Structure Factor
MSE 421/521 Structural Characterization
Structure Factor
dh00 = d = a/h
EM wave: E = E0eiφ
for constructive interference:
δ2’1’ = 2dsinθ = λ
δ3’1’ = 2xsinθ = (x/d)λ = (xh/a)λ
∴ E = E0exp[2πi(hu + kv + lw)]
Now remove λ by switching from
path difference to phase difference:
φ = (δ/λ)2π [radians]…
∴ φ3’1’ = (2πxh/a)
Express x as fractional
coordinate u = x/a, then:
φ3’1’ = 2πhu
In 3D this expression expands to:
φ = 2π(hu + kv + lw)
MSE 421/521 Structural Characterization
E0 = amplitude
= atomic scattering factor
= scattering due to one atom
=f
Now add up all such waves for all
atoms in a unit cell:
N
Fhkl = ∑ fne 2 πi(hu
n
+ kv n + lw n )
1
Complex – amplitude & phase
Intensity of diffracted beam is
proportional to |F|2, so imaginary part
always vanishes.
Laue Classes
Point groups: The 32 symmetries allowed about a fixed point in a crystal,
derived by combining all possible combinations of non-translational (point)
symmetry elements (rotations, inversions, reflections, etc.).
Triclinic
Monoclinic
1
1
2
2
m
m
Orthorhombic 222 mm2 mmm
4
Tetragonal
4
422
4
m
Trigonal
3
32
3m
3
6
Hexagonal
6
622
6
m
Cubic
23 m3
432
4 3m
Sym axis on c
4mm
3m
6mm
m3 m
4 2m
4
m
mm
Sym axes a, b, c
Sym axis || c
6m2
6
m
mm
Sym axis on [111]
Sym axis on c
3fold axes on <111>
Because diffraction always adds a centre of symmetry, we can only distinguish
between the 11 centrosymmetric groups (Laue classes), shown outlined in bold.
There are 11 Laue classes, each denoted by the highest point-group symmetry in
that class. Diffraction can only distinguish between these 11 Laue classes.
MSE 421/521 Structural Characterization