The scattering problem now reduces to calculating the matrix, TLL'λ E The first approximation made is to terminate the summations over L' and l yielding a matrix of finite dimensions. Most computer programs are capable of handling up to 36 × 36 matrices, which corresponds to a maximum I value of 5. The matrix TλLL' can then be obtained from the equation

[MATHEMATICAL EXPRESSION OMITTED] (5)

where

[MATHEMATICAL EXPRESSION OMITTED] (6)

and

[MATHEMATICAL EXPRESSION OMITTED] (7)

Equation (7) is a t-matrix for the elastic scattering of an incident electron of mass m and wave vector k from a potential characterized by a set of energy dependent phase shifts, δ(E). The only conditions on the potential at this point are that it be spherically symmetrical and that neighbouring potentials do not overlap. Laramore and Duke 9 have shown that lattice motion can be taken into consideration at this point by the renormalization of the site scattering vertices, tλ. The renormalized quantity bλ is expressed

[MATHEMATICAL EXPRESSION OMITTED] (8)

where Wλ (kt-ki) is the Debye–Waller factor for the λth layer. If we use the Debye model for the phonon spectrum, the resulting matrix bλLL' is diagonal, as was tLL'λ. The effect of finite temperature is thus to alter the phase shifts, δi(E) appearing in equation (7). This is done internally in the computer program by expanding the exponential term in equation (8) in spherical harmonics. Provision is made in equation (8) for a layer-dependent Debye–Waller factor. The quantity τλLL' in equation (6) represents a t- matrix for scattering from a single plane λ parallel to the surface. The matrix GL"L"'sp is a subplane propagator which is calculated by a summation over the site positions in plane λ. In order to include the non-zero temperature effect, the quantities bλLL' described above should be substituted for tλLL'. Equation (5) completes the definition of TλLL' employing GL"L"'λλ' as the propagator matrix for scattering from layer λ to layer λ'. Thus, TλLL'is composed of the scattering of the plane wave from the λth layer, τλLL', plus the contributions from all possible interplane scattering combinations ending on layer λ. The matrix equation (6) can be inverted to yield an exact expression for τλ,

τλ = [τλ-1 - Gsp](9)

Equation (5) can be similarly inverted, but the summation over the total number of layers, λ' ≠ λ, leads to a matrix of impractically large dimensions. In practice one truncates the calculation by considering only several of the outermost layers in the surface region. Satisfactory results are obtained by writing equation (5) as a perturbation expansion and iterating until the desired accuracy is obtained. Owing to the inelastic damping factor which appears in the expression for the interplane propagator Gλλ, this convergence is rapid and reliable.

The input parameters mentioned in the foregoing description of the computer program may be systematically varied. Among these are the number of phase shifts used to characterize the ion core site potential (maximum l value), the number of outer crystal layers included in the summation (λ), the order of multiple scattering considered in the iterative solution of equation (5), and the effect of the layer-dependent Debye–Waller factor. The calculations are performed in the energy range between 40 and 150 eV, where most of the diffraction information about the structure of the surface is obtained. At lower energies the inner potential changes rapidly, based on free electron gas calculations, and experimental difficulties can occur in determining the precise angle of incidence of the electron beam owing to small stray magnetic fields. At energies in excess of 150 eV, a larger number of phase shifts (lmax > 5) must be employed to characterize the ion core potential. The penetration depths of the beam are approximately proportional to [square root of E], giving rise to increased sampling of the bulk structure at higher energies.

Ion-core Potentials and Energy-dependent Phase Shifts. — The scattering potential at the lattice site is specified by a number of precalculated energy-dependent phase shifts, δi(E). These phase shifts may be obtained at present from self-consistent APW potentials or by the ab initio method of Pendry. Apparently the description of the scattering process by these two methods is of sufficient accuracy to yield qualitatively correct theoretical beam intensities vs. incident electron energy (I vs. eV) curves. However, as closer agreement with experiment is sought and as surface overlayers (especially molecular overlayers) are investigated, it may be necessary to investigate these scattering potentials in more detail.

The first numerical results obtained by Duke and Tucker were based on an s-wave approximation to the scattering amplitudes. Since that time, calculations for aluminium have appeared utilizing from 3 to 8 phase shifts (Figure 1). It is apparent from the I vs. eV curves calculated using only the first two phase shifts, that for aluminium they are poor approximations to the observed intensity patterns. There is a remarkable qualitative similarity between all the curves calculated and experimental I vs. eV curves which include more than two phase shifts. Tong and Rhodin have pointed out the dominance of d-wave scattering for energies in excess of 24 eV. The low scattering power of the l = 0 partial wave can be explained by the (2l + 1) weighting factor which appears in each term of the expansion for the scattering amplitude f(θ), even though the magnitude of the phase shift may be large. This feature will persist in all materials. In conclusion, five phase shifts give the best agreement with the experimental I vs. eV curves for all (111), (100), and (110) aluminium surfaces.

The Effect of including Various Numbers of Layers Parallel to the Surface in the Calculation. — Owing to the large inelastic damping parameter corresponding to an attenuation length of from 4 to 10 Å within the crystal at low electron energies (at less than 200 eV), the major portion of the elastic LEED scattering arises from events in the outermost layers of the sample. Not surprisingly, the consideration of only three surface layers is sufficient to yield diffraction beam intensities comparable with more exact treatments (Figure 2). The interlayer spacing for the aluminium (100) surface is 2.02 Å and the inelastic damping length is 6.4 Å at 50 eV, and 7.7 Å at 100 eV, for a free electron gas whose density is the same as that of the aluminium conduction electrons. Thus, the amplitude of the electrons scattered kinematically, the most favourable case, from the fourth layer, is diminished by a factor of 0.15 to 0.2 compared with one scattered from the surface layer.

Multiple Scattering: Order and Temperature Effects. — The iterative method used to solve equation (5) indicates that third-order diffraction yields results that are very close to those obtained from matrix inversion for the various aluminium crystal faces, differing by a maximum of 10% at high energies (Figure 3). Thus multiple scattering events that involve higher-order diffraction can be neglected. The elastic force constants between the outermost surface layer and the bulk can be different from those between two bulk layers. As a result, the Debye temperature assigned to the surface layer may be quite different from the bulk value. However, the peak intensities arc uniformly diminished owing to the larger average thermal displacement of the outermost layer, and no shifts in the beam positions are observed. At the present state of theoretical accuracy there is no reason to include layer-dependent Debye temperatures when working with a clean metal crystal, as the intensity ratios and the qualitative appearance of the patterns change only slightly. Separate surface values of θD should be included in the calculation of intensity curves from metals covered by overlayers of different materials since the Debye temperature may differ significantly and the effect of this difference may vary between integral and fractional order peaks. Studies of the temperature dependence of the diffraction beam intensity have given detailed information about the mean square displacement of surface atoms, via the Debye–Waller factor, and also about thermal diffuse scattering. The Debye–Waller factor determined for surface atoms has shown that the mean square displacement of these atoms in most face-centred cubic and body-centred cubic surfaces, perpendicular to the surface, is appreciably larger than the bulk atom mean square displacement. This mean square displacement is sensitive to the structure of the surface and to the presence of adsorbed layers at the surface. Thus a great deal of information about the lattice dynamics of surface atoms can be obtained from these measurements. It was found that near normal incidence, kinematic analysis of the data approximates well the analysis that is carried out in the multiple scattering framework. Future studies in this field will include investigation of the effect of adsorbed layers on the mean square displacement of surface atoms, and studies of the zero-point vibration of surface atoms by carrying out Debye–Waller measurements at low temperatures. It would be important to determine the mean square displacement of surface atoms parallel to the surface plane, a measurement that is quite difficult under present experimental circumstances. However, the anisotropy of the mean square displacement of surface atoms is an important piece of information for detailed surface lattice dynamics calculations.