Cross sections Integral

This differential cross section may be integrated over scattering angles to define an integral cross section a as follows ... 

Equation (A3.13.17) is a simple, usefiil fomuila relating the integrated cross section and the electric dipole transition moment as dimensionless quantities, in the electric dipole approximation [10, 100] ... 

The integral cross section for scattering over all directions is... 

The integral cross section is therefore the effective area presented by each field particle B for scattering of the test particles A into all directions. The probability that the test particles are scattered into a given direction v ( /, ([)) is the ratio... 

Their individual contributions to the integral cross sections are... 

The combined S-, P-wave (l = 0, 1) contributions to the differential and integral cross sections are... 

The optical theorem relates the integral cross section to the unaginary part of the forward scattering amplitude by... 

These combinations also hold for the integral cross sections. 

The integral cross section for the transition =. f= a ySyin the target atom, where a denotes the... 

Flere qiiantum-mechanical vibrational state-to-state differential cross sections were calculated for a translational energy of = 20 eV and compared with experiments, with very good agreement between experiment and theory. In another application of this approach, state-selected integral cross sections were... 

Zhang J Z H and Miller W H 1989 Quantum reactive scattering via the S-matrix version of the Kohn variational principle—differential and integral cross sections for D + Hj —> HD + H J. Chem. Phys. 91 1528... 

In addition to the DCS, we also need to consider the state-to-state integral cross-section (ICS), which is a measure of the total amount of scattered AC product in quantum state n, and is given by... 

One would expect that effects of similar magnitude to those shown in Fig. 8 should also appear in the corresponding state-to-state differential and integral cross-sections. However, this is not the case. As already mentioned, there is a considerable amount of cancellation of GP effects in these quantities, which we refer to as the cancellation puzzle. The unexpected cancellations appear in the state-to-state DCS at low impact parameters (i.e., low values of J), and in the state-to-state ICS (including all impact parameters). We now discuss each of these cancellations in turn. 

If the isotropic coefficient is specified to be unity, a is just the total (integrated) cross-section. In Appendix A, an alternative quantum mechanical expression for this cross-section is obtained in the electric dipole approximation. By comparing the two expressions, it can be seen that the Legendre polynomial coefficients in Eq. (11) may be obtained from the inner summation terms in Eq. (A.15). Hence, the Legendre polynomial coefficients are... 

First to be considered is the isotropic Pq coefficient. The parameter is proportional to the integrated cross-section, a [Eq. (11)]. In fact, the preceding arguments show that when j = 0, I m = Im, and so there are no interference cross terms in this case. Consequently, as is already widely recognized, the integrated cross-section displays no dependence on the relative phase of the final continuum channels. 

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