Chapter 22: Specific Heat of Solids and Electron Gas (Set-2)
For one mole of a solid, what does “3R” represent in Dulong–Petit law
A B. Fermi energy value
B A. Molar heat capacity
C C. Debye temperature
D D. Electron heat term
In the Dulong–Petit law, many crystalline solids at sufficiently high temperature have molar heat capacity close to 3R, meaning about three gas constants per mole due to three vibrational degrees of freedom.
Which assumption makes classical Dulong–Petit law give constant heat capacity
A B. States are discrete
B C. Phonons are absent
C A. Equipartition holds
D D. Electrons are bound
The classical result assumes equipartition of energy, where each quadratic degree of freedom contributes (1/2)kBT. For lattice vibrations, this leads to a temperature-independent molar heat capacity near 3R.
When temperature decreases, the lattice heat capacity usually
A B. Becomes infinite
B C. Stays at 3R
C D. Turns negative
D A. Decreases strongly
At low temperature, vibrational modes are not fully excited because energy levels are quantized. As fewer phonon modes are populated, lattice heat capacity falls well below the classical 3R value.
Einstein model treats each atom’s vibration energy as
A B. Continuous energy
B A. Quantized levels
C C. Pure rotation
D D. Random jumps
Einstein assumed atoms behave like quantum harmonic oscillators with discrete energy levels. This quantization explains why specific heat drops at low temperatures instead of staying constant as in classical theory.
Einstein temperature mainly sets the scale for
A A. Heat capacity change
B B. Electron drift speed
C C. Crystal density
D D. Electrical resistance
Einstein temperature indicates when quantum effects become important. When T is below this scale, fewer vibrational levels are populated, so the specific heat becomes smaller than the classical value.
A key reason Einstein model misses low-T data is that it assumes
A B. No lattice vibrations
B C. No Planck constant
C A. One vibration frequency
D D. Infinite atom mass
Real solids have many vibration frequencies. Einstein’s single-frequency assumption cannot reproduce the observed low-temperature behavior, especially the T³ dependence explained well by Debye theory.
In Debye model, vibrational modes are treated as
A B. Rotations only
B A. Waves in solid
C C. Electron orbits
D D. Gas collisions
Debye modeled the crystal as an elastic medium supporting waves. Quantizing these waves gives phonons with a range of frequencies, improving the heat capacity prediction over a wide temperature range.
The phonon concept is directly related to
A B. Quantized light waves
B C. Quantized electrons
C D. Quantized nuclei
D A. Quantized lattice waves
A phonon is the quantum of lattice vibration, just like a photon is the quantum of light. Phonons help explain temperature-dependent heat capacity and thermal conductivity in solids.
Debye model introduces a maximum frequency mainly to
A B. Remove optical modes
B C. Fix electron number
C A. Count 3N modes
D D. Stop scattering
A solid with N atoms has 3N vibrational modes. Debye chooses a cutoff (Debye frequency) so that the total number of allowed phonon states equals 3N.
At very low temperature, Debye heat capacity varies with temperature as
A B. T squared
B A. T cubed
C C. T inverse
D D. Constant
Debye theory predicts lattice heat capacity ∝ T³ at low temperature because only long-wavelength acoustic phonons are excited, and their number increases rapidly with temperature.
Debye T³ law is most applicable to
A A. Insulating crystals
B B. Hot metals only
C C. Liquids only
D D. Ideal gases
In insulators, there are no free electrons to add a linear term. So at low temperature, heat capacity is mainly from phonons and follows the Debye T³ law well.
A straight line in a C vs T³ graph indicates dominance of
A B. Electron term
B A. Lattice phonons
C C. Magnetic moments
D D. Chemical bonding
If lattice heat capacity follows C = βT³, then plotting C against T³ produces a straight line with slope β. This is useful to identify phonon behavior experimentally.
In metals, the low-T electronic heat capacity is proportional to
A B. Temperature T³
B C. Temperature 1/T
C A. Temperature T
D D. Temperature T²
Only electrons near the Fermi energy can be excited thermally. The number of such electrons is proportional to T, so the electronic heat capacity varies linearly with temperature at low T.
The coefficient of electronic heat capacity depends mainly on
A B. Debye temperature
B C. Lattice spacing
C D. Crystal shape
D A. DOS at EF
Electronic heat capacity coefficient (Sommerfeld coefficient) is proportional to the density of states at the Fermi energy. More available states near EF means more electrons can absorb heat.
At absolute zero, electron occupation is best described as
A B. Half filled all
B A. Filled below EF
C C. Empty below EF
D D. Randomly filled
At T = 0 K, all electron states with energy below the Fermi energy are occupied and all above are empty. This sharp boundary defines the Fermi energy in metals.
Fermi–Dirac distribution mainly gives
A B. Atomic vibration rate
B C. Sound velocity
C A. Occupation probability
D D. Lattice constant
The Fermi–Dirac function gives the probability that a state of energy E is occupied at temperature T. It is essential for calculating electronic energy and specific heat.
In a 3D free electron gas, density of states increases with energy as
A A. Square root E
B B. E to power 2
C C. Inverse E
D D. Constant with E
In 3D, the number of states grows with volume in momentum space. This leads to a density of states proportional to √E, which influences many electronic thermal properties.
Fermi temperature being very large implies at room temperature electrons are
A B. Non-degenerate gas
B A. Strongly degenerate
C C. Fully classical
D D. Not present
Since typical room temperature is much smaller than Fermi temperature, most electrons remain in filled states up to EF. Only a small fraction near EF gets thermally excited.
Fermi energy is mainly determined by
A B. Crystal color
B C. Magnetic field
C D. Sample length
D A. Electron density n
In the free electron model, the Fermi energy depends strongly on electron number density. Higher electron density requires filling more states, raising EF and associated Fermi momentum.
Fermi velocity is the speed of electrons associated with
A B. Drift motion only
B C. Phonon motion
C A. Highest filled state
D D. Ion vibration
Fermi velocity corresponds to electrons at the Fermi energy at T = 0 K. It represents typical electron speeds in the metal, much higher than drift speeds from an applied field.
A common reason electron specific heat is small is
A B. No electrons move
B A. Only near EF
C C. Phonons dominate EF
D D. Heat is negative
Only electrons within about kBT of the Fermi level can change occupancy and store thermal energy. Since this fraction is tiny, electronic specific heat is much smaller than lattice heat capacity.
The combined low-T heat capacity form for a metal is typically
A B. γ + βT
B C. γT² + βT
C D. γT³ + β
D A. γT + βT³
At low temperature, metals show a linear electronic term (γT) and a cubic lattice term (βT³). This separation helps experimentally determine both electronic DOS and lattice properties.
Plotting C/T versus T² for a metal gives a straight line because
A B. C/T = β + γT
B A. C/T = γ + βT²
C C. C/T = γT + β
D D. C/T = constant always
Using C = γT + βT³, dividing by T gives C/T = γ + βT². A straight line allows easy extraction of γ (intercept) and β (slope) from data.
Debye model low-T heat capacity arises mainly from
A A. Acoustic phonons
B B. Optical phonons
C C. Nuclear motion
D D. Electron drift
At low temperature, long-wavelength acoustic phonons dominate because they have low energy and are easiest to excite. Their density of states leads directly to the T³ heat capacity law.
Optical phonons generally have
A B. Zero frequency
B C. Same as drift
C A. Higher frequency
D D. No energy
Optical phonons involve out-of-phase motion of atoms in the basis and usually have higher frequencies than acoustic modes. They contribute more at higher temperatures when these modes become populated.
The Debye–Waller factor matters in diffraction because thermal motion
A B. Shifts electron charge
B A. Reduces peak intensity
C C. Changes Planck’s constant
D D. Creates new atoms
As atoms vibrate, their average positions fluctuate, reducing coherent scattering. This lowers diffraction peak intensity, described by the Debye–Waller factor, especially noticeable as temperature rises.
Lattice thermal conductivity decreases at higher temperature mainly due to
A B. Increased EF
B C. Reduced electron charge
C D. Larger crystal mass
D A. Phonon scattering
With rising temperature, phonon-phonon collisions (especially anharmonic processes) increase. This reduces phonon mean free path, lowering the ability of phonons to carry heat efficiently.
Anharmonicity in a lattice is important because it leads to
A B. Zero heat capacity
B C. Perfect elasticity
C A. Thermal expansion
D D. No phonon collisions
In a purely harmonic crystal, average atomic spacing would not change with temperature. Anharmonic potential makes average spacing increase as vibrations grow, causing thermal expansion and phonon scattering.
A “phonon dispersion” curve mainly shows relation between
A B. Charge and voltage
B A. Frequency and wavevector
C C. Pressure and volume
D D. Mass and density
Phonon dispersion gives how phonon frequency depends on wavevector in a crystal. It helps distinguish acoustic and optical branches and affects heat capacity and thermal conductivity trends.
In simple terms, Debye model treats phonons like
A A. Quantized sound waves
B B. Quantized light waves
C C. Electron drift packets
D D. Nuclear waves only
Debye’s picture is that low-energy vibrations behave like sound waves in an elastic medium. When these are quantized, they become phonons that store thermal energy and determine heat capacity.
For a fixed material, increasing temperature generally increases phonon population because
A B. EF decreases fast
B C. Density drops to zero
C D. Charge becomes larger
D A. More modes excite
Higher temperature supplies more thermal energy, allowing more phonon modes and higher frequencies to be excited. This increases internal energy and thus the measured heat capacity.
Debye model uses an integral for heat capacity mainly because it includes
A B. One frequency
B C. No quantization
C A. Many frequencies
D D. Only electrons
Debye assumes a continuous spectrum of phonon frequencies up to a cutoff. Heat capacity must sum contributions from all these modes, leading to an integral over phonon frequency distribution.
The electronic contribution dominates total heat capacity at low T because
A B. Electron term is cubic
B A. T³ term drops faster
C C. Phonons become electrons
D D. EF becomes zero
Lattice contribution falls as T³, which becomes extremely small at very low T. The electronic term falls only linearly as T, so it can dominate the total heat capacity in metals.
In a free electron gas, EF is the energy at
A A. Top filled states
B B. Bottom empty states
C C. Middle of band gap
D D. Ion vibration peak
Fermi energy is the highest occupied electron energy at T = 0 K. It sets the reference point for thermal excitations and many metal properties like electron specific heat.
The Fermi surface is a surface in
A B. Real space
B C. Time space
C D. Temperature space
D A. Momentum space
The Fermi surface is defined in k-space as the boundary between occupied and unoccupied electron states at T = 0 K. Its geometry affects transport and electronic behavior.
Degeneracy of electrons means at low temperature they
A B. Lose all charge
B A. Cannot all crowd
C C. Stop obeying Pauli
D D. Become phonons
Due to Pauli exclusion, electrons cannot occupy identical quantum states. Even at low temperature they fill different momentum states up to EF, producing degeneracy pressure and unique thermal behavior.
The main difference between optical and acoustic phonons is
A B. Electron number density
B C. Fermi temperature value
C A. Relative atom motion
D D. Heat capacity sign
Acoustic modes involve in-phase atomic motion and relate to sound waves. Optical modes involve out-of-phase motion (in multi-atom cells) and usually have higher frequency.
In the Drude picture, electrical resistance arises mainly from
A B. Phonon creation only
B C. Fixed EF value
C D. Zero mass electrons
D A. Electron collisions
Drude model explains resistivity by electrons scattering from ions, impurities, and phonons. Collisions limit mean free path and relaxation time, reducing electrical conductivity.
A larger effective mass usually implies for the same carrier density
A B. Higher Debye temperature
B A. Lower Fermi velocity
C C. Smaller lattice constant
D D. Zero phonon modes
For a given momentum, velocity is p/m*. If effective mass increases, electron velocity at comparable energies tends to decrease. This can influence transport and electronic specific heat estimates.
The Wiedemann–Franz idea supports that in metals
A B. Only phonons carry heat
B C. Ions carry current
C A. Same carriers carry heat
D D. Heat capacity is constant
The Wiedemann–Franz relationship reflects that conduction electrons carry both electrical current and thermal energy. This links thermal conductivity to electrical conductivity, especially in normal metals.
In semiconductors at low temperature, electronic heat capacity is usually
A A. Very small
B B. Always maximum
C C. Negative
D D. Much larger than lattice
At low temperature, few charge carriers are thermally excited across the band gap, so electronic contribution is tiny. Lattice phonons dominate heat capacity until carriers increase at higher temperature.
The “high-temperature limit” for lattice heat capacity means
A B. Approaches zero
B A. Approaches 3R
C C. Becomes negative
D D. Depends on EF only
When temperature is high compared to characteristic phonon temperatures, all vibrational modes are excited. The solid behaves classically and molar heat capacity approaches the Dulong–Petit value 3R.
The Sommerfeld model improves free electron theory by focusing on
A B. Lattice cutoff only
B C. Optical phonons only
C D. Atom rotations only
D A. Excitations near EF
Sommerfeld used Fermi–Dirac statistics and recognized only electrons near EF respond to temperature changes. This yields correct linear electronic specific heat and explains why most electrons remain inactive thermally.
A key signature of electronic specific heat in experiment is
A B. Cubic in T
B C. Constant with T
C A. Linear in T
D D. Inverse in T
In metals at low temperature, plotting heat capacity shows a term proportional to T from electrons. This linear behavior distinguishes electronic contribution from lattice phonon term that scales as T³.
The Debye model works best for low-T lattice heat capacity because it includes
A B. Only high modes
B A. Correct low modes
C C. No acoustic modes
D D. No cutoff modes
Debye theory correctly counts low-frequency acoustic modes that dominate at low temperature. Their excitation pattern produces the experimentally observed T³ heat capacity, unlike the Einstein single-frequency approach.
If impurities increase in a metal, electron mean free path usually
A A. Decreases
B B. Increases
C C. Becomes infinite
D D. Turns negative
Impurities act as scattering centers. More scattering reduces electron mean free path and relaxation time, increasing resistivity and affecting thermal transport, though electronic heat capacity depends mainly on DOS.
The “heat capacity jump” is often discussed in context of
A B. Dulong–Petit law
B A. Superconducting transition
C C. Crystal indexing
D D. Ideal gas law
At the superconducting critical temperature, heat capacity shows a noticeable jump because the electronic system changes state and an energy gap forms, altering how electrons store thermal energy.
Phonon contribution to heat capacity becomes small at very low T because
A B. EF becomes tiny
B C. Electrons vanish
C A. Few phonons exist
D D. Mass becomes zero
At very low temperature, thermal energy is insufficient to excite many vibrational quanta. The phonon population drops strongly, so lattice heat capacity becomes very small and follows a T³ dependence.
“Classical specific heat” usually refers to the prediction based on
A B. Fermi–Dirac rule
B C. Pauli exclusion only
C D. Band gap concept
D A. Equipartition theorem
Classical theory assumes each quadratic degree of freedom contributes kBT/2 to energy. For lattice vibrations this predicts constant heat capacity near 3R, which fails at low temperature due to quantization.
The best single statement about why solids need quantum theory for heat capacity is
A B. Atoms stop vibrating
B A. Vibrations are quantized
C C. Electrons become ions
D D. Heat has no units
In real solids, lattice vibrations have discrete energy levels. At low temperature, many modes cannot be excited, so heat capacity decreases. Quantum models (Einstein, Debye) match this behavior.