Crystal physics

CRYSTAL
PHYSICS
Haroon Hussain Moidu
Assistant Professor (FDP Substitute)
Department of Physics
MAMO College, Manassery

What is solid state physics?
 Solid state physics, also known as condensed matter
physics, is the study of the behaviour of atoms when
they are placed in close proximity to one another.
 In fact, condensed matter physics is a much better
name, since many of the concepts relevant to solids
are also applied to liquids.

Gases
 Gases have atoms or molecules that do not
bond to one another in a range of pressure,
temperature and volume.
 These molecules haven’t any particular order
and move freely within a container.

Liquids
 Similar to gases, liquids haven’t any
atomic/molecular order and they assume the
shape of the containers.
 Applying low levels of thermal energy can
easily break the existing weak bonds.

Solids
 Solids consist of atoms or molecules
executing thermal motion about an
equilibrium position fixed at a point in space.
 Solids can take the form of crystalline,
polycrstalline, or amorphous materials.
 Solids (at a given temperature, pressure, and
volume) have stronger bonds between
molecules and atoms than liquids.
 Solids require more energy to break the
bonds.

SOLID MATERIALS
CRYSTALLINE POLYCRYSTALLINE
AMORPHOUS
(Non-crystalline)
Single Crystal
 Each type is characterized by the size of ordered
region within the material.
 An ordered region is a spatial volume in which atoms
or molecules have a regular geometric arrangement
or periodicity.

Crystalline Solids
 In solid state physics we will deal with crystalline
solids, that is solids with an atomic structure based
on a regular repeated pattern.
 Many important solids are crystalline.
 More progress has been made in understanding the
behaviour of crystalline solids than that of non-
crystalline materials since the calculation are easier
in crystalline materials.

9
Polycrystalline Solid
Polycrystalline
Pyrite form
(Grain)
 Polycrystal is a material made up of an aggregate of many small single
crystals (also called crystallites or grains).
 Polycrystalline material have a high degree of order over many atomic or
molecular dimensions.
 These ordered regions, or single crytal regions, vary in size and orientation
wrt one another.
 These regions are called as grains ( domain) and are separated from one
another by grain boundaries. The atomic order can vary from one domain to
the next.
 The grains are usually 100 nm - 100 microns in diameter. Polycrystals with
grains that are <10 nm in diameter are called nanocrystalline

Amorphous Solid
 Amorphous (Non-crystalline) Solid is composed of randomly
orientated atoms , ions, or molecules that do not form defined
patterns or lattice structures.
 Amorphous materials have order only within a few atomic or
molecular dimensions.
 Amorphous materials do not have any long-range order, but they
have varying degrees of short-range order.
 Examples to amorphous materials include amorphous silicon,
plastics, and glasses.
 Amorphous silicon can be used in solar cells and thin film transistors.

Crystalline Solid
 Atoms or molecules are arranged in a definite,
repeating pattern in three dimension.
 Have a high degree of order, or regular geometric
periodicity, throughout the entire volume of the material.
 Anisotropic in nature : due to the different arrangement
of particles along different directions.
 Eg : NaCl, Kcl,Sugar, Quartz

CRYSTALLOGRAPHY
 The branch of science that deals with the geometric description
of crystals and their internal arrangement.
 Symmetry of a crystal can have a profound influence on its
properties.
 The atomic arrangement in a crystal is called crystal structure.
 Structures should be classified into different types according to
the symmetries they possess.

 A space lattice is defined as a
regular distribution of points in
space in which every point
has identical surrounding as
any other point in the array.
There are such 14 possible
lattices called Bravais lattices.
 Arrays are arranged exactly in
a periodic manner.
 Each point in a lattice is
called lattice point or lattice
site.
Crystal Lattice
α
a
b
CB ED
O A
y
x

Translational Lattice Vectors – 2D
A space lattice is a set of points such
that a translation from any point in the
lattice by a vector;
Rn = n1 a + n2 b
locates an exactly equivalent point,
i.e. a point with the same environment
as P . This is translational symmetry.
The vectors a, b are known as lattice
vectors and (n1, n2) is a pair of
integers whose values depend on the
lattice point.
P
Point D(n1, n2) = (0,2)
Point F (n1, n2) = (0,-1)

 The two vectors a and b
form a set of lattice
vectors for the lattice.
 The choice of lattice
vectors is not unique.
Thus one could equally
well take the vectors a
and b’ as a lattice vectors.
Lattice Vectors – 2D

Basis and Crystal Structure
 Crystal structure can be obtained by associating atoms, groups of
atoms or molecules which are called basis (pattern) to the lattice.
 Crystal structure is real while the lattice is imaginary
 Basis is identical in composition, arrangement and orientation.
 Basis is single atom in Na,Cu ; is diatomic in NaCl, CsCl ;
is triatomic in CaF2
Crystal Structure = Crystal Lattice ( ) + Basis( )

A two-dimensional Bravais lattice with
different choices for the basis
A group of atoms
which describe
crystal structure

Crystal structure
 Don't mix up atoms with
lattice points
 Lattice points are
infinitesimal points in
space
 Lattice points do not
necessarily lie at the
centre of atoms
Crystal Structure = Crystal Lattice + Basis

Unit Cells and lattice parameters
 Unit cell is the smallest portion of a crystal lattice which,
when repeated in different directions, generates the
entire lattice.
 Unit cell is the basic structural unit or building block of
the crystal structure.
 Lattice is made up of a repetition of unit cells.
A unit cell is characterised by 3
lattice vectors (a,b,c) and 3
interfacial angles (α , β , γ )
between them which are called
as lattice parameters.

Unit Cell in 2D
 The smallest component of the crystal (group of atoms,
ions or molecules), which when stacked together with
pure translational repetition reproduces the whole
crystal.
S
a
b
S
S
S
S
S
S
S
S
S
S
S
S
S
S

Crystal Structure 23
UNIT CELL
Primitive Conventional & Non-primitive
 Single lattice point per cell
 Smallest area in 2D, or
Smallest volume in 3D
 More than one lattice point per cell
 Integral multibles of the area of
primitive cell
Body centered cubic(bcc)
Conventional ≠ Primitive cell
Simple cubic(sc)
Conventional = Primitive cell

The Conventional Unit Cell
 A unit cell just fills space when
translated through a subset of
Bravais lattice vectors.
 The conventional unit cell is
chosen to be larger than the
primitive cell, but with the full
symmetry of the Bravais lattice.
 The size of the conventional cell
is given by the lattice constant
a.

Primitive and conventional cells of FCC

1
2
3
1
ˆ ˆ ˆ( )
2
1
ˆ ˆ ˆ( )
2
1
ˆ ˆ ˆ( )
2
a x y z
a x y z
a x y z
  
   
  
r
r
r
Primitive and conventional cells of BCC
Primitive Translation Vectors:

a
b c
Simple cubic (sc):
primitive cell=conventional cell
Fractional coordinates of lattice points:
000, 100, 010, 001, 110,101, 011, 111
Primitive and conventional cells
Body centered cubic (bcc):
conventional primitive cell
a
b c
Fractional coordinates of lattice points in
conventional cell:
000,100, 010, 001, 110,101, 011, 111, ½ ½ ½

Body centered cubic (bcc):
primitive (rombohedron) conventional cell
a
b
c
Fractional coordinates:
000, 100, 101, 110, 110,101, 011, 211, 200
Face centered cubic (fcc):
conventional primitive cell
a
b
c
000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½
0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1
Primitive and conventional cells

Hexagonal close packed cell (hcp):
conventional primitive cell
100, 010, 110, 101,011, 111,000, 001
points of primitive cell
a
b
c
Primitive and conventional cells-hcp

 A primitive unit cell is made of primitive
translation vectors a1 ,a2, and a3 such
that there is no cell of smaller volume
that can be used as a building block for
crystal structures.
 A primitive unit cell will fill space by
repetition of suitable crystal translation
vectors. This defined by the parallelpiped
a1, a2 and a3. The volume of a primitive
unit cell can be found by
 V = a1.(a2 x a3) (vector products) Cubic cell volume = a3
Primitive Unit Cell and vectors

 The primitive unit cell must have only one lattice point.
 There can be different choices for lattice vectors , but the
volumes of these primitive cells are all the same.
P = Primitive Unit Cell
NP = Non-Primitive Unit Cell
Primitive Unit Cell
1a

 Primitive Unit cells :
 when constituent particles are present only on the corner
positions of a unit cell.
 Centred Unit cells:
 Body –centred Unit cells : constituent particle at its body
centre in addition to those at corners.
 Face - centred Unit cells : constituent particles at the centre
of each face in addition to those at corners.
 End - centred unit cells : constituent particles at the centre
of any two opposite faces in addition to those at corners.
2 Categories of Unit cells

Seven Primitive Unit Cells and their
possible variations

Unit cells of 14 types of Bravais Lattices

Coordinatıon Number
 Coordinatıon Number (CN) : the number of equidistant
nearest neighbours that an atom has in the given structure
 The Bravais lattice points closest to a given point are the
nearest neighbours.
 Because the Bravais lattice is periodic, all points have the
same number of nearest neighbours or coordination
number. It is a property of the lattice.
 A simple cubic has coordination number 6; a body-
centered cubic lattice, 8; and a face-centered cubic
lattice,12.

1-CUBIC CRYSTAL SYSTEM
 Simple Cubic has one lattice point so its primitive cell.
 In the unit cell on the left, the atoms at the corners are cut
because only a portion (in this case 1/8) belongs to that cell.
The rest of the atom belongs to neighboring cells.
 Coordinatination number of simple cubic is 6.
 polonium is one of the example.
a- Simple Cubic (SC)
a
b
c

b-Body Centered Cubic (BCC)
 BCC has two lattice points so BCC
is a non-primitive cell.
 BCC has eight nearest neighbors.
Each atom is in contact with its
neighbors only along the body-
diagonal directions.
 Many metals (Fe,Li,Na..etc),
including the alkalis and several
transition elements choose the
BCC structure.
a
b c

c- Face Centered Cubic (FCC)
 There are atoms at the corners of the unit cell and at the
center of each face.
 Face centered cubic has 4 atoms so its non primitive cell.
 Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC
structure.
Or Cubic Close Packed (ccp)

Atoms Shared Between: Each atom counts:
corner 8 cells 1/8
face centre 2 cells 1/2
body centre 1 cell 1
end centre 2 cells 1/2
Unit Cell cell contents
Primitive Cubic 1 [=8 x 1/8]
Body centered 2 [=(8 x 1/8) + (1 x 1)]
Face Centered 4 [=(8 x 1/8) + (6 x 1/2)]
End centered 2 [=(8 x 1/8) + (2 x 1/2)]
Unit cell contents
Counting the number of atoms within the unit cell

 Atomic Packing Factor (APF) is defined as the volume
of atoms within the unit cell divided by the volume of the
unit cell.
 Nearest Neighbour distance (2r) : the distance
between the centres of two nearest neighbouring atoms.
It will be 2r if r is the radius of atom.
 Atomic radius(r) : it is half the distance between nearest
neighbours in a crystalline solid without impurity.
 Coordination Number : Number of equidistant nearest
neighbours that an atom has in the structure. Greater
coordination no, the more closely packed.
Terms :

Close packed structures:
Square close packing in 2D
hexagonal close packing in 2D
3D close packing from 2D square
close packed layers can be made
possible in two ways which produces
(AAA pattern) and (ABAB pattern)
3D close packing from 2D close
hexagonal close packed layers can
be made possible in two ways which
produces (ABAB pattern) and
ABCA pattern)

Voids
 If No of close
packed spheres =
N, then
 No of Octahedral
void = N
 No of tetrahedral
voids = 2N

Intro to close packed (Cubic)
structures
3D close packing from 2D
square close packed layers
AAA pattern ABAB pattern
-Simple cubic -Body centred
3D close packing from 2D
hexagonal close packed layers
ABCA patternABAB pattern
- hcp structure - ccp or fcc

3D close packing from 2D square close
packed layers
 AAA pattern
- Simple cubic
- CN = 6
- APF = 52%
 ABAB pattern
-Body centred
- CN = 8
- APF = 68%

3D close packing from 2D hexagonal close
packed layers - ABAB pattern (hcp structure)
 Tetrahedral voids of
the second layer may
be covered by the
spheres of the third
layer.
 The spheres of the
third layer are
aligned with those
the first layer. So
ABAB pattern.
 APF = 74%
 CN = 12

3D close packing from 2D hexagonal close
packed layers - ABCA pattern (ccp or fcc)
 The third layer may
be placed above the
second layer in a
manner such that its
spheres cover the
octahedral voids.
 The spheres of the
third layer are not
aligned with those
of either the first or
the second layer.So
ABCA patterm
 APF = 74%
 CN = 12

Summary of close packed structures

Metallic Crystal structures:
Most of common metals are found in :
 Simple cubic structure(sc)
 Face centred cubic structure(fcc)
 Body centred cubic structure(bcc)
 Hexagonal close packed structure(hcp)

 Simple Cubic has one lattice point so its primitive cell.
 In the unit cell on the left, the atoms at the corners are cut
because only a portion (in this case 1/8) belongs to that cell.
The rest of the atom belongs to neighboring cells.
 Share of each corner atom to a unit cell is 1/8 th of an atom.
 Atoms touches each other along edges.
 Coordinatination number of simple cubic is 6.
 polonium is one of the example.
(a) Simple Cubic (sc)
a
b
c

a- Simple Cubic (SC)
Coordination Number , N = 6
Nearest neighbour distance , 2r = a
Lattice constant , a = 2r
Number of atoms per cell , n = 1/8 *8 = 1
No of lattice points = 1
Volume of all atoms in a unit cell , v = 4/3×𝜋r3
Volume of unit cell , V = a3 = (2r)3

(b)Body Centered Cubic (bcc)
 BCC has two lattice points so
BCC is a non-primitive cell.
 BCC has eight nearest
neighbors. Each atom is in
contact with its neighbors only
along the body-diagonal
directions.
 Many metals (Li,Na,K,Cr..etc),
including the alkalis and several
transition elements choose the
BCC structure.
a
b c

Think of this unit cell as
made by stuffing another
atom into the centre of the
simple cubic lattice,
slightly spreading the
corners. Thus, the corner
atoms do not touch each
other, but do touch the
centre along the bodily
diagonal.
Nearest neighbour distance,2r = (a√3)/2
Lattice constant, a = 4r/√3
No of atoms per unit cell , n= 2
Volume of all atoms in unit cell v = 2 ×4/3×𝜋r3
Volume of unit cell V = a3 = (64 r3)/(3 √3)

0.68=
V
V
=APF
3
R4
=a
cellunit
atoms
BCC
2 (0,433a)
Atomic Packing Factor of bcc

(c) Face Centered Cubic (fcc)
 There are atoms at the corners of the unit cell and at the
center of each face.
 Face centered cubic has 4 atoms so its non primitive cell.
 Many of common metals (Cu,Al,Pb,Ag..etc) crystallize in
FCC structure.

 The corner atoms touch
the one in the centre of
the face.
 No other packing can
exceed this efficiency.
 If we stack the cells into a
lattice we notice that the
atoms form diagonal
layers
Face Centered Cubic (fcc)
Nearest neighbour distance,2r = (a√2)/2
Lattice constant, a = 4r/√2
Volume of all atoms in unit cell v = 4 ×4/3×𝜋r3
Volume of unit cell V = a3 = (64 r3)/(2√2)

4 (0,353a)
0.68=
V
V
=APF
3
R4
=a
cellunit
atoms
BCCFCC
0,74
Atomic Packing Factor of fcc

(c) Hexagonal close packed(hcp)
 Conventional unit cell
 The unit cell contains one atom at each corner , one atom
each at the centre of hexagonal face and three more atoms
within the body of the cell.
 Hexagonal close packed has 6 atoms in a unit cell. so its
non primitive cell.
 Many of common metals (Magnesium, Zinc, cadmium)
crystallize in this structure.

 Each atom touches three
atoms in the layer below
its pane, six atoms in its
own plane and three
atoms in the layer above.
So, CN= 12
 The top layer contains
seven atoms . Each
corner atom is shared by
6 surrounding hexagon
cells and the centre atom
is shared by 2
surrounding cells. The
three atoms within the
body of the cell are fully
contributing to the cell.
So, No of atoms in a unit
cell is 6.
Hexagonal close packed

 The hcp structure is characterised by two
nested hexagonal lattice that are shifted by
the vector (2/3,1/3,1/2) (in the conventional
unit cell basis) against each other.
 The lattice is not a Bravais lattice since the
individual lattice points are not equivalent
with respect to their environments.
 But it can be looked at as a
hexagonal Bravais lattice with a two-atomic
basis with the atoms sitting at the
positions (0,0,0,) and (2/3,1/3,1/2)
 Even though the term hcp is used for all those
lattices, a close-packing of equal atoms is
only obtained for a certain ratio
of the lattice constants.
Position of the additional atom
within the conventional unit
cell: (2/3,1/3,1/2)
The hexagonal layers form a
close-packed structure.
c 8
a 3


Calculation of APF
APF = Volume of all atoms in a unit cell , v = 6  4/3 r3
Volume of unit cell, V = base area * height
30 
O
30 
X
B
A
O
A
C
a

Nearest neighbour distance, 2r = a (Each and every corner
atom touches each other, Therefore a = 2r ; i.e., The atomic
radius, r = a/2)
Lattice constant, a = 2r
Volume of all atoms in unit cell v = 6 ×4/3×𝜋r3 = 𝜋a3
Volume of unit cell V = area of base × height = (3/2)(a2 c)√3
a
a

Volume of unit cell:
AB = AC = BO = a (lattice constant)
Also CX = c/2 where c  height of the hcp unit cell.( because The atom at point C
is midway between the top and bottom faces of the unit cell )
Area of the base = 6  area of the triangle ABO
= 6  1/2  AB  OO
= 6  1/2  a  OO
In triangle OBO ;
Angle O OB = 30º
Cos 30º =
OO = a cos 30º = a
Now, substituting the value of OO in the eqn of the area of the base,
Area of the base = 6   a  a
=
OO' OO'
BO a

3
2
2
3 3a
2
3
2

Therefore,
Volume of the cell ,V = area of the base  height of the hcp unit cell(c)
=  c
Volume of All atoms in the unit cell:
Volume of all atoms in unit cell, v = 6 ×4/3× 𝜋r3
= 𝜋a3
Atomic packing fraction
APF = Volume of all atoms in a unit cell , v = 𝜋a3
Volume of unit cell, V =  c
=
2
3 3a
2
2
3 3a
2
2𝜋
3 3
𝑎
𝑐

Now we have to find out the ratio c/a:
APF =
= 0.74
Question:
1. Rhenium has an HCP crystal structure, an atomic radius of
0.137 nm, and a c/a ratio of 1.615. Compute the volume of
the unit cell for Re.
2. Zinc has HCP structure. The height of the unit cell is
4.935Å. Find (i). How many atoms are there in a unit cell?
and (ii). What is the volume of the unit cell ?
c 8
a 3

2𝜋
3 3
𝑎
𝑐

Relation between the density of crystal
material and lattice constant in a cubic
lattice:
 Consider a cubic crystal of lattice constant ,a
 Let the number of atoms per unit cell be ‘n’
 Density of the material be ‘ρ’
 Atomic weight of the material is MA
 NA be the avogadro’s number.
 Then MA /ρ m3 volume contain NA atoms.
 Volume of one atom =
 Volume occupied by n atoms in a unit cell =
𝑀𝐴
𝜌 𝑁𝐴
𝑀𝐴 𝑛
𝜌 𝑁𝐴

 ---
 ---
 ---
𝑎3
=
𝑀𝐴 𝑛
𝜌 𝑁𝐴
𝜌 =
𝑀𝐴 𝑛
𝑎3 𝑁𝐴
𝑎 =
𝑀𝐴 𝑛
𝜌 𝑁𝐴
1
3
Relation between the density of crystal
material and lattice constant in a cubic
lattice:

Diamond Cubic Structure
 Crystal structure which is formed by many elements of the 4th main group of
the periodic table; Carbon , silicon and germanium crystallises in this
structure.
 Diamond (and silicon and germanium) have a variant of the face-centered
cubic crystal structure with extra atoms in the unit cell.
 The diamond lattice is formed by two interpenetrating two face centered
bravais lattices along the body diagonal by ¼ th cube edge.
 The bonds in diamond are stronger than in silicon or germanium because
atomic orbitals tend to be smaller higher up on the periodic table
 Its a three-dimensional structure in which
every atom has four uniformly distributed
nearest neighbours as binding partners
 There are eight atoms in the structure of
diamond.
 Each carbon atom is at the centre of a
regular tetrahedron formed by four other
carbon atoms, the coordination number
is four.

The underlying structure
is fcc with a two-atomic
basis. One of the two atoms
is sitting on the lattice
point and the other one is
shifted by 1/4 along each
axes. This forms a
tetrahedrical structure
where each atom is
surrounded by four equal-
distanced neighbours.
Each atom forms bonds with four nearest
neighbours
Conventional unit cell of the diamond structure:
Tetrahedrical structure of diamond

The structure is not a Bravais lattice by itself because there are two types of
lattice points with different environments. The underlying structure is
actually a fcc structure with a two-atomic basis. Thus there are two atoms
attached to each fcc lattice point: One located just at the position of
the lattice point and one being shifted by the vector (1/4,1/4,1/4). Thereby
the number of atoms per conventional unit cell is doubled from 4 to 8.

Calculation of packing factor
APF=34%

Cubic zinc sulphide or zinc blende
structure
 Identical to the diamond structure except that the two
interpenetrating fcc sublattices are of different atoms and
displaced from each other by one quarter of the body
diagonal.
 Zinc atoms are placed in one fcc lattice and S atoms on
other fcc lattice.
 Conventional cell is a cube.
 Four molecules per conventional cell.
 For each atom, there are four equally distant atoms of
opposite kind arranged at a regular tetrahedron.
 CuCl, ZnS, InSb(indium antimonide) and CdS( Cadmium
Sulphide) have similar structures.

Zinc atoms are placed on one fcc lattice and S atoms on the
other fcc lattice.
There are 4 molecules per conventional cell.
For each atom, there are four equally distant atoms of
opposite kind arranged at regular tetrahedron.
Site Zn S
Central 4 0
Face 0 6(1/2) = 3
Corner 0 8(1/8) = 1
Total 4 4

Sodium Chloride Structure
 The bravais lattice is a face centred cube, with a basis of one
Na ion and one Cl ion seperated by one half the body
diagonal of the unit cell.
 There are 4 molecules of NaCl in Unit cell.
 Each ion is surrounded by six nearest neighbours of the
opposite kind. There fore CN is 6.
 There are 12 next nearest neighbours of same kind ions.
 KCL,Kbr,MgO,AgBr etc have NcCL structure.
Ions positions 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

Caesium Chloride
 The space lattice is simple cubic.
 The basis has one Cs+ ion at 000 and one Cl- ion at ½ ½ ½ .
 The lattice points of CsCl are two interpenetrating simple cubic lattices. ,
the corner of one sublattice is the body center of the other. One sublattice is
occupied by Cs+ and other by Cl-
 Each ion is at the centre of a cube of ions of the opposite kind,
Thus CN = 8.
 CsCl, RbCl crystallises in this structure.

Lattice Sites in Cubic Unit Cell

Crystal Directions
Fig. Shows
[111] direction
 We choose one lattice point on the line as
an origin, say the point O. Choice of origin
is completely arbitrary, since every lattice
point is identical.
 Then we choose the lattice vector joining O
to any point on the line, say point T. This
vector can be written as;
R = n1 a + n2 b + n3c
 To distinguish a lattice direction from a
lattice point, the triple is enclosed in square
brackets [ ...] is used.[n1n2n3]
 [n1n2n3] is the smallest integer of the same
relative ratios.

210
X = 1 , Y = ½ , Z = 0
[1 ½ 0] [2 1 0]
X = ½ , Y = ½ , Z = 1
[½ ½ 1] [1 1 2]
Examples

Negative directions
 When we write the direction
[n1n2n3] depend on the origin,
negative directions can be
written as
 R = n1 a + n2 b + n3c
Direction must be smallest integers.
Y direction
(origin) O
- Y direction
X direction
- X direction
Z direction
- Z direction
][ 321 nnn
][ 321 nnn

X = -1 , Y = -1 , Z = 0 [110]
Examples of crystal directions
X = 1 , Y = 0 , Z = 0 [1 0 0]

Examples
X =-1 , Y = 1 , Z = -1/6
[-1 1 -1/6] [6 6 1]
We can move vector to the origin.

Miller Indices (h,k,l)
Miller Indices are a symbolic vector representation for the
orientation of an atomic plane in a crystal lattice and are defined
as the reciprocals of the fractional intercepts which the plane
makes with the crystallographic axes.
To determine Miller indices of a plane, take the following steps;
1. Determine the coordinates (p,q,r) of the intercepts of the plane
along the three crystallographic directions.
2. Express the intercepts as multiples of the unit cell dimension or
lattice parameters.
3. Take the reciprocals of the intercepts
4. If fractions result, multiply each by the denominator of the
smallest fraction

Features of miller indices
 All the parallel equidistant planes have the same miller
indices.
 A plane parallel to one of the coordinate axes has an
intercept of infinity.
 If the miller indices of two planes have the same ratio;
for example (8 4 4) and (4 2 2), then the planes are
parallel to each other.
 If (h,k,l) are the miller indices of a plane, then the plane
cuts the axes into h,k and l equal segments resp.

Axis X Y Z
Intercept
points 1 ∞ ∞
Reciprocals 1/1 1/ ∞ 1/ ∞
Smallest
Ratio 1 0 0
Miller İndices (100)
Example-1
(1,0,0)

Axis X Y Z
Intercept
points 1 1 ∞
Reciprocals 1/1 1/ 1 1/ ∞
Smallest
Ratio 1 1 0
Example-2
(1,0,0)
(0,1,0)

Axis X Y Z
Intercept
points 1 1 1
Reciprocals 1/1 1/ 1 1/ 1
Smallest
Ratio 1 1 1
Miller İndices (111)(1,0,0)
(0,1,0)
(0,0,1)
Example-3

Axis X Y Z
Intercept
points 1/2 1 ∞
Reciprocals 1/(½) 1/ 1 1/ ∞
Smallest
Ratio 2 1 0
(1/2, 0, 0)
(0,1,0)
Example-4

Axis a b c
Intercept
points 1 ∞ ½
Reciprocals 1/1 1/ ∞ 1/(½)
Smallest
Ratio 1 0 2
Example-5

Axis a b c
Intercept
points -1 ∞ ½
Reciprocals 1/-1 1/ ∞ 1/(½)
Smallest
Ratio -1 0 2
Example-6

Miller Indices
Reciprocal numbers are:
2
1
,
2
1
,
3
1
Plane intercepts axes at cba 2,2,3
Indices of the plane (Miller): (2,3,3)
(100)
(200)
(110)
(111)
(100)
Indices of the direction: [2,3,3]
a
3
2
2
b
c
[2,3,3]

1. The miller indices of a plane in a simple cubic crystal are 1 2 3 . Find the
coordinates of the plane.
2. Obtain the miller indices of a plane which intercepts a, b/2, 3c in a simple cubic
unit cell.
3. A plane makes intercepts of 1,2,0.5 A0 on the crystallographic axis of an
orthorhombic crystal with a:b:c = 3:2:1. determine the miller indices of this
plane.
4. Determine the miller indices of the crystal planes shown below and compare
with the same direction.
Questions:

 A symmetry operation is a transformation performed on the body
which leaves it unchanged or invariant.
 If the body becomes indinguishable from its initial configuration
after performing an operation on the body , then the body is said
to posses a symmetry element corresponding to that particular
symmetry operation.
 Symmetry operations performed about a point or plane or line are
called point group symmetry operations. Point group symmetry
elements are Centre of symmetry or inversion centre, Reflection
symmetry and Rotation symmetry
 Point group symmetry operations combined with compatible
translational symmetry elements are called space group
symmetry operations. There for it is the group of all symmetry
elements of a crystal structure.
SYMMETRY ELEMENTS in crystals

 There are 32 point groups in a 3D lattice which
produce 14 distinct bravais lattices.
 There are 230 space groups exhibited by crystals.
Operation Element
Inversion Point
Reflection Plane
Rotation Axis
Rotoinversion Axes

23 symmetry elements
in a cubic crystal
-Centre of symmetry -------- 1
- straight planes -------- 3
-Diagonal planes -------- 6
-Tetrad axes -------- 3
-Triad axes -------- 4
-Diad axes -------- 6
9 planes
13 planes
Total symmetry elements of a cubic crystal = 23

(1) Inversion Centre of a
cubic crystal
 A center of symmetry: A point at the center of the molecule.
(x,y,z) --> (-x,-y,-z)
 This symmetry element implies that each point located at r
relative to a lattice point has an identical point located at –r
relative to the same lattice point.
 Cube has three pairs of parallel and opposite faces all of
them of same size and shape.

(2) Reflection symmetry (planes of
symmetry )of cubic crystal
 There 3 straight planes and 6 diagonal planes
of symmetry in a cube.

Reflection Plane (example)
 A lattice is said to posses reflection symmetry if
there exists a plane in the lattice which divides it into
two identical halves which are mirror images of each
other.

Examples
 Triclinic has no reflection plane.
 Monoclinic has one plane midway between and
parallel to the bases, and so forth.

• A body is said to posses rotational symmetry about an axis if
the body left invariant after a rotation of the body about an
by some angle.
• The axis is called `n-fold, axis’
if the angle of rotation =
• If equivalent configuration occurs after rotation of
180º, 120º and 90º, the axes of rotation are known as
two-fold, three-fold and four-fold axes of symmetry
respectively.
• only one, two, three, four and six fold rotations axes are
possible in a crystal lattice.
(3) Rotation Symmetry
n
360

During one complete rotation about an
axis normal to one of its faces at its
mid point, at four positions the cube is
coincident with its original position (
ie, for each 90 degree) . Such an axis
is called four-fold axes of symmetry
or tetrad axis. There will be three such
tetrad axes.
Rotation symmetry of a cubic crystal
If a cube is rotated about a body
diagonal through 120 degree ,
cube will remain invariant. A
cube posses four such axes.

If a cube is rotated about a
line joining the midpoints of
a pair of opposite parallel
edges through 180, the cube
is indistinguishable form the
position it occupied
originally. A cube posses six
such axes
-Centre of symmetry -------- 1
- straight planes -------- 3
-Diagonal planes -------- 6
-Tetrad axes -------- 3
-Triad axes -------- 4
-Diad axes -------- 6
9 planes
13 planes
Total symmetry elements of a cubic crystal = 23

Crystal Structure
Rotation Axis
 This is an axis such that, if the cell is rotated around it
through some angles, the cell remains invariant.
 The axis is called n-fold if the angle of rotation is 2π/n.
90
°
120° 180°

• There are 32 compatible combinations of the three point group
symmetry elements , called point groups.
Rotation - inversion Axis
 Crystal is said to process a rotation – inversion axis if it is
brought back to its original condition by rotation
followed by an inversion about a lattice point through
which the rotation axis passes.
 Crystals can posses 1,2,3,4 and 6 – fold roto inversion
axes.

Translation Symmetry Elements
 There are two translation symmetry elements :
 Screw axis
 Glide planes
Srew axis
 Rotation axis coupled with a translation parallel to the
rotation axis will give rise to a new symmetry element called
screw axis.
 A screw axis is represented by the symbol nm.
 This is performed by a rotation of 2𝜋/n and translation of
[m/n] times the translation vectors parallel to the rotation
axis.

Twofold screw axis.
A screw axis consists of a rotation
followed by a translation

Glide planes
 The operation in a glide plane involves a translation t/2
parallel to a reflection plane followed by reflection across the
plane. t denotes the distance between successive atoms.
 Glide planes are divided into three :
 (i) axial glides (a,b,c) - glide component is parallel to
crystollographic axis a,b,c and equal in length to a/2, b/2,
c/2.
 (ii) diagonal glides (n) – corresponds to planes with the
vector sum of any two of the vectors a/2, b/2, c/2 as
glide components.
 (iii) diamond glides(d) – this plane has the glide
component equal to the vector sum of any two of the a/4,
b/4, c/4.

Glide plane.
A glide plane consists of a reflection followed
by a translation

Crystal physics

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