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The other, more satisfactory, way to think about diffraction from lattices is the method due to von Laue, which does emphasise scattering from atoms and their constituent electrons and nuclei. The scattering angle is 2q. The planes of the crystal are characterised by the Fourier coefficients of the electron density r(r) (for X-ray scattering) or of the interaction potential V(r) (for electron scattering). In either case we can write the scattering agent in terms of a Fourier series based on the reciprocal lattice g, i.e.
We can visualise this relation using the following figure 3.1, which shows the traditional Bragg's law picture (on its side, figure 3.1(a)) and the von Laue picture, leading to the Ewald sphere construction (figure 3.1(b)). The incoming wave is described by a vector k and the outgoing wave by k', where |k| = |k'| = 2p/l. From the geometry of the Ewald sphere there is a one-to-one relation between the two pictures provided that d = 2p/|g|, and that the plane described by g is drawn as a vector normal to the planes in the Bragg picture. This picture can be filled out, for any real crystal lattice, by expressing g as the hkl point of the reciprocal lattice, such that
where h, k, and l are the Miller indices, and the b's are the unit vectors of the reciprocal lattice. Just to keep you on your toes, g is written G (Sutton, Pettifor sometimes) or capital K (Ashcroft and Mermin, chapter 6). The last possibility is a nightmare for lecturers who have to repeat sentences like k-prime equals k plus K many times!
Note that the traditions in the crystallography and solid state literature differ, in that 2p enters the equations at different points. The solid state literature typically uses exp(ig.r) as above, with |k| = 2p/l and |g| = 2p/d. The crystallography literature has exp(2pig.r), but with 2p absent from the other definitions.
Here . represents the scalar and x the vector product, and the indices i, j and k have to be taken cyclically. Again we have to be consistent about the 2p, which is needed for the 'solid state' definition of bk used here, but not for the 'crystallography' definition where the 2p is in the exponential. If you are not familiar with these ideas, check out for yourself the relation between the two spaces via problem 3.4.1.
A case which is particularly instructive is when only one g-vector (one reciprocal lattice point) is involved, as this reduces to a 2x2 matrix which can be solved analytically, as in problem 3.4.3. This case is discussed in some detail by Sutton and by Pettifor. When Vg is small relative to the electron energy E, this corresponds to the Nearly Free Electron (NFE) model, which we will discuss in the lecture. The model is widely used, not only for conduction electrons, but also in (transmission) electron microscopy, where it was instrumental in providing the first understanding of contrast from crystal defects in thin crystalline samples. In the latter case, the k-vectors are much longer, corresponding to energies of 100 keV or more, but the angle q is correspondingly smaller, so that the same low order g-vectors are involved.
The plane in the reciprocal lattice perpendicular to g through the point g/2 is known as the Brillouin zone boundary, sometimes called the BZB, and any k-vector which ends on the BZB satisfies the condition for Bragg scattering via k' = k + g. In 2D or 3D a series of such planes for the shortest g-vectors forms a polygon called the first Brillouin zone (BZ). Because of way the reciprocal lattice has been set up, all electron states can be referred back to the first BZ, which is called the reduced as opposed to extended zone scheme. The mathematics behind this was outlined in section 2.3, and is described by both Sutton and Pettifor, amongst many others.
Show, for one of these lattices, that the intensity of diffraction Ig, in the weak scattering (kinematic) limit, is proportional to |Vg|2.
If you haven't studied this model before, now is a good time to do so. You could produce a 'working model' which allows you to vary the parameters and see the result, maybe viewable on the web, or on the computer projection system. See me if this appeals to you. The 1D programs developed by Edward Hernandez a few years ago may be studied in the computation sessions if there is time.