Low-temperature resistivity of a perfect metal crystal Assertion (A): The resistivity of a perfect single crystal of a metal tends to zero as temperature T approaches 0 K. Reason (R): Electron scattering arises from deviations from perfect lattice periodicity (e.g., phonons and defects), which vanish in an ideal, defect-free crystal at 0 K.
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ABoth A and R are true and R is correct explanation of A
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BBoth A and R are true but R is not correct explanation of A
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CA is true but R is false
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DA is false but R is true
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EBoth A and R are false
Answer
Correct Answer: Both A and R are true and R is correct explanation of A
Explanation
Introduction / Context:Understanding metallic resistivity at low temperature requires distinguishing between temperature-dependent phonon scattering and temperature-independent defect/impurity scattering (residual resistivity). The idealized “perfect crystal” clarifies the role of each mechanism.
Given Data / Assumptions:
- “Perfect single crystal” means no impurities, no vacancies, no dislocations, and perfect periodicity.
- Only lattice vibrations (phonons) remain as a scattering source at finite temperature.
- As T → 0 K, the phonon population tends to zero.
Concept / Approach:Resistivity ρ can be considered as ρ = ρres + ρph(T). The residual term ρres is due to static defects and impurities; ρph(T) stems from electron–phonon scattering and decreases with temperature. In the strictly perfect crystal, ρres = 0. Therefore, ρ → 0 as T → 0 K.
Step-by-Step Solution:Identify scattering sources: phonons (thermal), defects/impurities (structural).For a perfect crystal, structural scattering is absent ⇒ ρres = 0.As T decreases, phonon occupation falls; at 0 K, phonons vanish ⇒ ρph(0) = 0.Hence, total ρ(0) = 0, explaining A via R.
Verification / Alternative check:Real crystals exhibit a finite residual resistivity plateau at low T due to unavoidable impurities/defects, confirming that deviations from periodicity dominate when phonons freeze out.
Why Other Options Are Wrong:Any option denying R fails to recognize the two-term decomposition; denying A ignores the ideal limit definition.
Common Pitfalls:Confusing the idealized perfect-crystal limit with practical samples; attributing low-T saturation to phonons rather than impurities.
Final Answer:Both A and R are true and R is correct explanation of A