Finding nearest neighbour equilibrium distance

[Cocoleia]

Homework Statement

The energy per ion in for CsCl is nearly – (αe 2 /(4πε0)) + 8Ae -(R/ρ) , where α is the Madelung constant and A = 5.64 x 103 eV and ρ = 0.34 Å. Calculate the nearest neighbour equilibrium distance.

Homework Equations

alpha = 2 ln 2

The Attempt at a Solution

I think that CsCl is a simple cubic structure
I found online

For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a. Therefore, for a simple cubic lattice there are six (6) nearest neighbors for any given lattice point.

so then my answer would be 0.34 A ? Is this correct ?

 

[Charles Link]

The first line of the problem statement appears incomplete. I don't see any definition of $R$ and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.

 

[Cocoleia]

The first line of the problem statement appears incomplete. I don't see any definition of $R$ and also what do you take a derivative of to set it equal to zero? It looks like you may have a typo or two in your equation.

That's the entire problem copied and pasted from the assignment.

I'm not sure what you mean by what do I take the derivative of ?

 

[Charles Link]

That's the entire problem copied and pasted from the assignment.

I'm not sure what you mean by what do I take the derivative of ?

The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that $\frac{dV}{dR}=0$. $\\$ Consider for example a mass on a spring in a gravitational field.: $U=\frac{1}{2}kx^2+mgx$ . Taking derivative and setting equal to zero: $kx+mg=0$ ==>> $x_{equilibrium}=-\frac{mg}{k}$, which the spring constant equation also tells you the forces are balanced there.

 

[Cocoleia]

The equilibrium distance for a system is normally found as the position where the potential energy is a minimum, so that $\frac{dV}{dR}=0$. $\\$ Consider for example a mass on a spring in a gravitational field.: $U=\frac{1}{2}kx^2+mgx$ . Taking derivative and setting equal to zero: $kx+mg=0$ ==>> $x_{equilibrium}=-\frac{mg}{k}$, which the spring constant equation also tells you the forces are balanced there.

Ok so it is not enough to say
For a simple cubic lattice, it is clear that the nearest neighbor distance is just the lattice parameter, a.

I would have to derive the energy equation that is given to me and set =0