Non-relativistic theories

Creagh, D. C.

doi:10.1107/97809553602060000592

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 244-245

Section 4.2.6.2.2. Non-relativistic theories

D. C. Creagh^b

4.2.6.2.2. Non-relativistic theories

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The matrix element for Rayleigh scattering from an atom having a radially symmetric charge distribution may be written as $[M=M_1({\boldvarepsilon}_i\cdot{\boldvarepsilon}^*_f)+M_2({\boldvarepsilon}_i\cdot{\boldkappa }_f)({\boldvarepsilon}^*_f\cdot{\boldkappa}_i),\eqno (4.2.6.16)]$ where $[{\boldvarepsilon}_i]$ and $[{\boldvarepsilon}_f]$ represent the initial and final states of the photon. The matrix element [M_1] represents scattering for polarizations $[{\boldvarepsilon}_i]$ and $[{\boldvarepsilon}_f]$ perpendicular to the plane of scattering and [M_2] represents scattering for polarization states lying in the plane of scattering.

Averaged over polarization states, the differential scattering cross section takes the form $[{{\rm d}\sigma\over {\rm d}\Omega}={ r^2_e\over2}(|M_1|{}^2+|M_2|{}^2).\eqno (4.2.6.17)]$ Here σ is the photoelectric scattering cross section, which is related to the mass attenuation coefficient $[\mu_m]$ by $[\sigma=(M/N_A)\mu_m\times10^{-24},\eqno (4.2.6.18)]$ where M is the molecular weight and [N_A] is Avogadro's number.

Using the vector potential of the wavefield A, an expression for the perturbed Hamiltonian of a hydrogen-like atom coupled to the radiation field may be written as $[\hat{\scr H}=\hat{\scr H}_0 - r_ec{\bf A}\cdot{\bf P}+{r^2_e\over 2}{\bf A}^2,\eqno (4.2.6.19)]$ where $[\hat{\scr H}_0]$ is the Hamiltonian for the unperturbed atom and $[{\bf P}=i\hbar\nabla]$ .

After application of the second-order perturbation theory, the matrix element may be deduced to be $[M=({\boldvarepsilon}_i\cdot{\boldvarepsilon}_f^*)f_0(\Delta)+{1\over m}\langle1|T_1|1\rangle+{1\over m}\langle1|T_2|1\rangle. \eqno (4.2.6.20)]$ In this equation, the initial and final wavefunctions are designated as $[\langle1|]$ and $[|1\rangle]$ , respectively, and the terms [T_1] and [T_2] are given by $[T_1={\boldvarepsilon}_f\cdot{\bf P}\exp(-i{\bf k}_f\cdot {\bf r}){1\over E_1-{\scr H}_0+\hbar\omega+i\xi}{\boldvarepsilon}_i\cdot{\bf P}\exp(-i{\bf k}_i\cdot{\bf r})]$ and $[T_2={\boldvarepsilon}_i\cdot{\bf P}\exp(-i{\bf k}_i\cdot {\bf r}){1\over E_1-{\scr H}_0+\hbar\omega+i\xi}{\boldvarepsilon}_f\cdot{\bf P}\exp(-i{\bf k}_f\cdot{\bf r})]$ where ξ is an infinitesimal positive quantity.

The first term of equation (4.2.6.20) corresponds to the atomic scattering factor and is identical to the value given by classical theory. The terms involving [T_1] and [T_2] correspond to the dispersion corrections. Equation (4.2.6.20) contains no terms to account for radiation damping. More complete theories take the effect of the finite width of the radiating level into account.

It is necessary to realize that the atomic scattering factor depends on both the photon's frequency $[\omega]$ and the momentum vector $[{\boldDelta}]$ . To emphasize this dependence, equation (4.2.6.7) is rewritten as $[f(\omega,{\boldDelta})=f_0({\boldDelta})+f'(\omega,{\boldDelta})+if''(\omega,{\boldDelta}). \eqno (4.2.6.21)]$ In the dipole approximation, it can be shown that $[f'(\omega,0)={\textstyle2\over\pi}P\int\limits^\infty_0\displaystyle{\omega'f''(\omega',0)\over \omega^2-\omega'^2}\,{\rm d}\omega', \eqno (4.2.6.22)]$ which may be compared with equation (4.2.6.15) and $[f''(\omega,0)={\omega\over4\pi r_ec}\sigma(\omega), \eqno (4.2.6.23)]$ which may be compared with equation (4.2.6.14).

There is a direct correspondence between the predictions of the classical theory and the theory using second-order perturbation theory and non-relativistic quantum mechanics.

The extension of Hönl's (1933a,b) study of the scattering of X-rays by the K shell of atoms to other electron shells has been presented by Wagenfeld (1975).

In these calculations, the energy of the photon was assumed to be such that relativistic effects do not occur, nor do transitions within the discrete states of the atom occur. Transitions to continuum states do occur, and, using the analytical expressions for the wavefunctions of the hydrogen-like atom, analytical expressions may be developed for the photoelectric scattering cross sections. By expansion of the retardation factor $[\exp(-i{\bf k}\cdot{\bf r})]$ as the power series $[1-i{\bf k}\cdot{\bf r}-{1\over2}({\bf k}\cdot{\bf r}){}^2+\ldots]$ , it is possible to determine dipolar, quadrupolar, and higher-order terms in the analytical expression for the photoelectric scattering cross section.

The values of the cross section so obtained were used to calculate the values of $[f'(\omega,{\boldDelta})]$ using the Kramers–Kronig transform [equation (4.2.6.22)] and $[f''(\omega,{\boldDelta})]$ using equation (4.2.6.23). The work of Wagenfeld (1975) predicts that the values of $[f'(\omega,{\boldDelta})]$ and $[f''(\omega,{\boldDelta})]$ are functions of $[{\boldDelta}]$ . Whether or not this is a correct prediction will be discussed in Subsection 4.2.6.3.

Wang & Pratt (1983) have drawn attention to the importance of bound–bound transitions in the dispersion relation for the calculation of forward-scattering amplitudes. Their inclusion is especially important for elements with small atomic numbers. In a later paper, Wang (1986) has shown that, for silicon at the wavelengths of Mo Kα and Ag Kα₁, values for $[f'(\omega,0)]$ of 0.084 and 0.055, respectively, are obtained. These values should be compared with those listed in Table 4.2.6.4.

References

Hönl, H. (1933a). Zur Dispersions theorie der Röntgenstrahlen. Z. Phys. 84, 1–16.Google Scholar

Hönl, H. (1933b). Atomfactor für Röntgenstrahlen als Problem der Dispersionstheorie (K-Schale). Ann. Phys. (Leipzig), 18, 625–657.Google Scholar

Wagenfeld, H. (1975). Theoretical computations of X-ray dispersion corrections. Anomalous scattering, edited by S. Ramaseshan & S. C. Abrahams, pp. 12–23. Copenhagen: Munksgaard.Google Scholar

Wang, M. S. (1986). Relativistic dispersion relation for X-ray anomalous scattering factor. Phys. Rev. A, 34, 636–637.Google Scholar

Wang, M. S. & Pratt, R. H. (1983). Importance of bound–bound transitions in the dispersion relation for calculation of soft-X-ray forward Rayleigh scattering from light elements. Phys. Rev. A, 28, 3115–3116.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 244-245