1. X-ray Neutron Electron
Diffraction
X-ray
Neutron
Electron

2. Diffraction
An atom in space will elastically scatter X-rays as a function of angle.
It will behave as a point source, but the intensity will decrease with angle.
The intensity is calculated from theoretical wave functions for the electron distribution and given as a polynomial expression called the scattering factor (f ).

3. Diffraction

4. X-Ray Diffraction
Atoms separated by distance d will scatter in phase when the path length difference is an integral number of wavelengths.
Path length difference B-C-D = nl
nl = 2d sin q

5. Diffraction Fhkl = square root of integrated intensity.
Intensities can be calculated knowing the position and scattering characteristics of each atom.
Fhkl = square root of integrated intensity.
fj = scattering of atom j at angle 2q
Atom j located at fractional coordinates xj, yj, zj.

6. Structure Factors
The structure factor, F, is the square root of the measured integrated intensity.

7. Structure Factors

8. Structure Factors
If, for every atom, j, at (xj, yj, zj), there is an identical atom at (-xj, -yj, -zj),
The imaginary term in the equation is zero.

9. Systematic Extinction
If, for every atom, j, at (xj, yj, zj), there is an identical atom at (1/2+xj, 1/2+yj, 1/2+zj), (body centering)
The structure factor is zero for h + k + l = 2n+1 (h+k+l odd) for all hkl
So systematic extinction of structure factors tells us which symmetry operators are present

10. Systematic Extinction
Lattice centering Operations
For all hkl
F: hkl all even or all odd
I: h+k+l = 2n
A: k+l = 2n
B: h+l = 2n
C: h+k = 2n

11. Systematic Extinction
Glide planes
a glide normal to c: for hk0, h = 2n
b-glide normal to c: for hk0, k = 2n
n-glide normal to c: for hk0, h+k = 2n
a-glide normal to b: for h0l, h = 2n
c-glide normal to b: for h0l, l = 2n
n-glide normal to b: for h0l, h+l = 2n
b-glide normal to a: for 0kl, k = 2n
c-glide normal to a: for 0kl, l = 2n
n-glide normal to a: for 0kl, k+l = 2n

12. Reciprocal Lattice
The reciprocal lattice is a mathematical construct of points each corresponding to a given Miller index, hkl.
It is a three dimensional lattice where the nodes are diffracted intensities and the spacing between points is inversely proportional to the unit cell parameters in real space.

13. Reciprocal Lattice Reciprocal axes are denoted For Orthorhombic
a*, b*, c*, a*, b*, g*
For Orthorhombic
a* = 1/a ; b* = 1/b ; c* = 1/c
Scale is arbitrary

14. Wadsleyite Imma hk0
a = 5.7Å
b = 11.5 Å
c = 8.3 Å

15. Four-Circle Diffractometer

16. Four-Circle Diffractometer

17. Four-Circle Diffractometer

18. Diffraction Experiment
Mount Crystal (~100mm)
Rotation photograph
Index and obtain matrix
Refine unit cell parameters (1/104)
Measure Intensities ( )
Determine or refine atom position and displacement parameters