Impact theory of rotational spectra

In quantum theory as in classical theory the isotropic Raman spectrum is expressed in terms of the average value of the polarizibility tensor a(0) = (1/3) Sp a randomly changing in time due to collisions  

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows 

Owing to space isotropy and isotropy of collisions in free space, only 

Here hk is a translational moment of a pair of reduced mass p. Averaging over kinetic energy Ek = (hk)2/2p is included in the operation 

The elements of S-matrices are determined in the basis of orbital angular momentum l and rotational moments jt,jf of vibrational states i,f and their projections (m,m,-,m/). Both S-matrices in Eq. (4.58) have to be calculated for the same energy Ek of colliding particles. 

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