1. X-ray Diffraction & Crystal Structure Analysis
Chapter 1. Introduction
X-ray Diffraction & Crystal Structure Analysis
Powder XRD pattern
Single-crystal XRD pattern
Intensity
Two-theta

2. Diffraction spots
on detector
x’tal
Incident X-ray
beam

3. Intensity
Two-theta

5. Type λ 對空氣穿透率
(10 dm path length)
Hard X-rays < 5 Å > 80%
Soft X-rays > 5 Å < 80%
Applications: (i) Diffraction for powder, single crystals
and partially crystalline materials
(ii) Spectroscopy
(iii) Spectrometry (medical and industrial)

6. Intensity of scattering I  |a|2  (q/m)2|E|2
When the particle is large compared to the wavelength of the radiation, therev will be several cycles of the waves contained in the same volume of space as that occupied by the particle. The particle will experience the average electric field close to zero.
If the particle is small compared to the wavelength of the radiation, it will “see” only a very small portion of a cycle of the wave, and will experience a
well-defined electric field E. It then, experience a force F given by F = qE. If the particle has a mass m, the force F will give rise to an acceleration a,
as F = ma = qE with a = (q/m)E.
incident wave
scattered wave
Intensity of scattering I  |a|2  (q/m)2|E|2
The above equation states that the intensity of the radiation scattered by
a particle of mass m and charge q depends on the ratio (q/m)2.

7. X-rays are scattered by electrons
Neutrons have no charge and will not scatter X rays.
Protons have the elemental charge e, and a mass mp, and electrons also have the elemental charge e, but a mass me. A proton, however, is 1837 times heavier than an electron, i. e.
mp = 1837 me . The intensities of radiation scattered by an electron Ie and that by a proton Ip are
Ie / Ip = (e/me)2(mp/e)2 = 18372
X-ray diffraction therefore looks at the electron distribution in a crystal, and not directly at the positions of the nuclei of atoms.

10. Chapter 2

11. – on the center of symmetry
Lattice: a set of identical points in identical surroundings. b a
Lattice point
– in identical enviroment
– on the center of symmetry
Translation vectors
intersecting at lattice points
2D lattice: a and b are non-parallel
3D lattice : a, b, and c are non-co-planar

12. a, b, c, , ,  V = a x b . c 2.1.1 The Unit cell b.
6 cell constants (parameters):
a, b, c, , , 
right-handed system
V = a x b . c
V = abc(1- cos2 - cos2  -cos2
+ 2cos - cos -cos)½ b.

13. 555
565
455
556

14. How many lattice points are in each unit?
Select a unit cell for each of the following lattices.
How many lattice points are in each unit?
(a)
(c)
(b)

17. 1  x  0
1  y  0
1  z  0

18. (i) inversion center i (x,y,z)  (-x,-y,-z)
Crystal Symmetry--
Three simple symmetry elements i, m, n
(i) inversion center i (x,y,z)  (-x,-y,-z)
(ii) mirror plane m (x,y,z)  (x,y,-z) …
(iii) rotation axes n, where n = 360o/α
(α= angle of rotation) (x,y,z)  (-x,-y,z) …
All two-dimensional lattices have two-fold rotation axis.
180o
Horizontally clockwise rotation

19. The periodicity (i.e. operation of translation) will impose
rotational symmetry to a lattice. P Q P’ Q’  a
If P and Q are lattice points. If the lattice possesses a rotation
of  angle, the resulting P’ and Q’ are also lattice points.
DPQ’ = ma = a + 2a sin ( - 90o)
(m must be an integer).

20. In all 2D and 3D lattices, only five kinds of rotation angles
DPQ’ = ma = a + 2a sin ( - 90o)
(m must be an integer).
After rearrangement,
cos  = (m-1)/2
m must fall in the range of 3 to -1:
m  n-fold rotation
o (one-fold)
o (six-fold)
o (four-fold)
o (three-fold)
o (two-fold)
In all 2D and 3D lattices, only five kinds of rotation angles
are allowed.

21. Five 2D Lattices cell parameters: a, b, and  2-fold  to plane
Rotational Symmetry
2-fold  to plane
one 2-fold  to plane
two 2-fold // plane

22. same as above
4-fold  to plane
6-fold or 3-fold
 to plane

23. The Seven Crystal Systems: cell parameters: a, b, c, , , 
no. independent
parameters. 6 4 3 2 1 *
The symbol  implies that equality is not required by symmetry.
*In hexagonal lattice. If in rhombohedral lattice: a = b = c;  =  =   90o
The Seven Crystal Systems:
cell parameters: a, b, c, , , 

24. Basic Lattice Types: P, I, C, F, and R

25. The 14 Bravais Lattices :

26. Relationship between
monoclinic B and P lattices
Choice of C or I unit cells in
monoclinic lattices

27. Relationship between
tetragonal C and P lattices
Possible choice of an orthorhombic unit cell
in the hexagoanl system

28. Ten basic symmetry elements in crystals
(i) Combination of simple symmetry elements of inversion and rotation m i
They also represent ten point groups.
(ii) Combination of symmetry axes – when two rotation axes are combined
a third rotation axis is created. There are only six possible combinations of
rotation axes of symmetry that can operate on crystal lattice.

29. -- the 32 crystallographic point groups
The the External Symmetry of Crystals
-- the 32 crystallographic point groups
-- the 32 crystal class

30. Three Translational vectors 10 basic symmetry elements
Five rotation axes
Four lattice types
32 symmetry point groups
10 basic symmetry elements
Seven crystal systems
14 Bravais lattices
73 symmorphic
space groups
Three simple
symmetry elements
Symmetry
combinations
External symmetry of crystals