Quantum number and energy scaling of rotationally inelastic scattering cross sections

A general formula for rotationally inelastic cross sections in atom-diatomic collisions is derived. This result is achieved by assuming the transition probability is a function of rotational quantum number differences, the kinetic energy in the upper states, and is inversely proportional to the number of accessible states within an effective hamiltonian formalism. The scaling law is able to predict all rows, or columns, of the inelastic cross section matrix, σ_jj, given any one row, or column, as a function of energy. Finally, we applied this scaling theory to a variety of collision systems; in general, good agreememt between predicted and exact results is exhibited. We have developed and tested a scaling law for rotationally inelastic transitions which predicts all rows or columns of the inelastic cross section matrix, σ_jj, given any one row or column. The agreement between exact and predicted cross sections was good for the HeHF [20], HeHCl [21], H₂CS [17], HeCO [16] and Li⁺H₂ [18] systems which vary in reduced mass from approximately 2-4 an and rotor constant from 0.82-60.0 cm^-1. However, for ArN₂ [13,14, 19,24] collisions (μ = 16.5 au, B_e = 2.01 cm^-1) the error increased with increasing j and/or Δ. In every case the disagreement was in the opposite direction from the prediction of eqs. (15a)-(15c) which are based upon fully degenerate statistics instead of the effective hamiltonian non-degenerate statistics of eqs. (2a) and (2b). Finally, we state two general conclusions: 1. (1) Whenever the initizil kinetic energy (E-ε{lunate}_j) is greater than approximately five times the energy gap (ε{lunate}_j+Δ-ε{lunate}_j), the scaling theoretic cross sections are accurate within 20%; and 2. (2) at total energies much larger than that of the internal states of interest, all cross sections with the same change in rotational quantum number become approxinutely equal.