Above $\pu{0 K}$ the atoms in a solid are vibrating. However, what kind of potential restores each atom in the starting position?
Consider the crystal lattice of NaCl. The potential energy of a $\ce{Na+}–\ce{Cl-}$ pair is
$$E(r) = -\frac{1}{4\pi \varepsilon_0}\frac{e^2M}{r^2} + \frac{B}{r^m},\tag{1}$$
where $M$ is the Madelung constant, $B$ and $m$ are the material-depending constants.
When a $\ce{Na+}$ ion and a $\ce{Cl-}$ ion get very close, the second term of the equation is increased very much and it becomes the restoring potential in the vibration of the ions.
In a crystal lattice made of $\ce{Si}$ atoms above $\pu{0 K}$ each atom oscillates at the Debye frequency from its starting position as part of a quantum harmonic oscillator:
$$\frac{-h^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x) + V(x)\Psi(x) = hf\left(\frac 1 2 + N\right)\Psi(x),\tag{2}$$
and $x=0$ will be the position of equilibrium of the atom.
What kind of expression is $V(x)?$ To which power do we raise the variable $x$?
