Harmonic crystalline solids

An example of a system that can be treated analytically is a crystalline solid at low temperature. We shall show that in this limit the potential energy acting on each particle of the crystal can be approximated by a sum of independent parabolic potentials, each with its own curvature:

$$U = U_0 + \sum_{i=1}^{3N} \frac{1}{2} M \omega_i^2 q_i^2,$$ (7.1)

with $M$ the mass of the particles, $q_i$ a set of normal coordinates, and $U_0$ the energy of the system in its ground state. This approximation is known as the harmonic approximation and the form of the potential in  is known as a quadratic form. The Newton's equations of motion for the normal coordinates are:

$$M\frac{d^2 q_i}{d t^2} = -\frac{\partial U}{\partial q_i} = -M \omega_i^2 q_i,$$ (7.2)

which have the general solution

$$q_i (t) = A \sin (\omega_i t + \phi),$$ (7.3)

where $\phi$ is an arbitrary phase. ^7.1

In the particular case of the potential of  one can easily obtain $Q$ (for a classical system), as the integral  factorises:

$$Q = \frac{1}{V^N}\int dq_1 \dots dq_{3N} e^{-\beta U} =$$

$$e^{-\beta U_0}\frac{1}{V^N}\int dq_1 e^{-\beta M \omega_1^2 q_1^2 / 2} \dots \int dq_{3N} e^{-\beta M \omega_{3N}^2 q_{3N}^2 / 2},$$ (7.4)

where the range of the integrals depends on the confining volume $V$. For a large enough volume, the range of integration over the variables $q_i$ can be extended from $-\infty$ to $+\infty$, because of the exponential decay of the integrand, which makes the integrals easy to compute using :

$$\int_{-\infty}^{+\infty} dq_i e^{-\beta M \omega_i^2 q_i^2 / 2} = \left (\frac{2\pi k_{\rm B}T}{M \omega_i^2}\right )^\frac{1}{2},$$ (7.5)

from which we obtain:

$$Q = e^{-\beta U_0}\frac{1}{V^N} \left (\frac{2\pi k_{\rm B}T}{M}\right )^\frac{3N}{2} \prod_{i=1}^{3N} \frac{1}{\omega_i}.$$ (7.6)

Combining this with  we obtain the total partition function:

$$Z = e^{-\beta U_0}\prod_{i=1}^{3N} \frac{k_{\rm B}T}{\hbar \omega_i},$$ (7.7)

where $\hbar = h/2\pi$ is the reduced Plank's constant and we have omitted the internal partition function $Z_{int}$ as we are dealing with structureless particles. Note that in  we have not included a $N!$ factor, as the oscillators are uniquely identified by the positions of their parabolic potentials in the space of the normal modes and by their frequencies $\omega_i$. The Helmholtz free energy is

$$F = -k_{\rm B}T\ln Z = U_0 + k_{\rm B}T \sum_{i=1}^{3N} \ln \frac{\hbar \omega_i}{k_{\rm B}T}.$$ (7.8)

These expressions for the partition function and the Helmholtz free energy are consequence of us taking the classical limit. If the oscillators cannot be described as classical particles, then the integrals in  have to be replaced by sums over the quantum energies, which are the eigenvalues of the time independent Schrödinger equation. For a system described by a sum of harmonic potentials they are given by $E^i_n = (n+1/2)\hbar \omega_i$. For a system of independent harmonic oscillator the partition function is still obtained by a product of single particle partition functions,

$$Z = e^{-\beta U_0} \prod_{i=1}^{3N} Z_i,$$ (7.9)

with each term in the product having the form:

$$Z_i = \sum_{n=0}^\infty e^{-E^i_n/k_{\rm B}T} = \sum_{n=0}^\infty... ...2} \frac{\hbar\omega_i}{k_{\rm B}T}}}{1-e^{-\frac{\hbar\omega_i}{k_{\rm B}T}}}.$$ (7.10)

The Helmholtz free energy is:

$$F = -k_{\rm B}T\ln Z = U_0 + \sum_{i=1}^{3N}\frac{\hbar \omega_i}{2} + k_{\rm B}T \sum_{i=1}^{3N} \ln (1-e^{-\frac{\hbar\omega_i}{k_{\rm B}T}}).$$ (7.11)

The classical limit  can be easily be obtained from a Taylor expansion with $\hbar \omega_i \ll k_{\rm B}T$ ^7.2. The first term on the r.h.s. in  does not have a classical analog. Note that this term does not depend on temperature and so it is present also at $T=0$. For this reason it is called zero point energy.