This document contains information about crystal structures, including: - Unit cells can be primitive or non-primitive, with primitive cells containing one lattice point and non-primitive cells containing additional points. - Crystal structures are defined by their lattice type, lattice parameters, and motif. Lattice types define lattice point locations and parameters define unit cell size and shape. - Miller indices describe plane orientations in a lattice relative to the unit cell. They allow accurate definition of planes and quantitative analysis in materials science. - Exercises are provided on identifying Miller indices of planes in different crystal structures.

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Engineering Physics
Semester 2
Academic year 2021-22
G. S. Mandal’s
Maharashtra Institute of Technology, Aurangabad
(An autonomous institute)
7/7/2022

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Lattice and unit cell
Miller indices
Atomic radius
Coordination number
Packing fraction
Calculation for SC, BCC, FCC, diamond structure, NaCl
Relation between lattice constant and density

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1. The structure of a crystal can be seen to be composed of a repeated element in
three dimensions. This repeated element is known as the unit cell. It is the
building block of the crystal structure.
2. A unit cell is a representative unit of the structure which when translationally
repeated (by the basis vector(s)) gives the whole structure.
3. The term unit should not be confused with ‘having one’ lattice point or motif
(The term primitive or sometimes simple is reserved for that).
4. If the lattice points are only at the corners, the cell is primitive. Primitive unit
cell has one lattice point per cell.
5. Instead of full atoms (or other units) only a part of the entity may be
present in the unit cell (a single unit cell).

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We abstracted points from the shape:
Now we abstract further:

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There are two distinct types of unit cell: primitive and non-primitive.
Primitive unit cells contain only one lattice point, which is made up from the
lattice points at each of the corners.
Non-primitive unit cells contain additional lattice points, either on a face of the
unit cell or within the unit cell, and so have more than one lattice point per unit
cell.

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The structure of a crystal is a combination of following elements:
The lattice type, the lattice parameters, and the motif.
The lattice type defines the location of the lattice points within the
unit cell.
The lattice parameters define the size and shape of the unit cell.
The motif is a list of the atoms associated with each lattice point
along with their fractional coordinates relative to the lattice point.
https://www.doitpoms.ac.uk/tlplib/crystallography3/structure.php
A crystal is defined to be a repeating, regular array of atoms.
The fundamental repeating unit is referred to as the
basis.
How the basis repeats is specified by Identifying the
underlying lattice.
Crystal structure = lattice + basis

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Miller Indices are a method of describing the orientation of a plane or set of planes
within a lattice in relation to the unit cell. They were developed by William
Hallowes Miller.
These indices are useful in understanding many phenomena in materials science,
such as explaining the shapes of single crystals, the form of some materials'
microstructure, the interpretation of X-ray diffraction patterns, and the movement
of a dislocation, which may determine the mechanical properties of the material.
Treatise on Crystallography (1839)
Miller Indices are the convention
used to label lattice planes.
This mathematical description
allows us to define accurately,
planes within a crystal, and
quantitatively analyse many
problems in materials science.

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Interplanar angle is given by
1 2 1 2 1 2
2 2 2 2 2 2
1 1 1 2 2 2
h h k k l l
cos
h k l h k l

 

   

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises

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Lattice planes
Exercises
https://www.doitpoms.ac.uk/tlplib/miller_indices/lattice_index.php

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Crystallographic Planes
z
x
y
a b
c
4. Miller Indices (110)
example a b c z
x
y
a b
c
4. Miller Indices (100)
1. Intercepts 1 1 
2. Reciprocals 1/1 1/1 1/
1 1 0
3. Reduction 1 1 0
1. Intercepts 1/2  
2. Reciprocals 1/½ 1/ 1/
2 0 0
3. Reduction 2 0 0
example a b c

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Crystallographic Planes
z
x
y
a b
c 


4. Miller Indices (634)
example
1. Intercepts 1/2 1 3/4
a b c
2. Reciprocals 1/½ 1/1 1/¾
2 1 4/3
3. Reduction 6 3 4
(001)
(010),
Family of Planes {hkl}
(100), (010),
(001),
Ex: {100} = (100),

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Thermal properties of graphene: Fundamentals and
applications Eric Pop , Vikas Varshney , and Ajit K.
Roy, MRS BULLETIN, VOLUME 37, DECEMBER 2012,
1273, 10.1557/mrs.2012.203

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Metals deform along the
closest pack directions or
closest packed slip planes.

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Why are planes in a lattice important?
Determining crystal structure
Diffraction methods measure the distance between parallel lattice planes of atoms.
This information is used to determine the lattice parameters in a crystal.
Diffraction methods also measure the angles between lattice planes.
Plastic deformation
Plastic deformation in metals occurs by the slip of atoms past each other in the crystal.
This slip tends to occur preferentially along specific crystal-dependent planes.
Transport Properties
In certain materials, atomic structure in some planes causes the transport of electrons
and/or heat to be particularly rapid in that plane, and relatively slow not in the plane.
Graphite: heat conduction is more in sp2-bonded plane.
YBa2Cu3O7 superconductors: Cu-O planes conduct pairs of electrons(Cooper pairs)
responsible for superconductivity, but perpendicular insulating.

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Miller indices does the following:
 Avoids infinities in the indices (intercepts of (1, , ) becomes
(100) index).
 Avoids dimensioned numbers
 Instead we have multiples of lattice parameters along the a, b, c
directions (this implies that 1a could be 10.2Å, while 2b could be 8.2Å).

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Coordination number –the number of nearest neighbour atoms or ions surrounding
an atom or ion.
1. The simple cubic has a coordination number of 6
2. The body-centered cubic (bcc) has a coordination number of 8
3. The face-centered cubic (fcc) has a coordination number of 12

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There are 8 atoms present in a unit cell on every corner, number of
atoms per unit cell
Let ‘a’ be the edge length of the unit cell and r be the radius of sphere.
The spheres are touching each other along the edge
Therefore a = 2r
No. of spheres per unit cell = 1/8 × 8 = 1
Volume of the sphere = 4/3 πr3
Volume of the cube = a3= (2r)3= 8r3
% occupied = 52.4 %
Polonium
1
8
1
*
8 
Packing fraction of sc

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There are 8 atoms present in a unit cell on every corner + one atom at
the centre
2
1
8
1
*
8 

Unit cell of bcc structure side is a.
The radius of atom present in unit cell = r
FD surface diagonal = b and body diagonal AF = c
Packing fraction of bcc

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Chromium, tungsten, Molybdenum, and niobium

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4
6
*
2
1
8
1
*
8 

The side of an unit cell = a
And diagonal AC = b
There are 8 atoms present in a unit cell on
every corner + one atom at the surface

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Number of atoms contributed by the corner atoms to an unit cell is 1/8×8 =1
Number of atoms contributed by the face centred atoms to the unit cell is 1/2 × 6 = 3
The atoms inside the structure =4
Total number of atoms present in a diamond cubic unit cell is 1 + 3 + 4 = 8
Since each carbon atom is surrounded by four more carbon atoms, the co-ordination
number is 4

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Packing fraction = 8 x 4/3 × π × r3/ a3
The radius of an atom r is equal to √3 x a/8
√3 x π/16 = 0.3401
2 r
4
a
4
a
4
a
A B
C
D

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