Chapter 22: Specific Heat of Solids and Electron Gas (Set-1)
What does Dulong–Petit law predict for molar heat capacity at high temperature
A 3R value
B R/2 value
C 2R value
D R value
Explanation: Dulong–Petit law says many crystalline solids (especially metals) have molar heat capacity near 3R at high temperature, based on classical equipartition giving 3 degrees of vibrational freedom per atom.
Dulong–Petit law fails mainly at which condition
A High pressure
B High temperature
C Low temperature
D Liquid state
Explanation: At low temperature, heat capacity drops below 3R because vibrational energy levels are quantized. Classical equipartition no longer holds, so fewer modes are thermally excited.
Dulong–Petit law works best for which solids
A Very light atoms
B Amorphous solids
C Molecular gases
D Heavy atom solids
Explanation: Solids with heavier atoms generally have lower characteristic vibration frequencies, so classical behavior (near 3R) appears at more accessible temperatures. Light atoms show stronger quantum effects.
In Einstein model, atoms in a solid are treated as
A Identical oscillators
B Free particles
C Rigid rotators
D Colliding spheres
Explanation: Einstein assumed each atom vibrates independently with the same frequency (single frequency). Energy is quantized, leading to a temperature-dependent heat capacity that approaches 3R at high T.
Einstein model introduces which key idea
A Continuous spectrum
B Energy quantization
C Classical collisions
D Perfect conductivity
Explanation: Einstein used quantized oscillator energy levels (using Planck’s constant). This explains why heat capacity decreases at low temperature, unlike classical predictions.
At very high temperature, Einstein heat capacity approaches
A Zero
B R/3
C 3R limit
D 6R
Explanation: When temperature is much larger than Einstein temperature, many vibrational levels are populated and classical equipartition is recovered, giving the Dulong–Petit high-T limit of 3R.
Main limitation of Einstein model is
A Single frequency assumption
B Uses classical energy
C Ignores quantization
D Predicts constant C always
Explanation: Real solids have many vibration frequencies. Einstein’s single-frequency model does not match experimental low-temperature behavior well, especially the observed T³ dependence.
Debye model improves Einstein model by assuming
A Single fixed frequency
B No lattice vibrations
C Only electronic motion
D Spectrum of frequencies
Explanation: Debye treated lattice vibrations as collective modes (phonons) with a range of frequencies up to a cutoff. This gives better agreement with experiments, especially at low temperatures.
In Debye theory, lattice vibrations are described as
A Photons
B Protons
C Phonons
D Positrons
Explanation: Phonons are quantized lattice vibration modes. Debye model uses them to calculate heat capacity from the phonon density of states and temperature-dependent mode occupation.
Debye temperature represents roughly
A Characteristic vibration scale
B Melting point
C Fermi energy
D Band gap
Explanation: Debye temperature sets the scale where quantum effects become important for lattice heat capacity. Below it, heat capacity falls strongly; above it, it approaches 3R.
Low-temperature Debye heat capacity varies as
A T
B T³
C T²
D Constant
Explanation: For insulating crystals at low temperature, only long-wavelength acoustic phonons are excited. Their number grows with T³, so lattice heat capacity follows the Debye T³ law.
Debye model high-temperature limit is
A 2R
B R
C 3R
D Zero
Explanation: At high temperature, phonon modes are fully excited and classical equipartition is recovered. Debye theory then matches Dulong–Petit, giving molar heat capacity close to 3R.
Debye cutoff frequency is chosen to
A Match total modes
B Remove electrons
C Fix lattice constant
D Stop conduction
Explanation: Debye introduces a maximum frequency so the total number of phonon modes equals 3N (for N atoms). This keeps the model physically consistent with degrees of freedom.
A C vs T³ plot is useful because
A Electronic term vanishes
B Density becomes constant
C Phonons stop forming
D Lattice term becomes linear
Explanation: At low temperature, lattice heat capacity ∝ T³, so plotting C versus T³ gives a straight line for the phonon contribution. This helps separate lattice and electronic terms in metals.
In metals at low temperature, electronic heat capacity varies as
A T³
B T²
C T
D Constant
Explanation: Only electrons near the Fermi energy can be thermally excited. The number of excited electrons is proportional to T, so electronic specific heat in metals shows a linear temperature dependence.
Total low-T heat capacity of a metal is often written as
A aT + bT³
B a + bT
C aT² + bT
D aT³ + bT⁴
Explanation: Metals have both electronic (∝T) and lattice phonon (∝T³) contributions at low temperature. Fitting C = aT + bT³ lets experiments extract electronic and lattice coefficients.
Free electron model assumes electrons behave as
A Bound oscillators
B Non-interacting particles
C Rigid dipoles
D Lattice ions
Explanation: In the simplest metal model, conduction electrons move freely inside the solid and interactions are neglected. This helps derive density of states, Fermi energy, and basic thermal properties.
Fermi–Dirac distribution gives
A Lattice spacing
B Sound velocity
C Occupation probability
D Heat conduction path
Explanation: Fermi–Dirac distribution tells the probability that an energy state is occupied by an electron at temperature T. It is essential for calculating electron energy and heat capacity in metals.
At T = 0 K, electrons fill states up to
A Debye energy
B Band gap
C Work function
D Fermi energy
Explanation: At absolute zero, all electron states below the Fermi energy are occupied and those above are empty. This defines the Fermi energy as the highest filled energy level at T = 0.
Density of states g(E) in 3D free electrons varies as
A √E
B E²
C 1/E
D Constant
Explanation: For 3D free electrons, the number of states increases with momentum-space volume, leading to g(E) proportional to √E. This shape strongly affects electronic properties and heat capacity.
Fermi temperature is defined by
A 3R/kB
B θD/kB
C EF/kB
D h/kB
Explanation: Fermi temperature TF is EF divided by Boltzmann constant. It is usually very high for metals, so ordinary temperatures are much smaller than TF and electrons remain strongly degenerate.
Fermi energy depends on electron number density as
A n^(2/3)
B n^(1/2)
C n^(1/3)
D n^(−1/3)
Explanation: In the free electron model, EF increases with electron density. The relation EF ∝ n^(2/3) comes from filling states in 3D momentum space up to the Fermi momentum.
Fermi velocity is related to Fermi energy by
A vF = EF/m
B vF = √(2EF/m)
C vF = m/EF
D vF = EF²
Explanation: Using kinetic energy EF = (1/2)mvF² for free electrons, Fermi velocity becomes vF = √(2EF/m). It is typically much larger than drift velocity in a conductor.
Drift velocity compared to Fermi velocity is
A Nearly equal
B Much larger
C Exactly zero
D Much smaller
Explanation: Drift velocity comes from an applied electric field and is slow. Fermi velocity reflects electrons’ high quantum kinetic energy. In metals, vF is typically orders of magnitude larger.
Only electrons near which energy contribute to heat capacity
A Near EF
B Near zero energy
C Near band gap
D Near work function
Explanation: At low temperature, most electrons are frozen in filled states below EF. Only those within about kBT of EF can change occupancy, so they dominate electronic heat capacity.
Sommerfeld coefficient is connected to
A Debye temperature
B Lattice constant
C DOS at EF
D Optical phonons only
Explanation: The electronic heat capacity coefficient (γ) is proportional to the density of states at the Fermi energy. Larger DOS at EF means more electrons can be thermally excited.
In insulators at low temperature, dominant heat capacity is
A Lattice phonons
B Free electrons
C Nuclear spins
D Photons only
Explanation: Insulators lack free conduction electrons, so the low-temperature heat capacity mainly comes from phonons. Debye theory predicts C ∝ T³ in this region.
Debye model treats solid as
A Isolated atoms
B Ideal gas
C Liquid lattice
D Elastic continuum
Explanation: Debye approximates the crystal as a continuous elastic medium supporting sound waves. Quantizing these waves gives phonons with a range of frequencies, improving low-temperature predictions.
Optical phonons are generally associated with
A In-phase motion
B Electron drift only
C Out-of-phase motion
D Heat flow only
Explanation: In crystals with more than one atom per unit cell, optical phonons involve neighboring atoms moving out of phase. They typically have higher frequencies than acoustic modes.
Acoustic phonons correspond to
A Sound waves
B Light waves
C Electron waves
D Nuclear decay
Explanation: Acoustic phonons are lattice vibrations where atoms move nearly in phase, producing long-wavelength modes that behave like sound waves. These dominate low-temperature heat capacity.
Lattice thermal conductivity mainly involves
A Proton drift
B Phonon transport
C Photon emission
D Nuclear fusion
Explanation: In nonmetals, heat is carried mainly by phonons. Their mean free path and scattering (defects, boundaries, phonon-phonon collisions) control lattice thermal conductivity.
Anharmonic effects in lattices are linked with
A Thermal expansion
B Zero heat capacity
C Perfect harmonic motion
D Infinite conductivity
Explanation: Pure harmonic vibrations give no thermal expansion. Real crystals are anharmonic, so average bond length increases with temperature, producing thermal expansion and enabling phonon-phonon scattering.
Calorimetry is used to measure
A Electron charge
B Lattice spacing
C Heat capacity
D Magnetic moment
Explanation: Calorimetry measures heat exchanged for a given temperature change. From Q = CΔT, one can determine specific heat and study how it varies with temperature in solids.
Debye–Waller factor is related to
A Electron spin
B Nuclear charge
C Band gap size
D Atomic vibrations
Explanation: Debye–Waller factor describes reduction in diffraction intensity due to thermal motion of atoms. Larger vibrations smear scattering positions, weakening diffraction peaks as temperature increases.
Debye T³ law is most valid for
A T ≪ θD
B T ≫ θD
C T = θD exactly
D All temperatures
Explanation: The T³ law comes from low-temperature excitation of long-wavelength phonons. It holds well when temperature is much smaller than Debye temperature; at higher T it deviates.
At extremely low T in some metals, C/T vs T² plot helps find
A EF and vF
B θE and θD
C γ and β
D n and m only
Explanation: Using C = γT + βT³, dividing by T gives C/T = γ + βT². Plotting C/T vs T² yields a straight line: intercept gives γ and slope gives β.
Fermi surface is defined in k-space as
A Boundary of filled states
B Ion core boundary
C Phonon cutoff
D Crystal edge only
Explanation: At T = 0 K, all electron states inside the Fermi surface are filled and outside are empty. Its shape affects conductivity, heat capacity, and many electronic properties.
Degeneracy pressure arises because of
A Coulomb attraction
B Pauli exclusion
C Gravity only
D Lattice defects
Explanation: Pauli exclusion prevents electrons from occupying the same state, forcing them into higher momentum states. This creates a pressure-like effect even at zero temperature, called degeneracy pressure.
The electronic specific heat is small at room temperature mainly because
A No electrons exist
B EF is small
C Phonons absent
D Only few excite
Explanation: Room temperature is much smaller than Fermi temperature, so only a tiny fraction of electrons near EF can be thermally excited. Thus the electronic heat capacity is much smaller than lattice heat capacity.
Debye model uses which type of phonons to explain T³ law
A Acoustic modes
B Optical modes
C Surface plasmons
D Magnons only
Explanation: Low-temperature heat capacity is dominated by long-wavelength acoustic phonons. Their density of states and excitation lead directly to C ∝ T³ in the Debye model.
Einstein temperature mainly depends on
A Electron density
B Crystal size
C Oscillator frequency
D Work function
Explanation: Einstein temperature θE is set by the assumed vibration frequency of atoms: θE = ħω/kB. Higher frequency means larger θE, causing stronger quantum suppression of heat capacity at low T.
Debye temperature depends on
A Sound velocity
B Electron charge
C Nuclear radius
D Photon energy
Explanation: Debye temperature is linked to the maximum phonon frequency, which depends on sound speed and atomic density. Faster sound velocity generally implies higher θD.
Phonon density of states is important because it controls
A Electron drift rate
B Nuclear decay
C Chemical bonding only
D Heat capacity integral
Explanation: Heat capacity depends on how many phonon modes exist at each frequency and how they are occupied. Density of states provides the weighting needed to compute energy and C(T).
In 2D, electronic density of states is approximately
A √E dependent
B Constant
C E² dependent
D 1/√E
Explanation: For free electrons in two dimensions, the number of states increases linearly with energy spacing, making DOS roughly constant. This changes thermal and transport behavior compared to 3D.
“Quantum correction” to classical specific heat mainly means
A Quantized vibrations
B More collisions
C Higher pressure
D Larger atoms
Explanation: Classical models assume continuous energy and equipartition. Quantum corrections account for discrete vibrational energy levels, reducing heat capacity at low temperature and matching experimental trends.
Wiedemann–Franz law connects
A Pressure and volume
B Heat and mass
C Thermal and electrical
D Charge and spin
Explanation: In metals, the same electrons carry heat and charge. Wiedemann–Franz states that thermal conductivity divided by electrical conductivity is proportional to temperature, linking transport to electron theory.
Electron mean free path affects mainly
A Conductivity value
B Debye temperature
C Atomic mass
D Phonon frequency only
Explanation: Mean free path sets how far electrons travel between collisions. A longer mean free path generally increases electrical and thermal conductivity, while impurities and defects reduce it by scattering electrons.
Superconductor heat capacity shows near Tc a
A Always zero
B Pure T³ law
C Linear decrease only
D Jump at transition
Explanation: In superconductors, heat capacity changes abruptly at the critical temperature due to formation of Cooper pairs and an energy gap. This produces a characteristic jump compared with normal metal behavior.
Best reason Debye fits low-T data better than Einstein is
A No quantization used
B Ignores phonons
C Many frequencies included
D Uses constant DOS
Explanation: Real solids have a range of vibrational modes. Debye includes a continuous spectrum and acoustic modes, giving the correct low-T behavior (T³), unlike Einstein’s single-frequency model.
For a good metal at very low T, dominant heat capacity term is often
A Lattice T³ only
B Electronic linear term
C Constant term
D Nuclear term only
Explanation: As temperature becomes very low, the lattice term βT³ drops faster than γT. Therefore, the electronic contribution can dominate, allowing experiments to measure γ and infer DOS at EF.