Metal Resistivity at Absolute Zero (Ideal Crystal) At T = 0 K, what is the electrical resistivity of a perfect single-crystal metal (no impurities, no defects)?

Introduction / Context:
Metallic resistivity arises from scattering of conduction electrons by phonons (lattice vibrations), impurities, and defects. Understanding which mechanisms vanish at very low temperatures clarifies why high-purity crystals show dramatically reduced resistivity at cryogenic temperatures and why a residual resistivity often remains in real samples.

Given Data / Assumptions:

• Perfect single crystal with no impurities or defects.
• Temperature T = 0 K (no thermal phonons).
• Normal metallic (non-superconducting) behavior assumed.

Concept / Approach:

In a perfect crystal at T = 0 K, phonon scattering vanishes because lattice vibrations are frozen out. With no impurities or structural defects, there are no scattering centers. In the Drude–Sommerfeld view, electron mean free path tends to infinity, so resistivity ρ (proportional to 1/mean free path) tends to zero. Real metals show a finite residual resistivity due to defects/impurities, but the ideal limit is ρ → 0.

Step-by-Step Solution:

Verification / Alternative check:

Experimental evidence: ultra-pure metals approach very small resistivities as T → 0 K, limited by residual defect scattering; superconductors show exactly zero due to Cooper pairing (a separate phenomenon), but the ideal Drude limit for a flawless metal is also zero.

Why Other Options Are Wrong:

• “Low” or “finite set by impurities” presupposes imperfections, which are excluded.
• “High” contradicts metallic band structure and absence of scattering.

Common Pitfalls:

Confusing ideal metal with real sample (residual resistivity), or conflating normal-metal behavior with superconductivity.

Final Answer:

Zero