# Finding nearest neighbour equilibrium distance

## 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.

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 ?

## Answers and Replies

Homework Helper
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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.

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 ?

Homework Helper
Gold Member
2020 Award
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.

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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