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Modern Physics
Metals and Free Electron Theory
Metals are crystalline solids characterized by high electrical and thermal conductivities arising from the presence of valence electrons moving freely through the periodic lattice of positive ions. The quantum mechanical treatment of these electrons overcomes the failures of the classical Drude-Lorentz model by incorporating Fermi-Dirac statistics and the Pauli exclusion principle.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Properties of metallic bonding, classical free electron theory (Drude model), Ohm's law derivation, electrical and thermal conductivity, Wiedemann-Franz law.
PGT extension
Quantum free electron theory (Sommerfeld model), Fermi-Dirac distribution function, Fermi energy, density of states, contact potential, thermionic emission, and band theory of solids.
Prerequisites
Pauli's exclusion principle
Bohr's theory of hydrogen atom and quantum numbers
Maxwell-Boltzmann and Fermi-Dirac statistics
Classical Free Electron Theory (Drude-Lorentz Model)
Treats valence electrons in a metal as an ideal gas of particles obeying classical Maxwell-Boltzmann statistics.
Assumes collisions with fixed positive ion cores are instantaneous and random, characterized by a mean free time between collisions.
Fails to explain the specific heat capacity of metals at room temperature and the temperature dependence of conductivity.
Quantum Free Electron Theory (Sommerfeld Model)
Replaces classical statistics with Fermi-Dirac statistics, recognizing that electrons are fermions governed by Pauli's exclusion principle.
Electrons fill available energy states up to a maximum energy level at absolute zero, known as the Fermi energy (EF).
Successfully resolves the specific heat paradox by showing that only a small fraction of electrons near the Fermi surface participate in thermal excitation.
Fermi-Dirac Distribution and Fermi Energy
The probability f(E) of an electron energy state being occupied at temperature T is given by the Fermi-Dirac function.
At absolute zero (T = 0 K), f(E) = 1 for E < EF and f(E) = 0 for E > EF.
At temperatures greater than 0 K, the distribution curve smears out around the Fermi energy within an energy range of roughly kBT.
Density of States and Electrical Conduction
The density of states g(E) represents the number of allowed quantum states per unit volume per unit energy interval.
Electrical conduction occurs via the net drift of electrons near the Fermi level when an external electric field is applied.
The Wiedemann-Franz law relates thermal conductivity (κ) and electrical conductivity (σ) through the Lorenz number.
Derivations
Drude Derivation of Electrical Conductivity
Conditions: Uniform electric field E, constant relaxation time τ, classical electron gas
Write the equation of motion for an electron of mass m and charge e in an electric field E: m(dv/dt) = -eE - m(v/τ).
Under steady-state conditions, the acceleration dv/dt = 0, yielding the drift velocity vd = (-eτ/m)E.
Express electric current density J as $J = -nev_{d}$, where n is the electron concentration.
Substitute vd into J to get $J = (ne^{2}\tau/m)E$, and use Ohm's law $J = \sigmaE$ to identify conductivity.
Result: $\sigma = \frac{ne^2\tau}{m}$
Fermi Energy at Absolute Zero
Conditions: T = 0 K, free electrons in a three-dimensional box of volume V
State the density of states for electrons including spin degeneracy factor 2: g(E)d$E = \frac{V}{2\pi^2} \left(\frac{2m}{\hbar^2}\right)^{3/2} E^{1/2} dE$.
Integrate the density of states from E = 0 to the Fermi energy EF to equate to the total number of electrons N: $N = \int_{0}^{E_F} g(E)dE$.
Evaluate the integral: $N = \frac{V}{3\pi^2} \left(\frac{2mE_F}{\hbar^2}\right)^{3/2}$.
Rearrange the expression to solve explicitly for the Fermi energy EF.
h = Planck's constant, m = electron mass, n = electron concentration Joules (J) or Electron-volts (eV)
T = 0 K, quantum free electron gas
Fermi-Dirac Distribution Function
$$f(E) = \frac{1}{e^{(E - E_F)/k_BT} + 1}$$
E = energy, $E_F = Fermi energy$, $k_B = Boltzmann constant$, T = absolute temperature Dimensionless probability
Thermal equilibrium
Wiedemann-Franz Law
$$\frac{\kappa}{\sigma T} = L$$
\kappa = thermal conductivity, \sigma = electrical conductivity, T = temperature, L = Lorenz number W·Ω·K-2
Free electron metals at standard temperatures
Total Electronic Specific Heat (Sommerfeld)
$$C_v = \frac{\pi^2 \k_B^2 T \n}{2 E_F}$$
$C_v = specific heat capacity$, $k_B = Boltzmann constant$, T = temperature, n = electron density, $E_F = Fermi energy$ J·mol-1·K-1
Quantum model at low temperatures
Fermi Velocity
$$v_F = \sqrt{\frac{2E_F}{m}}$$
$v_F = velocity of electrons at Fermi level$, $E_F = Fermi energy$, m = mass m/s
Absolute zero
Mean Free Path
$$\lambda = v_F \tau$$
\lambda = mean free path, $v_F = Fermi velocity$, \tau = relaxation time meters (m)
Quantum free electron model
Graphs and experiments
Fermi-Dirac Distribution Function vs Energy
Draw: Plot of f(E) on the y-axis against energy E on the x-axis, showing a step function at T = 0 K and a smeared curve with a pivot at $E = E_F$ for T > 0 K.
Interpret: Illustrates how thermal energy excites electrons across the Fermi level, allowing states slightly below E_F to be unoccupied and states above to be occupied.
Exam use: Frequently tested in conceptual MCQs regarding occupation probability at and around the Fermi energy.
Density of States for 3D Free Electron Gas
Draw: Parabolic curve of g(E) proportional to E1/2 plotted against energy E.
Interpret: Shows that the number of available quantum energy states increases with the square root of energy in a three-dimensional metal.
Exam use: Used in PGT derivations for total energy, number of electrons, and specific heat capacity.
Electrical Resistivity vs Temperature in Metals
Draw: Linear increase of resistivity (ρ) with temperature T at high temperatures, dropping to a residual resistivity at T → 0 K.
Interpret: At high temperatures, resistivity is proportional to phonon scattering (lattice vibrations); at low temperatures, impurity scattering dominates.
Calculate the relaxation time of conduction electrons in a metal having an electrical conductivity of 5.8 × 107 S/m, given that the free electron concentration is 8.5 × 1028 m-3 and mass of electron is 9.1 × 10-31 kg.
Recall the formula for conductivity: σ = ne2τ / m.
Rearrange to solve for relaxation time τ: τ = (σ m) / (n e2).
Calculate the numerical value: τ ≈ 2.46 × 10-14 s.
Answer: 2.46 × 10^-14 s
PGT extension
Find the Fermi energy of copper at 0 K, assuming each copper atom contributes one free electron. Given the density of coppe$r = 8960 kg/m^{3}$, atomic weigh$t = 63.5 g/mol$, and Avogadro's numbe$r = 6.022 \times 10^{23} mol^{-1}$.
Calculate the electron concentration $n = (Density \times N_{A}) / (Atomic Weight)$.
Calculate the final value: vF ≈ 1.39 × 106 m/s.
Answer: 1.39 × 10^6 m/s
Assumptions and limitations
The Sommerfeld model assumes a constant potential inside the metal (potential well approximation), ignoring the periodic potential of the ion cores which requires band theory for complete description.
Inter-electronic repulsion is neglected in the independent electron approximation, treated only in an averaged mean-field manner.
Common misconceptions
Common error: All free electrons in a metal contribute to the electronic specific heat at room temperature.
Correct: Only a small fraction of electrons within an energy range of k_B·T of the Fermi surface are thermally excited.
Pauli's exclusion principle prevents electrons deeply seated in the Fermi sea from accepting thermal energy because nearby states are fully occupied.
Common error: The drift velocity of electrons is comparable to the speed of light or thermal velocity when current flows.
Correct: Drift velocity is extremely small, typically on the order of millimeters per second.
Current is carried by the massive aggregate flow under an applied field, whereas the random thermal or Fermi velocity is immensely higher (~10^6 m/s).
Quick revision
Sommerfeld's quantum model replaces classical Maxwell-Boltzmann statistics with Fermi-Dirac statistics.
Fermi energy is the highest occupied energy level at absolute zero temperature.
Wiedemann-Franz law states that the ratio of thermal to electrical conductivity is directly proportional to temperature.
Drude conductivity is proportional to the square of the electronic charge and the relaxation time.
Must remember
Classical free electron theory treats valence electrons using Maxwell-Boltzmann statistics, while the quantum model uses Fermi-Dirac statistics.
The Fermi energy ($E_F$) is the maximum energy level occupied by electrons at absolute zero ($T = 0$ K).
Only a small fraction of electrons near the Fermi surface participate in thermal excitation, resolving the specific heat paradox.
The Wiedemann-Franz law relates thermal conductivity ($\kappa$) and electrical conductivity ($\sigma$) through the Lorenz number $L$.
Drift velocity ($v_d = \mu E$) is extremely small, typically on the order of millimeters per second, despite high Fermi velocities.
Glossary
Fermi EnergyThe maximum energy state occupied by electrons in a metal at absolute zero temperature.
Relaxation TimeThe average time between successive collisions of an electron with ion cores in a metal.
Density of StatesThe number of allowed quantum states per unit volume per unit energy interval.
Wiedemann-Franz LawA law stating that the ratio of thermal conductivity to electrical conductivity is directly proportional to temperature.
Classical vs Quantum Free Electron Theory
Parameter
Classical (Drude-Lorentz)
Quantum (Sommerfeld)
Statistical Distribution
Maxwell-Boltzmann statistics
Fermi-Dirac statistics
Specific Heat Capacity
Predicts $3R$ per mole (fails)
Matches experimental values using only electrons near Fermi surface
Electron Velocity
Thermal velocity ($v_{th} \propto \sqrt{T}$)
Fermi velocity ($v_F$, independent of $T$)
Practice Quiz
Choose an option to reveal the explanation.
1. Which statistics govern the behavior of electrons in the quantum free electron theory of metals?
Explanation: Electrons are fermions with half-integer spin that obey the Pauli exclusion principle, and therefore follow Fermi-Dirac statistics.
2. What is the probability of an electron energy state being occupied at absolute zero ($T = 0$ K) for energies $E < E_F$?
Explanation: At absolute zero, all available energy states up to the Fermi energy ($E_F$) are completely filled, making the occupancy probability $f(E) = 1$ for $E < E_F$.
3. According to the classical Drude model, electrical conductivity $\sigma$ is given by:
Explanation: The classical expression derived from the Drude model is $\sigma = \frac{ne^2\tau}{m}$, where $n$ is electron density, $e$ is charge, $\tau$ is relaxation time, and $m$ is mass.
4. The Fermi energy $E_F$ of a free electron gas at $T = 0$ K depends on the electron concentration $n$ as:
Explanation: From the formula $E_F = \frac{h^2}{2m} \left( \frac{3n}{8\pi} \right)^{2/3}$, the Fermi energy is proportional to $n^{2/3}$.
5. What is the primary failure of the classical Drude-Lorentz model regarding electronic specific heat?
Explanation: Classical theory treats every electron as contributing $k_B$ to specific heat, leading to an electronic specific heat value much larger than experimentally observed.
6. The Wiedemann-Franz law relates thermal conductivity $\kappa$ and electrical conductivity $\sigma$ by stating that $\frac{\kappa}{\sigma T}$ is:
Explanation: The Wiedemann-Franz law states that the ratio $\frac{\kappa}{\sigma T}$ is a constant called the Lorenz number $L$, which is approximately independent of the metal.
7. In the Fermi-Dirac distribution function $f(E) = \frac{1}{e^{(E - E_F)/k_BT} + 1}$, what is the value of $f(E)$ when $E = E_F$ for any $T > 0$ K?
8. What is the typical order of magnitude of the drift velocity of conduction electrons in a metal carrying a steady current?
Explanation: Drift velocity is very small, typically on the order of millimeters per second ($10^{-4}$ m/s to $10^{-3}$ m/s), unlike the high thermal or Fermi velocities.
9. The density of states $g(E)$ represents:
Explanation: Density of states $g(E)$ quantifies how many quantum energy states are available per unit volume and unit energy range.
10. According to Sommerfeld's quantum theory, the electronic specific heat $C_v$ at low temperatures is proportional to:
Explanation: Sommerfeld's formula shows $C_v = \frac{\pi^2 k_B^2 T n}{2 E_F}$, demonstrating that electronic specific heat is directly proportional to temperature $T$ at low temperatures.
11. What determines the mean free path $\lambda$ of a conduction electron in the quantum free electron model?
Explanation: The mean free path is the product of the Fermi velocity $v_F$ and the relaxation time $\tau$.
12. Why do only electrons near the Fermi surface participate in thermal and electrical transport phenomena?
Explanation: States below $E_F$ are fully occupied; an electron cannot absorb energy unless an empty adjacent state is available, which only happens near the Fermi surface.
13. The unit of electrical conductivity $\sigma$ in the International System of Units (SI) is:
Explanation: Electrical conductivity is measured in Siemens per meter (S/m) or $\Omega^{-1} \cdot \text{m}^{-1}$.
14. What is the Fermi velocity $v_F$ expression in terms of Fermi energy $E_F$ and mass $m$?
Explanation: Equating kinetic energy at the Fermi level to $\frac{1}{2} m v_F^2 = E_F$ yields $v_F = \sqrt{\frac{2E_F}{m}}$.
15. Which of the following phenomena is best explained by the quantum free electron model rather than the classical model?
Explanation: The linear dependence of electronic specific heat on temperature at low temperatures is successfully resolved by Sommerfeld's quantum model.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The Fermi energy for a metal depends on which of the following parameters?
Explanation: Fermi energy depends solely on the electron concentration $n$ at absolute zero: $E_F = \frac{h^2}{2m} \left( \frac{3n}{8\pi} \right)^{2/3}$.
2. In the classical Drude model, the electrical resistivity of a metal is expected to vary with absolute temperature $T$ as:
Explanation: Classical theory predicts resistivity proportional to $\sqrt{T}$ through thermal velocity, whereas quantum theory combined with lattice vibrations gives linear dependence at high temperatures, but classical collision frequency gives $\rho \propto T$.
3. If the electron concentration in a metal is doubled, how does its Fermi energy at $T = 0$ K change?
Explanation: Since $E_F \propto n^{2/3}$, doubling $n$ multiplies $E_F$ by $2^{2/3}$.
4. The Lorenz number $L$ appearing in the Wiedemann-Franz law is defined as:
Explanation: The Wiedemann-Franz ratio is $\frac{\kappa}{\sigma T} = L$, defining the Lorenz number $L$.
5. What is the value of the Fermi-Dirac distribution function $f(E)$ for an energy state located $2k_B T$ above the Fermi energy ($E = E_F + 2k_B T$)?
6. Which model successfully resolved the specific heat paradox of metals at room temperature?
Explanation: Sommerfeld incorporated Fermi-Dirac statistics and Pauli's exclusion principle, resolving the specific heat paradox.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the probability of finding an electron in an energy state equal to the Fermi energy at any temperature $T > 0$ K?
Explanation: At any temperature greater than zero, the Fermi-Dirac distribution function evaluates to exactly $0.5$ when $E = E_F$.
2. If the relaxation time $\tau$ of electrons in a metal decreases due to increased impurity scattering, what happens to the electrical conductivity $\sigma$?
Explanation: Since $\sigma = \frac{ne^2\tau}{m}$, conductivity is directly proportional to relaxation time $\tau$.
3. The Fermi temperature $T_F$ is defined by the relation $E_F = k_B T_F$. For typical metals, $T_F$ is of the order of:
Explanation: Fermi temperatures for typical metals are very high, typically around $10^4$ to $10^5$ K.
4. In the context of free electron theory, the drift velocity $v_d$ is related to the electric field $E$ and mobility $\mu$ by:
Explanation: By definition, drift velocity is $v_d = \mu E$ in the low electric field regime.
5. What does the classical Maxwell-Boltzmann distribution assume about identical electrons in a metal?
Explanation: Classical Maxwell-Boltzmann statistics treat particles as distinguishable and place no restriction on multiple occupancy of energy states.
6. How does the density of states $g(E)$ vary with energy $E$ for a free electron gas in three dimensions?
Explanation: In 3D, the density of states for free electrons is proportional to the square root of energy: $g(E) \propto E^{1/2}$.
7. Which parameter links thermal conductivity $\kappa$, electrical conductivity $\sigma$, and temperature $T$ in metals?
Explanation: The Wiedemann-Franz law states $\frac{\kappa}{\sigma T} = L$, where $L$ is the Lorenz number.
8. What is the physical significance of the mean free path $\lambda$ in metals?
Explanation: Mean free path $\lambda$ represents the average distance an electron travels between two successive collisions with ion cores.
9. If the temperature $T$ of a metal is increased from $0$ K to a room temperature value, how does the Fermi-Dirac distribution function change near $E_F$?
Explanation: Thermal energy $k_B T$ causes the sharp step function at $T = 0$ K to smear out around $E_F$ within an interval of approximately $k_B T$.
10. In the free electron model, the total number of valence electrons $N$ in a volume $V$ can be obtained by integrating which quantity from $E = 0$ to $E = E_F$ at $T = 0$ K?
Explanation: The total electron concentration is found by integrating the product of the density of states and the Fermi-Dirac function: $n = \int_{0}^{E_F} g(E) f(E) dE$.