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 $x$?

  • $\begingroup$ Please edit this and typeset your equations. The GIF images are less amenable to searching. $\endgroup$ Mar 3, 2022 at 16:10
  • $\begingroup$ I don't know how to do it. $\endgroup$
    – Jun Seo-He
    Mar 3, 2022 at 16:14
  • 1
    $\begingroup$ Come on, you've been a member on this site for 8 months now. Read the docs. You give, you get. $\endgroup$ Mar 3, 2022 at 16:19
  • $\begingroup$ 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. $\endgroup$
    – andselisk
    Mar 3, 2022 at 16:35
  • 1
    $\begingroup$ @JunSeo-He You have just given two examples for $V(x)$. So what is exactly your question? $\endgroup$
    – Greg
    Mar 4, 2022 at 3:25


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