Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 4.3, pp. 541-542   | 1 | 2 |

Section 4.3.2. Inelastic scattering

J. M. Cowleya and J. K. Gjønnesb

aArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287–1504, USA, and  bInstitute of Physics, University of Oslo, PO Box 1048, N-0316 Oslo 3, Norway

4.3.2. Inelastic scattering

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In the kinematical approximation, a general expression which includes inelastic scattering can be written in the form quoted by Van Hove (1954[link])[\eqalignno{ I ({\bf u}, \nu) &= {m^{3}\over 2^{2}h^{6}} {k\over k_{o}}\cr &\quad \times W ({\bf u}) \sum P_{n_{o}} {\sum\limits_{\eta}} {\sum\limits_{j=1}^{Z}} | \langle n_{o}| \exp \{2\pi i{\bf u} \cdot {\bf R}_{j}\} |n \rangle |^{2}\cr &\quad \times \delta \left(\nu + {E_{n} - E_{n_{o}}\over h}\right) &(}]for the intensity of scattering as function of energy transfer and momentum transfer from a system of Z identical particles, [{\bf R}_{j}]. Here m and h have their usual meanings; [k_{o}] and k, [E_{n_{o}}] and [E_{n}] are wavevectors and energies before and after the scattering between object states [n_{o}] and n; [P_{n_{o}}] are weights of the initial states; W(u) is a form factor (squared) for the individual particle.

In equation ([link], u is essentially momentum transfer. When the energy transfer is small [(\Delta E/E \ll \theta)], we can still write [|{\bf u}| = 2 \sin \theta / \lambda], then the sum over final states n is readily performed and an expression of the Waller–Hartree type is obtained for the total inelastic scattering as a function of angle:[I_{\rm inel} ({\bf u}) \propto {S\over u^{4}},]where[S(u) = Z - {\textstyle\sum\limits_{j=1}^{Z}} |\,f_{jj} ({u})|^{2} - {\textstyle\sum\limits_{\ \ j}^{Z}} {\textstyle\sum\limits_{\neq k\ \ }^{Z}} |\,f_{jk} ({u}) |^{2}, \eqno(]and where the one-electron f's for Hartree–Fock orbitals, [f_{jk} ({\bf u}) = \langle j| \exp (2\pi i{\bf u} \cdot {\bf r})|k \rangle], have been calculated by Freeman (1959[link], 1960[link]) for atoms up to [Z = 30]. The last sum is over electrons with the same spin only.

The Waller–Hartree formula may be a very good approximation for Compton scattering of X-rays, where most of the scattering occurs at high angles and multiple scattering is no problem. With electrons, it has several deficiencies. It does not take into account the electronic structure of the solid, which is most important at low values of u. It does not include the energy distribution of the scattering. It does not give a finite cross section at zero angle, if u is interpreted as an angle. In order to remedy this, we should go back to equation ([link] and decompose u into two components, one tangential part which is associated with angle in the usual way and one normal component along the beam direction, [u_{n}], which may be related to the excitation energy [\Delta E = E_{n} - E_{n_{o}}] by the expression [u_{n} = \Delta E_{k}/2E]. This will introduce a factor [1/(u^{2} + u_{n}^{2})] in the intensity at small angles, often written as [1/(\theta^{2} + \theta_{E}^{2})], with [\Delta E] estimated from ionization energies etc. (Strictly speaking, [\Delta E] is not a constant, not even for scattering from one shell. It is a weighted average which will vary with u.)

Calculations beyond this simple adjustment of the Waller–Hartree-type expression are few. Plasmon scattering has been treated on the basis of a nearly free electron model by Ferrel (1957[link]):[{\hbox{d}^{2}\sigma\over \hbox{d}(\Delta E)\, \hbox{d}\Omega} = (1/\pi^{2} a_{H} mv^{2}N) (-\hbox{Im} \{1/\varepsilon\}) / (\theta^{2} + \theta_{E}^{2}),\eqno (]where m, v are relativistic mass and velocity of the incident electron, N is the density of the valence electrons and [\varepsilon (\Delta E, \theta)] their dielectric constant. Upon integration over [\Delta E]:[{\hbox{d}\sigma\over \hbox{d}\Omega} = {E_{p}\over 2\pi a_{H} mv N} [1 / (\theta^{2} + \theta_{E}^{2}) G (\theta, \theta_{\rm c})], \eqno(]where [G(\theta, \theta_{\rm c})] takes account of the cut-off angle [\Theta_{\rm c}]. Inner-shell excitations have been studied because of their importance to spectroscopy. The most realistic calculations may be those of Leapman et al. (1980[link]) where one-electron wavefunctions are determined for the excited states in order to obtain `generalized oscillator strengths' which may then be used to modify equation ([link].

At high energies and high momentum transfer, the scattering will approach that of free electrons, i.e. a maximum at the so-called Bethe ridge, [E = h^{2}u^{2}/2m].

A complete and detailed picture of inelastic scattering of electrons as a function of energy and angle (or scattering variable) is lacking, and may possibly be the least known area of diffraction by solids. It is further complicated by the dynamical scattering, which involves the incident and diffracted electrons and also the ejected atomic electron (see e.g. Maslen & Rossouw, 1984a[link],b[link]).


Ferrel, R. A. (1957). Characteristic energy loss of electrons passing through metal foils. Phys. Rev. 107, 450–462.
Freeman, A. J. (1959). Compton scattering of X-rays from non-spherical charge distributions. Phys. Rev. 113, 169–175.
Freeman, A. J. (1960). X-ray incoherent scattering functions for non-spherical charge distributions. Acta Cryst. 12, 929–936.
Leapman, R. D., Rez, P. & Mayers, D. F. (1980). L and M shell generalized oscillator strength and ionization cross sections for fast electron collisions. J. Chem. Phys. 72, 1232–1243.
Maslen, W. V. & Rossouw, C. J. (1984a). Implications of (e, 2e) scattering electron diffraction in crystals: I. Philos. Mag. A, 49, 735–742.
Maslen, W. V. & Rossouw, C. J. (1984b). Implications of (e, 2e) scattering electron diffraction in crystals: II. Philos. Mag. A, 49, 743–757.
Van Hove, L. (1954). Correlations in space and time and Born approximation scattering in systems of interacting particles. Phys. Rev. 95, 249–262.

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