Summary of lecture from 09.03.2018
Structure factors, systematic abscences, and XRD plots
Form factor and structure factor
I
Structure factor
S
k= X
atomj∈unit cell
f
j(k) e
ik·rjProportional to peak intensity: |S
k|
2∝ I
k IAtomic form factor f
j(k)
"Strength of scattering from atom j "
f
j(k) = Z
e
ik·rV
j(r ) dr
Fourier transform of scattering potential V
j(r )
Monoatomic bases
S
k= j · f (k) X
j
e
ik·rj= j · f (k) X
j
e
i2π(
hxj+kyj+lzj) = j · f · S
klatticewith S
klattice= P
j
e
ik·rjthe structure factor for the lattice and f (k) the form factor
lattice points S
klatticesystematic absence sc [0, 0, 0] 1 all h, k, l allowed bcc [0, 0, 0] 2 h + k + l must be even
12
,
12,
12fcc [0, 0, 0] 4 h, k , l all even or all odd
12
,
12, 0
12
, 0,
12Systematic absences
I
For some combinations of h, k, l : S
k= 0
I
(hkl ) not allowed: destructive interference
I
absence of peak
Polyatomic bases
S
k= S
klattice· S
kbasisexample crystal basis S
latticeS
basisCsCl sc Cs: [0, 0, 0] 1 f
Cs+ f
Cl(−1)
h+k+lCl:
12
,
12,
12NaCl fcc Na: [0, 0, 0] 4 f
Na+ f
Cl(−1)
h+k+lCl:
12