Q$1.$ Predicting density using the concepts from crystal structures

Q$1.($i$)$ Copper has face$-$centered cubic $($fcc$)$ crystal structure. Using this information, atomic radius of

copper $(0\mathrm{.}1278$ nm $),$ and its atomic mass $(63\mathrm{.}5),$ determine the theoretical density of copper. Compare

this value to the empirical value of $8\mathrm{.}93\frac{g}{c}{m}^2.$

$($ii$)$ Assume that instead of fcc structure, copper had a bec structure having the same atomic radii. What

would be its density?

$($iii$)$ Would it be same ductile as foc copper $($explain briefly in no more than $2-3$ lines$)?$

Q$2.$ Phase transformation $-$ volume change, linear change

On heating from room temperature, iron changes from bec to foc metal structure at $912C.$ At this

temperature, the atomic radii of the iron in the two structures are $0\mathrm{.}126$ nm and $0\mathrm{.}129$ nm $,$ respectively.

$($i$)$ What is the relative volume change $d\frac{V}{V}$ when the structure changes from bcc to foc at $912C?$

$($ii$)$ What is the magnitude of the abrupt linear change dL$/$L at $912C?$

$($iii$)$ In contrast to the above calculated abrupt linear change, explain the continuous linear change vs

temperature for boc iron from room temperature to $912C$ using the concepts of cohesive

energy. Also explain why the rate of linear change vs temperature for fcc iron differs from that

for the boc structure.

$($ii$)$ What is the APF for NaCl $?$

$($iii$)$ Why is this APF less than the APF of $0\mathrm{.}74$ for a foc unit cell having a single atom as its basia.

Q$4.$ Cohesive energies

The net potential energy $E_N$ between two adjacent ions is sometimes represented by the expression

$E_N=-\frac{C}{r}+$Dexp$(-\frac{r}{p})$

In which $r$ is the interatomic separation and C$,$ D$,$ and $$ constants whose values depend on the specific

materials.

Derive the expression for bonding energy $E_0$ in terms of the equilibrium interatomic separation $r_0$ and

the constants $D$ and $.$

Q$5.$ Determination of distance between planes and the atomic radius.

The metal iridium has fec structure. If the angle of diffraction for the $(220)$ set of planes occur at $69\mathrm{.}22$

$($for $n=1)$ when $x-$ray wavelength of $0\mathrm{.}1542$ nm $,$ compute:

$($i$)$ The interplanar spacing of the $(220)$ planes.

$($ii$)$ The atomic radius for the inidium atom.

$($iii$)$ The theoretical density of iridium.

Q$3.$ Atomic packing factor $($APF$)$ of crystals

NaCl has FCC structure with a basis of $1N{a}^{+}$and $1$ Cr ion. The atomic radii of Na ion is $0\mathrm{.}097$ nm and

atomic radii of Cl ion is $0\mathrm{.}181$ nm $.$

$($i$)$ What is the lattice constant of the foc NaCl unit cell?