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Lennard$-$Jones potential$*$

A commonly used potential energy function to describe the in

teraction between two atoms is the Lennard$-$Jones $6-12$ potential

given by

U $=$ r

$0$

r

$12$

$2$ r

$0$

r

$6.$

$($a$)$ Find the position of the potential minimum and its value.

$($b$)$ Near the minimum the atoms execute simple harmonic mo

tion. Find the frequency of oscillation.