Solid State

Space Lattice or Crystal Lattice:

SPACE LATTICE may be defined as a regular three dimensional arrangement of identical points in space



Unit Cell:

Smallest unit of the crystal lattice which when repeated forms the whole crystal.



Bravais Lattice:
Bravais (1848) showed from geometrical considerations that there are only seven shapes in which unit cells can exist. These are :

(i) Cubic,

(ii) Orthorhombic,

(iii) Rhombohedral,

(iv) Hexagonal,

(v) Tetragonal,

(vi) Monoclinic

(vii) Triclinic.

Moreover he also showed that there are basically four types of unit cells depending on the manner in which they are arranged in a given shape. These are : Primitive, Body Centered, Face Centered and End Centered. He also went on to postulate that out of the possible twenty eight unit cells (i.e. seven shapes × four types in each shape = 28 possible unit cells), only fourteen actually would exist. He postulated these based only on symmetry considerations.

These fourteen unit cells that actually exist are called Bravais Lattices.












Packing Fraction:

It is defined as ratio of the volume of the unit cell that is occupied by the spheres of a unit cell to the volume of the unit cell.

i) Primitive Cubic Unit Cell:

Atoms are present at the corners of the cube. Each of the eight atoms present at the eight corners is shared amongst eight unit cells. Hence,

Number of atoms per unit cell = 8

Atoms touch each other along edges. Hence,

r = a/2

Therefore, PF = .


ii) Body-Centred Cubic Unit Cell :

Besides atoms present at the corners of the cube, there is one atom in the centre of cube which belongs exclusively to this unit cell. Hence,

Number of atoms per unit cell = 8

Atoms touch each other along the cross diagonal of the cube. Hence



The packing fraction in this case is =


iii) Face-Centred Cubic Unit Cell:

Besides atoms present at the corners of the cube, there are atoms at the centres of six faces, each of which is shared between two unit cells. Hence

Number of atoms per unit cell = 4

Atoms touch each other along the face diagonal. Hence







Hexagonal Primitive Unit Cell :
Each corner atom would be common to 6 other unit cells, therefore their contribution to one unit cell would be 1/6. Therefore, the total number of atoms present per unit cell effectively

is 6. The height of the unit cell 'c' is and the length of the unit cell 'a' is 2r. Therefore the area of the base is equal to the area of six equilateral triangles, = . The

volume of the unit cell =
.
P.F. = –– 0.74



Density of Cubic Crystals

r =

Where z is the number of molecules (or atoms) per unit cell, 'a' is the edge length of the unit cell, M is the molar mass of the substance, and N_A is Avogadro's constant.

Crystal structure	Brief description and examples	Co-ordination number	No. of formula units per unit cell	Fig.
1. Rock salt (NaCl- type)	Cl- ions in CCP, Na+ions occupy all the octahedral voids (or vice versa) Examples Halides of Li, Na, K and Rb, AgCl, AgBr, NH4Cl etc.	Na+ - 6 Cl- - 6	4
2. CsCl - type	Cl- ions at the corners of a cube and Cs+ ions in the cubic void (or vice-versa) Examples: CsCl, CsBr, CsI etc.	Cs+ - 8 Cl - - 8	1
3. Zinc - blende (ZnS - type)	S2- ions in CCP, Zn+2 ions occupy alternate tetrahedral voids. i.e. only half of the total number of tetrahedral voids are occupied. Examples: ZnS, CuCl, CuBr, CuI, AgI etc.	Zn+2 - 4 S2- - 4	4
Crystal structure	Brief description and examples	Co-ordination number	No. of formula units per unit cell	Fig.
4. Fluorite structure (CaF2 - type)	Ca+2 ions in CCP and F- ions in all the tetrahedral voids Examples: CaF2, SrF2, BaF2, BaCl2 etc.	Ca+2 - 8 F- - 4	4
5. Antifluorite structure	Negative ions in CCP and positive ions in all the tetrahedral voids Examples Na2O	Na+ - 4 O2 - 8	4

Note:
(1) In the above figures, a black circle would denote a cation and a white circle would denote an anion.

(2) In any solid of the type AxBy, the ratio of the co-ordination of A to that of B would be Y : X.

Tetrahedral Voids And Octahedral Voids:

These voids are only found in either FCC or Hexagonal primitive unit cells.

1. Tetrahedral Voids:- In close packing arrangement each sphere in the second layer rests on the hollow (triangular void) in the three touching spheres in the first layer. The centres of

these four spheres are at the corners of a regular tetrahedron. The vacant space between these four spheres is called tetrahedral void.




2. Octahedral Voids:- The interstitial void formed by combination of two triangular voids of the first and second layer is called octahedral void, because this is enclosed between six

spheres, centres of which occupy corners of a regular octahedron.




Position of Octahedral and Tetrahedral Void in a Unit Cell:

1. Let us assume that there is an atom (different from the one that forms the FCC) at the center of an edge. Let it be big enough to touch one of the corner atoms of the FCC. In that

case, it can be easily understood that it would also touch six other atoms at the same distance. Such voids in an FCC unit cell in which if we place an atom it would be in contact with

six spheres at equal distance (in the form of an octahedron). This void is called Octahedral void.



2. Let us again consider a FCC unit cell. If we assume that one of its corners is an origin, we can locate a point having Co-ordinates . If we place an atom (different from the
ones that form the FCC) at this point and if it is big enough to touch the corner atom, then it would also touch three atoms which are at the face centers of all those faces which meet at

that corner. Moreover, it would touch all these atoms at the corners of a regular tetrahedron.



Note: 1. On calculation, it can be found out that FCC unit cell has four octahedral voids effectively. The number of octahedral voids of a unit cell is equal to the effective no. of atoms of

that unit cell.

2. As there are eight corners, there are eight tetrahedral voids in a FCC unit cell, therefore the no. of tetrahedral voids is double the no. of octahedral void. Radius Ratio:

The ratio of radii of cation and that of anion in ionic crystal is called as radius ratio .

Limiting radius ratio	Co - ordination number	Shape	Example
x < 0.155	2	Linear	BeF2
0.155 x < 0.225	3	Planar Triangle	AlCl3
0.225 x < 0.414	4	Tetrahedron	ZnS
0.414 x < 0.732	4	Square planar	PtCl42-
0.414 x < 0.732	6	Octahedron	NaCl
0.732 x < 0.999	8	Body centered cubic	CsCl