Given the lattice energies $(U_{\text{l}\text{a}\text{t}\text{t}\text{i}\text{c}\text{e}})$ and lattice parameters $(a)$ for the alkali halides with the

NaCl structure listed in the table below, calculate the Madelung energy $($U $\{$:$U_{\text{M}\text{a}\text{d}\text{e}\text{l}\text{u}\text{n}\text{g}}),$ assuming

each ion has formal charge $($i$.$e$.$ the charges given by the valence, so that the cation charge is

$1$ unit of positive electron charge, and the anion charge is $-1$ unit of electron charge$),$ and

condition for equilibrium, calculate the value of $$ for each case, and hence the coefficients $B$

in the Born$-$Mayer interaction. For each crystal, calculate the value of the bulk modulus $(),$

and compare with experimental values. $($Hint: $(1)$ The bulk modulus is given as $=$

$\frac{1}{{?}_9{N}_{A}R}(\frac{U_{\text{M}\text{a}\text{d}\text{e}\text{l}\text{u}\text{n}\text{g}}}{R^2}+\frac{U_{\text{R}\text{e}\text{p}\text{u}\text{l}\text{s}\text{i}\text{v}\text{e}}}{}),$ and $=2$ for the NaCl structures. $(2)$ You can use spreadsheet

to speed up the calculations.$)$