crystal structure and x ray diffraction

Crystal Structure and X ray
Diffraction
Unit I
Dr Md Kaleem
Department of Applied Sciences
Jahangirabad Institute of Technology (JIT),
Jahangirabad, Barabanki(UP) - 225203
1/31/2017 1DR MD KALEEM/ ASSISTANT PROFESSOR

• 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

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

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

1/31/2017 DR MD KALEEM/ ASSISTANT PROFESSOR 6
Lattice  where to repeat
Motif  what to repeat
Lattice: Translationally periodic arrangement of points
Crystal: Translationally periodic arrangement of motifs

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

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.

• 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.

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

• 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.

Coordination Number: It is defined as the number of nearest
neighbors around any lattice point in the crystal lattice.

•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

• 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

• 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 )
3

• 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.

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.

• 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

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.

• 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)

 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

The path difference between rays = 2d Sin
 For constructive interference: n = 2d Sin

• 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

Substitute in Bragg’s equation

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.

• 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.

• 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

• 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.

• When light encounters charged particles, the particle
interact with light and cause some of the light to be
scattered. This is called Compton Scattering.

• 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.
  cos1
cm
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crystal structure and x ray diffraction

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