Specific heats of solids

Consider a simple solid containing $N$ atoms. Now, atoms in solids cannot translate (unlike those in gases), but are free to vibrate about their equilibrium positions. Such vibrations are called lattice vibrations, and can be thought of as sound waves propagating through the crystal lattice. Each atom is specified by three independent position coordinates, and three conjugate momentum coordinates. Let us only consider small amplitude vibrations. In this case, we can expand the potential energy of interaction between the atoms to give an expression which is quadratic in the atomic displacements from their equilibrium positions. It is always possible to perform a normal mode analysis of the oscillations. In effect, we can find $3\,N$ independent modes of oscillation of the solid. Each mode has its own particular oscillation frequency, and its own particular pattern of atomic displacements. Any general oscillation can be written as a linear combination of these normal modes. Let $q_i$ be the (appropriately normalized) amplitude of the $i$th normal mode, and $p_i$ the momentum conjugate to this coordinate. In normal mode coordinates, the total energy of the lattice vibrations takes the particularly simple form

$$E = \frac{1}{2}\sum_{i=1}^{3 N} (p_i^{~2} + \omega_i^{~2} q_i^{~2}),$$ (499)

where $\omega_i$ is the (angular) oscillation frequency of the $i$th normal mode. It is clear that in normal mode coordinates, the linearized lattice vibrations are equivalent to $3\,N$ independent harmonic oscillators (of course, each oscillator corresponds to a different normal mode).

The typical value of $\omega_i$ is the (angular) frequency of a sound wave propagating through the lattice. Sound wave frequencies are far lower than the typical vibration frequencies of gaseous molecules. In the latter case, the mass involved in the vibration is simply that of the molecule, whereas in the former case the mass involved is that of very many atoms (since lattice vibrations are non-localized). The strength of interatomic bonds in gaseous molecules is similar to those in solids, so we can use the estimate $\omega\sim \sqrt{\kappa/m}$ ($\kappa$ is the force constant which measures the strength of interatomic bonds, and $m$ is the mass involved in the oscillation) as proof that the typical frequencies of lattice vibrations are very much less than the vibration frequencies of simple molecules. It follows from ${\mit\Delta} E = \hbar\, \omega$ that the quantum energy levels of lattice vibrations are far more closely spaced than the vibrational energy levels of gaseous molecules. Thus, it is likely (and is, indeed, the case) that lattice vibrations are not frozen out at room temperature, but, instead, make their full classical contribution to the molar specific heat of the solid.

If the lattice vibrations behave classically then, according to the equipartition theorem, each normal mode of oscillation has an associated mean energy $k\,T$ in equilibrium at temperature $T$ [$(1/2)\,k\,T$ resides in the kinetic energy of the oscillation, and $(1/2)\,k\,T$ resides in the potential energy]. Thus, the mean internal energy per mole of the solid is

$$\overline{E} = 3 \,N\,k\,T = 3\,\nu\, R\,T.$$ (500)

It follows that the molar heat capacity at constant volume is

$$c_V = \frac{1}{\nu}\left(\frac{\partial\overline{E}}{\partial T}\right)_V = 3\,R$$ (501)

for solids. This gives a value of $24.9$ joules/mole/degree. In fact, at room temperature most solids (in particular, metals) have heat capacities which lie remarkably close to this value. This fact was discovered experimentally by Dulong and Petite at the beginning of the nineteenth century, and was used to make some of the first crude estimates of the molecular weights of solids (if we know the molar heat capacity of a substance then we can easily work out how much of it corresponds to one mole, and by weighing this amount, and then dividing the result by Avogadro's number, we can obtain an estimate of the molecular weight). Table 4 lists the experimental molar heat capacities $c_p$ at constant pressure for various solids. The heat capacity at constant volume is somewhat less than the constant pressure value, but not by much, because solids are fairly incompressible. It can be seen that Dulong and Petite's law (i.e., that all solids have a molar heat capacities close to $24.9$ joules/mole/degree) holds pretty well for metals. However, the law fails badly for diamond. This is not surprising. As is well-known, diamond is an extremely hard substance, so its intermolecular bonds must be very strong, suggesting that the force constant $\kappa$ is large. Diamond is also a fairly low density substance, so the mass $m$ involved in lattice vibrations is comparatively small. Both these facts suggest that the typical lattice vibration frequency of diamond ( $\omega\sim \sqrt{\kappa/m}$) is high. In fact, the spacing between the different vibration energy levels (which scales like $\hbar \,\omega$) is sufficiently large in diamond for the vibrational degrees of freedom to be largely frozen out at room temperature. This accounts for the anomalously low heat capacity of diamond in Tab. 4.

Table 4: Values of $c_p$ (joules/mole/degree) for some solids at $T= 298^\circ$ K. From Reif.

Solid	$c_p$	Solid	$c_p$
Copper	24.5	Aluminium	24.4
Silver	25.5	Tin (white)	26.4
Lead	26.4	Sulphur (rhombic)	22.4
Zinc	25.4	Carbon (diamond)	6.1

Dulong and Petite's law is essentially a high temperature limit. The molar heat capacity cannot remain a constant as the temperature approaches absolute zero, since, by Eq. (488), this would imply $S\rightarrow \infty$, which violates the third law of thermodynamics. We can make a crude model of the behaviour of $c_V$ at low temperatures by assuming that all the normal modes oscillate at the same frequency, $\omega$, say. This approximation was first employed by Einstein in a paper published in 1907. According to Eq. (499), the solid acts like a set of $3N$ independent oscillators which, making use of Einstein's approximation, all vibrate at the same frequency. We can use the quantum mechanical result (479) for a single oscillator to write the mean energy of the solid in the form

$$\overline{E} = 3\,N \,\hbar\,\omega\left(\frac{1}{2} + \frac{1}{\exp(\beta\,\hbar\,\omega) - 1} \right).$$ (502)

The molar heat capacity is defined

$$c_V = \frac{1}{\nu}\left(\frac{\partial \overline{E}}{\parti... ...} \left( \frac{\partial\overline{E}}{\partial \beta}\right)_V,$$ (503)

giving

$$c_V = - \frac{3 \,N_A \,\hbar \,\omega}{k\, T^2} \left[ -\fr... ...\hbar\, \omega} {[\exp(\beta\,\hbar\, \omega) - 1]^2} \right],$$ (504)

which reduces to

$$c_V = 3\,R \left(\frac{\theta_E}{T}\right)^2 \frac{\exp(\theta_E / T)} {[\exp(\theta_E/T) - 1]^2}.$$ (505)

Here,

$$\theta_E = \frac{\hbar \,\omega}{k}$$ (506)

is called the Einstein temperature. If the temperature is sufficiently high that $T\gg \theta_E$ then $k\,T \gg \hbar\, \omega$, and the above expression reduces to $c_V = 3\,R$, after expansion of the exponential functions. Thus, the law of Dulong and Petite is recovered for temperatures significantly in excess of the Einstein temperature. On the other hand, if the temperature is sufficiently low that $T\ll \theta_E$ then the exponential factors in Eq. (505) become very much larger than unity, giving

$$c_V \sim 3 \,R \left(\frac{\theta_E}{T}\right)^2\,\exp(-\theta_E / T).$$ (507)

So, in this simple model the specific heat approaches zero exponentially as $T\rightarrow 0$.

In reality, the specific heats of solids do not approach zero quite as quickly as suggested by Einstein's model when $T\rightarrow 0$. The experimentally observed low temperature behaviour is more like $c_V \propto T^3$ (see Fig. 6). The reason for this discrepancy is the crude approximation that all normal modes have the same frequency. In fact, long wavelength modes have lower frequencies than short wavelength modes, so the former are much harder to freeze out than the latter (because the spacing between quantum energy levels, $\hbar \,\omega$, is smaller in the former case). The molar heat capacity does not decrease with temperature as rapidly as suggested by Einstein's model because these long wavelength modes are able to make a significant contribution to the heat capacity even at very low temperatures. A more realistic model of lattice vibrations was developed by the Dutch physicist Peter Debye in 1912. In the Debye model, the frequencies of the normal modes of vibration are estimated by treating the solid as an isotropic continuous medium. This approach is reasonable because the only modes which really matter at low temperatures are the long wavelength modes: i.e., those whose wavelengths greatly exceed the interatomic spacing. It is plausible that these modes are not particularly sensitive to the discrete nature of the solid: i.e., the fact that it is made up of atoms rather than being continuous.

Consider a sound wave propagating through an isotropic continuous medium. The disturbance varies with position vector ${\bf r}$ and time $t$ like $\exp[-{\rm i}\,({\bf k}\!\cdot{\bf r} - \omega \,t)]$, where the wave-vector ${\bf k}$ and the frequency of oscillation $\omega$ satisfy the dispersion relation for sound waves in an isotropic medium:

$$\omega = k \,c_s.$$ (508)

Here, $c_s$ is the speed of sound in the medium. Suppose, for the sake of argument, that the medium is periodic in the $x$-, $y$-, and $z$-directions with periodicity lengths $L_x$, $L_y$, and $L_z$, respectively. In order to maintain periodicity we need

$$k_x\, (x+ L_x) = k_x\, x + 2\,\pi\, n_x,$$ (509)

where $n_x$ is an integer. There are analogous constraints on $k_y$ and $k_z$. It follows that in a periodic medium the components of the wave-vector are quantized, and can only take the values

$$k_x = \frac{2\pi}{L_x} \,n_x,$$ (510)

$$k_y = \frac{2\pi}{L_y} \,n_y,$$ (511)

$$k_z = \frac{2\pi}{L_z} \,n_z,$$ (512)

where $n_x$, $n_y$, and $n_z$ are all integers. It is assumed that $L_x$, $L_y$, and $L_z$ are macroscopic lengths, so the allowed values of the components of the wave-vector are very closely spaced. For given values of $k_y$ and $k_z$, the number of allowed values of $k_x$ which lie in the range $k_x$ to $k_x + d k_x$ is given by

$${\mit\Delta} n_x = \frac{L_x}{2\pi} \,dk_x.$$ (513)

It follows that the number of allowed values of ${\bf k}$ (i.e., the number of allowed modes) when $k_x$ lies in the range $k_x$ to $k_x + d k_x$, $k_y$ lies in the range $k_y$ to $k_y + d k_y$, and $k_z$ lies in the range $k_z$ to $k_z + d k_z$, is

$$\rho\, d^3{\bf k} = \left(\frac{L_x}{2\pi}\,dk_x\right) \le... ...{2\pi}\,dk_z\right)= \frac{V}{(2\pi)^3}\, dk_x\, dk_y \,dk_z,$$ (514)

where $V = L_x L_y L_z$ is the periodicity volume, and $d^3{\bf k}\equiv dk_x\, dk_y \,dk_z$. The quantity $\rho$ is called the density of modes. Note that this density is independent of ${\bf k}$, and proportional to the periodicity volume. Thus, the density of modes per unit volume is a constant independent of the magnitude or shape of the periodicity volume. The density of modes per unit volume when the magnitude of ${\bf k}$ lies in the range $k$ to $k+dk$ is given by multiplying the density of modes per unit volume by the ``volume'' in ${\bf k}$-space of the spherical shell lying between radii $k$ and $k+dk$. Thus,

$$\rho_k \,dk = \frac{4\pi k^2\,dk}{(2\pi)^3} = \frac{k^2}{2 \pi^2} \,dk.$$ (515)

Consider an isotropic continuous medium of volume $V$. According to the above relation, the number of normal modes whose frequencies lie between $\omega$ and $\omega+d \omega$ (which is equivalent to the number of modes whose $k$ values lie in the range $\omega/c_s$ to $\omega/c_s + d\omega/c_s$) is

$$\sigma_c(\omega)\,d\omega = 3\, \frac{k^2\,V}{2\pi^2} \, dk = 3 \,\frac{V} {2\pi^2 \,c_s^{~3}} \,\omega^2\,d\omega.$$ (516)

The factor of $3$ comes from the three possible polarizations of sound waves in solids. For every allowed wavenumber (or frequency) there are two independent torsional modes, where the displacement is perpendicular to the direction of propagation, and one longitudinal mode, where the displacement is parallel to the direction of propagation. Torsion waves are vaguely analogous to electromagnetic waves (these also have two independent polarizations). The longitudinal mode is very similar to the compressional sound wave in gases. Of course, torsion waves can not propagate in gases because gases have no resistance to deformation without change of volume.

The Debye approach consists in approximating the actual density of normal modes $\sigma(\omega)$ by the density in a continuous medium $\sigma_c(\omega)$, not only at low frequencies (long wavelengths) where these should be nearly the same, but also at higher frequencies where they may differ substantially. Suppose that we are dealing with a solid consisting of $N$ atoms. We know that there are only $3\,N$ independent normal modes. It follows that we must cut off the density of states above some critical frequency, $\omega_D$ say, otherwise we will have too many modes. Thus, in the Debye approximation the density of normal modes takes the form

$$\sigma_D(\omega)= \sigma_c(\omega) {\rm for~} \omega<\omega_D$$

$$\sigma_D(\omega)= 0 {\rm for~} \omega>\omega_D.$$ (517)

Here, $\omega_D$ is the Debye frequency. This critical frequency is chosen such that the total number of normal modes is $3\,N$, so

$$\int_0^\infty \sigma_D(\omega)\, d\omega = \int_0^{\omega_D }\sigma_c(\omega)\, d\omega = 3\, N.$$ (518)

Substituting Eq. (516) into the previous formula yields

$$\frac{3 V}{2\pi^2 \,c_s^{~3}}\int_0^{\omega_D} \omega^2\, d\omega = \frac{V} {2\pi^2\, c_s^{~3}}\,\omega_D^{~3} = 3\,N.$$ (519)

This implies that

$$\omega_D = c_s\left(6 \pi^2 \,\frac{N}{V}\right)^{1/3}.$$ (520)

Thus, the Debye frequency depends only on the sound velocity in the solid and the number of atoms per unit volume. The wavelength corresponding to the Debye frequency is $2\pi\,c_s/\omega_D$, which is clearly on the order of the interatomic spacing $a\sim (V/N)^{1/3}$. It follows that the cut-off of normal modes whose frequencies exceed the Debye frequency is equivalent to a cut-off of normal modes whose wavelengths are less than the interatomic spacing. Of course, it makes physical sense that such modes should be absent.

Figure 5: The true density of normal modes in diamond compared with the density of normal modes predicted by Debye theory. From C.B. Walker, Phys. Rev. 103, 547 (1956).

Figure 5 compares the actual density of normal modes in diamond with the density predicted by Debye theory. Not surprisingly, there is not a particularly strong resemblance between these two curves, since Debye theory is highly idealized. Nevertheless, both curves exhibit sharp cut-offs at high frequencies, and coincide at low frequencies. Furthermore, the areas under both curves are the same. As we shall see, this is sufficient to allow Debye theory to correctly account for the temperature variation of the specific heat of solids at low temperatures.

We can use the quantum mechanical expression for the mean energy of a single oscillator, Eq. (479), to calculate the mean energy of lattice vibrations in the Debye approximation. We obtain

$$\overline{E} = \int_0^\infty \sigma_D(\omega) \,\hbar\,\omeg... ...{2} + \frac{1}{\exp(\beta\,\hbar\, \omega)-1}\right)\,d\omega.$$ (521)

According to Eq. (503), the molar heat capacity takes the form

$$c_V = \frac{1}{\nu\, k\, T^2} \int_0^\infty \sigma_D (\omega... ...r\,\omega} {[\exp(\beta\,\hbar\,\omega)-1]^2}\right]\,d\omega.$$ (522)

Substituting in Eq. (517), we find that

$$c_V = \frac{k}{\nu}\int_0^{\omega_D} \frac{\exp(\beta\,\hbar... ...a)-1]^2}\, \frac{3 \,V}{2\pi^2 \,c_s^{~3}}\,\omega^2\,d\omega,$$ (523)

giving

$$c_V = \frac{3\,V\, k}{2\pi^2 \,\nu\,(c_s \,\beta \,\hbar)^3}... ...eta\,\hbar\,\omega_D} \frac{\exp x}{(\exp x - 1)^2}\,x^4\,dx,$$ (524)

in terms of the dimensionless variable $x=\beta\,\hbar\,\omega$. According to Eq. (520), the volume can be written

$$V = 6\,\pi^2\, N \left(\frac{c_s}{\omega_D}\right)^3,$$ (525)

so the heat capacity reduces to

$$c_V = 3R\,f_D(\beta \,\hbar\,\omega_D)= 3\,R\,f_D(\theta_D/T),$$ (526)

where the Debye function is defined

$$f_D(y) \equiv \frac{3}{y^3}\int_0^y \frac{\exp x} {(\exp x -1)^2}\,x^4\,dx.$$ (527)

We have also defined the Debye temperature $\theta_D$ as

$$k\,\theta_D = \hbar \,\omega_D.$$ (528)

Consider the asymptotic limit in which $T\gg \theta_D$. For small $y$, we can approximate $\exp x$ as $1+x$ in the integrand of Eq. (527), so that

$$f_D(y) \rightarrow \frac{3}{y^3} \int_0^y x^2\,dx = 1.$$ (529)

Thus, if the temperature greatly exceeds the Debye temperature we recover the law of Dulong and Petite that $c_V = 3\,R$. Consider, now, the asymptotic limit in which $T\ll \theta_D$. For large $y$,

$$\int_0^y \frac{\exp x}{(\exp x -1)^2}\,x^4 \, dx \simeq \int... ...y \frac{\exp x}{(\exp x -1)^2}\,x^4 \, dx = \frac{4\pi^4}{15}.$$ (530)

The latter integration is standard (if rather obscure), and can be looked up in any (large) reference book on integration. Thus, in the low temperature limit

$$f_D(y) \rightarrow \frac{4\pi^4}{5} \frac{1}{y^3}.$$ (531)

This yields

$$c_V \simeq \frac{12 \pi^4}{5} R \left(\frac{T}{\theta_D}\right)^3$$ (532)

in the limit $T\ll \theta_D$: i.e., $c_V$ varies with temperature like $T^3$.

Table 5: Comparison of Debye temperatures (in degrees kelvin) obtained from the low temperature behaviour of the heat capacity with those calculated from the sound speed. From C. Kittel, Introduction to solid-state physics, 2nd Ed. (John Wiley & Sons, New York NY, 1956).

Solid	$\theta_D$ from low temp.	$\theta_D$ from sound speed
NaCl	308	320
KCl	230	246
Ag	225	216
Zn	308	305

The fact that $c_V$ goes like $T^3$ at low temperatures is quite well verified experimentally, although it is sometimes necessary to go to temperatures as low as $0.02 \,\theta_D$ to obtain this asymptotic behaviour. Theoretically, $\theta_D$ should be calculable from Eq. (520) in terms of the sound speed in the solid and the molar volume. Table 5 shows a comparison of Debye temperatures evaluated by this means with temperatures obtained empirically by fitting the law (532) to the low temperature variation of the heat capacity. It can be seen that there is fairly good agreement between the theoretical and empirical Debye temperatures. This suggests that the Debye theory affords a good, thought not perfect, representation of the behaviour of $c_V$ in solids over the entire temperature range.

Figure 6: The molar heat capacity of various solids.

Finally, Fig. 6 shows the actual temperature variation of the molar heat capacities of various solids as well as that predicted by Debye's theory. The prediction of Einstein's theory is also show for the sake of comparison. Note that 24.9 joules/mole/degree is about 6 calories/gram-atom/degree (the latter are chemist's units).