Scattering theory reciprocal lattice vectors

The lattice wave numbers defined in Eq. (16-16) arc the reciprocal lattice vectors familiar in diffraction theory. Only if the change in wave number resulting from the diffraction is equal to a lattice wave number can a wave, whether it be an X-ray or an electron, be diffracted otherwise the wavelets scattered by the different ions interfere with each other and reduce the diffracted intensity to zero. Only for diffraction by q equal to a lattice wave number do the scattered waves add in phase. Thus a wave having wave number k can only be diffracted to final states of wave number k that can be written as k = k -I- q, where q is a lattice wave number. Furthermore, the diffracted wave will have the same frequency as the incident wave if it is an X-ray, or the same energy if it is an electron, from which it follows that k = k. Combining the two conditions, k -I- qp = gives the Bragg condition for diffraction. 

We now examine the DCA against the set of conditions to be met by a satisfactory alloy theory. By construction, the DCA yields an analytic self-energy and a Green function that take account of statistical fluctuations in the fictitious real-space cluster corresponding to the set of reciprocal-lattice vectors K. The self-energy is periodic with the point symmetry of the real lattice, and vanishes in the limit c — 0 and as the scattering strength approaches zero. Its behavior for small but non-zero concentration is not known. This behavior would be of relevance in applications of the theory to ordered systems. 

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

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