Physics Formulae: Energy and Force
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Questions and Answers

What is the Lorentz Force acting on an electron in an electric field?

eE

Thermal conductivity is defined as the amount of heat flowing per unit time through the material having unit area of cross section and maintaining at unit temperature gradient (dT/dx). The formula is Q = __________.

KdT

Which statement is true about the classical free electron theory?

  • It is used to derive Wiedemann-Franz law. (correct)
  • It successfully explains the electrical conductivity of semiconductors.
  • It fails to explain certain quantum phenomena.
  • All free electrons absorb the supplied energy.

    • Quantum theory explains the phenomenon of photo-electric effect better than the classical theory.

    <p>True</p> Signup and view all the answers

    What is the term used to describe the number of electrons per unit volume?

    <p>Carrier concentration</p> Signup and view all the answers

    What is the formula to calculate the carrier concentration of electrons in energy bands?

    <p>nc = 4π(2m)^(3/2)/(h^3) ∫ E^2 dE F(E)</p> Signup and view all the answers

    What is Fermi energy level at 0 Kelvin also known as?

    <p>Fermi energy level (Ef)</p> Signup and view all the answers

    In the free electron approximation, the potential energy of the electron is assumed to be greater than its total energy.

    <p>False</p> Signup and view all the answers

    Which approximation assumes the atoms are free while the electrons are tightly bounded?

    <p>Tight binding approximation</p> Signup and view all the answers

    What does equation (10) represent?

    <p>The effective mass of an electron in a periodic potential</p> Signup and view all the answers

    In which special case is the effective mass positive?

    <p>When d^2E / dk^2 is positive</p> Signup and view all the answers

    True or False: In the upper energy band, the effective mass is negative.

    <p>True</p> Signup and view all the answers

    The electron with negative effective mass is called _______.

    <p>hole</p> Signup and view all the answers

    What is the concept of forbidden energy levels?

    <p>Energy levels that are not allowed between certain radii</p> Signup and view all the answers

    How are materials classified based on the width of the forbidden band gap?

    <p>Metals, Semiconductors, Insulators</p> Signup and view all the answers

    Calculate the temperature at which there is 1% probability of a state with energy 0.5 eV above Fermi energy.

    <p>1261.58 K</p> Signup and view all the answers

    Calculate the carrier concentration of electrons in an energy interval of 0.01 eV above the Fermi level of sodium metal.

    <p>1.1786 x 10^26 m^-3</p> Signup and view all the answers

    Write the Fermi - Dirac distribution function.

    <p>F (E) = 1 / (1 + e^((E - Ef) / (kT)))</p> Signup and view all the answers

    Explain how Fermi function varies with temperature.

    <p>At T = 0K, if E &lt; Ef, F(E) = 1; if E &gt; Ef, F(E) = 0. At T &gt; 0K, Fermi function varies with E such that F(E) approaches 0.5 when E = Ef. At very high temperatures, the function approaches a classical Boltzmann distribution.</p> Signup and view all the answers

    What is the density of states for a metal?

    <p>Z(E) = 4π * (2m)^1.5 * a^3 * E^0.5 / h^3</p> Signup and view all the answers

    How is the density of states used to calculate the Fermi energy of metals?

    <p>The density of states (Z(E)) is used in conjunction with the Fermi-Dirac distribution function to calculate the Fermi energy of metals. The probability of electron filling in energy states is determined by the Fermi function.</p> Signup and view all the answers

    What happens to the density of states for non-zero temperature?

    <p>The density of states remains the same for non-zero temperature, but the Fermi function changes with temperature influencing the probability of electron occupation at different energy levels.</p> Signup and view all the answers

    Discuss the average energy of electrons at zero Kelvin.

    <p>At zero Kelvin, the average energy of electrons is aligned with the Fermi energy. Electrons exhibit quantum mechanical behavior and occupy energy levels below the Fermi energy level exclusively.</p> Signup and view all the answers

    Calculate the number of electrons per unit volume in copper.

    <p>8.46 x 10^25 m^-3</p> Signup and view all the answers

    Find the relaxation time in the given scenario.

    <p>2.43 x 10^-11 sec</p> Signup and view all the answers

    Calculate the mobility of electrons in copper.

    <p>4.27 m^2 V^-1 S^-1</p> Signup and view all the answers

    Determine the electrical conductivity of copper at 300K.

    <p>6.4 x 10^7 Ω^-1 m^-1</p> Signup and view all the answers

    Calculate the electrical conductivity of copper based on mean free path, electron density, and thermal velocity.

    <p>5.9 x 10^7 Ω^-1 m^-1</p> Signup and view all the answers

    Compute the electrical conductivity, thermal conductivity, and Lorentz number for a metal with a given relaxation time.

    <p>Electrical conductivity: 1.686 x 10^7 Ω^-1 m^-1, Thermal conductivity: 123.79 Wm^-1K^-1, Lorentz number: 2.45 x 10^-8 WΩK^-2</p> Signup and view all the answers

    Calculate the drift velocity and thermal velocity of conduction electrons in copper at 300K.

    <p>Drift velocity: 0.645 x 10^-3 m s^-1, Thermal velocity: 1.168 x 10^5 m s^-1</p> Signup and view all the answers

    Determine the drift velocity of free electrons in a copper wire with a specific cross-sectional area.

    <p>7.35 x 10^-5 m s^-1</p> Signup and view all the answers

    Calculate the average drift velocity and the mean collision time in a metallic wire with given parameters.

    <p>Average drift velocity: 1.023 x 10^-3 m s^-1, Mean collision time: 4.17 x 10^-14 sec</p> Signup and view all the answers

    Find the probability of an energy level being occupied by an electron above the Fermi level.

    <p>0.39</p> Signup and view all the answers

    Evaluate the Fermi function for an energy kT above the Fermi energy.

    <p>0.2689</p> Signup and view all the answers

    Calculate the Fermi energy of copper at 0 K.

    <p>The Fermi energy is calculated using the formula ne^2/2m.</p> Signup and view all the answers

    Study Notes

    Electrical Properties of Materials

    • Metals are composed of atoms with electrons in permissible orbits, which are free to move in all directions like molecules in a perfect gas container
    • The force between conduction electrons and ion cores is neglected, and the total energy of the electron is assumed to be kinetic energy (potential energy is zero)
    • When an electric field is applied, the free electrons move in a direction opposite to the direction of the applied field with a drift velocity (Vd)
    • The Lorentz force acting on the electron is F = eE, which accelerates the electrons, causing them to collide with positive ion cores and other free electrons elastically
    • The relaxation time (τ) is the time between collisions, and the drift velocity (Vd) is proportional to the electric field (E) and inversely proportional to the relaxation time (τ)

    Electrical Conductivity

    • Electrical conductivity (σ) is the amount of electrical charge (Q) conducted per unit time (t) per unit area (A) of a solid along a unit applied electrical field (E)
    • Expression for electrical conductivity: σ = ne²τ / m, where ne is the density of conduction electrons, e is the charge of an electron, τ is the relaxation time, and m is the mass of an electron

    Thermal Conductivity

    • Thermal conductivity (K) is the amount of heat (Q) flowing per unit time through a material having unit area of cross-section and maintaining a unit temperature gradient (dT/dx)
    • Expression for thermal conductivity: K = nvkλ / 2, where nv is the density of conduction electrons, k is the Boltzmann constant, λ is the mean free path of electrons, and v is the velocity of electrons
    • Wiedemann-Franz law: K / σ = LT, where L is the Lorentz number, a constant that depends on the material

    Fermi-Dirac Distribution Function

    • The probability (F(E)) of an electron occupying a given energy level at absolute temperature T is given by the Fermi-Dirac distribution function: F(E) = 1 / (1 + e^(E-Ef)/kT), where E is the energy of the level, Ef is the Fermi energy, k is the Boltzmann constant, and T is the absolute temperature
    • Effect of temperature on Fermi function:
      • At T = 0 K, F(E) = 1 for E < Ef, and F(E) = 0 for E > Ef
      • At T > 0 K, F(E) varies with E, and the Fermi function reduces to the classical Boltzmann distribution at high temperatures

    Density of States

    • The density of states (Z(E)) is the number of available electron states per unit volume in an energy interval E and E+dE
    • Expression for density of states: Z(E) = (1/8) * (4πn³/3) / (V * Ea³), where V is the volume of the metal piece, Ea is the energy of the electron, and n is the quantum number
    • Importance of density of states:
      • It gives the probability of electron occupation at a given energy state at a given temperature
      • It is used to calculate the number of free electrons per unit volume at a given temperature
      • It is used to calculate the Fermi energy of the metal### Density of States
    • The density of states (N(E)) is the number of energy states between the shell of radius 'n' and 'n+dn'
    • The formula for N(E) is: N(E) = (1/8) × (4πn^2 dn) / (3) = (π/2) × (n^2 dn) / (3)
    • Neglecting higher powers of dn, N(E) = (π/2) × (4n^2 dn) / (3)
    • The density of states is used to calculate the carrier concentration in metals and semiconductors

    Carrier Concentration

    • The carrier concentration (n_c) is the number of electrons per unit volume
    • The formula for carrier concentration is: n_c = ∫[Z(E)F(E)dE]
    • The Fermi distribution function (F(E)) is used to calculate the probability of filling an electron state
    • The carrier concentration is calculated by integrating the product of the density of states (Z(E)) and the Fermi distribution function (F(E)) over the entire energy range

    Fermi Energy

    • The Fermi energy (E_f) is the maximum energy level that can be occupied by an electron at 0K
    • The Fermi energy level is dependent on the density of states and the Fermi distribution function

    Bloch's Theorem

    • Bloch's theorem states that the wave function of an electron in a crystal can be written as: ψ(x) = e^(ikx)u_k(x)
    • The theorem is used to solve the Schrödinger equation for an electron in a periodic potential
    • The solution leads to the formation of allowed energy bands separated by forbidden energy gaps

    Energy Bands

    • The energy bands are allowed energy ranges for an electron in a crystal
    • The energy bands are separated by forbidden energy gaps
    • The width of the allowed energy bands increases with the increase in the interaction between the electrons and the lattice points

    Free Electron Approximation

    • The free electron approximation assumes that the potential energy of an electron is less than its total energy
    • The approximation is used to describe the behavior of electrons in metals and semiconductors
    • The approximation is valid when the interaction between the electrons and the lattice points is weak

    Tight Binding Approximation

    • The tight binding approximation assumes that the potential energy of an electron is almost equal to its total energy
    • The approximation is used to describe the behavior of electrons in insulators
    • The approximation is valid when the interaction between the electrons and the lattice points is strong

    Brillouin Zones

    • Brillouin zones are the boundaries marked by the values of the propagation vector k
    • The Brillouin zones are used to describe the energy bands of an electron in a crystal
    • The first Brillouin zone is the range of allowed energy values between -π/a and π/a

    Effective Mass of an Electron

    • The effective mass of an electron (m*) is the mass of an electron in a periodic potential
    • The effective mass is dependent on the energy of the electron and the curvature of the energy band
    • The effective mass can be positive or negative depending on the energy band

    Concept of a Hole

    • A hole is a positively charged particle with a negative effective mass
    • The concept of a hole is used to describe the behavior of electrons in the upper band of the energy spectrum
    • The presence of a hole is attributed to an empty state in the energy band### Valence Electrons and Energy Levels
    • Valence electrons possess more energy than inner orbit electrons, and the larger the orbit, the greater is its energy.
    • Energy levels (E1, E2, E3, ...) represent the energy of different orbits (K, L, M, ... shells).
    • Electrons can only revolve in certain permitted orbits (r1, r2, r3, ...) and not in arbitrary orbits, resulting in forbidden energy levels.

    Energy Bands

    • When atoms are brought together, interatomic force of attraction modifies the energy levels of a solid, forming energy bands.
    • Energy levels of single free atoms split into multiple levels, forming energy bands in solids.
    • Energy bands can be defined as the range of energies possessed by an electron in a solid.

    Forbidden Gap

    • Energy bands are separated by small regions, called forbidden gaps or forbidden energy gaps (Eg), which do not allow any energy levels.

    Classification of Materials

    • Based on band theory and the presence of forbidden band gaps, materials are classified into:
      • Metals (or) Conductors: no forbidden band gap, valence and conduction bands overlap.
      • Semiconductors: small forbidden band gap (0.5 to 1.5 eV), free electrons in valence band are relatively less.
      • Insulators: wide forbidden band gap (3 to 5.47 eV), requiring a large energy for conduction.

    Calculations

    • Mobility (μ) and average time of collision (τ) can be calculated using the Wiedemann-Franz (WKF) law.
    • Electrical conductivity (σ) can be calculated using the Lorentz number (L) and thermal conductivity (K).
    • Drift velocity (vd) and thermal velocity (v) can be calculated using the electron density (n), charge (e), and mass (m).

    Problems and Solutions

    • Several problems and their solutions are provided, covering topics such as electrical conductivity, thermal conductivity, Lorentz number, drift velocity, and Fermi energy.

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