*Please wait for the animation to completely load.*

The measure of the change in temperature with the change in internal energy (heating
an object up, for example) is called the specific heat (at a constant volume), *c*_{V}, and is given by

*c*_{V} = *dE*/*dT *.
(15.9)

To find the specific heat of a solid, then, we simply need to find an expression of the total energy as a function of temperature. Restart. One way is to imagine a solid as composed of atoms in a lattice, each individually sitting in its own three dimensional simple harmonic well in which it vibrates with quantized energies of vibration. In this picture, we can use Maxwell-Boltzmann statistics (and the corresponding distributions) and find the following expression for the specific heat:

*c*_{V }= 3*R* (*ħ*ω/*k*_{B}*T*)^{2}
exp(*ħ*ω/*k*_{B}*T*)/(exp(*ħ*ω/*k*_{B}*T*) − 1)^{2}
, (15.15)

where we pick the value of *ħ*ω to match the
experimental data. Note that for high temperatures (*k*_{B}*T *>> *ħ*ω), the value of the specific heat is consistent with the equipartition of
energy theorem: *c*_{V} = (*f*/2)*R* = 3*R* (since *f* = 6, why?).

At low temperatures, however, the previous model fails. In
this case, the coupling between the oscillators cannot be ignored. The
vibration frequency of one atom does have an impact on its neighbor. So,
instead of treating the oscillators individually, we must treat them as a group
of points on an elastic spring that oscillates up and down.
These oscillations of the group as a whole create elastic standing waves in the
solid called phonons. Phonon statistics are then treated as photons in blackbody radiation
except that there is a limit to the phonon energy (the vibration wavelength and
thus, frequency, is limited by the spacing between atoms). This gives a specific heat as
follows:^{5}

c_{V} = − 9*R*
(Θ/*T*) [1/(exp(Θ/*T*) − 1)] + 36*R* (*T*/Θ)^{3} ∫_{0}^{Θ/T} *x*^{3}/(exp(*x*)
− 1) *dx* ,
(15.16)

where Θ is the Debye temperature and can be measured
independently. At low temperatures (*T *<< Θ), the integral can be evaluated
analytically giving

*c*_{V} = (12π^{4}*R*/5) (*T*/Θ)^{3 }.
(15.17)

Compare a plot of this expression with the classically derived
version for silver from 0 to 30K. The **red plot
**is the Debye model at low
temperatures. At high temperatures, both expressions give the same result.
So, in general we use the following

*c*_{V }= 3*R* (*A*/*T*)^{2} exp(*A*/*T*)/(exp(*A*/*T*)
− 1)^{2}
and *c*_{V} = (12π^{4}*R*/5) (*T*/Θ)^{3}
, (15.18)

for high temperatures (derived classically) and for low (*T* < 10-20% of Θ) temperatures (derived
quantum mechanically), respectively. What determines a high and low temperature of a material?
It depends on Θ, the Debye temperature. In order to compare the two
functions, for materials of interest here, *A* = 0.65 Θ. Looking at the data in the temperature ranges given, what is
considered *low temperature* for iron? Does it have a lower or
higher Debye temperature than silver?

___________________

^{5} See See R. Eisberg and R. Resnick, *Quantum Physics*, Wiley (1974), pp. 421-425 for a detailed analysis of both
the classical and proper quantum-mechanical derivation of the specific heat of
solids.

Section by Anne J. Cox and William F. Junkin III