Chapter 22: Specific Heat of Solids and Electron Gas (Set-5)
In classical harmonic solid, why does equipartition give molar heat capacity 3R rather than 3R/2
A Only kinetic terms
B Only potential terms
C Two quadratic terms
D Electron degeneracy
Each vibrational direction has kinetic and potential quadratic energy terms. Equipartition gives (1/2)kBT to each, so one direction contributes kBT, three directions give 3kBT per atom, hence 3R per mole.
For a metal with low-T fit C=2.0T+0.10T3C=2.0T+0.10T3 (same units), at what temperature do electronic and phonon terms become equal
A 4.5
B 0.45
C 2.0
D 20
Set 2.0T=0.10T32.0T=0.10T3. Then T2=20T2=20 so T=20≈4.47T=20≈4.47. Below this, 2.0T2.0T dominates; above it, 0.10T30.10T3 dominates
In Debye model, the low-T heat capacity depends on θD mainly as
A θD⁺³ scaling
B θD⁻³ scaling
C θD⁻¹ scaling
D θD independent
Debye low-T form is Cv ≈ (12π⁴/5)R(T/θD)³. For fixed T, Cv decreases strongly if θD increases, giving the key dependence Cv ∝ θD⁻³.
For Debye acoustic phonons in 3D, which pair is correct
A g(ω)∝ω, C∝T²
B g(ω)∝const, C∝T
C g(ω)∝ω³, C∝T⁴
D g(ω)∝ω², C∝T³
Mode counting in 3D gives number of states up to k proportional to k³. With ω ∝ k for acoustic modes, states up to ω go as ω³, hence g(ω)=dN/dω ∝ ω², producing C ∝ T³.
Einstein model low-T heat capacity is much smaller than Debye mainly because Einstein has
A No low-ω modes
B Too many low-ω modes
C Constant g(EF)
D No cutoff ωD
Einstein assumes all oscillators have one frequency ωE. There are no arbitrarily low-energy modes to excite at low T, so excitation is strongly suppressed (nearly exponential), unlike Debye’s many low-ω acoustic modes.
In a 2D crystal (idealized), acoustic phonon DOS at low frequency roughly gives heat capacity scaling closest to
A C ∝ T³
B C ∝ constant
C C ∝ T²
D C ∝ T
In 2D, the number of acoustic modes up to k grows as k². With ω ∝ k, states up to ω go as ω², so g(ω) ∝ ω. This leads to lattice heat capacity scaling approximately as T² at low T.
A good low-T signature that phonon term is dominant in a metal is that C/T versus T² plot has
A Large positive slope
B Negative slope
C Zero intercept
D Curved intercept
C/T = γ + βT². If phonon term dominates, βT² is relatively large in the measured range, so the line has a noticeably steep positive slope, while intercept γ reflects electronic contribution.
For free electrons in 3D, which correct dependence links DOS and energy
A g(E) ∝ E²
B g(E) ∝ 1/√E
C g(E) ∝ constant
D g(E) ∝ √E
The number of states with wavevector less than k scales as k³. Since E ∝ k², N(E) ∝ E^(3/2), so the density of states g(E)=dN/dE ∝ √E.
A metal has higher electron density n. Which change must occur in the free-electron model
A EF unchanged
B EF decreases
C EF increases
D θD decreases
In 3D free-electron theory, EF ∝ n^(2/3). Higher density means more states must be filled, pushing the Fermi level higher. This typically also increases Fermi momentum and Fermi velocity.
Which statement correctly connects Fermi velocity and density n for free electrons
A vF ∝ n^(1/3)
B vF ∝ n^(−1/3)
C vF ∝ n^(2/3)
D vF ∝ n^(1/2)
For free electrons, kF ∝ n^(1/3). Since vF = ħkF/m, vF also scales as n^(1/3). EF scales as kF², so EF ∝ n^(2/3), but vF is weaker.
In Sommerfeld theory, why is Ce much smaller than classical 3NkB at room temperature
A No electrons present
B Only kBT/EF fraction
C EF equals kBT
D DOS vanishes
Only electrons within about kBT of EF can change occupancy. The active fraction is roughly kBT/EF, which is very small for metals. Hence electronic heat capacity is tiny compared with lattice at room temperature.
If g(EF) doubles while other factors stay similar, the Sommerfeld coefficient γ will
A Halve
B Stay same
C Double
D Become zero
Sommerfeld theory gives γ ∝ g(EF). If the density of states at the Fermi energy increases by a factor of two, the number of thermally excitable electrons rises accordingly, doubling γ and Ce at fixed T.
Which relation for Lorenz number L0 is correct in ideal Wiedemann–Franz law
A (π²/3)(kB/e)²
B (kB/e)
C (π²/2)(e/kB)²
D (3/π²)(e/kB)²
Sommerfeld theory predicts κ/(σT) approaches a constant L0 at low or moderate temperatures in many metals, with L0 = (π²/3)(kB/e)², reflecting that electrons carry both heat and charge.
A metal shows Ce = γT and lattice Cph = βT³. At what temperature do they become equal
A T = γ/β
B T = β/γ
C T = √(β/γ)
D T = √(γ/β)
Equality requires γT = βT³. Dividing by T gives γ = βT², hence T = √(γ/β). This crossover helps decide when electronic or phonon heat capacity dominates.
Which statement about Einstein and Debye high-temperature limits is correct
A Einstein approaches 0
B Both approach 3R
C Debye approaches 0
D Both stay below R
At high temperatures, quantization becomes less important and classical equipartition is recovered. Both Einstein and Debye models then predict the Dulong–Petit limit: molar heat capacity approaching 3R.
The Debye model assumes a linear dispersion ω = vk mainly for
A Zone boundary only
B All k exactly
C Long wavelengths
D Optical branches only
For small k (long wavelength), lattice vibrations behave like sound waves with linear dispersion. Near the Brillouin-zone boundary dispersion deviates, so Debye’s linear assumption is an approximation valid mainly at low ω.
Why does the Debye cutoff not represent a real physical “maximum frequency” for all branches
A Real dispersion differs
B It removes phonons
C Electrons set cutoff
D It is temperature-only
Debye cutoff is chosen to match the total mode count 3N. Real crystals have multiple branches and non-linear dispersion; ωD is an effective cutoff, not the exact highest phonon frequency of every branch.
A solid with a very small β (from C = γT + βT³) most likely has
A Low Debye temperature
B Zero phonon modes
C High Debye temperature
D Huge lattice expansion
β is roughly proportional to 1/θD³. A smaller β indicates fewer low-energy phonon states and a stiffer lattice, which corresponds to a higher Debye temperature.
Which statement is best about why Cp − Cv is small for solids
A No lattice vibrations
B Large compressibility
C Large volume change
D Small expansion coefficient
Cp − Cv involves thermal expansion and compressibility. Solids have small thermal expansion and are relatively stiff, so little work is done against pressure during heating, making Cp close to Cv.
In the Einstein model, which factor makes C approach 3R at high T
A kBT ≫ ħω
B ω → 0 always
C kBT ≪ ħω
D ħ → 0 always
When thermal energy far exceeds quantum spacing ħω, many oscillator levels are populated and quantization effects become negligible. The energy per mode approaches the classical value, giving molar heat capacity close to 3R.
A C/T³ vs T plot for an insulator is nearly flat in a certain range. That implies
A Einstein exponential holds
B Debye T³ holds
C Electronic term dominates
D Negative heat capacity
If C ∝ T³, then C/T³ is approximately constant. A flat region in C/T³ vs T indicates a temperature interval where Debye low-T phonon behavior is valid and other contributions are small.
In a metal, why does impurity scattering strongly change resistivity but barely change γ
A γ depends on mean free path
B γ depends on τ
C γ depends on DOS
D γ depends on drift speed
Resistivity depends on scattering time and mean free path, so impurities matter a lot. But γ is tied mainly to density of states at EF, which impurities usually do not change much in simple metals.
A reason phonon–phonon scattering increases with temperature is that anharmonic interactions cause
A More Umklapp events
B Lower phonon energy
C Less mode population
D Perfect periodicity
At higher temperature, phonon population rises, and anharmonicity allows momentum-nonconserving Umklapp processes (momentum shifted by a reciprocal lattice vector). These processes strongly limit thermal conductivity at high T.
Which statement about Debye–Waller factor is correct
A Depends on EF
B Depends on ⟨u²⟩
C Depends on drift speed
D Depends on γ only
Debye–Waller factor involves the mean-square atomic displacement ⟨u²⟩. Larger thermal vibrations increase ⟨u²⟩, reducing diffraction peak intensity by diminishing coherent scattering from well-defined lattice sites.
In the free electron model, why is the Fermi surface a sphere for an isotropic metal
A Energy depends on k
B Energy depends on 1/k
C Energy depends on T³
D Energy depends on k²
For free electrons, E = ħ²k²/2m, which depends only on magnitude of k, not direction. Therefore the constant-energy surface E = EF is a sphere in k-space for isotropic free-electron metals.
Which statement about degeneracy pressure is most accurate
A Needs high temperature
B Needs phonon scattering
C Exists at T=0
D Needs drift current
Degeneracy pressure arises from Pauli exclusion forcing electrons to occupy higher momentum states as density increases. This pressure exists even at absolute zero, independent of thermal motion or applied fields.
For a normal metal where C=γT+βT3C=γT+βT3, why does C/TC/T increase as temperature rises in the low-T region
A βT² lattice term
B Zero-point term
C γ intercept term
D EF shift term
Dividing by T gives C/T=γ+βT2C/T=γ+βT2. The γγ term is constant, while βT2βT2 grows with temperature, so C/TC/T rises as T increases in the low-temperature range.
Which choice best explains why Einstein model can fit some solids better at intermediate T than at very low T
A No quantization needed
B Optical-like modes matter
C Only electrons contribute
D Cp equals Cv always
Some solids have prominent vibrational frequencies resembling optical modes. A single characteristic frequency can approximate their behavior over an intermediate temperature range, even though it fails to capture the full low-frequency spectrum needed at low T.
For a 3D free electron gas, which scaling is correct for TF and n
A TF ∝ n^(1/3)
B TF ∝ n^(1/2)
C TF ∝ n^(2/3)
D TF ∝ n^(−1/3)
Fermi temperature TF = EF/kB. Since EF ∝ n^(2/3) for 3D free electrons, TF has the same scaling. Higher carrier density means higher EF and higher TF.
A larger effective mass m* generally increases γ because it
A Increases g(EF)
B Removes phonons
C Decreases g(EF)
D Makes EF negative
In many band structures, larger effective mass increases the density of states near EF. Since γ is proportional to g(EF), a heavier effective mass usually raises the electronic specific heat coefficient.
When using Debye model, which property most directly links to θD experimentally besides heat capacity
A Drift velocity
B Work function
C Nuclear spin
D Sound velocity
θD is tied to the maximum acoustic phonon frequency, which depends on sound speed in the material and atomic density. Measuring sound velocity can provide an independent estimate of θD.
In a metal, why is Ce ∝ T while κ/σT tends to a constant at low T
A Fermi edge physics
B Classical equipartition
C Zero electron density
D Phonon-only transport
Both results come from Fermi–Dirac statistics near EF. Only states near the sharp Fermi edge contribute to heat capacity, giving Ce ∝ T, and the same carriers carry heat and charge, leading to Wiedemann–Franz behavior.
Which statement correctly describes “Debye frequency” ωD in the model
A True max phonon ω
B Electron plasma ω
C Effective cutoff ω
D Optical-only ω
ωD is chosen so total phonon modes equals 3N. It is an effective cutoff representing the highest frequency in the simplified model, not necessarily the true maximum frequency across all real phonon branches.
A strong deviation from Debye T³ at very low T in glasses is often linked to
A Fermi surface sphere
B Two-level systems
C Dulong–Petit limit
D Optical phonons only
Amorphous solids can show extra low-energy excitations (often modeled as two-level systems) that add non-T³ contributions at very low temperatures. This causes deviations from the simple crystalline Debye behavior.
In the Debye model, why is the continuum approximation less accurate near zone boundary
A Phonons vanish
B Electrons dominate
C R changes value
D Discreteness matters
Near the Brillouin-zone boundary, wavelengths become comparable to lattice spacing, so atomic discreteness and detailed crystal structure strongly affect dispersion. The continuum elastic wave picture becomes a poorer approximation there.
The “historical importance” of Dulong–Petit law is mainly that it supported
A Atomic nature matter
B Superconductivity gap
C Fermi–Dirac statistics
D XRD indexing
Dulong–Petit suggested many solids share a common molar heat capacity near 3R, supporting the idea of atoms and consistent degrees of freedom. It helped early development of atomic theory and thermodynamics of solids.
If a crystal has very high θD, which of these is most likely true
A Large thermal expansion
B Low sound speed
C High phonon energies
D Soft lattice bonds
High θD corresponds to high characteristic phonon frequencies, meaning vibrational quanta are large. This usually occurs in stiff lattices with strong bonding and higher sound velocity, not soft bonds.
Which statement about “electronic term separation” in experiments is correct
A Use optical color
B Use low-T fitting
C Use melting point
D Use XRD peaks
At low temperature, heat capacity data can be fitted to C = γT + βT³. This method separates electronic and lattice contributions cleanly because their temperature dependences are different and well predicted.
Why does the Debye model naturally predict a universal curve when plotting Cv/3R versus T/θD
A θD sets scale
B Work function sets scale
C EF sets scale
D Drift speed sets scale
Debye theory expresses Cv as a function of T/θD through the Debye integral. Scaling temperature by θD collapses different solids onto a similar curve, explaining why normalized heat capacity shows near-universal behavior.
For electrons, why does “degenerate” mean classical Maxwell–Boltzmann statistics fail
A kBT much larger
B No quantization exists
C Pauli blocking strong
D Collisions absent
In a degenerate Fermi gas, many low-energy states are already filled. Pauli exclusion blocks occupation changes, so classical statistics fail. Fermi–Dirac statistics are required, especially when T is much less than TF.
In Debye model, which quantity is used to approximate the number of modes between ω and ω+dω
A f(E)dE
B n(T)dT
C σ(T)dT
D g(ω)dω
The phonon density of states g(ω) tells how many modes exist per frequency interval. Multiplying by dω gives the number of vibrational modes in that small range, used in energy and heat capacity integrals.
Which choice best explains why electronic heat capacity is sensitive to band structure details
A θD depends bands
B g(EF) depends bands
C R depends bands
D Cp depends bands only
The electronic heat capacity coefficient γ is proportional to g(EF), which depends on the band structure and effective mass. Therefore materials with different band shapes can have very different γ even if they are all metals.
A system with Ce ∝ T and Cph ∝ T² at low T is most consistent with
A 2D phonons
B Einstein oscillators
C 3D phonons
D Dulong–Petit limit
2D acoustic phonons give Cph ∝ T². Metals still give Ce ∝ T from Fermi statistics. This combination suggests a system with effectively two-dimensional phonon behavior, such as thin films or layered materials.
In Debye model, the integral form for heat capacity arises mainly because phonon energies are
A Fixed at one ω
B Classical continuous energy
C Distributed over ω
D Negative at low ω
Debye assumes phonons exist with a continuum of frequencies up to ωD. The total energy is obtained by summing contributions across all ω, which becomes an integral weighted by g(ω) and Bose occupation.
Which statement about “low-T experiments overview” is most relevant for extracting θD
A Measure below 0.1θD
B Measure near melting
C Measure above 5θD
D Measure only at 300K
To see clean T³ behavior and accurately fit β, measurements should be in the low-T regime where Debye approximation holds well, typically well below θD. This minimizes higher-mode and anharmonic effects.
A metal shows a larger-than-expected γ compared to free-electron prediction. The best explanation is
A Lower atomic mass
B Higher Debye cutoff
C Smaller lattice term
D Enhanced effective mass
Interactions and band structure can increase effective mass and density of states at EF, raising γ above the simple free-electron value. This is common in transition metals and strongly correlated materials.
Why does the Debye model give better overall agreement than Einstein for many solids
A Ignores acoustic modes
B Correct DOS shape
C Avoids quantization
D Uses constant ω
Debye captures the low-frequency phonon density of states and includes a spectrum of modes, giving correct low-T scaling and smoother crossover. Einstein’s single-frequency model misses the important low-energy modes.
In Sommerfeld model, the correction to internal energy at low T scales as
A T³ term
B exp(−T) term
C T² term
D constant only
Sommerfeld expansion shows the electronic internal energy differs from its T=0 value by a term proportional to T². Differentiating with respect to T gives Ce proportional to T, matching observed linear behavior.
Which condition ensures a solid is in the Dulong–Petit regime
A T ≫ θD
B T independent
C T ≈ 0
D T ≪ θD
When temperature is much larger than the Debye temperature, almost all vibrational modes are excited and quantization effects are small. The heat capacity approaches the classical limit close to 3R.
If a metal has very low θD but normal γ, the best expectation at modest low T is that total heat capacity will
A Show smaller βT³
B Lose linear γT
C Show larger βT³
D Become constant
Low θD implies more low-energy phonon modes, increasing β since β ∝ 1/θD³. Thus the lattice T³ contribution becomes larger at the same low temperature, even if γ stays normal.