Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 2.5, pp. 299-301   | 1 | 2 |

Section The interactions of electrons with matter

J. M. Cowleya The interactions of electrons with matter

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  • (1) The elastic scattering of electrons results from the interaction of the charged electrons with the electrostatic potential distribution, [\varphi ({\bf r})], of the atoms or crystals. An incident electron of kinetic energy eW gains energy [e\varphi ({\bf r})] in the potential field. Alternatively it may be stated that an incident electron wave of wavelength [\lambda = h / mv] is diffracted by a region of variable refractive index [n ({\bf r}) = k / K_{0} = \{[W + \varphi ({\bf r})] / W\}^{1/2} \simeq 1 + \varphi ({\bf r}) / 2 W.]

  • (2) The most important inelastic scattering processes are:

    • (a) thermal diffuse scattering, with energy losses of the order of 2 × 10−2 eV, separable from the elastic scattering only with specially devised equipment; the angular distribution of thermal diffuse scattering shows variations with [(\sin \theta) / \lambda] which are much the same as for the X-ray case in the kinematical limit;

    • (b) bulk plasmon excitation, or the excitation of collective energy states of the conduction electrons, giving energy losses of 3 to 30 eV and an angular range of scattering of 10−4 to 10−3 rad;

    • (c) surface plasmons, or the excitation of collective energy states of the conduction electrons at discontinuities of the structure, with energy losses less than those for bulk plasmons and a similar angular range of scattering;

    • (d) interband or intraband excitation of valence-shell electrons giving energy losses in the range of 1 to 102 eV and an angular range of scattering of 10−4 to 10−2 rad;

    • (e) inner-shell excitations, with energy losses of 102 eV or more and an angular range of scattering of 10−3 to 10−2 rad, depending on the energy losses involved.

    • (3) In the original treatment by Bethe (1928)[link] of the elastic scattering of electrons by crystals, the Schrödinger equation is written for electrons in the periodic potential of the crystal; i.e. [\nabla^{2} \psi ({\bf r}) + K^{2}_0 [1 + \varphi ({\bf r}) / W] \psi ({\bf r}) = 0, \eqno (]where [\eqalignno{\varphi ({\bf r}) &= \textstyle\int V ({\bf u}) \exp \{-2 \pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf u}\cr &= \textstyle\sum\limits_{\bf h} V_{\bf h} \exp \{-2 \pi i{\bf h} \cdot {\bf r}\}, &(}]K0 is the wavevector in zero potential (outside the crystal) (magnitude [2\pi / \lambda]) and W is the accelerating voltage. The solutions of the equation are Bloch waves of the form [\psi ({\bf r}) = \textstyle\sum\limits_{\bf h} C_{\bf h} ({\bf k}) \exp \{-i ({\bf k}_{0} + 2 \pi {\bf h}) \cdot {\bf r}\}, \eqno (]where [{\bf k}_{0}] is the incident wavevector in the crystal and h is a reciprocal-lattice vector. Substitution of ([link] and ([link] in ([link] gives the dispersion equations [(\kappa^{2} - k_{\bf h}^{2}) C_{\bf h} + \textstyle\sum\limits_{\bf g}^{}{}^{\prime}\,V_{\bf h-g} C_{\bf g} = 0. \eqno (]Here κ is the magnitude of the wavevector in a medium of constant potential [V_{0}] (the `inner potential' of the crystal). The refractive index of the electron in the average crystal potential is then [n = \kappa / K = (1 + V_{0} / W)^{1/2} \simeq 1 + V_{0} / 2 W. \eqno (]Since [V_{0}] is positive and of the order of 10 V and W is [10^{4}] to [10^{6}] V, [n - 1] is positive and of the order of [10^{-4}].

      Solution of equation ([link] gives the Fourier coefficients [C_{\bf h}^{(i)}] of the Bloch waves [\psi^{(i)} ({\bf r})] and application of the boundary conditions gives the amplitudes of individual Bloch waves (see Chapter 5.2[link] ).

    • (4) The experimentally important case of transmission of high-energy electrons through thin specimens is treated on the assumption of a plane wave incident in a direction almost perpendicular to an infinitely extended plane-parallel lamellar crystal, making use of the small-angle scattering approximation in which the forward-scattered wave is represented in the paraboloidal approximation to the sphere. The incident-beam direction, assumed to be almost parallel to the z axis, is unique and the z component of k is factored out to give [\nabla^{2} \psi + 2k \sigma \varphi \psi = \pm i2k {\partial \psi \over \partial z}, \eqno (]where [k = 2\pi / \lambda] and [\sigma = 2\pi me\lambda / h^{2}]. [See Lynch & Moodie (1972)[link], Portier & Gratias (1981)[link], Tournarie (1962)[link], and Chapter 5.2[link] .]

      This equation is analogous to the time-dependent Schrödinger equation with z replacing t. Retention of the ± signs on the right-hand side is consistent with both ψ and [\psi^{*}] being solutions, corresponding to propagation in opposite directions with respect to the z axis. The double-valued solution is of importance in consideration of reciprocity relationships which provide the basis for the description of some dynamical diffraction symmetries. (See Section 2.5.3[link].)

    • (5) The integral form of the wave equation, commonly used for scattering problems, is written, for electron scattering, as [{\psi ({\bf r}) = \psi^{(0)} ({\bf r}) + (\sigma / \lambda) \int {\exp \{- i{\bf k}| {\bf r - r}'|\} \over |{\bf r - r}'|} \varphi ({\bf r}') \psi ({\bf r}') \,\hbox{d}{\bf r}'}. \eqno (]

      The wavefunction [\psi ({\bf r})] within the integral is approximated by using successive terms of a Born series [\psi ({\bf r}) = \psi^{(0)} ({\bf r}) + \psi^{(1)} ({\bf r}) + \psi^{(2)} ({\bf r}) + \ldots .\eqno (]

      The first Born approximation is obtained by putting [\psi ({\bf r}) = \psi^{(0)} ({\bf r})] in the integral and subsequent terms [\psi^{(n)} ({\bf r})] are generated by putting [\psi^{(n - 1)} ({\bf r})] in the integral.

      For an incident plane wave, [\psi^{(0)} ({\bf r}) = \exp \{- i{\bf k}_{0} \cdot {\bf r}\}] and for a point of observation at a large distance [{\bf R} = {\bf r} - {\bf r}'] from the scattering object [(|{\bf R}| \gg |{\bf r}'|)], the first Born approximation is generated as [\psi^{(1)} ({\bf r}) = {i\sigma \over \lambda R} \exp \{- i{\bf k} \cdot {\bf R}\} \int \varphi ({\bf r}') \exp \{i{\bf q} \cdot {\bf r}'\} \,\hbox{d}{\bf r}',]where [{\bf q} = {\bf k} - {\bf k}_{0}] or, putting [{\bf u} = {\bf q}/2\pi] and collecting the pre-integral terms into a parameter μ, [\Psi ({\bf u}) = \mu \textstyle\int \varphi ({\bf r}) \exp \{2\pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf r}. \eqno (]This is the Fourier-transform expression which is the basis for the kinematical scattering approximation. It is derived on the basis that all [\psi^{(n)} ({\bf r})] terms for [n \neq 0] are very much smaller than [\psi^{(0)} ({\bf r})] and so is a weak scattering approximation.

      In this approximation, the scattered amplitude for an atom is related to the atomic structure amplitude, [f({\bf u})], by the relationship, derived from ([link], [\eqalignno{\psi ({\bf r}) &= \exp \{- i{\bf k}_{0} \cdot {\bf r}\} + i {\exp \{- i{\bf k} \cdot {\bf r}\} \over R\lambda} \sigma f({\bf u}),&\cr f({\bf u}) &= \textstyle\int \varphi ({\bf r}) \exp \{2\pi i{\bf u} \cdot {\bf r}\} \,\hbox{d}{\bf r}.&(\cr}]For centrosymmetrical atom potential distributions, the [f({\bf u})] are real, positive and monotonically decreasing with [|{\bf u}|]. A measure of the extent of the validity of the first Born approximation is given by the fact that the effect of adding the higher-order terms of the Born series may be represented by replacing [f({\bf u})] in ([link] by the complex quantities [f({\bf u}) = |{\bf f}| \exp \{i\eta ({\bf u})\}] and for single heavy atoms the phase factor η may vary from 0.2 for [|{\bf u}| = 0] to 4 or 5 for large [|{\bf u}|], as seen from the tables of IT C (2004[link], Section 4.3.3[link] ).

    • (6) Relativistic effects produce appreciable variations of the parameters used above for the range of electron energies considered. The relativistic values are [\eqalignno{m &= m_{0} (1 - v^{2} / c^{2})^{-1/2} = m_{0} (1 - \beta^{2})^{-1/2}, &(\cr \lambda &= h[2m_{0}|e| W (1 + |e| W / 2m_{0}c^{2})]^{-1/2} &(\cr &= \lambda_{\rm c} (1 - \beta^{2})^{1/2} / \beta, &(}%(]where [\lambda_{\rm c}] is the Compton wavelength, [\lambda_{\rm c} = h / m_{0}c = 0.0242\,\hbox{\AA}], and [\eqalignno{\sigma &= 2\pi me\lambda / h^{2} = (2\pi m_{0}e / h^{2})(\lambda_{\rm c} / \beta)\cr &= 2\pi / \{\lambda W[1 + (1 - \beta^{2})^{1/2}]\}. &(}]Values for these quantities are listed in IT C (2004[link], Section 4.3.2[link] ). The variations of λ and σ with accelerating voltage are illustrated in Fig.[link]. For high voltages, σ tends to a constant value, [2\pi m_{0}e\lambda_{\rm c} / h^{2} = e / \hbar c].


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      The variation with accelerating voltage of electrons of (a) the wavelength, λ and (b) the quantity [\lambda [1 + (h^{2} / m_{0}^{2} c^{2} \lambda^{2})] = \lambda_{\rm c} / \beta] which is proportional to the interaction constant σ [equation ([link]]. The limit is the Compton wavelength [\lambda_{\rm c}] (after Fujiwara, 1961[link]).


International Tables for Crystallography (2004). Vol. C. Mathematical, Physical and Chemical Tables, edited E. Prince, 3rd ed. Dordrecht: Kluwer Academic Publishers.
Bethe, H. A. (1928). Theorie der Beugung von Elektronen an Kristallen. Ann. Phys. (Leipzig), 87, 55–129.
Lynch, D. F. & Moodie, A. F. (1972). Numerical evaluation of low energy electron diffraction intensity. I. The perfect crystal with no upper layer lines and no absorption. Surf. Sci. 32, 422–438.
Portier, R. & Gratias, D. (1981). Diffraction symmetries for elastic scattering. In Electron Microscopy and Analysis. Inst. Phys. Conf. Ser. No. 61, pp. 275–278. Bristol, London: Institute of Physics.
Tournaire, M. (1962). Recent developments of the matrical and semi-reciprocal formulation in the field of dynamical theory. J. Phys. Soc. Jpn, 17, Suppl. B11, 98–100.

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