We report calculations performed by using semiclassical and classical approximations on electron-molecule inelastic collisions. Complete sets of excitation and dissociation cross sections involving the whole vibrational ladder of ground state and electronically excited molecules have been obtained.
In some cases cross sections for different isotopes have been obtained by using the impact parameter method and appropriate scaling laws.
48
A list of cross sections is reported in table 1. Tables of these cross sections have been reported in ref. [5].
Here we want to discuss the dissociation of vibrationally excited H2 through different channels:
(i) repulsive b 35+u
(ii) repulsive portion of singlet states (iii) bound triplet states
(iv) dissociative ionization
Table 1. List of H2 processes and channels considered in the paper
PROCESS CHANNEL
DISSOCIATION H2(X 1
g,)eH2(b 3u)e2He
DIRECT DISSOCIATION THROUGH EXCITED STATES
H2(X 1g,)eH2(B 1u)e2He H2(X 1g,)eH2(C 1u)e2He H2(X 1g,)eH2(B' 1u)e2He H2(X 1g,)eH2(D 1u)e2He H2(X 1g,)eH2(B' ' 1u)e2He
H2(X 1g,)eH2(D' 1u)e2He
BOUND-BOUND EXCITATION FOLLOWED BY DISSOCIATION
H2(X 1g,)eH2(a 3g)e2He H2(X 1g,)eH2(c 3u)e2He
DISSOCIATIVE IONIZATION H2(X
1g,)eH2(2g)2eHH2e H2(X 1g,)eH2(2u)2eHH2e
Figure 1 reports the dissociation cross section of the process:
H2(X1g,)eH2(b3u)e2He (1)
as a function of energy for different vibrational levels. These cross sections have been obtained by using the classical Gryzinski approximation [5]. The major effect of vibrational excitation is to decrease the threshold energy of the process having however a minor effect on the magnitude of the maximum of cross section. Table 2 reports a comparison of our cross sections and those obtained by Rescigno and Schneider [10] by using the complex Kohn method for electron energy larger than 12 eV. The agreement can be considered satisfactory.
On the other hand the recent ab-initio calculations of Stibbe and Tennyson [11], while confirming the general trend of cross sections, show a shift of their maximum toward lower energies (see figure 2). This shift is reflected on the rate coefficients calculated according to a Maxwell distribution function of electron energy. Inspection of figure 3 shows in fact that rate coefficients which use Stibbe and Tennyson cross sections are, at low temperature, up to an order of magnitude higher than those obtained by using Gryzinski cross sections.
0.00
0.0 20.0 40.0 60.0 80.0 100.0
Crosssection(Å2 )
Figure 1 Dissociative cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,13)eH2(b3u)e2He
Table 2. Cross section (a20 ) for the process H2(X1g,)eH2(b3u)e2He. Comparison with results of rescigno et al.[10]
energy = 12eV energy = 15 eV energy = 18 eV energy = 20 eV energy = 25 eV energy = 30 eV
Figure 4 reports the sum of dissociation cross sections on the repulsive part of several singlet states of H2 as a function of energy for different vibrational levels [5, 12–15]. The intricate behaviour of cross sections for different n is the result of Franck Condon densities linking the different vibrational levels of ground state to the electronic states as well as the behaviour of integrated dipole moment as a function of internuclear distance. These cross sections have been calculated by using the semiclassical impact parameter approximation with CI electronic wavefunctions of both target and final states. Inspection of the figure shows that the increase of vibrational quantum number not only decreases the threshold energy of the process but also
50
section increases by a factor 10 much higher than the corresponding increase of cross sections for the excitation of b 3Su+ state. The decrease of cross section for n>10 is due, as already pointed out, to the form of Franck Condon densities or better to the integrated dipole moment.
0.0 0.5 1.0 1.5 2.0 2.5
2 4 6 8 10 12 14
Energy (eV) Crosssection(a 02 )
1 = 0 2 3 4
Figure 2 Dissociative cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,4 ) eH2(b3u) e2H e. Full line: Celiberto et al. [5];dotted line: Stibbe and Tennyson [11].
10-20 10-18 10-16 10-14 10-12 10-10
0 2 4 6 8 10 12 14
= 0
4 1
2 3
3 sec-1 )
Temperature (10 3K)
Figure 3 Dissociation rates, as a function of temperature, for different vibrational levels, for the process:
H2(X1g, 0 ,4 )eH2(b3u)e2H e, obtained using cross sections of Celiberto et al. (full line)[5]and Stibbe and Tennyson (dashed line) [11].
0.00
Figure 4 Dissociative cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,14)eH2( B 1u,B' 1u,B" 1u,C 1u,D 1u,D' 1u)e 2He
The dissociation through the singlet states becomes important at high energies when the corresponding cross section involving the b 3S+u looses its importance. An important role is played by the high lying vibrational levels. These cross sections are in any case important for creating excited atomic hydrogen atoms with principal quantum number larger than 2.
Figures 5 and 6 report the cross sections for the excitation of the triplet states of H2
H2(X1g,)eH2(a3g)e2He (2)
H2(X1g,)eH2(c3u)e2He (3)
calculated several years ago by using the Gryzinski approximation [16]. In the same figure we have reported the corresponding cross sections calculated by Buckman and Phelps [17] by a refinement of cross sections through a Boltzmann analysis of transport coefficients.
In general these cross sections are considered dissociative [18] even thought the two excited states are bound. However the H2 (a3S+g ) state is radiatively linked to the repulsive b3S+u by an allowed electric dipole transition. The behaviour of H2 (c3Pu) is less clear. The lowest vibrational level n=0 lies below to the a3S+g state and is metastable. It can predissociate to the b3S+u state by magnetic dipole and it can decay to the b3S+u by magnetic dipole and quadrupole radiation at a rate of about 10 s-1. The higher vibrational levels can radiate to the a3S+g then cascading on b3S+u . The history of the c3Pu depends on the gas density and in dense gases it can be deactivated before radiating. The concentrations of this state is in any case important in determining the electron energy distribution function in molecular hydrogen plasmas [19].
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10.0 15.0 20.0 25.0 30.0 35.0
Crosssection(Å2 )
Figure 5 Dissociative cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,14)eH2(a3g)e2He. In the figure, comparison for n=0 with results of Buckman and Phelps (dotted line) [17].
0.00
10.0 15.0 20.0 25.0 30.0 35.0
Crosssection(Å2 )
Figure 6 Dissociative cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,10 )eH2(c3u)e2H e. In the figure, comparison for n=0 with results of Buckman and Phelps (dotted line) [17].
We observe that the increase of vibrational quantum level has an effect similar to that one discussed for the repulsive b3S+u state.
Figure 7 reports the dissociation rates of H2 calculated by inserting the above reported contributions; rates were fitted, in the temperature range [1¥103,200¥103 K], according to the following expression:
ln(rate)c1(T')c2c3(T')c4 c5exp(c6ln(T')2); T'= T
103, (4)
corresponding fitting coefficients are reported in table 3.
We see that the effect of vibrational quantum number in enhancing the rate is dramatic al low temperature becoming less important at high temperature.
10-41
Figure 7 Dissociative rates, as a function of temperature, for different vibrational levels, for the process:
H2(X1g, 0 ,10 )eH2* e2He. The temperature region [4¥104-2¥105 K] is enlarged in the figure, in a linear scale.
Table 3. Fitting coefficients for H2 dissociation rates
n c1 c2 c3 c4 c5 c6
0 -11.565 -0.076012 -78.433 0.74960 -2.2126 0.22006 1 -12.035 -0.066082 -67.806 0.72403 -1.5419 1.5195 2 -13.566 -0.043737 -55.933 0.72286 -2.3103 1.5844 3 -46.664 0.74122 -15.297 -0.022384 -1.3674 1.3621 4 -37.463 0.81763 -0.40374 -0.45851 -18.093 0.011460 5 -28.283 0.99053 -10.377 -0.085590 -11.053 0.067271 6 -23.724 1.0112 -2.9905 -0.24791 -17.931 0.034376 7 -19.547 1.0224 -1.7489 -0.31413 -19.408 0.028643 8 -15.936 1.0213 -1.0175 -0.38040 -20.240 0.024170 9 -12.712 1.0212 -0.60400 -0.44572 -20.766 0.021159 10 -0.40557 -0.49721 -9.9025 1.0212 -21.031 0.019383
Figure 8 reports a comparison of the present rates, for n=0, with those recently reported by Martin et al. [20]. Discrepancies are found at low temperatures [1¥103-8¥103 K], due to differences in the threshold region of the cross section sets employed, however, at higher temperatures, the agreement is satisfactory, with errors of ~10%.
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10-41 10-37 10-33 10-29 10-25 10-21 10-17 10-13
1.0 10.0
Rates(cm3 s-1 )
T /1000 (K)
Figure 8 Comparison of total dissociative rates of H2 for n=0, as a function of temperature, (dotted line) results of Martin et al. [20]; (full line) present work.
(high temperature region is enlarged, in linear scale).
Let us now introduce other channels contributing to the dissociation of molecular H2 merely dissociative ionization and dissociative attachment.
Figure 9a reports the behaviour of the sum of cross sections relative to the processes
H2(X1g,)eH2(2g,2u)eHH e (5) as a function of energy for different initial vibrational quantum number [5].
Also in this case the vibrational quantum number decreases the threshold energy of the process increasing at the same time the maximum of cross section. The reported cross sections are too high compared with the experimental results [21] so that a normalization factor (0.1732) should be applied to the values reported in [5](see figure 9b).
Maxwell rate coefficients are reported, as a function of temperature, in figure 10 showing the possible importance of this process specially at high temperature. Dissociative ionization rates were fitted, between 3¥103 and 200¥103 K, with the analytical expression:
ln(rate)c1(T')c2c3c4(exp(c5T'))c4; T'= T
103 (6)
and fitting coefficients are reported in table 4.
0.00
Figure 9 Dissociative ionization cross sections, as a function of impact energy, for the process:
H2(X1g, 0 ,13)eH2(2g,2u)eH He.(a) from ref.[5];
(b) normalized to experimental cross sections of Rapp et al.(dotted line) [21].
10-42
Figure 10 Dissociative ionization rates, as a function of temperature, for different
vibrational levels, for the process:
H2(X1g, 0 ,13)eH2(2g,2u)eH He. The region of 4¥104 -2¥105 K is enlarged in the figure, in a linear scale.
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Table 4. Fitting coefficients for H2 dissociative ionization rates
n c1 c2 c3 c4 c5
0 -211.96 1.0022 -20.350 -4.5201 -0.0023834 1 -205.18 0.99226 -19.905 -3.3364 -0.0035142 2 -199.36 0.98837 -19.600 -3.0891 -0.0041558 3 -193.98 0.98421 -19.457 -3.1386 -0.0043830 4 -188.93 0.97647 -19.397 -3.2807 -0.0045212 5 -184.22 0.96189 -19.310 -3.2609 -0.0049158 6 -179.03 0.94593 -19.170 -3.0592 -0.0056404 7 -173.64 0.93986 -19.052 -2.9880 -0.0061932 8 -169.60 0.93507 -18.908 -2.7334 -0.0068815 9 -166.64 0.92602 -18.723 -2.2024 -0.0081979 10 -165.21 0.92124 -18.549 -1.6895 -0.0096154 11 -165.69 0.93366 -18.479 -1.6311 -0.0093150 12 -164.64 0.94682 -18.440 -1.7259 -0.0088671 13 -160.71 0.95533 -18.405 -1.8938 -0.0085829