If some of the $U^{4+}$ ions turn into only $U^{6+}$ ions, the fraction of $U^{6+}$ ion in hyperstoichiometic uranium

dioxide, i$.$e$.,U{O}_{2+x}$ must satisfy the charge neutrality.

a$)$ Write down the equation for the charge neutrality of the total positive and negative charges in

$U_{2+x},$ if the fraction of $U^{6+}$ ions is given as $f_6.$ Based on this, find out the relation between $f_6$ and the

additional oxygen composition $x$ in $U{O}_{2+x}.$ Assume that no point defect other than oxygen interstitials

and $U^{6+}$ ions forms inside the material.

b$)$ Describe all possible point defects in $U{O}_{2+x}$ using Kroger$-$Vink notation at $500C.$

c$)$ Write down a balanced defect equation in Kroger$-$Vink notation for $U{O}_{2+x},$ if oxygen gas gets

absorbed into pristine $U{O}_2.$