10. Appendix

618 Appendix B
approximation we can expand the term º(v) as a function of v about the equilibrium
volume v0 and keep only terms up to the quadratic term:
º(v) ∼ º(v0) (dº/dv)v0 (v v0) (1/2)(d 2 º/dv 2 2
)V0 (v v0)
(3.17.9)
By definition, the linear term (dº/dv)V0 vanishes at the equilibrium volume so
we can take the derivative of º(v) with respect to v and obtain:
(dº/dv) ∼ (d 2 º/dv 2 )v0 (v v0) . (3.17.10)
On the other hand, if we have started from (3.17.8) and take the derivative
with respect to v at equilibrium volume then we would obtain:
(dP/dv)V0 ∼ (d2 º/dv 2 )v0 (3.17.11)
by noting that U is proportional to v. (3.17.11) can be simplified by introducing
the isothermal compressibility Î0 (1/v0)(dv/dP)T. The bulk modulus B
defined on Page 139 is related to Î by: B 1/Î. With this simplification we
arrived at:
(dP/dv)T 1/Î0v0 . (3.17.12)
Combining (3.17.10) – (3.17.12) we obtain:
(dº/dv) (dP/dv)V0 (v v0) (v v0)/Î0v0
(3.17.13)
On substituting (3.17.13) back into (3.17.8) we obtain the equation-of-state
within the quasi-harmonic approximation:
P (v v0)/Î0v0 〈Á〉U/v . (3.17.14)
This equation was first derived by Mie and later by Grüneisen.
To obtain the coefficient of thermal expansion ‚ (1/v)(v/T)P we take
the partial derivative of (3.17.14) with respect to T while keeping P constant:
0 ‚v
〈Á〉
Î0v0
‚U
U
‚
v v T
1
U
. (3.17.15)
v T v
The quantities inside the braces can be expressed in terms of measurable
quantities, such as the heat capacity. For example, the constant volume heat
capacity Cv is equal to (U/T)V. By differentiating the equation (3.17.7) for
U with respect to v, we can show that:
(U/v)T (〈Á〉CvT/v) (〈Á〉U/v) . (3.17.16)
Actually this calculation is very similar to the one used to derive (3.17.5) by
taking the derivative of Fvib in (3.17.4) with respect to v. Substituting back
(3.17.16) into (3.17.15) we obtain:
‚V
Î0v0
〈Á〉‚U
CVT U
‚〈Á〉2
v v v
〈Á〉Cv
(3.17.17)
v
Since we have assumed a weak anharmonic term we expect that terms
quadratic in 〈Á〉 can be neglected. This means that at P 0 (when v v0)