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Physics C Ultimate Exam
Question 1.Which of the following correctly describes a Bravais lattice?A) A set of points generated by translating a single primitive vectorB) A periodic arrangement of atoms that can be described by a lattice plus a basisC) A lattice that contains only one atom per unit cellD) Any crystal structure that possesses cubic symmetryAnswer: BExplanation: A Bravais lattice is the periodic array of points obtained by translating a primitive cell; the actual crystal structure is obtained by attaching a basis (one or more atoms) to each lattice point. Question 2.The Wigner-Seitz cell is defined as:A) The smallest volume that contains exactly one lattice pointB) The primitive cell bounded by planes perpendicular to the vectors joining a lattice point to its nearest neighborsC) The unit cell with the highest symmetry in the crystalD) The reciprocal space analogue of the conventional cellAnswer: BExplanation: The Wigner-Seitz cell is constructed by drawing lines from a lattice point to its nearest neighbors, bisecting them with planes, and taking the region enclosed; it is the primitive cell with maximal symmetry in real space. Question 3.In a face-centered cubic (FCC) lattice, how many atoms are associated with one conventional unit cell?A) 1B) 2C) 4D) 8Answer: CExplanation: An FCC cell has 8 corner atoms (each contributes 1/8) and 6 face atoms (each contributes 1/2), giving 8×1/8 + 6×1/2 = 4 atoms per cell. Question 4.The structure factor for a BCC crystal with identical atoms at (0,0,0) and (½,½,½) is zero for which set of Miller indices?A) All h + k + l evenB) All h + k + l oddC) h, k, l all evenD) h, k, l all oddAnswer: BExplanation: For BCC, the two atoms give a phase factor exp[πi(h+k+l)]; when h + k + l is odd the factor is –1, causing total cancellation (structure factor =0). Question 5.The Bragg law nλ = 2d sinθ predicts a diffraction peak when the scattering vector magnitude equals:A) 2π/dB) 2π sinθ/dC) 2π/d sinθD) 2π/d cosθAnswer: AExplanation: In reciprocal space the Laue condition is G = k_out – k_in; for Bragg diffraction |G| = 2π/d, which corresponds to the condition nλ = 2d sinθ. Question 6.The reciprocal lattice vector b₁ is defined by the relation a₂·b₁ = 2πδ₂₁. If the primitive vectors are orthogonal with lengths a, b, c, what is the magnitude of b₁?A) 2π/aB) 2π/bC) 2π/cD) 2π/(abc)Answer:
Physics C Ultimate Exam
AExplanation: For orthogonal primitive vectors, b₁ = 2π a₁/a², so its magnitude is 2π/a. Question 7.The first Brillouin zone of a simple cubic lattice is:A) A sphere of radius π/aB) A cube of side 2π/aC) An octahedronD) A rhombic dodecahedronAnswer: BExplanation: The Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice; for a simple cubic reciprocal lattice (also cubic) it is a cube extending from –π/a to +π/a in each direction, i.e., side length 2π/a. Question 8.In a diatomic linear chain with masses M₁ ≠ M₂, the optical branch at the Brillouin-zone centre (k = 0) has a frequency ω₀ given by: A) √(2K/M₁)B) √(2K/M₂)C) √[K(M₁+M₂)/(M₁M₂)]D) √[K/(M₁+M₂)]Answer: CExplanation: At k = 0 the two atoms move out of phase, giving ω₀² = K(M₁+M₂)/(M₁M₂). Question 9.For a mono-atomic one-dimensional chain, the dispersion relation is ω(k)=2√(K/M) |sin(ka/2)|. The group velocity v_g = dω/dk is maximal at which k value?A) k = 0B) k = π/aC) k = π/2aD) k = π/4aAnswer: CExplanation: Differentiating yields v_g = a√(K/M) cos(ka/2); the cosine is maximal (=1) at ka/2 = 0, but the derivative of sin gives zero at k=0. The first non-zero maximum occurs at ka/2 = π/ → k = π/2a. Question 10.The Einstein model predicts the heat capacity C_V of a solid to approach zero exponentially at low temperature because:A) Phonon frequencies are all equal, giving a gapB) The Debye temperature is infiniteC) Electrons dominate the low-T heat capacityD) Anharmonicity is ignoredAnswer: AExplanation: In the Einstein model every atom vibrates with the same frequency ω_E, so at T ≪ ℏω_E/k_B the occupation of phonon states is Boltzmann-suppressed, giving C_V ∝ e^(−ℏω_E/k_BT). Question 11.In the Debye model the low-temperature heat capacity varies as C_V ∝ T³. This behavior arises from:A) The linear dispersion of acoustic phonons at small kB) The existence of an optical phonon branchC) A constant density of states for phononsD) The quantization of electron energy levelsAnswer: AExplanation: For acoustic phonons ω ≈ v_s k, the number of modes with frequency less than ω scales as ω³, leading to C_V ∝ T³.
Physics C Ultimate Exam
Fermi surface collapses at low TAnswer: AExplanation: At T ≪ T_F only electrons in a thin shell of width ∼k_BT around the Fermi energy can be thermally excited, giving a linear term in C_e. Question 18.The Drude model predicts the electrical conductivity σ = ne²τ/m. Which of the following statements is not a limitation of the Drude model?A) It treats electrons as classical particlesB) It assumes a single relaxation timeC) It neglects the Pauli exclusion principleD) It correctly predicts the temperature dependence of σ in most metalsAnswer: DExplanation: The Drude model fails to reproduce the observed temperature dependence (σ ∝ T⁻¹) because it ignores electron-electron interactions and quantum statistics; therefore statement D is false (i.e., not a limitation). Question 19.The Sommerfeld model improves on Drude by incorporating:Fermi-Dirac statistics, leading to a temperature-independent electron density at low T.Answer: The Sommerfeld model uses quantum statistics (Fermi-Dirac) for the electron gas, which yields a temperature-independent electron density and correctly predicts the linear electronic heat capacity and the temperature dependence of conductivity. Question 20.In a Hall-effect experiment on a metal with negative charge carriers, the Hall coefficient R_H is negative. The magnitude of R_H is given by:R_H = −1/(ne). Which assumption is implicit in this expression?A) Only one type of carrier dominatesB) Carrier scattering is isotropicC) The magnetic field is weak enough for linear responseD) All of the aboveAnswer: DExplanation: The simple formula assumes a single carrier type, isotropic scattering, and a low magnetic field so that the Hall voltage is linear in B. Question 21.The Wiedemann-Franz law states that the ratio κ/σT equals the Lorenz number L₀ = π²k_B²/3e². This law is most accurate for:A) Pure metals at low temperatureB) Pure metals at high temperatureC) All semiconductorsD) All insulatorsAnswer: AExplanation: In pure metals at low T, electron scattering is dominated by impurities (elastic) and the same electrons carry both heat and charge, making the Lorenz number approach the theoretical L₀. Question 22.In the nearly free electron model, an energy gap opens at the Brillouin-zone boundary because:A) The periodic potential couples plane waves with wavevectors differing by a reciprocal lattice vectorB) Electron-electron interactions
Physics C Ultimate Exam
become dominantC) Phonon scattering freezes outD) Spin-orbit coupling splits the bandsAnswer: AExplanation: At the zone boundary, two degenerate free-electron states are coupled by the periodic crystal potential, leading to level repulsion and an energy gap. Question 23.Bloch’s theorem states that the electron wavefunction in a periodic potential can be written as ψ_k(r)=u_k(r) e^{ik·r}. The function u_k(r) has:A) The same periodicity as the latticeB) No particular symmetryC) A wavelength twice that of the latticeD) Zero at the atomic positionsAnswer: AExplanation: u_k(r) is a lattice-periodic function, i.e., u_k(r+R)=u_k(r) for any lattice vector R. Question 24.In the Kronig-Penney model, the condition for allowed energy bands is:cos(k a)=cos(α a)+ (P/α a) sin(α a). Here α = √(2mE)/ℏ and P ∝ potential strength. Which of the following statements is correct?A) Larger P widens the gapsB) Larger P narrows the gapsC) Gaps disappear when P = 0D) Both A and CAnswer: DExplanation: When P = 0 the potential vanishes and the equation reduces to cos(k a)=cos(α a), giving a free-electron dispersion (no gaps). Increasing P strengthens the periodic potential, enlarging the forbidden gaps. Question 25.The tight-binding approximation for a simple cubic lattice with one s-orbital per atom gives the dispersion E(k)=E₀ − 2t[cos(k_x a)+cos(k_y a)+cos(k_z a)]. The bandwidth (maximum-minimum energy) equals: A) 4tB) 6tC) 8tD) 12tAnswer: BExplanation: The sum of cosines ranges from +3 (k = 0) to −3 (k = π/a each component). Thus ΔE = 2t(3 − (-3)) = 12t? Wait: E(k)=E₀ − 2t Σcos. Maximum at Σcos=3 → E_min = E₀ − 6t. Minimum at Σcos=- 3 → E_max = E₀ + 6t. Bandwidth = E_max − E_min = 12t. So answer is D. Question 26.The effective mass m* of an electron near the bottom of a conduction band is given by ℏ²/(∂²E/∂k²). For a parabolic band E = ℏ²k²/2m, the effective mass equals: A) The free-electron mass m_eB) Twice the free-electron massC) The curvature-derived m (identical to m_e only if the band is free-electron like)D) ZeroAnswer: CExplanation: By definition m* is the curvature mass; if the band is exactly parabolic with coefficient ℏ²/2m, then m equals that coefficient, which may differ from the free-electron mass. Question 27.In an intrinsic semiconductor at temperature T, the product of electron and hole concentrations satisfies:n p = n_i². The intrinsic carrier concentration n_i depends on temperature as: A) exp(−E_g/2k_BT)B) exp(−E_g/k_BT)C)
Physics C Ultimate Exam
Question 33.The forward I-V characteristic of an ideal diode follows the Shockley equation I = I_s (e^{qV/k_BT} − 1). The saturation current I_s depends exponentially on: A) The band-gap energy B) The doping concentration C) The junction area D) All of the aboveAnswer: DExplanation: I_s ∝ A T³ exp(−E_g/k_BT) · (N_D N_A)^{1/2}, so it is strongly affected by the band gap, doping, and device area. Question 34.Diamagnetism in a solid arises from: A) Alignment of permanent magnetic moments B) Induced currents opposing the applied field C) Exchange interaction D) Spin-orbit couplingAnswer: BExplanation: Lenz’s law causes circulating electron currents that generate a magnetic moment opposite to the external field, producing a weak negative susceptibility. Question 35.Paramagnetism follows Curie’s law χ = C/T. This law assumes: A) Non-interacting magnetic moments B) Strong exchange coupling C) Saturation of magnetization D) Temperature-independent susceptibilityAnswer: AExplanation: Curie’s law is derived for a collection of independent magnetic dipoles that align partially with the field; no interactions are considered. Question 36.Ferromagnetism can be described by the Weiss molecular-field theory, which introduces an internal field proportional to magnetization: H_int = λM. The Curie-Weiss law for susceptibility above the Curie temperature is χ = C/(T − θ_C). The parameter θ_C equals: A) λC B) λM C) λ C / k_B D) λ C / μ₀Answer: CExplanation: In Weiss theory θ_C = λ C/k_B, linking the molecular-field constant λ to the Curie constant C. Question 37.The exchange interaction responsible for ferromagnetism originates from: A) Direct Coulomb repulsion B) Overlap of electron wavefunctions leading to a lowering of energy for parallel spins C) Spin-lattice coupling D) Magnetic dipole-dipole interactionAnswer: BExplanation: The Pauli principle forces antisymmetric spatial wavefunctions for parallel spins, reducing Coulomb repulsion (exchange energy) and favoring alignment. Question 38.The Meissner effect is the expulsion of magnetic flux from a superconductor when it transitions below T_c. Which of the following statements is true? A) It is a consequence of perfect conductivity alone B) It requires the formation of a macroscopic quantum state C) It only occurs in type-II
Physics C Ultimate Exam
superconductors D) The expelled field penetrates a distance of order the London penetration depthAnswer: DExplanation: The magnetic field decays exponentially inside a superconductor over the London penetration depth λL; the Meissner effect is distinct from mere zero resistance. Question 39.Type-I superconductors are characterized by a single critical field H_c. Above H_c the material: A) Remains superconducting but with vortices B) Becomes normal conducting C) Transitions to a mixed state D) Exhibits a partial Meissner effectAnswer: BExplanation: Type-I superconductors undergo a first-order transition to the normal state when the applied field exceeds H_c. Question 40.In a type-II superconductor, the lower critical field H{c1} marks: A) The field at which the material becomes normal B) The onset of vortex penetration C) The field where the Meissner state is restored D) The maximum field the superconductor can sustainAnswer: BExplanation: For H > H_{c1} magnetic flux enters the superconductor as quantized vortices while superconductivity persists up to H_{c2}. Question 41.The BCS energy gap Δ at zero temperature is related to the critical temperature T_c by: Δ(0) ≈ 1.76 k_B T_c. This factor originates from: A) The phonon frequency spectrum B) The Cooper-pair binding energy C) The electron-phonon coupling constant D) The density of states at the Fermi levelAnswer: BExplanation: In BCS theory the energy required to break a Cooper pair is 2Δ; the ratio Δ(0)/k_B T_c ≈ 1.76 follows from solving the gap equation with a constant interaction over the Debye window. Question 42.The London penetration depth λ_L is given by λ_L = √(m/μ₀ n e²). Increasing the carrier density n will: A) Increase λ_L B) Decrease λ_L C) Not affect λ_L D) Make λ_L infiniteAnswer: BExplanation: λ_L varies inversely with the square root of the superfluid density n; a larger n yields a shorter penetration depth. Question 43.In the free-electron model, the plasma frequency ω_p is ω_p = √(n e²/ε₀ m). Metals with ω_p in the visible range appear: A) Transparent B) Opaque C) Colored due to interband transitions D) Reflective in the infrared onlyAnswer: BExplanation: When ω < ω_p, the metal reflects electromagnetic waves; visible light frequencies are below typical plasma frequencies of metals, leading to high reflectivity (opaque appearance).
Physics C Ultimate Exam
BExplanation: A larger hopping integral broadens the band, reduces effective mass, and enhances carrier mobility, thus increasing conductivity. Question 50.The Lorentz oscillator model for the dielectric function predicts a resonance at the phonon frequency ω_0. Near this frequency the real part of ε(ω) can become negative, leading to: A) Metallic reflection B) Transparency C) Reststrahlen band (high reflectivity) D) SuperconductivityAnswer: CExplanation: In the Reststrahlen region the dielectric constant is negative, causing strong reflection of infrared radiation. Question 51.In a crystal with a primitive lattice vector a₁, the reciprocal vector b₁ satisfies a₁·b₁ = 2π. If the lattice is hexagonal with a = b ≠ c, the magnitude of b₁ equals: A) 2π/a B) 4π/(√3 a) C) 2π/(√3 a) D) 4π/aAnswer: BExplanation: For a hexagonal lattice, |b₁| = 2π/(a sin 60°) = 4π/(√3 a). Question 52.The phonon dispersion in a three-dimensional crystal often shows a linear acoustic branch at low k. The slope of this branch is: A) The speed of sound v_s B) The group velocity of electrons C) The Debye frequency D) The lattice constantAnswer: AExplanation: The acoustic phonon frequency ω ≈ v_s k, so the slope dω/dk equals the sound velocity. Question 53.In the Einstein model, the characteristic temperature θ_E is defined by ℏω_E = k_Bθ_E. If θ_E is much larger than room temperature, the heat capacity at 300 K will be: A) Close to the Dulong-Petit limit B) Much lower than Dulong-Petit C) Independent of θ_E D) ZeroAnswer: BExplanation: When θ_E ≫ T, the exponential factor suppresses phonon excitation, so C_V is far below the classical 3k_B per atom value. Question 54.The specific heat of a metal at low temperature can be expressed as C = γT + βT³. The T³ term originates from: A) Electrons B) Acoustic phonons C) Optical phonons D) Magnetic excitationsAnswer: BExplanation: The T³ contribution is the Debye phonon term; the linear term γT is the electronic contribution. Question 55.The relaxation time τ in the Drude model can be experimentally obtained from: A) The slope of the resistivity vs. temperature B) The Hall coefficient C) The plasma frequency D) The optical conductivity linewidthAnswer:
Physics C Ultimate Exam
DExplanation: In optical measurements the width of the Drude peak is ∝ 1/τ, allowing direct extraction of the scattering time. Question 56.In a semiconductor under illumination, the excess carrier concentration Δn is proportional to: A) Light intensity I_ph B) Square root of I_ph C) 1/I_ph D) Log(I_ph)Answer: AExplanation: Generation rate G ∝ I_ph; under steady state Δn = G τ, so Δn scales linearly with light intensity. Question 57.The built-in electric field in the depletion region of a p-n junction points: A) From n to p B) From p to n C) Opposite to the diffusion direction D) Zero in equilibriumAnswer: AExplanation: Electrons diffuse from n to p, leaving positive charge on the n side; the resulting field points from the positively charged n-side toward the p-side. Question 58.The minority-carrier diffusion length L_n is given by L_n = √(D_n τn). If the diffusion coefficient D_n is increased by a factor of 4 while τn stays constant, L_n will: A) Double B) Quadruple C) Remain unchanged D) HalveAnswer: AExplanation: L_n scales with the square root of D_n; √4 = 2, so the diffusion length doubles. Question 59.The Curie temperature T_C of a ferromagnet can be estimated from the exchange integral J and the number of nearest neighbours z by: k_B T_C ≈ (2/3) z J S(S+1). Which assumption is implicit? A) Classical spins B) Mean-field approximation C) Quantum fluctuations are dominant D) Only dipolar interactions matterAnswer: BExplanation: This expression follows from Weiss mean-field theory, which replaces the many-body problem by an average molecular field. Question 60.In a type-II superconductor, the upper critical field H{c2} is related to the coherence length ξ by: H{c2} = Φ₀/(2πξ²). Reducing ξ (e.g., by impurity scattering) will: A) Increase H_{c2} B) Decrease H_{c2} C) Not affect H_{c2} D) Make H_{c2}=0Answer: AExplanation: Since H_{c2} ∝ 1/ξ², a shorter coherence length raises the upper critical field. Question 61.The Cooper pair wavefunction in BCS theory is symmetric in momentum space and antisymmetric in spin, leading to: A) Spin-triplet, s-wave pairing B) Spin-singlet, s-wave pairing C) Spin-singlet, p-wave pairing D) Spin-triplet,
Physics C Ultimate Exam
mass m* B) The band gap C) The lattice constant D) The Fermi velocityAnswer: AExplanation: ω_c depends on the charge-to-mass ratio; thus the measured resonance yields the effective mass of carriers. Question 68.In a crystal, the phonon density of states g(ω) at low frequencies varies as: A) ω² B) ω³ C) ω⁰ (constant) D) ω⁻¹Answer: AExplanation: In three dimensions the number of modes with frequency less than ω scales as ω³, so the differential density g(ω) ∝ d(ω³)/dω ∝ ω². Question 69.The Hall angle θ_H is defined by tan θ_H = σ_xy/σ_xx. In a metal where scattering is isotropic, θ_H is proportional to: A) ω_c τ B) (ω_c τ)² C) 1/(ω_c τ) D) √(ω_c τ)Answer: AExplanation: σ_xy ≈ ne²τ (ω_c τ)/m, σ_xx ≈ ne²τ/m, giving tan θH = ωc τ. Question 70.The Moss–Burstein shift in heavily doped n-type semiconductors leads to: A) An apparent increase in the optical band gap B) A decrease in the effective mass C) Enhanced phonon scattering D) Suppression of the Hall effectAnswer: AExplanation: Filling of the conduction band states pushes the absorption edge to higher energies, making the measured gap appear larger. Question 71.The Kondo effect observed in metals with magnetic impurities manifests as: A) A resistivity minimum at low temperature B) A superconducting transition C) Negative magnetoresistance D) Enhanced Hall coefficientAnswer: AExplanation: Spin-flip scattering off localized magnetic moments increases resistivity upon cooling, producing a characteristic minimum. Question 72.In a quantum well with infinite barriers, the allowed energy levels are given by E_n = (ℏ²π² n²)/(2m* L²). Reducing the well width L will: A) Decrease the spacing between levels B) Increase the spacing between levels C) Not affect the levels D) Collapse all levels to zeroAnswer: BExplanation: Energy scales as 1/L²; a smaller L pushes levels farther apart. Question 73.The effective mass tensor m*{ij} is defined as ℏ²/(∂²E/∂k_i∂k_j). In an isotropic band, the tensor reduces to: A) A scalar m* times the identity matrix B) Zero matrix C) Diagonal matrix with different elements D) Off-diagonal matrixAnswer: AExplanation: Isotropy implies the second derivative is the same in every direction, yielding m* δ{ij}.
Physics C Ultimate Exam
Question 74.The Lifshitz transition refers to: A) A change in the topology of the Fermi surface B) A magnetic ordering transition C) The onset of superconductivity D) A structural phase changeAnswer: AExplanation: When a band extremum crosses the Fermi level, the Fermi surface topology changes without symmetry breaking, constituting a Lifshitz transition. Question 75.In a semiconductor laser, population inversion is achieved by: A) Electrical pumping only B) Optical pumping only C) Both electrical and optical pumping to create more electrons in the conduction band than holes in the valence band D) Thermal excitation at high temperatureAnswer: CExplanation: Inversion requires more carriers in the upper laser level (conduction band) than in the lower level (valence band), achievable by either electrical injection or optical pumping. Question 76.The Debye–Waller factor describes: A) Attenuation of X-ray scattering due to thermal vibrations B) Enhancement of magnetic susceptibility C) Increase of band gap with temperature D) Reduction of carrier mobility Answer: AExplanation: It quantifies the reduction in coherent diffraction intensity caused by the mean-square displacement of atoms. Question 77.The structure factor for an FCC lattice with basis atoms at (0,0,0) and (¼,¼,¼) will be zero for reflections where: A) h, k, l all even but not all multiples of 4 B) h + k + l is odd C) h, k, l all odd D) h, k, l all multiples of 2Answer: AExplanation: The additional basis introduces a phase factor exp[πi/2 (h+k+l)]; when h, k, l are all even but not all divisible by 4, the factor leads to cancellation. Question 78.The Bragg condition for electron diffraction (using de Broglie wavelength λ_e) is identical in form to X-ray diffraction. For electrons accelerated through a potential V, λ_e is given by: A) h/√(2m_e eV) B) h c/eV C) h/2π √(2m_e eV) D) √(2eV/m_e)Answer: AExplanation: The de Broglie wavelength of a non-relativistic electron is λ = h/p = h/√(2m_e eV). Question 79.The phonon mean free path at low temperature is limited primarily by: A) Umklapp scattering B) Boundary scattering C) Electron-phonon scattering D) Impurity scatteringAnswer: BExplanation: At very low T, phonon-phonon processes freeze out, and scattering from sample boundaries or defects dominates.
Physics C Ultimate Exam
Question 86.In a superlattice consisting of alternating layers A and B, the mini-band formation is a consequence of: A) Brillouin zone folding due to the larger periodicity B) Strong electron-phonon coupling C) Magnetic exchange D) Surface states Answer: AExplanation: The superlattice introduces a new periodicity, leading to Brillouin zone folding and the emergence of mini-bands. Question 87.The photoconductivity of a semiconductor increases when: A) Light generates electron-hole pairs that add to the dark carrier concentration B) The temperature is lowered C) The material is heavily doped D) The band gap widens Answer: AExplanation: Absorbed photons create additional free carriers, raising the conductivity. Question 88.The Landau–Lifshitz–Gilbert equation describes: A) Magnetization dynamics in ferromagnets B) Phonon dispersion C) Electron tunneling D) Superconducting phase coherence Answer: AExplanation: The LLG equation incorporates precession and damping of the magnetization vector under effective fields. Question 89.In a type-I superconductor, the critical field H_c is related to the thermodynamic critical field by: A) H_c = (Φ₀/2πξλ) B) H_c² = (μ₀/2) (ΔF) C) H_c = √(2μ₀ ΔF) D) H_c = Φ₀/(2πλ²) Answer: CExplanation: The critical field equals the square root of twice the magnetic energy density associated with the condensation free-energy difference ΔF. Question 90.The Josephson effect predicts a supercurrent I = I_c sin φ across two superconductors separated by an insulating barrier. The phase difference φ evolves in time according to: A) dφ/dt = 2eV/ℏ B) dφ/dt = eV/ℏ C) dφ/dt = ℏ/2eV D) φ = constant Answer: AExplanation: The AC Josephson relation links the rate of change of the phase to the voltage across the junction. Question 91.In a crystal with a glide plane symmetry, the allowed reflections in X-ray diffraction obey: A) Systematic absences for h + k = odd B) No systematic absences C) Only even l indices allowed D) h − k = 0 only Answer: AExplanation: Glide planes introduce a translation combined with reflection, causing certain hkl reflections (where the sum of specific indices is odd) to be extinct.
Physics C Ultimate Exam
Question 92.The Raman active phonon modes in a crystal are determined by: A) The point group symmetry and selection rules B) The electronic band gap C) The Debye temperature D) The magnetic ordering Answer: AExplanation: Group theory predicts which vibrational representations are Raman-active based on the crystal’s symmetry. Question 93.The effective dielectric constant for a composite material can be estimated by the Maxwell-Garnett formula. This approximation assumes: A) Spherical inclusions embedded in a host matrix B) Perfectly aligned fibers C) Random percolation D) No interfacial polarization Answer: AExplanation: Maxwell-Garnett treats inclusions as isolated spheres within a continuous matrix, leading to an analytical expression for ε_eff. Question 94.In a two-dimensional electron gas (2DEG) formed at a semiconductor heterointerface, the subband energy spacing is proportional to: A) 1/L² where L is the confinement width B) L C) √L D) L³ Answer: AExplanation: Quantum confinement in the direction normal to the interface yields a particle-in-a-box spectrum ∝ 1/L². Question 95.The Kramers-Kronig relations connect the real and imaginary parts of a response function. For the optical conductivity σ(ω), the real part σ₁(ω) can be obtained from σ₂(ω) via: A) Principal value integral over all frequencies B) Simple derivative C) Direct multiplication D) No relation Answer: AExplanation: Causality imposes that σ₁ and σ₂ are Hilbert transforms of each other, requiring a principal-value integral. Question 96.The Seebeck coefficient of a degenerate semiconductor at low temperature varies as: A) T B) T² C) √T D) Constant Answer: AExplanation: For degenerate carriers the Mott formula gives S ∝ T, similar to metals. Question 97.The Hall mobility μ_H can differ from the drift mobility μ_d because: A) Scattering may be anisotropic, affecting the Hall factor r_H = μ_H/μ_d B) The magnetic field changes carrier charge C) Temperature dependence is opposite D) Hall mobility includes phonon drag Answer: AExplanation: The Hall factor accounts for the angular dependence of scattering; for isotropic scattering r_H ≈ 1, but anisotropy makes μ_H ≠ μ_d.
Physics C Ultimate Exam
Question 105.In a semiconductor, the Auger recombination rate R_A scales with carrier concentration as: A) n² B) n³ C) n⁴ D) n⁵ Answer: B Explanation: Auger processes involve three carriers (two recombining, one receiving energy), giving R_A ∝ n³ for intrinsic material. Question 106.The optical phonon frequency in a diatomic crystal is given by ω_opt = √[2K/(μ)], where μ is the reduced mass. Replacing the lighter atom with a heavier isotope will: A) Decrease ω_opt B) Increase ω_opt C) Not affect ω_opt D) Double ω_opt Answer: A Explanation: ω_opt inversely depends on √μ; a larger reduced mass lowers the optical phonon frequency. Question 107.The Debye–Waller factor appears in the intensity expression I = I₀ e^{−2M}. The exponent 2M is proportional to: A) ⟨u²⟩ |G|² where ⟨u²⟩ is the mean-square displacement B) Temperature only C) The square of the scattering angle D) The electron charge Answer: A Explanation: Thermal vibrations reduce coherent scattering; the factor involves the mean-square atomic displacement and the magnitude of the reciprocal vector G. Question 108.In a ferromagnet, the spin-wave (magnon) dispersion at long wavelength is ω(k) = D k². The stiffness constant D is proportional to: A) Exchange integral J times the lattice constant squared B) The magnetic field C) The electron effective mass D) Temperature Answer: A Explanation: D ∝ J a², reflecting that the energy cost of a slowly varying spin twist is set by the exchange interaction and lattice spacing. Question 109.The quantum Hall plateau resistance is quantized in units of h/e². The presence of disorder is essential because: A) It localizes states between Landau levels, allowing plateaus B) It enhances carrier mobility C) It increases the band gap D) It creates superconductivity Answer: A Explanation: Disorder broadens Landau levels and localizes states in the gaps, leading to quantized Hall conductance with robust plateaus. Question 110.The Rashba spin splitting energy Δ_R at wavevector k is given by Δ_R = α_R k. The parameter α_R is determined by: A) Structural inversion asymmetry and spin-orbit coupling B) Magnetic ordering C) Electron-phonon interaction D) Carrier concentration Answer: A Explanation: α_R arises from the
Physics C Ultimate Exam
electric field gradient due to lack of inversion symmetry combined with spin-orbit interaction. Question 111.The phonon drag contribution to the Seebeck coefficient becomes significant when: A) Phonon mean free path exceeds the electron mean free path B) Electron–phonon coupling is weak C) Temperature is much higher than θ_D D) The material is a metal Answer: A Explanation: When phonons can transfer momentum efficiently to charge carriers, they “drag” them, enhancing S. Question 112.The Kramers degeneracy theorem states that in a system with time-reversal symmetry and half-integer spin, each energy level is at least doubly degenerate. This degeneracy can be lifted by: A) Applying a magnetic field B) Increasing temperature C) Adding non-magnetic impurities D) Changing pressure Answer: A Explanation: A magnetic field breaks time-reversal symmetry, removing Kramers degeneracy. Question 113.In a superconductor, the coherence length ξ is related to the energy gap Δ and Fermi velocity v_F by: ξ ≈ ℏv_F/πΔ. A larger Δ therefore: A) Shortens ξ B) Lengthens ξ C) Has no effect D) Inverts the relation Answer: A Explanation: ξ is inversely proportional to Δ; a larger gap means Cooper pairs are more tightly bound, reducing the spatial extent. Question 114.The optical absorption coefficient α near the band edge of a direct semiconductor varies as: α ∝ (hν − E_g)^{½}. For an indirect gap, α varies as: A) (hν − E_g ± ℏω_ph)^{2} B) (hν − E_g)^{½} C) (hν − E_g)^{3/2} D) (hν − E_g)^{2} Answer: A Explanation: Indirect transitions require phonon assistance, leading to a quadratic dependence on the excess photon energy. Question 115.The Hall voltage in a thin rectangular slab of thickness t carrying current I in a magnetic field B is given by V_H = IB/(nqt). If the carrier density n is doubled, V_H will: A) Halve B) Double C) Remain unchanged D) Quadruple Answer: A Explanation: V_H is inversely proportional to n; doubling n reduces the Hall voltage by a factor of two. Question 116.The Bloch-Grüneisen formula for the temperature dependence of resistivity due to electron-phonon scattering predicts ρ(T) ∝ T⁵ at low temperatures (T ≪ θ_D). This T⁵ law arises because: A) Only phonons with energy < k_BT
