crystalline solids :-in crystalline solids the atoms are arranged in some regular periodic...

Crystalline Solids :-In Crystalline Solids the atoms are arranged in some regular periodic geometrical pattern in three dimensions- long range order

Eg :- NaCl, CuSo4 ,CsCl, ZnS, etcAmorphous Solids :- In an amorphous solids the

atoms are not arranged in regular periodic geometrical pattern –short range order

Eg :- Boran trioxide (B2O3),Lamp Soot, glass etcUnit Cell :- A Unit cell is the volume of a solid

from which the entire crystal can be constructed by a translation repetition in three directions in sp

Crystal lattice :- An array of points in the space (2D,3D) in which every point has the same environment with respect

to all other points is called space lattice

Basis :- Atoms or molecules attached to each

lattice point in the crystal system is called basis.

Lattice+ Basis = Crystal

CsCl structure

Lattice

n-dimensional, infinite, periodic array of points,

each of which has identical surroundings.

Lattice

n-dimensional, infinite, periodic array of points,

each of which has identical surroundings.

use this as test for lattice points

lattice points

Crystallographic axes

Crystallographic axes:- The lines drawn parallel to the line of intersection of any three faces of the unit Cell which do not lie in the same plane are called Crystallographic axes X,Yand Z

Lattice ParametersInterfacial angles :- The angles between

three Crystallographic axes represented by

α,β,Ƴ are called interfacial angles * Primitives :- The intercepts a,b and c,

which define the dimensions of the unit cell on the respective Crystallographic axes are called as primitives of the Unit Cell

An array of points such that every point has identical surroundings

In Euclidean space infinite array

We can have 1D, 2D or 3D arrays (lattices)

Space Lattice

Translationally periodic arrangement of points in space is called a lattice

or

A lattice is also called a Space Lattice

Note: points are drawn with finite size for clarity in reality they are 0D (zero dimensional)

Crystals are grouped into seven crystal systems, according to characteristic symmetry of their unit cell.

The characteristic symmetry of a crystal is a combination of one or more rotations and inversions.

Crystal SystemsThere are Seven Basic Crystal SystemsCubicTetragonalOrthorhombicMonoclinicTriclinicRhombohedral (Trigonal)Hexagonal

Shape of UC Used as UC for crystal: Lattice Parameters

Cube Cubic (a = b = c, = = = 90)

Square Prism Tetragonal (a = b c, = = = 90)

Rectangular Prism Orthorhombic (a b c, = = = 90)

Parallelogram Prism Monoclinic (a b c, = = 90 )

Parallelepiped (general) Triclinic (a b c, )

Parallelepiped (Equilateral, Equiangular)

Rhombohedral (Trigonal)

(a = b = c, = = 90)

120 Rhombic Prism Hexagonal (a = b c, = = 90, = 120)

Latti

In 1848, Auguste Bravais demonstrated that in a 3-dimensional system there are fourteen possible lattices

A Bravais lattice is an infinite array of discrete points with identical environment

seven crystal systems + four lattice centering types = 14 Bravais lattices

Lattices are characterized by translation symmetry

Auguste Bravais (1811-1863)

Bravais showed that there are only 14 independent ways of arranging points in space so that the environment looks the same from each point. These lattices are called Bravais lattices

Cubic space lattices

Summary: Fourteen Bravais Lattices in Three Dimensions

Fourteen Bravais Lattices …

1 Cubic Cube P I F C

Lattice point

PPII

FF

a b c 90

4 23

m m

Symmetry of Cubic latticesSymmetry of Cubic lattices

P I F C

2 Tetragonal Square Prism (general height)

II

PP

a b c

90

Symmetry of Tetragonal latticesSymmetry of Tetragonal lattices

4 2 2

m m m

P I F C

3 Orthorhombic Rectangular Prism (general height)

PPII

FFCC

a b c

90

Symmetry of Orthorhombic latticesSymmetry of Orthorhombic lattices

2 2 2

m m m

Note the position of ‘a’ and ‘b’

a b c One convention

Why is Orthorhombic called Ortho-’Rhombic’?Why is Orthorhombic called Ortho-’Rhombic’?Is there a alternate possible set of unit cells for OR?Is there a alternate possible set of unit cells for OR?

P I F C

4 Hexagonal 120 Rhombic Prism

What about the HCP?(Does it not have an additional atom somewhere in the middle?)

What about the HCP?(Does it not have an additional atom somewhere in the middle?)

A single unit cell (marked in blue) along with a 3-unit cells forming a

hexagonal prism

Note: there is only one type of hexagonal lattice (the primitive one)

a b c

90 , 120

Symmetry of Hexagonal latticesSymmetry of Hexagonal lattices

6 2 2

m m m

P I F C

5 Trigonal Parallelepiped (Equilateral, Equiangular)

90

a b c

Symmetry of Trigonal latticesSymmetry of Trigonal lattices

Rhombohedral

23

m

Note the position of the origin and of ‘a’, ‘b’ & ‘c’

P I F C

6 Monoclinic Parallogramic Prism

90

a b c

Symmetry of Monoclinic latticesSymmetry of Monoclinic lattices

2

m

a b c

Note the position of ‘a’, ‘b’ & ‘c’

One convention

P I F C

7 Triclinic Parallelepiped (general)

a b c

Symmetry of Triclinic latticesSymmetry of Triclinic lattices

1

Arrangement of lattice points in the Unit Cell

& No. of Lattice points / Cell

Position of lattice points Effective number of Lattice points / cell

1 P 8 Corners = [8 (1/8)] = 1

2 I8 Corners + 1 body centre

= [1 (for corners)] + [1 (BC)] = 2

3 F8 Corners +

6 face centres= [1 (for corners)] + [6 (1/2)] = 4

4

A/

B/

C

8 corners +2 centres of opposite faces

= [1 (for corners)] + [2 (1/2)] = 2

PH 0101 UNIT 4 LECTURE 2 29

MILLER INDICES

d

DIFFERENT LATTICE PLANES


MILLER INDICES

The orientation of planes or faces in a crystal can be

described in terms of their intercepts on the three

axes.

Miller introduced a system to designate a plane in a

crystal.

He introduced a set of three numbers to specify a

plane in a crystal.

This set of three numbers is known as ‘Miller Indices’

of the concerned plane.


MILLER INDICES

Procedure for finding Miller Indices

Step 1: Determine the intercepts of the plane along the axes X,Y and Z in terms of the lattice constants a,b and c.

Step 2: Determine the reciprocals of these numbers.


Step 3: Find the least common denominator (lcd) and multiply each by this lcd.

Step 4:The result is written in paranthesis.This is called the `Miller Indices’ of the plane in the form (h k l).

This is called the `Miller Indices’ of the plane in the form (h k l).

MILLER INDICES


ILLUSTRATION

PLANES IN A CRYSTAL

Plane ABC has intercepts of 2 units along X-axis, 3

units along Y-axis and 2 units along Z-axis.


DETERMINATION OF ‘MILLER INDICES’

Step 1:The intercepts are 2,3 and 2 on the three axes.

Step 2:The reciprocals are 1/2, 1/3 and 1/2.

Step 3:The least common denominator is ‘6’. Multiplying each reciprocal by lcd, we get, 3,2 and 3.

Step 4:Hence Miller indices for the plane ABC is (3 2 3)

ILLUSTRATION


IMPORTANT FEATURES OF MILLER INDICES

For the cubic crystal especially, the important features of Miller indices are,

A plane which is parallel to any one of the co-ordinate axes has an intercept of infinity (). Therefore the Miller index for that axis is zero; i.e. for an intercept at infinity, the corresponding index is zero.

MILLER INDICES


EXAMPLE

( 1 0 0 ) plane

Plane parallel to Y and Z axes


EXAMPLE

In the above plane, the intercept along X axis is 1 unit.

The plane is parallel to Y and Z axes. So, the intercepts along Y and Z axes are ‘’.

Now the intercepts are 1, and .

The reciprocals of the intercepts are = 1/1, 1/ and 1/.

Therefore the Miller indices for the above plane is (1 0 0).


MILLER INDICES


A plane passing through the origin is defined in terms of a parallel plane having non zero intercepts.

All equally spaced parallel planes have same ‘Miller indices’ i.e. The Miller indices do not only define a particular plane but also a set of parallel planes. Thus the planes whose intercepts are 1, 1,1; 2,2,2; -3,-3,-3 etc., are all represented by the same set of Miller indices.


MILLER INDICES


It is only the ratio of the indices which is important in this notation. The (6 2 2) planes are the same as (3 1 1) planes.

If a plane cuts an axis on the negative side of the origin, corresponding index is negative. It is represented by a bar, like (1 0 0). i.e. Miller indices (1 0 0) indicates that the plane has an intercept in the –ve X –axis.


MILLER INDICES OF SOME IMPORTANT PLANES

l + k + h

a = d

222hkl

Spacing between planes in a cubic crystal

where dhkl = inter-planar spacing between planes with Miller indices h,k,and l.a = lattice constant (edge of the cube)h, k, l = Miller indices of cubic planes being considered.


PROBLEMS

Worked Example:Calculate the miller indices for the plane with

intercepts 2a, - 3b and 4c the along the crystallographic axes.

The intercepts are 2, - 3 and 4

Step 1: The intercepts are 2, -3 and 4 along the 3 axes

Step 2: The reciprocals are

Step 3: The least common denominator is 12.

Multiplying each reciprocal by lcd, we get 6 -4 and 3

Step 4: Hence the Miller indices for the plane is

1 1 1, and

2 3 4

6 4 3


Worked Example

The lattice constant for a unit cell of aluminum is 4.031Å Calculate the interplanar space of (2 1 1) plane.

a = 4.031 Å(h k l) = (2 1 1)Interplanar spacing

d = 1.6456 Å

PROBLEMS

10

2 2 2 2 2 2

4.031 10ad

h k l 2 1 1


PROBLEMS

Worked Example:Find the perpendicular distance between the two planes

indicated by the Miller indices (1 2 1) and (2 1 2) in a unit cell of a cubic lattice with a lattice constant parameter ‘a’.We know the perpendicular distance between the origin and the plane is (1 2 1)

and the perpendicular distance between the origin and the plane (2 1 2),

1 2 2 2 2 2 21 1 1

a a ad

6h k l 1 2 1

2 2 2 2 2 2 22 2 2

a a a ad

39h k l 2 1 2


PROBLEMS

The perpendicular distance between the planes (1 2 1) and (2 1 2) are,

d = d1 – d2 =

(or) d = 0.0749 a.

3a 6a a(3 6)a a

36 3 6 3 6

crystalline solids :-in crystalline solids the atoms are arranged in some regular periodic...

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