b$)$ In a linear chain of neon $(Ne,$ each atom interacts with its nearest neighbors on either side through the Lenard$-$Jones potential $U=4lon|(\frac{}{r}{)}^{12}-(\frac{}{r}{)}^6|.$ The atomic mass of Ne is $20\mathrm{.}18$ amu, ind Taking the force constant in the hamonic limit to be C $=\frac{de{l}^{2}v}{del{r}^2}{|}_r||=a,$ find the following:

$($i$)$ The equilibrium spacing $($a$)$ between the atoms.

$($ii$)$ The sound speed;

$($iii$)$ The maximum frequency.