1. XRD allows Crystal Structure Determination
POSITION OF PEAKS
LATTICE TYPE
WIDTH OF PEAK
PERFECTION OF LATTICE
INTENSITY OF PEAKS
POSITION OF ATOMS IN BASIS
What do we need to know in order to define the crystal structure?
The size of the unit cell and the lattice type
(this defines the positions of diffraction spots)
The atom type at each point
(these define the intensity of diffraction spots)
Conclusion: If we measure positions and intensities of many spots, then we should be able to determine the crystal structure.
Need to change K’s to G’s in accordance with Kittel
After our diffraction topic you should be able to:
Calculate the structure factor for simple cubic, bcc, fcc, diamond, rock salt, cesium chloride

2. Structure Factor Shkl gives intensity of peaks
To get a diffraction peak, K has to be a reciprocal lattice vector, but even if K is,  f(r)e-irK might still be zero!
Structure Factor Shkl gives intensity of peaks
The scattered x-ray amplitude is proportional to:
Structure factor
Structure factor helps give intensity of peaks, and sometimes that can be zero. Technically, SS*
There is some angle dependence on top of this, but we won’t focus on that.
Can break this sum into a sum over all lattices and a sum over all of the atoms within the basis.

3. r ○ K r = n1 a1 + n2 a2 + n3 a3 (real space) Cubic form:
Where xi, yi and zi are the lattice positions of the atoms in the basis.
h, k and l are the miller indices of different planes in the crystal.

4. In Class: Simple Cubic Lattice
Simplify the structure factor for the simple cubic lattice for a one atom basis. Just let f be a constant.
Sure, you could define the atom to be at a different spot, but then you would still get a phase factor that pulled out and when you take S times its complex conjugate, you’d get the same thing.
Where xi, yi and zi are the lattice positions of the atoms in the basis.

5. How Do We Determine The Lattice Constant?
For the simple cubic lattice with a one atom basis:
So the x-ray intensity is nonzero for all values of (hkl), subject to the Bragg condition, which can be expressed
We know for cubic lattices (a=b=c):
Substituting and squaring both sides:
Ignoring the angle dependence, it would look like all of the peaks should have the same intensity. In reality, there is angle dependence.
Compare sine squared theta ratios. They should be integer values.
What will be the smallest angle peak? 100
Thus, if we know the x-ray wavelength and are given (or can measure) the angles at which each diffraction peak occurs, we can determine a for the lattice! How?

6. Missing Spots in the Diffraction Pattern
In some lattices, the arrangement and spacing of planes produces diffractions from planes that are always exactly 180º out of phase causing a phenomenon called extinction.
For the BCC lattice the (100) planes are interweaved with an equivalent set at the halfway position, giving a reflection exactly out of phase, which exactly cancel the signal.
Can you think of an example?
=/2

7. Extinction (out of phase) for 100 family of planes in BCC
(-101)
Extinction (out of phase) for 100 family of planes in BCC
(001)
Also 110 and 011
What about the 101 family of planes?

8. Group: The Structure Factor of BCC What values of hkl do not have diffraction peaks?
Analysis of more than one lattice point per conventional unit cell
E.g: bcc and fcc lattices
We COULD do it this other way with the primitive lattice vectors. The math would get a little worse, but you get the same result. This is where the conventional cell is of great value.
bcc lattice has two atoms per unit cell located at r1 = (0,0,0) and r2 = (1/2,1/2,1/2)

9. Group: Find the structure factor for BCC
Group: Find the structure factor for BCC. Under what h,k,l is it non-zero?
Euler's identity
Does the minus sign matter? No, you just have (-1)^(-1) or 1/-1 which is still -1

10. Visualizing the structure factor for BCC
Allowed low order reflections are:
110, 200, 112, 220, 310, 222, 321, 400, 330, 411, 420 …
Draw lowest on this cube ->
Forbidden reflections are:
100, 111, 210
Due to identical plane of atoms halfway between causes destructive interference
Real bcc lattice has an fcc reciprocal lattice (this is a good trick for remembering the rule)
002
022
112
101
011
110
211
121
202
000
020
200
220
This kind of argument leads to rules for identifying the lattice symmetry from "missing" reflections.

11. How to determine lattice parameter this time?
For a bcc lattice with a one atom basis, the x-ray intensity is nonzero for all planes (hkl), subject to the Bragg condition, except for the planes where h+k+l is odd. Thus, diffraction peaks will be observed for the following planes:
(100) (110) (111) (200) (210) (211) (220) (221) (300) …
Just as before, if we are given or can measure the angles at which each diffraction peak occurs, we can graphically determine a for the lattice!
What if you don’t know the h k l values?
If you don’t know. You have to guess and find ratios until you get something that makes sense. (More discussion on next slide)
A similar analysis can be done for a crystal with the fcc lattice with a one atom basis. For materials with more than one of the same atom type per basis in a cubic lattice, the rules for the structure factor can be modified.

12. Allowed Diffraction Peaks (See a trend?)
More lattice points in conventional cell  less peaks 2 3 4 5 6 8
If you are reading this, please say so in your homework question on the structure factor of diamond. I’d just like to know if anyone is paying close attention to these notes. Thanks! 
Discuss the simple cubic and the planes the lines refer to: 100, 110, 111, 200, 210, 211, 220, 221…families of planes
The correct angular separations are not reproduced in this diagram.
You have to go out pretty fair to tell between bcc and sc. (Though, note that the only single element material to form in sc is polonium.) 2 3 4 5 6 7
8/3 4 / 3 11 / 3 16 / 3 19 / 3 4

13. Group: Find the structure factor and extinctions for FCC.
Draw on board f*[1 + (-1) ^(h+k) + (-1)^ (k+l) + (-1)^ (h+l)] What rules can we develop? If having trouble, try picking some planes. Note that there is a lot of symmetry so the 100 and 001 will give same result, as would 110 to 011, and so on.

14. Group: Find the structure factor for FCC.
002
022
220
020
200
202
000
111
Allowed low order reflections are:
111, 200, 220, 311, 222, 400, 331, 310
Forbidden reflections:
100, 110, 210, 211

15. When more than one element Ni3Al Nickel aluminide structure
Simple cubic lattice, with a four atom basis
( )
Again, since simple cubic, intensity at all points.
But each point is ‘chemically sensitive’.

16. What Would You Expect for Diamond?
HW: Calculate the structure factor for the diamond structure. Lattice = FCC. Primitive Basis = (000), (¼ ¼ ¼)
I expect you to show your work to calculate the structure factor. Obviously you can check your answers with published work, but I want to make sure you can do it.
3.3 If time, discuss how to deal with the basis.

17. Or I could work this backwards! Just angles.
Use Bragg’s law and structure factor, calculate the diffraction angles (2) for the first three peaks in an aluminum powder pattern if a = nm and the x-ray source wavelength is nm.
How to start?
Or I could work this backwards! Just angles.
Bring back to the idea of plotting squared sine and h k l squared sum to get lattice parameter a.

18. Common to see an average decrease in intensity of the diffraction peaks despite rules for peak intensities

19. Atomic Scattering Factor f (key points) (aka Form Factor)
Atoms are of a comparable size to the wavelength of the x-rays and so the scattering is not point like. There is a small path difference between waves scattered at either side of the electron cloud
This effect increases with angle
For x-rays, scattering strength depends on electron density
All electrons in atom (Z of them) participate, core e- density ~spherical
Also, thermal effects increase the effective size of atom
I could go into this in much more detail, but doesn’t seem necessary or particularly useful.
Only at 2=0 does f=Z

20. Structure Factor with Different Atoms NaCl (rock salt) structure
Don’t make them do, but get them to think about
Similar to diamond, except this time with different atoms.
Due to this intensity depending on the atoms, we often say this is chemically (or elementally) sensitive
FCC Reminder:

21. Structure Factor with Different Atoms NaCl (rock salt) structure
Don’t make them do, but get them to think about
Similar to diamond, except this time with different atoms.
Due to this intensity depending on the atoms, we often say this is chemically (or elementally) sensitive
What would happen if you offset one of the atoms in the basis?

22. Extra slides
There is a lot of useful information on diffraction. Following are some related slides that I have used or considered using in the past.
A whole course could be taught focusing on diffraction so I can’t cover everything here.

23. The X-ray Shutter is the most important safety device on a diffractometer
X-rays exit the tube through X-ray transparent Be windows.
X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays.
Being aware of the status of the shutters is the most important factor in working safely with X rays.

24. If the wavelength of the incident x-rays and the scattering angle are known, then one can deduce the distance (already done) between the planes, dhkl, responsible for each scattering peak. The following twelve lines were obtained from a crystalline powder, known to belong to a cubic system.
Line d(Å) relative intensity
Index the lines in terms of their Miller indices (hkl) and calculate the lattice constant of the cubic lattice. Establish the type of cubic lattice.
This is a process of trial and error! (Computers are good at this)
Conditions for Peak SC
All
points
BCC
Sum =
even
FCC
Even/odd
Diam
Odd or sum4n
NaCl
Even/odd FCC
Ni3Al
Points (SC)
If I ever have time, I’d like to convert this back to angle because that’s how people normally see it. I think it will make more sense this way—even though it still takes some thought. I like the approach of trying a structure and seeing if lattice constants come out consistently (can us formula from simple cubic section).
Expect intensity to drop off, because atomic form factor decreases with increasing angle
{Data were obtained using nickel Ka radiation.}

25. Non-xray Diffraction Methods (more in later chapters)
Any particle will scatter and create diffraction pattern
Beams are selected by experimentalists depending on sensitivity
X-rays not sensitive to low Z elements, but neutrons are
Electrons sensitive to surface structure if energy is low
Atoms (e.g., helium) sensitive to surface only
For inelastic scattering, momentum conservation is important
X-Ray
Neutron
Electron
Replace top figure (not so informative, probably more confusing)
λ = 1A°
E ~ 104 eV
interact with electron
Penetrating
λ = 1A°
E ~ 0.08 eV
interact with nuclei
Highly Penetrating
λ = 2A°
E ~ 150 eV
interact with electron
Less Penetrating

26. Group: Consider Neutron Diffraction
Qualitatively discuss the atomic scattering factor (e.g., as a function of scattering angle) for neutron diffraction (compared to x-ray) by a crystalline solid.
For x-rays, we saw that f is related to Z and has a strong angular component. For neutrons?
The same equation applies, but since the neutron scatters off a tiny nucleus, scattering is more point-like, and f is ~ independent of .
3.9

27. Systematic Extinction
Systematic extinction is a consequence of lattice type
At right is table of systematic extinctions for symmetry elements
Other extinctions can occur as a consequence of screw axis and glide plane translations (Dove, Ch.6 Structure and Dynamics)
Accidental Extinctions may occur resulting from mutual interference of other scattering vectors
Symmetry
Extinction Conditions
simple
none C
hkl; h + k = odd B
hkl; h + l = odd A
hkl; k + l = odd
body
hkl; h + k + l = odd
All faces
hkl; h, k, l mixed even and odd
HCP has basis: /3, 2/3, ½ gives really weird conditions. Not instructive to show in class in my opinion.
Key: C, B, A = side-centered on c-, b-, a-face; I = body centered; F = face centered (001)

28. Symmetry within the Structure Factor
Structure factors are subject to the same point symmetry of the crystal.
For mirror symmetry (y’=-y, x’=x, z’=z):
For 2-fold rotation axis about [001], (y’=-y, x’=-x, z’=z):
Thus, the point group symmetry operations are reflected in the diffraction pattern.
Alternative method to figure out the symmetry
Here I remind you of the structure factor of BCC. Does it make a difference if we change x to minus x (or h here)? No. Now is it surprising that not sensitive to changes in sign? No. All bravais lattices have inversion symmetry but not always when you add the basis. Ok, so let’s consider a multiple atom example.