QMMS Lecture #3 (Venables/Heggie)

Notes for QMMS Lecture 3 (Venables/Heggie)

Lecture notes by John A. Venables. Latest version 23rd March 2008, ex 31 Aug 2005.

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3. Crystallography in real and reciprocal space

3.1 Diffraction and Bragg's law in real and reciprocal space

The elementary way of deriving Bragg's law uses a drawing of 'planes' in the real lattice, spacing d, and shows that for constructive interference we have to have the (glancing) angle q given by 2dsinq = nl. Here n is known as the order of diffraction, but we can simplify this idea by always referring to first order diffraction from planes of spacing d/n. Note that the angle between the diffracted and incident beam is 2q, and also that there don't actually have to be any atoms, which actually do the scattering, on the planes. The picture is a bit artificial in this respect.

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.

V(r) = Σ_g V_gexp(ig.r). (3.1)

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

g = hb₁ + kb₂ + lb₃. (3.2)

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!

Figure 3.1: a) Real space picture of Bragg's law from the planes of spacing d, via 2dsinq = nl; b) Reciprocal space picture via the Ewald sphere construction: the diffraction condition is k' = k + g.

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.

3.2 Examples of real and reciprocal lattice structures

The reciprocal lattice is just as 'real' to the crystallographer or 'solid stater' as the real lattice, as it is the natural space in which to discuss waves in lattices, and to enumerate the available states, as we saw in section 2.2. Formally, the reciprocal lattice b_{i, j, k} is related to the real lattice a_{i, j, k} as

b_k = 2p a_i x a_j/t, where t is the volume of the unit cell, i.e. t = a_i . a_j x a_k. (3.3)

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 b_k 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.

3.3 Brillouin zones and electron scattering in 3D

A soluble problem, given in many places, is the Kronig-Penney model, in which diffraction arises in a 1D chain of d-function potentials. This model can be investigated further via problem 3.4.2. Electron band structures in 2D and 3D can be viewed as arising from diffraction from the lattice planes, or more specifically from the Fourier coefficients of the scattering potential V_g associated with the specific reciprocal lattice vector g, as illustrated graphically in figure 3.1(b).

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 V_g 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.

3.4 Problems relating to this topic

3.4.1: Reciprocal lattices and kinematic diffraction

You may have already understood from undergraduate classes that real and reciprocal lattices really are reciprocal, in the sense that the reciprocal lattice of the reciprocal lattice is the real lattice. Show this to be the case for the face-centered cubic (f.c.c) and body-centered cubic (b.c.c) lattices.

Show, for one of these lattices, that the intensity of diffraction I_g, in the weak scattering (kinematic) limit, is proportional to |V_g|².

3.4.2: Band theory in the Kronig-Penney model

The Kronig-Penney model is instructive, in that it allows one to solve a 1D problem relating to coupled barriers precisely. By reducing the strength of the barrier, one can follow the transition from the single potential well with atomic-like states to tight-binding-like states. Starting from the other limit, one can go from free electrons to nearly free electrons to tight-binding by varying only one parameter.

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.

3.4.3: NFE Band theory near a single Bragg plane

We will go through this example in the lecture, but if time runs out, you should look at it in some detail, and solve the particular example given by Pettifor in section 5.4, page 118-121. The question of the values of the potentials needed to explain real band structures is the next topic, so this reasonably realistic example of aluminium, with V_g = 0.5 eV for the 200 plane, is a good one.

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