The partition function of a Debye crystal is $$ \ln Z=-\frac{9}{(\Theta / T)^{3}} \int_{0}^{\theta / T} x^{2} \ln \left(1-e^{-x}\right) d x $$ (a) Show that $$ \ln Z=-3 \ln \left(1-e^{-\theta / T}\right)+\frac{9}{(\Theta / T)^{3}} \int_{0}^{\Theta / T} \frac{x^{3} d x}{e^{x}-1} $$ (b) Calculate the Helmholtz function. (c) Show that the equation of state of the crystal is given by $$ P V+f(V)=\Gamma\left(U-U_{0}\right) $$ where \(U_{0}\) is the zero-point energy.

The partition function is a central concept in statistical mechanics. It helps us understand how particles within a given system distribute themselves across various energy states at a particular temperature. For a Debye crystal, the partition function encapsulates information about the vibrational modes of the crystal lattice. It's given by:
\( \ln Z = -\frac{9}{(\Theta / T)^{3}} \int_{0}^{\theta / T} x^{2} \ln \left(1-e^{-x}\right) dx \).
Here, \ \Theta \ is the Debye temperature and \ T \ is the temperature.
This expression allows us to quantify the contribution of phonons (quantized lattice vibrations) to the thermodynamic properties of the crystal.
Phonons obey Bose-Einstein statistics, and their distribution is given by the term \ \left(1 - e^{-x}\right) \. By integrating this expression, we can find various thermodynamic properties such as the Helmholtz free energy and internal energy.

The Debye temperature \( \Theta \) is a characteristic temperature that provides information about the crystal's vibrational properties. It represents the temperature above which all the phonon modes are excited. Below the Debye temperature, only a fraction of these modes will have enough energy to be excited.
It is defined as:
\[ \Theta = \frac{h u_D}{k_B} \]
where \ h \ is the Planck constant, \ u_D \ is the Debye frequency, and \ k_B \ is the Boltzmann constant. The Debye temperature is crucial as it dictates the behavior of the heat capacity at low temperatures. When \ T << \Theta \, the heat capacity follows a \ T^3 \ dependence (Debye's T^3 law). In this context, the Debye temperature helps us relate the vibrational properties of the lattice to the macroscopic thermodynamic properties of the crystal.

The Helmholtz free energy \( F \) is one of the fundamental thermodynamic potentials. It is useful for understanding the work obtainable from a system at constant temperature and volume. For a Debye crystal, the Helmholtz free energy can be derived from the partition function:
\[ F = -k_B T \ln Z \]
By substituting the expression of the logarithm of the partition function, we get:
\[ F = k_B T \left[ 3 \ln \left(1-e^{-\Theta / T}\right) - \frac{9}{(\Theta / T)^{3}} \int_{0}^{\Theta / T} \frac{x^{3} dx}{e^{x}-1} \right] \]
This encapsulates the contribution of vibrational modes (phonons) to the free energy of the crystal. The Helmholtz free energy allows us to determine the equilibrium properties and stability of a system under constant temperature and volume conditions.

The equation of state describes the relationship between state variables such as pressure (P), volume (V), and internal energy (U) of a thermodynamic system. For a Debye crystal, the equation of state provides crucial insights into its thermodynamic behavior. It is given by:
\[ PV + f(V) = \Gamma(U - U_0) \]
Here, \ U_0 \ is the zero-point energy of the crystal, and \ \Gamma \ is a constant. To derive this equation, we use the Helmholtz free energy (F) and relate it to internal energy (U) and pressure (P). The equation highlights how the internal energy changes with volume at constant temperature. By manipulating the thermodynamic identities and derivatives of the Helmholtz free energy, we obtain this equation state.
Understanding the equation of state helps in determining how external conditions like pressure and volume changes affect the internal energy and other properties of the crystal.