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where the cross section
for the
transition is given by
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q(L,M,M'|E). |
Here, the sum is finite, limited by triangle inequalities on J, J' and L,
and each of M and M' can independently assume the values ranging from
-L to +L.
The interaction potential is expressed in terms of spherical harmonics, and
therefore, the scattering cross section is expressed here in terms of a
parameter
q(L,M,M'|E) (Hutson & Green 1995).
The spectroscopic coefficients,
,
depend on the
wave-functions of the molecule and on angular momentum coupling factors:
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These spectroscopic coefficients are obviously independent of collision dynamics. Thus, the rate coefficient is given by
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Q(L,M,M'|T) |
where
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Now, owing to symmetries of
,
S(L, M, M') = S(L, M', M),
so that only the real part of Q(L, M, M') is required, and the
cross sections are real. - Green et al. (1987) calculated the
Q(L,M,M'|T)
for C3H2. These calculations were based on a potential energy surface
describing the interaction of C3H2 with He and used the infinite order
sudden approximation (IOSA) according to Green (1979). They derived the
Q(L,M,M'|T)
for
and T=30, 60, 90, and 120 K. Despite the larger
parameter space for which the
Q(L,M,M'|T) were calculated, Green et al.
give the state-to-state rate coefficients only for T=60 K and for transitions between the lowest
20 energy levels of ortho-C3H2 and the lowest 22 of para-C3H2. - Similar
calculations were performed by Palma & Green (1987) for SiC2. They computed the
Q(L,M,M'|T) for
and T=25, 50, 75, 100, and 125 K.
State-to-state coefficients were given only for T=100 K and for transitions between
the lowest 18 energy levels.
As mentioned in the introduction, in astrophysical applications, i.e. for actual NLTE calculations, one needs to know the state-to-state rate coefficients for a large number of transitions in a sufficiently large range of temperatures. We therefore extended the work of Green et al. and Palma and Green, respectively, by using the Q(L,M,M'|T) computed by them to calculate state-to-state rate coefficients for an extended set of transitions and for all temperatures for which the Q(L,M,M'|T) were given.
Transition | Rate | l | u |
1(0, 1)==> 1(1, 0) | 7.800D-12 | 1 | 2 |
1(0, 1)==> 2(1, 2) | 3.041D-11 | 1 | 3 |
1(0, 1)==> 2(2, 1) | 1.666D-11 | 1 | 4 |
1(0, 1)==> 3(0, 3) | 3.489D-12 | 1 | 5 |
1(1, 0)==> 2(1, 2) | 2.409D-12 | 2 | 3 |
1(1, 0)==> 2(2, 1) | 1.143D-11 | 2 | 4 |
1(1, 0)==> 3(0, 3) | 4.431D-11 | 2 | 5 |
2(1, 2)==> 2(2, 1) | 1.217D-11 | 3 | 4 |
2(1, 2)==> 3(0, 3) | 9.661D-12 | 3 | 5 |
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