reciprocal lattice.pdf
TRANSCRIPT
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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
real space reciprocal space
a
b
2 /a
2 /b(0,0)
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Lattice waves
real space reciprocal space
a
b
2 /a
2 /b(0,0)
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Lattice waves
real space reciprocal space
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.