Charge Transport – Relaxation Time vs. Mean Time Between Collisions Assertion (A): The relaxation time τ equals the mean time between collisions Tc when scattering of carriers is isotropic. Reason (R): The mean free path λ, carrier speed v, and Tc are related by λ = v Tc.

Introduction / Context:
In semiclassical transport, mobile carriers undergo random scattering events. Two characteristic times appear: Tc, the average interval between successive collisions, and τ, the momentum relaxation time that governs how quickly drift momentum decays toward equilibrium. Their equality is not guaranteed in general; it depends on the angular distribution of scattering events.

Given Data / Assumptions:

• Isotropic (angle-independent) scattering.
• Low-field transport in a parabolic band (Drude-like model).
• Steady-state small-signal regime (linear response).

Concept / Approach:

Momentum relaxation accounts for how effectively each collision randomizes the carrier momentum. For isotropic scattering, each collision, on average, reverses momentum components symmetrically, causing the average drift momentum to decay with the same time constant as the collision rate: τ ≈ Tc. For forward-peaked scattering, τ > Tc because many collisions barely alter momentum; for large-angle–dominated scattering, τ ≈ Tc. The kinematic relation λ = v Tc is always true by definition of mean free path but does not by itself explain why τ equals Tc; the explanation requires angular weighting of post-collision momentum (1 − cos θ) in the transport integral.

Step-by-Step Solution:

Verification / Alternative check:

Boltzmann transport with relaxation-time approximation shows 1/τ = ∫ W(θ) (1 − cos θ) dΩ; for constant W(θ), integral yields 1/Tc, confirming τ = Tc.

Why Other Options Are Wrong:

• A true, R false: R is true as a definition of mean free path.
• A false, R true: τ equals Tc under isotropic scattering; A is not false.
• True/true with correct explanation: R does not explain A; it lacks the angular-momentum argument.

Common Pitfalls:

Equating τ and Tc in all contexts; overlooking that forward-scattering dominance increases τ relative to Tc.

Final Answer:

Both A and R are true but R is not correct explanation of A