This is an educational ppt of Physics 2 for engineering first year students. The presentation is according to APJKTU, Lucknow syllabus

1. Crystal Structure and X ray
Diffraction
Unit I
Dr Md Kaleem
Department of Applied Sciences
Jahangirabad Institute of Technology (JIT),
Jahangirabad, Barabanki(UP) - 225203
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2. • Relationship between structures of engineering
materials
• To understand the classification of crystals
• To understand mathematical description of ideal
crystal
• To understand Miller indices for directions and
planes in lattices and crystals
• To understand how to use X-Ray Diffraction for
determination of crystal geometry
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4. Solid can be divided in two categories on the basis of
periodicity of constituent atoms or group of atoms
• Crystalline solids consists of atoms, ions or molecules
arranged in ordered repetitive array
e.g: Common inorganic materials are crystalline
– Metals : Cu, Zn, Fe, Cu-Zn alloys
– Semiconductors: Si, Ge, GaAs
– Ceramics: Alumina (Al2O3), Zirconia (Zr2O3), SiC, SrTiO3.
• Non crystalline or Amorphous consists of atoms, ions
or molecules arranged in random order
e.g: organic things like glass, wood, paper, bone, sand; concrete walls, etc
Crystalline Solids  grains  crystals
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Crystal = Lattice + Motif
Lattice : regular repeated three-dimensional arrangement of
points
Motif/ Basis: an entity (typically an atom or a
group of atoms) associated with each lattice
point

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Lattice  where to repeat
Motif  what to repeat
Lattice: Translationally periodic arrangement of points
Crystal: Translationally periodic arrangement of motifs

7. Space lattice: An array of points in space such that every point
has identical surroundings
Unit Cell: It is basic structural unit of crystal, with an atomic
arrangement which when repeated three dimensionally gives
the total structure of the crystal
Lattice Parameters: It defines shape and size of the unit cell
Three lattice vector (a, b, c) and
interfacial angle (, , ) are
known as lattice parameters
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8. Unit cell with lattice points at the corners only, called
primitive cell. Unit cell may be primitive cell but all primitive
cells are not essentially unit cells.
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9. • Crystallographers classified the unit cells into
seven possible distinct types of unit cells by
assigning specific values to lattice vector (a, b,
c) and interfacial angle (, , ) called seven
crystal system.
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10. Crystal System Lattice
Vector
Interfacial Angle Example
1 Cubic a = b = c  =  = = 90o NaCl, CaF2, Au, Ag, Cu, Fe
3 Tetragonal a = b ≠ c  =  = = 90o TiO2, NiSO4, SnO2
3 Orthorhombic a ≠ b ≠ c  =  = = 90o KNO3, BaSO4, PbCO3, Ga
4 Monoclinic a ≠ b ≠ c  =  = 90o≠  CaSO4.2H2O (Gypsum),
FeSO4
5 Triclinic a ≠ b ≠ c  ≠  ≠ ≠ 90o CuSO4, K2Cr2O7
6 Trigonal a = b = c  =  = ≠ 90o As, Sb, Bi, Calcite
7 Hexagonal a = b ≠ c  =  = 90o,
=120o
SiO2, AgI, Ni, As, Zn, Mg
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12. • A. J. Bravais in 1948 shown that with the
centering (face, base and body centering) added
to these, 14 kinds of 3D lattices, known as Bravais
lattices.
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13. Coordination Number: It is defined as the number of nearest
neighbors around any lattice point in the crystal lattice.
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14. •Miller indices for crystallographic
planes
•Miller notation system (hkl)
•Miller index – the reciprocals of
the fractional intercepts that the
plane makes with the x, y, and z
axes of the three nonparallel edges
of the cubic unit cell
William Hallowes Miller
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15. • Choose a plane not pass through (0, 0, 0)
• Determine the intercepts of the plane with x,
y, and z axes
• Form the reciprocals of these intercepts
• Find the smallest set of whole numbers that
are in the same ratio as the intercepts
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16. • Find the Miller Indices of the plane which cuts off intercepts in the ratio
1 a:3b:-2c along the three co-ordinate axes, where a, b and c are the
primitives.
• If pa, qb and rc are the intercepts of the given set of planes on X-, Y-, and
Z- axes respectively then,
pa: qb: rc= 1 a:3b:-2c
or p:q:r=1:3:-2
so 1/p : 1/q : 1/r = 1/1 :1/3 : -1/2
LCM of 1, 3 and 2 = 6, so multiply by it
1/p : 1/q : 1/r = 6:2:-3
Thus the Miller Indices of the plane is (6 2 )
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21. • It is infinite periodic three dimensional array
of reciprocal lattice points whose spacing
varies inversely as the distances between the
planes in the direct lattice of the crystal.
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22. Take some point as an origin
From this origin, lay out the
normal to every family of parallel
planes in the direct lattice;
Set the length of each normal
equal to 2p times the reciprocal of
the interplanar spacing for its
particular set of planes;
Place a point at the end of each
normal.
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23. • Any diffraction pattern of a crystal is a map of the reciprocal lattice of the
crystal whereas the microscopic image is a map of the direct lattice.
• While the primitive vectors of a direct lattice have the dimensions of
length those of the reciprocal lattice have the dimensions of (length)− 1.
• Direct lattice or crystal lattice is a lattice in ordinary space or real space.
Reciprocal lattice is in reciprocal space or k-space or Fourier space.
• The direct lattice is the reciprocal of its own reciprocal lattice.
• The reciprocal lattice of a simple cubic lattice is also a simple cubic lattice.
• The reciprocal lattice of a face centered cubic lattice is a body centered
cubic lattice.
• The reciprocal lattice of a body centered cubic lattice is a face centered
cubic lattice, and
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24. NaCl has a cubic unit cell. It is
best thought of as a face-
centered cubic array of anions
with an interpenetrating fcc
cation lattice (or vice-versa)
The cell looks the same
whether we start with anions
or cations on the corners. Each
ion is 6-coordinate and has a
local octahedral geometry.
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25. • The Bravais space lattice of NaCl is truly fcc with a
basis of one Na+ ion one Cl- ion separated by one half
the body diagonal (a√3/2) of a unit cube.
• There are four pair of Na+ and Cl- ions present
per unit cell.
• The position of ions in unit cell are
• Na+ : (½, ½, ½), (0,0, ½), (0, ½,0), (½,0,0)
• Cl- : (0,0,0), (½, ½,0), (½,0, ½), (0, ½, ½)
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26.  For electromagnetic radiation to be diffracted
the spacing in the grating should be of the
same order as the wavelength
 In crystals the typical inter-atomic spacing ~
2-3 Å so the suitable radiation is X-rays
 Hence, X-rays can be used for the study of
crystal structures
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27. The path difference between rays = 2d Sin
 For constructive interference: n = 2d Sin
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28. • Q. A beam of X-rays of wavelength 0.071 nm is diffracted
by (110) plane of rock salt with lattice constant of
0.28 nm. Find the glancing angle for the second-order
diffraction.
• Given data are:
• Wavelength (λ) of X-rays = 0.071 nm, Lattice constant
(a) = 0.28 nm
Plane (hkl) = (110), Order of diffraction = 2
Glancing angle θ = ?
Bragg’s law is 2d sin θ = nλ
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29. Substitute in Bragg’s equation
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30. Bragg’s spectrometer method is
one of the important method for
studying crystals using X-rays. The
apparatus consists of a X-ray tube
from which a narrow beam of X-
rays is allowed to fall on the crystal
mounted on a rotating table. The
rotating table is provided with scale
and vernier, from which the angle
of incidence, θ can be measured.
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31. • Bragg’s spectrometer is used to determine the
structure of crystal.
• The ratio of lattice spacing for various groups
of planes are obtained by using Bragg’s Law.
• The ratio would be different for different
crystals
• By comparing those known standard ratios
with experimentally determined ratios, crystal
structure can be obtained.
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32. • If for a particular crystal having interplaner
spacing d1, d2, d3 strong Bragg’s reflection
occur at glancing angle θ1, θ2, θ3 then from
Bragg’s law
• 2d1sin θ1=λ, 2d2sin θ2=λ, 2d3sin θ3=λ
• So, d1: d2: d3 = 1/sin θ1= 1/sin θ2=1/sin θ3
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33. • For KCl Crystal, Bragg’s obtained strong Bragg’s reflection at
θ1= 5o23’, θ2=7o37’, θ3=9o25’’for planes (100), (110) and (111)
• So, d100: d110: d111= 1/sin 5o23’= 1/sin 7o37’=1/sin 9o25’
= 1:1/√2:1/√3
• This corresponds to theoretical result for simple cubic lattice .
Therefore it is concluded that KCl crystal has simple cubic
structure.
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34. • When light encounters charged particles, the particle
interact with light and cause some of the light to be
scattered. This is called Compton Scattering.
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35. • Arthur H. Compton in 1923 observed that
when electromagnetic wave of short
wavelength (X ray) strikes an electron, an
increase in wavelength of X-rays or gamma
rays occurs when they are scattered.
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