$(1)$ As$-$deposited NiO thin films are often p$-$type, how does Li$+$ ion doping enhance further p$-$type doping? How does annealing in vacuum lead to n$-$type conversion? Write down a detailed mechanism using Kroger$-$Vink notations. How can we make a practical oxide p$-$n junction devices from this? What are the advantages?

$(2)$ Splitting of water by using oxygen vacancies in TiO$2$ is possible, write down the reactions leading to generation of hydrogen. How can we make a practical device using these concepts?

$(3)$ In donor doping of chromium oxide $($Cr$2$O$3)$ by TiO$2,$ discuss how does n$-$type conductivity vary with oxygen partial pressure? Derive the dependence of n$-$type conductivity with oxygen partial pressure and describe any assumptions made for the derivation.

$(4)$ In Ni$-$doped MgO, write down equations for generation of defects and formation of nickel precipitates during reduction treatment in hydrogen ambient at high temperatures. How does this reduction treatment compare with current induced effects?

$(5)$ The diamond unit cell consists of four tetrahedra with central atoms and four adjacent tetrahedra with missing central atoms. Can we fill these missing atoms and create a new material with another crystal structure? Calculate the number density of atoms in this new unit cell.