Reciprocal lattice.pdf

The The Reci Recipr proc oca al Latti attice ce Lecture 2 Summary

The The reci recipr proc ocal al latt lattic ice e • Aim: To become familiar with the concept of the reciprocal lattice. It is very important and frequently used in the rest of the course to develop models and describe the physical properties of solids. • 1. Another description of crystal structure (H p. 1617). • Firs Firstt we trea treatt an alte altern rnat ativ ive e des descr crip ipti tion on of of a lat latti tice ce 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 artificial but we will see later that it is a natural way to interpret diffraction experiments in order to characterize the crystal structure.

Lattice planes • Set of parallel planes • Every lattice point belongs to one plane in the set. • There is an infinite number  of such families of planes that can be associated with a Bravais lattice. • Three such families of  planes are drawn in the figure. • This is an alternative description of a lattice.

Miller indices • The orientation of a plane can be specified by three integers, the so called Miller indices. In the figure above, the Miller indices for the three sets of ”planes” are given as an example. Note also that the planes in the sets with lower Miller 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 same symmetry 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). • See next how to determine the Miller indices!

Labelling crystal planes (Miller indices) 1. determine the intercepts with the crystallographic axes in units of the lattice vectors 2. take the reciprocal of each number 

step 1: (2,1,2) step 2: ((1/2),1,(1/2)) step 3: (1,2,1)

3. reduce the numbers to the smallest set of integers having the same ratio. These are then called the Miller  indices.

Example

Exercise: • Determine the Miller indices of  the indicated family of lattice planes. • In which points does the plane closest to the origin cut the axes?

2. Real lattices and reciprocal lattices •

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Lattices are periodic structures and position-dependent physical properties that depend on the structural arrangement of the atoms are also periodic. For example the electron density in a solid is a function of position vector r , and its periodicity can be expressed as ρ(r +T)=ρ(r ), where T is a translation vector of the lattice. The periodicity means that the lattice and physical properties associated with it can be Fourier transformed. Since space is three dimensional, the Fourier analysis transforms it to a threedimensional reciprocal space. Physical properties are commonly described not as a function of r , but instead as a function of wave vector (“spatial frequency” or kvector) k. This is analogous to the 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 reciprocal lattice, also in the reciprocal space.

Periodicity of physical properties 1D chain of atoms

example of charge density 1

example of charge density 2

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Greatly simplifies the description of lattice-periodic functions (charge density, one-electron potential...).

3. 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 instead of 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 propagating through the lattice with a certain periodicity can be represented by points in reciprocal space.

Fourier analysis: 1dim example: charge density in the chain

Fourier series

alternatively

Fourier components of the lattice

Reciprocal lattice 1D

3D

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

Lattice waves real space

reciprocal space

b a 2π/a (0,0) 2π/b

4. Reciprocal lattice vectors (H p. 19-20). • The reciprocal lattice points are described by the reciprocal lattice vectors, starting from the origin. • G = m´ b1 + n´ b2 + 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, it can be shown that: • exp (iGR) = 1. • This condition is actually taken as the starting point by Hoffman; it is required by the condition that the Fourier  series must have the periodicity of the lattice. • Hence, the Fourier series of a function with the periodicity of the lattice can only contain the lattice vectors (spatial frequencies) G.

The reciprocal lattice

|a1|=a |a2|=b |b1|=2π/a |b2|=2π/b example 1: in two dimensions

The reciprocal lattice example 2:

The fcc lattice is the reciprocal of the bcc lattice and vice versa.

Reciprocal lattice and Miller indices • Since the Miller indices of sets of parallel lattice planes in real space are integers, we can interpret the coefficients (m´n´o´) 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 + k b3. 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 are necessary 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 by the vectors (a1/i) - (a2/j) and analogously for the other two sides. Now the scalar  product of Gijk with any of these sides is easily seen to be zero. • Hence the vector Gijk is directed normal to the (ijk) planes. The distance between two lattice planes is given by the projection of for example a1/i onto the unit vector  normal to the planes, i.e. d ijk = (a1/i)•nijk, where dijk is the interplanar distance between two (ijk)-planes. • The unit normal is just Gíjk divided by its length. Hence the magnitude of the reciprocal lattice vector is given by 2π/ dijk.

5. Brillouin Zones (H p. 46-47, 51-53). • The (first) Brillouin zone is defined as the Wigner-Seitz cell of the reciprocal lattice. It is constructed exactly in the same way as the Wigner-Seitz cell of the real lattice. The zone boundaries are planes normal to each of the G’s, 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 obtained from the simple relation • k.G = G2/2 • This follows because the projection of a k-vector from the origin to a zone boundary must be G/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 –π/a and π/a. It is also easy to draw Brillouin zones of two dimensional lattices. • In three dimensions things become a little more complicated, though: • The reciprocal lattices to a simple cubic lattice is also simple cubic but with side length of the cube 2π/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 .

Examples of Brillouin Zones

For the most common lattice structures.

Higher Brillouin Zones •

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The zone boundaries are planes normal to each G at its midpoint. They partition reciprocal space into a number  of fragments. These fragments 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 the same. This might look like a very artificial formalism, but it is actually used in the discussion of electronic structure of  structure, the so called energy bands.