Bonding in Solid State Physics

[RedMech]

1. The problem statement

The potential energy between two ions is
u(r) = -α/r² + β/r⁸

Determine:

(i) The intermolecular distance r_o for which the potential energy is minimum
(ii) The inter-atomic distance for which the potential energy is zero is

R= (2)^-1/6r_o

Homework Equations

du/dr=o at equilibrium distance, r_o

The Attempt at a Solution

i) du/dr= 2α/r³ - 8β/r⁹ = 0 at equilibrium distance r_o

we end up with

r_o = (4β/α)^1/6



ii) -α/r² + β/r⁸ = 0

which give us

R^-6 = α/β

We can express the ratio α/β interms of r_o so that

R^-6 = 4r_o^-6

and finally

R = (4)^-1/6r_o

My question now is why does my answer have a 4 instead of a 2 in it?

 

[ehild]

Your result is correct, and it can be written also as R=2^-1/3 r₀.

ehild