Refer again to the result of Problem 29. (a) Use the fact that the average net force on...

Refer again to the result of Problem 29. (a) Use the fact that the average net force on an ion must be zero (if there is no overall translation of the crystal) to show that the average values of r and (δr)² are related by . (b) According to the equipartition theorem the mean potential energy of an oscillator is Use this to show that and thereby show that the coefficient of thermal expansion is approximately . (c) Use the result of (b) to evaluate the coefficient of thermal expansion for NaCl and compare with the experimental value of 4 × 10^-6 K_-1.

Problem 29

Starting with the result of Problem 28, approximate r r₀ + δ r (where δ r << r₀) and show that

Perform a Taylor series expansion about r = r₀, keeping terms up to and including order (δ r)². Show that

Where

and