2. Wave Diffraction and Reciprocal Lattice

• Diffraction of Waves by Crystals

• Scattered Wave Amplitude

• Brillouin Zones

• Fourier Analysis of the Basis

• Quasicrystals

Diffraction Of Waves By Crystals

Bragg’s Law

2 sind n

Reflectance of each plane is about 103 to 105 .

Monochromator

X-ray Diffractometer on Powdered Si

1.16A neutron beam on CaF2

Relative intensities are due to basis.

Scattered Wave Amplitude

Fourier Analysis n n r T r integersi i ii

l l T a

→ in n e GG

r

G

rwhere 2 integer TG T

1 i

cellC

n dV n eV

G rG r

i ii

mG b mi integers

bi is called the primitive vectors of the reciprocal lattice, and G a reciprocal lattice vector.

2i j i j b aDefine then

*n real n n G Gr

1 2 3

2 j ki

a

a ab

a

a i,j,k cyclic

1 2 3jj k

k

j ki

a a

a a a

1

1 if , , is permutation of 1, 2,3

0i j k

even

i j k odd

not

,i

VdV e V k

rk0 →

Diffraction Conditions

Difference in phases between waves scattered at r and O

rk k k r

k k k

Scattering amplitude

iF dV n e k k rr idV n e k rr

Scattering vector

in dV e G kG

G

r

,n V G G kG

some reciprocal lattice vector if

0

V n

otherwise

G k G

Diffraction condition: k k k G

k k 22 0G k G 22 G k G(G G)

From Problem 1: 2d hkl n

G

where 1 2 3h k l G b b b

22 2 sinG

k G

4 2sin G n

d

Diffraction condition can be written as

2 sind n Bragg’s law

k k G

Laue EquationsDiffraction condition: j j

j

n G bk

→ j i jij

n a ba k 2 in

k lies in the intersection of 3 cones about the crystal axes.

Ewald construction

• White dots are reciprocal lattice points.• Incident k drawn with end at lattice point.• Scattered k obtained by drawing a circle.

Brillouin Zones

Brillouin Zone Wigner-Seitz cell of reciprocal lattice.

Diffraction condition22 G k G →

2

2 2

G

Gk

→ k is on boundary of BZ.

k k G

Square lattice

Reciprocal Lattice to SC Lattice

Primitive lattice vectors: 1 ˆaa x 2 ˆaa y 3 ˆaa z

Primitive cell volume:

1

2ˆ

a

b x 2

2ˆ

a

b y 3

2ˆ

a

b zPrimitive reciprocal lattice vectors:

3V a

Reciprocal lattice is also SC.

Reciprocal Lattice to BCC Lattice

Primitive lattice vectors: 1 ˆ ˆ ˆ2

a a x y z

Primitive cell volume:

1

2ˆ ˆ

a

b y zPrimitive reciprocal lattice vectors:

31 1 1

1 1 18

1 1 1

aV

Reciprocal lattice is FCC.

2 ˆ ˆ ˆ2

a a x y z 3 ˆ ˆ ˆ

2

a a x y z

31

2a

2

2ˆ ˆ

a

b x z 3

2ˆ ˆ

a

b x y

Reciprocal lattice vector: j jj

nG b 2 3 1 3 1 2

2, ,n n n n n n

a

bcc

1st BZ

rhombic dodecahedron

Cartesian coord

Reciprocal Lattice to FCC Lattice

Primitive lattice vectors: 1 ˆ ˆ2

a a y z

Primitive cell volume:

1

2ˆ ˆ ˆ

a

b x y z

Primitive reciprocal lattice vectors:

30 1 1

1 0 18

1 1 0

aV

Reciprocal lattice is BCC.

2 ˆ ˆ2

a a x z 3 ˆ ˆ

2

a a x y

31

4a

2

2ˆ ˆ ˆ

a

b x y z 3

2ˆ ˆ ˆ

a

b x y z

Reciprocal lattice vector: j jj

nG b 1 2 3 1 2 3 1 2 3

2, ,n n n n n n n n n

a

fcc

1st BZ

Cartesian coord

Fourier Analysis of the Basis

Scattering amplitude F N SG G

Structure factor i

cellS dV n e G r

G r

For a basis with s atoms 1

s

j jj

n n

r r r

ij jcel

jl

S dV n e G rG r r ji i

jcellj

e dV n e G r G ρρj ρ r r

j

j

i

jS e f G r

G G ij jcell

f dV n e G ρG ρ atomic form factor

cellV n G

Structure Factor of BCC Lattice

With respect to the SC lattice, the BCC has a basis of 2 atoms at

1 0,0,0r and 2 1,1,12

ar

→ 1 2 3

1 2 3, , 1i n n n

S S n n n f e G

1 2 3

2, ,n n n

a

G

1 2 3

0

2

oddfor n n n

f even

E.g., metallic Na: no (100), (300), (111), or (221) lines (cancelled by extra plane at half separation)

Structure Factor of FCC Lattice

With respect to the SC lattice, the FCC has a basis of 4 atoms at

1 0,0,0r 2 0,1,12

ar

→ 1 3 1 22 3

1 2 3, , 1i n n i n ni n nS S n n n f e e e G

1 2 3

2, ,n n n

a

G

4

0jf n all odd or all even

forotherwise

3 1,0,12

ar 4 1,1,0

2

ar

f K f Cl

Atomic Form Factor

For a spherical distribution of electron density

i

cellf dV n e G ρρ

12 cos

0 1

2 cos i G rr dr d n r e

2

0

2i G r i G re e

r dr n ri G r

2

0

sin4

Grr dr n r

G r

For n Z r r f Z

For forward scattering, G 0 , so that f Z.

For X-ray diffraction, f Z. ( X-ray not sensitive to change in n(r) caused by bonding)