Transcript Crystal Diffraction - UNL | Department of Physics & Astronomy

Crystal Diffraction
Laue 1912
1914 Nobel prize
Max von Laue
(1879-1960)
Lattice spacing
typically
10
10
o
m  1
o
  1A
Today X-ray diffraction supplemented by electron and neutron diffration
Energies X-ray, electrons and neutrons wave-particle
E  h 
hc


X-ray:
hc
E
o
  1A
E 12 k eV
o
  1A
Electrons: p  k 
h

h
 
p
me  9.1 10-31 kg
h
2mE
E 150 eV
o
  1A
Neutrons:

h

p
h
2mE
mn  1.6749 10-27 kg
E
0.08 eV
Typical Laue X-ray diffraction pattern
symmetry of the pattern
symmetry of the crystal
Laue X-ray diffraction
YAlO3
c-axis normal to picture
Complementarity of the three types of radiation
X-ray diffraction
•Photon energies 10keV-100keV
large penetration depth
3D crystal structure
•scattering by electron density
best results for
atoms with high Z
Electron diffraction
•Charged particle
“strong” interaction
with matter
low penetration depth
Study of: surfaces
thin films
Neutron diffraction
•Interaction with nuclei
Improved efficiency
for light atoms
Inelastic scattering:
phonons
•Magnetic moment interacts
with moment of electrons
Magnetic scattering:
Structure, magnons
Bragg Diffraction Law
Law describing the minimum condition for diffraction
Applicable for photons, electrons and neutrons
Bragg’s law
Condition for efficient specular reflection
(click for java applet)
2d sin   n 
2dhkl sin   
n: integer
Spacing dhkl between successive (hkl) planes
In cubic systems: dhkl 
a
h2  k 2  l2
Top view
y
a
x
d110
2 d110  a
2
2
dhkl for non cubic lattice
d110
a

2
later in the framework
of the reciprocal lattice
•structure factor
Bragg’s law necessary condition
Intensity of particular
(hkl) reflection
•atomic form factor
General theory of Diffraction
P
r
R
X-ray source
R’
B
AP  A0 ei k0 ( Rr )it
Plane wave incoming at P
P
r
k0
k
R’
R
X-ray source
B
Scattered wave contribution from P
incoming at B
i k ( R r )
AB  AP  (r ) e
AB  A0 e
 A0 e
i k 0 ( R  r ) it
i ( k 0 R  k R t )
 (r ) e
i k ( R  r )
 (r ) e i ( k  k
Electron density at P
0 )r
Total scattering from the entire volume:
A    (r ) e
i ( k k 0 ) r
d3r
Diffraction experiment measures the intensity I of the scattered waves
2
I( )  A   (r ) e
  (r ) e
i  r
3
d r
i( k k 0 )r
3
d r
2
2
where   k  k 0 is the scattering vector
Diffracted intensity is the square of the Fourier transform of the electron density
In crystals (r ) is periodic
1D example ( x)  ( x  na), n  0,  1, ,  2,  3,...
( x )   n e
n
i
2
nx
a
Fourier series expansion
2π periodic function decomposed into cos kx and sin kx
or
a0 
f(x) 
  ak cos kx  bk sin kx
2 k 1
a
f(x) 

c e
ikx
k 
k
where

for
k

0
2



1

c k   (ak  ibk ) for k  0 
2

1

(a

ib
)
for
k

0
k
 2 k



0

1

f(x)eikx dx

2 
1dimensional case
( x )   n e
i
3dimensional case
2
nx
a
(r )   G ei G r
G
n
( x  ma )   n e
i
2
n ( x  ma )
a
with respect to lattice vector
n
  n e
i
translational invariance of (r )
2
nx
a
e
i
2
n ma
a
n
r n  n1 a1  n2 a2  n3 a3
(r  r n )  (r )
with
e
i
2
n ma
a
e
G  r n  2m
i 2 n m
 cos2mn   i sin(2mn )  1
( x  ma )   n e
i
n
2
nx
a
Reciprocal lattice vectors
 ( x)
Remember:
Diffracted intensity is the square of the Fourier transform of the electron density
I( )   (r ) e
(r )   G ei G r
i r
3
dr
2
periodic electron density
G
I( )   G  e
i( G  )r
2
3
d r
G
with
1 i( G  )r 3
(G   )   e
d r
VV
(click for information about -functions)
2
I(   G)  G V
2
2
I(   G)  G V
Scattering condition
G
2
is nothing but Bragg´s law
!
The reciprocal lattice
G  r n  2m
G  h g1  k g2  l g3
with h, k, l integers
r n  n1 a1  n2 a2  n3 a3
decomposition into so far unknown basis vectors
The basis vectors
g1, g2 , g3
of the reciprocal lattice are determined by:
a 2  a3
g1  a  2
a1  (a2  a3 )

1
a3  a1
g2  a2  2
a1  (a2  a3 )


g3  a3  2
These
gi
a1  a 2
a1  (a 2  a3 )
fulfill the condition
G  r n  2m
gi  a j  2ij
holds, where
G  h g1  k g2  l g3
Examples for reciprocal lattices
3 dimensions
2 dimensions
Important properties of the reciprocal lattice vectors Ghkl  h g1  k g2  l g3
Ghkl lies perpendicular to the lattice plane with Miller indices (hkl)
simple example for the (111) plane in the cubic structure
a1  a2  (a,a,0)
and
a3  a2  (0,a,a)
a3  (0,0, a)
a3  a2  (0,a,a)
span the (111) lattice plane
vector (a1  a2 )  (a3  a2 )
(111) plane
(a1  a2 )  (a3  a2 ) 
a1  (a,0,0)
a2  (0, a,0)
(a1  a3 )  (a2  a3 )  (a1  a2 )  (a2  a2 )
 (a3  a1 )  (a2  a3 )  (a1  a2 )
a1  a2  (a,0,0)  (0, a,0)  (a,a,0)
0
 g1  g2  g3  G111
Distance dhkl between lattice planes (hkl) related to Ghkl according to
Ghkl
2

dhkl
gi  a j  2ij
d111
 cos (a1, G111 )
a
a1 G111

a1 G111
d111
d111 
G111

2
a G111
2
G111
Equivalence between the scattering condition   Ghkl
and Bragg´s law 2dhkl sin   
k  
-k0
k0
Ө
k
Ө
lattice plane (hkl)
Elastic scattering: k=k0
k 2
  k 

 k 2  2kk 0 cos ( 2)  k 0

 k 2 cos   sin   cos   sin 
2
2
1
2
cos 2
2

2
 k 21  cos 2
2
4

G

sin 
 2k sin  
hkl
dhkl

2dhkl sin   
Geometrical interpretation of the scattering condition
  Ghkl
Ewald construction
reciprocal lattice
k
2Ө
G
k0
(000)
(click for animation)
Crystal in random orientation not necessarily reflection
polychromatic radiation
rotation of the crystal
Rotating crystal arrangement
determine unknown structure
Powder method / Debye Scherrer
incoming monochromatic beam
Precise measurement of
lattice constants
Laue method
transmission
Polychromatic X-rays
reflection
Orientation of crystal with known structure
The structure factor
I(   Ghkl )  G
hkl
2
V
2
Controls the actual intensity of the (hkl)-reflex
Scattering condition (
necessary condition
Remember:
Bragg’s law )
(r )   G ei G
Ghkl
Fourier-coefficients
hkl
hkl
r
because crystal periodic
(r  r n )  (r )
1
i G r 3
hkl 
cell(r ) e d r
Vc
hkl
Majority of the electrons are centered in a small region around the atoms
core electrons
Scattering from valence electrons can be neglected
Atom in n-th unit cell is located at position
1
1
i G r 3
hkl 
cell(r ) e d r 
Vc
Vc
hkl
r
e

i Ghkl r 
i G r  3

  (r ) e d r
hkl
r  r   r
atomic scattering factor fα
Fhkl   f e

 i Ghkl r 
Structure factor
f    (r ) e  i G r d3 r 
hkl
atomic scattering factor
Spherically symmetric
i G

f    (r ) e
hkl r  cos
where

2

r sin dr dd

G , r’
    (r ) e
r  dr d(cos )d
 i Ghkl r  cos 
2
sin Gr  2
 4  (r )
r  dr  
Gr 
G  2k 0 sin 
atomic scattering factor
sin 4r (sin ) / )) 2
f  4  (r )
r  dr 
4r(sin ) / )
Maximum at Ө=0 (forward scattering)
2


f (  0)  4  (r ) r dr 
Z
number of electrons/atom
(Click for calculations of
atomic scattering factors)
Structure factor of a lattice with basis
Structure factor of the bcc lattice:
Conventional cell contains two atoms at
r2=(1/2,1/2,1/2)
r1=(0,0,0)
Both atoms have the same atomic scattering factor f1 = f2 = f
Reciprocal unit cell: cube with cell side of 2π/a
Fhkl   f e

 i Ghkl r 
2
Ghkl 
(h, k, l)
a
 f 1  e
 i ( hk l )

0 for h  k  l odd

i ( h k l )
 
Fhkl  f 1  e
2f for h  k  l even

ei (hk l )  cos( (h  k  l ))  i sin( (h  k  l ))
We observe e.g. diffraction peaks from (110), (200), (211) planes
but no peaks from (100), (111), (2,1,0) planes
If f1 = f2 like CsI
peaks like (100), (111), (2,1,0) appear
Similar situation in the case of fcc KCl/KBr: f(K+)=f(Cl-) = f(Br-)
KCl: Non-zero if all
indices even
KBr: all fcc-peaks
present
•Shape and dimension of the unit cell can be deduced from Bragg peaks
•Content of the unit cell (basis) determined from intensities of reflections