Transcript 2. Wave Diffraction and Reciprocal Lattice

2. Wave Diffraction and Reciprocal Lattice
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Diffraction of Waves by Crystals
Scattered Wave Amplitude
Brillouin Zones
Fourier Analysis of the Basis
Quasicrystals
Diffraction Of Waves By Crystals
Bragg’s Law
2d sin   n
Reflectance of each plane is about 103 to 105 .
Monochromator
1.16A neutron beam on CaF2
Relative intensities
are due to basis.
X-ray Diffractometer on Powdered Si
Scattered Wave Amplitude
n  r  T  n  r 
Fourier Analysis
→
n  r    nG e

V
G  T  2  integer
where
nG 
→
Define bi  a j  2  i j
li  integers
i
i G r
G
dV e i k r  V k ,0
 T   li ai
1
VC
then

cell
dV n  r  e i G r
G   mi bi
 T
n  r  real  nG *  nG
mi integers
i
bi is called the primitive vectors of the reciprocal lattice,
and G a reciprocal lattice vector.
bi  2

a j  ak
a1  a 2  a3

jk
a j  ak
i jk
a1  a 2  a3
i,j,k cyclic
i jk
even
1

 1 if i, j, k is odd permutation of 1, 2,3
0
not

Diffraction Conditions
k  k   k
Scattering vector
Difference in phases between waves scattered at r and O   k  k  r  k  r
Scattering amplitude  F   dV n  r  e
i  k  k   r
  nG  dV e i  G  k r
G
  dV n  r  e  i k r
  nG V  G , k
G
k  some reciprocal lattice vector G
V n
  G if
otherwise
 0
k  k  k  G
Diffraction condition:
k  k

From Problem 1:
2k  G  2

2k  G  G 2  0
d  hkl  
2

2
n
G
2k  G  G

G  hb1  k b2  l b3
where
G sin 
Diffraction condition can be written as
2d sin   n
2
4

sin   G 
Bragg’s law
2
n
d
(G  G)
k  k  G
Diffraction condition:
Laue Equations
k  G   n b
j
j
j
→
ai  k   n j ai  b j  2  ni
j
 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 condition
2k  G  G
2
→
G G
k  
2 2
2
k  k  G
→ k is on boundary of BZ.
Square lattice
Reciprocal Lattice to SC Lattice
Primitive lattice vectors:
a1  a xˆ
Primitive cell volume:
V  a3
Primitive reciprocal lattice vectors:
Reciprocal lattice is also SC.
a3  a zˆ
a2  a yˆ
2
b1 
xˆ
a
b2 
2
yˆ
a
b3 
2
zˆ
a
Reciprocal Lattice to BCC Lattice
Primitive lattice vectors:
a
 xˆ  yˆ  zˆ 
2
a1 
Primitive cell volume: V 
3
a
8
Primitive reciprocal lattice vectors:
1
1
1
1
1 1
1 1
b1 
a2 
1

a
 xˆ  yˆ  zˆ 
2
a3 
a
 xˆ  yˆ  zˆ 
2
1 3
a
2
2
 yˆ  zˆ 
a
b2 
2
 xˆ  zˆ 
a
b3 
2
 xˆ  yˆ 
a
Reciprocal lattice is FCC.
Reciprocal lattice vector:
G   n j b j  2  n2  n3 , n1  n3 , n1  n2 
j
a
Cartesian coord
1st BZ
bcc
rhombic
dodecahedron
Reciprocal Lattice to FCC Lattice
a1 
Primitive lattice vectors:
V
Primitive cell volume:
Primitive reciprocal
lattice vectors:
b1 
a
 yˆ  zˆ 
2
3
a
8
0 1 1
1 0 1
1 1 0
2
 xˆ  yˆ  zˆ 
a
a2 

a
 xˆ  zˆ 
2
a3 
a
 xˆ  yˆ 
2
1 3
a
4
b2 
2
 xˆ  yˆ  zˆ 
a
b3 
2
 xˆ  yˆ  zˆ 
a
Reciprocal lattice is BCC.
Reciprocal lattice vector:
G   n j b j  2  n1  n2  n3 , n1  n2  n3 , n1  n2  n3 
j
a
Cartesian coord
1st BZ
fcc
Fourier Analysis of the Basis
FG  N SG
Scattering amplitude
Structure factor
SG   dV n  r  ei G  r  Vcell nG
cell
n  r    n j r  rj 
s
For a basis with s atoms
j1
SG    dV n j  r  rj  ei G  r   e
j
SG   e
j
cell
i G  r j
j
f j G 
i G  r j

cell
dV n j  ρ  ei G  ρ
ρ  r  rj
f j  G    dV n j  ρ  ei G  ρ  atomic form factor
cell
Structure Factor of BCC Lattice
With respect to the SC lattice, the BCC has a basis of 2 atoms at
r1   0,0,0 
G
→
and
r2 
a
1,1,1
2


2
 n1 , n2 , n3 
a

SG  S  n1 , n2 , n3   f 1  e
 i  n 1  n2  n3

odd
0

for n1  n2  n3 
even
2 f
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
r1   0,0,0 
r2 
G
a
 0,1,1
2
r3 
a
1, 0,1
2
r4 
2
n1 , n2 , n3 

a

→ SG  S  n1 , n2 , n3   f 1  e
 i   n2  n3 
e

 i  n 1  n3

e

 i  n 1  n2

n all odd or all even
4 f

for j
otherwise
0
f K 
f  Cl  

a
1,1, 0 
2
Atomic Form Factor
For a spherical distribution of electron density
f   dV n  ρ  e
cell
i G  ρ

 2  r dr
2
0

e i G r  ei G r
 2  r dr n  r 
iGr
0
2
For
n r   Z  r 
1
 i G r cos
d
cos

n
r
e



1

 4  r 2 dr n  r 
0
sin Gr
Gr
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)