J . Phys. Chem. 1984,88, 4854-4862
4854
i - ’
correction factor 1 2(1 c,,? AE should represent this; in this case it is 29 X 10- hartree. Using this wave function, we calculated the dipole moment he as -5.86 D, compared to -5.83 D, obtained from experimental valuesI9 by extrapolating to v = -1/2. There were several computational difficulties involved with this calculation of 168 basis functions yielding 26 250 124 integrals in C2,symmetry. The 4-index transformation from A 0 to M O integrals involved ten magnetic tapes (two primary, five main file, and three final). The graphical unitary group (GUGA) CI program’J was adapted so that (ijlxx) integral blocks, rather than (ixlxx) blocks, could be read in, block by block, in the iterative CI procedure. The CI vector was too long to hold in the machine, and therefore paging procedures had to be. used. By careful choice, it was found best to hold vector segments of length up to 40 000 in the memory, but the CPU/IO ratio was approximately 1/5 in the C I iterations on the IBM 3081D machine with 1.8 Mbytes available. The total time for the calculation was approximately 9 h, made up from integral evaluation (Cray) 700s Cray (approximately 1 h IBM time), SCF and CASSCF calculation (Cray) 600s (approximately 40 min IBM time), 4-index transformation (IBM) (5 h), and C I calculation (2 h 45 min).
Discussion This large Gaussian type basis set, variational MRSDCI calculation on LiH at R = 3.015 bohrs, gives an energy 1.5 X low3 hartree above the exact eigenvalue. To some extent, this must represent the best that current state of the art calculations can achieve without using the special symmetry and other circumstances of this molecule. It is appropriate to compare this calculation with recent quantum Monte Carlo calculations, which have recently produced some impressive results. Moskowitz and Schmidt4 have presented variational domain Greens function Monte Carlo results on LiH at three internuclear distances including R = 3.015 bohrs. At R = 3.015 bohrs, dependent upon whether either an S C F or GVB trial wave function is used, Moskowitz and Schmidt4 results are (19) L.Wharton, L.Gold, and W. Klemperer, J . Chem. Phys., 37,2149 (1962).
-8.071 or -8.068 hartree, each of which is associated with an error bar u of 0.002 hartree. The statistical interpretation of the error bar is that, “after an infinite amount of time, they are 66% confident that the resulting variational energy would be within 0.002 hartree of the energy quoted”. However, there must also be a large error of about 25%22 in the variance. Moskowitz, Schmidt, Lee, and Kalos2’ have published results using the short-time approximation at R = 3.015 bohrs of a similar accuracy. Reynolds et ale5have also published results using the short-time approximation. The above two paragraphs state the current position of ab initio and Monte Carlo results on LiH, and perhaps no further comment is necessary. As biased observers, we might suggest that the Monte Carlo workers will have to reduce their u values by a factor of 10 before their results are competitive with current ab initio methods. This is especially important when it is recalled that chemists are always interested in smooth surfaces, and energy differences, rather than absolute energies, and it is for these reasons why ab initioists are so very rarely interested in performing calculations which correlate all the electrons. This is not to say that the Monte Carlo results on systems such as H 2 0 are not impressive, where an energy of -76.377 k 0.007 hartree was obtained by Reynolds et al.s and -76.402 f 0.015 hartree by M o s k o w i t ~but , ~ ~Meyer” obtained -76.3683 hartree in 1971, and it must surely be possible to improve substantially on this number, if it was thought necessary, by using the latest procedures (the experimetal energy is -76.43 hartree).
Acknowledgment. We are grateful to G. J. Sexton for his help with the 4-index transformation, and R.J.H. and P.J.K. acknowledge an SERC Research Grant. H.F.S. received partial support from the US. National Science Foundation, Grant CHE-8218785. Registry No. LiH, 7580-67-8. (20) W. Meyer, Int. J. Quantum Chem., Quantum Chem. Symp., No. 5 , 341 (1971). (21) J. W. Moskowitz, K. W. Schmidt, M. A. Lee, and M. H. Kalos, J . Chem. Phys., 77, 349 (1982). (22) D. M. Ceperley, to be submitted for publication in J . Comput. Phys. (23) J. M. Moskowitz, private communication.
Electron Distribution Functions and Thermalization Times in Inert Gas Moderators B. Shizgal* and D. R. A. McMahon Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T 1 Y6 (Received: October 5, 1983)
The energy relaxation of electrons in inert gases is examined with a discrete ordinate method of solution of the Boltzmann equation. Realistic electron moderator momentum transfer cross sections are employed. The time evolution of the electron velocity distribution function is obtained as well as particular averages such as the energy and the directed velocity. Each property can be expressed as a small sum of exponential terms each characterized by an eigenvalue of the Boltzmann collision operator.
I. Introduction The study of the thermalization of energetic electrons via elastic collisions with inert gas moderators has received considerable attention during the past several years.’-* In a very recent paper,
Ternbe and Mozumder9 have discussed the importance of the study of the time evolution of the electron thermalization in the analysis of swarm experiments, electron scavenging, delayed luminescence in gaseous mixtures, and transient conductivity in pulsed experiments. Other important applications include the thermalization
(1) A. Mozumder, J. Chem. Phys., 72, 1657 (1980). (2) A. Mozumder, J. Chem. Phys., 72, 6289 (1980). (3) A. Mozumder, J. Chem. Phys., 74, 6911 (1981). (4) K. D. Knierim, M. Waldmann, and E. A. Mason, J . Chem. Phys., 77, 943 (1982). (5) B. Shizgal, J . Chem. Phys., 7,, 5741 (1983).
(6) B. Shizgal, Chem. Phys. Letr. 100, 41 (1983). (7) J. W. Warman and M. C. Sauer, J . Chem. Phys., 62, 1971 (1975). (8) G. L. Braglia, G. L. Caraffini, and M. Diligenti, Nuovo Cimento, 62B, 139 (1981). (9) B. L. Tembe and A. Mozumder, Phys. Reu. A, 27, 3274 (1983).
0022-3654/84/2088-4854$01.50/0
0 1984 American Chemical Society
Electron Thermalization of photoelectrons in the ionosphere,l0>"the determination of the velocity distribution function of solar wind electron^,^*,^^ and other applications. The development of an efficient method of calculation of electron distributions in diverse physical situations is clearly an important endeavor. Recently, Shizga15q6employed an efficient discrete ordinate method to study the rate of thermalization of electrons with an initial nonequilibrium distribution function chosen to be a 6 function. The first paper considered electrons moderated by a hard-sphere gas whereas the second paper considered helium as the moderator with an energy-dependent electron-helium cross section. The present work is concerned with the study of electron thermalization in the other inert gases. The important aspect that is introduced by considering the heavier inert gases (Ar, Kr, and Xe) is that the momentum transfer cross sections for these gases with electrons exhibit large Ramsauer-Townsend minima vs. energy. The situation for H e was found to be somewhat similar to the hard-sphere case owing to the slow variation of the e-helium cross section with energyS6The basic objective of the present work is to study the relaxation of the velocity distribution function from some initial nonequilibrium epithermal form (a &function in this case) and the time evolution of certain moments of the distribution function such as the energy and the directed velocity as thermal equlibrium is approached. The electron velocity distribution function approaches thermal equilibrium with two very different time scales which differ by On the the electron-moderator mass ratio of the order of short time scale, anisotropies of the electron distribution and the directed velocity decay well before the onset of appreciable energy relaxation. Energy relaxation and the approach of the speed distribution function to Maxwellian occur on the longer time scale. The decay of anisotropic distributions has been discussed in a general way by Shizgal and B1a~kmore.l~ In a series of p a p e r ~ l - Mozumder ~,~ studied the relaxation of electrons with a displaced pseudo-Maxwellian characterized by an effective temperature and directed mean velocity, both of which are time dependent. Differential equations for the temperature and the mean velocity, with time-dependent coefficients, obtained from appropriate averages with the assumed form of the distribution function are integrated numerically. The use of a Maxwellian with a rapidly increasing temperature over the short time scale is questionable. The speed distribution function should not change during this time period whereas the distribution employed by Mozumder is rapidly broadening as the velocity direction is randomized. With regard to the use of a Maxwellian for the energy relaxation over the long time scale, Mozumder's method is analogous to a method introduced by KeizerI5 in the study of hot atom reactions. For the thermalization and reaction of hot atoms, Kiezer and later Grant et a1.16 assumed that the velocity distribution function is a Maxwellian characterized by a time-dependent temperature. The motivation for this assumption, either for the relaxation of hot atoms or energetic electrons, is the expectation that after some very short initial transient, and well before any appreciable energy exchange, the velocity distribution function attains a local Maxwellian form. The Maxwellian for the nonequilibrium gas would have a temperature different from the moderator. However, for the systems under consideration here, there is no a priori reason for this to happen. The way in which a local Mawellian could be established is to allow for collisions between the nonequilibrium "test" particles (electrons or hot atoms as the case may be). If the electron-electron collision rate is rapid relative to the electron-moderator collision rate, then it is conceivable that, on the short time scale, the electron distribution (10) G. P. Mantas, Planet. Space Sci., 29, 1319 (1981). (11) J. Jasperse, Planet. Space Sci., 24, 33 (1976). (12) J. D. Scudder, and S. Olbert, J . Geophys. Res., 84, 603 (1979). (13) R. Rouseel-Dupre, Solar Phys., 68, 265 (1980). (14) B. Shizgal and R. Blackmore, Chem. Phys., 56, 249 (1983). (15) J. Keizer, J . Chem. Phys., 58, 4524 (1973). (16) E. R. Grant, D. Feng, J. Keizer, K. D. Knierim, and J. W. Root, Am. Chem. SOC.Symp. Ser., No. 66, 314 (1978).
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4855 will tend to a Maxwellian characterized by the electron temperature which is different from the moderator temperature. The approach to equilibrium, that is, the equilibration of the electron temperature to that of the moderator, will then occur on the longer time scale. A discussion of the conditions for this "epochal" relaxation has appeared elsewhere." Since the electron thermalization process considered in this paper and by other worke r ~concerns ~ ~ ,electrons ~ dilutely dispersed in a large excess of moderator, epochal relaxation does not occur since there is only one (energy) relaxation time scale determined by the electronmoderator collision cross section. Consequently, the electron velocity distribution function will remain non-Maxwellian throughout the thermalization process until thermal equilibrium is attained. Details of the electron velocity distribution function are shown in this paper and the results confirm this point. This is consistent with similar results pertaining to the thermalization and reaction of hot a t ~ r n s . ~ * , ' ~ The present work is based on the solutions of the LorentzFokker-Planck equation for the electron velocity distribution function with a discrete ordinate method employed in the earlier paper^.^,^ The method has proven to be very efficient and superior to the traditional moment m e t h ~ d . ~ .The ' ~ discrete ordinate method involves the expansion of the velocity distribution function in the eigenfunctions of the Lorentz-Fokker-Planck operator, determined by diagonalizing the matrix representative in the discrete ordinate basis. A detailed study of the method as applied t,a the solution of boundary value and eigenvalue problems is described elsewhere.20 The reciprocal of the eigenvalues of the operator are then the characteristic relaxation times for the thermalization process and the time evolution of the velocity distribution function and average properties can be written as sum of a small number of exponential terms each characterized by a different eigenvalue. By contrast, the moment method4 involves the numerical integration of a set of coupled differential equations (moment equations) with time-dependent coefficients. The solutions in this case cannot be written in terms of characteristic relaxation times. In addition, there are numerical problems associated with the evaluation of the two temperature matrix elements of the collision operator and with the convergence of the moment equations. The discrete ordinate method does not suffer from these difficulties. 11. Relaxation of the Directed Velocity
As discussed in the Introduction, there are two very different time scales in the approach of an arbitrary anisotropic electron velocity distribution function to thermal equilibrium. The distribution function for electrons becomes isotropic well before any appreciable energy exchange occurs. A useful way to follow the relaxation of the anisotropic portion of the distribution function is to follow the decay of the directed velocity, that is, the average of u,, the z component of the electron velocity, v. The average of the directed velocity involves only the L = 1 componentf(')(u,t), of the expansion of the distribution f(v,r) in Legendre polynomial~!*~ The time dependence of this component of the distribution function is obtained from the Boltzmann equation and is given by4 df ') / d t = -NU(,a ~ ) f ( l ) (1) where N is the moderator density and urnis the electron moderator momentum transfer cross section. The solution of eq 1 is elementary and we have immediately f(l)(u,t) = f(l)(u,O) exp[-Nua,t] The average of the directed velocity is then4s5 (u,) = (4a/3) Jmf(l)(u,O) exp[-Nuu,t]u (17) (18) (19) (1981). (20)
(2)
du
(3)
J. M. Fitzpatrick and B. Shizgal, Phys. Fluids, 22, 436 (1979). B. Shizgal, J. Chem. Phys., 74, 1401 (1981). K. D. Knierim, S. L. Lin, and E. A. Mason, J. Chem. Phys., 54, 341
B. Shizgal and R. Blackmore, J. Comput. Phys., 55, 313 (1984).
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
4856
I
I
,
,
Ar
4k 4
>'
u, = 4.8
3
A
v
TABLE I: Convergence of Eigenvalues n N=lO N=20 N=30 1 2 3 4 5 6 7 8 9 10
\
9
Shizgal and McMahon
2
5.587 9.049 13.63 21.97 31.37 41.60 87.97 149.6 242.8
5.579 8.948 13.15 18.24 24.18 31.76 40.66 52.54 69.55 85.56
1
S
10
Nt
1s
20
2s
Figure 1. Decay of the directed velocity, Tb = 290.1 K; Nt in units of io7 s/cm3.
4
2
3
\
A
$
2
1
1 2 3 4 5 6 7 8 9 10
91.84 151.7 224.1 354.4 603.5 1011 1561 2642 4012
87.62 117.4 151.1 185.9 223.6 283.9 357.9 446.6 622.1 835.8
1 2 3 4 5 6 7 8 9 10
27.05 54.85 92.56 184.5 328.9 510.9 1034 2022 3649
26.85 50.31 81.06 120.6 178.8 244.1 377.0 491.0 722.9 914.9
A,"
5.579 8.948 13.15 18.19 24.04 30.70 38.13 46.81 56.08 68.92
N=40
N=50
18.19 24.04 30.67 38.04 46.15 54.95 64.76
30.67 38.04 46.12 54.88 64.31
A,b
87.61 117.4 149.9 183.2 217.1 256.0 304.8 366.6 431.8 538.7
149.9 183.2 216.7 255.2 300.4 352.3 408.9 477.7
216.7 255.1 300.3 351.0 406.8 467.1
50.24 80.50 117.5 161.3 211.7 271.2 337.7 424.4 521.3
161.2 211.5 268.3 332.5 403.3 487.9
A,c
26.84 50.24 80.51 117.7 162.4 216.1 288.2 366.5 500.8 612.0
"Argon, Tb = 290.1 K. *Xenon, Tb = 290.1 K. 'Xenon, Tb = 700 K. A, in units of 1/i- with unit cross section, uo = 1 A.
Figure 2. Decay of the directed velocity, Tb = 290.1 K.
With the choice of an initial 6 f ~ n c t i o n , ' *that ~ - ~is~fl)(u,O) ~ = 6(u - uo), the decay of (u,) is a pure exponential Mozumder's approach is based on the use of the displaced pseudo-Maxwellian with a time-dependent temperature. The temperature employed in the Maxwellian is determined from
where ( E ) is the mean kinetic energy of the electrons which is essentially constant (equal to mu2/2) during this short time period. Consequently, as (0,) decays to zero, the temperature rises from zero to tlie value as determined by uo. This approach is questionable, since with this formalism the isotropic portion of the distribution function is varying over this time scale. However, during this time scale, the isotropic portion of the distribution function (the& = 0 component off; see section 111) is essentially constant. Where eq 3 only samples urnat the initial speed, uo, Mozumdbr's approach samples the cross section over a wide range in energy as the distribution function broadens with increasing tempemtpre. Although, the decay of (u,) is a simple exponential decay, the variation with time is shown in Figure 1 for argon for different uo andi in Figure 2 for the different gases with uo = 4.8, to be compared with the analogous figures in Mozumder's paper. (The speed uQis in units of the thermal speed uth = 1.148 X 10' cm/s defined by Mozumder' corresponding to the rms energy of 0.025 eV at a temperature of 290.lK.l) The values of urnat uo are those
0 4
1 2
0 0
1 6
E(eV)
Figure 3. Energy dependence of electron momentum transfer cross sec-
tions. listed by Mozumder in Table I. The variation of the cross sections with energy is shown in Figure 3. These are drawn from an analytic fit to the numerical cross sections, as discussed in section 111. The results in Figure 1, show clearly that the variation in the rate of decay of (u,) follows the variation of u, with uo. The rate of decay exhibits a minimum in going from uo = 1 to uo = 4.8. The results shown by Mozumder are qualitatively similar to the present results for uo = 4.8 and 4, but the rate of decay for uo = 3 is considerably faster. The comparison of the rates of decay for the four moderators, shown in Figure 2, is consistent with the ordering of urn(ug = 4.8). Since the decay is a pure exponential, the different curves cannot cross. The results shown by Mozumder for Ar and Xe are
The Journal of Physical Chemistry, Vol. 88, No, 21, 1984 4851
Electron Thermalization qualitatively in agreement with those in Figure 2. However, the rates of decay for Ne and Kr, calculated by Mozumder, are slower for Ne and faster for Kr, and the curves for these gases actually intersect. For an initial distribution with a finite width, such as a Gaussian, the decay of (u,) would no longer be a pure exponential and the cross section at speeds other than uo would influence the results. The important point is that at the end of this time period the speed distribution has not changed in a significant manner from its initial form.
If the components of the vector a are the coefficients of the expansion of some arbitrary functionfb) in B, polynomials and f is the vector whose components arefb,), then the matrix Twith elements (wi) 1/2B,(yi)gives the transformation between the polynomial and discrete ordinate representations, that is
111. Electron Energy Relaxation
The representation of the derivative operator in the discrete ordinate basis given by
f = T-a
The basic methodology has been discussed in the previous
N
paper^.^.^ However, since there are a few changes in several definitions and since the development is rather short, it is repeated here for completeness. For energy relaxation, only the spherical or isotropic portion of the spatially homogeneous VDF,fio), is required. This is the L = 0 term in the expansion of the VDF in Legendre polynomials! It is convenient to writeyo) dv = 3 exp(-y)J, dy where the reduced electron speedy = (m/2kTb)'/2u, where v is the electron velocity, m is the electron mass, and Tb is the moderator temperature. The electron VDF in the presence case is given by the LFP
aq/at' = ZJ,
(6)
where
(15)
N
Dij = C C Tim(mld/dyln)TnJ
(16)
(mld/dyln) = J=wb)BrnB'n dv
(17)
m=ln=l
where
are the matrix elements of the derivative operator in the polynomial basis. The matrix elements of I in the polynomial basis are given by (mllln) = -Jm
wb)yt?.B',B',,
dy
(18)
Consequently, the matrix representative of the LFP operator in the discrete ordinate basis is found to be given by N
Lij is the definition of the collision operator in dimensionless form. In eq 7, 8 = um/uowhere urnis the e-inert gas momentum transfer cross section. The dimensionless time in eq 6 is given by t' = t / r where T
= [Nm(2kTb/m) '~2a,/2M]-'
where a, is some convenient hard sphere cross section. N a n d M are the moderator density and mass, respectively. The formal solution of eq 6 can be written
J,bJ?= exp(It? J,b,O)
(8)
where w(y)J,(y,O)is the initial distribution which is taken to be S(y -yo) where yo is the initial reduced speed of the electrons and w b ) = y2 exp(-y*). The time evolution of the energy is given by E(t?/EO = Yo-2
Am
WWY2
J,b8dv
(9)
where Eo = E(0). With the expansion
2
bo - Y) = w b ) n-0 + n b o ) + n b )
(10)
where +,b) are the eigenfunctions of the LFP operator with eigenvalues A,, that is
I+nb)= - h + n b )
(11)
the energy relaxation is given by m
E ( t ? / E o = C c, exp(-Ant? n=O
(12)
In eq 12, c, = b,+,(yo)/yo2 where
The eigenvalue problem defined by eq 11 with the operator given by eq 7 is to be solved. A discrete ordinate method based on the use of weights and points associated with a Gaussian quadrature rule is employed, as discussed elsewhere.z0 The particular quadrature that is used is based on speed polynomials &Cy), orthogonal on the interval [O,m] with weight function w b ) = y 2 exp(-y2),21~Z2 that is
-C Yk2bk)DkjDki k= 1
(19)
where the transformation T has been used. The eigenvalues A, and eigenfunctions +n are determined by diagonalizing the matrix Li,. The coefficients b, and c, are determined from the eigenfunctions and the energy relaxation is determined from eq 12. The momentum transfer cross sections employed with either Ar, Kr, or Xe as moderator are those listed by Mozumder2 in Table I of his paper. Initially an interpolation based on a spline fit was used to calculate the cross section between the tabulated values. However, the convergence of the expansion (eq 12) was not entirely satisfactory, owing to the occurrence of the derivative of the cross section in the collision operator. This difficulty was eliminated when an alternate fit based on the functional form a(u) = u(0) exp[(u/u,~)~a,(v/u,~)"] was employed. A least-squares fit was used to determine the a, coefficients. The result was a good overall fit to the tabulated values up to 5.2 eV. The convergence with this fit was much better. The cross sections for helium and neon were calculated from the phase shifts of N e ~ b e t ~ ~ and of O'Malley and C r o m p t ~ n respectively, ,~~ which gave no difficulties with convergence. The numerical calculations involve setting up the matrix Lijin eq 19 and determining the eigenvalues. The convergence of the eigenvalues vs. the number of quadrature points is shown in Table I, for several representative cases. As is clear from the results in the table, the convergence is extremely rapid, particularly for the lower eigenvalues which dominate the relaxation at long times. It appears that the convergence is rapid for cases where the cross section exhibits a moderate variation with energy. The energy range sampled in the calculation depends on the temperature. For example, in Table I, the convergence improves slightly going from Xe at 700 K to Xe at 290.1 K and to Ar at 290.1 K. These situations correspond to cross sections that vary increasingly less with energy. However, the convergence is very good in all cases. It is particularly rapid for H e and N e as moderators, although these results are not shown. These features are further illustrated in Table 11, where the eigenvalues for the different gases are listed normalized such that the first eigenvalue coincides with the hard-sphere value which (21) B. Shizgal, J . Chem. Phys., 70, 1984 (1979). (22) B. Shizgal, J . Compur. Phys., 41, 309 (1981). (23) R. K. Nesbet, Phys. Reu. A, 20, 58 (1979). (24) T. F. O'Malley and R. W. Crompton, J . Phys. B, 13, 3451 (1980).
Shizgal and McMahon
4858 The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
h’
TABLE II: Comparison with a Hard-Sphere Gas
n
hard sphere
He
Ne
Ar
Xe
Kr
1 2
4.683 10.11 16.43 23.57 31.47 40.05 49.28 59.11 69.51 80.45
4.683 10.13 16.56 23.91 32.12 41.12 50.86 61.30 72.40 84.14
4.683 9.688 16.02 23.61 32.42 42.38 53.47 65.64 78.87 93.12
4.683 7.511 11.04 15.27 20.18 25.75 31.93 38.74 46.13 54.36
4.683 6.275 8.013 9.793 11.58 13.64 16.05 18.76 21.74 24.97
4.683 5.908 7.048 8.437 10.17 12.16 14.34 16.72 19.28 22.01
3 4 5 6 7 8 9 10
I
1
- Xe _ _ _ Ne(xi0)
A‘,O
‘A‘,, = 4.683Xn/A,; Tb = 290.1 K. TABLE III: Coefficients in the Energy Relaxation‘ Cn
n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
ug
=2
0.2500 0.4034 0.4913 0.1420 -0.1484 -0.15 13 -0.0349 0.0238 0.0142 -0.0005 0.0045 0.0108 0.0053 -0.0041 -0.0066 -0.0028 0.001 1 0.00 18 0.0004 -0.0001 0.0001
ug = 3
0.1111 4.0221 -6.41 19 4.4244 -0.4746 -1.5351 1.0923 -0.1126 -0.1466 -0.0 128 0.003 1 0.1301 -0.106 1 -0.0371 0.0886 -0.0228 -0.0297 0.0171 0.0035 -0.0022 0.0001 -0.0018 0.0001 0.0010
ug
=4
0.0625 5.1352 -14.8220 26.3620 -34.7480 37.2480 -32.6060 22.5460 -1 0.0960 -0.8929 7.7451 -9.5366 7.3697 -3.4615 -0.0198 1.937 1 -2.2292 1.4200 -0.4606 0.0288 -0.0539 0.1855 -0.1520 -0.01 50 0.0835 -0.0217 0.0003 -0.0107 -0.0023 0.0052 -0.0001 0.0012 -0.0003
‘Argon, Tb = 290.1 K.
is also shown. A comparison of the results for He and N e show clearly that these moderators behave nearly as hard-sphere moderators. The other gases Ar, Xe, and Kr show reasonable deviations from the hard-sphere behavior. This is clearly due to the strong variation of the cross sections for these moderators and the Ramsauer-Townsend minima that occur. The ratio of each eigenvalue to the corresponding hard-sphere eigenvalue in Table I1 follows with increasing n the same qualitative trend as the cross sections with increasing energy. Notice the minimum in this ratio for argon for n = 5 . Thus each eigenvalue and eigenfunction is sensitive to the cross section on a different energy range, with AI sensitive to the cross section at the lowest energies. The behavior shown in Table I1 will vary with the temperature. Figure 1 compares the temperature dependence of the first three eigenvalues for Xe and Ne. The strong variation with temperature for Xe is indicative of the rapidly decreasing cross section in the low-energy region. In view of the minimum in this cross section, it is reasonable to expect that a corresponding minimum in the An will occur at sufficiently high temperature. By contrast, the results for Ne show only a small variation vs. temperature, consistent with the slowly varying cross section. It is important to
TABLE IV: Definition of Time Scales
moderator He Ne Kr Xe
Ar Ar Ar
T, K
uo A*
290.1 290.1 290.1 290.1 290.1 450.0 700.0
5.457 0.366 10.53 18.71 1.191 0.7 13 0.473
7, 10”
s/cm3
28.52 2143 309.0 27 3 .O 1303 1748 2113
keep in mind that the reciprocal of these eigenvalues are the relaxation times for these moderators. The relaxation time for Xe is then strongly increasing with temperature whereas for Ne it is almost independent of temperature. The behavior for the other moderators can be qualitatively understood by interpolating between these extreme cases. The convergence of the energy expansion, eq 12 depends on the choice of the initial distribution function. The choice of a 6 function for the initial condition is a stringent test of the method. Other choices of the initial distribution with a finite width should converge faster than the initial 6 function. The convergence of the expansion is shown for argon at 290.1 K in Table I11 for several different values of uo. It is evident that with increasing uo the convergence gets progressively slower. Quite generally, the convergence worsens for the more rapidly varying cross sections and higher values of uo. Clearly, the convergence will get intolerably slow if uo is chosen sufficiently large. This is inevitable as each eigenfunction is sensitive to a different energy range and to incorporate higher energies requires either a rescaling of the or an quadrature points for fixed N as employed increase of N . However, as will be shown, the interesting features of the relaxation are those that are influenced by the Ramsauer-Townsend minima. These features can be studied if the initial speed is placed just above the minima in the cross sections. The present method was successful in this objective. In all the calculations reported here, 50 quadruature points were retained. The computer costs involved were relatively minor. A summary of the relaxation of electrons in the inert gases as moderators is shown in Figures 5-7. The temperature ratio shown in these figures is given by m
T ( t ? / T , = C(c,/cd n=0
exp(-A,t?
(20)
A time-dependent eigenvalue (or pseudo-first-order rate coefficents) can be defined by A ( t ? / A , = -d In [T(t?/T,,- l ] / d t ’ (21)
and approaches unity as thermal equilibrium is attained. This parameter is a useful probe of the details of the momentum transfer cross section. It is also related to the energy exchange
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4859
Electron Thermalization
20
Ke
16 I=
> - 12 1
I-
8
4 I
0 5
1
t
1. s
I
I
I
0.3
2
0.6 t‘
0.9
1.2
I
I
Ne
16
w
0.5
1
f
1.5
2
i
Ill
0 1
t’
0 2
0.3
Figure 5. Temperature relaxation.
rate coefficient measured experimentally.’ It is important to note that the time scale T varies with the electron moderator mass ratio, the effective hard-sphere cross section, uo, and the temperature. Some representative values of T are listed in Table IV. The real time is t = 7 f ’ . The temperature relaxation for the four moderators shown in Figure 5 shows a distinct change in going from Ne to Xe. The curves for the higher values of uo tend to exhibit a somewhat linear portion for the moderators with deeper minima in their cross sections. There is a distinct change in the rate of relaxation during the course of the approach to equilibrium. This is clearly seen in the variation of A(t?/A, shown in Figure 6. While the behavior for N e is completely monotonic for all uo, the behavior for the other moderators (with Ramsauer-Townsend minima) shows strong extremum values. In the case of argon, uo = 2 ( E = 0.150 eV) is below the minimum in the cross section whereas uo = 4 ( E = 0.600 eV) and 4.8 ( E = 0.864 eV) represent initial speeds above the minimum. The intermediate value, uo = 3 ( E = 0.338 eV) occurs almost precisely at the minimum. The relaxation rate increases for uo = 2 and 3 since the distribution function samples the cross section below the minimum which increases with decreasing energy. For uo = 4 and 4.8, the relaxation rate decreases initially as the distribution function samples the cross section above the minimum, which decreases with decreasing energy. The relaxation rate attains a minimum value as the region of the minimum in the cross section is sampled and then increases with increasing time as the lower energy portion of the cross section is then sampled. The results for the other gases, Kr and Xe, can be understood in the same way. The effects are somewhat larger due to the deeper minima in the cross sections for these gases. It is useful to notice the similarity in the curves for argon with
uo = 2 and for Xe with uo = 4.8. Both situations correspond to the initial speed at or very close to the minimum in the cross section. Figure 7 shows the effect of varying the temperature in the case of argon. The changes with temperature are not dramatic and there are only a few changes in the form of the curves for A(t’)/A, (compare the curves for uo = 2 and 3). The relaxation times are longer at the higher temperatures due to the decreased average cross section over the energy range for the relaxation. It is important to note that the time scale varies with temperature (see Table IV), so that although the peak in A ( t ? / A , in Figure 7 appears sharper for uo at 700 K than at 450 K,the effect is in reality not as large. Figures 8 and 9 show the details of the electron distribution function during the approach to equilibrium for a representative case. The distributions shown here correspond to the relaxation shown in Figures 5 and 6 for argon. The two figures are drawn for two different initial speeds as indicated by the vertical line representing the initial 6 function distribution function. The dashed curves drawn at particular times are Maxwellian distributions evaluated with the electron temperature at the indicated time. The results in Figure 8 show clearly the broadening of the distribution function and the energy degradation of the electrons with time. The distribution functions are also clearly non-Maxwellian, even for long times, well into the relaxation. This is evident from the curves in Figure 8 for t’ = 0.5. For t’ = 0.7, the distribution function even appears to have a bimodal form and is non-Maxwellian. The results in Figure 9 for a lower initial speed are somewhat similar. In this case, there is a significant amount of upscattering initially (see t’ = 0.1). The bimodal form of the distribution
4860
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984
Shizgal and McMahon
2
12 25
x
1
I
I
3.
4
1
0.3
0.6
0.5
1
f
0 3
1.2
15
2
1
/
r
Sx I 0
0
t
Figure 6. Time dependence of the effective eigenvalue or energy exchange rate coefficient. TABLE V: Relaxation Times vs. Initial Speed" uoc He Ne Ar Xe 7( 1.Ol)* 2.0 29.64 1905 1430 152.7 3.0 31.49 1963 2277 346.6 4.0 32.34 1986 2505 636.9 4.8 32.73 1994 2547 741.2
2.0 3.0 4.0 4.8
15.83 17.70 18.54 18.94
903.7 963.6 985.4 994.0
T(1.1) 826.6 1626 1853.1 1894.6
88.73 238.0 496.5 598.9
Kr
TABLE VI: Temperature Dependence of Relaxation Times for Argon'
267.7 493.1 764.7 872.2 158.2 349.2 599.5 705.0
'Tb = 290.1 K. br(l.Ol) is the time required for the temperature to
relax to l.OITb; in units of 10" s/cmJ. cuO= 2, 3, 4, and 4.8 corresponds to electron energies of 0.0375, 0.1500, 0.3375, 0.6000, and 0.8640 eV, respectively. function occurs here as in Figure 8. Notice that the distribution function is approximated by a Maxwellian only for long times into the relaxation (t' = 0.9) when thermalization is essentially complete (see Figure 5 ) . A summary of the relaxation times and their variation with initial speed and temperature is given in Tables V-VII. The increase in the thermalization time with increasing initial speed is a rather large effect, somewhat larger than indicated by Mozumder's results. The effect is larger for a moderator such a Xe than for Ne. This is again a manifestation of the different energy dependence of the these two cross sections. The temperature dependence of the thermalization times shown in Table VI show conflicting trends depending on uo and which definition is used. The times ~(1.01)increase with increasing
2.0 3.0 4.0 4.8 'See
1561 2534 2766 2809
748.7 1672 1905 1947
1279 2639 2884 2925
390.7 1600 1844 1887
caption to Table V.
TABLE VII: Comparison of Relaxation Times"
moderator Xe Kr Ar Ne
He
~(1.01) ~(1.1) ~(1.01) T(1.1) r(1.01) r(1.1) r(1.01) ~(1.1) ~(1.01) ~(1.1)
this work
ref 2f
1913 1492 2297 1801 7526 5567 5967 2960 97.15 55.71
739 640 1400 1100 3990 3000 3080 1600 73.9 40
expt 1670d
1300,b 5860,' 4580' 670b 26b
= 4; Tb = 290.1 K; relaxation times are in units of 10" ~s torr. bReference7. cReference 26. dT. Takahashi, J. Ruon, S. Kubota, and F. Shiraishi, Phys. Rev. A, 25, 600 (1982). eT. Takahashi, J. Ruon, S. Kubota, and F. Shiraishi, Phys. Rev. A, 25, 2820 (1982). IMozumder. O u 0
temperature except the case uo = 2. The times T ( 1.1) decrease with increasing temperature for all values of uo studied. The increase and/or decrease of these thermalization times is a re-
The Journal of Physical Chemistry, Vol. 88, No. 21, 1984 4861
Electron Thermalization
P ' *1 9
+
Y
6
1
3 1
I
I
I
0.1
0.2
0.3
I
0,4
I
0 5
0.1
0 2
0.3
04
0.5
5
f
H
I
I
I
I
0,1
0.2
0.3
I
0.4
I
I
i
I
0.5
I
01
t
0 2
0 3
I
04
t
I
05
f
Figure 7. Temperature relaxation and time variation in the effective eigenvalue; temperature dependence. Tb equal to (a) 450 and (b) 700 K.
- 1.2
12
*%
h
5
7
.5.
z 0 3
^ -
0.8
+
06 0.4
03
1
3
2
4
V/Vih
Figure 8. Time evolution of the speed distribution during thermalization: argon, Tb = 290.1 K,uo = 4; (---) Maxwellian with T(t9.
flection of the c, coefficients and the temperature dependence of the eigenvalues in eq 12. The temperature dependence of the thermalization times T( 1.1) and T( 1.01) are influenced by the rate of convergence of the expansion eq 12 determined by uo. One must also keep in mind the fact that a single exponential decay is not completely adequate even far into the relaxation process as shown in Figures 6 and 7. The results shown in Table VI are a reflection of this combination of effects that is somewhat difficult to unravel.
Figure 9. Same as Figure 8; uo = 3.
The thermalization times calculated in the present paper are compared in Table VI1 with the values reporteg by Mozumder and the small number of experimental results that are available. Our calculated relaxation times are consistently larger than those reported by Mozumder by a factor close to 2. The smaller relaxation times determined by Mozumder are probably due to his use of a broad Maxwellian distribution. The thermalization times calculated in the present work are obtained with the actual nonequilibrium distribution function which is considerably narrower (seb Figures 8 and 9). The experimental relaxation times
J . Phys. Chem. 1984,88, 4862-4867
4862
by Warman and Sauer7 are probably too short due to their assumption of a constant A(t ?/Al throughout the energy relaxation whereas Figures 6 and 7 clearly indicate a time dependence sensitive to the initial condition. In addition, assumptions were made with regard.to the form of the energy distribution in order to estimate the mean electron energy at each time. A more recent observation for argonz6measures the transient conductivity and gives a more direct estimate of the relaxation time. This new result is in agreement with the present calculated value as shown in Table VII. Calculations of the transient condutivity and diffusion coefficient show a similar behavior as the energy r e l a x a t i ~ n . ~ ~ (25) G . N. Haddad and T. F. O’Malley, Aust. J . Phys., 35, 35 (1982). (26) U. Sowada and J. M. Warman. J . Electrostatics. 12. 37 (1982). . , (27) D. R. A. McMahon and B. Shizgal, Phys. Reu. A,’ in press.
IV. Summary The present paper has considered gn efficient discrete ordinate method of solution of the Boltzmaqn equation and the determination of electron thermalization times in inert gas moderators. The method is superior to traditional moment methods and is being employed in the calculation of time-dependent diffusion and conductivity coefficients for electrons during t h e r m a l i ~ a t i o n .It~ ~ is anticipated that the present method of solution of the Boltzmann equation for both-time dependent and steady-state electron velocity distribution functions will find important applications to the analysis of swarm experiments and to atmospheric applications.
Acknowledgment. This research is supported by a grant from the Natura1 Sciences and Engineering Research Council of Canada.
Aspects of Nuclear Dynamics In Short-Lived Negative Ion States W. Domcke,* M. Berman, H. Estrada, C. Mundel, and L. S. Cederbaum Theoretische Chemie, Physikalisch- Chemisches Institut, Universitat Heidelberg, Im Neuenheimer Feld 253, 0-6900 Heidelberg, West Germany (Received: January 23, 1984)
The nonlocal potential theory of nuclear motion in resonance states and weakly bound states of negative ions is reviewed. The ab initio calculation of the electronic width and level-shift functions, which determine the nonlocal part of the potential for the nuclear motion, is briefly discussed. Methods to solve the nuclear dynamical problem taking full account of the nonlocality of the potential are described. Illustrative calculations are presented for vibrational excitation of Nz via the 2.3-eV shape resonance, as well as for a realistic model of dissociative attachment. It is shown that the widely used local complex potential approximation (boomerang model) is of excellent accuracy for vibrational excitation of N2. In other situations the local approximation is less reliable; this is demonstrated for shape resonances with repulsive potential energy curves such as in the halogen anions.
Introduction Inelastic and reactive collision processes between low-energy electrons and molecules as well as low-energy ion-atom collisions leading to electron detachment have been of experimental and theoretical interest for many decades.’ Examples are the vibrational excitation of molecules by electron impact e AB(v=O) AB(u>O) e (1)
The accurate calculation of cross sections for the inelastic and reactive collision processes 1-3 from first principles is still a largely unsolved theoretical and computational problem. In this article we sketch the present status of the theory as far as it is based on the Feshbach projection operator approach and the treatment of nuclear dynamics in the nonlocal complex potential. The emphasis is on those topics where progress has been made recently.
dissociative attachment
General Theory of Nuclear Dynamics in Resonance States The projection operator formalism of FeshbachZ provides a general and t r h p a r e n t description of resonances in terms of discrete closed-channel states interacting with the open-channel continua. As a result of the interaction with the continua, the discrete states acquire a width and are shifted in energy. The formalism is tailored to the description of so-called core-excited or Feshbach resonance^,^ e.g., doubly excited electronic states embedded in single-excitation continua. For electronic resonances of the single-particle or shape type,3 on the other hand, the choice of the appropriate discrete state may be sometimes a m b i g u o ~ s . ~ , ~ In addition, one has to enforce explicitly the orthogonality of the continuum to the chosen discrete ~ t a t e . ~ f ’ . ~ Important aspects in electron-atom and electron-molecule scattering are the ubiquituous long-range potentials which cause
-
+
e
+
+ AB - + A + B-
(2)
and the reverse process, associative detachment A
+ B--+
AB
+e
(3) These processes constitute a major part of the physics and chemistry of outer space, the higher atmosphere, and in plasmas and discharges. They violate the Born-Oppenheimer principle in the sense that electronic energy is converted into kinetic energy of nuclear motion or vice versa. As is well known, resonances, Le., temporarily formed electronic states of the negative ion, play an essential role in this interconversion of electronic and nuclear kinetic energy.l The dissociative attachment process 2, for example, proceeds via an electronic resonance state of AB- with a repulsive potential energy curve, which crosses the potential energy curve of the target molecule AB and dissociates into bound electronic states of the fragments A pnd B-. In processes of this type we have to deal, therefore, with nuclear dynamics in both bound electronic states as well as short-lived resonance states of the negative ion. (1) H. Massey, “Negative Ions”, 3rd ed, Cambridge University Press, Cambridge, 1976; P. G . Burke, Adu. A t . Mol. Phys., 15, 471 (1979).
0022-3654/84/2088-4862$01.50/0
(2) H. Feshbach, Ann. Phys. (N.Y.),19, 287 (1962). (3) H. S. Taylor, Adu. Chem. Phys., 18, 91 (1970). (4) A. U. Hazi in “Electron-Molecule and Photon-Molecule Collisions”, T. N. Rescigno, V. McKoy, and B. I. Schneider, Ed., Plenum Press, New York, 1979, p 281. (5) R. K. Nesbet, Comments A t . Mol. Phys., 11, 25 (1981). (6) R. K. Nesbet, “Variational Methods in Electron-Atom Scattering Theory”, Plenum Press, New York, 1980, section 3.2. (7) W. Domcke, Phys. Reu. A , 28, 2777 (1983).
0 1984 American Chemical Society
