where ©r = hcBr/k is a characteristic temperature of rotation and depends on the rotational constant Br. Thus, we see that since hc/k = 1.439 cm K and Br is typically about 2 cm"1, then ©r is about 3 K, and hence for atmospheric temperatures ©r/T ^ 1 and hence the sum in the above equation can be replaced by an integral to give
while for polyatomic molecules of atmospheric interest, the rotational partition function varies as T3/2 (Herzberg 1945). The partition function for vibration of a diatomic molecule, neglecting anharmonicity effects, to a very good approximation for the lower levels (equally spaced vibrational levels), is
where ©v = hcuo/k is a characteristic temperature of vibration, typically about 3000 K. Since ©r ^ ©v we can write, ignoring the zero-point energy and the ©r value when v=0, ni = gjexp(-^©v/T) ^
We can sum over all the rotational lines, m, in the band to obtain the temperature dependence of the band strength. The value of the band strength k at some temperature T, compared to its value at some reference temperature Ts, is then given by k(T ) = K(Ts)RrRvRbRst, (4.89)
where the ratios R are associated with the rotational and vibrational partition functions, the Boltzmann distribution, and the correction for stimulated emission, are given by
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