Chapter 22: Specific Heat of Solids and Electron Gas (Set-3)
For a monoatomic solid, why does Dulong–Petit give 3R at high temperature for one mole
A Two rotational modes
B One translational mode
C Three vibrational modes
D Zero-point only
Each atom vibrates in three perpendicular directions. At high temperature, classical equipartition gives kB per direction for a harmonic oscillator (kinetic + potential), leading to 3NkB = 3R per mole.
Which experimental trend directly shows Dulong–Petit breakdown at low temperature
A C decreases with T
B C stays constant
C C becomes negative
D C becomes infinite
Experiments show heat capacity falls sharply as temperature decreases, instead of staying near 3R. This indicates that energy is not shared classically among all vibrational modes at low temperature.
For a light-atom crystal like diamond, Dulong–Petit fails at ordinary temperatures mainly because
A Weak chemical bonds
B Too many electrons
C No lattice motion
D High vibrational frequency
Light atoms and stiff bonds produce high characteristic phonon frequencies (high θD). At ordinary temperatures, T may still be below the vibration energy scale, so many modes remain unexcited and C < 3R.
In Einstein theory, the Einstein temperature θE is defined using
A kB/ħω
B 3R/kB
C ħω/kB
D EF/kB
Einstein model uses one oscillator frequency ω. The characteristic temperature is θE = ħω/kB, setting when quantum effects become important. For T ≫ θE, the model approaches the 3R limit.
In Einstein model, the specific heat approaches zero at very low T mainly because
A Equipartition increases
B Few levels excited
C Frequency goes to zero
D Mass becomes infinite
When T is very low, thermal energy kBT is too small to excite most oscillators to higher quantized levels. With minimal energy absorption, the heat capacity becomes very small.
Compared to Einstein, Debye model matches low-T data better because Debye assumes
A No quantization at all
B Only optical modes
C Continuous frequency spectrum
D Only electronic heat
Debye includes a range of phonon frequencies up to a cutoff, especially low-frequency acoustic modes. This produces the observed low-temperature T³ behavior, which Einstein’s single-frequency model cannot reproduce.
The Debye density of states for low-frequency acoustic phonons in 3D varies approximately as
A ω² dependence
B ω⁻² dependence
C Constant in ω
D Exponential in ω
In 3D, the number of vibrational modes grows with the volume of k-space, giving mode count ∝ k³. Since ω ∝ k for acoustic phonons, the density of states scales as g(ω) ∝ ω².
In Debye model, the phonon cutoff is set so that total vibrational modes equal
A N modes
B 2N modes
C 6N modes
D 3N modes
Each atom contributes three vibrational degrees of freedom, giving 3N normal modes. Debye selects ωD so the integral of g(ω) up to ωD equals 3N.
Debye temperature θD is related to Debye frequency ωD by
A kB/ħωD
B 3R/ωD
C ħωD/kB
D EF/ωD
Debye temperature is defined as θD = ħωD/kB. It represents the temperature scale where most phonon modes become thermally populated, controlling the crossover to the high-T 3R limit.
In a low-T insulator, why does heat capacity follow T³ rather than T
A Phonons dominate
B Electrons dominate
C Photons dominate
D Ions drift
Insulators have negligible free electrons, so electronic linear-in-T heat capacity is absent. Low-temperature heat capacity mainly comes from acoustic phonons, and their mode population leads to C ∝ T³.
A practical method to separate electronic and lattice heat capacity in a metal is plotting
A C vs T
B T/C vs T
C C/T vs T²
D C vs 1/T
With C = γT + βT³, dividing by T gives C/T = γ + βT². A straight-line plot yields γ from intercept (electrons) and β from slope (phonons).
In the free electron model, why is electronic heat capacity proportional to T at low temperature
A All electrons excite
B Excitations near EF
C DOS becomes zero
D Lattice stops vibrating
Only electrons within about kBT of the Fermi energy can change occupancy. The fraction of active electrons is proportional to T, so the extra energy and heat capacity scale linearly with temperature.
At T = 0 K, the Fermi–Dirac distribution becomes
A Step function
B Gaussian curve
C Linear function
D Sine wave
At absolute zero, all states below EF are occupied and all above EF are empty. This produces a sharp step at EF, which becomes smoother only when temperature increases.
In 3D, the free-electron density of states g(E) varies with energy as
A E proportional
B 1/E proportional
C Constant with E
D √E proportional
The number of states grows with momentum-space volume. Since E ∝ k², the number of states up to E goes as k³ ∝ E^(3/2), giving g(E) = dN/dE ∝ √E.
Fermi momentum pF is related to electron density mainly through
A Lattice defects
B Optical phonons
C Filling k-space
D Heat conduction
Electrons occupy quantum states in k-space up to a maximum radius kF. For a given density n, the filled volume determines kF and thus pF = ħkF.
Why is Fermi velocity typically much larger than drift velocity in metals
A Quantum kinetic energy
B Strong electric field
C No scattering
D Zero resistance
Fermi velocity comes from the large kinetic energy of electrons filling states up to EF even without an applied field. Drift velocity is a small net motion due to a weak field and frequent scattering.
The Sommerfeld coefficient γ increases if the density of states at EF
A Decreases
B Becomes imaginary
C Increases
D Becomes negative
γ is proportional to g(EF). If more electronic states exist near EF, more electrons can be thermally excited for a given temperature rise, increasing electronic heat capacity.
In Debye theory, increasing sound velocity generally causes Debye temperature to
A Decrease
B Increase
C Stay zero
D Become random
Debye cutoff frequency depends on how fast waves propagate in the solid. Higher sound velocity raises ωD and thus θD = ħωD/kB, increasing the characteristic temperature scale.
In metals, why does lattice heat capacity become less important compared to electronic heat capacity at very low temperature
A T term drops faster
B DOS becomes zero
C Phonons increase quickly
D T³ drops faster
Lattice heat capacity goes as βT³ and falls rapidly when T is small. The electronic term γT decreases more slowly, so electrons can dominate the total heat capacity at sufficiently low T.
The Debye specific heat integral is needed because phonons have
A One frequency
B No quantization
C Many frequencies
D Only electrons
Debye model includes a distribution of phonon frequencies up to ωD. Total energy is found by summing contributions from all frequencies, which naturally becomes an integral using the phonon density of states.
A C vs T curve for many solids approaches 3R at high temperature because
A All modes excite
B Only low modes excite
C EF changes strongly
D Phonons disappear
At high temperature, thermal energy is enough to populate many vibrational levels, so nearly all 3N vibrational modes contribute. Classical equipartition is recovered, giving molar heat capacity near 3R.
In Einstein model, what feature produces a faster rise of heat capacity with T compared to classical at low T
A Linear DOS
B Constant occupancy
C Exponential suppression
D Negative energy
Einstein heat capacity at low temperature is strongly suppressed because excitations require energy quanta. As temperature increases, occupation rises rapidly, giving a steep increase toward the 3R limit.
Why does Debye theory predict a power-law T³ instead of Einstein’s stronger suppression
A Low ω modes exist
B Only optical modes exist
C Single ω dominates
D No cutoff needed
Debye includes many low-frequency acoustic phonons that can be excited even at small temperatures. Their increasing number with ω produces a smooth power-law heat capacity C ∝ T³ at low T.
In a metal, measuring heat capacity at very low temperature mainly probes
A Band gap value
B Ion charge state
C States near EF
D Crystal color
Low-temperature heat capacity is sensitive to electronic states around EF because only those electrons can be excited. This allows experiments to estimate g(EF) and infer information about the electronic structure.
For a 2D electron gas, the density of states is approximately
A √E dependent
B E² dependent
C Exponential in E
D Constant in E
In two dimensions, the number of states increases linearly with k, leading to a nearly constant g(E). This changes thermal and transport behavior compared with 3D metals.
The Debye–Waller factor increases with temperature mainly because atomic vibrations
A Stop completely
B Increase amplitude
C Become static
D Lose mass
As temperature rises, atoms vibrate more strongly around equilibrium positions. This reduces coherent diffraction intensity and is captured by the Debye–Waller factor, important in X-ray and neutron scattering.
The main carriers of thermal conductivity in an insulator are
A Free electrons
B Positrons
C Phonons
D Neutrons
Insulators lack free electrons, so heat is mainly transported by lattice vibrations. Phonon mean free path and scattering control the thermal conductivity, especially its temperature dependence.
In a crystal with two atoms per unit cell, optical phonons appear because atoms can move
A Out of phase
B With equal phase
C Without forces
D Without mass
With multiple atoms in a basis, additional normal modes exist. Optical phonons involve atoms moving opposite to each other within the unit cell, typically giving higher frequencies than acoustic modes.
Which statement best explains why electron specific heat is much smaller than lattice specific heat at room temperature in many metals
A No electrons present
B EF equals kBT
C Only small fraction
D DOS is zero
Even at room temperature, kBT is much smaller than EF. Only a tiny fraction of electrons near EF can be thermally excited, so electronic heat capacity remains small compared to phonon contribution.
The work function is connected to which concept more directly
A Debye frequency cutoff
B Phonon scattering rate
C Lattice heat constant
D Fermi level position
Work function depends on the energy difference between vacuum level and the Fermi level in a metal. It is related to electronic structure, not directly to lattice heat capacity, but both share EF as a key reference.
In Debye model, the low-temperature heat capacity constant β depends strongly on
A Electron charge
B Debye temperature
C Work function
D Drift speed
β is inversely related to θD³. Higher Debye temperature implies stiffer lattice and fewer low-energy phonons, reducing the low-T heat capacity coefficient for the T³ term.
Phonon–phonon scattering becomes significant mainly due to
A Perfect harmonicity
B Electron degeneracy
C Anharmonic forces
D Static lattice
In a purely harmonic lattice, phonons do not interact. Anharmonicity allows phonons to scatter and exchange energy, limiting thermal conductivity and contributing to realistic temperature-dependent behavior.
Why does Dulong–Petit often work well for many metals at room temperature
A θD is moderate
B Electrons dominate C
C Phonons are absent
D Crystal is amorphous
Many metals have Debye temperatures not too high, so room temperature is comparable to or higher than θD. This excites many vibrational modes, making heat capacity close to the classical 3R value.
In Sommerfeld theory, the internal energy change with temperature comes mostly from electrons
A At bottom states
B At all energies
C Near EF only
D At vacuum level
Electrons deep below EF remain filled and cannot change occupancy. Only electrons near the Fermi surface can be excited, so they contribute to temperature-dependent energy and linear specific heat.
Which plot is most directly used to confirm T³ law experimentally in an insulator
A C vs 1/T
B T vs C
C C² vs T
D C vs T³
If C ∝ T³ at low temperature, C plotted against T³ yields a straight line. This provides a clear experimental test of the Debye prediction for insulating crystals.
The reason Debye uses a “continuum approximation” is to treat long-wavelength phonons as
A Localized electrons
B Isolated atoms
C Elastic waves
D Nuclear rotations
For wavelengths much larger than lattice spacing, the discrete atomic nature can be approximated by a continuous elastic medium. This simplifies counting modes and deriving low-frequency phonon behavior.
At very low temperature, which heat capacity term is usually absent in a perfect insulator
A Cubic phonon term
B Linear electronic term
C Quadratic boundary term
D Zero-point energy
Insulators have no free conduction electrons, so the γT electronic term is negligible. The remaining heat capacity is mainly phonon-based, often following T³ behavior over a low temperature range.
Why does zero-point energy not contribute to heat capacity
A It is negative
B It grows with T
C It is constant
D It vanishes at T
Zero-point energy exists even at T = 0 K, but it does not change with temperature. Heat capacity measures how internal energy changes with temperature, so a constant energy term gives no contribution.
In a metal, which measurement helps estimate density of states at EF most directly
A Low-T heat capacity
B Room-T expansion
C Optical absorption
D Crystal density
The electronic heat capacity coefficient γ is proportional to g(EF). Measuring C at low T and extracting γ provides an experimental method to estimate the density of states at the Fermi energy.
Which statement correctly compares Einstein and Debye models at high temperature
A Both give zero
B Einstein gives T³
C Debye gives exponential
D Both give 3R
At high temperatures, quantization becomes less important and classical equipartition is recovered. Both Einstein and Debye models then approach the Dulong–Petit limit for molar heat capacity, close to 3R.
In a C/T vs T² plot, the intercept represents
A Debye temperature θD
B Fermi velocity vF
C Electronic coefficient γ
D Work function φ
Since C/T = γ + βT², the intercept at T² = 0 gives γ, the electronic heat capacity coefficient. It reflects density of states at EF and is key for metals.
In the free electron gas, why is EF weakly affected by temperature at ordinary conditions
A T ≪ TF
B DOS is zero
C Phonons dominate EF
D n changes fast
The Fermi temperature is extremely high for metals. Since ordinary temperatures are much smaller, the distribution changes only slightly near EF, causing negligible change in EF itself.
The relation EF ∝ n^(2/3) comes mainly from states filling in
A Two dimensions
B Three dimensions
C One dimension
D Zero dimensions
In 3D, the number of states up to kF is proportional to kF³. Because energy goes as kF², the Fermi energy becomes proportional to n^(2/3), linking EF strongly to electron density.
A solid showing strong deviation from T³ at extremely low temperature could indicate additional contribution such as
A Ideal gas motion
B Photon pressure
C Magnetic moments
D Nuclear fission
At very low temperatures, extra contributions like magnetic ordering, impurities, or nuclear terms can modify the simple phonon T³ law. Such effects produce deviations from a pure C ∝ T³ curve.
Why does Debye theory connect θD with elastic properties of the solid
A Electron charge sets ωD
B Density sets EF only
C Planck constant cancels
D Sound speed sets ωD
Debye cutoff frequency depends on how fast acoustic waves travel through the solid. Since sound speed depends on elastic constants and density, θD reflects the solid’s stiffness and bonding.
If a metal has larger effective mass, the Sommerfeld coefficient γ tends to
A Decrease
B Become zero
C Increase
D Oscillate randomly
A larger effective mass generally increases the density of states near EF. Since γ is proportional to g(EF), enhanced effective mass often increases the electronic specific heat coefficient.
Which statement correctly describes the role of phonon dispersion in heat capacity
A Sets mode spectrum
B Fixes electron density
C Removes quantization
D Makes C constant
Phonon dispersion determines allowed phonon frequencies for each wavevector. This spectrum controls the density of states and how energy is stored in vibrations, affecting the temperature dependence of heat capacity.
Lattice specific heat separation in metals is often done because lattice term behaves approximately as
A Constant at low T
B 1/T at low T
C T³ at low T
D T² at low T
Debye theory predicts phonon heat capacity ∝ T³ at low temperature. By fitting experimental data with γT + βT³, one can separate the lattice part from electronic contribution.
The idea of “degeneracy” in electron gas mainly refers to
A Zero phonon modes
B Constant heat capacity
C No electric charge
D Many filled states
Electron degeneracy means that even at low temperature, many quantum states are occupied up to EF due to Pauli exclusion. This is very different from classical gases and drives Fermi statistics behavior.
If θD is very high, room-temperature heat capacity is more likely to be
A Exactly 3R
B Above 6R
C Below 3R
D Negative value
A high Debye temperature means lattice vibration energies are large. At room temperature, not all modes are fully excited, so heat capacity may not reach the classical 3R limit and stays lower.