1. Factors that affect the diffracted intensity
structure factor
polarization factor
Lorentz factor
multiplicity factor
temperature factor
absorption factor

2. Single electron scattering
The term is
called the polarization factor

3. Single electron scattering
The magnitude of the single electron scattering is very small
Since there are ~1023 atoms/cm3, and even more electrons, the small magnitude of electron scattering can be detected
(e2/mc) is known as the classical electron radius

4. The Compton effect
Compton scattering occurs when an X-ray photon suffers an inelastic collision with a loosely-bound electron
scattered X-ray photon
hn1
hn2
mc2(g-1) q f
incident X-ray photon
relativistic recoil electron
where

5. Scattering by an atom
An atom with atomic number Z is surrounded with Z electrons, all of which can scatter X-rays
Since an atom has a finite size, there will be a difference in the path traveled (i.e. a phase difference) in the X-rays scattered from different locations around the atom

6. Scattering by an atom The atomic scattering factor f is defined as:
amplitude scattered by Z electrons in an atom
amplitude scattered by a single electron
f =
The atomic scattering factor f = Z for any atom in the forward direction (2q = 0) Þ I(2q=0) =Z2
As q increases f decreases  functional dependence of the decrease depends on the details of the distribution of electrons around an atom (sometimes called the form factor)
f is calculated from quantum mechanics

7. Scattering by an atom
note that f is plotted as a function of sinq/l so that the same data can be used for different radiations

8. Scattering by a unit cell
scattering by an single electron
is modified by
scattering by an collection of electrons
is modified by
scattering by an collection of atoms
The structure factor is the amplitude scattered per unit structure, or the amplitude scattered by a unit cell

10. The structure factor
amplitude scattered by all atoms in a unit cell
amplitude scattered by a single electron
Fhkl =
The structure factor contains the information regarding the types (f) and locations (u, v, w) of atoms within a unit cell
The intensity of a Bragg reflection is then proportional to the square of the structure factor:
F may be complex, but “I” must be real!

11. The structure factor and complex numbers
Fhkl can be (and often is) a complex number
It is very useful to be able to do some mathematical manipulation of complex numbers
Hint: don’t try to use the trigonometric form – using exponentials is much easier!

12. A simple example – primitive unit cell with one atom
Let the atom be placed at 000 (remember that we can define the unit cell any way we want)
In this case the structure factor is independent of h, k and l; it will decrease with f as sinq/l increases (higher-order reflections)

13. Base centered unit cell
The bcc crystal structure has atoms at 000 and ½½0:
the atoms are the same, so the atomic scattering factor can be factored out
•If h and k are both odd or even, then F=2f and F2=4f2
• If h and k are mixed, F=0 and F2=0.
• Example:
– 111, 112, 113, and 021, 022, 023 have the same reflection
– 011, 012, 013 and 101, 102, 103 have zero reflections

14. Body centered cubic unit cell
The bcc crystal structure has atoms at 000 and ½½½:
the atoms are the same, so the atomic scattering factor can be factored out
So Fhkl =
2f for h+k+l = an even integer
0 for h+k+l = an odd integer
(200), (400), (220)  
(100), (111), (300)  
“forbidden” reflections

15. CsCl unit cell
The CsCl crystal structure is primitive cubic with a two-atom basis: Cs at 000 and Cl at ½½½:
atoms are different, so we cannot factor out the atomic scattering factor
fCs + fCl for h+k+l = an even integer
fCs - fCl for h+k+l = an odd integer
So Fhkl =
(200), (400), (220)  
(100), (111), (300)  
if Cs and Cl were the same, this result would be the same as BCC

16. Face centered cubic unit cell
The fcc crystal structure has atoms at 000, ½½0, ½0½ and 0½½:
If h, k and l are all even or all odd numbers (“unmixed”), then the exponential terms all equal to +1  F = 4f
If h, k and l are mixed even and odd, then two of the exponential terms will equal -1 while one will equal +1  F = 0
16f 2, h, k and l unmixed even and odd
0, h, k and l mixed even and odd

17. Diamond cubic unit cell
The diamond cubic structure is fcc with a two atom basis (000 and ¼¼¼)
Eight atoms per unit cell: 000, ½½0, ½0½, 0½½ ¼¼¼, ¾¾¼, ¼¾¾, ¾¼¾
The structure factor can be written:
This could get very messy; however, there is a trick…

18. “Factoring the structure factor”
We can re-write the structure factor as following
[fcc term]
the two-atom basis modulates the fcc structure factor, but if the fcc term is zero, Fhkl is zero no matter what the two-atom term is!
4f for h, k and l unmixed
0 for h, k and l mixed

19. Dealing with a complex structure factor
Now we multiply by the fcc term (42 =16 for hkl unmixed and zero for hkl mixed even and odd)
and we have the following conditions on h, k and l
when (h + k + l) is a even multiple of 2
when (h + k + l) is an odd multiple of 2
when (h + k + l) is odd
h, k and l mixed

20. Dealing with a complex structure factor
Neglecting the fcc term the diamond cubic structure factor is
The exponential term is complex, but remember that we’re interested in FF*, so we multiply by the complex conjugate:

21. Multiple atom unit cells -- NaCl
NaCl is fcc with a two atom basis:
Na+ at ½½0 ½0½ 0½½ fcc translations
Cl- at ½½½ 00½ 0½0 00½ fcc translations
We can immediately factor out the common fcc term:
so the structure factor becomes:
h, k and l mixed
h, k and l unmixed
(h + k + l) even
(h + k + l) odd

22. The structure factor for the hexagonal close-packed structure
The hcp structure is primitive hexagonal with a two-atom basis: 000 and 1/3 2/3 1/2 :

23. The structure factor for the hexagonal close-packed structure
h + 2k l
3m odd 0
3m even 4f 2
3m1 odd 3f 2
3m1 even f 2

24. The multiplicity factor
The multiplicity factor arises from the fact that in general there will be several sets of hkl-planes having different orientations in a crystal but with the same d and F2 values
Evaluated by finding the number of variations in position and sign in h, k and l and have planes with the same d and F2
The value depends on hkl and crystal symmetry
For the highest cubic symmetry we have:
p100 = 6
p110 = 12
p111=8

26. The Lorentz factor

27. The Lorentz factor
In a polycrystalline sample, the diffracted intensity is spread over a cone with a base radius of Rsin2qB so the relative intensity per unit length of line is proportional to 1/sin2qB

28. The Lorentz-polarization factor
The combination of geometric corrections are lumped together into a single Lorentz-polarization (LP) factor:
The effect of the LP factor is to decrease the intensity at intermediate angles and increase the intensity in the forward and backwards directions

29. Thermal Motion

30. The temperature factor
As atoms vibrate about their equilibrium positions in a crystal, the electron density is spread out over a larger volume
This causes the atomic scattering factor to decrease with sinq/l (or |S| = 4psinq/l )more rapidly than it would normally:

31. The temperature factor
The temperature factor is given by:
where the thermal factor B is related to the mean square displacement of the atomic vibration:
This is incorporated into the atomic scattering factor:

33. The absorption factor
Angle-dependent absorption within the sample itself will modify the observed intensity
Absorption corrections are complicated in the Debye-Scherrer geometry
tables of absorption correction factors are given for different geometries in the International Tables for Crystallography

34. The absorption factor
For a flat diffractometer sample in a symmetric geometry, the absorption factor is independent of angle
Let A be the cross-section area of the incident beam; the intensity diffracted at the Bragg angle by an infinitesimal thickness dx at depth x:
 then integrate from x = 0:

35. Expressions for the integrated intensity in a powder pattern
The relative integrated intensities in the lines in an X-ray powder diffraction experiment are given by:
approximate (but suitable for this class)
exact, for diffractometer
exact, for Debye- Scherrer camera

36. Complicating factors in intensity measurements
Preferred orientation
we explicitly assume that the polycrystalline sample contains randomly oriented grains
the presence of preferred orientation can cause enormous headaches
careful sample preparation is a must!
Extinction
the simple theory assumes the sample is “ideally imperfect”: small crystals, single diffraction events
large perfect crystals (“ideally perfect”) diffract more weakly than ideally-imperfect crystals
extinction can be minimized by grinding, liquid N2, etc.