Restoring potential of atoms in a solid

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 $$x$$?

• Please edit this and typeset your equations. The GIF images are less amenable to searching. Mar 3, 2022 at 16:10
• I don't know how to do it. Mar 3, 2022 at 16:14
• Come on, you've been a member on this site for 8 months now. Read the docs. You give, you get. Mar 3, 2022 at 16:19
• Zero kelvin is not OK, it's $\pu{0 K}$. Please visit this page, this page and this one on how to format your future posts better with MathJax and Markdown. Mar 3, 2022 at 16:35
• @JunSeo-He You have just given two examples for $V(x)$. So what is exactly your question?
– Greg
Mar 4, 2022 at 3:25