non-relativistic region and the data generated by equation (2) in the relativistic region. We do not include vibrational and rotational cross sections in this work as these have been treated separately by Green and Mann (1987) and can be used in conjunction with the cross section set reported here. For dissociative attachment cross sections, which are of some consequence in the limited energy range from about 4 to 16eV, it seems reasonable to use the analytic form given by Green et al. (1971) as there has not been significant progress in the measurement of these cross sections since then.
+d)
R mc*fi=
For fitting with equation (8) we used the total ionization cross section data assembled by Berger (1984) for use in his OGRE code. These data are based on the experimental results of Mark and Egger (1976) and of Schutten et al. (1966) in the non-relativistic region and are extended to relativistic region by equation (2) using the parameters a and b
cross section parameters
for excitations
d
Y
P
n
10.0
Direct Excitations 7.913 x 10-Z -7.710
x 10-1
1.1
0.8
11.8
4.957 x 10-Z
-7.945
x 10-l
1.1
0.8
11.0
1.450 x 10-l
-8.027
x IO-’
I.1
0.8
12.05
1.397 x 10-1
-8.170
x IO-’
1.1
0.8
13.32 7.4
5.142 x 10-l 9.922 x lO-2
-8.114 -7.225
x 10-i x 10-l
1.1 1.1
0.8 0.8
9.67
1.148 x 10-i
-5.191
x 10-l
1.1
0.8
3.0 1.0
1.0 1.0
2.2 2.2 2.2 2.2 2.2 2.2 2.2 0.9663 2.749
1.876 I .a78 1.677 2.161 2.043 1.862 1.515 x 10-Z 2.455 3.084 x 10-I
4.5 2.116 x IO-* 9.81 1.959 x 10-Z (12.5 for fitting) 18.5 18.5 18.6 18.9 19.1 17.3 17.6 9.1 11.0
3.0 3.0
Dissociative Excitations 3.720 x IO-* 1.604 x 7.784 x 10-3 1.603 x 1.493 x 10-j 1.458 x 6.382 x IO-’ 1.356 x 2.453 x IO-’ 1.274 x 6.607 x 1O-4 1.349 x 8.247 x lOm6 3.797 x 8.355 x 10-Z - 1.795 1.331 x 10-J 3.526 x
(1) to be used for these states with a-f.
p-*/I’
IO’ 10’ IO’ IO’ 10’ 10’ 102 10’
Analytic electron cross sections
based on the Fano plot procedure. Similarly, partial ionization cross section data for la1 (01s) state (K-shell of oxygen) are based on the experimental data of Glupe and Mehlhorn (1971) and equation (2). The partial ionization cross section data for other ionization states are based on the relative probabilities of ionization estimated by S. M. Seltzer using the Weizacker-Williams method (Berger, 1984). Table 2 gives the best fit parameters for total ionization and for partial ionization cross section of various states. With the fitting parameters known for each ionization state, the differential ionization cross section for each state can be represented by the expression proposed by Green and Sawada (1972) as
w,
T) = NE)~*(E)/t{ T - To(E)}2 + r*(E)] (9)
where T is the secondary electron energy and f and To are adjustable parameters which may depend on the incident energy. The normalizing parameter A(E) is given by
A(E)
2Z[R+d]
R rnc*fl*
tan-‘-
T, - T,
/[ + tan-’ :
csd”
TS(E, T) dT,
including in the summation the integrated expressions for all the ionization states. The integrated expression for a given state has the form (T,,, - T,)*+r* T;+Z-*
TS(E, T) dT = A(E)r*
Table 2. Ionization state
Tkshold I (eV)
lb1 301 lb2 2ul la1
12.6 14.7 18.4 32.2 539.7
For differential all states.
ionization
cross section parameters
0
d
Y
D
0.9176 0.7170 0.6099 0.2184 0.03204
I .I65
I.148 I.261 1.189 1.360 0.8883
10.07 7.513 5.383 3.219 2.250
2.157 2.106 2.544 0.9850
cross soztion r =
11.03 and
By fitting the data it turned out that the parameter r and To are almost independent of incident energy E and have the values r = 11.03 and To = 1.045. With these parameters the differential cross section for each ionization state is clearly represented in analytic form. This form is very convenient for speedy sampling of secondary electron energies in Monte Carlo calculations. ENERGY DEPOSITION
As a consistency check on the analytic cross section set presented here, we have carried out a Monte Carlo simulation study of electron energy deposition using this set. In this simulation, history of an electron was followed in a collision by collisioin manner until the primary electron as well as the secondaries of all generations were degraded to an energy below 4.5 eV, which is the lowest electronic excitation threshold. The contribution of Auger electrons resulting from the K-shell ionization of oxygen was included on the basis of measured Auger spectrum reported by Moddeman et al. (1971). However, as a simplification, for computational convenience the actual Auger spectrum was approximated by a line spectrum corresponding to the peak energies of 498.6, 493.8, 486.8, 482.2, 474.6, 469.2 and 475.4eV with probabilities 0.417, 0.251, 0.091, 0.066, 0.119, 0.033 and 0.023 respectively. The computed excitation and ionization yields for various states expressed as the number of events per 100 eV, are given in Table 3 for a few selected electron energies. It may be mentioned that the yields obtained by using equations (1) and (2) are found to be close to those obtained by using equation (4) in the entire energy region from low to high energies. It is seen from Table 3 that most of the yields become almost constant at high energies. The gross ionization yield commonly expressed in terms of the quantity W, the mean energy expended per ion pair formed, is also given in Table 3. While experimental data on the excitation yields are not available, the ionization yields are in good agreement with the experimental measurements (Combecker, 1980; ICRU, 1979), as shown in Fig. 1. The computed constant W-value at energies above about 2 keV is 30.6 eV while the recommended value based on a composite of experimental measurements is 29.6eV. Also shown in Table 3 and Fig. 1 are the computed values of the Fano factor, which is a measure of the ionization yield fluctuations, defined as the ratio of the variance of the distribution of number of ion pairs produced to that expected from a Poisson distribution. Similar to the energy dependence of the W-value, the Fano factor is also found to decrease with increasing energy, reaching a constant value of 0.23 for electrons of energy above about 2 keV. Turner et al. (1982) and Paretzke ef al. (1986) have reported computations of the Fano factor for energies only up to 10 keV, giving an asymptotic value of 0.25, which is somewhat
1(10)
where T, = (E - Z)/2 is the maximum energy of secondary electrons, based on the convention that slower outgoing electron is the secondary. Here I represents the ionization threshold. The normalizing parameter ensures that the expression for differential ionization cross secion is consistent with the integrated cross section. The adjustable parameters r(E) and T,(E), taken to be the same for all the ionization states, were determined by requiring consistency with the independently assigned stopping power values as given by Berger (1984). This was done by fitting the ionization loss function data by the quantity
To = I .045 for
525
DAYASHANKAR and A. E. S. GREEN
526
Table 3. Excitation and ionization yields and Fano factor in water vapor Yield, number of events per IOOeV
state
E=50
Rydberg (A + B), n = 3 Rydberg (A + B), n > 4 Rydberg (C + D), n = 3 Rydberg (C+D), n ,4 Diffuse bands
0.255
Electron 500 e;;gy,
E (eV) 104
105
106
Direct Excitations 0.216 0.174 0.168
0.163
0.166
0.166
0.073
0.074
0.075
100
0.104
0.094
0.350
0.305
0.247
0.242
0.237
0.240
0.245
0.261
0.240
0.201
0.193
0.194
0.1%
0.198
Dissoc. cont. 2 Triplet 1 Triplet 2
0.761 0.773 0.569 0.189 0.030
0.736 0.62 I 0.473 0.156 0.020
0.627 0.501 0.367 0.138 0.016
0.605 0.490 0.355 0.133 0.016
0.609 0.474 0.345 0.134 0.016
0.623 0.478 0.348 0.133 0.016
0.632 0.478 0.349 0.132 0.016
Lyman OL Balmer do Balmer b Balmer y Balmer 6 018441 01 7774 OH 3064 OH 2800
0.156 0.032 0.0067 0.0020 0.0007 0.0033 0.0019 0.596 0.019
Dissociative Excifatiom 0.229 0.242 0.226 0.048 0.051 0.045 0.0094 0.0098 0.0085 0.0036 0.0036 0.0033 0.0014 0.0012 0.0014 0.0045 0.0039 0.0040 0.0022 0.0014 0.0016 0.467 0.378 0.367 0.013 0.009 0.009
0.190 0.040 0.0075 0.0028 0.0010 0.0035 0.0012 0.364 0.008
0.182 0.038 0.0070 0.0025 0.0009 0.0036 0.0011 0.366 0.008
0.178 0.038 0.0068 0.0027 0.0010 0.0032 0.0011 0.367 0.008
HO_
0.413 0.056
Dissociative Attachment 0.346 0.309 0.306 0.046 0.040 0.040
0.305 0.040
0.303 0.040
0.301 0.040
lb1
1.114 0.745 0.459 0.035
1.299 0.998 0.744 0.220 0.0060 30.55 0.23
1.303 0.985 0.744 0.219 0.0097 30.59 0.23
Dissoc. cont.
301 lb2 2al 101 W-value (eV) Fano factor
I
42.50 0.39
1.176 0.868 0.589 0.096 36.66 0.32
ELECTRON
0.075
0.073
Ionization 1.253 1.282 0.983 0.995 0.712 0.728 0.190 0.207 0.0006 31.87 31.13 0.25 0.24
ENERGV
1.297 0.985 0.740 0.217 0.01 I2 30.66 0.23
(oV )
Fig. 1. W-value and Fano factor as a function of incident electron energy. Circles are experimental of Cornbecker et al. (1980) for W-value.
data
Analytic electron cross sections
521
Tabk 4. The aatkd subexcitation electroo spectrum for IO keV cla%rons TIE, O-O.1 0.3-0.4 0.6-0.7 0%I.0
EI WYN 0.638 0.674 0.547 4.271
TIE,
E, W-)/Y
TIE,
EI N(T)/&
0.1-0.2 0.818 0.2-0.3 0.808 0.4-0.5 0.587 OS-O.6 0.568 0.7-0.8 0.536 0.8-0.9 0.585 (E, = 4.5 eV, Ni = 327.4 for 10 keV ekctrons)
higher than our value. Experimental data on the Fano factor for water vapor are not available. Recently there has been a good deal of interest in studies of energy deposition in subexcitation domain, i.e. the energy region below the lowest electronic state threshold. In this region vibrational and rotational excitations play an important role. The energy spectrum of subexcitation electrons (also sometimes called the entry spectrum) is a useful quantity in this connection. The results of our computations show that the subexcitation spectra scaled by the total ionization yield are more or less invariant with respect to initial electron energy. As a typical example, the subexcitation spectrum, N(T), for 10 keV electrons, expressed in the scaled form by the quantity E, N( T)/Ni , is given in Table 4 as a function of T/E, . Here El is the lowest electronic state threshold, equal to 4.5 eV and Ni is the total ionization yield, equal to 327.4 for 10 keV electrons. In a recent review article Kaplan and Miterev (1987), based on the computations of their group, have stated that the sub-excitation spectrum in water vapor follows the Platxman form (Platxman, 1955), which is a monotonically decreasing function, and is similar to that of helium. (The lowest electronic state threshold in their work was taken to be 7.4 eV.) The present work contradicts this statement in that the computed spectrum does not follow the Platzman form and is not like that of helium. The main reason for these differences is total neglect of the vibrational excitations in the earlier work even though these excitations are important in the low energy region. It is pertinent to note that the Platzman form is essentially meant for atomic systems, where vibrational excitations do not occur; although quite often it has been used for molecular systems also. The results of this study suggest that, unlike in the case of atomic systems, the subexcitation spectrum is quite sensitive to the electronic structure of the molecular systems. SUMMARY
We have proposed simple analytic forms for electron impact inelastic cross sections of water vapor. The forms represent the cross sections in nonrelativistic as well as relativistic energy region and are consistent with proper behavior of the ionization yield and the Fano factor. Some interesting aspects of the subexcitation electron spectra have also been brought out. The cross sections in the analytic forms may serve as convenient inputs to energy deposition calculations for the modeling of radiation effects that
are relevant in many fields such as microdosimetry, radiation chemistry and planetary aeronomy. Acknowledgemenf-The authors are grateful to Dr Martin J. Berger for supplying details of the cross section data used in his OGRE code and also for useful discussions. RRFERENCES
Beenakker C. I. M., De Heer F. J., Krop H. B. and Mohlmann G. R. (1974) Dissociative excitation of water by electron impact. Chem. Phys. 6,445. Berger M. J. (1984) Discussion of electron cross sections for transport calculations. In Proceedings of rhe Workshop on Electronic and Ionic Collision Cross Sections Needed in the Modeling of Radiation Interactions with Matter, p. 1. Report ANL-84-28, Argonne National Laboratory, Argonne, IL. Combeclcer D. (1980) Measurement of W values of lowenergy electrons in-several gases. Radiat. Res. 84, 189. Glupe G. and Mehlhom W. (1971) Absolute electron impact ionization cross sections of N, 0, and Ne. 1. Phys. (X,40. Green A. E. S., Dayashankar, Schippnick P. F., Rio D. E. and Schwartz J. M. (1985) Yield and concentration microplumes for electron impact on water vapor. Radiat. Res. 104, 1. Green A. and Mann A. (1987) The big bands of H,O. Radiat. Res. 112, 11. Green A. E. S., Oliver0 J. J. and Stagat R. W. (1971) Microdosimetry of low-energy electrons. In Biophysical Aspects of Radiarion Quality, p. 79. International Atomic Energy Agency, Vienna. Green A. E. S. and Sawada T. (1972) Ionization cross sections and secondary electron distributions. J. Atmos. Terr. Phys. 34, 1719. ICRU (1979) Average energy required to produce an ion pair. Report 31, International Commission on Radiation Units and Measurements, Washington, DC. Kaplan I. G. and Miterev A. M. (1987) Interaction of charged particles with molecular medium and track effects in radiation chemistry. In Advances in Chemical Physics, Vol. 68 (Edited by I. Prigogine and S. A. Rice), p. 255. Wiley, New York. Kaplan I. G., Miterev A. M. and Sukhonosov V. Ya. (1986) Comparative study of yields of primary products in tracks of fast electrons in liquid water and in water vapor. Radiat. Phys. Chem. 27, 83. Mark T. D. and Egger F. (1976) Cross section for single ionization of H,O and D,O bv electron imoact from threshold up to i70 eV. Int: J. Mass Spectrom.-Ion Phys. 20, 89. Moddeman W. E., Carlson T. A., Krause M. O., Pullen B. P., Bull W. E. and Schweitzer G. K. (1971) Determination of the K-LL Auger spectra of N,, Or, CO, NO, H,O and CO,. J. Chem. Phys. 55, 2317. Oliver0 J. J., Stagat R. W. and Green A. E. S. (1972) Electron dew&ion in water vaoor. with atmosnheric applications. J. Geophys. Res. 77; 4791. Paretzke H. G.. Turner J. E.. Hamm R. N.. Wriaht H. A. and Ritchie R. H. (1986) ‘Calculated yields aid fluctuations for electron degradation in liquid water and water vapor. J. Chem. Phys. 84, 3182. Plataman R. L. (1955) Subexcitation electrons. Radiat. Res.
528
DAYASHANKAR and A. E. S. GREEN
Porter H. S., Jackman C. H. and Green A. E. S. (1976) Efficiencies for production of atomic nitrogen and oxygen by relativistic proton impact in air. J. C/rem. Phys. 65, 154. Schutten J., De Heer F. J., Moustafa H. R., Boerboom A. J. H. and Kistemaker J. (1966) Gross- and partialionization cross sections for electrons on water vapor in the energy range 0.1-20 keV. J. Chem. Phys. 44, 3924.
Turner J. E., Paretxke H. G., Hamm R. N., Wright H. A. and Ritchie R. H. (1982) Comparative study of electron energy deposition and yields in water in the liquid and vapor phases. Radiat. Rex 92, 47. Zaider M, Brenner D. J. and Wilson W. E. (1983) The applications of track calculations to radiobiology. I. Monte Carlo simulation of proton tracks. Radiat. Res. 95, 231.
OM-5724/92 $5.00+ 0.00 copyright 0 1992Pergamon PrcS Ltd
ANALYTIC ELECTRON IMPACT INELASTIC CROSS SECTIONS FOR WATER VAPOR AND A STUDY OF ENERGY DEPOSITION DAYASHANKAR’
and A. E. S. GREEN*
‘Division of Radiological Protection, Bhabha Atomic Research Ccntre, Bombay 400 085, India and *Department of Nuclear Engineering Sciences, University of Florida, Gainesville, FL 32611, U.S.A. (Received I5 January 1992; accepted 6 April 1992) Abstrac-Electron impact cross sections for various electronic excited states and for ionization of water vapor are represented by analytic forms that cover non-relativistic as well as relativistic energy region. This inelastic cross section set may serve as a convenient input for electron transport calculations relevant in several fields. With this cross section input an electron energy deposition study has been carried out using a Monte Carlo simulation giving excitation and ionization yields, the Fano factor and the subexcitation spectra for initial energies up to 1 MeV.
EXCITATION
INTRODUCTION
gas phase cross section inputs used by various groups (Green et al., 1985; Berger, 1984;.Tumer et al., 1982; Paretxke et al., 1986; Zaider et al., 1983; Kaplan et al., 1986) in electron energy deposition studies for the modeling of radiation effects in water are very similar, resembling for the most part the detailed atomic cross section set first assembled in 1971 by the Florida group (Green et al., 1971; Oliver0 et al., 1972). There are, however, at this time, individual group variations and updating and, in this connection, there appears to be a need to establish a base set of water vapor cross section inputs. Then, based upon the same inputs, it would be possible to genuinely compare outputs of different calculating machinery used for energy deposition problems that are relevant in several fields such as microdosimetry, radiation chemistry and planetary aeronomy. In the present work we avail ourselves of the relativistic cross section set used by Berger (1984) in conjunction with his OGRE code and the earlier non-relativistic cross section set assembled by the Florida group (Green et al., 1971; Oliver0 et al., 1972). To avoid any ambiguities in input cross section which might be. reflected in the output of a transport calculation, we have attempted to seek an analytic expression which covers the entire energy range, non-relativistic as well as relativistic. Successful functions of this nature were proposed earlier by Porter, Jackman and Green (PJG) (Porter et al., 1976) who demonstrated the possibility of achieving this objective for N, and 02. Representation of cross sections in analytic forms is very convenient for efficient storage of information as well as for interpolation and extrapolation that are generally necessary in the computational studies.
The
RPC 40/c-”
CROSS SECTIONS
For excitation cross sections, Green et al. (1971) have introduced the approximate non-relativistic characterization
where q. = 4xuiR* = 6.514 x 10-‘4cm2 eV*, a,, being the Bohr radius and R the Rydberg energy. Herefis the optical oscillator strength and W, is the excitation energy, E represents the incident energy and Cl and /I’ are fitting parameters. This form was chosen over twenty years ago to facilitate analytic integrations and other conveniences. These purposes are now largely superceded by recent trends towards strict numerical computations. One of the principal devices for these earlier purposes was the approximation ln(E/ W,A x c(E/ We&’ where c is a constant and q is a small power between 0.25 and 0.33. Since this approximation does not serve a useful purpose in transport calculations carried out by strict numerical methods we abandon equation (1) in developing broad range analytic cross sections for the purpose of unambiguous input. Berger (1984) in his relativistic water vapor energy degradation studies has chosen a relativistic form somewhat similar to that of PJG: u ex
(E)=% R-&w+4
where /I = v/c and H=ln-
fi2 l-82
-82
Berger has evaluated the constants a and b for many excitations by the Fano plot graphical procedure. 523
DAYASHANKAR and A. E. S. GREEN
524
Since his work reflects a recent analysis of experimental data we consider only analytic forms which extrapolate to equations (2) and (3) while going at low energies to reasonable non-relativistic forms. After exploring many such possibilities, including variations of those proposed by PJG, we have settled on the expression %X(E) = (&If/ KX) (2 we, lmc ‘B*)[I? +
w
(4)
where
d=b_ln$ a
(6) ex
and Z is a sigmoid function which in the nonrelativistic region goes to 0 as E + W,, and in the high energy region goes to 1. This form has the property of going over exactly to Berger’s relativistic form if we identify f = aWex/R. The form of j? in the non-relativistic region goes to ln(E/ WA, neglecting the term b*. This is satisfactory in that it almost vanishes as E --, W,. Among the family of Z’s the best one which we have found to have reasonable low energy behavior is the function Z=
(E - w,J K W:,
with K = lpd 1, where p, d and v are fitting parameters. Table 1 contains the best fit parameters to the excitation cross sections based upon equations (4) and (7). In the case of direct excitations we used for fitting the cross section “data” set generated by equation (1) together with the relativistic cross
State Rydtxrg (A + B), n = 3 Rydberg (A+B), n ,4 Rydberg (C+D), n =3 Rydberg (C+D),n,4 Diffuse bands Dissociative continuum 1 Dissociative continuum 2 Triplet It Triplet 2t
Lyman D: Balmer OL Balmer jj Balmer y Balmer 6 018441 01 7114 OH 3064 OH 2800 Wquation
1. Analytic
Threshold W, (ev)
IONIZATION
CROSS SECTION
The total ionization cross section as well as the partial ionization cross section for various ionization states, namely 161, 3a1, 162, 2al and lal, from thresholds to relativistic energies can be fitted very well by using a slight modification in the formula that we have used for discrete excitations. In particular we propose for the ionization cross section
a,=%a 2 -Z(A ’
(7)
+ (E - W,,)
Table
sections generated by equation (2) using Berger’s a and 6 parameters. In the case of dissociative excitations we fitted the experimental data of Beenaker ef al. (1974) in the non-relativistic region and the data generated by equation (2) in the relativistic region. We do not include vibrational and rotational cross sections in this work as these have been treated separately by Green and Mann (1987) and can be used in conjunction with the cross section set reported here. For dissociative attachment cross sections, which are of some consequence in the limited energy range from about 4 to 16eV, it seems reasonable to use the analytic form given by Green et al. (1971) as there has not been significant progress in the measurement of these cross sections since then.
+d)
R mc*fi=
For fitting with equation (8) we used the total ionization cross section data assembled by Berger (1984) for use in his OGRE code. These data are based on the experimental results of Mark and Egger (1976) and of Schutten et al. (1966) in the non-relativistic region and are extended to relativistic region by equation (2) using the parameters a and b
cross section parameters
for excitations
d
Y
P
n
10.0
Direct Excitations 7.913 x 10-Z -7.710
x 10-1
1.1
0.8
11.8
4.957 x 10-Z
-7.945
x 10-l
1.1
0.8
11.0
1.450 x 10-l
-8.027
x IO-’
I.1
0.8
12.05
1.397 x 10-1
-8.170
x IO-’
1.1
0.8
13.32 7.4
5.142 x 10-l 9.922 x lO-2
-8.114 -7.225
x 10-i x 10-l
1.1 1.1
0.8 0.8
9.67
1.148 x 10-i
-5.191
x 10-l
1.1
0.8
3.0 1.0
1.0 1.0
2.2 2.2 2.2 2.2 2.2 2.2 2.2 0.9663 2.749
1.876 I .a78 1.677 2.161 2.043 1.862 1.515 x 10-Z 2.455 3.084 x 10-I
4.5 2.116 x IO-* 9.81 1.959 x 10-Z (12.5 for fitting) 18.5 18.5 18.6 18.9 19.1 17.3 17.6 9.1 11.0
3.0 3.0
Dissociative Excitations 3.720 x IO-* 1.604 x 7.784 x 10-3 1.603 x 1.493 x 10-j 1.458 x 6.382 x IO-’ 1.356 x 2.453 x IO-’ 1.274 x 6.607 x 1O-4 1.349 x 8.247 x lOm6 3.797 x 8.355 x 10-Z - 1.795 1.331 x 10-J 3.526 x
(1) to be used for these states with a-f.
p-*/I’
IO’ 10’ IO’ IO’ 10’ 10’ 102 10’
Analytic electron cross sections
based on the Fano plot procedure. Similarly, partial ionization cross section data for la1 (01s) state (K-shell of oxygen) are based on the experimental data of Glupe and Mehlhorn (1971) and equation (2). The partial ionization cross section data for other ionization states are based on the relative probabilities of ionization estimated by S. M. Seltzer using the Weizacker-Williams method (Berger, 1984). Table 2 gives the best fit parameters for total ionization and for partial ionization cross section of various states. With the fitting parameters known for each ionization state, the differential ionization cross section for each state can be represented by the expression proposed by Green and Sawada (1972) as
w,
T) = NE)~*(E)/t{ T - To(E)}2 + r*(E)] (9)
where T is the secondary electron energy and f and To are adjustable parameters which may depend on the incident energy. The normalizing parameter A(E) is given by
A(E)
2Z[R+d]
R rnc*fl*
tan-‘-
T, - T,
/[ + tan-’ :
csd”
TS(E, T) dT,
including in the summation the integrated expressions for all the ionization states. The integrated expression for a given state has the form (T,,, - T,)*+r* T;+Z-*
TS(E, T) dT = A(E)r*
Table 2. Ionization state
Tkshold I (eV)
lb1 301 lb2 2ul la1
12.6 14.7 18.4 32.2 539.7
For differential all states.
ionization
cross section parameters
0
d
Y
D
0.9176 0.7170 0.6099 0.2184 0.03204
I .I65
I.148 I.261 1.189 1.360 0.8883
10.07 7.513 5.383 3.219 2.250
2.157 2.106 2.544 0.9850
cross soztion r =
11.03 and
By fitting the data it turned out that the parameter r and To are almost independent of incident energy E and have the values r = 11.03 and To = 1.045. With these parameters the differential cross section for each ionization state is clearly represented in analytic form. This form is very convenient for speedy sampling of secondary electron energies in Monte Carlo calculations. ENERGY DEPOSITION
As a consistency check on the analytic cross section set presented here, we have carried out a Monte Carlo simulation study of electron energy deposition using this set. In this simulation, history of an electron was followed in a collision by collisioin manner until the primary electron as well as the secondaries of all generations were degraded to an energy below 4.5 eV, which is the lowest electronic excitation threshold. The contribution of Auger electrons resulting from the K-shell ionization of oxygen was included on the basis of measured Auger spectrum reported by Moddeman et al. (1971). However, as a simplification, for computational convenience the actual Auger spectrum was approximated by a line spectrum corresponding to the peak energies of 498.6, 493.8, 486.8, 482.2, 474.6, 469.2 and 475.4eV with probabilities 0.417, 0.251, 0.091, 0.066, 0.119, 0.033 and 0.023 respectively. The computed excitation and ionization yields for various states expressed as the number of events per 100 eV, are given in Table 3 for a few selected electron energies. It may be mentioned that the yields obtained by using equations (1) and (2) are found to be close to those obtained by using equation (4) in the entire energy region from low to high energies. It is seen from Table 3 that most of the yields become almost constant at high energies. The gross ionization yield commonly expressed in terms of the quantity W, the mean energy expended per ion pair formed, is also given in Table 3. While experimental data on the excitation yields are not available, the ionization yields are in good agreement with the experimental measurements (Combecker, 1980; ICRU, 1979), as shown in Fig. 1. The computed constant W-value at energies above about 2 keV is 30.6 eV while the recommended value based on a composite of experimental measurements is 29.6eV. Also shown in Table 3 and Fig. 1 are the computed values of the Fano factor, which is a measure of the ionization yield fluctuations, defined as the ratio of the variance of the distribution of number of ion pairs produced to that expected from a Poisson distribution. Similar to the energy dependence of the W-value, the Fano factor is also found to decrease with increasing energy, reaching a constant value of 0.23 for electrons of energy above about 2 keV. Turner et al. (1982) and Paretzke ef al. (1986) have reported computations of the Fano factor for energies only up to 10 keV, giving an asymptotic value of 0.25, which is somewhat
1(10)
where T, = (E - Z)/2 is the maximum energy of secondary electrons, based on the convention that slower outgoing electron is the secondary. Here I represents the ionization threshold. The normalizing parameter ensures that the expression for differential ionization cross secion is consistent with the integrated cross section. The adjustable parameters r(E) and T,(E), taken to be the same for all the ionization states, were determined by requiring consistency with the independently assigned stopping power values as given by Berger (1984). This was done by fitting the ionization loss function data by the quantity
To = I .045 for
525
DAYASHANKAR and A. E. S. GREEN
526
Table 3. Excitation and ionization yields and Fano factor in water vapor Yield, number of events per IOOeV
state
E=50
Rydberg (A + B), n = 3 Rydberg (A + B), n > 4 Rydberg (C + D), n = 3 Rydberg (C+D), n ,4 Diffuse bands
0.255
Electron 500 e;;gy,
E (eV) 104
105
106
Direct Excitations 0.216 0.174 0.168
0.163
0.166
0.166
0.073
0.074
0.075
100
0.104
0.094
0.350
0.305
0.247
0.242
0.237
0.240
0.245
0.261
0.240
0.201
0.193
0.194
0.1%
0.198
Dissoc. cont. 2 Triplet 1 Triplet 2
0.761 0.773 0.569 0.189 0.030
0.736 0.62 I 0.473 0.156 0.020
0.627 0.501 0.367 0.138 0.016
0.605 0.490 0.355 0.133 0.016
0.609 0.474 0.345 0.134 0.016
0.623 0.478 0.348 0.133 0.016
0.632 0.478 0.349 0.132 0.016
Lyman OL Balmer do Balmer b Balmer y Balmer 6 018441 01 7774 OH 3064 OH 2800
0.156 0.032 0.0067 0.0020 0.0007 0.0033 0.0019 0.596 0.019
Dissociative Excifatiom 0.229 0.242 0.226 0.048 0.051 0.045 0.0094 0.0098 0.0085 0.0036 0.0036 0.0033 0.0014 0.0012 0.0014 0.0045 0.0039 0.0040 0.0022 0.0014 0.0016 0.467 0.378 0.367 0.013 0.009 0.009
0.190 0.040 0.0075 0.0028 0.0010 0.0035 0.0012 0.364 0.008
0.182 0.038 0.0070 0.0025 0.0009 0.0036 0.0011 0.366 0.008
0.178 0.038 0.0068 0.0027 0.0010 0.0032 0.0011 0.367 0.008
HO_
0.413 0.056
Dissociative Attachment 0.346 0.309 0.306 0.046 0.040 0.040
0.305 0.040
0.303 0.040
0.301 0.040
lb1
1.114 0.745 0.459 0.035
1.299 0.998 0.744 0.220 0.0060 30.55 0.23
1.303 0.985 0.744 0.219 0.0097 30.59 0.23
Dissoc. cont.
301 lb2 2al 101 W-value (eV) Fano factor
I
42.50 0.39
1.176 0.868 0.589 0.096 36.66 0.32
ELECTRON
0.075
0.073
Ionization 1.253 1.282 0.983 0.995 0.712 0.728 0.190 0.207 0.0006 31.87 31.13 0.25 0.24
ENERGV
1.297 0.985 0.740 0.217 0.01 I2 30.66 0.23
(oV )
Fig. 1. W-value and Fano factor as a function of incident electron energy. Circles are experimental of Cornbecker et al. (1980) for W-value.
data
Analytic electron cross sections
521
Tabk 4. The aatkd subexcitation electroo spectrum for IO keV cla%rons TIE, O-O.1 0.3-0.4 0.6-0.7 0%I.0
EI WYN 0.638 0.674 0.547 4.271
TIE,
E, W-)/Y
TIE,
EI N(T)/&
0.1-0.2 0.818 0.2-0.3 0.808 0.4-0.5 0.587 OS-O.6 0.568 0.7-0.8 0.536 0.8-0.9 0.585 (E, = 4.5 eV, Ni = 327.4 for 10 keV ekctrons)
higher than our value. Experimental data on the Fano factor for water vapor are not available. Recently there has been a good deal of interest in studies of energy deposition in subexcitation domain, i.e. the energy region below the lowest electronic state threshold. In this region vibrational and rotational excitations play an important role. The energy spectrum of subexcitation electrons (also sometimes called the entry spectrum) is a useful quantity in this connection. The results of our computations show that the subexcitation spectra scaled by the total ionization yield are more or less invariant with respect to initial electron energy. As a typical example, the subexcitation spectrum, N(T), for 10 keV electrons, expressed in the scaled form by the quantity E, N( T)/Ni , is given in Table 4 as a function of T/E, . Here El is the lowest electronic state threshold, equal to 4.5 eV and Ni is the total ionization yield, equal to 327.4 for 10 keV electrons. In a recent review article Kaplan and Miterev (1987), based on the computations of their group, have stated that the sub-excitation spectrum in water vapor follows the Platxman form (Platxman, 1955), which is a monotonically decreasing function, and is similar to that of helium. (The lowest electronic state threshold in their work was taken to be 7.4 eV.) The present work contradicts this statement in that the computed spectrum does not follow the Platzman form and is not like that of helium. The main reason for these differences is total neglect of the vibrational excitations in the earlier work even though these excitations are important in the low energy region. It is pertinent to note that the Platzman form is essentially meant for atomic systems, where vibrational excitations do not occur; although quite often it has been used for molecular systems also. The results of this study suggest that, unlike in the case of atomic systems, the subexcitation spectrum is quite sensitive to the electronic structure of the molecular systems. SUMMARY
We have proposed simple analytic forms for electron impact inelastic cross sections of water vapor. The forms represent the cross sections in nonrelativistic as well as relativistic energy region and are consistent with proper behavior of the ionization yield and the Fano factor. Some interesting aspects of the subexcitation electron spectra have also been brought out. The cross sections in the analytic forms may serve as convenient inputs to energy deposition calculations for the modeling of radiation effects that
are relevant in many fields such as microdosimetry, radiation chemistry and planetary aeronomy. Acknowledgemenf-The authors are grateful to Dr Martin J. Berger for supplying details of the cross section data used in his OGRE code and also for useful discussions. RRFERENCES
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DAYASHANKAR and A. E. S. GREEN
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