In Section 2.6 it was noted that the net bonding energy EN between two isolated positive and negative ions is a function of interionic distance r as follows:

where A, B, and n are constants for the particular ion pair. Equation 6.25 is also valid for the bonding energy between adjacent ions in solid materials. The modulus of elasticity E is proportional to the slope of the interionic force–separation curve at the equilibrium interionic separation; that is,

Derive an expression for the dependence of the modulus of elasticity on these A, B, and n parameters (for the two-ion system) using the following procedure:

1. Establish a relationship for the force F as a function of r, realizing that is

2. Now take the derivative dF/dr.

3. Develop an expression for r0, the equilibrium separation. Since r0 corresponds to the value of r at the minimum of the EN-versus-r curve (Figure), take the derivative dEN/dr, set it equal to zero, and solve for r, which corresponds to r0.

4. Finally, substitute this expression for r0 into the relationship obtained by takingdF/dr.

EN %3D (6.25) dF Ecc dr