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Energy spectra of subexcitation electrons
CHEMICAL
Volume 157, number 6
ENERGY
SPECTRA
OF SUBEXCITATION
Mitio INOKUTI, Mineo KIMURA d,:gonne Nutional Laborarorv, Argonne, IL Rrcrivcd
2 July 198X; in final form
ELECTRONS
and Ken-ichi 60439,
25 November
PHYSICS LETTEKS
1988
*
KOWARI
USA
I6 Sep~rmber I988
SubexcItation electrons are those electrons that are produced in irradiated matter and have kmctx cncrgics (T) lower than the first elccrroniccxcltation threshold ofthe major component. The abundant subexcitation electrons play an Important role in many contexts such as the excitation of minor-component molecules. For quantitative assessment of effects of the subexcitation electrons, their energy spectra must be known. Often they are assumed to be of the Platzman form: A+B/( T+I)‘, where I is the ionization threshold. We point out that this form is justified for He and HI, but not in general. For Ar we have carried out an accuratecalculation and have obtained a spectrum not representable by the Platzman form.
All ionizing radiations directly or indirectly generate in matter abundant electrons having widely differing kinetic energies. These electrons are characterized by degradation spectra, which were originally introduced by Spencer and Fano [ 11. At kinetic energies (T) exceeding the first electronic excitation threshold (E, ), the electrons degrade rapidly, chiefly through electronic excitation and ionization. However, at kinetic energies (T) below E,, electrons degrade much more slowly because they transfer only much smaller quanta to molecules through vibrational excitation, rotational excitation, or momentum transfer to translation upon elastic collisions with molecules. Thus, it is suitable to treat those electrons with TcE, as a distinct entity, as was first pointed out by Platzman [ 21, who named them subexcitation electrons. They are always abundant and are often important as precursors of radiationchemical products, e.g. negative ions resulting from electron attachment to gaseous molecules, solvated electrons in liquids, and excited states of impurity molecules lying below E,. Quantitative analyses of subexcitation-electron effects, exemplified by recent studies [ 3-5 1, must begin with the energy spectrum resulting from the rapid * Work supported
by the US Department of Energy, Off& of Health and Environmental Research, under Contract W-31. l09-Eng-38.
504
degradation at T> E, . In other words, one needs to know the number N,, (7”) dT of subexcitation electrons that reach energies between T and T+dT in the subexcitation domain (0 < T-c E, ). The function N,, ( 7’) has often been referred to as the initial spectrum of subexcitation electrons: here the term initial is used to imply that the electrons first arrive in the subexcitation domain and then degrade further, and that their moderation is described by another spectrum. We propose to call N,,(T) the entry spectrum of subcxcitation electrons, because the term initial is used in radiation chemistry to refer to different time scales, depending upon the context, and is imprecise. The entry spectrum N,,(T) is precisely determined by the solution of the Spencer-Fan0 equation at T>E,, as was first demonstrated by Douthat [ 6,7 1. He fitted [ 8 ] his results for He and Hz to Platzman’s semi-empirical expression [ 2 ] N,,(T)=A+B/(T+Z)3,
(1)
where I is the first ionization threshold, and A and B are constants. The total number of subexcitation electrons is equal to the total ionization as given by the W value [ 91, and the mean energy of subexcitation electrons is a fraction of I in general (about 0.31 in rare gases [ lo] ). These two quantities enable determination of values of A and B for commonly studied gases. (To be more precise, the fit of the H, result [ 71 to eq. (1) is globally good; however, the
0 009-2614/88/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 152. number 6
minor
flattening
CHEMICAL
of
the
T/E, (0.1 seen in fig. 6 of Douthat
spectrum at [7] cannot be
reproduced by eq. ( 1) .) There may have been an impression that N,,(T) should always be closely approximated by the Platzman form. The purpose of the present note is to point out that this is not the case. We have determined N,,(T) for Ar from the Spencer-Fan0 theory (fig. 1). In this example, IV,, ( T) differs qualitatively from the Platzman form. The mean energy of subexcitation electrons turns out to be 0.281, according to our calculation. (A full account of the calculation will be presented in a forthcoming article [ 111. ) For the solution of the Spencer-Fan0 equation we use a straightforward numerical method similar to that in the 1979 work of Douthat [ 71. The use of the Cray computer (at the Livermore Laboratory, provided through the Office of Health and Environmental Research, US Department of Energy) has enabled us to carry out the calculations much more rapidly and extensively. For the input, we have used the cross-section data determined by Eggarter [ 12,13 1. To show why eq. ( 1) applies to He and Hz (but not to Ar), we may recall how eq. (1) was conceived. The entry spectrum N,,(T) consists of two major contributions in general. The first contribution comes from those secondary electrons that re-
0.4
r 1
:‘: *:
0.0
0.2
0.4
0.6
0.8
PHYSICS LETTERS
25 November
1988
sult directly from an ionizing collision of high-energy electrons. The spectrum of the secondary-electron energy E resulting from an individual ionizing collision is insensitive to the primary-electron energy and is approximately given in the form I/ ( t + I)’ for Hz and He, as was fully explained by Inokuti et al. [ 141. This contribution is represented by the B term of eq. ( 1). The second contribution arises indirectly, i.e. as a result of many collisions of degrading electrons. This second, indirect contribution is roughly uniform in the domain 0.3 < T/E,-e1, as shown by calculations [ 111 on AT. This contribution is represented by the A term of eq. ( 1). Consequently, combining the direct and indirect contributions gives the Plattman form, eq. ( 1). Returning to Ar, we note that the secondary-electron spectrum is not a simple, monotonically decreasing function of E, but has a maximum at low E followed by a steep decline [ 12,131. Thus, the direct contribution to N,,( 1’) in Ar differs qualitatively from that in He and H2. The reason for the rualitative difference in the secondary-electron spectrum is the delayed resonance followed by the Cooper minimum, well known in the study of photoionization spectra, which is closely connected with secondary-electron spectra. (See Fano and Cooper [ 15 ] and Dillon and Inokuti [ 161 for the basic discussions of this topic.) We may add that Ar is not a special case in our context. Our general knowledge of secondary-electron spectra [ 171 indicates that qualitatively the same results for the entry spectra of subexcitation electrons are expected for Kr and Xe and for many common molecules, including N2 and CO. Finally, the entry spectrum in argon (fig. 1) shows wavy structures at T/E,~0.3,which arise from an interplay of discrete excitation and ionization in the last few collisions prior to the entry of electrons into the subexcitation domain. A full discussion will be given in ref. [ 111.
1.0
T/ E,
Fig. 1. The entry spectrum of subexcitation electrons in argon. The horizontal axis represents T/E,, the subexcitation electron energy measured in the first electronic threshold. The vertical axis represents the function N,,( T)E, /N;, where _f$ N,,(T) dT= T, is the total number of ion pairs produced by a source electron of 2 keV. The solid curve represents the result obtained by the solution of the Spencer-Fan0 equation. The dashed curve represents the values obtained from the Platzman expression
References [I ] L.V. Spencer and U. Fano. Phys. Rev. 93 ( 19.54) I 172. [2] R.L. Platzman, Radial. Res. 2 (1955) I. [3] C. Naleway, M. Inokuti. M.C. Sauer Jr. and R. Cooper, J. Phys Chem. 90 (1986) 6154. [4] M.A. Dillon. M. lnokuti and M. Kimura. Radiat. Phys. Chem. 32 (1988) 43.
505
Volume 152, number 6 [ 51 [b] [ 71 [S] [ 91
CHEMICAL
D.A. Douthar. Astrophys. .I. 314 (1987) 419. D.A. Douthat. Radiat. Res. 61 (1975) I. D.A. Douthat. J. Phys. B. 12 (1979) 663. D.A. Douthat. J. Chem. Phys. 79 (1983) 4599. International Commission on Radiation Units and Measurements Average Energy Rcquircd to Product an Ion Pair, ICRU Report 31 (Washington, 1979). IO] R.L. Platzman, Intern. J. Appl. Radiat. Isot. 10 (1961) 116. I I ] K. Kowari, M. Kimura and M. Inokuti, manuscript in preparation.
506
PHYSICS LETTERS
25 November
1988
[ 121 E. Eggarter, J. Chem. Phys. 62 (I 975) 833. [ 131 E. Eggarter and M. Inokuti, Argonne National Laboratory Report. ANL-80-58 (I 980). [ 141 M. Inokuti, M.A. Dillon, J.ll. Miller and K. Omidvar, J. Chem. Phys. 87 (1987) 6967. [IS] U. Fano and J.W. Cooper, Rev. Mod. Phys, 40 ( 1968) 441. [ 16j M.A. Dillon and M. Inokuti, J. Chcm. Phys. 82 (I 985) 4415. [ 17 ] C.B. Opal, E.C. Beaty and W.K. Peterson, At. Data 4 ( I Y72) 209.
Volume 157, number 6
ENERGY
SPECTRA
OF SUBEXCITATION
Mitio INOKUTI, Mineo KIMURA d,:gonne Nutional Laborarorv, Argonne, IL Rrcrivcd
2 July 198X; in final form
ELECTRONS
and Ken-ichi 60439,
25 November
PHYSICS LETTEKS
1988
*
KOWARI
USA
I6 Sep~rmber I988
SubexcItation electrons are those electrons that are produced in irradiated matter and have kmctx cncrgics (T) lower than the first elccrroniccxcltation threshold ofthe major component. The abundant subexcitation electrons play an Important role in many contexts such as the excitation of minor-component molecules. For quantitative assessment of effects of the subexcitation electrons, their energy spectra must be known. Often they are assumed to be of the Platzman form: A+B/( T+I)‘, where I is the ionization threshold. We point out that this form is justified for He and HI, but not in general. For Ar we have carried out an accuratecalculation and have obtained a spectrum not representable by the Platzman form.
All ionizing radiations directly or indirectly generate in matter abundant electrons having widely differing kinetic energies. These electrons are characterized by degradation spectra, which were originally introduced by Spencer and Fano [ 11. At kinetic energies (T) exceeding the first electronic excitation threshold (E, ), the electrons degrade rapidly, chiefly through electronic excitation and ionization. However, at kinetic energies (T) below E,, electrons degrade much more slowly because they transfer only much smaller quanta to molecules through vibrational excitation, rotational excitation, or momentum transfer to translation upon elastic collisions with molecules. Thus, it is suitable to treat those electrons with TcE, as a distinct entity, as was first pointed out by Platzman [ 21, who named them subexcitation electrons. They are always abundant and are often important as precursors of radiationchemical products, e.g. negative ions resulting from electron attachment to gaseous molecules, solvated electrons in liquids, and excited states of impurity molecules lying below E,. Quantitative analyses of subexcitation-electron effects, exemplified by recent studies [ 3-5 1, must begin with the energy spectrum resulting from the rapid * Work supported
by the US Department of Energy, Off& of Health and Environmental Research, under Contract W-31. l09-Eng-38.
504
degradation at T> E, . In other words, one needs to know the number N,, (7”) dT of subexcitation electrons that reach energies between T and T+dT in the subexcitation domain (0 < T-c E, ). The function N,, ( 7’) has often been referred to as the initial spectrum of subexcitation electrons: here the term initial is used to imply that the electrons first arrive in the subexcitation domain and then degrade further, and that their moderation is described by another spectrum. We propose to call N,,(T) the entry spectrum of subcxcitation electrons, because the term initial is used in radiation chemistry to refer to different time scales, depending upon the context, and is imprecise. The entry spectrum N,,(T) is precisely determined by the solution of the Spencer-Fan0 equation at T>E,, as was first demonstrated by Douthat [ 6,7 1. He fitted [ 8 ] his results for He and Hz to Platzman’s semi-empirical expression [ 2 ] N,,(T)=A+B/(T+Z)3,
(1)
where I is the first ionization threshold, and A and B are constants. The total number of subexcitation electrons is equal to the total ionization as given by the W value [ 91, and the mean energy of subexcitation electrons is a fraction of I in general (about 0.31 in rare gases [ lo] ). These two quantities enable determination of values of A and B for commonly studied gases. (To be more precise, the fit of the H, result [ 71 to eq. (1) is globally good; however, the
0 009-2614/88/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 152. number 6
minor
flattening
CHEMICAL
of
the
T/E, (0.1 seen in fig. 6 of Douthat
spectrum at [7] cannot be
reproduced by eq. ( 1) .) There may have been an impression that N,,(T) should always be closely approximated by the Platzman form. The purpose of the present note is to point out that this is not the case. We have determined N,,(T) for Ar from the Spencer-Fan0 theory (fig. 1). In this example, IV,, ( T) differs qualitatively from the Platzman form. The mean energy of subexcitation electrons turns out to be 0.281, according to our calculation. (A full account of the calculation will be presented in a forthcoming article [ 111. ) For the solution of the Spencer-Fan0 equation we use a straightforward numerical method similar to that in the 1979 work of Douthat [ 71. The use of the Cray computer (at the Livermore Laboratory, provided through the Office of Health and Environmental Research, US Department of Energy) has enabled us to carry out the calculations much more rapidly and extensively. For the input, we have used the cross-section data determined by Eggarter [ 12,13 1. To show why eq. ( 1) applies to He and Hz (but not to Ar), we may recall how eq. (1) was conceived. The entry spectrum N,,(T) consists of two major contributions in general. The first contribution comes from those secondary electrons that re-
0.4
r 1
:‘: *:
0.0
0.2
0.4
0.6
0.8
PHYSICS LETTERS
25 November
1988
sult directly from an ionizing collision of high-energy electrons. The spectrum of the secondary-electron energy E resulting from an individual ionizing collision is insensitive to the primary-electron energy and is approximately given in the form I/ ( t + I)’ for Hz and He, as was fully explained by Inokuti et al. [ 141. This contribution is represented by the B term of eq. ( 1). The second contribution arises indirectly, i.e. as a result of many collisions of degrading electrons. This second, indirect contribution is roughly uniform in the domain 0.3 < T/E,-e1, as shown by calculations [ 111 on AT. This contribution is represented by the A term of eq. ( 1). Consequently, combining the direct and indirect contributions gives the Plattman form, eq. ( 1). Returning to Ar, we note that the secondary-electron spectrum is not a simple, monotonically decreasing function of E, but has a maximum at low E followed by a steep decline [ 12,131. Thus, the direct contribution to N,,( 1’) in Ar differs qualitatively from that in He and H2. The reason for the rualitative difference in the secondary-electron spectrum is the delayed resonance followed by the Cooper minimum, well known in the study of photoionization spectra, which is closely connected with secondary-electron spectra. (See Fano and Cooper [ 15 ] and Dillon and Inokuti [ 161 for the basic discussions of this topic.) We may add that Ar is not a special case in our context. Our general knowledge of secondary-electron spectra [ 171 indicates that qualitatively the same results for the entry spectra of subexcitation electrons are expected for Kr and Xe and for many common molecules, including N2 and CO. Finally, the entry spectrum in argon (fig. 1) shows wavy structures at T/E,~0.3,which arise from an interplay of discrete excitation and ionization in the last few collisions prior to the entry of electrons into the subexcitation domain. A full discussion will be given in ref. [ 111.
1.0
T/ E,
Fig. 1. The entry spectrum of subexcitation electrons in argon. The horizontal axis represents T/E,, the subexcitation electron energy measured in the first electronic threshold. The vertical axis represents the function N,,( T)E, /N;, where _f$ N,,(T) dT= T, is the total number of ion pairs produced by a source electron of 2 keV. The solid curve represents the result obtained by the solution of the Spencer-Fan0 equation. The dashed curve represents the values obtained from the Platzman expression
References [I ] L.V. Spencer and U. Fano. Phys. Rev. 93 ( 19.54) I 172. [2] R.L. Platzman, Radial. Res. 2 (1955) I. [3] C. Naleway, M. Inokuti. M.C. Sauer Jr. and R. Cooper, J. Phys Chem. 90 (1986) 6154. [4] M.A. Dillon. M. lnokuti and M. Kimura. Radiat. Phys. Chem. 32 (1988) 43.
505
Volume 152, number 6 [ 51 [b] [ 71 [S] [ 91
CHEMICAL
D.A. Douthar. Astrophys. .I. 314 (1987) 419. D.A. Douthat. Radiat. Res. 61 (1975) I. D.A. Douthat. J. Phys. B. 12 (1979) 663. D.A. Douthat. J. Chem. Phys. 79 (1983) 4599. International Commission on Radiation Units and Measurements Average Energy Rcquircd to Product an Ion Pair, ICRU Report 31 (Washington, 1979). IO] R.L. Platzman, Intern. J. Appl. Radiat. Isot. 10 (1961) 116. I I ] K. Kowari, M. Kimura and M. Inokuti, manuscript in preparation.
506
PHYSICS LETTERS
25 November
1988
[ 121 E. Eggarter, J. Chem. Phys. 62 (I 975) 833. [ 131 E. Eggarter and M. Inokuti, Argonne National Laboratory Report. ANL-80-58 (I 980). [ 141 M. Inokuti, M.A. Dillon, J.ll. Miller and K. Omidvar, J. Chem. Phys. 87 (1987) 6967. [IS] U. Fano and J.W. Cooper, Rev. Mod. Phys, 40 ( 1968) 441. [ 16j M.A. Dillon and M. Inokuti, J. Chcm. Phys. 82 (I 985) 4415. [ 17 ] C.B. Opal, E.C. Beaty and W.K. Peterson, At. Data 4 ( I Y72) 209.