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

As there are 8 corners in a cube and gold atoms are present at the corners of the cube, Thus, 8 gold atoms at the eight corners contribute 8 x 1/8 = 1 [Contribution of an 8 atom at the corner per unit cell is 1/8] As there are 6 face centers in a cube and cadmium atoms fit into six face centers of the cube, Thus, 6 cadmium atoms at the six faces contribute 6 x 1/2 =3 [Contribuion per unit cell of an atom at face center is 1/2]. Thus, formula of this alloy is AuCd³

Suppose the edge length of the unit cell = a and radius of the sphere = r Since the spheres are touching each other, a = 2r Now, there are eight spheres at the corners of the cube: Each sphere at the corner is shared by eight unit cells and the contribution per unit cell is 1/8 so that number of spheres per unit cell is 8 x 1/8 = 1 Volume of cube = a³ = (2r)³ = 8r³. Since simple cubic unit cell has only one atom per unit cells the volume of space occupied is \(\frac { 4 }{ 3 } \pi { r }^{ 3 }\)

\(Packing \ efficiency=\frac { Volume \ of \ one \ sphere }{ Volume \ of \ cubic \ unit \ cell } \times 100\)
\(\\ \frac { \frac { 4 }{ 3 } \pi { r }^{ 3 } }{ 8{ r }^{ 3 } } \times 100\)
= 52.37 or   52.4

Ferromagnetism: When magnetic moments are aligned in the same direction, it gives rise to ferromagnetism, e.g. iron, cobalt and nickel show ferromagnetism.
Antiferromagnetism: When magnetic moments are aligned in such a way that net magnetic moment is zero, then magnetism is called antiferromagnetism, e.g. MnO shows antiferromagnetism.

(i) The unit cell is the smallest portion of a crystal lattice which when repeated over and over again in different directions results in the entire lattice. For example, simple cubic unit cell.
(ii) The number of nearest neighbors in a packing is called coordination number.For example,in a body-centered cubic structure, the coordination number is eight.

Let us assume, Z = 4 (copper has fcc)
\(d={Z\times M\over a^3\times N_4}\)
\(\Rightarrow\ 8.95={4\times63.54\over a^3\times6.02\times10^{23}}\)
\(\Rightarrow\ a^3={4\times63.54\over 8.95\times6.02\times10^{23}}\)
\(={254.16\over 53.88}\times10^{-23}\)
= 4.7171 x 10^-23cm³
a³ = 47.171 x 10^-24 crrr³
\(\Rightarrow\ a=\sqrt[3]{47.171}\times10^{-24}cm^3\)
= 3.613 x 10^-8 cm = 361.2 pm
For fcc,
\(4r=\sqrt{2a}\)
\(\Rightarrow\ r={a\over 2\sqrt{2}}={361.\over 2\times1.414}\)
= 127.76 pm
It shows that it has fcc structure as given radius is nearly equal to calculated radius by taking Z = 4.

(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.

For a fcc structure, length of the unit cell is related to radius of atom as
\(r=\frac { a }{ 2\sqrt { 2 } } \)
\(\\ or \ a=2\sqrt { 2 } r\)
\(=2\times 1.414\times 127.8 \ pm=361.42pm\)
\(Density, \ \rho =\frac { Z\times M }{ { a }^{ 3 }\times { N }_{ A } } \)
\(=Z=4(for \ fcc),M=63.55 \ g \ { mol }^{ -1 }\)
\(a=361.42\times { 10 }^{ -12 }m=361.42\times { 10 }^{ -10 }cm\)
\({ N }_{ A }=6.02\times { 10 }^{ 23 }\)
\(\rho =\frac { 4\times 63.55 }{ { \left( 361.42\times { 10 }^{ -10 } \right) }^{ 3 }\times 6.02\times { 10 }^{ 23 } }\)
\(=8.94 \ g{ cm }^{ -3 }\)
\(Density, \ \rho =\frac { Z\times M }{ { a }^{ 3 }\times { N }_{ A } } \)
\(a=286.65pm=286.65\times { 10 }^{ -12 }m\)
\(=286.65\times { 10 }^{ -10 }cm\)
\(\ Z=2(for \ bcc), \ M=56.0\)
\(\rho =7.87 \ g{ cm }^{ -3 }\)
\(7.87=\frac { 2\times 56.0 }{ { \left( 286.65\times { 10 }^{ -10 } \right) }^{ 3 }\times { N }_{ A } }\)
\( { N }_{ A }=\frac { 2\times 56.0 }{ { \left( 286.65\times { 10 }^{ -10 } \right) }^{ 3 }\times 7.87 } \)
\(=6.04\times { 10 }^{ 23 }\)

\(d=\frac { Z\times M }{ { a }^{ 3 }\times { N }_{ A } } \)
\(\\ M=\frac { d\times { a }^{ 3 }\times { N }_{ A } }{ Z }\)
\( Z=\frac { 8.0g{ cm }^{ -3 }\times { \left( 250 \right) }^{ 3 }\times { 10 }^{ -30 }{ cm }^{ 3 }\times 6.023\times { 10 }^{ 23 } }{ 2 }\)
\(Z=\frac { 8\times 15.625\times 6.023 }{ 20 } \)
\(=\frac { 752.875 }{ 20 } \)
\(\\ 37.64 \ g{ \ mol }^{ -1 }\)
\(For \ bcc, \ 4r=\sqrt { 3 } a\)
\(\ r=\frac { \sqrt { 3 } a }{ 4 } =\frac { 1.732\times 250 }{ 4 } =108.25 \ pm\)

\(1.86\overset{0}{A}\)

(i) Ferrites are magnetic material. 
(ii) They are used in telephones and memory loops in computers.

(i) Piezoelectric crystals behaves as mechanical electrical transducer. 
(ii) Piezoelectric crystals are used as pickups in record players where they produce electric signals by application of pressure. 
(iii) We generally become tense and pessimistic under stress. But we should face every situation, accept every worst situation as challenge and maintain our mental balance. It will definitely result in our performance just like in piezoelectric crystals. We can learn from these non-living crystals to be optimistic as worst situations can be converted into an opportunity.