Electron transport in metals: relation between relaxation time and mean free time If τ is the relaxation time and τc is the average time between successive electron–scattering collisions in an isotropic metal, which relation holds under the Drude model assumptions?

Introduction / Context:In electrical conduction models, the relaxation time τ characterizes how quickly the average electron drift velocity decays toward zero after removal of an applied field. Understanding its connection to microscopic collision statistics is foundational in solid-state physics and materials engineering.

Given Data / Assumptions:

• Isotropic metal with random, uncorrelated scattering events.
• Drude–Sommerfeld picture: free electrons undergoing instantaneous collisions.
• Average time between collisions is τc (also called the mean free time).

Concept / Approach:In the Drude model, momentum relaxation is described statistically by an exponential decay of drift velocity with time constant τ. If collisions randomize momentum completely and occur as a Poisson process, the mean free time τc equals the relaxation time τ. This leads directly to standard relations for mobility μ and conductivity σ.

Step-by-Step Solution:Define mobility: μ = e * τ / m, where e is charge magnitude and m effective mass.Relate conductivity: σ = n * e^2 * τ / m with carrier density n.With isotropic, memoryless scattering, τ equals the average inter-collision time τc.Hence, the correct relation is τ = τc.

Verification / Alternative check:Kinetic theory gives mean free path ℓ = v̄ * τ, where v̄ is average electron speed. Experimental σ and Hall measurements yield τ that matches τc inferred from ℓ and v̄, validating τ = τc in simple metals at moderate temperatures.

Why Other Options Are Wrong:Statements like τ < τc or τc < τ imply systematic momentum relaxation faster/slower than collisions without justification; “T = 0.01 τe” is unrelated; τ = 2 τc is not a general result.

Common Pitfalls:Confusing momentum relaxation time with energy relaxation time (they can differ in semiconductors with specific scattering mechanisms). Here, the question concerns the basic isotropic Drude picture.

Final Answer:τ = τc