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.

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