Example 9.6: Estimate of Einstein temperature Consider a solid consisting of a cubic lattice with lattice constant a (shortest distance between neighbouring lattice points), and assume a spring-like interaction between neighbouring atoms at lattice points. Show that the compressibility κ of the solid and the spring constant κ0 are related such that κκ0 =

a. Taking one atom per lattice cube, express the lattice constant a in terms of atomic weight mA, the density ρ of the material and Avogadro’s number NA = 6.0 × 1023.

Then estimate roughly the Einstein temperature θE of Eq. (9.24) for such a solid made of copper

(atomic weight mA = 63.5, κ = 4.5 × 10−7bar−1 (1 bar = 106 g /cm s2 or 105 newton /m2) and density ρ = 8.9g cm−3).

[Hint: κ is defined as κ = [ρ(dP/dρ)]−1, where P is the pressure, and κ0 by the equation m¨x+κ0x =

0 to be compared with m¨x +4π2mν2x = 0]. (Answer: θE = (/k)

(NA/κmA)(mA/ρNA)1/3).‡‡