What is the diffusion length L_d and how is it defined in terms of D and Σ_a?

• A. L_d = D Σ_a; characterizes distance neutrons diffuse before absorption.
• B. L_d = sqrt(D / Σ_a); it characterizes the distance neutrons diffuse before absorption.
• C. L_d = sqrt(Σ_a / D); it characterizes the distance neutrons diffuse before absorption.
• D. L_d = sqrt(D × Σ_a); it characterizes the distance neutrons diffuse before absorption.

Answer: (B)

Neutrons diffuse a characteristic distance before being absorbed, and that distance is called the diffusion length. In the diffusion approximation for a uniform medium, the steady-state equation -D d^2φ/dx^2 + Σ_a φ = 0 leads to a decaying flux φ ~ e^{-x/L_d}. Matching the form gives L_d^2 = D/Σ_a, so the diffusion length is L_d = sqrt(D/Σ_a).

This length combines how far neutrons can spread (D) with how likely they are to be absorbed (Σ_a). A larger diffusion coefficient means neutrons spread farther before absorption, increasing L_d, while a larger absorption cross-section reduces L_d.

Other proposed forms don’t yield a true distance scale for diffusion. For example, D Σ_a has wrong units for a length, and the other radical combinations don’t produce the correct length dimension or the correct dependence on D and Σ_a. Only the square root of D divided by Σ_a gives the appropriate diffusion distance.