1. Determination of Crystal Structures by X-ray Diffraction
Lec. 6,7
Determination of Crystal Structures by X-ray Diffraction

2. X-Ray Diffraction
Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
Can’t resolve spacings  
Spacing is the distance between parallel planes of atoms.  

3. X-Ray Diffractıon Methods
Von Laue
Rotating Crystal
Powder
Lattice
Parameters
Poly Crystal
Monchromatic
Beam, Variable
Angle Many s
(orientations)
Orientation
Single Crystal
Polychromatic
Beam, Fixed
Angle single 
Lattice constant
Single Crystal
Monchromatic
Beam, Variable
Angle 
Varied by
rotation

4. Laue Method The Bragg angle is fixed
The Laue method is mainly used to determine the orientation of large single crystals while radiation is reflected from, or transmitted through a fixed crystal.
The diffracted beams form arrays of spots, that lie on curves on the film.
The Bragg angle is fixed
for every set of planes in the crystal. Each set of planes picks out & diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d & θ involved.

5. Transmission Laue Method
In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal.
In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal.
One side of the cone of Laue
reflections is defined by the
transmitted beam.
The film intersects the cone,
with the diffraction spots
generally lying on an ellipse.
Single
Crystal
Film
X-Rays

6. Crystal Structure Determination
by the Laue Method
The Laue method is mainly used to determine the crystal orientation.
Although the Laue method can also be used to determine the crystal structure, several wavelengths can reflect in different orders from the same set of planes, with the different order reflections superimposed on the same spot in the film. This makes crystal structure determination by spot intensity diffucult.
The rotating crystal method overcomes this problem.

7. Rotatıng Crystal Method
In the rotating crystal method, a single crystal is mounted with an axis normal to a monochromatic x-ray beam. A cylindrical film is placed around it & the crystal is rotated about the chosen axis. 
As the crystal rotates, Sets of lattice planes will at some point make the correct Bragg angle
for  the monochromatic incident beam, & at that point a diffracted beam will be formed.  

8. Rotatıng Crystal Method
The Lattice constant of the crystal can be determined with this method. For a given wavelength λ if the angle θ at which a reflection occurs is known, d can be determined by using Bragg’s Law.

9. Rotatıng Crystal Method
The reflected beams are located on the surfaces of imaginary cones. By recording the diffraction patterns (both angles & intensities) for various crystal orientations, one can determine the shape & size of unit cell as well as the arrangement of atoms inside the cell.
Film

10. Hence, X-rays can be used for the study of crystal structures
For electromagnetic radiation to be diffracted the spacing in the grating should be of the same order as the wavelength
In crystals the typical interatomic spacing ~ 2-3 Å so the suitable radiation is X-rays
Hence, X-rays can be used for the study of crystal structures
Target
X-rays
Beam of electrons
An accelerating (/decelerating) charge radiates electromagnetic radiation

11. Relationship of the Bragg angle (θ) and the experimentally measured diffraction angle (2θ).
X-ray
intensity
(from
detector) q c
d =
n l
2 sin

12. K K Mo Target impacted by electrons accelerated by a 35 kV potential
Characteristic radiation → due to energy transitions in the atom K
White radiation
Intensity
0.2
0.6
1.0
1.4
Wavelength ()

13. Target Metal
 Of K radiation (Å) Mo
0.71 Cu
1.54 Co
1.79 Fe
1.94 Cr
2.29

14. d dSin The path difference between ray 1 and ray 2 = 2d Sin
BRAGG’s EQUATION
Deviation = 2
Ray 1
Ray 2     d 
dSin
The path difference between ray 1 and ray 2 = 2d Sin
For constructive interference: n = 2d Sin

17. Note that in the Bragg’s equation:
The interatomic spacing (a) along the plane does not appear
Only the interplanar spacing (d) appears
 Change in position or spacing of atoms along the plane should not affect Bragg’s condition !! d
Note: shift (systematic) is actually not a problem!

18. Diffraction = Reinforced Coherent Scattering
Bragg’s equation is a negative law  If Bragg’s eq. is NOT satisfied  NO reflection can occur  If Bragg’s eq. is satisfied  reflection MAY occur
Diffraction = Reinforced Coherent Scattering
Reflection versus Scattering
Reflection
Diffraction
Occurs from surface
Occurs throughout the bulk
Takes place at any angle
Takes place only at Bragg angles
~100 % of the intensity may be reflected
Small fraction of intensity is diffracted
X-rays can be reflected at very small angles of incidence

19. n is an integer and is the order of the reflection
n = 2d Sin, n= 1, 2, 3, …
n is an integer and is the order of the reflection
For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å n
Sin  1
0.34
20.7º
First order reflection from (110) 2
0.69
43.92º
Second order reflection from (110)
Also written as (220)

20. In XRD nth order reflection from (h k l) is considered as 1st order reflection from (nh nk nl)

22. The Powder Method
If a powdered crystal is used instead of a single crystal, then there is no need to rotate it, because there will always be some small crystals at an orientation for which diffraction is permitted. Here a monochromatic X-ray beam is incident on a powdered or polycrystalline sample.
Useful for samples that are difficult to obtain in single crystal form.
The powder method is used to determine the lattice parameters accurately. Lattice parameters are the magnitudes of the primitive vectors a, b and c which define the unit cell for the crystal.

23. The Powder Method
For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction.
Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line.

24. The Powder Method
If a monochromatic X-ray beam is directed at a single crystal, then only one or two diffracted beams may result. See figure
For a sample of several randomly orientated single crystals, the diffracted beams will lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards See figure
For a sample of hundreds of crystals (powdered sample), the diffracted beams form continuous cones. A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film. See figure

25. THE POWDER METHOD
Cone of diffracted rays

26. Diffraction cones and the Debye-Scherrer geometry
POWDER METHOD
Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections
Film may be replaced with detector

27. Debye Scherrer Camera
A small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract X-Rays.
The sample is placed in the Debye Scherrer camera and is accurately aligned to be in the center of the camera. X-Rays enter the camera through a collimator.
The powder diffracts the X-Rays in accordance with Braggs Law to produce cones of diffracted beams. These cones intersect a strip of photographic film located in the cylindrical camera to produce a characteristic set of arcs on the film.

28. Powder Diffraction Film
When the film is removed from the camera, flattened & processed, it shows the diffraction lines & the holes for the incident & transmitted beams.

29. Some Typical Measurement Results
Laue - “white” X-rays
Yields stereoscopic projection of reciprocal lattice
Rotating-Crystal method: monochromatic X-rays
Fix source & rotate crystal to reveal reciprocal lattice
Powder diffraction - monochromatic X-rays
Powder sample to reveal “all” directions of RL

30. (Courtesy H&M Services.)
Photograph of a
XRD Diffractometer
(Courtesy H&M Services.)

31. (a) Diagram of a
diffractometer
showing a powdered
sample, incident &
diffracted beams.
(b) Diffraction Pattern
from a sample of
gold powder.

32. Example (From the Internet)
The results of a diffraction experiment using X-Rays with λ = Å (radiation obtained from a molybdenum, Mo, target) show that diffracted peaks occur at the following 2θ angles:
Find: The crystal structure, the indices of the plane producing each peak, & the lattice parameter of the material.

33. Example (Solution)
First calculate the sin2 θ value for each peak, then divide through by the lowest denominator,

34. Example (Solution Continued) Picking Peak 8: 2θ = 59.42° or θ = 29.71°
Then use the 2θ values for any of the peaks to calculate the interplanar spacing & thus the lattice parameter.
Picking Peak 8: 2θ = 59.42° or θ = 29.71°
So, for example: Ǻ Ǻ
This is the lattice parameter for body-centered cubic iron.

35. Note: XRD is a nondestructive technique! Some uses of XRD include:
Applications of XRD
Note: XRD is a nondestructive technique!
Some uses of XRD include:
Distinguishing between crystalline & amorphous materials.
Determination of the structure of crystalline materials.
Determination of electron distribution within the atoms, & throughout the unit cell.
Determination of the orientation of single crystals.
Determination of the texture of polygrained materials.
Measurement of strain and small grain size…..etc.

36. Advantages & Disadvantages of XRD Compared to Other Methods
X-Rays are the least expensive, the most convenient & the most widely used method to determine crystal structures.
X-Rays are not absorbed very much by air, so the sample need not be in an evacuated chamber.
Disadvantages
X-Rays do not interact very strongly with lighter elements.

37. Diffraction Methods