1. UNIT-II
X-ray Diffraction: Basic Principles, Bragg’s Law, Powder Method, Applications of X-ray Diffraction.
Defects in Crystals: Point Defects: Vacancies, Substitutional, Interstitial, Frenkel and Schottky Defects-Calculation of concentration, Qualitative treatment of line (Edge and Screw Dislocations) Defects, Burger’s Vector.

2. X-ray Diffraction
Basic Principles, Bragg’s Law, Powder Method, Applications of X-ray Diffraction.

3. Introduction:
X-ray diffraction techniques are very useful for crystal structure analysis and identification of different types of crystals.
Experimental study of crystalline materials became possible only after the discovery of X-rays.
Diffraction occurs when waves traveling through an object whose dimensions are order of wavelength.
Typical inter atomic spacing in crystals is 2-5°A.
The x-rays have wavelengths (0.02°A to 100°A) in this range . Hence x-ray diffraction is utilized to study the crystal structures.

4. Basic Principle:
X-rays are the electromagnetic waves like visible light. Hence , x-rays exhibit the phenomenon like interference and diffraction. But x-rays have very short wavelength of the order of 1A0 . Therefore it is not possible to produce diffraction of x-rays by using diffraction grating. The reason is that grating spacing is more (20,000 A0 ) compared to x-rays wavelength.
However it is possible to obtain x-rays diffraction by making use of crystals in which the atoms are uniformly spaced in planes and separated by a distance of order of 2A0 to 5A0 . Therefore the diffraction of x-rays takes place when they incident on the surface of crystals.
Hence, X-rays can be used for the study of crystal structures

5. Bragg’s Law:
Bragg’s law states the x-rays reflected from different parallel planes of a crystal interfere constructively when the path difference is integral multiple of the wavelength of x-rays. A C D F
Consider a set of parallel lattice planes of a crystal separated by spacing ‘d’ between each other as shown in figure. B θ θ G θ θ H d E

6. Consider narrow beam of x-rays AB and DE are incident on the surface of crystal at angle ‘θ’.
The beam will be reflected in all directions by the atoms of various atomic planes.
After reflections these rays reflected (BC & EF) and are in phase and interfere each other.
Now extended the rays AB to cut EF at ‘H’ and CB to cut DE at ‘G’.
The path length of the ray DEF is greater than the path length of ABC by an amount (GE+EH)

7. Therefore, the path difference between two rays reflected from adjacent planes
GE + EH (1)
Let us consider GBE , EBH and BE = ‘d’
From angle GBE, Sin θ = GE/BE GE= BE Sin θ
= d Sin θ ----(2)
From angle EBH, Sin θ = EH/BE EH= BE Sin θ
= d Sin θ ----(3)
Therefore, the path difference = GE + EH
= d Sin θ + d Sin θ
= 2d Sin θ

8. But, for constructive interference , the path difference must be an integral multiple of wavelength i.e., nλ
where n= 0,1,2,3…….
this relations is know as Bragg’s Law.
2d Sin θ = nλ
Note:
Bragg’s law gives the diffraction maxima condition for reflected x-rays from several planes, where θ is the angle of diffraction and ‘n’ is the order of diffraction.
If n=0, it is called zero order diffraction i.e., it has not experienced any diffraction.
If n=1, First order Maxima
θ1 = Sin-1 [ λ/2d]

9. If n=2, Second order Maxima
θ2 = Sin-1 [ 2λ/2d]
If n=3, Third order Maxima
θ3 = Sin-1 [ 3λ/2d]
and so on
Sin θ1 : Sin θ2 : Sin θ3 =1:2:3
The diffraction angles θ1 , θ2 & θ3 are measured by Bragg’s Spectrometer.

10. Importance of Braggs law
The Bragg’s equation can be used for determining the lattice parameters of cubic crystals.
Let us consider the first order spectrum from three planes of cubic crystal system.
Hence by knowing the values of glancing angles ratio,
the ratio of inter-planar spacing
hence the type of lattice can be identified.

11. X-ray Diffraction Methods:
There are three main x-ray diffraction methods by which the crystal structures can be analyzed.
They are
i). Laue Method – Applicable to Single Crystal
ii). Powder Method(Debye-Scherrer Method)
–finely divided crystalline or polycrystalline specimen
powder
iii). Rotating Crystal Method --- Single Crystal

12. Powder (Debye-Scherrer) Method:
The powder method is used to determine the inter planar spacing.
The Powder Method is applicable to finely divided Crystalline powder.
A monochromatic X-ray beam of radiation is incident on randomly oriented crystals in the powder form(fine grained poly crystalline specimen) taken in a thin walled capillary tube or pasted on a thin fiber of glass.
The specimen is located at the center of circular camera with photographic film at the inner circumference ( Debye-Scherrer camera)

13. r Incident X-Ray beam Crystal Powder l1 l2 l32θ
Incident X-Ray beam
Crystal Powder l1 l2 l3
Lines on the powder photographic film

14. Raddi of these rings can be used to find glancing angle.
A narrow pencil of monochromatic X-Ray is diffracted from the powder and recorded by the Photographic film as a series of concentric circular rings.
Raddi of these rings can be used to find glancing angle.
Due to the narrow width of the film, only parts of circular rings registered on it(as shown in above figure).
The curvature of arcs reverses when the angle of diffraction exceeds 900 .
Let l1 , l2 & l3 … etc be the distance between symmetrical lines on stretched photograph and ‘D’ is the diameter of cylindrical film or circular film,
where ‘θ’ is glancing angle

15. Applications of Powder Method Study of d-spacing. Study of mixtures.
Using the values of ‘θ’ in Bragg’s formula inter planar spacing ‘d’ can be determined.
Applications of Powder Method
Study of d-spacing.
Study of mixtures.
Study of alloys.
Stress determination in metals.
Determination of particle size.

16. Applications of X-ray Diffraction:
I). XRD analysis providers:
a). Analysis of solid and liquid samples.
b). Reflection, transmission or capillary geometry in high temperature and reactive environment.
c). Orientation/ texture data
d). Grazing incidence
e). Reflectometry
f). Kralky small angle X-ray scattering (SAXS)
I). XRD analysis Applications:
a). Determination of lattice parameters
b). Determination of grain size in polycrystalline blocks.

17. c). Nano materials: phase composition, crystallite size and shape , lattice distortions and faulting, composition variations , orientation. d). New materials development e). Polymer & Composites: Crystalline form, crystallinity, crystalline perfection, orientation. f). Pharmaceuticals and organics g). Catalyst for nano-materials, with monitoring of structure through manufacturing and use to develop structure- property relationships. Limitations of XRD Analysis: i). Cannot identify amorphous materials. ii). No depth profile information iii). Minimum spot size of ~50um

18. Defects in Crystals Point Defects: Vacancies, Substitutional, Interstitial, Frenkel and Schottky Defects-Calculation of concentration, Qualitative treatment of line (Edge and Screw Dislocations) Defects, Burger’s Vector.

19. Introduction
In an ideal crystal, the atomic arrangement is perfectly regular and continuous but real crystals never perfect.
They always contain a considerable density defects and imperfections that affect their physical, chemical ,mechanical and electronic properties.
Crystalline imperfections can be classified on the basis of their geometry under four main divisions namely

20. Defects Point defects (0-dimensional) Line defects 1.Edge dislocation
1.Vacancies or Schottky
2.Interstitialcies or Frenkel
3.Compositional defects.
a. Substitutional
b. interstitial
4.Electronic defects
Defects
Point defects
(0-dimensional)
Line defects
(1-dimensional)
Surface defects
(2-dimensional)
Volume defects
(3-dimensional)
1.Edge dislocation
2.Screw dislocation
1.Grain boundaries
2.Tilt boundaries
3.Twin boundaries
4.Stacking faults
1.Cracks
2.Voids or air bubbles

21. Point imperfections are also called zero dimensional imperfections.
Point Defects:
Point imperfections are also called zero dimensional imperfections.
One or two atomic diameters is the typical size of a point imperfection.
Perfect Crystal

22. Vacancy:
A Vacancy refers to an atomic site from where the atom is missing.

23. Compositional defects:
A Substitutional impurity is a point imperfection and it refers to a foreign atom that substitutes for or replaces a parent atom in the crystal.
A small sized atom occupying the void space in the parent crystal disturbing the parent atoms from their regular sites is a interstitial impurity.

24. Schottky defect:
A pair of one cataion and one anion can be missing from an ionic crystal as shown in a figure. such a pair of vacant ion sites is called Schottky defect.

25. Frenkel Defect: An atom leaves the regular site and occupies interstitial position. Such defects are called Frenkel defects.
The point imperfections in silver halides and CaF2 are of the Frenkel type.

26. Electronic defects
Errors in charge distribution in solids are called electronic defects.
These defects are produced, when the composition of an ionic crystal does not correspond to the exact Stoichiometric formula.

27. Calculation of number of vacancies at a given temperature.
All most in all crystals vacancies are present and the main cause for these defects is thermal agitation.
Let us consider ‘ Ev’ is the energy required to move an atom from lattice site inside the crystal to lattice site on the surface.
Therefore the amount of energy required to produce ‘n’ number of isolated vacancies can be written as

28. The total number of ways to move ‘n’ number of atoms out of ‘N’ number of atoms in a crystal on to its surface will be
The increase in entropy due to formation of ‘n’ vacancies can be written as

29. But the free energy
Using Sterling approximation, log x! = x log x - x

30. At thermal equilibrium, free energy is constant and minimum with respect to ‘n’, hence
Hence equilibrium concentration of vacancies decreases with temperature.

31. Calculation of number Schottky defects at a given temperature:
In ionic crystals, the number of Schottky defects at a given temperature, can be calculated assuming an equal number of positive and negative ion vacancies are present.
Let us consider ‘ Es’ is the energy required to move an ion Pair from lattice site inside the crystal to a lattice site on the surface.
Therefore the amount of energy required to produce ‘n’ number of isolated ion pair vacancies will be

32. The total number of ways that to move ‘n’ numbers of ion pairs out of ‘N’ number of ionic molecules in a crystal on to the surface will be

33. The free energy
Using sterling approximation

34. At thermal equilibrium, free energy is constant and minimum with respect to ‘n’, hence

35. Calculation of number of Frenkel Defects at given temperature:
In ionic crystal an ion may be displaced from the regular lattice into an interstitial site or void space.
If it is so, then a vacancy and an interstitial defect will be formed.
A Frenkel imperfections in silver halides and calcium fluoride are of the Frenkel type.
Frenkel and Schottky defects together are called Intrinsic defects.

36. Let us consider Ei is the energy required to move an atom from lattice site inside the crystal to a lattice site on the surface.
The amount of energy required to produce ‘n’ number of isolated vacancies…

37. The total number of ways to move n numbers of ions out of N number ionic molecules in a crystal on to the surface will be,

39. At equilibrium, the free energy is constant and minimum with respect to n, hence

40. Hence it is concluded that number of Frenkel defects, is proportional (NNi)1/2

41. Line defects Line defects
Line defects are one dimensional imperfections in the geometrical sense.
There are in general two types of dislocations:
1. Edge dislocation
2. Screw dislocation

42. Edge dislocation
In a perfect crystal, atoms are arranged in both vertical and horizontal planes parallel to the side faces.
If one of these vertical planes does not extended to full length but ends in between, within the crystal as shown in figure, it is called edge dislocation.
Edge dislocations are symbolically represented by ┴ or ┬ or depending on whether the incomplete plane starts from the top or from the bottom of the crystal.
These two configurations are referred to as positive and negative edge dislocations.

43. Perfect Crystal
An incomplete plane in a
Crystal results in an edge dislocation

44. Perfect crystal
Extra half plane
Slip plane
Edge dislocated crystal

45. The edge dislocation containing an extra plane of atoms lying above the positive slip plane (or) Burgers plane are conventionally called the positive edge dislocation.
If the extra half plane of atoms containing below the slip plane called the negative edge dislocation.

46. Positive and negative dislocations

47. Burgers vector
The magnitude and the direction of the displacement are defined by a vector called the Burgers vector.
Consider two crystals one perfect and another with edge dislocation.

48. Fig 2. Fig 1. An incomplete plane in a Crystal results in an edgeQ b P
An incomplete plane in a
Crystal results in an edge
dislocation
Perfect crystal

49. The Burgers vector is perpendicular to the edge dislocation line.
From fig. 1.
Starting from the point P, we go up by 6 steps, then move towards right by 5 steps, and move down by 6 steps and finally move towards left by 5 steps to reach the starting point P, the burgers circuit gets closed.
From fig 2.
We end up at Q instead of the starting point P.
Now we have to move an extra step QP to return to ‘P’ in order to close the burgers circuit.
The magnitude and the direction of the step defines the Burgers vector (BV)
BV = QP = b
The Burgers vector is perpendicular to the edge dislocation line.

50. Screw dislocation:
The second basic type of dislocation is the Screw or Burgers dislocation.
In this, the atoms are displaced in two separate planes perpendicular to each other.
In a figure the plane ABCD is the slipped area.
The upper portion of the crystal has been sheared by an atomic distance to the right relative to the lower portion.
No slip has taken place to the right of AD and AD is a dislocation line.

51. A B C D
Shear vector
Here, the dislocation is parallel to its Burgers vector or shear vector.
The designation ‘screw’ for this lattice defect is derived from the fact that the lattice planes of the crystal spiral the dislocation line AD.