1. The Debye-Waller factor Start w/ structure factor
Fhkl =  fn exp (2πi r*  rn)
unit
cell
hkl
So/
S/
(S -So)/ 2
|r*| = |s - so|/ = (2 sin )/
r* = (s - so)/
= ha* + kb* + lc*
r = xa + yb + zc

2. For small rn
exp (2πi r*  rn) = 1 + 2πi r*  rn + 1/2 (2πi r*  rn)2 + …
hkl
The Debye-Waller factor
Averaging over time
exp (2πi r*  rn) ≈ 1 - 2π2 |r* |2 <ux> = 1 - 8π2(sin /)2 <ux>
hkl 2
exp (-
8π2(sin /)2 <ux>) = exp (-B(sin /)2)
Debye-Waller
factor 2
B = 8π2 <ux> 2

3. For anisotropic motion
exp {-(11h2 + 22k2 + 33l2 + 212hk + 213hl + 223kl)}
exp {-1/4(B11h2a*2 + B22k2b*2 + B33l2c*2 + 2B12hk a*b*
+ 2B13hl a*c* + 2B23kl b*c*)}
exp {-2π2(U11h2a*2 + U22k2b*2 + U33l2c*2 + 2U12hk a*b*
+ 2U13hl a*c* + 2U23kl b*c*
The Debye-Waller factor
 = 2π2 Ua* B = 8π2 U

4. For anisotropic motion
exp {-(11h2 + 22k2 + 33l2 + 212hk + 213hl + 223kl)}
exp {-1/4(B11h2a*2 + B22k2b*2 + B33l2c*2 + 2B12hk a*b*
+ 2B13hl a*c* + 2B23kl b*c*)}
exp {-2π2(U11h2a*2 + U22k2b*2 + U33l2c*2 + 2U12hk a*b*
+ 2U13hl a*c* + 2U23kl b*c*
The Debye-Waller factor
Bii are lengths of thermal ellipsoid semi-major and semi-minor axes
All Bs describe orientation of ellipsoids wrt lattice vectors

5. The Debye-Waller factor
Bii are lengths of thermal ellipsoid semi-major and semi-minor axes
All Bs describe orientation of ellipsoids wrt lattice vectors
Need:
Bii > 0
Bii Bjj > Bij2
B11 B22 B33 + B122 B132 B232 > B11 B232 + B22B132 + B33 B122

6. Thermal diffuse scattering
(see extensive discussion in James: Optical Principles
of the Diffraction of X-rays, chapter V)
Following Egami & Billinge, p. 33:
Define |Q| = 4π (sin )/
rn = rno + un n
Then, scatt ampl due to displacement only is
1/<b> iQunbn exp (iQrno)
un(rno) = 1/N1/2  uq exp (iqrno) q
i/<b>N1/2  bnQuq exp (i(Q-q)rno)
q,n

7. Thermal diffuse scattering
i/<b>N1/2  bnQuq exp (i(Q-q)rno)
q,n
Summing over all unit cells, and separating out
Bragg peak scattering
Idiffuse(Q) = |1/N  bj exp (2πir*rjo)|2  |QuQ-2πr*|2 j