The free electron model simplifies how electrons behave in metals. It treats them as a gas of non-interacting particles moving in a uniform background. This approach helps explain many metal properties, like electrical conductivity and heat capacity.

Despite its usefulness, the model has limitations. It fails to fully explain some materials, like transition metals and semiconductors. This led to more advanced theories, such as band theory, which consider the complex interactions between electrons and the crystal lattice.

Free electron gas model

• The free electron gas model a simplified approach to understanding the behavior of electrons in metals and their contribution to various physical properties
• Treats electrons as a gas of non-interacting particles moving in a uniform positive background potential created by the metal ions
• Provides a foundation for understanding electronic properties of metals and serves as a starting point for more advanced models in solid state physics

Electrons in periodic potential

• Electrons in a metal experience a periodic potential due to the regularly arranged positive ions in the crystal lattice
• The periodic potential influences the motion and energy states of the electrons
• Bloch's theorem states that electron wavefunctions in a periodic potential can be expressed as the product of a plane wave and a periodic function with the same periodicity as the lattice

Assumptions of model

• Electrons are treated as independent particles and electron-electron interactions are neglected
• The positive ions in the metal are assumed to form a uniform background potential in which the electrons move
• The electron-ion interactions are represented by a weak periodic potential
• The model assumes a free electron dispersion relation $E = \frac{\hbar^2k^2}{2m}$, where $E$ is the electron energy, $\hbar$ is the reduced Planck's constant, $k$ is the electron wavevector, and $m$ is the electron mass

Electron density

• The electron density $n$ represents the number of electrons per unit volume in the metal
• For a free electron gas, the electron density is related to the Fermi wavevector $k_F$ by $n = \frac{k_F^3}{3\pi^2}$
• The Fermi wavevector $k_F$ is the maximum wavevector occupied by electrons at absolute zero temperature
• The electron density determines various electronic properties of the metal, such as the Fermi energy and electronic heat capacity

Fermi energy

• The Fermi energy $E_F$ is the highest occupied energy level in the electron gas at absolute zero temperature
• For a free electron gas, the Fermi energy is given by $E_F = \frac{\hbar^2k_F^2}{2m}$
• The Fermi energy depends on the electron density and increases with increasing electron concentration (e.g., in metals with higher valence electron count, like aluminum)
• The Fermi energy plays a crucial role in determining the electronic properties of metals, such as electrical conductivity and heat capacity

Density of states

• The density of states $g(E)$ represents the number of electron states per unit energy interval
• For a free electron gas in three dimensions, the density of states is given by $g(E) = \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$
• The density of states increases with increasing energy, following a square root dependence
• The density of states is essential for calculating various thermodynamic properties, such as the electronic heat capacity and thermal conductivity

Electron dynamics

• Electron dynamics in the free electron gas model describes the motion and response of electrons to external fields and perturbations
• Understanding electron dynamics is crucial for explaining the electrical and thermal properties of metals
• The free electron gas model provides a simplified framework to analyze electron behavior and derive important relationships

Equation of motion

• The motion of electrons in the free electron gas model is governed by the classical equation of motion $m\frac{d\vec{v}}{dt} = -e\vec{E}$, where $m$ is the electron mass, $\vec{v}$ is the electron velocity, $e$ is the electron charge, and $\vec{E}$ is the applied electric field
• In the presence of an electric field, electrons experience a force and accelerate in the opposite direction of the field
• The equation of motion allows the calculation of electron drift velocity and current density in response to an applied electric field

Electron effective mass

• The electron effective mass $m^$ is a concept introduced to account for the influence of the periodic potential on electron motion
• In the free electron gas model, the effective mass is equal to the free electron mass $m$
• However, in real metals, the periodic potential can modify the electron dispersion relation, leading to a different effective mass
• The effective mass can be derived from the curvature of the electron energy band structure and is given by $\frac{1}{m^} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}$

Electron mobility

• Electron mobility $\mu$ is a measure of how easily electrons can move through a metal under the influence of an electric field
• It is defined as the ratio of the electron drift velocity $v_d$ to the applied electric field $E$, given by $\mu = \frac{v_d}{E}$
• Higher electron mobility indicates that electrons can move more freely through the metal, resulting in higher electrical conductivity
• Electron mobility depends on factors such as electron scattering by lattice vibrations (phonons) and impurities

Conductivity of metals

• Electrical conductivity $\sigma$ is a measure of a metal's ability to conduct electric current
• In the free electron gas model, conductivity is given by $\sigma = \frac{ne^2\tau}{m}$, where $n$ is the electron density, $e$ is the electron charge, $\tau$ is the average time between electron collisions (relaxation time), and $m$ is the electron mass
• The conductivity depends on the electron density, electron mobility, and the average time between collisions
• Metals with higher electron density and longer relaxation times exhibit higher electrical conductivity

Matthiessen's rule

• Matthiessen's rule states that the total resistivity of a metal $\rho_{total}$ is the sum of the resistivity contributions from different scattering mechanisms, such as electron-phonon scattering $\rho_{phonon}$ and electron-impurity scattering $\rho_{impurity}$
• Mathematically, Matthiessen's rule is expressed as $\rho_{total} = \rho_{phonon} + \rho_{impurity}$
• Each scattering mechanism independently contributes to the total resistivity, and the dominant scattering mechanism determines the overall resistivity
• Matthiessen's rule is useful for understanding the temperature dependence of resistivity in metals and the effects of impurities on electrical conductivity

Heat capacity

• Heat capacity is a measure of the amount of heat required to raise the temperature of a substance by a certain amount
• In the free electron gas model, the heat capacity of metals consists of contributions from both the electrons and the lattice vibrations (phonons)
• The electronic heat capacity is a consequence of the quantum nature of electrons and differs from the classical prediction

Classical vs quantum behavior

• Classical physics predicts that the heat capacity of a metal should be independent of temperature, known as the Dulong-Petit law
• However, experiments show that the heat capacity of metals decreases at low temperatures, deviating from the classical prediction
• Quantum mechanics is necessary to explain the temperature dependence of the electronic heat capacity
• The quantum behavior arises from the Fermi-Dirac distribution, which describes the occupation of electron energy states at different temperatures

Electronic heat capacity

• The electronic heat capacity $C_e$ is the contribution to the total heat capacity from the conduction electrons in a metal
• In the free electron gas model, the electronic heat capacity is given by $C_e = \gamma T$, where $\gamma$ is the Sommerfeld coefficient and $T$ is the absolute temperature
• The Sommerfeld coefficient $\gamma$ is proportional to the density of states at the Fermi energy, $\gamma = \frac{\pi^2k_B^2}{3}g(E_F)$, where $k_B$ is the Boltzmann constant
• The electronic heat capacity exhibits a linear dependence on temperature, in contrast to the classical prediction

Low temperature approximation

• At low temperatures (typically below the Debye temperature), the electronic heat capacity dominates over the lattice contribution
• In this regime, the electronic heat capacity can be approximated by the linear term $C_e = \gamma T$
• The low temperature approximation is valid when $k_BT \ll E_F$, where $E_F$ is the Fermi energy
• Measuring the electronic heat capacity at low temperatures allows the determination of the Sommerfeld coefficient and the density of states at the Fermi energy

Linear temperature dependence

• The linear temperature dependence of the electronic heat capacity, $C_e = \gamma T$, is a key prediction of the free electron gas model
• This linear behavior arises from the Fermi-Dirac distribution and the density of states near the Fermi energy
• Experimental measurements of the heat capacity of metals at low temperatures confirm the linear temperature dependence
• Deviations from the linear behavior can indicate the presence of additional heat capacity contributions or limitations of the free electron gas model

Thermal conductivity

• Thermal conductivity $\kappa$ is a measure of a material's ability to conduct heat
• In metals, heat is primarily carried by conduction electrons, making the free electron gas model relevant for understanding thermal transport
• The free electron gas model provides a framework for relating thermal conductivity to electrical conductivity and for understanding the temperature dependence of thermal conductivity

Wiedemann-Franz law

• The Wiedemann-Franz law states that the ratio of the thermal conductivity $\kappa$ to the electrical conductivity $\sigma$ of a metal is proportional to the absolute temperature $T$
• Mathematically, the Wiedemann-Franz law is expressed as $\frac{\kappa}{\sigma} = LT$, where $L$ is the Lorenz number
• The law arises from the fact that both thermal and electrical conductivity in metals are dominated by conduction electrons
• The Wiedemann-Franz law is a consequence of the free electron gas model and holds well for most metals at room temperature

Lorenz number

• The Lorenz number $L$ is a fundamental constant that appears in the Wiedemann-Franz law
• In the free electron gas model, the Lorenz number is given by $L = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2$, where $k_B$ is the Boltzmann constant and $e$ is the electron charge
• The theoretical value of the Lorenz number is $L = 2.44 \times 10^{-8}$ W$\Omega$K$^{-2}$
• Experimental measurements of the Lorenz number for various metals agree well with the theoretical prediction, supporting the free electron gas model

Electron mean free path

• The electron mean free path $l$ is the average distance an electron travels between collisions in a metal
• It is related to the electron velocity $v$ and the relaxation time $\tau$ by $l = v\tau$
• The mean free path depends on various scattering mechanisms, such as electron-phonon scattering and electron-impurity scattering
• A longer mean free path indicates fewer collisions and contributes to higher thermal conductivity

Phonon contribution

• In addition to the electronic contribution, lattice vibrations (phonons) also contribute to the thermal conductivity of metals
• The phonon contribution to thermal conductivity becomes significant at higher temperatures, typically above the Debye temperature
• Phonons can scatter electrons and reduce the electronic mean free path, leading to a decrease in thermal conductivity
• The total thermal conductivity of a metal is the sum of the electronic and phonon contributions, $\kappa_{total} = \kappa_{electronic} + \kappa_{phonon}$

Failures of free electron model

• While the free electron gas model provides valuable insights into the behavior of electrons in metals, it has limitations and fails to explain certain properties of real materials
• The model's assumptions, such as neglecting electron-electron interactions and considering a uniform background potential, lead to discrepancies with experimental observations
• Understanding the limitations of the free electron gas model motivates the development of more advanced theories, such as the band theory of solids

Alkali metals

• Alkali metals (e.g., lithium, sodium) have a single valence electron per atom and are often considered the closest approximation to a free electron gas
• However, even in alkali metals, the free electron gas model fails to accurately predict certain properties, such as the electronic heat capacity at low temperatures
• The discrepancies arise from the non-parabolicity of the electron energy bands and the presence of electron-electron interactions

Transition metals

• Transition metals (e.g., iron, copper) have partially filled d-orbitals, which lead to complex electronic structures
• The free electron gas model fails to capture the behavior of electrons in transition metals due to the presence of strong electron-electron interactions and the localized nature of d-electrons
• Transition metals exhibit properties such as magnetism and catalytic activity, which cannot be explained by the free electron gas model

Semiconductors and insulators

• The free electron gas model assumes a metallic behavior with a partially filled conduction band
• However, semiconductors and insulators have a filled valence band and an empty conduction band separated by an energy gap
• The free electron gas model cannot describe the electronic properties of semiconductors and insulators, such as the temperature dependence of electrical conductivity and the existence of the band gap

Band theory of solids

• The band theory of solids is a more comprehensive framework that builds upon the limitations of the free electron gas model
• It considers the periodic potential of the lattice and the wave-like nature of electrons, leading to the formation of energy bands and band gaps
• The band theory successfully explains the electronic properties of metals, semiconductors, and insulators
• It takes into account the complex interactions between electrons and the lattice, providing a more accurate description of electronic behavior in solids