In Section 2.6, it was noted that the net bonding energy EN between two isolated positive...

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 τ as follows:

where A, B, and n are constants for the particular ion pair. Equation 6.31 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 τ, realizing that

2. Now take the derivative dF/dr.

3. Develop an expression for τ0, the equilibrium separation. Because τ0 corresponds to the value of τ at the minimum of the EN-versus-τ curve (Figure 2.10b), take the derivative dEN/dr, set it equal to zero, and solve for r, which corresponds to τ0.

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

Figure 2.10 (b) The dependence of repulsive, attractive, and net potential energies on interatomic separation for two isolated atoms.

 (6.31)