Chapter 22: Specific Heat of Solids and Electron Gas (Set-4)
In classical theory, why does a vibrating atom contribute kB per direction to internal energy
A Only kinetic part
B Only potential part
C Kinetic plus potential
D Electron drift energy
A harmonic vibration has both kinetic and potential energy terms. Equipartition gives (1/2)kBT to each quadratic term, so one direction gives kBT total, leading to 3NkB (≈3R per mole).
For most solids, why is Cp nearly equal to Cv at ordinary temperatures
A Expansion work small
B Heat loss dominates
C Electrons freeze out
D Phonons vanish fully
Solids expand very little when heated, so the extra work term that makes Cp exceed Cv is tiny. Therefore measured heat capacity at constant pressure is usually close to the constant-volume value.
Dulong–Petit “atomic heat” refers to heat capacity per
A Unit cell only
B Gram of solid
C One conduction electron
D Mole of atoms
Atomic heat is essentially the molar heat capacity of a solid per mole of atoms. Dulong–Petit suggested this value is near 3R at high temperature for many crystalline solids.
Einstein theory explains low-T drop mainly by introducing
A Variable atom number
B Energy level spacing
C Random bond breaking
D Classical collisions
Einstein treated each atom as a quantum oscillator with discrete energy levels separated by ħω. At low T, thermal energy is insufficient to cross these gaps, so heat capacity falls strongly.
In Einstein model, the low-temperature heat capacity decreases roughly like
A Linear in T
B Cubic in T
C Exponential in T
D Constant with T
Einstein heat capacity at low T is suppressed because excitations require a quantum ħω. The occupation of excited levels becomes very small, giving an approximately exponential decrease rather than a power law.
In Debye theory, why does low-frequency spectrum matter most at low temperature
A Easily excited modes
B Highest energy modes
C Electron-bound states
D Optical-only spectrum
At low T, only low-energy excitations are thermally accessible. Long-wavelength acoustic phonons have small energies, so they dominate energy storage and set the observed low-temperature heat capacity behavior.
Debye’s key step that gives T³ law is assuming for acoustic modes
A ω proportional to k²
B ω is constant
C ω is random
D ω proportional to k
For long wavelengths, acoustic phonons behave like sound waves with ω = vk. This linear relation plus 3D mode counting produces density of states ∝ ω² and leads to lattice heat capacity ∝ T³.
The Debye low-T molar heat capacity coefficient contains the factor
A 3π²/2
B 2π/3
C 12π⁴/5
D π⁴/12
Debye theory gives Cv ≈ (12π⁴/5)R(T/θD)³ at low temperature. The numerical factor comes from evaluating the Debye integral in the limit T ≪ θD.
A higher Debye temperature usually indicates a lattice that is
A Softer bonds
B Stiffer bonds
C Fully amorphous
D Electron-free always
Higher θD corresponds to higher characteristic phonon frequencies, which usually occur in stiffer, strongly bonded lattices. Such solids require higher temperature to excite most modes and reach the 3R limit.
When extracting θD from low-T data, which quantity is most directly fitted
A β in βT³
B γ in γT
C EF value
D Work function
In the low-T region, lattice heat capacity behaves like βT³. Since β is related to θD (roughly β ∝ 1/θD³), fitting β from experiments allows estimating Debye temperature.
In a metal, which term dominates heat capacity at sufficiently low temperature
A βT³ term
B Constant term
C γT term
D Negative term
As temperature decreases, βT³ becomes extremely small faster than γT. Therefore the linear electronic contribution can dominate at very low temperatures, enabling extraction of γ and electronic density of states.
Sommerfeld theory predicts the electronic heat capacity coefficient is proportional to
A θD only
B Crystal volume only
C Phonon cutoff only
D g(EF)
Sommerfeld theory gives Ce = γT with γ proportional to the density of states at the Fermi level. More available states near EF means more electrons can be thermally excited.
The standard relation between γ and density of states is
A γ ∝ (π²/3)kB²g(EF)
B γ ∝ 3RθD
C γ ∝ ħωD
D γ ∝ e²/ħ
In Sommerfeld theory, the temperature dependence comes from the sharp Fermi edge. The result is γ = (π²/3)kB²g(EF) (with appropriate units), linking electronic heat capacity to states near EF.
Which expression correctly shows EF dependence on electron density for free electrons
A EF ∝ n^(1/3)
B EF ∝ n^(−1/3)
C EF ∝ n^(2/3)
D EF ∝ n^(−2/3)
Filling quantum states in 3D gives kF ∝ n^(1/3). Since EF ∝ kF², it follows that EF ∝ n^(2/3). This explains why denser electron gases have higher Fermi energy.
The free-electron Fermi energy formula can be written as
A (ħ/2m)(3πn)
B (ħ²/2m)(3π²n)^(2/3)
C (kB/ħ)(3π²n)
D (e²/ħ)(πn)
For a 3D free electron gas, EF comes from the highest filled k-state: EF = ħ²kF²/2m and kF = (3π²n)^(1/3). Combining gives the stated expression.
The Lorenz number in Wiedemann–Franz law (ideal case) equals
A (3/π²)(e/kB)²
B (π/2)(kB/e)
C (π²/2)(e/kB)²
D (π²/3)(kB/e)²
Wiedemann–Franz states κ/(σT) ≈ L0 for many metals at moderate temperatures. Sommerfeld theory gives L0 = (π²/3)(kB/e)², linking heat and charge transport by electrons.
In a C/T vs T² plot, the slope physically represents mainly
A Fermi velocity vF
B Drift velocity vd
C Lattice coefficient β
D Work function φ
Since C/T = γ + βT², the slope of C/T versus T² is β. This slope reflects the phonon (lattice) contribution and is related to Debye temperature and vibrational properties.
Why is the electronic heat capacity not proportional to total electron number in a metal
A Most electrons frozen
B No Pauli principle
C EF equals zero
D Phonons carry charge
At low T, almost all electrons remain in filled states below EF and cannot change occupancy. Only a small fraction within about kBT of EF can be excited, so heat capacity depends on EF-edge states.
The “Fermi surface” is most correctly described as a surface of constant
A Phonon energy ħω
B Electron energy EF
C Atomic potential V
D Lattice spacing a
The Fermi surface is the boundary in k-space between occupied and unoccupied electron states at T = 0 K. Points on it have energy equal to the Fermi energy EF.
In Debye model, why does one use a cutoff in k-space
A Remove electrons
B Fix bond length
C Limit mode count
D Create optical branch
The continuum model would allow infinite wavevectors, giving infinite modes. Debye imposes a maximum k (or ωD) so the total number of modes matches 3N, consistent with a crystal of N atoms.
A material showing C ≈ γT at low temperature is most likely
A A good metal
B An insulator
C A noble gas
D A liquid crystal
A dominant linear term indicates significant electronic contribution, which is typical in metals due to conduction electrons. Insulators lack such electrons, so their low-T heat capacity is mainly phonon T³.
For a pure insulator, which low-temperature plot best highlights Debye behavior
A C/T vs T²
B C vs 1/T
C T/C vs T
D C/T³ vs T
If C ∝ T³, then C/T³ should be roughly constant in the Debye region. Plotting C/T³ versus T helps see deviations and the temperature range over which T³ law holds.
Which statement about Einstein vs Debye low-T predictions is correct
A Debye drops faster
B Both constant at low T
C Einstein drops faster
D Both linear at low T
Einstein model gives an exponential-like suppression at low T because all oscillators share one energy quantum. Debye includes many low-frequency modes, giving a power-law T³ decrease, which is slower than exponential.
The phonon density of states for acoustic modes in 3D increases mainly because
A Charges pile up
B k-space volume grows
C Bonds break with T
D Electrons become ions
The number of allowed wavevectors within radius k grows as k³. With ω ∝ k, the number of modes grows as ω³, giving density of states g(ω) ∝ ω² and shaping heat capacity trends.
If θD is very large, at room temperature the lattice heat capacity is more likely
A Below 3R
B Exactly 3R
C Above 3R
D Negative value
Large θD means many phonon modes have high energies. At room temperature, not all modes are thermally populated, so the lattice heat capacity may not reach the classical 3R limit.
Why does Debye–Waller factor reduce diffraction intensity with increasing temperature
A Smaller lattice spacing
B Higher electron charge
C Larger atomic motion
D Lower phonon count
Thermal vibrations increase the mean-square displacement of atoms, which blurs their positions for coherent scattering. This reduces Bragg peak intensity, quantified by the Debye–Waller factor in diffraction.
In metals, electron mean free path mainly influences which transport property strongly
A Debye temperature
B Einstein temperature
C Molar gas constant
D Electrical resistivity
Resistivity depends on how often electrons scatter. Shorter mean free path (more impurities or phonons) increases scattering and reduces conductivity. Heat capacity depends mostly on DOS, not mean free path.
Which process mostly limits lattice thermal conductivity at high temperature
A Electron degeneracy
B Phonon–phonon scattering
C Nuclear transitions
D Optical absorption
At high temperature, phonon population is large and anharmonic interactions cause frequent phonon-phonon collisions. This shortens phonon mean free path, reducing lattice thermal conductivity.
The reason zero-point energy does not appear in measured heat capacity is that it
A Does not change
B Is always negative
C Depends on pressure
D Equals thermal energy
Heat capacity is the derivative of internal energy with temperature. Zero-point energy is present even at 0 K but remains essentially constant with temperature, so it contributes nothing to heat capacity.
In a crystal with a basis, optical phonons exist because atoms can
A Move only together
B Stop vibrating
C Move oppositely
D Lose mass entirely
With more than one atom in the unit cell, additional normal modes appear. Optical phonons correspond to out-of-phase motion of atoms in the basis, typically at higher frequencies than acoustic modes.
A simple experimental signature of superconductivity in heat capacity near Tc is a
A Smooth linear rise
B Constant value
C Negative divergence
D Sudden jump
At the superconducting transition, electronic states reorganize and an energy gap forms. This causes a characteristic discontinuity (jump) in heat capacity at Tc compared to the normal-state trend.
Electron–phonon coupling can affect the electronic heat capacity mainly by changing
A Effective mass m*
B Lattice constant a
C Gas constant R
D Photon wavelength
Interactions with phonons can renormalize electron properties, often increasing effective mass. Since density of states near EF depends on m*, stronger coupling can increase γ and thus electronic heat capacity.
For free electrons, which relation correctly links Fermi momentum to number density
A pF = ħ(πn)
B pF = (ħ²/2m)n
C pF = ħ(3π²n)^(1/3)
D pF = kBθD
In 3D, electrons fill a sphere in k-space up to kF with volume set by n. This gives kF = (3π²n)^(1/3) and therefore pF = ħkF.
A good reason Fermi energy barely changes with temperature in metals is
A DOS is zero at EF
B T much smaller TF
C Electrons become classical
D Lattice loses all modes
Fermi temperature TF is extremely high, so everyday temperatures are tiny compared with TF. The distribution only smears slightly around EF, so EF and related quantities change negligibly.
In an insulator, a deviation from pure T³ law at very low temperature can be caused by
A Higher electron density
B Larger drift velocity
C Higher work function
D Defects or spins
At very low T, extra contributions from defects, paramagnetic impurities, or two-level systems can appear. These add terms beyond phonon T³ and cause visible deviations in experimental plots.
Which condition best describes the temperature range for clear Debye T³ behavior
A T much greater θD
B T equals θD
C T much less θD
D Any temperature range
Debye T³ law is derived using the low-temperature approximation where only low-frequency acoustic phonons are excited. This is valid when temperature is much smaller than Debye temperature.
In calorimetry, the heat capacity is most directly found from measuring
A Q and ΔT
B Voltage and current
C Pressure and volume
D Frequency and k
Calorimetry supplies a known heat Q and measures temperature rise ΔT. Heat capacity is computed using C = Q/ΔT (with suitable corrections), allowing precise study of C(T) at low and high temperatures.
The best physical meaning of “phonon mean free path” is the average distance a phonon travels
A Between unit cells
B Between scatterings
C Between electrons
D Between nuclei
Phonon mean free path is the typical distance a phonon travels before scattering from defects, boundaries, or other phonons. It strongly influences lattice thermal conductivity and its temperature dependence.
In the free electron model, why does g(E) increase with energy in 3D
A Fewer electrons exist
B Charge increases with E
C Phonons create electrons
D More k-states available
Higher energy corresponds to larger k. The number of available quantum states inside a sphere of radius k increases with k³, so the density of states grows with energy as √E for 3D free electrons.
A metal with unusually large γ is most likely to have
A Very small θD
B Zero electron density
C Large g(EF)
D No phonon modes
Since γ is proportional to g(EF), a large measured γ indicates many available electronic states near the Fermi level. This often occurs in materials with heavy effective mass or strong interactions.
Which statement correctly compares lattice and electronic contributions at low temperature in metals
A One is T, other T³
B Both are T³
C Both are constant
D One is 1/T
At low temperature, the electronic contribution is approximately γT while the phonon lattice contribution is approximately βT³. This difference in temperature dependence enables clean separation using low-T fitting methods.
The Debye model treats the solid as an elastic continuum mainly to simplify
A Removing phonons
B Counting long waves
C Fixing electron charge
D Breaking unit cells
For wavelengths much larger than atomic spacing, the discrete lattice can be approximated as a continuum. This makes it easier to count acoustic modes, derive g(ω), and predict low-temperature heat capacity.
If the measured heat capacity tends toward 3R only at very high temperature, the solid likely has
A Very low electron density
B No lattice vibrations
C High θD value
D Negative expansion
A high Debye temperature means phonon energies are large. It takes higher temperature to excite most vibrational modes and reach the classical 3R limit, so the approach to 3R is delayed.
Which short statement best describes the physical origin of degeneracy pressure in an electron gas
A Pauli exclusion filling
B Coulomb attraction only
C Thermal agitation only
D Phonon drag only
Pauli exclusion forces electrons into higher momentum states as density increases, even at T = 0 K. This creates a pressure-like effect independent of temperature, called degeneracy pressure.
When a metal is cooled, why does the Wiedemann–Franz ratio often approach a constant value
A Phonons carry charge
B EF becomes zero
C Heat becomes quantized
D Same carriers transport
In the normal state, electrons carry both heat and charge. As temperature decreases and scattering becomes more elastic, the ratio κ/(σT) tends toward the Lorenz number predicted by Sommerfeld theory.
The Einstein model is sometimes associated with “optical modes idea” mainly because it uses
A Many acoustic ω
B One characteristic ω
C No quantized ω
D Random ω values
Einstein assumes all atoms oscillate with the same frequency, resembling a single characteristic vibration like an optical mode conceptually. Real crystals have many modes, so Debye’s spectrum better matches experiments.
In Debye theory, why does dimensionality matter for low-T heat capacity
A EF becomes negative
B R becomes variable
C DOS power changes
D Phonons become electrons
The number of vibrational modes available at low frequency depends on dimension. Since heat capacity depends on how DOS scales with frequency, changing dimensionality changes the power-law temperature dependence.
In a semiconductor at low temperature, electronic heat capacity is small mainly because
A DOS is infinite
B θD is zero
C Phonons are absent
D Few carriers exist
At low temperature, very few electrons are thermally excited across the band gap, so electronic contribution to heat capacity is tiny. Lattice phonons dominate until temperature becomes high enough to create carriers.
Which relation best shows why vF is large in metals even without any applied field
A vF from EF
B vF from drift
C vF from phonons
D vF from pressure
Fermi velocity is determined by the high kinetic energy of electrons filling states up to EF. It exists even in equilibrium, unlike drift velocity which needs an electric field and is much smaller.
If a crystal shows a good linear C vs T³ plot only over a limited range, the best interpretation is
A C is constant always
B T³ valid range
C Electron term absent
D Phonons are classical
Debye T³ law holds when T is sufficiently low compared to θD and other effects are negligible. Outside that range, higher-frequency phonons or extra contributions cause deviations, so linearity appears only over part of data.