Lattice and electronic specific heats of a solid

In a solid the contribution to the heat capacity arises both from the vibrations of the lattice and from the free electrons. Thus, the total heat capacity is a sum of the two contributions, as discussed below. 

1. Lattice Specific Heat

Dulong and Petit  measured the specific heat of 13 solid elements near room temperature and found out that the product of specific heat per unit mass at constant pressure and atomic weight of the element is nearly constant and about $\rm6cal.(g-at)^{-1}deg^{-1}$. Thus, based on these observations they formulated the ‘Dulong-Petit law’ which states that the heat capacity per atom is about the same for different elements. This law was generalized in the form of ‘Kopp-Newmann’ law’ according to which the heat capacity per gram molecular weight of a chemical compound is equal to the sum of the heat capacities  per gram atomic weights of the constituent atoms. Thus the diatomic and triatomic solids were expected to have specific heats of about $\rm 12 cal. (g-at)^{-1}.deg^{-1}$ and $\rm 18 cal. (g-at)^{-1} deg^{-1}$, respectively.

   A theoretical explanation of the Dulong and Petit’s law was first given in 1871 by Boltzmann on the basis of his law of equipartition of energy. This law states that for a system in thermal equilibrium, each degree of freedom contributes $\rm 1/2 k_bT$ to the average energy of the system. If an atom is considered to be a three dimensional harmonic oscillator then its average kinetic energy is $\rm 3/2 k_bT$  According to Virial theorem, a particle moving in a parabolic potential well has an average total energy twice its kinetic energy, in this case $\rm 3k_bT$  Hence, the specific heat of an atom at constant volume is $\rm3k_b$ and per mole it is $\rm3Nk_b$  (N is the Avogadro number) or 3R ( $\rm\approx 6 cal. (g-at)^{-1}deg^{-1}$). For a compound containing n atoms per molecule, the above argument can be extended to yield $\rm  C_v=3nR$. Thus, for diatomic and triatomic solids with n equal to 2 and 3 respectively, the molar specific heat has the value 12 and 18 $\rm cal.(mol)^{-1}deg^{-1}$, respectively. Nevertheless, exceptions to the Dulong-Petit rule do exist like in the case of silicon, boron and diamond which have specific heats of about 20, 11 and 8 $\rm J. Mol^{-1}.K^{-1}$, respectively, near 300 K. Later measurements by Weber on diamond over a temperature range of 200 to 1300 K established that specific heat of all the substances approaches the Dulong-Petit value at sufficiently high temperatures. Experiments at low temperature led Nernst  and others to suggest that specific heat of solids must tend towards zero as absolute zero of temperature is approached. It was thus realised that the classical theory which predicts a constant specific heat down to low temperatures was not sufficient to describe the behaviour of a solid. This must be explained by the quantum theory. Einstein assumed that a crystal containing N atoms can be treated as a combination of 3N one dimensional oscillators. He assumed that the atoms vibrate independently of each other and with the same frequency because of the assumed identical surroundings. Einstein’s theory gave the specific heat at constant volume as,

\begin{equation}C_v=3R(\theta_E/T)^2exp(\theta_E/T)[exp(\theta_E/T)-1]^2\end{equation}

where $\rm\theta_E=\hbar\omega/k_B$ is the characteristic temperature known as the Einstein temperature, $\rm\omega$  is the frequency of the oscillating atom, $\rm k_B$ is the Boltzmann’s constant and $\rm\hbar$ is the reduced Planck’s constant.  At high temperature equation 1 reduces to the Dulong-Petit value. Further, at low temperatures the expression for specific heat reduces to  

\begin{equation}C_v=3R\left(\frac{\hbar\omega}{k_B T}\right)exp\left(\frac{-\hbar\omega}{k_b T}\right)\end{equation}

   Therefore, as $\rm T\rightarrow 0$, $\rm C_v\rightarrow 0$ because of the exponential term. This theory was able to explain the decrease in specific heat with temperature, but the work of Nernst and others showed that while the low temperature behaviour predicted by Einstein model was qualitatively correct, the specific heat of real solids did not decrease as rapidly with decreasing temperature as predicted by Einstein. The major drawback in this theory was that Einstein had assumed each atom to be an independent harmonic oscillator, oscillating with a frequency $\rm\omega$. But, actually, the atoms oscillate relative to their neighbours in the lattice. For wavelengths which are long relative to lattice spacings, the motion of atoms is hardly independent and large regions of crystal move together coherently. The long wavelength motions have low frequencies and these are particularly important at low temperatures.

   Debye considered this situation and realised that it was possible to propagate waves through solids covering a wavelength region extending from low frequencies (Sound waves) upto short waves (infrared absorption). The essential difference between the Einstein and Debye model is that Debye considered the vibrational motion of the crystal as a whole. He assumed that the continuum model could be employed for all possible vibrational modes of the crystal. For wavelengths which are long as compared to the interatomic spacings the crystals can be considered as a elastic continuum from the point of view of the wave. The fact that the crystal consists of atoms is taken into account in Debye theory by limiting the total number of vibrational modes to 3N. Thus, the frequency spectrum corresponding to perfect continuum is cut off so as to comply with the condition of the total number of modes being 3N. The cut off procedure leads to a maximum frequency $\rm\omega_D$   common to the transverse and longitudinal waves. Debye obtained the specific heat per gram-atom as

\begin{equation}C_v=9R(T/\theta_D)^3\int_0^{x_{max}}\frac{e^{x} x^4}{(e^x-1)^2}dx\end{equation}

where $\rm\theta_D=hbar/k_B$ is known as the Debye temperature and $\rm x_{max}=\theta_d/T$. At high temperatures, $\rm x=\hbar\omega/k_B T$ is small and so the integral reduces to         $\int x^2dx$ yielding the Dulong and Petit value of 3R for the specific heat. At low temperatures,  x is large and so the upper limit $\rm x_{max}$ can be considered to be infinity. The integral in Eq. 3 with $\infty$ as the upper limit has the value $\rm 4\pi^4/15$. Accordingly, the following expression is obtained for the low temperature specific heat

\begin{equation} C_v=\frac{12}{5}\pi^4 R\left(\frac{T}{\theta_D}\right)^3\end{equation}

The cubic expression should hold up to a temperature of $\rm\approx\theta_D/10$. As mentioned,  in the  Debye treatment the range of frequencies is limited to some maximum or cut-off frequency $\rm\nu_D$ and corresponding to this frequency is defined the Debye temperature $\rm\theta_D=h\nu_D/k_B$. According to the Debye theory the Debye temperature should have a constant value. However, in order to accurately fit the Debye model to the experimental specific heat data of a solid, the $\rm\theta_D$ value has to be varied with temperature which implies that one or more of the approximations made in the derivation of the Debye theory may not be valid.  In general the $\rm\theta_D(T)$ functions for different solids exhibit a number of similar features. At temperatures below $\rm\theta_D$, the Debye temperature becomes essentially constant and is approximately equal to its limiting value at absolute zero. At very high temperatures, all the vibrational modes are excited and thus, $\rm\theta_D$ is expected to be constant.  In many solids, this is approximately true at temperatures above $\rm\theta_D/2$. At intermediate temperatures, however, $\rm\theta_D$ varies somewhat, often passing through a minimum.

2. Electronic Specific Heat     

   

The conduction electrons in normal metals are fermions and they obey Fermi-Dirac statistics. These statistics play a fundamental role determining the temperature dependence of the electronic contribution to the specific heat.  As the temperature is decreased, only those electrons which have energy very close to the Fermi energy $\rm E_F$ are able to change their state. Therefore, there is only a very slight increase in the electron energy distribution, implying thereby that electronic specific heat very small. Thus, the contribution of electrons to the total specific heat of a metal is negligible at room temperatures when the lattice contribution is quite high. The electronic specific heat may be expressed to a first approximation as 

\begin{equation}C_v=\frac{\pi^2}{3}n(E_f)kb^2T=\gamma T\end{equation}

In the above equation $\rm n(E_F)$ is the electron density of states (EDOS) at $\rm E_F$ and is called the Sommerfeld constant. A linear variation of electronic specific heat is observed in all normal metals. At room temperature electronic specific heat is only about 1 percent of the lattice specific heat. But, at low temperatures it becomes important because then lattice specific heat has a $\rm T^3$variation. Thus, the lattice specific heat decreases much faster than the linearly varying electronic specific heat.