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msm

Unit: 1

STRUCTURE OF CRYSTALLINE SOLIDS

1.1      Structure of crystalline solids

All solid materials are classified into two broad groups:

1.  Crystalline solids

2.  Non-crystalline solids or Amorphous solids

**(Basis of classification is building blocks)

1.2      Differences

	Crystalline solids		Amorphous solids
1.	Basic structural unit is a crystal and a number of crystal come together to form crystalline solids.	1.	Basic structural unit is a molecule and chain of these molecules come together to form amorphous solids.
2.	Crystals/grains are made of number of repetitive blocks called unit cell which are arranged in a particular manner.	2.	Chain molecules are random and irregular in symmetry.
3.	Metal, alloys, salt (NaCl), diamond, etc.	3.	Glass, polymers, etc.
4.	Density is generally high as arranged and high M.P and strength.	4.	Density is generally low as molecules are not arranged and compact, and low M.P and strength.
5.	Structures are stable and materials are stronger.	5.	Less stable, and less strong.

1.3      Aggregates

As all materials are grouped under crystalline solid and amorphous solid or non- crystalline solid, but some materials are found in both the forms such as silicate [crystalline form (quartz) and Amorphous form (silicate glass)], and some time found in combination of both crystalline and amorphous solids. Those materials which have short range order but no long range order are termed as “Aggregates”.

Ex: concrete, rocks, minerals, etc.

1.4      Unit Cell

Unit cell (made up of groups atoms) is the repeatable units of a crystalline solid arranged in definite order.

	UNIT CELL (Crystal system)	SPACE LATTICE (atomic arrangement)		ABB.	EXAMPLES
1.	Cubic (C)	1.	Simple cubic (8-atoms are arranged on the corner of the unit cell.)	SC	Mn,NaCl
		2.	Body Centred cubic. (8-atoms at corner and 1 atom at the centre.	BCC	Na,V,CaCl,Fe*
		3.	Face centre cubic (8-atoms are arranged on the corner & 1-atom each at the centre of 6 faces.)	FCC	Ni,Ag,Au,Al,Pb,Pt

2.	Tetragonal(T)	4.	Simple tetragonal (8-atoms at each corners)	ST	Pa,In
		5.	Body centred tetragonal (8-atoms at each corner and 1-atom at the centre	BCT	Sn*,Cl

3.	Orthorhombic(O)	6.	Simple orthorhombic (8-atoms in the corner )	SO	As, Bi
		7.	End centred orthorhombic (8-atoms at each corner & 2-atoms at face centre opposite to each other)	ECO	MgS­O4, KNO3
		8.	Body centred orthorhombic (8-atoms at each corner & 1-atom at body centre)	BCO	Cementite
		9.	Face centred orthorhombic (8-atoms at each corner & 6-atoms at each face centred)	FCO	Ga

4.

Rhombohedral (R)

10.

Simple Rhombohedral (8-atoms at each corner )

SR

B,CaCO3,SiO2

5.

Hexagonal(H)

11.

Hexagonal close packed (6-atoms at corner of each face top & bottom,2-atoms hexagonal face centred & 3-atoms within the body.

HCP

Mg,Zn,Hf

 fig:HCP-1

 fig HCP-2

 FIG: HCP-3

6.	Monoclinic (M)	12.	Simple monoclinic (8-atoms at the each corner)	SM	FeSO4
		13.	End centred monoclinic (8-atoms at the each corner & 2-atoms at face centred opposite to each other)	ECM	S*

fig:monoclinic

7.

Triclinic(T)

14.

Simple triclinic (8-atoms at each corner)

ST

CuSO4, K2Cr2O7

SPACE LATTICE IN CRYSTALS

In a unit cell atoms are regularly spaced and arranged in various directions. This three dimensional pattern where the atoms arranged themselves in an orderly manner along various directions is known as “space lattice”.

There are 14 possible types of space lattice and they fall under 7 crystal system.

These 14 possible type of space lattice is called Bravis` space lattice.

The Bravais lattices

The Bravais lattice are the distinct lattice types which when repeated can fill the whole space. The lattice can therefore be generated by three unit vectors, a₁, a₂ and a₃ and a set of integers k, l and m so that each lattice point, identified by a vector r, can be obtained from:

r = k a₁ + l a₂ + m a₃

In two dimensions there are five distinct Bravais lattices, while in three dimensions there are fourteen. These fourteen lattices are further classified as shown in the table below where a₁, a₂ and a₃ are the magnitudes of the unit vectors and a, b and g are the angles between the unit vectors.

Name	Number of Bravais lattices	Conditions
Triclinic	1	a1 ¹ a2 ¹ a3 a ¹ b ¹ g
Monoclinic	2	a1 ¹ a2 ¹ a3 a = b = 90° ¹ g
Orthorhombic	4	a1 ¹ a2 ¹ a3 a = b = g = 90°
Tetragonal	2	a1 = a2 ¹ a3 a = b = g = 90°
Cubic	3	a1 = a2 = a3 a = b = g = 90°
Trigonal	1	a1 = a2 = a3 a = b = g < 120° ¹ 90°
Hexagonal	1	a1 = a2 ¹ a3 a = b = 90° g = 120°

Cubic lattices

Cubic lattices are of interest since a large number of materials have a cubic lattice. There are only three cubic Bravais lattices. All other cubic crystal structures (for instance the diamond lattice) can be formed by adding an appropriate base at each lattice point to one of those three lattices. The three cubic Bravais lattices are the simple cubic lattice, the body cantered cubic lattice and the face cantered cubic lattice. A summary of some properties of cubic lattices is listed in the table below:

Lattice type	Number of lattice points/atoms per unit cell	Nearest distance between lattice points	Maximum packing density	Example
Simple cubic	1/1	A	p/6 = 52 %	Phosphor
Body cantered cubic	2/2	aÖ3/2	pÖ3/8 = 68 %	Tungsten
Face cantered cubic	4/4	aÖ2/2	pÖ2/3 = 74 %	Aluminium
Diamond	4/8	aÖ2/2 Nearest distance between atoms: aÖ3/4	pÖ3/16 = 34 %	Silicon

Cubic lattices have the highest degree of symmetry of any Bravais lattice. They belong to the (m3m) symmetry group which contains the following symmetry groups and operations:

Identity	1
Three equivalent axis of two-fold rotation	3[2|]	[100], [010], [001]
Six equivalent axis of four-fold rotation	6[4|]	[100], [010, [001], [-100], [0-10], [00-1]
Six equivalent axis of two-fold rotation	6[2]	[110], [101], [011], [1-10], [10-1], [01-1]
Eight equivalent axis of three-fold rotation	8[3]	[111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [-11-1], [1-1-1]
Inversion	-1
Three equivalent mirror planes	3[m|]	[100], [010], [001]
Six equivalent axis of four-fold rotation with inversion	6[-4]	[100], [010, [001], [-100], [0-10], [00-1]
Six equivalent mirror planes	6[m]	[110], [101], [011], [1-10], [10-1], [01-1]
Eight equivalent axis of three-fold rotation with inversion	8[-3]	[111], [11-1], [1-11], [-111], [-1-1-1], [-1-11], [-11-1], [1-1-1]

Note that the (m3m) symmetry group is the highest possible symmetry group associated with a cubic crystal. A limited symmetry of the basis (the arrangement of atoms associated with each lattice point) can yield a lower overall symmetry group of the crystal.

Simple cubic lattice

The simple cubic lattice consists of the lattice points identified by the corners of closely packed cubes.

sc.gif

The simple cubic lattice contains 1 lattice point per unit cell. The unit cell is the cube connecting the individual lattice points. The atoms in the picture are shown as an example and to indicate the location of the lattice points. The maximum packing density occurs when the atoms have a radius which equals half of the side of the unit cell. The corresponding maximum packing density is 52 %.

Body cantered cubic lattice

The body cantered lattice equals the simple cubic lattice with the addition of a lattice point in the centre of each cube.

bcc.gif

The body cantered cubic lattice contains 2 lattice points per unit cell. The maximum packing density occurs when the atoms have a radius which equals one quarter of the body diagonal of the unit cell. The corresponding maximum packing density is 68 %.

Face cantered cubic lattice

The face cantered lattice equals the simple cubic lattice with the addition of a lattice point in the centre of each of the six faces of each cube.

fcc.gif

The face cantered cubic lattice contains 4 lattice points per unit cell. The maximum packing density occurs when the atoms have a radius which equals one quarter of the diagonal of one face of the unit cell. The corresponding maximum packing density is 74 %. This is the highest possible packing density of any crystal structure as calculated using the assumption that atoms can be treated as rigid spheres.

Diamond lattice

The diamond lattice consists of a face cantered cubic Bravais point lattice which contains two identical atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of diamond, germanium and silicon.

diamond.gif

The diamond lattice contains also 4 lattice point per unit cell but contains 8 atoms per unit cell. The maximum packing density occurs when the atoms have a radius which equals one eighth of the body diagonal of the unit cell. The corresponding maximum packing density is 34 %.

Zincblende lattice

The zincblende lattice consists of a face cantered cubic Bravais point lattice which contains two different atoms per lattice point. The distance between the two atoms equals one quarter of the body diagonal of the cube. The diamond lattice represents the crystal structure of zincblende (ZnS), gallium arsenide, indium phosphide, cubic silicon carbide and cubic gallium nitride.

zincblen.gif

CO-ORDINATION NUMBERS

Each atom in a crystal structure is surrounded by a number of atoms, and these surrounding atoms are located at different definite distances.

Number of nearest and equidistant neighbouring atoms that each atom has in a space lattice is called “co-ordination number”.

SC-6, BCC-8, FCC-12, HCP-12

Atomic pacing factor (APF)

vedio for SC APF