Recipr Lattice

7/27/2019 Recipr Lattice

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3. RECIPROCAL LATTICE.

Aim: To become familiar with the concept of the reciprocal lattice. It is very important andfrequently used in the rest of the course to develop models and describe the physical properties ofsolids.

A. A description of crystal structures (H p. 16-17).

First we treat an alternative description of a lattice is in terms of a set of parallel identical planes.They are defined so that every lattice point belongs to one of these planes. The lattice can actually

be described with an infinite number of such sets of planes. This description may seem artificialbut we will see later that it is a natural way to interpret diffraction experiments in order tocharacterize the crystal structure. The figure below shows a 2-D lattice with three different sets of1-D planes drawn.

The orientation of a plane can be specified by three integers, the so called Miller indices. Theintercepts of one plane with the coordinate axes (the translation vectors of the lattice) aredetermined and normalized by division by the corresponding lattice constants. Then the reciprocalsare taken and reduced to the set of smallest integers. In the figure above, the Miller indices for thethree sets of planes are given as an example. Note also that the planes in the sets with lowerMiller indices are spaced further apart (shaded areas depict the interplanar spacing).

The Miller indices of a single plane or a set of planes are denoted (ijk). All planes with the samesymmetry are denoted {ijk}. A direction in a crystal is denoted [uvw] and all equivalent directions(same symmetry) are . In cubic crystals [ijk] is orthogonal to (ijk).


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Example: Determine the Miller indices of the set of lines in the two-dimensional case illustratedbelow. Note that every lattice point will belong to one of the lines in the infinite set. Where doesthe plane closest to the origin cut the axes?

B. Real lattices and reciprocal lattices

Lattices are periodic structures and position-dependent physical properties that depend on thestructural arrangement of the atoms are also periodic. For example the electron density in a solid is

a function of position vectorr, and its periodicity can be expressed as (r+T)=(r), where T is atranslation vector of the lattice.

The periodicity means that the lattice and physical properties associated with it can be Fouriertransformed. Since space is three dimensional, the Fourier analysis transforms it to a three-

dimensional reciprocal space. Physical properties are commonly described not as a function ofr,

but instead as a function of wave vector (spatial frequency or k-vector) k. This is analogous tothe familiar Fourier transformation of a time-dependent function into a dependence on (temporal)frequency. The lattice structure of real space implies that there is a lattice structure, the reciprocallattice, also in the reciprocal space.

C. Fourier transformation (H p. 20-22).

We start with a repetition of the basics of the Fourier transformation in one and three dimensions.It is very convenient to specify a periodic function with one or a few Fourier components insteadof specifying its value throughout real space. We write the Fourier series in the complex form (H,eq. 2.21), which can easily be generalized to the three dimensional case (eq. 2.24). Waves


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propagating through the lattice with a certain periodicity can be represented by points in reciprocalspace.

D. Reciprocal lattice vectors (H p. 19-20).

The reciprocal lattice points are described by the reciprocal lattice vectors, starting from the origin.

G = mb1 + nb2 + o

b3, where the coefficients are integers and the bi are the primitive

translation vectors of the reciprocal lattice. From the definition of the bi (H p. 19-20), it can be

shown that exp (iGT) = 1. This condition is actually taken as the starting point by Hoffman; it isrequired by the condition that the Fourier series must have the periodicity of the lattice. Hence, theFourier series of a function with the periodicity of the lattice can only contain the lattice vectors

(spatial frequencies) G.

Since the Miller indices of sets of parallel lattice planes in real space are integers, we can interpretthe coefficients (mno) as Miller indices (ijk). This leads to a physical relation between sets of

planes in real space and reciprocal lattice vectors. Hence we write G = i b1

+ j b2

+ kb3

. The

reciprocal lattice vectors are labelled with Miller indices Gijk. Each point in reciprocal space

comes from a set of crystal planes in real space (with some exceptions, for example (n00), (nn0)and (nnn) with n>1 in cubic systems; nevertheless they occur in the Fourier series and arenecessary for a correct physical description).

Consider the (ijk) plane closest to the origin. It cuts the coordinate axes in a1/i, a2/j and a3/k. A

triangular section of the plane has these points in the corners. The sides of the triangle are given bythe vectors (a1/i) - (a2/j) and analogously for the other two sides. Now the scalar product ofGijk

with any of these sides is easily seen to be zero. Hence the vectorGijk is directed normal to the

(ijk) planes. The distance between two lattice planes is given by the projection of for example a1/ionto the unit vector normal to the planes, i.e. dijk= (a1/i)nijk, where dijkis the interplanar

distance between two (ijk)-planes. The unit normal is just Gjk divided by its length. Hence the

magnitude of the reciprocal lattice vector is given by 2/ dijk.

E. Brillouin zones (H p. 46-47, 51-53).

The (first) Brillouin zone is defined as the Wigner-Seitz cell of the reciprocal lattice. It isconstructed exactly in the same way as the Wigner-Seitz cell of the real lattice. The zone

boundaries are planes normal to each of the Gs, at half the distance to the nearest neighbours

points of the reciprocal lattice. The values of the k-vector at the zone boundaries can be obtainedfrom the simple relation

,2/2G=Gk

which follows because the projection of a k-vector from the origin to a zone boundary must beG/2. We will have use for this equation in the next section on diffraction.

Examples: The Brillouin zone in one dimension (H, p. 47) consists of the interval between /a

and /a. It is also easy to draw Brillouin zones of two dimensional lattices. In three

dimensions things become a little more complicated, though:


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The reciprocal lattices to a simple cubic lattice is also simple cubic but with side

length of the cube 2/a. The reciporocal lattices of b) the fcc lattice is bcc (H p. 52),c) the bcc lattice is fcc , d) the hexagonal lattice is also hexagonal.

Brillouin zones of the three latter structures are illustrated below and can also be found in PhysicsHandbook.

It is also customary to define higher Brillouin zones. The zone boundaries are planes normal to

each G at its midpoint. They partition reciprocal space into a number of fragments. Thesefragments are labelled with the number of a higher Brillouin zone. Each of these zones (the second,the third and so on) consist of several fragments, but the volume of a Brillouin zone is always thesame. This might look like a very artificial formalism, but it is actually used in the discussion ofelectronic structure of structure, the so called energy bands. The higher zones are illustrated belowfor a two-dimensional quadratic lattice.


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