Crystal Diffraction - UNL | Department of Physics & Astronomy
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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 ( Rr )it
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 ) it
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 2m
i 2 n m
cos2mn i sin(2mn ) 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 2m
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 2m
gi a j 2ij
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 2ij
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 21 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 dd
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 4r (sin ) / )) 2
f 4 (r )
r dr
4r(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 ( hk l )
0 for h k l odd
i ( h k l )
Fhkl f 1 e
2f for h k l even
ei (hk 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