Cross sections angular momentum

This completes our introduction to the subject of rotational and vibrational motions of molecules (which applies equally well to ions and radicals). The information contained in this Section is used again in Section 5 where photon-induced transitions between pairs of molecular electronic, vibrational, and rotational eigenstates are examined. More advanced treatments of the subject matter of this Section can be found in the text by Wilson, Decius, and Cross, as well as in Zare s text on angular momentum. 

Here p is the radius of the effective cross-section, (v) is the average velocity of colliding particles, and p is their reduced mass. When rotational relaxation of heavy molecules in a solution of light particles is considered, the above criterion is well satisfied. In the opposite case the situation is quite different. Even if the relaxation is induced by collisions of similar particles (as in a one-component system), the fraction of molecules which remain adiabatically isolated from the heat reservoir is fairly large. For such molecules energy relaxation is much slower than that of angular momentum, i.e. xe/xj > 1. 

Table 1.1. Cross-sections of rotational energy and angular momentum...

Fig. 1.23. Density-dependence of angular momentum relaxation rate. Points correspond to experimental data presented in Fig. 1.17. The straight solid line is a binary estimation of this rate with the cross-section Oj = 3 x 10-15 cm2 and the broken curve presents the result obtained in the rough-sphere approximation used in [72, 80].

Storozhev A. V., Strekalov M. L. Relaxation cross sections for transfer of rotational angular momentum in a semiclassical approximation, Chem. Phys. 153, 99-113 (1991). 

Green S., Monchik L., Goldflam R., Kouri D. J. Computational tests of angular momentum decoupling approximations for pressure broadening cross sections, J. Chem. Phys. 66, 1409-12 (1977). 

There does not seem to be any selection rule such as conservation of spin or orbital angular momentum which this reaction does not satisfy. It is also not clear that overall spin conservation, for example, is necessary in efficient reactions (5, 16, 17, 20). Further, recent results (21) seem to show a greatly enhanced (20 times) reaction rate when the N2 is in an excited vibrational state (vibrational temperature 4000 °K. or about 0.3 e.v.). This suggests the presence of an activation energy or barrier. A barrier of 0.3 e.v. is consistent with the low energy variation of the measured cross-section in Figure 1. 

The primary reason it is difficult to treat angular momentum rigorously is due to the angular momentum catastrophe [58]. As noted in Section III, cross sections and other experimental observables are sums over all relevant total angular momentum quantum numbers, J. Each J represents a quantum dynamics problem to be solved, and the size of the problem increases dramatically with J. For each J, there are Nk projections of K, where Nts = fmax — min + 1- For a fliree-atom system, the minimum value of K, is a function of both J and p, such that = 0 when J and p are... 

An example of this kind, in which the energy and angular momentum of the two critical points coincide, occurs for the hydrogen atom in crossed electric and magnetic helds (see Section IVC). The pinched torus then has two pinch points. 

Fig. 4. Computed partial cross-sections in A2 for the F + HD (v = 0, j = 0) —> HF + D reaction as a function of the total angular momentum quantum number, J, up to collision energies of 3 kcal/mol.

Fig. 10. The partial cross-section summed over all final states in A2 for F + H2(0,0) —> H + HF for several values of the total angular momentum J. The partial cross-section shows a double peak structure which J-shift to higher energy with J, and eventually merge at about J = 10.

For a diatom-diatom reaction, the reaction cross-section from a specific initial state (vi, V2, ji, ji) is obtained by summing the reaction probabilities over all the partial waves with total angular momentum J46 ... 

Darwin, 1929 Mott, 1930). The incident particle has momentum HKg before any interaction its momentum after exciting atoms 1 and 2 respectively into the nth and mth states is represented by hKnm. Mott showed that the entire process has negligible cross section unless the angular divergences are comparable to or less than (K a)-1, where a denotes the atomic size. As Darwin (1929) correctly conjectured, the wavefunction of the system before any interaction is the uncoupled product of the wavefunctions of the atom and of the incident particle. After the first interaction, these wavefunctions get inextricably mixed and each subsequent interaction makes it worse. Also, according to the Ehrenfest principle, the wavefunction of the incident particle is localized to atomic dimensions after the first interaction therefore, the subsequent process is adequately described in the particle picture. 

Fig. 2.5. Dependence of total reaction cross-section on the angular momentum quantum number.

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