Cross Sections and Matrix Elements

The scattering coefficients (4.53) can be simplified somewhat by introducing the Riccati-Bessel functions  

If we take the permeability of the particle and the surrounding medium to be the same, then 

Note that an and bn vanish as m approaches unity this is as it should be when the particle disappears, so does the scattered field. 

As far as notation for the scattering coefficients is concerned, we have followed as much as possible van de Hulst (1957) and Kerker (1969), with the exception of the opposite sign convention for the time-harmonic factor exp( — iwt). Kerker (1969, p. 60) gives a table comparing the notation of various authors who have written on the theory of scattering by a sphere. 

Although we considered only scattering of x-polarized light in the preceding section, the scattered field for arbitrary linearly polarized incident light, and 

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We use a basis set ij, A I, 5, G, N, F, M) where // is taken to represent different vibrational levels of the ground electronic state Jefferts measurements involved the v = 4 to 8 levels, these being the ones with the optimum populations and photodissociation cross-sections. The matrix elements of each term in (11.79) are now readily calculated. The Fermi contact interaction is found to be diagonal in the chosen basis ... 

Let us turn now to the corresponding inelastic cross-section. The matrix elements (/, M L + 2S J, M) vanish except for / —/ = 1. Hence, near the forward direction we observe only the dipole-allowed transitions, i.e., the /—> / 1 transitions out of the Hund s rule ground state. Beyond the limit of small K, higher-order transitions contribute to the cross-section, and these are the main subject of the subsequent theory valid for arbitrary values of k. The small-K result we present for inelastic events, / = / 1, is of limited practical value since the minimum value of c is usually quite large owing to the kinematic constraints on the scattering process. Even so, the result is a useful guide to the size of the cross-section, and a welcome check on a complete calculation. 

SiHcon carbide s relatively low neutron cross section and good resistance to radiation damage make it useful in some of its new forms in nuclear reactors (qv). SiHcon carbide temperature-sensing devices and stmctural shapes fabricated from the new dense types are expected to have increased stabiHty. SiHcon carbide coatings (qv) may be appHed to nuclear fuel elements, especially those of pebble-bed reactors, or siHcon carbide may be incorporated as a matrix in these elements (153,154). 

Thus, if we have in hand the scattering coefficients an and bn, we can determine all the measurable quantities associated with scattering and absorption, such as cross sections and scattering matrix elements. 

A computer program for calculating the scattering coefficients (8.38) and the corresponding cross sections and scattering matrix elements is described in Appendix C all the examples in this section were obtained with this program. 

In Chapter 3 we considered briefly the photoexcitation of Rydberg atoms, paying particular attention to the continuity of cross sections at the ionization limit. In this chapter we consider optical excitation in more detail. While the general behavior is similar in H and the alkali atoms, there are striking differences in the optical absorption cross sections and in the radiative decay rates. These differences can be traced to the variation in the radial matrix elements produced by nonzero quantum defects. The radiative properties of H are well known, and the radiative properties of alkali atoms can be calculated using quantum defect theory. 

A new R-matrix approach for calculating cross-sections and rate coefficients for electron-impact excitation of complex atoms and ions is reviewed in [307]. It is found that accurate electron scattering calculations involving complex targets, such as the astrophysically important low ionization stages of iron-peak elements, are possible within this method. 

To understand an electron—atom collision means to be able to calculate correctly the T-matrix elements for excitations from a completely-specified entrance channel to a completely-specified exit channel. Quantities that can be observed experimentally depend on bilinear combinations of T-matrix elements. For example the differential cross section (6.55) is given by the absolute squares of T-matrix elements summed and averaged over magnetic quantum numbers that are not observed in the final and initial states respectively. This chapter is concerned with differential and total cross sections and with quantities related to selected magnetic substates of the atom. 

Abstract. The Chebyshev operator is a diserete eosine-type propagator that bears many formal similarities with the time propagator. It has some unique and desirable numerical properties that distinguish it as an optimal propagator for a wide variety of quantum mechanical studies of molecular systems. In this contribution, we discuss some recent applications of the Chebyshev propagator to scattering problems, including the calculation of resonances, cumulative reaction probabilities, S-matrix elements, cross-sections, and reaction rates. 

Resonance phenomena have been shown to play a significant role in many electron collision and photoionization problems. The long lived character of these quasi-stationary states enables them to influence other dynamic processes such as vibrational excitation, dissociative attachment and dissociative recombination. We have shown it is possible to develop ab initio techniques to calculate the resonant wavefunctions, cross sections and dipole matrix elements required to characterize these processes. Our approach, which is firmly rooted in the R-matrix concept, reduces the scattering problem to a matrix problem. By suitable inversion or diagonalization we extract the required resonance parameters. 

The general conclusion from these comparisons for Hp is that the different DW theories give qualitatively similar results for relative quantities such as state-to-state differential cross sections and rotational product distributions. Howover. the absolute values of cross sections are different in the various DW theories, the order being SSDW < VADW < RADW < CADW s exact. This order Illustrates that as systematically better approximations (based on physical understanding) are made to the exact wavefunction the T matrix element (16). the magnitudes of the... 

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