Chapter Three: The Classical Free Electron Theory (Drude Theory) PDF

# Chapter Three: The Classical Free Electron Theory (Drude Theory) This theory was proposed by Drude and Lorentz, and it assumes that the free electrons in metals behave according to classical mechanics laws. Although this theory failed to explain many phenomena, it still plays a significant role i...

# Chapter Three: The Classical Free Electron Theory (Drude Theory) This theory was proposed by Drude and Lorentz, and it assumes that the free electrons in metals behave according to classical mechanics laws. Although this theory failed to explain many phenomena, it still plays a significant role in understanding the basic properties of metals. ## Properties of Metals 1. **Metals obey Ohm's law:** - $J = \sigma E$ - $J = \frac{I}{A}$ Where: - $\sigma$ is the electrical conductivity (Siemens) - $\rho$ is the resistivity (Ωm) - $E = \frac{V}{d}$ is the electric field (Volt/m) - V is the voltage and d is the distance - R is the resistance of the material. 2. **Metals are good conductors (electricity):** - In metals, $\sigma$ is very large, making them excellent conductors. - The conductivity of metals at room temperature is approximately: * **(2) x 10⁻⁶ (Ω cm)** for metals * **(10⁻¹⁶ - 10⁻¹⁰) (Ω cm)** for insulators 3. **Metals have a large thermal conductivity:** - Each atom in a metal has a high thermal conductivity, and as a result, metals are excellent conductors of heat as well. - This means that the ratio of thermal conductivity ($K_e$) to electrical conductivity ($\sigma$) is constant at a specific temperature: - $K_e/\sigma$ = Const = L (Jomt^2/cm) - L = 2.2 * 10⁻⁸ (V/K) 4. **As the metal is cooled down, the thermal conductivity ($K_e$) and the resistivity ($\sigma$) increase, but the increase in $\sigma$ is much faster.** - $K_e$ varies as (T) but at very low temperatures, it becomes constant. - $\sigma$ varies with (T) before becoming constant. - The reason behind this behavior is that the number of collisions responsible for thermal conductivity is not equal to the number of collisions responsible for electrical conductivity. 5. **Metals have a very small specific heat (Cv) and it is temperature-dependent (1/T).** - A smaller specific heat indicates that metals require less energy to raise their temperature. - (C_v)= (1/T)RT ≈ 18 T₀, where T₀ is the temperature in Kelvin. 6. **Metals have a large paramagnetic susceptibility.** 7. **Metals have other magnetic properties.** ## Drude model (Classical Free Electron - FEM) **Assumptions:** - The Drude model assumes that the metal is composed of positive ions fixed in a lattice, forming a positive background. - The model also assumes that the electrons in the metal are free, moving according to the classical model (free electron gas). - This model does not take into account the quantum aspects of electrons or the interactions between the electrons. ## Max-Boltzmann vs Drude Distribution (Model of electron gas) The Drude model assumes that the free electrons in the metal form a gas that follows the Maxwellian distribution. However, the electrons are not truly free but rather are confined by the positive background of the lattice. This leads to a slight difference in the velocity distribution of the electrons, which is known as the Drude distribution. ## Understanding the Drude model - **+ve Core**: The +ve core is composed of the nucleus and all the closed shells of electrons. The +ve core is responsible for creating a positive charge density, thus forming the positive array of positively charged ions. - **Free electrons**: The free electrons in the metal are those that are not bound to the nucleus and are responsible for the conduction of electricity in metals. They form a quasi-uniform distribution of charge. **Key points:** - The free electrons move in the metal with complete freedom, colliding with the +ve cores and changing their direction, causing the charge density to vary with the potential. - The classical free electron theory focuses on the movement of free electrons in the metal, neglecting the effect of the periodic potential, resulting in a negligible impact. The Drude model is a simplified model that provides an important foundation for understanding the behavior of electrons in metals. While it has its shortcomings, it is a helpful starting point for explaining many basic properties of metals.