Solid State Chemistry

Solid:

Matter which possess rigidity having definite shape & volume is called solid.

They are highly ordered three- dimensional structures are said to be crystalline.

Because the particles of a solid are not free to undergo long- range movement, solids are rigid.

Types of solid:

1. Crystalline Solid:

- Contain regular arrangement having short range & as well as long range order.

- Definite geometric shape, Sharpe melting point, they have definite heat fusion, they undergo clean cleavage, and they are true solid.

Eg:- Metals And Non metals

2. Amorphous Solid:

Containing irregular arrangement having short range order only, irregular shape, melting over range of temp, they do not have definite heat of fusion they undergo an irregular cut they are pseudo solid or super cooled solid.

Eg:- Rubber ,Glass ,Plastic

Classification of Crystalline Solid:

Crystal Lattice:

The main characteristic of crystalline solid is regular and repeating pattern of constituent particle.

If the three dimensional arrangement of constituent particle in a crystal is present diagrammatically, in which each particle is depicted as point, the arrangement is called crystal lattice.

Accordingly, a regular 3-D arrangement of constituent particles in space is known as crystal lattice or space lattice.

**A portion of three dimensional cubic lattice and its unit cell **

There are only 14 possible three dimensional arrangement of lattice corresponding to seven crystal system are known as Bravais Lattice.

The following are the characteristics of crystal lattice

a. Each point in a lattice is called lattice point or lattice site.

b. Each point in crystal lattice represent one constituent particle which may be an atom or molecule.

c. Lattice point join straight line to bring out the geometry of lattice.

Unit Cell:

The smallest three dimensional portion of a complete space lattice which when repeated over and again in different direction produces the complete space lattice is called unit cell.

**Parameters of a unit Cell **

Its dimensions along the three edges a, b and c, which may or may not me mutually perpendicular to each other.

The angle between the edges α (between b and c),β (between a and c)and γ(between a and b) therefore a unit cell is characterized by six parameters a, b, c, α,β and γ.

Unit cell can be broadly divided into two types:

A) Primitive Unit Cell

B) Center Unit Cell

A) Primitive Unit Cell : The particles are present only on the corner positions of a unit cell.

B) Center Unit Cell : When a unit cell contain one or more particles present at position other than corner in addition to those at corner is called center unit cell.

Generally this have three types.

a) Body center unit cell : Such unit cell contain one constituent particle at its body center besides the ones that are its corner.

b) Face- center unit cell : Such unit cell contain one particle present at center of each face, beside the ones that are corner.

c) End- center/Base-center unit cell: One constituent particle is present at the center of any two opposite faces along with the ones present at its corner.

Basic Shapes of Unit Cell :

There are seven basic shapes of unit cell of crystal which give rise to seven crystal system.

Crystal system characterized on the basis of angle between edges of unit cell and length of edge which determine the shape and size of the crystal.

Number of Atoms per Unit Cell:

a) Simple cubic unit cell: 1 (1/8th part at corner x 8) = 1

b) Body center unit cell: 1 (1/8th part at corner x 8) + 1 (center of the body) =2

c) Face center unit cell: 1 (1/8th part at corner x 8) + 3 (1/2 atom at each face x 6) = 4

d) Base center unit cell: 1 (1/8th part at corner x 8) + 1(1/2 atom at each face x 2) =2

Atomic Packing Factor:

Space occupied by atom in a unit cell is called atomic packing factor (APF) or atomic packing density or efficiency.

** APF = (Volume of the atom in unit cell )/(volume of the
unit cell) **

Volume of the atom in unit cell = no. of atoms present in unit cell
x volume of atoms i.e. ( 4/3 πr^{3}) and simply volume of
the cubic unit cell is a^{3} i.e. (edge length)^{3}

a) Simple Cubic Unit Cell:

No. of Atom in SC = 1

Volume of Atoms = 4/3 πr^{3}

Volume of Unit Cell = a^{3}

Relation between r and a is a = 2r

APF = (1 ×4/3 πr^{3})/a^{3}

= (1 ×4/3 πr^{3})/(2r)^{3}

= 0.52

APF of SC system is 0.52 or in percentag 52%.

b) Body Center Cubic Unit Cell (BCC) :

No. of atom in BCC = 2

Volume of atoms = 4/3 πr^{3}

Volume of unit cell = a^{3}

Relation between r and a is 4r =√(3 )a ( apply Pythagoras theorem)

APF = (2 ×4/3) πr^{3})/(a)^{3}

= (2 ×4/3 πr^{3})/(4/√3 a)^{3}

= 0.68

APF of BCC system is 0.68 or in percentag 68%.

c) Face Center Cubic Unit Cell (FCC) :

No. of atom in BCC = 4

Volume of atoms = 4/3 πr^{3}

Volume of unit cell = a3

Relation between r and a is 4r =√(2 )a (apply Pythagoras theorem)

APF = (4 ×4/3 πr^{3})/(a)^{3}

= (2 ×4/3 πr^{3})/(4/√2 a)^{3}

= 0.74 %

d) Hexagonal Closed Packed (HCP):

C = 1.633 a

Numbe rof atoms in HCP Unit Cell = (12 x (1/6)) + (2 x (1/2) + 3) = 6 atoms

Volume of HCP Unit Cell = Area of the hexagonal face x Height of the hexagonal

Area of the Hexagonal Face = Area of each triangle x 6

Area of Triangle = bh/2 = ah/2 = 1/2 a (a√3/2)

Area of Hexagon = 6 ((a^{2}√3)/4)

Volume of HCP = 6 ((a^{2}√3)/4) x C = 6 ((a^{2}√3)/4) x 1.633a

APF = 6 x ((4πr^{3})/3)/ (√3/4) x 6 x 1.633 . a^{3}

Note : a = 2r

APF = 0.74

e) Cubic Unit Cell Summary:

In the diamond cubic unit cell, there are eight corner atoms, six face centred atoms and four more atoms.

xz = 2r = √((a√2/4)^{2} + (a/4)^{2}) = a√3/4

Atomic Radius (r) is: r = a√3/8

It becomes possible to calculate the APF as follows:

APF = (8 x [(4/3)π(√3a/8)^{3}])/a^{3}

= π√3/16 = 0.34

Summary:

Atom Positions in Cubic Unit Cells :

Cartesian coordinate system is use to locate atoms.

In a cubic unit cell :

- y axis is the direction to the right.

- x axis is the direction coming out of the paper.

- z axis is the direction towards top.

- Negative directions are to the opposite of positive directions.

Atom positions are located using unit distances along the axes.

Atomic position in BCC Unit Cell

Direction in Cubic Unit Cell :

In cubic crystals, direction indices are vector components of directions resolved along each axes, resolved to smallest integers.

Direction indices are position coordinates of unit cell where the direction vector emerges from cell surface, after converted to integers.

Some directions in Cubic Unit Cell

Procedure for finding direction Indices:

Lattice Plane

Miller Indices are used to refer to specific lattice planes of atoms.

(The method of making the plane symbolically was made by English crystallographer Miller and hence the indices of planer called Miller indices).

They are reciprocals of the fractional intercepts (with fractions cleared) that the plane makes with the crystallographic x, y and z axes of three nonparallel edges of the cubic unit cell. It write in a bracket (h k l)and read as e.g. (1 1 1) one one one.

Important features of miller indices of crystal plane

1. All the parallel equidistance plane have the same Miller indices. Thus Miller indices define set of parallel planes.

2. If the Miller indices of two planes have same ratio (i.e.) (8 4 4) and (2 1 1 ) then plane are parallel to each other.

3. Any two plane with indices (h_{1} k_{1} l_{1}) and
(h_{2} k_{2} l_{2}) will be perpendicular to each other if : h_{1}h_{2} + k_{1}k_{2} + l_{1}l_{2}
=0

4. Angle between any two planes is given by formulae:

Procedure for finding Miller indices

Examples:

Miller Bravais Indices for Hexagonal System:

Directions and planes in hexagonal lattices and crystals are
designated by the 4-index **(h k i l)** Miller-Bravais notation.

The first three indices are a symmetrically related set on the basal plane, the third index is a redundant one (which can be derived from the first two as in the formula i = - (h+k)) and is introduced to make sure that members of a family of directions or planes have a set of numbers which are identical this is because in 2D two indices suffice to describe a lattice (or crystal) the fourth index represents the 'c' axis (I to the basal plane).

Hence the first three indices in a hexagonal lattice can be permuted to get the different members of a family; while, the fourth index is kept separate.

Basal Plane a_{1} = h , a_{2}= k, a_{3}=i and l=c Hexagonal system
with (h k i l)

Examples:

Interplaner Distance :

To find the interplanar spacing for the set of planes with Miller indices (hkl), take the origin on any one plane of this set and erect axes in the directions of crystal axes a, b and c.

The interplanar spacing d for the planes of indices (hkl) is equal to the distance from origin to the nearest plane of the set under question.

The distance d is measured along the normal drawn from origin to the plane (hkl). Let the normal to the plane make angles 1, 2 and 3 with a-axis, b-axis and c-axis respectively.

The plane (hkl) intersects a-axis at a/h, b-axis at b/k and c-axis at c/l.

From fig we see that:

cosδ_{1} = d/(a/h) , cosδ_{2} = d/(b/k) , cosδ_{3} = d/(c/l)

For orthogonal axes:

cos^{2}δ_{1} + cos^{2}δ_{2} + cos^{2}δ_{3} = 1

Hence:

d^{2}[h^{2}/a^{2} + k^{2}/b^{2} + l^{2}/c^{2} ] = 1

Cubic:

1/d^{2} = (h^{2} + k^{2} + l^{2})/a^{2}

Tetragonal:

1/d^{2} = (h^{2} + k^{2})/a^{2} + l^{2}/c^{2}

Orthorhombic:

1/d^{2} = h^{2}/a^{2} + k^{2}/b^{2} + l^{2}/c^{2}

Hexagonal:

1/d^{2} =4/3((h^{2} + hk + k^{2})/a^{2} ) + l^{2}/c^{2}

Monoclinic:

1/d^{2} = 1/(sin^{2} β) (h^{2}/a^{2} + (k^{2} sin^{2} β)/b^{2} + l^{2}/c^{2} -2hlcosβ/ac)

Triclinic:

Most probably used for cubic system after rearranging equation

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