crystal structure and x ray diffraction

$Page 1: crystal structure and x ray diffraction$
Crystal Structure and X ray Diffraction

Unit I

Dr Md KaleemDepartment 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 + MotifLattice : regular repeated three-dimensional arrangement of

points

Motif/ Basis: an entity (typically an atom or agroup of atoms) associated with each latticepoint

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 pointhas identical surroundings

Unit Cell: It is basic structural unit of crystal, with an atomicarrangement which when repeated three dimensionally givesthe 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 thecentering (face, base and body centering) addedto 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 arrayof reciprocal lattice points whose spacingvaries inversely as the distances between theplanes in the direct lattice of the crystal.


Take some point as an originFrom this origin, lay out thenormal to every family of parallelplanes in the direct lattice;Set the length of each normalequal to 2p times the reciprocal ofthe interplanar spacing for itsparticular set of planes;Place a point at the end of eachnormal.


• 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 isbest thought of as a face-centered cubic array of anionswith an interpenetrating fcccation lattice (or vice-versa)The cell looks the samewhether we start with anionsor cations on the corners. Eachion is 6-coordinate and has alocal octahedral geometry.


• The Bravais space lattice of NaCl is truly fcc with abasis of one Na+ ion one Cl- ion separated by one halfthe 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 diffractedthe spacing in the grating should be of thesame 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 ofcrystal 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 isone of the important method forstudying crystals using X-rays. Theapparatus consists of a X-ray tubefrom which a narrow beam of X-rays is allowed to fall on the crystalmounted on a rotating table. Therotating table is provided with scaleand vernier, from which the angleof 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 interplanerspacing 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 thatwhen electromagnetic wave of shortwavelength (X ray) strikes an electron, anincrease in wavelength of X-rays or gammarays occurs when they are scattered.


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Engineering

crystal structure and x ray diffraction