1. The theory of diffraction
Playing with waves
The theory of diffraction
Saulius Gražulis 2007

2. Scattering of waves (X-rays)
∣𝐤∣ = 2π λ
ω= 2π 𝑇 𝐤′ 𝐤
𝐸= 𝐸 0 cos ω𝑡−𝐤𝐱−Δϕ
When a flat wave hits charged particles, those start emitting spherical scattered waves.

3. Wave from the particle 2 lags with the phase difference 2rS.
Scattering of waves (2) 𝐤′ θ 𝐤 θ 𝐤 𝐫 𝑞 𝑝
Wave from the particle 2 lags with the phase difference 2rS.

4. Sin waves can be represented as rotating vectors.
Addition of sin waves 20
-20
-15
-10 -5 5 10 15 20 ϕ
ϕ=ω𝑡
-20
Sin waves can be represented as rotating vectors.

5. Addition of sin waves (2)
-30
-20
-10 10 20 30 20 ω𝑡
-20
A sum of sin waves is again a sin wave with the same frequency but with different amplitude and phase.

6. Adding many waves F ϕ
𝐹= ∣𝐹∣ 𝑒 𝑖ϕ
Multiple sin waves add up, giving a sin wave with some new amplitude and some new phase

7. Single Isomorphous Replacement (SIR)
𝐹 𝑃
𝐹 𝐻1
𝐹 𝑃𝐻1

8. What happens during MIR
𝐹 𝑃𝐻2
𝐹 𝐻2
𝐹 𝑃
𝐹 𝐻1
𝐹 𝑃𝐻1

9. The real life...
𝐹 𝑃𝐻2
𝐹 𝐻2
𝐹 𝑃
𝐹 𝐻1
𝐹 𝑃𝐻1

10. Friedel's law 𝐤′ θ 𝐤 1 θ F 𝐤 𝐫 ϕ −ϕ 𝑞 θ 𝑝 θ 2
𝐤′ Friedel
𝐤 Friedel
A reflection with -S scattering vector will have the same amplitude but an opposite phase.

11. Anomalous scattering
𝑓 𝑛 𝑓
f 'a
f ''a
f a
f a
f ''a 𝑓
f 'a
𝑓 𝑛

12. Anomalous difference 𝑓 ′ ℎ𝑘𝑙 , 𝑓′ ˉ ℎ ˉ 𝑘 ˉ 𝑙 ˉ 𝑓′ ′ ℎ𝑘𝑙 𝑓′ ′ ℎ𝑘𝑙
𝑓 ′ ℎ𝑘𝑙 , 𝑓′ ˉ ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝑓′ ′ ℎ𝑘𝑙
𝑓′ ′ ℎ𝑘𝑙
𝑓′ ˉ ′ ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝑓 ′ ℎ𝑘𝑙
𝐹 ℎ𝑘𝑙
𝐹 ℎ𝑘𝑙
𝐹 ˉ ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝐹 ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝑓′ ′ ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝑓 ′ ℎ ˉ 𝑘 ˉ 𝑙 ˉ

13. Dependence of the anomalous signal from energy
-10 -8 -6 -4 -2 2 4
10000
12000
14000
16000
18000
20000
f'' f' Se
E, Ev

14. Anomalous phasing (SAD)
𝐹 ℎ𝑘𝑙
𝑓′ ′ ℎ𝑘𝑙
𝑓′ ˉ ′ ℎ ˉ 𝑘 ˉ 𝑙 ˉ
𝐹 ℎ𝑘𝑙 𝑁
𝐹 ˉ ℎ ˉ 𝑘 ˉ 𝑙 ˉ

15. Crystals x a b y
A crystal is a periodic array of molecules; its lattice is defined by three lattice vectors a, b and c.

16. Reflection from a tilted crystal
𝐒≠ 𝐚×𝐜 𝑉 ℎ;𝑆≠ ℎ 𝑑 𝐤′
Δϕ=2π𝐒𝐚𝑛 𝐤 θ θ 𝐒 a 𝑑 b
In a general orientation of a crystal, for every atom in a reflecting plane there is an equivalent atom that scatters out of phase

17. Reflection from a tilted plain
...
𝑛=3
𝑛=2
𝐒𝐚∉ℤ
𝑛=1 ϕ
𝑛=0
ϕ=2π𝐒𝐚𝑛
Waves from the out-of-planes atoms cancel each other
(Adapted from J. Drenth, “Principles of Protein X-ray Crystallography”)

18. Reflections from a crystal𝐤′
𝐒= 𝐛×𝐜 𝑉 ℎ= 𝐚 ∗ ℎ 𝐤
Δϕ=2π𝐒𝐚𝑛 θ
𝐚 ∗ = 𝐛×𝐜 𝑉 θ 𝐒
𝐚 ∗ a
𝑑= 𝑉 𝐴
𝑆= ℎ 𝑑 b
When a lattice plane coincides with the reflecting plane, all equivalent atoms scatter in phase.

19. Scattering from a crystal plane
𝐒𝐚=ℎ;ℎ∈ℤ
𝑛=0
𝑛=1
𝑛=2
...
ϕ=2π𝐒𝐚𝑛=2πℎ𝑛=2π𝑚;𝑚∈ℤ
Waves from such the in-phase scattering atoms add up to give an appreciable amplitude.

20. Example of sin wave addition
-10 -5 5 10 15 20 25 -3 -2 -1 1 2 3
-10 -5 5 10 15 20 25 -3 -2 -1 1 2 3
𝑛=1 𝑁 cos 2π𝐒𝐚𝑛
𝑛=1
𝑛=3 𝐒𝐚
-10 -5 5 10 15 20 25 -3 -2 -1 1 2 3
-10 -5 5 10 15 20 25 -3 -2 -1 1 2 3
𝑛=10
𝑛=21
Only for integral values of Sa the waves add up to give an appreciable intensity.

21. Scattering from the crystal𝐤′ 𝐤 𝐒 θ θ
𝐚 ∗
𝐚 ∗ a
𝑑= 𝑉 𝑆 b
1) Laue conditions:
2) Bragg's law:
∣𝐒∣ = 𝑛 𝑑 = 2sinθ λ
𝐒𝐚=ℎ;𝐒𝐛=𝑘;𝐒𝐜=𝑙
𝐒=ℎ 𝐚 ∗ +𝑘 𝐛 ∗ +𝑙 𝐜 ∗
𝐒= 𝐤−𝐤′ 2 π
To get a beam reflected from a crystal, Laue conditions must be satisfied for the chosen Bragg reflection

22. Ewald's construct diffracted beam Evald sphere
𝐒=ℎ 𝐚 ∗ +𝑘 𝐛 ∗ +𝑙 𝐜 ∗
𝐤′ 2π
𝐛 ∗ 𝐒
𝐤 2π 2θ
𝐚 ∗
1 λ
A scattered beam is reflected when a node of the reciprocal lattice lands on the Ewald sphere