Past exam of the condensed matter physics course of the University of Oxford / 2013 / 2 / b

For a repulsive short-range potential and a stable finite minimum, assume $A>0$ and integer $n>1$. Setting $u^{\prime }(R)=C/R^2-2nA/R^{n+1}=0$ givesAt this point $2A/R_0^n=C/(n{R}_0)$, soHere $u^{\prime \prime }$ denotes the second derivative of the energy $u$, not a displacement variable. The negative equilibrium energy is the cohesive energy per pair relative to infinite separation.

The qualifiers on $A,n$ matter: an arbitrary real coefficient and integer exponent do not automatically describe stabilizing repulsion. For instance $A<0,n>1$ gives collapse as $R\to 0$, whereas $n=1$ gives $u=(2A-C)/R$ and has no isolated finite equilibrium. The displayed minimum therefore applies to the physically repulsive, faster-than-Coulomb branch of the stated model.