The scattering matrix formalism

Creagh, D. C.

doi:10.1107/97809553602060000592

International
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Volume C
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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 246-248

Section 4.2.6.2.3.2. The scattering matrix formalism

D. C. Creagh^b

4.2.6.2.3.2. The scattering matrix formalism

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Kissel, Pratt & Roy (1980) have developed a computer program based on the second-order S-matrix formalism suggested by Brown, Peierls & Woodward (1955). Their aim was to provide a prescription for the accurate (∼1%) prediction of the total-atom Rayleigh scattering amplitudes.

Their model treats the elastic scattering as the sum of bound electron, nuclear, and Delbrück scattering cross sections, and treats the Rayleigh scattering by considering second-order, single-electron transitions from electrons bound in a relativistic, self-consistent, central potential. This potential was a Dirac–Hartree–Fock–Slater potential, and exchange was included by use of the Kohn & Sham (1965) exchange model. They omitted radiative corrections.

In principle, the observables in an elastic scattering process are momentum $[(\hbar{\bf k})]$ and polarization $[{\boldvarepsilon}]$ . The complex polarization vectors $[{\boldvarepsilon}]$ satisfy the conditions $[{\boldvarepsilon}^*\cdot{\boldvarepsilon}=1'\semi\quad{\boldvarepsilon}\cdot{\bf k}=0. \eqno (4.2.6.29)]$

In quantum mechanics, elastic scattering is described in terms of a differential scattering amplitude, M, which is related to the elastic cross section by equation (4.2.6.16).

If polarization is not an observable, then the expression for the differential scattering cross section takes the form of equation (4.2.6.17). If polarization is taken into account, as may be the case when a polarizer is used on a beam scattered from a sample irradiated by the linearly polarized beam from a synchrotron-radiation source, the full equation, and not equation (4.2.6.17), must be used to compute the differential scattering cross section.

The principle of causality implies that the forward-scattering amplitude $[M(\omega,0)]$ should be analytic in the upper half of the $[\omega]$ plane, and that the dispersion relation $[{\rm Re}\ M(\omega,0)={2\omega^2\over\pi}{\int\limits^\infty_0}{{\rm Im}\ M(\omega',0)\over \omega'(\omega'^2-\omega^2)}{\rm d}\omega'\eqno (4.2.6.30)]$ should hold, with the consequence that $[{\rm Re}\ M(\infty,0)=-{2\over\pi}{\int\limits^\infty_0}{{\rm Im}\ M(\omega',0)\over \omega'}{\rm d}\omega'.\eqno (4.2.6.31)]$ This may be rewritten as $[M(\omega,0)-M(\infty,0)=f'(\omega,0)+if''(\omega,0),\eqno (4.2.6.32)]$ with the value of $[f'(\omega,0)]$ defined by equation (4.2.6.15). Using the conservation of probability, $[{\rm Im}\ M(\omega,0)={\omega\over4\pi r_ec}\sigma_{\rm tot},\eqno (4.2.6.33)]$ which is to be compared with equation (4.2.6.23).

Starting with Furry's extension of the formalism of quantum mechanics proposed by Feynman and Dyson, the total Rayleigh amplitude may be written as $[M_n= {\sum_p} \bigg[{\langle n|T_1^*|p\rangle\langle p|T_1|n\rangle\over E_n-E_p+\hbar\omega}+{\langle n|T_2|p\rangle\langle n|T_2^*|p\rangle\over E_n-E_p+\hbar\omega}\bigg], \eqno (4.2.6.34)]$ where $[T_1={\boldalpha}\cdot{\boldepsilon}_i\cdot\exp(i{\bf k}_i\cdot{\bf r})]$ and $[T_2={\boldalpha}\cdot{\boldepsilon}_f^*\cdot\exp(-i{\bf k}_f\cdot{\bf r}).]$ The $[|p\rangle]$ are the complete set of bound and continuum states in the external field of the atomic potential. Singularities occur at all photon energies that correspond to transitions between bound $[|n\rangle]$ and bound state $[|p\rangle]$ . These singularities are removed if the finite widths of these states are considered, and the energies E are replaced by $[iE\Gamma/2]$ , where $[\Gamma]$ is the total (radiative plus non-radiative) width of the state (Gavrila, 1981). By use of the formalism suggested by Brown et al. (1955), it is possible to reduce the numerical problems to one-dimensional radial integrals and differential equations. The required multipole expansions of [T_1] and the specification of the radial perturbed orbitals that are characterized by angular-momentum quantum numbers have been discussed by Kissel (1977). Ultimately, all the angular dependence on the photon scattering angle is written in terms of the associated Legendre functions, and all the energy dependence is in terms of multipole amplitudes.

Solutions are not found for the inhomogeneous radial wave equations, and Kissel (1977) expressed the solution as the linear sum of two solutions of the homogeneous equation, one of which was regular at the origin and the other regular at infinity.

Because excessive amounts of computer time are required to use these direct techniques for calculating the amplitudes from all the subshells, simpler methods are usually used for calculating outer-shell amplitudes. Kissel & Pratt (1985) used estimates for outer-shell amplitudes based on the predictions of the modified form-factor approach. A tabulation of the modified relativistic form factors has been given by Schaupp, Schumacher, Smend, Rullhusen & Hubbell (1983).

Because of the generality of their approach, the computer time required for the calculation of the scattering amplitudes for a particular energy is quite long, so that relatively few calculations have been made. Their approach, however, does not confine itself solely to the problem of forward scattering of photons as does the Cromer & Liberman (1970) approach. Using their model, Kissel et al. (1980) have been able to show that it is incorrect to assign a dependence of the dispersion corrections on the scattering vector $[{\boldDelta}]$ . This is at variance with some established crystallographic practices, in which the dispersion corrections are accorded the same dependence on $[{\boldDelta}]$ as $[f_0({\boldDelta})]$ , and also at variance with the predictions of Wagenfeld's (1975) non-relativistic model.

References

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Kohn, W. & Sham, L. S. (1965). Self consistent equations including exchange and correlation effects. Phys. Rev. A, 140, 1133–1138.Google Scholar

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 246-248