International
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Volume C
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© International Union of Crystallography 2006

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International Tables for Crystallography (2006). Vol. C. ch. 4.3, p. 260

Section 4.3.1.3. Approximations of restricted validity

J. M. Cowley^b ^‡

4.3.1.3. Approximations of restricted validity

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(a) Kinematical approximation. In the limiting case of a vanishingly weak interaction of the incident electrons with the scattering potential of the crystal, the Born series (4.3.1.5) may be terminated at the term $[\psi_1]$ , corresponding to single scattering. Then the diffracted wave is given for a potential $[\varphi(r)]$ as $[\psi({\bf s})(\exp-ikR)/R]$ , with $[\psi({\bf s})=K\textstyle\int\limits\varphi({\bf r})\exp\{i[{\bf r}\cdot{\bf s}]\}\,{\rm d}\tau_{\bf r},\eqno (4.3.1.16)]$ where R is the distance to the point of observation. For a periodic potential, $[\varphi({\bf r})]$ , the scattering amplitude for the h beam is $[\psi({\bf h})=NK\textstyle\int\limits\varphi({\bf r})\exp\{2\pi i{\bf h}\cdot{\bf r}\}\,{\rm d}\tau_{\bf r},\eqno (4.3.1.17)]$ where the integral is taken over one unit cell and N is the number of unit cells. From (4.3.1.16), it then follows that the scattering amplitude $[\psi({\bf h})]$ is proportional to the structure amplitude, $[V({\bf h})]$ ; $[\eqalignno{ \psi({\bf h}) &=NKV(h) &(4.3.1.18)\cr &=NK\textstyle\sum\limits_i\,f_{{\rm el},i}({\bf h})\exp\{2\pi i{\bf h}\cdot {\bf r}_i\}. &(4.3.1.19)}%fd4.3.1.19]$ The intensity of the h diffracted beam is then proportional to $[\psi({\bf h})\psi^*({\bf h})]$ , and so to $[|V({\bf h})|{}^2]$ .

Similarly, we may write the differential scattering cross section for the scattering from a single isolated atom as $[|\,f^B(s)|{}^2=K^2|V_i(s)|{}^2.\eqno (4.3.1.20)]$
(b) Two-beam approximation. For some specific orientations of a crystal of relatively simple structure, the incident beam may be close to the Bragg angle for a strong, inner reflection but not for any other reflection. Then the approximation may be made that only those beams with indices 0 and h have appreciable intensity. The intensities of these beams for a parallel-sided, plate-shaped, centrosymmetric crystal are given in MacGillavry's (1940) development of the theory of Bethe (1928) as $[I({\bf h})=I_0\{\sigma V({\bf h})\}^2{\sin^2\{\pi t(\zeta^2_{\bf h}+\xi^{-2}_{\bf h})^{1/2}\}\over \pi^2(\zeta^2_{\bf h}+\xi^{-2}_{\bf h})}\eqno (4.3.1.21)]$ and $[I({\bf 0})=I_0-I({\bf h})]$ , where is the incident-beam intensity, t is the crystal thickness, $[\xi_{\bf h}]$ is the extinction distance given by $[\xi_{\bf h}=\pi/\sigma V({\bf h})]$ , and $[\zeta_{\bf h}]$ is the excitation error which measures the distance of the reciprocal-lattice point h from the Ewald sphere.

A formula due to Blackman (1939), obtained by integrating (4.3.1.21) over $[\xi_{\bf h}]$ , provides a useful first approximation for the intensities of ring or arc patterns given by polycrystalline material (see Section 2.5.2 ).
(c) Phase-grating approximations. For extremely thin crystals, the scattering can be approximated by that of a two-dimensional potential distribution given by projection of the three-dimensional distribution in the beam direction. Then, by analogy with (4.3.1.6), the emerging wave is $[\psi(xy)=\exp\{-i\sigma\varphi(xy)\}\eqno (4.3.1.22)]$ when $[\varphi(xy)=\textstyle\int\limits^H_0\varphi(xyz)\,{\rm d} z\eqno (4.3.1.23)]$ and the diffraction amplitudes are given by the Fourier transform of this expression.

For thicker crystals, this approximation applies in the limit of very high electron-accelerating voltage, with the value of σ appropriate for the Compton wavelength, λ = 0.024262 Å, viz σ = 0.0005068.

It may be noted that for the special case of a single layer of atoms the solution of the wave equations (4.3.1.2) or (4.3.1.4), with the real potential (4.3.1.1) inserted, leads to a form equivalent to the Moliere high-energy approximation for the scattering by single atoms, namely $[KV(s)=-{i\over\lambda} \int\limits^\infty_{-\infty}\, \{\exp[-i \sigma\varphi({\boldrho})]-1\}\exp\{i{\boldrho}\cdot{\bf s}\}\,{\rm d}^2{\boldrho},\eqno (4.3.1.24)]$ where $[\boldrho]$ is a two-dimensional vector with components x, y, and $[\varphi({\boldrho})=\textstyle\int\limits^\infty_{-\infty}\,\varphi({\boldrho},z)\,{\rm d} z,\eqno (4.3.1.25)]$ and this, in the low-angle approximation, is the same as (4.3.1.23). Then the scattered amplitude can be considered as made up from contributions from individual atoms that are equal (apart from bonding effects) to the complex atomic scattering amplitudes tabulated in connection with the diffraction of electrons by gases.

References

First citation

Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. Leipzig, 87, 55–129.Google Scholar

First citation

Blackman, M. (1939). On the intensities of electron diffraction rings. Proc. R. Soc. London Ser. A, 173, 68–82.Google Scholar

First citation

MacGillavry, C. H. (1940). Zur Prufung der Dynamischen Theorie der Elektronenbeugung an Kristallgitter. Physica (Utrecht), 7, 329–343.Google Scholar

© International Union of Crystallography 2006

International Tables for Crystallography (2006). Vol. C. ch. 4.3, p. 260

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