# Mechanics Question - Atoms Modelled As A Chain of Masses Connected By A Spring

For a chain of masses lying on a horizontal frictionless surface, with each mass connected to its neighbour mass by a spring of force constant s, the equation of motion for the nth mass is:

m(x_n)'' = -s[2(x_n) - x(n-1) - x(n=1)]
Where: x_n is the displacement of the nth mass from its equilibrium position

This model can be used to represent the 1-D propogation of waves in a crystal of lattice spacing a between the atoms. The potential between two atoms, distance r apart is:
U(r) = e{(a/r)^12 - 2(a/r)^6], where a is the equilibrium spacing between the atoms.

Show s = 72e/a^2 for small oscillations about the equilibrium spacing.
NB: e = epsilon.

## The Attempt at a Solution

I've tried tons of things with this and have gone round in circles tbh, getting all sorts of answers. I think I'm missing a piece of understanding of the problem.

I tried to use the general equation for the system of n masses but applied to this case. So x_(n-1) = -a, x_(n+1) = a, x_n = (+/-)e:

Consider LHS of equlibrium: F_L = -s{-e - x_(n-1)} = -s{-e + a}
Consider RHS of equilibrium: F_R = -s{e - x_(n+1} = -s{e - a}

So to get the total force we sum: F = -s{-e + a + e - a} = 0 ??? No good.

For some reason I'm thinking I need a 2e in that bracket but I can't see how I get that.

From this: F = -du/dr = -e{12a^6r^-7 - 12a^12r^(-13)} = 0 but the S shouldn't dissapear.

Even with F = -2es I don't get their answer anyway, it's a mess.

Help appreciated. :)

Why aren't any of you helping me? I thought people were supposed to help? I have an exam tomorrow so I need help with this. I see other people getting help so why aren't I.

Physics Monkey