\(d=\frac { Z\times M }{ a^{ 3 }\times N_{ A } } \)

\(\Rightarrow 8.92=\frac { 4\times 63.5 }{ 8.92\times 6.02\times 10^{ 23 } } \)
\(\Rightarrow a^{ 3 }=\frac { 4\times 63.5 }{ 8.92\times 6.02\times 10^{ 23 } } \)
\(=\frac { 254 }{ 53.6984\times 10^{ 23 } } \)
\(=4.73\times 10^{ -23 }\)
\(=0.0473\times 10^{ -21 }\)
\(a=0.36\times 10^{ -7 }\ cm\)
For fcc crystal
Radius (r) =\(\frac { \sqrt { 2 } a }{ 4 } \)
\(=\frac { 1.414\times 36\times 10^{ -7 } }{ 4 } \)
\(=0.127\times 10^{ -8 }\ cm\)

\(d=\frac { Z\times M }{ a^{ 3 }\times N_{ A } } \)
\(\Rightarrow Z=\frac { d\times a^{ 3 }\times N_{ A } }{ M } \)
\(\Rightarrow Z=\frac { (2 \ g \ cm^{ -3 })(5\times 10^{ -8 })^{ 3 }(6\times 10^{ 23 }mol^{ -1 }) }{ 75gmol^{ -1 } } \)
Z = 2
Since the number of atoms per unit cell is 2, it indicates that the metal has body centred cubic (bcc) lattice.
For bcc lattice, body diagonal of the unit cell
 d = 2r
\(=\frac { \sqrt { 3a } }{ 2 } =\frac { 4\times Atomic \ radius }{ \sqrt { 3 } xedge \ length } \)
\(\Rightarrow 4r=\sqrt { 3 } a\)
\(\Rightarrow 4r=\frac { \sqrt { 3 } }{ 4 } \times 5\)
=216.5 pm.

Non-stoichiometric compounds: Non-stoichiometric compounds are those in which the ratio of the number of positive ions and the negative ions does not correspond to the simple whole number as suggested by the formula. There is a large number of non-stoichiometric ionic solids which contain an excess or deficiency of one of the components. As a result, the crystal structure becomes irregular and this leads to the presence of a defect. Such defects are also called non-stoichiometric defects.
For example, vanadium oxide has the formula VO_x where x lies between 0.6 and 1.3. Similarly, iron (II) oxide samples contain more oxygen atoms than iron atoms while samples contain more oxygen atoms than iron atoms while samples of zinc oxides, ZnO usually has excess of zinc atoms than oxygen atoms. However, it may be noted that in each case, the electrical neutrality of the crystal is maintained.

When imperfections or defects in a crystal are caused by the departure from the definite arrangement in the vicinity of an atom or a group of atoms, the imperfections are called point defects. The imperfections are caused either by missing or dislocations of a constituent particle to a position meant for another particle or shifting to an interstitial position.
Point defects are of two common types:
(i) Schottky defect
(ii) Frenkel defect.
For Schottky and Frenkel defects.

(i) Dislocation: Lattice imperfections which are extended along lines are called dislocations or line defects.
(ii) F-centre: The electrons trapped in anion vacancies are called F-centres.
(iii) Anti-ferroelectric: If in a piezoelectric crystal, the dipoles in alternative polyhedra pointing up and down so that there is no net dipole moment, the crystal is reffered to as anti-ferroelectric. For example, PbZrO₃ (lead zirconate).

(i) When there is spontaneous alignment of magnetic moments of domains in the same direction, we get ferromagnetism. These have strong magnetic effects and ordering of domains persists even when magnetic field is removed and ferromagnetic substance becomes permanent magnet.
(ii) When the magnetic moments of the domains are aligned in parallel or anti-parallel directions in unequal numbers resulting in net magnetic moment, we get ferromagnetism.

Physical properties such as electrical conductivity, thermal conductivity, refractivity index etc. show different values when measured along different directions in the same crystal. This phenomenon is called anisotropy. It arises because of different arrangement of particles in different directions. quartz shows this property because it is crystalline.

7.64 x 10²⁰

A₇B₂₄

Quartz is three dimensional crystalline solids having regular arrangement of silicon and oxygen.

The ordinary glass which is metal silicate is amorphous solid. When quartz is melted and the melt is cooled rapidly, it forms a glass or amorphous solid.

\(4{ Ca }^{ 2+ }and{ 8F }^{ - }ions \ means \ { 4CaF }_{ 2 }units\)
\(Z=4, \ d=\frac { Z\times M }{ { a }^{ 3 }\times { N }_{ A } } \)
\({ N }_{ A }=\frac { Z\times M }{ d\times { a }^{ 3 } }\)
\( =\frac { 4\times 78.08g{ mol }^{ -1 } }{ 3.18g{ cm }^{ -3 }\times { \left( 5.46 \right) }^{ 3 }\times { 10 }^{ -24 }{ cm }^{ 3 } } \)
\(=\frac { 312.32\times { 10 }^{ 24 } }{ 3.18\times { \left( 5.46 \right) }^{ 3 } } =\frac { 3123.2 }{ 517.61 } \times { 10 }^{ 23 }\)
\({ N }_{ A }=6.0338\times { 10 }^{ 23 }{ mol }^{ -1 }\)

If a is the edge length of fcc unit cell, then
face diagonal = 4r
\(4r=\sqrt { 2 } a\)
\( r=\frac { \sqrt { 2 } }{ 4 } a\)
\(=\frac { \sqrt { 2 } \times 400pm }{ 4 } \)
\(=1.414\times 100=141.4pm\) 
​

(i) Ferromagnetism: Those substances which remain magnetized even after the magnetic field is removed are called ferromagnetic substances. They have unpaired electrons, e.g. iron, cobalt, nickel, CrO_2, etc. These substances are strongly attracted by a magnetic field. The phenomenon is called Ferromagnetism.
(ii) Paramagnetism: Those substances. which have unpaired electrons are called paramagnetic substances. They are weakly attracted by a magnet, e.g. O₂, Cu²⁺, Fe²⁺ are paramagnetic in nature. The phenomenon is called Paramagnetism.
(iii) Ferrimagnetism: It is observed when the magnetic moments of the domains are aligned parallel and anti-parallel but unequal in number is called ferromagnetic substances, e.g.Fe₃O₄, CuFep.vetc. The phenomenon is called Ferrimagnetism.

Crystalline solids: Benzoic acid, naphthalene, potassium nitrate, copper.
Amorphous solids: Polyurethane, teflon, cellophane, polyvinyl chloride, fibre glass.

Hexagonal closed-packed lattice has maximum efficiency; i.e 74%, whereas body centred cubic has 68% and simple cubic has 52.4%.