Reciprocal lattice.pdf

7/27/2019 Reciprocal lattice.pdf

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The Reciprocal Lattice

Lecture 2

Summary


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The reciprocal lattice Aim: To become familiar with the concept of the

reciprocal lattice. It is very important and frequently usedin the rest of the course to develop models and describethe physical properties of solids.

1. Another description of crystal structure (H p. 16-17).

First we treat an alternative description of a lattice interms 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 withan infinite number of such sets of planes. Thisdescription may seem artificial but we will see later that it

is a natural way to interpret diffraction experiments inorder to characterize the crystal structure.


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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 planesthat can be associated witha Bravais lattice.

Three such families of

planes are drawn in thefigure. This is an alternative

description of a lattice.


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Labelling crystal planes

(Miller indices)1. determine the intercepts

with the crystallographicaxes in units of the latticevectors

2. take the reciprocal of eachnumber

3. reduce the numbers to thesmallest set of integershaving the same ratio. These

are then called the Miller indices.

step 1: (2,1,2)

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

step 3: (1,2,1)


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Example


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Exercise:

Determine theMiller indices of the indicatedfamily of latticeplanes.

In which pointsdoes the planeclosest to the

origin cut theaxes?


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2. Real lattices and reciprocal lattices

Lattices are periodic structures and position-dependent physicalproperties that depend on the structural arrangement of the atomsare also periodic. For example the electron density in a solid is afunction of position vector r , and its periodicity can be expressed as (r +T)= (r ), where T is a translation vector of the lattice.

The periodicity means that the lattice and physical propertiesassociated with it can be Fourier transformed. 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 of r ,

but instead as a function of wave vector (spatial frequency or k-vector) 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 latticestructure, the reciprocal lattice , also in the reciprocal space.


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Periodicity of physical properties

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

1D chain of atoms

example of charge density 1

example of charge density 2


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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 veryconvenient to specify a periodic function with one or afew Fourier components instead of specifying its valuethroughout real space.

We write the Fourier series in the complex form (H, eq.2.21), which can easily be generalized to the threedimensional case (eq. 2.24).

Waves propagating through the lattice with a certainperiodicity can be represented by points in reciprocalspace.


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Fourier analysis: 1dim

example: charge density in the chain

Fourier series

alternatively


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Fourier components of the lattice


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Reciprocal lattice

1D

3D


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Lattice waves

real space reciprocal space

a

b

2 /a

2 /b(0,0)


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Lattice wavesreal space reciprocal space

a

b

2 /a

2 /b(0,0)


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Lattice waves


a

b

2 /a

2 /b(0,0)


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Lattice waves


a

b

2 /a

2 /b(0,0)


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Lattice waves


a

b

2 /a

2 /b(0,0)


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

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

G = m b 1 + n b 2 + o b 3, where the coefficients areintegers and the b i are the primitive translation vectors of the reciprocal lattice.

From the definition of the b i, it can be shown that: exp (i GR ) = 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 theperiodicity of the lattice can only contain the latticevectors (spatial frequencies) G.


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The reciprocal lattice

example 1: in two dimensions

|a 1 |=a

|a 2 |=b

|b 2 |=2 /b

|b 1|=2 /a


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The reciprocal latticeexample 2:

The fcc lattice is the reciprocal of the bcc lattice andvice versa.


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Reciprocal lattice and Miller indices

Since the Miller indices of sets of parallel lattice planes inreal space are integers, we can interpret the coefficients(mno) as Miller indices (ijk). This leads to a physicalrelation between sets of planes in real space andreciprocal lattice vectors.

Hence we write G = i b 1 + j b 2 + k b 3. The reciprocallattice vectors are labelled with Miller indices G ijk . 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 cubicsystems; nevertheless they occur in the Fourier seriesand are necessary for a correct physical description).


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Consider the (ijk) plane closest to the origin. It cuts thecoordinate axes in a 1/i, a 2/j and a 3/k. A triangular sectionof the plane has these points in the corners. The sides of the triangle are given by the vectors ( a 1/i) - (a 2/j) andanalogously for the other two sides. Now the scalar product of G ijk with any of these sides is easily seen tobe zero.

Hence the vector G ijk is directed normal to the (ijk)planes. The distance between two lattice planes is givenby the projection of for example a 1/i onto the unit vector normal to the planes, i.e. d ijk = (a 1/i)n ijk , where d ijk is theinterplanar distance between two (ijk)-planes.

The unit normal is just G jk divided by its length. Hencethe magnitude of the reciprocal lattice vector is given by2 / d ijk.


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Examples

The Brillouin zone in one dimension (H, p. 47)consists of the interval between /a and /a. Itis also easy to draw Brillouin zones of twodimensional lattices.

In three dimensions things become a little more

complicated, though: The reciprocal lattices to a simple cubic lattice isalso 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) thehexagonal lattice is also hexagonal .


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Examples of Brillouin Zones

For the most common

lattice structures.


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Higher Brillouin Zones

The zone boundaries areplanes normal to each G at itsmidpoint. They partitionreciprocal space into a number of fragments.

These fragments are labelledwith the number of a higher Brillouin zone. Each of thesezones (the second, the thirdand so on) consist of severalfragments, but the volume of aBrillouin zone is always thesame.

This might look like a veryartificial formalism, but it isactually used in the discussionof electronic structure of

structure, the so called energybands.

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Reciprocal lattice.pdf