The Condensed State

The thermal properties of solids reflect the quantum mechanical excitation of their crystal lattice vibrations and conduction electrons.

1. The High-Temperature Classical Limit

At high temperatures ($T \gg \Theta_D$), the thermal energy $k_B T$ is far larger than the quantum energy of any lattice vibration ($\hbar \omega_D$). All $3N$ normal modes of vibration are fully excited. According to the classical equipartition theorem, each harmonic mode contributes $k_B T$ of internal energy: $$U \to 3 N k_B T \implies C_V \to 3 N k_B = 3 R \approx 24.94\text{ J/(mol}\cdot\text{K)}$$ This is the classical Dulong-Petit value, which is identical for all non-magnetic monoatomic solids regardless of chemical identity.

2. The Low-Temperature Quantum Freezing

As temperature drops below the Debye temperature ($T \ll \Theta_D$), thermal energy becomes insufficient to excite high-frequency vibrational modes:

• In **Insulators:** Lattice acoustic phonons dominate completely, yielding the universal cubic temperature dependence: $$C_V^{\text{insulator}}(T) = A T^3 = \frac{12\pi^4}{5} N k_B \left( \frac{T}{\Theta_D} \right)^3$$
• In **Metals:** Both conduction electrons and acoustic phonons contribute: $$C_V^{\text{metal}}(T) = \gamma T + A T^3$$ Plotting $C_V / T$ versus $T^2$ yields a straight line whose $y$-intercept gives the electronic Sommerfeld constant $\gamma$ and whose slope gives the lattice coefficient $A$!

Peter Debye synthesized the low- and high-temperature regimes into a single, universal dimensionless function that describes virtually all insulating crystals.

1. The Debye Function ($D(y)$)

The internal energy of a crystal is expressed in terms of the **Debye Function**: $$U(T) = 3 N k_B T \, D\left( \frac{\Theta_D}{T} \right)$$ where $y = \Theta_D / T$, and $D(y)$ is defined as: $$D(y) \equiv \frac{3}{y^3} \int_0^y \frac{x^3 dx}{e^x - 1}$$

2. The Universal Specific Heat Expression

Differentiating $U(T)$ gives the molar heat capacity: $$C_V(T) = 3 R \left[ 4 D(y) - \frac{3 y}{e^y - 1} \right] = 9 R \left( \frac{T}{\Theta_D} \right)^3 \int_0^{\Theta_D / T} \frac{x^4 e^x dx}{(e^x - 1)^2}$$ Notice that:

The reduced heat capacity $\frac{C_V}{3R}$ is a universal function of the single dimensionless ratio $\frac{T}{\Theta_D}$.

When experimental specific heat data for diverse solids (lead with $\Theta_D = 105\text{ K}$, copper with $\Theta_D = 343\text{ K}$, and diamond with $\Theta_D = 2230\text{ K}$) are plotted against $T / \Theta_D$, all data points collapse onto the exact same theoretical Debye master curve!

Why does a solid expand when heated? A perfectly harmonic crystal lattice ($V(x) = \frac{1}{2}c x^2$) **does not expand at all**! Thermal expansion is a direct consequence of **anharmonicity** in the interatomic potential.

1. Proof that Harmonic Oscillators Have Zero Thermal Expansion

Consider an atom displaced by $x$ from its equilibrium position. In a symmetric parabolic potential $V(x) = c x^2$: $$\langle x \rangle = \frac{\int_{-\infty}^\infty x e^{-\beta c x^2} dx}{\int_{-\infty}^\infty e^{-\beta c x^2} dx} = 0$$ Because the potential is strictly symmetric, the thermal average displacement $\langle x \rangle$ vanishes identically at all temperatures. A purely harmonic crystal never expands!

2. Anharmonicity and Thermal Expansion

The real interatomic potential (such as the Lennard-Jones potential) is asymmetric, resisting compression more strongly than expansion. Expanding about the minimum: $$V(x) = c x^2 - g x^3 - f x^4, \quad (g, f > 0)$$ Treating the cubic anharmonic term $-g x^3$ as a perturbation: $$e^{-\beta V(x)} \approx e^{-\beta c x^2} (1 + \beta g x^3)$$ $$\langle x \rangle = \frac{\int_{-\infty}^\infty x e^{-\beta c x^2} (1 + \beta g x^3) dx}{\int_{-\infty}^\infty e^{-\beta c x^2} dx} = \frac{\beta g \int_{-\infty}^\infty x^4 e^{-\beta c x^2} dx}{\int_{-\infty}^\infty e^{-\beta c x^2} dx} = \frac{3 g}{4 c^2} k_B T$$ The average interatomic spacing increases linearly with temperature: $\langle x \rangle \propto T$.

3. The Grüneisen Parameter and Equation of State

As a solid expands ($V$ increases), phonon vibrational frequencies shift downward: $$\gamma_G \equiv -\frac{d\ln \omega_D}{d\ln V} = -\frac{V}{\omega_D}\frac{d\omega_D}{dV}$$ where $\gamma_G$ is the dimensionless **Grüneisen Parameter** (typically $\sim 1$ to $2$). The volumetric thermal expansion coefficient $\beta_V = \frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_P$ satisfies the **Mie-Grüneisen Equation**: $$\beta_V = \frac{\gamma_G C_V}{V K_T}$$ where $K_T$ is the isothermal bulk modulus. Thermal expansion is directly proportional to lattice heat capacity: as $T \to 0$, $\beta_V \propto T^3$, vanishing at absolute zero in agreement with the Third Law of Thermodynamics.

Liquid Helium-4 ($^4\text{He}$) is the prototypical strongly interacting **quantum Bose liquid**.

1. The Elementary Excitation Spectrum

Lev Landau (1941) proposed that elementary excitations in Liquid $^4\text{He}$ are single quasiparticles whose energy $\epsilon(p)$ depends continuously on momentum $p$:

1. Phonon Branch ($p/\hbar < 0.6\text{ Å}^{-1}$): Longitudinal sound waves with linear dispersion: $$\epsilon(p) = c_s p$$ where $c_s \approx 238\text{ m/s}$ is the speed of first sound.
2. Maxon Peak ($p/\hbar \approx 1.1\text{ Å}^{-1}$): A local maximum around $\epsilon / k_B \approx 14\text{ K}$.
3. Roton Minimum ($p/\hbar \approx 1.92\text{ Å}^{-1}$): A parabolic dip around characteristic momentum $p_0$: $$\epsilon(p) \approx \Delta + \frac{(p - p_0)^2}{2\mu}$$ with energy gap $\Delta / k_B \approx 8.65\text{ K}$, $p_0 / \hbar \approx 1.92\text{ Å}^{-1}$, and effective mass $\mu \approx 0.16 m_4$.

2. Bogoliubov's Microscopic Theory

Nikolay Bogoliubov (1947) derived this excitation spectrum microscopically for a weakly interacting Bose gas using canonical transformation: $$E(k) = \sqrt{\epsilon_k^2 + 2 n U_0 \epsilon_k}$$ where $\epsilon_k = \frac{\hbar^2 k^2}{2m}$ is the free-particle kinetic energy, and $U_0 = \frac{4\pi \hbar^2 a}{m}$ is the repulsive contact interaction.

• At low wavevector ($k \to 0$): $E(k) \approx \sqrt{2 n U_0 \frac{\hbar^2 k^2}{2m}} = \hbar c_s k$, reproducing linear sound waves with sound speed $c_s = \sqrt{n U_0 / m}$.
• At high wavevector ($k \to \infty$): $E(k) \to \epsilon_k + n U_0$, recovering quadratic single-particle behavior.

Liquid Helium-3 ($^3\text{He}$, a spin-1/2 fermion) and conduction electrons in heavy fermion metals represent strongly interacting **quantum Fermi liquids**.

1. Landau Fermi Liquid Theory

Lev Landau (1956) recognized that strong repulsive interactions do not destroy the sharpness of the Fermi surface. Instead, there is a continuous one-to-one correspondence between the eigenstates of a non-interacting Fermi gas and the low-energy excitations of the interacting system, called **quasiparticles**.

2. Quasiparticles and Effective Mass ($m^*$)

A quasiparticle is an individual fermion dressed by a surrounding cloud of interactions with other particles. Near the Fermi surface, the quasiparticle energy is: $$\epsilon(p) \approx \epsilon_F + v_F^* (p - p_F) = \epsilon_F + \frac{p_F}{m^*} (p - p_F)$$ where $m^*$ is the **quasiparticle effective mass**: $$\frac{m^*}{m} = 1 + \frac{1}{3} F_1^s$$ In Liquid $^3\text{He}$ at ambient pressure, $m^* \approx 3.0 m_3$; under high pressure ($30\text{ bar}$), $m^* \approx 6.0 m_3$.

3. Quasiparticle Lifetime Divergence

By Pauli exclusion, scattering between quasiparticles near the Fermi surface requires both initial and final states to lie within energy $k_B T$ of $\epsilon_F$. The scattering rate scales as $\Gamma \propto (k_B T)^2 + (\epsilon - \epsilon_F)^2$. Therefore, the quasiparticle lifetime diverges at low temperatures: $$\tau \propto \frac{1}{T^2} \to \infty$$ As $T \to 0$, quasiparticles become infinitely well-defined quantum excitations, explaining why free-electron models work remarkably well in metals!

In a crystalline metal, conduction electrons move in the periodic electrostatic potential created by the ionic lattice: $V(\mathbf{r} + \mathbf{R}) = V(\mathbf{r})$.

1. Bloch's Theorem and Energy Bands

According to Bloch's Theorem, single-electron wavefunctions are plane waves modulated by the lattice periodicity: $$\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}} u_{n\mathbf{k}}(\mathbf{r}), \quad u_{n\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})$$ The energy spectrum separates into continuous **energy bands** $\epsilon_n(\mathbf{k})$ separated by **bandgaps**.

2. The Electronic Hallmark of a Metal

A material is an electrical conductor (metal) if and only if:

The highest occupied energy level—the Fermi Energy $\epsilon_F$—lies inside a partially filled electronic energy band.

Because empty quantum states are available immediately above $\epsilon_F$ with infinitesimal energy spacing $d\epsilon \to 0$:

• An applied electric field $\mathbf{E}$ accelerates electrons into adjacent unoccupied states, producing a net electrical current (Drude-Sommerfeld conductivity $\sigma = \frac{n e^2 \tau}{m^*}$).
• The density of states at the Fermi level $g(\epsilon_F) > 0$ is finite, yielding linear electronic heat capacity $C_V = \gamma T$ and Pauli paramagnetism $\chi = \mu_B^2 g(\epsilon_F)$.

In solid dielectrics (insulators and semiconductors), the Fermi level lies within a forbidden energy gap separating filled and empty electronic bands.

1. Band Structure of Insulators and Semiconductors

In an intrinsic dielectric at $T = 0\text{ K}$:

• The **Valence Band** is $100\%$ completely filled with electrons.
• The **Conduction Band** is completely empty.
• They are separated by a finite **Bandgap** $E_g$: $$E_g = \epsilon_c - \epsilon_v$$ For semiconductors, $E_g \lesssim 3\text{ eV}$ (Silicon: $1.12\text{ eV}$, Gallium Arsenide: $1.42\text{ eV}$). For insulators, $E_g \gtrsim 4\text{ eV}$ (Diamond: $5.47\text{ eV}$, Silicon Dioxide: $9.0\text{ eV}$).

2. Thermal Carrier Generation and Chemical Potential

At finite temperature $T > 0$, thermal fluctuations promote electrons across the bandgap into the conduction band, leaving empty states (**holes**) in the valence band. The electron density $n$ and hole density $p$ are: $$n = N_c e^{-(\epsilon_c - \mu)/k_B T}, \quad p = N_v e^{-(\mu - \epsilon_v)/k_B T}$$ where $N_c = 2\left(\frac{m_e^* k_B T}{2\pi \hbar^2}\right)^{3/2}$ and $N_v = 2\left(\frac{m_h^* k_B T}{2\pi \hbar^2}\right)^{3/2}$ are the effective densities of states. Equating $n = p = n_i$ (intrinsic semiconductor): $$n_i = \sqrt{N_c N_v} \exp\left( -\frac{E_g}{2 k_B T} \right)$$ $$\mu = \frac{\epsilon_c + \epsilon_v}{2} + \frac{3}{4} k_B T \ln\left( \frac{m_h^*}{m_e^*} \right)$$ The Fermi level in an intrinsic dielectric sits essentially at the mid-gap! Electrical conductivity grows exponentially with temperature: $\sigma(T) \propto e^{-E_g / (2 k_B T)}$, in stark contrast to metals where resistance increases with temperature due to phonon scattering.