General scattering expression for X-rays

Coppens, P.

doi:10.1107/97809553602060000550

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 10 | 1 | 2 |

Section 1.2.2. General scattering expression for X-rays

P. Coppens^a ^*

^aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

1.2.2. General scattering expression for X-rays

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The total scattering of X-rays contains both elastic and inelastic components . Within the first-order Born approximation (Born, 1926) it has been treated by several authors (e.g. Waller & Hartree, 1929; Feil, 1977) and is given by the expression $[I_{\rm total} ({\bf S}) = I_{\rm classical} {\textstyle\sum\limits_{n}} \left|{\textstyle\int} \psi_{n}^{*} \exp (2\pi i{\bf S}\cdot {\bf r}_{j}) \psi_{0}\; \hbox{d}{\bf r}\right|^{2}, \eqno(1.2.2.1)]$ where $[I_{\rm classical}]$ is the classical Thomson scattering of an X-ray beam by a free electron, which is equal to $[(e^{2}/mc^{2})^{2} (1 + \cos^{2} 2\theta)/2]$ for an unpolarized beam of unit intensity, ψ is the n-electron space-wavefunction expressed in the 3n coordinates of the electrons located at $[{\bf r}_{j}]$ and the integration is over the coordinates of all electrons. S is the scattering vector of length $[2\sin \theta/\lambda]$ .

The coherent elastic component of the scattering, in units of the scattering of a free electron, is given by $[I_{\rm coherent, \, elastic} ({\bf S}) = \left|{\textstyle\int} \psi_{0}^{*}\right| {\textstyle\sum\limits_{j}} \exp (2\pi i{\bf S}\cdot {\bf r}_{j}) |\psi_{0} \;\hbox{d}{\bf r}|^{2}. \eqno(1.2.2.2)]$

If integration is performed over all coordinates but those of the jth electron, one obtains after summation over all electrons $[I_{\rm coherent, \, elastic} ({\bf S}) = |{\textstyle\int} \rho ({\bf r}) \exp (2\pi i{\bf S}\cdot {\bf r}) \;\hbox{d}{\bf r}|^{2}, \eqno(1.2.2.3)]$ where $[\rho({\bf r})]$ is the electron distribution. The scattering amplitude $[A({\bf S})]$ is then given by $[A({\bf S}) = {\textstyle\int} \rho ({\bf r}) \exp (2\pi i{\bf S}\cdot {\bf r}) \;\hbox{d}{\bf r} \eqno(1.2.2.4a)]$ or $[A({\bf S}) = \hat{F} \{\rho ({\bf r})\}, \eqno(1.2.2.4b)]$ where $[\hat{F}]$ is the Fourier transform operator.

References

Born, M. (1926). Quantenmechanik der Stoszvorgänge. Z. Phys. 38, 803.Google Scholar

Feil, D. (1977). Diffraction physics. Isr. J. Chem. 16, 103–110.Google Scholar

Waller, I. & Hartree, D. R. (1929). Intensity of total scattering X-rays. Proc. R. Soc. London Ser. A, 124, 119–142.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 10