(*a*) thermal diffuse scattering, with energy losses of the order of eV, separable from the elastic scattering only with specially devised equipment; the angular distribution of thermal diffuse scattering shows variations with 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 to 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 eV and an angular range of scattering of to rad;

(*e*) inner-shell excitations, with energy losses of eV or more and an angular range of scattering of to rad, depending on the energy losses involved.

(3) In the original treatment by Bethe (1928) of the elastic scattering of electrons by crystals, the Schrödinger equation is written for electrons in the periodic potential of the crystal; *i.e.* where **K** _{0} is the wavevector in zero potential (outside the crystal) (magnitude ) and *W* is the accelerating voltage. The solutions of the equation are Bloch waves of the form where is the incident wavevector in the crystal and **h** is a reciprocal-lattice vector. Substitution of (2.5.2.2) and (2.5.2.3) in (2.5.2.1) gives the dispersion equations Here κ is the magnitude of the wavevector in a medium of constant potential (the `inner potential' of the crystal). The refractive index of the electron in the average crystal potential is then Since is positive and of the order of 10 V and *W* is to V, is positive and of the order of .

Solution of equation (2.5.2.4) gives the Fourier coefficients of the Bloch waves and application of the boundary conditions gives the amplitudes of individual Bloch waves (see Chapter 5.2
).

(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 where and . [See Lynch & Moodie (1972), Portier & Gratias (1981), Tournarie (1962), and Chapter 5.2
.]

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 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.)

(5) The integral form of the wave equation, commonly used for scattering problems, is written, for electron scattering, as

The wavefunction within the integral is approximated by using successive terms of a Born series

The first Born approximation is obtained by putting in the integral and subsequent terms are generated by putting in the integral.

For an incident plane wave, and for a point of observation at a large distance from the scattering object , the first Born approximation is generated as where or, putting and collecting the pre-integral terms into a parameter μ, This is the Fourier-transform expression which is the basis for the *kinematical scattering approximation* . It is derived on the basis that all terms for are very much smaller than and so is a weak scattering approximation.

In this approximation, the scattered amplitude for an atom is related to the atomic structure amplitude, , by the relationship, derived from (2.5.2.8), For centrosymmetrical atom potential distributions, the are real, positive and monotonically decreasing with . 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 in (2.5.2.10) by the complex quantities and for single heavy atoms the phase factor η may vary from 0.2 for to 4 or 5 for large , as seen from the tables of *IT* C (2004, Section 4.3.3
).

(6) Relativistic effects produce appreciable variations of the parameters used above for the range of electron energies considered. The relativistic values are where is the Compton wavelength, , and Values for these quantities are listed in *IT* C (2004, Section 4.3.2
). The variations of λ and σ with accelerating voltage are illustrated in Fig. 2.5.2.1. For high voltages, σ tends to a constant value, .