1. Diffraction from cystals
Ps: I put things on the comment block below so please make sure you are in edit status
Fan Ziyi

2. DP: diffraction pattern Reciprocal lattice Bragger’s law
Main topic:
DP: diffraction pattern
Reciprocal lattice
Bragger’s law
Structure factor
Amplitude of electron beam scattered by a unit cell
We use DP to visualize diffraction from crystals.
Why some points in DP are weaker or stronger than others?
In order to explain that, we must know some basic knowledge:
1.reciprocal lattice & bragger’s law;
2.structure factor & Amplitude of electron beam scattered by a unit cell

3. How can Diffraction occur
Using bragger’s law and reciprocal lattice

4. Something about X-RAY
When x ray hit the electrons, it makes electrons to vibrate in the same frequency.
those electrons become the new source of X-ray.
We call beams which given by electrons SCATTERING BEAM.

5. Bragger’s law
Bragger’s law gives us the certain condition under which diffraction can happen.

6. Bragger’s law nl=2dsinø d Incident beam Scattering beam
Because the intensity of scattering beam from a single electron is weak, so the DP we see is actually the result of diffraction and interference.
Interference occur only when the optical path difference is nl.
Bragger’s law
nl=2dsinø

7. Reciprocal lattice
We can see reciprocal lattice as a different version of normal lattice. It‘s useful in interpreting diffraction function.
we may not go that deep into diffraction function since it’s the same as bragger’s law.
What we do now it’s to give you some ideas about reciprocal lattice and how to use it in bragger’s law.

8. In the reciprocal lattice, A point represents sets of parallel (hkl)
a*,b*,c* are base vectors of reciprocal lattice
Through math derivation, we have
In the reciprocal lattice,
A point represents sets of parallel (hkl)
g is the vector from origin to a single point.
|g|=1/dhkl

9. constructive interference will only occur when
Based on former definition of reciprocal lattice, we now have the conclusion that constructive interference will only occur when K=g

10. What affects the intensity
Using amplitude equation and structure factor

11. Ri is the vector which defines each atoms in the lattice
Multiply ri and k remember a×a*=1
Remember the definition of g and the condition K=g.
I now give you the equation of Acell which is the amplitude. If you want to know where it comes from please check chapter 3.
A^2=I, which gives the intensity.
Ignore the value before ∑
We now have Structure factor.
The value of structure factor is linked with crystallology
It can predict each plane’s DP intensity

12. Examples Important structures
Now we begin calculating structure factors for real lattice structures like BCC,FCC,HCP.

13. BCC: 2 atoms in one lattice (x,y,z)=(0,0,0) & (1/2,1/2,1/2)
we need to choose xi,yi,zi：It’s important to know how many atoms are in one lattice, different structure differs. (x,y,z) are the indices of those atoms.
put h,k,l in the equation
BCC: 2 atoms in one lattice
(x,y,z)=(0,0,0) & (1/2,1/2,1/2)
FCC:4 atoms in one lattice
(x,y,z)=(0,0,0),(1/2,1/2,0),(1/2.0,1/2),(0,1/2,1/2)
HCP:2 atoms
(x,y,z)=(0,0,0),(1/3,2/3,1/2)
Plane (h,k,l)can be any plane in the lattice, we want to check the intensity of the diffraction pattern of one certain plane.

14. BCC: 2 atoms in one lattice (x,y,z)=(0,0,0) & (1/2,1/2,1/2)
FCC:4 atoms in one lattice
(x,y,z)=(0,0,0),(1/2,1/2,0),(1/2.0,1/2),(0,1/2,1/2)
HCP:2 atoms
(x,y,z)=(0,0,0),(1/3,2/3,1/2)
If F=0, then there is no diffraction pattern. You can compare the value of F or F^2 to have a notion of intensity.

15. Examples
Extended
We just discuss the basic structures. So what about more complicated fcc structures? Like NaCl?
The Structures I’m going to talk about share something in common,they all can been seen with one element occupy FCC lattice points and other element occupy a specific position.

16. Cl related to it by the basis vector [1/2, 1/2, 1/2]
Na atoms on an fcc site
Cl related to it by the basis vector [1/2, 1/2, 1/2]
Structure factor for NaCl.

17. Ga located on the fcc lattice
As related to it by the basis vector [1/4, 1/4, 1/4]
We can see that if the basis vector is [1/2, 1/2, 1/2],then the additional face is π.
If the basis vector is [1/4, 1/4, 1/4] ,then the additional phase is π/2

18. Reference
Wiiliams,D.B.,&Barry Carter,C.,(1996).transmission electron microscopy.new york: Springer ScienceþBusiness Media

19. THANKS