Semi-reciprocal space

Moodie, A. F.; Cowley, J. M.; Goodman, P.

doi:10.1107/97809553602060000570

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 553 | 1 | 2 |

Section 5.2.6. Semi-reciprocal space

A. F. Moodie,^a J. M. Cowley^b ^‡ and P. Goodman^c ^‡

^a Department of Applied Physics, Royal Melbourne Institute of Technology, 124 La Trobe Street, Melbourne, Victoria 3000, Australia,^bArizona State University, Box 871504, Department of Physics and Astronomy, Tempe, AZ 85287-1504, USA, and ^cSchool of Physics, University of Melbourne, Parkville, Australia 3052

5.2.6. Semi-reciprocal space

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In the derivation of electron-diffraction equations, it is more usual to work in semi-reciprocal space (Tournarie, 1962). This can be achieved by transforming equation (5.2.2.1) with respect to x and y but not with respect to z, to obtain Tournarie's equation $[{\hbox{d}^{2} |U\rangle\over \hbox{d}z^{2}} = - {\bf M}_{b}(z)|U\rangle. \eqno(5.2.6.1a)]$ Here $[|U\rangle]$ is the column vector of scattering amplitudes and $[{\bf M}_{b} (z)]$ is a matrix, appropriate to LEED, with k vectors as diagonal elements and Fourier coefficients of the potential as nondiagonal elements.

This equation is factorized in a manner parallel to that used on the real-space equation [equation (5.2.3.1)] (Lynch & Moodie, 1972) to obtain Tournarie's forward-scattering equation $[{\hbox{d}|U^{\pm}\rangle\over \hbox{d}z} = \pm i{\bf M}^{\pm} (z) |U^{\pm}\rangle, \eqno(5.2.6.1b)]$ where $[\eqalign{ {\bf M}^{\pm} (z) &= \pm [{\bf K} + (1/2) {\bf K}^{-1} {V} (z)],\cr [K_{ij}] &= \delta_{ij} K_{i},}]$ and $[[V_{ij}] = 2k_{z} {\textstyle\sum\limits_{l}} V_{i - j} \exp \{ - 2\pi ilz\},]$ where $[V_{i}\equiv\sigma{}v_i]$ are the scattering coefficients and $[v_{i}]$ are the structure amplitudes in volts. In order to simplify the electron-diffraction expression, the third crystallographic index `l' is taken to represent the periodicity along the z direction.

The double solution involving M of equation (5.2.6.1b) is of interest in displaying the symmetry of reciprocity, and may be compared with the double solution obtained for the real-space equation [equation (5.2.3.2)]. Normally the $[{\bf M}^{+}]$ solution will be followed through to give the fast-electron forward-scattering equations appropriate in HEED. $[{\bf M}^{-}]$ , however, represents the equivalent set of equations corresponding to the z reversed reciprocity configuration. Reciprocity solutions will yield diffraction symmetries in the forward direction when coupled with crystal-inverting symmetries (Section 2.5.3 ).

Once again we set out to solve the forward-scattering equation (5.2.6.1a,b) now in semi-reciprocal space, and define an operator $[{\bf Q}(z)]$ [compare with equation (5.2.4.1a)] such that $[|U_{z}\rangle = {\bf Q}_{z}| U_{0} \rangle \quad\hbox{with}\quad U_{0} = |0\rangle \hbox{;}]$ i.e., $[{\bf Q}_{z}]$ is an operator that, when acting on the incident wavevector, generates the wavefunction in semi-reciprocal space.

Again, the differential equation can be transformed into an integral equation, and once again this can be iterated. In the projection approximation, with $[{\bf M}]$ independent of z, the solution can be written as $[{\bf Q}_{p} = \exp \{i{\bf M}_{p} (z - z_{0})\}.]$ A typical off-diagonal element is given by $[V_{i-j}/\cos\theta_i]$ , where $[\theta_{i}]$ is the angle through which the beam is scattered. It is usual in the literature to find that $[\cos \theta_{i}]$ has been approximated as unity, since even the most accurate measurements are, so far, in error by much more than this amount.

This expression for $[{\bf Q}_{p}]$ is Sturkey's (1957) solution, a most useful relation, written explicitly as $[|U\rangle = \exp \{i{\bf M}_{p}{T}\}|0\rangle \eqno(5.2.6.2)]$ with T the thickness of the crystal, and $[|0\rangle]$ , the incident state, a column vector with the first entry unity and the rest zero. $[{\bf S} = \exp \{i{\bf M}_{p}{T}\}]$ is a unitary matrix, so that in this formulation scattering is described as rotation in Hilbert space.

References

Lynch, D. F. & Moodie, A. F. (1972). Numerical evaluation of low energy electron diffraction intensities. Surf. Sci. 32, 422–438.Google Scholar

Sturkey, L. (1957). The use of electron-diffraction intensities in structure determination. Acta Cryst. 10, 858–859.Google Scholar

Tournarie, M. (1962). Recent developments of the matrical and semi-reciprocal formulation on the field of dynamical theory. J. Phys. Soc. Jpn, 17, Suppl. B11, 98–100.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 5.2, p. 553