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Chapter 2 Reciprocal Lattice

An important concept for analyzing periodic structures

• Theory of crystal diffraction of x-rays, neutrons, and electrons.

Where are the diffraction maximum? What is the intensity?

• Abstract study of functions with the periodicity of a Bravais latticeFourier transformation. Reciprocal lattice is closely related to Fourier transformation

• Consideration of how momentum conservation will be modified in periodic potential

Reasons for introducing reciprocal lattice

Basic knowledge: rkie

rr⋅ is a plane wave

( ) ( )rkirkerki rrrrrr

⋅+⋅=⋅sincos

Wave vector nk ˆ2

λ

π=

r

a periodic function

2

One dimensional reciprocal lattice

)()( xfaxf =+

)(xf

∑

∑

∑

>

−

>

−−

>

−+++=

−

++

+=

++=

0

22

0

0

2222

0

0

0

)22

()22

(

22

)2

sin()2

cos()(

n

a

nxi

nna

nxi

nn

n

a

nxi

a

nxi

n

a

nxi

a

nxi

n

n

nn

ei

sce

i

scf

i

ees

eecf

a

nxs

a

nxcfxf

ππ

ππππ

ππ

One dimensional periodic function

na na−0aRedefine coefficients

∑∑ ==n

iGx

n

n

a

nxi

n eaeaxf

π2

)(

We have

n runs through all integers

*

nn aa −= Constraint that ensures f(x) is a real function

a

nG

π2= is one dimensional reciprocal lattice

Expand f(x) into Fourier series

The set of discrete points

f(x) is expanded into a series of plane waves, with a set of wave vectors G’s.

Each plane wave preserve the periodicity of f(x).iGxe

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• Reciprocal lattice points of a Bravais lattice indicate the allowed terms in the Fourier series of a function

with the same periodicity as the Bravais lattice

•But reciprocal lattice does not indicate the magnitude of the each Fourier term an, which depends on the exact

form of f(x).

• Reciprocal lattice depends on the Bravais lattice, but does not depend on the particular form of f(x), as long as

f(x) has the periodicity of the Bravais lattice

∫∫−

−

==a

iGxa

a

nxi

n exfdxa

exfdxa

a00

2

)(1

)(1

π

• Reciprocal space is Fourier space

real space

reciprocal space

Expand to three dimensional case:

Three dimensional periodic function )(rfr

that satisfies )()( Rrfrfrrr

+=

Fourier expansion: ∑ ⋅=G

rGi

Gearf

r

rr

rr)( ∫

⋅−=cell

rGi

c

GerfdV

Va

rr

rr)(

1

Volume of a primitive cell

0a1+a1−a

2+a2−a

4

General definition of reciprocal lattice

Rr

rGie

rr⋅

Gv

Bravais lattice

a plane wave

The set of all wave vectors that yield plane waves with

the periodicity of a given Bravais lattice is known as its

reciprocal lattice

rGie

rr⋅

Rr

Analytically, the definition is expressed as rGiRrGiee

rrrrr⋅+⋅ =)(

The set of satisfyingGv

1=⋅RGie

rr

, for all in a Bravais latticeRr

• A reciprocal lattice is defined with reference to a particular Bravais lattice

• A set of vectors satisfying is called a reciprocal lattice only if the set of is a Bravais lattice

•Reciprocal lattice is a Bravais lattice (proven in the following)

Gv

1=⋅RGie

rr

Rr

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Construction of a reciprocal lattice

and Proof that is a Bravais lattice

)(2

321

32

1aaa

aab rrr

rrr

×⋅

×= π )(

2321

13

2aaa

aab rrr

rrr

×⋅

×= π

)(2

321

21

3aaa

aab rrr

rrr

×⋅

×= π

1ar

2ar

3ar

Are primitive vectors of a Bravais lattice

The reciprocal lattice can be generated by the primitive vectors1

br

2br

3br

ijji ab πδ2=⋅rr

Apparently

332211bvbvbvGrrrr

++=Reciprocal lattice vector

An arbitrary vector in reciprocal space can be written as a linear combination of

332211bgbgbggrrrr

++=

332211 anananRrrrr

++=

ig

To qualify for a reciprocal lattice, 1=⋅Rgie

rr

For all Rr

has to be integers

vi is a integer

Apparently, it is also a Bravais lattice with as its primitive vectors

)( 321 aaaVrrr

×⋅=

volume of primitive cell

ibr

Gr

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The reciprocal of the reciprocal lattice

The reciprocal lattice is a Bravais lattice, one can construct its reciprocal lattice, which turns out to

be nothing but the original direct lattice.

Rr

Gr

1=⋅RGie

rr

Direct lattice

Reciprocal lattice

Look for the reciprocal of the reciprocal lattice: Kr

which satisfies 1=⋅GKie

rr

The set of vectors is a subset of the set of vectors Rr

Kr

KRrr

⊆

For any vector that does not belong to the direct lattice R:

332211axaxaxrrrrr

++= At least one xi is non-integer

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≠===⋅⋅⋅ iiiii xibaixbriGri eeee

πrrrrrr

For ibGrr

= we have

So r does not belong to K

So the set of vectors R is identical to the set of vectors K

for all Rr

Or, every satisfies for allRr GRi

err

⋅Gr

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Examples of reciprocal lattice

xaa ˆ1 =r

3

3214

1)( aaaaV =×⋅=

rrr

yaa ˆ2 =r

zaa ˆ3 =r

Simple cubic Bravais lattice

xa

b ˆ2

1

π=

r ya

b ˆ2

2

π=

r

za

b ˆ2

3

π=

r

Reciprocal lattice

Also a simple cubic lattice

Face centered cubic

)ˆˆ(2

1zy

aa +=r )ˆˆ(

22 xz

aa +=r

)ˆˆ(2

3 yxa

a +=r

Primitive vectors of Reciprocal lattice

)ˆˆˆ(2

1zyx

ab ++−=

πr)ˆˆˆ(

22 zyx

ab +−=

πr

)ˆˆˆ(2

3 zyxa

b −+=πr

They are the primitive vectors of a body centered cubic lattice

Cubic unit cell with an edge of a

π2

Size of the cubic cell: 3)

4(

a

π

8

Body centered cubic

)ˆˆˆ(2

1 zyxa

a ++−=r

)ˆˆˆ(2

2 zyxa

a +−=r )ˆˆˆ(

23 zyx

aa −+=r

Primitive vectors

Primitive vectors for reciprocal lattice

)ˆˆ(2

1zy

ab +=

πr

)ˆˆ(2

2zx

ab +=

πr)ˆˆ(

23 yx

ab +=

πr

They are the primitive vectors of a fcc lattice

Fcc and bcc are reciprocal to each other

Brillouin Zones

The Wigner Seitz primitive cell of the reciprocal lattice is known as the first Brillouin zone.

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It is a regular 12 faced solid

The first Brillouin zone is enclosed by

perpendicular bisectors between the central lattice

point and its 12 nearest neighbors

First Brillouin zone for bcc structure

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First Brillouin zone of fcc structure

Primitive vectors of fcc structureBrillouin zones of fcc structure

The reciprocal lattice is bcc lattice

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Lattice planes and reciprocal lattice vectors

Lattice planes: any plane that contains at least three noncollinear Bravais latice points

Due to trnaslational symmetry, any lattice plane will contain infinitely many lattice points, which form a

two dimensional Bravais lattice

Family of lattice planes:

A set of parallel, equally spaced lattice planes which together contains all the points of the three

dimensional Bravais lattice.

Theorem:

For any family of lattice planes separated by a distance d, there are reciprocal lattice vectors

perpendicular to the planes, the shortest of which have a length of . Conversely, for any

reciprocal lattice vector , there is a family of lattice planes normal to and separated by a

distance d, where is the length of the shortest reciprocal lattice vector parallel to

d

π2

Gr

Gr

d

π2 Gr

The theorem is a direct consequence of

(1) The definition of reciprocal lattice

(2) the fact that a plane wave has the same value at all points lying in a family of planes that are

perpendicular to its wave vector and separated by an integral number of wavelength.

1=⋅RGie

rr

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Proof of the first part of the theorem:

Given a family of lattice planes with a separation of d , let be a unit vector normal to the planes. Let’s make up a

wave vector:n̂

nd

K ˆ2π

=r

We need to prove that is a reciprocal lattice vector, and it is the shortest one in that directionKr

To qualify for a reciprocal vector, has to be satisfied for all of the Bravais lattice1=⋅RKie

rr

Rr

We know this condition is satisfied for the origin of the Bravais lattice 0=rr

because 10 =⋅Ki

er

Then must be true for all the lattice points in the lattice plane that contains the origin1=⋅rKie

rr

0=rrr

r

is a periodic function with a wavelength in the direction of its wave vector

Because is perpendicular to that plane, and should be a constant in the planeKr

rKrr

⋅

rKie

rr⋅

dK

== rπ

λ2

The wavelength happens to be the separation of the planesd=λ

Therefore for all the lattice points in the family of planes, is satisfiedRr

1=⋅RKie

rr

So is a reciprocal vector, and we can write it asKr

nd

G ˆ2π

=r

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If there is another vector shorter than G′r

Gr

The wavelength of isG′r

dG

>′

=′ rπ

λ2

GGrr

<′

It is impossible for to have the same value on two planes with a distance shorter than the wavelength1=⋅′ rGie

rr

So can not be a reciprocal vectorG′r

Proof of the converse of the theorem

Given a reciprocal lattice vector, let be the shortest parallel reciprocal lattice vectorGr

Let’s construct a set of real space planes (not necessarily lattice planes) on which 1=⋅rGie

rr

Now we need to prove that this set of planes are lattice planes and contains all the Bravais lattice points

These planes must contain . They must be all perpendicular to and separated by a distance 0=rr

Gr

Gd

πλ

2==

For all Bravais lattice vector , must be true for any reciprocal vector1=⋅RGie

rr

Rr

So the set of planes must contain all the Bravais lattice points.

Lastly, we need to prove that each of these planes contain lattice points instead of every nth of them

Suppose only every nth of the planes contain lattice points. According to the first part of the theorem, the shortest

reciprocal vector perpendicular to the planes will be, , which contradict with assmuption that G is the

shortest reciprocal vector in that directionn

G

nd=

π2

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Consider a plane with miller indices h, k, l. is the shortest reciprocal vector normal

to that plane. Let’s prove that

Reciprocal lattice vectors and Miller indices of lattice planes

The miller indices of a lattice plane are the coordinates of the shortest reciprocal lattice vector normal to that plane,

with respect to a specified set of primitive reciprocal lattice vectors. Thus a plane with Miller indices h, k, l, is normal

to the reciprocal lattice vector 321blbkbhGrrrr

++=

1ar

2ar

3ar

11axr

22axr

33axr

Gr

321blbkbhGrrrr

′+′+′=

Proof

llkkhh =′=′=′

For any point on plane,rr

ArG =⋅rr

AaxG

AaxG

AaxG

=⋅

=⋅

=⋅

)(

)(

)(

33

22

11

rr

rr

rr

3

2

1

2

2

2

x

Al

x

Ak

x

Ah

π

π

π

=′

=′

=′

So

Then it follows that321

1:

1:

1::

xxxlkh =′′′

Since G is the shortest reciprocal vector perpendicular to the plane. There should be no common factors

between . These parameters satisfy the original definition of miller indices

Therefore llkkhh =′=′=′

h′ k′ l′