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 termed 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 corresponding 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 that 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 corresponding momentum. In normal-mode coordinates, the internal energy of the lattice vibrations takes the particularly simple form

$$U = \frac{1}{2}\sum_{i=1,3N}\left(p_i^{\,2} + \omega_i^{\,2}\,q_i^{\,2}\right),$$ (5.487)

where $\omega_i$ is the (angular) oscillation frequency of the $i$th normal mode. It is clear that, when expressed 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 (because 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 $\omega\simeq\! \sqrt{\kappa/m}$ ($\kappa$ is the force constant that 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. (See Section 5.5.8.)

If the lattice vibrations behave classically then, according to the equipartition theorem (see Section 5.5.5), each normal mode of oscillation has an associated mean energy $k_B\,T$, in equilibrium at temperature $T$ [ $(1/2)\,k_B\,T$ resides in the kinetic energy of the oscillation, and $(1/2)\,k_B\,T$ resides in the potential energy]. Thus, the internal energy of the solid is

$$U = 3 \,N\,k_B\,T = 3\,\nu\, R\,T,$$ (5.488)

where $N=\nu\,N_A$. It follows that the molar heat capacity at constant volume is

$$c_V = \frac{1}{\nu}\left(\frac{\partial U}{\partial T}\right)_V = 3\,R$$ (5.489)

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 that lie remarkably close to this value. This fact was discovered experimentally by Pierre Dulong and Alexis 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 then obtain an estimate of the molecular weight.)

Table 5.3 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 fairly 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 interatomic 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 ( $\omega\simeq\! \sqrt{\kappa/m}$) is high. In fact, the spacing between the different vibrational 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 Table 5.3.

Table: 5.3 Values of $c_p$ (joules/mole/degree) for some solids at $T= 298^\circ$ K.

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. We can make a crude model of the behavior of $c_V$ at low temperatures by assuming that all of 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 Equation (5.487), the solid acts like a set of $3\,N$ independent oscillators which, making use of Einstein's approximation, all vibrate at the same frequency. We can use the quantum mechanical result (5.390) for the mean energy of a single oscillator to write the internal energy of the solid in the form

$$U = 3\,N \left[\frac{1}{2} + \frac{1}{\exp(\hbar\,\omega/k_B\,T) - 1} \right]\hbar\,\omega,$$ (5.490)

giving

$$c_V \frac{1}{\nu}\left(\frac{\partial U}{\partial T}\right)_V= 3\... ...c{\theta_E}{T}\right)^2 \frac{\exp(\theta_E / T)} {[\exp(\theta_E/T) - 1]^{2}}.$$ (5.491)

Here,

$$\theta_E = \frac{\hbar \,\omega}{k_B}$$ (5.492)

is termed the Einstein temperature. If the temperature is sufficiently high that $T\gg \theta_E$ then the previous 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 appearing in Equation (5.491) become very much larger than unity, giving

$$c_V \simeq 3 \,R \left(\frac{\theta_E}{T}\right)^2\,\exp\left(-\frac{\theta_E}{ T}\right).$$ (5.493)

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 behavior is more like $c_V \propto T^{3}$. 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 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 that really matter at low temperatures are the long-wavelength modes; more explicitly, 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. In other words, they are not sensitive to the fact that the solid is made up of atoms, rather than being continuous.

According to Equation (5.461), the density of sound wave states in a continuous solid is

$$\rho_c(\omega) = \frac{3\,V}{2\pi^2}\,\frac{\omega^2}{v_{s}^{\,3}},$$ (5.494)

where $v_s$ is the average sound speed. The Debye approach consists in approximating the actual density of sound wave states, $\rho(\omega)$, by the density in a continuous medium, $\rho_c(\omega)$, not only at low frequencies (long wavelengths) where these should be nearly the same, but also at high 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

$$\rho_D(\omega) = \left\{ \begin{array}{lll}\rho_c(\omega) &\m... ...eq \omega_D\\ [0.5ex] 0& & \omega>\omega_D \end{array}\right. .$$ (5.495)

Here, $\omega_D$ is termed the Debye frequency, and is chosen such that the total number of normal modes is $3\,N$:

$$\int_0^\infty \rho_D(\omega)\, d\omega = \int_0^{\omega_D }\rho_c(\omega)\, d\omega = 3\,N.$$ (5.496)

Substituting Equation (5.494) into the previous formula yields

$$\frac{3\, V}{2\pi^{2} \,v_s^{\,3}}\int_0^{\omega_D} \omega^{2}\, d\omega = \frac{V} {2\pi^{\,2}\, v_s^{\,3}}\,\omega_D^{\,3} = 3\,N.$$ (5.497)

This implies that

$$\omega_D = v_s\left(6 \pi^{2} \,\frac{N}{V}\right)^{1/3}.$$ (5.498)

Thus, the Debye frequency depends only on the sound speed in the solid, and the number of atoms per unit volume. The wavelength corresponding to the Debye frequency is $2\pi\,v_s/\omega_D$, which is clearly on the order of the interatomic spacing, $d\simeq (V/N)^{1/3}$. 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.

We can use the quantum-mechanical expression for the mean energy of a single oscillator, Equation (5.390), to calculate the internal energy associated with lattice vibrations in the Debye approximation. We obtain

$$U = \int_0^\infty \rho_D(\omega) \left[\frac{1}{2} + \frac{1}{\exp(\hbar\, \omega/k_B\,T)-1}\right]\hbar\,\omega\,d\omega.$$ (5.499)

Hence, the molar heat capacity takes the form

$$c_V =\frac{1}{\nu}\left(\frac{\partial U}{\partial T}\right)_V = ... ...,\omega} {[\exp(\hbar\,\omega/k_B\,T)-1]^{\,2}}\right\} \hbar\,\omega\,d\omega.$$ (5.500)

Making use of Equations (5.494) and (5.495), we find that

$$c_V = \frac{k_B}{\nu}\int_0^{\omega_D} \frac{\exp(\hbar\,\omega/k... ...\omega/k_B\,T)-1]^{2}}\, \frac{3 \,V}{2\pi^2 \,v_s^{\,3}}\,\omega^{2}\,d\omega,$$ (5.501)

giving

$$c_V = \frac{3\,V\, k_B}{2\pi^{\,2} \,\nu\,(v_s \,\hbar/k_B\,T)^{3}}\int_0^{\hbar\,\omega_D/k_B\,T} \frac{x^4\,{\rm e}^x}{({\rm e}^x - 1)^{2}}\,dx,$$ (5.502)

in terms of the dimensionless variable $x=\hbar\,\omega/k_B\,T$. According to Equation (5.498), the volume can be written

$$V = 6\pi^{2}\, N \left(\frac{v_s}{\omega_D}\right)^3,$$ (5.503)

so the heat capacity reduces to

$$c_V = 3\,R\,f_D\left(\frac{\hbar\,\omega_D}{k_B\,T}\right)= 3\,R\,f_D\left(\frac{\theta_D}{T}\right),$$ (5.504)

where the Debye function is defined

$$f_D(y) \equiv \frac{3}{y^{3}}\int_0^y \frac{x^4\,{\rm e}^x} {({\rm e}^x -1)^{2}}\,dx.$$ (5.505)

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

$$k_B\,\theta_D = \hbar \,\omega_D.$$ (5.506)

Consider the asymptotic limit in which $T\gg \theta_D$. For small $y$, we can approximate ${\rm e}^x$ as $1+x$ in the integrand of Equation (5.505), so that

$$f_D(y) \rightarrow \frac{3}{y^{3}} \int_0^y x^{2}\,dx = 1.$$ (5.507)

Thus, if the temperature greatly exceeds the Debye temperature then 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 $y$,

$$\int_0^y \frac{x^4\,{\rm e}^x}{({\rm e}^x -1)^{2}} \, dx \simeq \... ...^\infty \frac{x^4\,{\rm e}^x}{({\rm e}^x -1)^{2}}\, dx = \frac{4\pi^{\,4}}{15}.$$ (5.508)

The latter integral can be looked up in standard reference books on integrals. Thus, in the low-temperature limit,

$$f_D(y) \rightarrow \frac{4\pi^{4}}{5} \frac{1}{y^{3}},$$ (5.509)

which yields

$$c_V \simeq \frac{12 \pi^{4}}{5} \,R \left(\frac{T}{\theta_D}\right)^{3}$$ (5.510)

in the limit $T\ll \theta_D$, Note that $c_V$ varies with temperature as $T^{3}$, in accordance with experimental observation.

Table 5.4: Comparison of Debye temperatures (in degrees kelvin) obtained from the low temperature behavior of the heat capacity with those calculated from the sound speed.

Solid	$\theta_D$ (low temperature)	$\theta_D$ (sound speed)
Na Cl	308	320
K Cl	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 behavior. Theoretically, $\theta_D$ should be calculable from Equation (5.498) in terms of the sound speed in the solid, and the molar volume. Table 5.4 shows a comparison of Debye temperatures evaluated by this means with temperatures obtained empirically by fitting the law (5.510) 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, though not perfect, representation of the behavior of $c_V$ in solids over the entire temperature range.

Figure: 5.9 The molar heat capacity of various solids. The solid curve shows the prediction of Debye theory. The dotted curve shows the prediction of Einstein theory (assuming that $\theta _E = \theta _D$).

Finally, Figure 5.9 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 shown, for the sake of comparison.