Problem $\backslash$#$1$

hexagonal close$-$packed $($HCP$)$ has two lattice parameters $(\backslash ($ a $\backslash ),$ and $\backslash ($ c $\backslash )).$ Show that the $\backslash (\backslash$frac$\{$c$\}\{$a$\}\backslash )$ ratio for an ideal HCP structure is $\backslash (\backslash$sqrt$\{\backslash$frac$\{8\}\{3\}\}=1\mathrm{.}633\backslash ).$

Problem $\backslash$#$2$

Determine whether Rhodium has an FCC or BCC crystal structure if it has an atomic radius of $0\mathrm{.}1345$ nm $,$ a density of $\backslash (12\mathrm{.}41\backslash$mathrm$\{$~g$\}/\backslash$mathrm$\{$cm$\}^\{3\}\backslash ),$ and an atomic weight of $\backslash (102\mathrm{.}9055\backslash$mathrm$\{$~g$\}/\backslash$mathrm$\{$mol$\}\backslash ).$

Problem $\backslash$#$3$

Answer the following questions$-$

a$)$ What family of directions has the highest linear density in the FCC system?

b$)$ What family of directions has the highest linear density in the BCC system?

c$)$ What family of planes has the highest planar density in the FCC system?

d$)$ What family of planes has the highest planar density in the BCC system?

e$)$ What family of planes has the highest planar density in the HCP system

Problem $\backslash$#$4$

Derive the planar density expression for the HCP $(0001)$ plane in terms of the atomic radius $\backslash ($ R $\backslash )$