Structure of SolidsObjectives

By the end of this section you should be able to:

• Construct a reciprocal lattice• Interpret points in reciprocal space• Determine and understand the Brillouin zone

Reciprocal Space

Also called Fourier space, k (wavevector)-space, or momentum space in contrast to real space or direct space.The reciprocal lattice is composed of all points lying at positions from the origin. Thus, there is one point in the reciprocal lattice for each set of planes (hkl) in the real-space lattice.

This abstraction seems unnecessary. Why do we care?

hklK

1. The reciprocal lattice simplifies the interpretation of x-ray diffraction from crystals

2. The reciprocal lattice facilitates the calculation of wave propagation in crystals (lattice vibrations, electron waves, etc.)

Why Use The Reciprocal Space?

A diffraction pattern is not a direct representation of the crystal lattice

The diffraction pattern is a representation of the reciprocal lattice

b2

b1

The Reciprocal Lattice

Crystal planes (hkl) in the real-space (or the direct lattice) are characterized by the normal vector and dhkl interplanar spacing

Practice has shown the usefulness of defining a different lattice in reciprocal space whose points lie at positions given by the vectors

hkln̂

x

y

z

hkld

hkln̂

hkl

hklhkl d

nK

ˆ2

This vector has magnitude 2/dhkl, which is a

reciprocal distance

[hkl]

Definition of the Reciprocal Lattice

Suppose K can be decomposed into reciprocal lattice vectors:321 blbkbhK

(h, k, l integers)

mRK n 2

ijji 2ab

The basis vectors bi define a reciprocal lattice: - for every real lattice there’s a reciprocal lattice- reciprocal lattice vector b1 is perpendicular to plane defined by a2 and a3

Note: a has dimensions of length, b has dimensions of

length-1

321

321 2

aaa

aa

b

+ cyclic permutations

321 aaa is volume of unit cell

Definition of a’s are not unique, but the volume is.

Rn = n1 a1 + n2 a2 + n3 a3 (lattice vectors a1,a2,a3)

2D Reciprocal Lattice

Khkl is perpendicular to (hkl) plane

Real lattice planes (hk0) K in reciprocal space

Identify these planes

a1

a2

A point in the reciprocal lattice corresponds to a set of planes planes (hkl) in the real-space lattice.

Magnitude of K is inversely proportional to distance between (hkl) planes

a

b

2π/a

2π/b(0,0)

Another Similar View: Lattice waves

real space reciprocal space

There is always a (0,0) point in reciprocal space.How do you expect the reciprocal lattice to look?

a

b

2π/a

2π/b(0,0)

Lattice waves

real space reciprocal space

Red and blue represent different amplitudes of the waves.

Lattice waves

real space reciprocal space

a

b

2π/a

2π/b(0,0)

Note that the vertical planes in real space correspond to points along the horizontal axis in reciprocal space.

a

b

2π/a

2π/b(0,0)

Lattice waves

real space reciprocal space

a

b

2π/a

2π/b(0,0)

Lattice waves

real space reciprocal space

The real horizontal planes relate to points along R.S. vertical.In 2D, reciprocal vectors are perpendicular to opposite axis.

a

b

2π/a

2π/b(0,0)

Lattice waves

real space reciprocal space

(11) plane

Examples of Image Fourier Transforms

Brightest side points relating to the frequency of the stripes

Real Image

Fourier Transform

Examples of Image Fourier Transforms

http://www.users.csbsju.edu/~frioux/diffraction/crystal-rot.pdf

Group: Reciprocal Lattice

• Determine the reciprocal lattice for:

a1

a2

Real space Fourier (reciprocal)

space

b1

b2

Group: Find the reciprocal lattice vectors of BCC

• The primitive lattice vectors for BCC are:

• The volume of the primitive cell is ½ a3(2 pts./unit cell)• So, the primitive translation vectors in reciprocal space

are:Good websites:http://newton.umsl.edu/run//nano/reltutor2.htmlhttp://matter.org.uk/diffraction/geometry/plane_reciprocal_lattices.htm

321

321 2

aaa

aa

b

Reciprocal Lattices to SC, FCC and BCCPrimitive Direct lattice Reciprocal latticeVolume of RL

SC

BCC

FCC

za

ya

xa

a

a

a

3

2

1

xza

zya

yxa

a

a

a

21

3

21

2

21

1

zyxa

zyxa

zyxa

a

a

a

21

3

21

2

21

1

zb

yb

xb

a

a

a

/2

/2

/2

3

2

1

yxb

zxb

zyb

a

a

a

23

22

21

zyxb

zyxb

zyxb

a

a

a

23

22

21

3/2 a

3/24 a

3/22 a

Direct Reciprocal

Simple cubic Simple cubic

bcc fcc

fcc bcc

We will come back to this if

time.

Volume of BZ

In general the volume of the BZ is equal to

(2 )3

Volume of real space primitive lattice

Discuss the reciprocal lattice in 1D

a

Weigner Seitz Cell: Smallest space enclosed when intersecting the midpoint to the neighboring lattice

points.

Why don’t we include second neighbors here (do in 2D/3D)?

Real lattice

Reciprocal lattice

k

0 2/a 4/a-2/a-4/a-6/a

x

-/a /a What is the range of unique environments?

The Brillouin Zone

• Is defined as the Wigner-Seitz primitive cell in the reciprocal lattice (smallest volume in RL)

• Its construction exhibits all the wavevectors k which can be Bragg-reflected by the crystal

Reciprocal lattice

k

0 2/a 4/a-2/a-4/a-6/a

-/a /a

Group: Draw the 1st Brillouin Zone of a sheet of graphene

Real Space

2-atom basis

a2

a1

a2*

a1*

Wigner-Seitz Unit Cell of Reciprocal Lattice= First Brillouin zone

The same perpendicular bisector logic applies in 3D

Group: PoloniumConsider simple cubic polonium, Po, which is the closest thing we can get to a 1D chain in 3 dimensions.

(a) Taking a Po atom as a lattice point, construct the Wigner-Seitz cell of polonium in real space. What is it’s volume?

(b) Work out the lengths and directions of the lattice translation vectors for the lattice which is reciprocal to the real-space Po lattice.

(c) The first Brillouin Zone is defined to be the Wigner-Seitz primitive cell of the reciprocal lattice. Sketch the first Brillouin Zone of Po.

(d) Show that the volume of the first Brillouin Zone is (2)3/V , where V is the volume of the real space primitive unit cell.

Square Lattice(on board)

Introduction of Higher Order BZs

Group: Determine the shape of the BZ of the FCC Lattice

FCC Primitive and Conventional Unit Cells

How many sides will it have and along what directions?

SC BCC FCC

# of nearest neighbors 6 8 12

Nearest-neighbor distance a ½ a 3 a/2

# of second neighbors 12 6 6

Second neighbor distance a2 a a

WS zone and BZLattice Real Space Lattice K-space

bcc WS cell Bcc BZ (fcc lattice in K-space)

fcc WS cell fcc BZ (bcc lattice in K-space)

The WS cell of bcc lattice in real space transforms to a Brillouin zone in a fcc lattice in reciprocal space while the WS cell of a fcc lattice transforms to a Brillouin zone of a bcc lattice in reciprocal space.

Nomenclature

Directions are chosen that lead aong special symmetry points. These points are labeled according to the following rules:

• Points (and lines) inside the Brillouin zone are denoted with Greek letters.

• Points on the surface of the Brillouin zone with Roman letters.

• The center of the Wigner-Seitz cell is always denoted by a G

Usually, it is sufficient to know the energy En(k) curves - the dispersion relations - along the major directions.

Ener

gy o

r Fre

quen

cy

Direction along BZ

Brillouin Zones in 3Dfcc

hcp

•The BZ reflects lattice symmetry•Construction leads to primitive unit cell in rec. space

bcc

Note: fcc lattice in reciprocal space is a bcc lattice

Note: bcc lattice in reciprocal space is a fcc lattice

Brillouin Zone of Silicon

Points of symmetry on the BZ are important (e.g. determining

bandstructure). Electrons in semiconductors are

perturbed by the potential of the crystal, which varies across unit cell.

Symbol DescriptionΓ Center of the Brillouin zone

Simple CubicM Center of an edgeR Corner pointX Center of a face

FCC

K Middle of an edge joining two hexagonal faces

L Center of a hexagonal face C6

U Middle of an edge joining a hexagonal and a square face

W Corner pointX Center of a square face C4

BCC

H Corner point joining 4 edges

N Center of a faceP Corner point joining 3 edges

Dirac cones

Brillouin zone representations of

grapheneReal space

Now that we are starting to understand reciprocal space, it’s time to take advantage of it.

X-Ray Diffraction (XRD)Why do we use it?

Reciprocal Lattices to SC, FCC and BCCPrimitive Direct lattice Reciprocal latticeVolume of RL

SC

BCC

FCC

za

ya

xa

a

a

a

3

2

1

xza

zya

yxa

a

a

a

21

3

21

2

21

1

zyxa

zyxa

zyxa

a

a

a

21

3

21

2

21

1

zb

yb

xb

a

a

a

/2

/2

/2

3

2

1

yxb

zxb

zyb

a

a

a

23

22

21

zyxb

zyxb

zyxb

a

a

a

23

22

21

3/2 a

3/24 a

3/22 a

Direct Reciprocal

Simple cubic Simple cubic

bcc fcc

fcc bcc

Back to this

NomenclatureWe use the following nomenclature: (red for fcc, blue for bcc):

The intersection point with the [100] direction is called X (H). The line G—X is called D.

The intersection point with the [110] direction is called K (N). The line G—K is called S.

The intersection point with the [111] direction is called L (P). The line G—L is called L.

Brillouin Zone for fcc is bccand vice versa.

Extra SlidesAlternative Approaches

If you already understand reciprocal lattices, these slides might just confuse you. But, they can help if you are lost.

Construction of the Reciprocal Lattice

1. Identify the basic planes in the direct space lattice, i.e. (001), (010), and (001).

2. Draw normals to these planes from the origin.

3. Note that distances from the origin along these normals proportional to the inverse of the distance from the origin to the direct space planes

Above a monoclinic direct space lattice is transformed (the b-axis is perpendicular to the

page). Note that the reciprocal lattice in the last panel is also monoclinic with * equal to 180°−.

The symmetry system of the reciprocal lattice is the same as the direct lattice.

Real space Fourier (reciprocal) space

Reciprocal lattice (Similar)

Consider the two dimensional direct lattice shown below. It is defined by the real vectors a and b, and the angle g. The spacings of the (100) and (010) planes (i.e. d100 and d010) are shown.

The reciprocal lattice has reciprocal vectors a* and b*, separated by the angle g*. a* will be perpendicular to the (100) planes, and equal in magnitude to the inverse of d100. Similarly, b* will be perpendicular to the (010) planes and equal in magnitude to the inverse of d010. Hence g and g* will sum to 180º.

Reciprocal Lattice

(01)

(10)(11)

(21)

10 20

11

221202

01 21

00

The reciprocal lattice has an origin!

1a

2a

1a1

1a

*11g *

21g*b2

*b1

1020

11

2212

02

01

21

00

(01)

(10)(11)

(21)

1a

2a

*b2

*b1

1a

(01)

(10)(11)

(21) Note perpendicularity of various vectors