Matter exists in three states due to changes taking place in temperature. Solids are formed when the liquids are cooled. Metals solidify in crystalline forms. Based on the geometrical order of atoms or ions in a solid, they are classified into amorphous (do not show regular shape and disorderly arrangement of the atoms) and crystalline (show regular shape and definite geometrical order). Amorphous solids are formed when rapid cooling of a liquid. They are hard and rigid. Their melting points are not sharp. They are unable to give diffraction bands. They are isotropic.
Crystalline solids are formed due to slow, careful cooling of a liquid. They are hard and rigid Their melting points are sharp. X-rays are diffracted by them. They are anisotropic. Crystalline solids are classified into 4 types based on different binding forces in them.
Thermodynamically all solids have tendency to become defective inorder to increase their entropy values. Density, heat capacity, entropy, mechanical strength, electrical conductivity, chemical activity, catalytic activity are effected by crystal defects. Any deviation from perfectly ordered arrangement of atoms or ions in a crystal are known as 'crystal defects'. These defects can be classified into intrinsic defects (seen in pure crystals), 'Extrinsic defects' (due to impurities), 'Extended defects' (present in more dimensions) and point defects (occurs at lattice points). Point defects may be Stoichiometric or Non stoichiometric or Impurity defects.
Schottky defect: A stoichiometric, point defect that occurs when the size of the ions are same and small, coordination number is high (6-8), compounds are more ionic if same number of cations and anions are missing from the lattice points. The density of a crystal decreases due to this defect.
e.g.: NaCl, CsCl, AgBr
Frenkel defect: A stoichiometric, point defect that occurs when the size of the cation is small and that occupies interstitial site instead of lattice sites. This defect is seen where the coordination number is low (4-6), compounds are less ionic. The density of the crystal remains the same.
e.g.: AgCl, AgBr
AgBr can have both Schottky as well as Frenkel defects.
The structure and properties of the metals are explained either by electron sea model or valence bond theory. Electron sea model was proposed by Drude and refined by Lorentz. According to this theory each metal atom contributes its valence electrons to the sea. As these electrons move freely in the interstices, they develop electrostatic attractions with metal ions to form a metallic bond. Valence bond theory was proposed by Linus Pauling. According to this theory the metallic bond could be a polar or non polar covalent bond (resonance).
The metal atoms are considered as spheres and these sphere are arranged in different layers in 3 dimensional space. This packing arrangement gives the structure of metallic elements. Empty spaces are formed during 3 dimentional arrangement of atoms (in different layers) are known as "Voids" or "Interstices" or "Holes. There are 2 types of holes.
Octahedral hole: A hole formed between 6 spheres due to combination of two triangular
voids.
If x spheres are present in a solid, they will form x octahedreal holes and 2 x tetrahedral
holes. The spheres in the solids could be arranged in many ways.
Simple cubic arrangement:
In this arrangement spheres are present at all the 8 corners of the cube. The coordination number of a sphere is 6. Packing efficiency (Volume occupied by the spheres) is 52.4%, 47.6% is void. The distance between two neighbours is "d", the length of the edge in unit cell is "a" atomic radius is "r".
d = a r = 
= 0.5a
e.g.: Po
Body Centred Cubic (BCC) arrangement:
In this arrangement one sphere is present at the centre of the unit cell,
Simple cubic lattice formed by AAA... arrangement
in addition to 8 spheres at all the corners of the unit cell. The coordination number of a sphere is 8. Packing efficiency is 68% (Void space is 32%)
r = 
a = 0.433 a d = (
a = 0.866a)
e.g.: Na, K, Cs, Ba, Cr, Mo, W, Fe
Face centred Cubic (FCC) arrangement: In this arrangement, one sphere is present on the centre of all the 6 faces of the unit cell, in addition to 8 spheres at all the corners of the unit cell. The coordination number of a sphere is 12. Packing efficiency is 74% (void space is 26%)
e.g.: Al, Au, Ag, Ca, Cu, Ni, Pt, Pd, Pb
FCC can be attained by repeating the 3 layers in the pattern ABC ABC ABC ABC...
Hexagonal Close Packed (HCP) arrangement:
HCP can be attained by repeating the 2 layers in the pattern AB AB AB AB...
The coordination number of a sphere is 12. Packing efficiency is 74% (Void space is 26%).
e.g.: Mg, Zn, Cd, Co, Be, Ti, Tl.
In the formation of crystals, the constituent particles try to pack as closely as possible so as to attain a state of maximum possible stability and density. The ratio of volume occupied by the particles (atoms, ions or molecules) in the Unit Cell and the Volume of the Unit Cell is called "packing fraction" and the percentage of total space filled by the particles is called "packing efficiency".
PACKING EFFICIENCY IN SIMPLE CUBIC LATTICE
Eight atoms (spheres) are occupied by all the 8 corners of the cube. Each atom makes 
contribution to the Unit Cell.
Simple cubic unit cell. The spheres are in contact with
each other along the edge of the cube.
Number of atoms in the Unit Cell = = 1 atom
Edge length of the cube = a
Radius of atom = r
Edge length a = r + r = 2r
Volume occupied by 1 atom = 
Volume of Cubic Unit cell = V = a3 = (2r)3 = 8r3
Packing efficiency = 
= 
= 52.4%
PACKING EFFICIENCY IN BODY - CENTRED CUBIC STRUCTURES
This type of unit cell contain 8 atoms at 8 corners and one atom at the centre of the body. Number of atoms in the Unit Cell =
corners + body centre (1) = 1 + 1 = 2
Body-centred cubic unit cell (sphere along the body
diagonal are shown with solid boundaries).
PACKING EFFICIENCY IN CUBIC CLOSE PACKING STRUCTURES
Packing efficiency in Face Centred Cubic (FCP), Hexagonal Close Packing (HCP) and Cubic Close Packing (CCP) is same. You have already learned about NaCl structure (FCP) in junior inter. In FCP 8 corners & 6 faces are occupied by 14 atoms.
Number of atoms in the Unit Cell = Corners
+ Faces 
= 1 + 3 = 4
Cubic close packing other sides are not provided with spheres for sake of clarity.
Density of Unit Cell
Molecular weight of solid crystalline substance = M
No. of atoms present per unit cell = Z
Mass of avogadro no. of atoms No --- M
Mass of 1 atom --- ?
Unit cell is "The smallest basic repeating 3 dimentional unit from which a crystal lattice is built". A space lattice is the representation of lattice points by "Points". The characteristics of Unit cell are utilized in calculating density, inter planar distances, in the crystals and wave length of X-rays. Internal structures of the crystalline solids are revealed by the unit cells and space lattices. From the radius ratio (rcation/ ranion), the shape of the crystal formed and coordination number of the ion are known. The coordination number of an atom or ion of a solid is the number of nearest neighbours for that atom or ion in a solid.
Unit Cells of 14 types of Bravais Lattices
The three cubic lattices: All sides of same length, angles between faces all 90º
The two tetragonal: One side different in length to the other, two angles between faces all 90º
The four orthorhombic lattices: unequals sides, angles between faces all 900
The two monoclinic lattices: Unequal sides, two faces have angles different to 900
Hexagonal lattice-one side different in length to the other two, the marked angles on two faces are 600
Rhombohedral lattice all sides of equal length, angles on two faces are less than 900
Triclinic lattice- unequal sides a, b, c
A, B, C are unequal angles with none equal to 900
X-ray diffraction Studies:
According to Max Von Laue, a crystal functions as three dimentional grating to X-rays. By the X-ray diffraction studies of the crystals, exact arrangement of atoms or ions are known. Size & shape of the unit cell can be measured. Waves of X-rays undergo either constructive interference (If waves are present in the same phase) or destructive interference (if waves are not in the same phase).
Derivation of Bragg's Equation:
A crystal has many planes. Atoms or ions are arranged in systematic geometry in these planes. When X-rays incident on these crystal planes, they undergo diffraction. When the waves are diffracted from these atoms or ions, they may have constructive interference or destructive interference.
From the figure it is very clear that 1st and 2nd X-rays travel the same distance till the wave front AD. Where as 2nd ray travels an extra distance DB + BC (path difference) than that of 1st X-ray. If the two waves are to present in the same phase (constructive interference), the path difference must be equal to the wave length λ or an integral multiple of wave length.
DB = d sin θ
BC = d sin θ
nλ = DB + BC = 2d sin θ
This relations is known as Bragg's equation
θ Values increases with n values. Knowing the values of θ & λ one can calculate d.
λ = wave length of X-ray
θ = angle of incident X-ray
n = order of diffraction
d = interplaner distance.
Bragg's method:
Bragg's spectrometer is used to determine the structures of the crystals. A fully developed large crystal is placed on a round table (which can be rotated 3600). When X -rays are indent on the crystal intensity of diffracted X-rays enter ionization chamber increases. Current produced due to ionization of CH3Br is measured by the electrometer. The values of 'd' in the X, Y, Z axes are compared an the structure of the crystal is determined from their ratio. The ratio of 'd' values in NaCl crystal is 1 : 0.703 : 1.134.
