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Brillouin Zones

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Contents

Main pages

Aims

Before you start

Introduction

Reciprocal lattice vectors

Brillouin Zone construction

The general case in three dimensions

Zone folding

Examples of Brillouin Zones in Three Dimensions

Summary

Questions

Going further

Additional pages

Bragg planes and Brillouin zone construction

Aims

On completion of this tutorial you should:

Understand how to generate the reciprocal lattice from the real
space lattice

Know how to construct Brillouin Zones in reciprocal space

Be familiar with the Brillouin Zones for some simple two- and
three-dimensional structures

Before you start

Though this package is largely self-contained, some familiarity with
simple crystal lattices is useful. If you want a reminder, look at the
teaching and learning package on Miller indices and
lattice planes.

Introduction

Leon Brillouin (1889-1969) first introduced Brillouin Zones in his work
on the general properties of periodic structures. The concept and
construction of the zones can appear quite abstract, but this is largely a
result of their wide range of useful applications.

Brillouin zones are polyhedra in reciprocal space in crystalline
materials and are the geometrical equivalent of Wigner-Seitz cells in real
space. Physically, Brillouin zone boundaries represent Bragg planes which
reflect (diffract) waves having particular wave vectors so that they cause
constructive interference.

Reciprocal lattice vectors

Every periodic structure has two lattices associated with it. The first
is the real space lattice, and this describes the periodic structure. The
second is the reciprocal lattice, and this determines how the periodic
structure interacts with waves. This section outlines how to find the
basis vectors for the reciprocal lattice from the basis vectors of the
real space lattice.

Reciprocal lattice vectors, K, are defined by the following
condition:

where R is a real space lattice vector. Any real
lattice vector may be expressed in terms of the lattice basis vectors,
a1, a2, a3.

in which the ci are integers. The
condition on the reciprocal lattice vectors may also be expressed as

where n is an integer. This expression can be
satisfied if K is expressed in
terms of the reciprocal lattice basis vectors
bi, which are defined as

Note that b2 and
b3 are given by cyclic permutations of the
expression for b1 . From this expression it
may be seen that the real lattice basis vectors and the reciprocal lattice
basis vectors satisfy the following relation:

where is the Kronecker delta, which takes the value 1 when
i is equal to j, and 0 otherwise. Any reciprocal lattice
vector may then be expressed as a linear sum of these reciprocal basis
vectors:

in which h, k and l are integers.
The set of all K vectors
defines the reciprocal lattice.

Brillouin Zone construction

The reciprocal lattice basis vectors span a vector space that is
commonly referred to as reciprocal space, or often in the context of
quantum mechanics, k space. This section covers the
construction of Brillouin zones in two dimensions.

The first step is to use the real space lattice vectors to find the
reciprocal lattice vectors and construct the reciprocal lattice. One of
the points in the reciprocal lattice is then designated to be the origin.
When constructing Brillouin zones, they are always centred on a reciprocal
lattice point, but it is important to keep in mind that there is nothing
special about this point as each reciprocal lattice point is equivalent
due to translation symmetry.

Draw a line connecting this origin point to one of its nearest
neighbours. This line is a reciprocal lattice vector as it connects two
points in the reciprocal lattice.

Then draw on a perpendicular bisector to the first line.
This perpendicular bisector is a Bragg
Plane.

Add the Bragg Planes corresponding to the other nearest
neighbours.

The locus of points in reciprocal space that have no Bragg
Planes between them and the origin defines the first Brillouin Zone. It is
equivalent to the Wigner-Seitz unit cell of the reciprocal lattice. In the
picture below the first Zone is shaded red.

Now draw on the Bragg Planes corresponding to the next
nearest neighbours.

The second Brillouin Zone is the region of reciprocal space
in which a point has one Bragg Plane between it and the origin. This area
is shaded yellow in the picture below. Note that the areas of the first
and second Brillouin Zones are the same.

The construction can quite rapidly become complicated as you move
beyond the first few zones, and it is important to be systematic so as to
avoid missing out important Bragg Planes. Click the animation below for an
interactive illustration that follows this process to show how to
construct the first six Brillouin Zones for the 2-D square reciprocal
lattice. Use the arrow buttons to navigate forwards and backwards through
the different steps.

2-D square lattice

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

As a further example, click the on the animation below for an
interactive illustration showing how to construct the first six zones for
a 2-D hexagonal reciprocal lattice.

2-D hexagonal lattice

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

The general case in three dimensions

After considering these simple examples, it should hopefully be clear
how to construct Brillouin Zones. Whilst in two dimensions this geometric
method is easy to apply, in three dimensions the lattice cannot be
represented on a piece of paper and in general it is much harder to
picture the shape of the Brillouin Zones beyond the first. This section
considers how the relevant Bragg planes for zone construction may be
generated in a systematic fashion.

In vector notation, the equation of a plane may be written as

In this expression, a is a vector from the origin to a
specified (but arbitrary) point in the plane, r is a
general point in the plane and is the unit vector normal to the plane.

For Brillouin Zones it is convenient to choose a so that it is the
perpendicular vector from the origin to the Bragg plane of interest, i.e.,
a vector of the form

also, the unit normal is then given by

Letting h, k and
l be integers (positive or negative) and excluding the
point where all three are equal to zero, the relevant Bragg Planes may be
generated in a systematic fashion. It is usually sufficient for finding
the first three or four zones to have the di range
between –3 and +3.

Then, given any point in reciprocal space, it may be allocated to a
Brillouin zone by determining the number, N, of Bragg Planes that
lie in between that point and the origin. The point is then in the
(N+1)th Brillouin Zone.

Whether a Bragg plane lies between a general point, r,
and the origin may be determined quite simply by considering the
projection of the position vector of the point in the direction of the
unit normal to the Bragg plane. The scalar product

a.(r –
a)

is positive if the plane is between the point and the
origin, and is negative when the plane does not lie between the point and
the origin.

The main complication when extending to Brillouin Zones beyond the
first zone is that it is very easy to overlook an important Bragg Plane.
This can only really be avoided by being careful and systematic. A useful
test to indicate if a mistake has been made is to check that all the zones
have the same symmetries as the reciprocal lattice itself. For example the
Brillouin Zones for the 2-D Square lattice always have fourfold-rotational
symmetry about the origin. A further useful check is to confirm that the
area (in 2D) or volume (in 3D) of each Brillouin zone is the same.

Some examples which show the Brillouin Zones for common 2-D lattices
may be found by clicking the links below.

2-D
Square lattice to 10th Brillouin zone (pdf file, 0.53 MB)

2-D
Hexagonal lattice to 6th Brillouin zone (pdf file, 0.60
MB)

Zone folding

An important property of the Brillouin Zones is that, because the
reciprocal lattice is periodic, there exists for any point outside the
first zone a unique reciprocal lattice vector that will translate that
point back inside the first zone. Each point in reciprocal space is only
unique up to a reciprocal lattice vector. Each Zone contains every single
physically distinguishable point, and so they all have the same area (in
2-D) or volume (in 3-D).

This is easiest to see by example. The links below are for interactive
illustrations which will show how the first six zones for the 2-D square
and hexagonal lattices can be translated or 'folded' back on top of the
first zone. Use the arrow buttons to navigate through the different
steps.

2-D square Zone folding

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

2-D hexagonal Zone folding

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

Examples of Brillouin Zones in Three Dimensions

The extension to three dimensions is straightforward by the method
outlined in the previous
section. However, it is important to remember not to overlook
important Bragg Planes. There is also a problem of representation, because
the structure of the Zones in 3-D can be quite complicated and hard to
visualise.

It is also important to remember that in 3-D the Zones all have the
same volume, and that the volume corresponding to the 3rd Zone is the
volume between the outer surface of the 2nd Zone and that of the 3rd
Zone.

Representations of the Brillouin Zones corresponding to Simple Cubic
(SC), Body Centred Cubic (BCC), Face Centred Cubic (FCC) lattices are
linked to below. Use the buttons on the left hand side to select which
Zone Surface to view, and use the arrow buttons to rotate the view about
the [001] axis.

Simple Cubic

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

Body Centred Cubic

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

Face Centred Cubic

Note: This animation requires Adobe Flash Player 8 and later, which
can be downloaded here.

Summary

This teaching and learning package has introduced Brillouin zones and
their construction in two and three dimensions.

The relationship between the construction of Brillouin zones and
diffraction from Bragg Planes has been shown by a consideration of the
Bragg equation.

The examples shown in the package itself demonstrate the principles of
the construction in two dimensions. In three dimensions the principles of
the construction are the same, but the Brillouin Zones are hard to
visualise. Rotating the shapes of Brillouin Zones about a cube axis as in
the examples we have given in this package helps to improve the
visualisation in comparison with simple line diagrams found in
textbooks.

Please follow this link if you would like to provide a short review for this TLP

Questions

Quick questions

You should be able to answer these questions without too much
difficulty after studying this TLP. If not, then you should go through it
again!

If a real space vector has length L, what is the magnitude
of the corresponding reciprocal lattice vector as defined in this TLP?

a

b

c

d

Which of the following statements about the Brillouin zones for a
particular reciprocal lattice are correct? (select yes for true and no
for false for each statement)

Yes

No

a

They all have the same area / volume.

Yes

No

b

They all have the same shape.

Yes

No

c

They all have the same symmetry.

Deeper questions

The following questions require some thought and reaching the answer
may require you to think beyond the contents of this TLP.

By either printing out the lattice in the file linked to here (pdf), or by drawing out a rectangular
lattice with the ratio between the short and long sides of 4:5. Use the
method outlined earlier to construct the first three Brillouin zones,
being careful not to miss any relevant Bragg planes.

By either printing out the lattice in the file linked to here (pdf), or by drawing out a
parallelogram lattice with the ratio between the short and long sides of
4:5 and an angle between the sides of about 75 degrees. Use the method
outlined earlier to construct the first three Brillouin zones, being
careful not to miss an relevant Bragg planes.

Confirm that when expressed in a Cartesian basis, possible lattice
vectors of the primitive unit cell (i.e., the unit cell with one lattice
point per unit cell) for a FCC lattice of unit side length are , and .

Confirm that when expressed in a Cartesian basis, possible lattice
vectors of the primitive unit cell for a BCC lattice of unit side length
are , and .

Using the definition of the reciprocal lattice basis vectors given in
the TLP, show that the reciprocal lattice of a FCC lattice is a BCC
lattice and vice versa.

For a conventional BCC unit cell with two lattice points per unit
cell of side L:

How many nearest neighbours does each point have?

What are their coordinates, and what distance away are they?

How many next nearest neighbours are there?

What are their coordinates and what distance away are
they?

Look at the picture below of the first Brillouin zone for a real
space FCC lattice. Using your answers to questions 5 and 6, try to index
the planes labelled (a) and (b).

Going further

Books

There a number of textbooks which discuss in varying levels of detail
Brillouin Zones and their construction and application. The classic text
on Brillouin Zones is

Wave Propagation in Periodic Structures, L. Brillouin (Dover 1953,
reprinted 2003)

Other texts are also useful, such as:

Solid State Physics, J.S. Blakemore (Saunders 1974)

Electronic Properties of Materials, R.E. Hummel (Springer-Verlag
1992)

Introduction to Solid State Physics, C. Kittel (Wiley 1996)

Solid State Physics, N.W. Ashcroft and N. D. Mermin. (Harcourt Brace
1976)

Websites

Wikipedia - Brillouin zones
This wikipedia page has
a number of useful external links which should help to reinforce and
confirm the concepts introduced in this teaching and learning
package.

Further links can be readily found by typing “Brillouin Zones” into
Google and searching the web. Examples of such links are:

http://www.tf.uni-kiel.de/matwis/amat/semi_en/kap_2/illustr/i2_1_3.html

http://britneyspears.ac/physics/crystals/wcrystals.htm

Bragg planes and Brillouin zone construction

The construction of Bragg Planes in the context of Brillouin zones can
be understood by considering Bragg’s Law

λ = 2dsinθ

where θ is the angle between the incident radiation and the diffracting
plane, λ is the wavelength of the incident radiation and d is the
interplanar spacing of the diffracting planes. (Further information on Bragg’s Law is found in the TLP on X-ray diffraction).

In reciprocal space this can be expressed in the form

k' – k =
g

where k is the wave vector of the incident wave of
magnitude 2π/λ, k' is the wave vector of the diffracted
wave, also of magnitude 2π/λ, and g is a reciprocal
lattice vector of magnitude 2π/d:

This can be shown graphically using the Ewald sphere construction:

Here 000 is the origin of the reciprocal lattice and O is the centre of
the sphere of radius . If the angle subtended at O between 000 and G on the
above diagram is 2θ, simple geometry shows that

which can be rearranged into the more familiar form

i.e., the Bragg equation.

The equation

k' – k =
g

can be rearranged in the form

k' = k +
g

Hence

k'.k' = (k +
g).(k + g) = k.k +
g.g + 2k.g

Since k'.k' = k.k because diffraction
is an elastic scattering event, it follows that

g.g + 2k.g = 0

To construct the Bragg Plane, it is convenient to replace
k by -k in this equation so that both
k and g begin at the origin, 000, of the
reciprocal lattice. Hence, the equation can be written in the form

Constructing the plane normal to g at the
midpoint, , then means that any vector
k drawn from the origin, 000, to a position on this plane
satisfies the Bragg diffraction condition:

Academic consultant: Kevin Knowles
Content
development: Chris Dunleavy
Web development:
Lianne Sallows

DoITPoMS is funded
by the UK Centre for
Materials Education and the Department of Materials Science and
Metallurgy, University of Cambridge

Additional support for the development of this TLP came from the
Worshipful Company of Armourers and Brasiers'

© University of Cambridge
DoITPoMS, Department of Materials Science and
Metallurgy, University of
Cambridge
Information provided by [email-masked].

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is a print-friendly version of the TLP at: http://www.doitpoms.ac.uk/tlplib/brillouin_zones