Mean free path

In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a result of one or more successive collisions with other particles.

Scattering theory

Imagine a beam of particles being shot through a target, and consider an infinitesimally thin slab of the target (see the figure). The atoms (or particles) that might stop a beam particle are shown in red. The magnitude of the mean free path depends on the characteristics of the system. Assuming that all the target particles are at rest but only the beam particle is moving, that gives an expression for the mean free path:

ℓ = ( σ n ) − 1 , {\displaystyle \ell =(\sigma n)^{-1},}

where ℓ is the mean free path, n is the number of target particles per unit volume, and σ is the effective cross-sectional area for collision per target particle. The area of the slab is L2, and its volume is L2 dx. The typical number of stopping atoms in the slab is the concentration n times the volume, i.e., n L2 dx. The probability that a beam particle will be stopped in that slab is the net area of the stopping atoms divided by the total area of the slab:

P ( stopping within d x ) = Area atoms Area slab = σ n L 2 d x L 2 = n σ d x , {\displaystyle {\mathcal {P}}({\text{stopping within }}dx)={\frac {{\text{Area}}_{\text{atoms}}}{{\text{Area}}_{\text{slab}}}}={\frac {\sigma nL^{2}\,dx}{L^{2}}}=n\sigma \,dx,}

where σ is the cross-sectional area (or, more formally, the "scattering cross-section") of one atom. The drop in beam intensity equals the incoming beam intensity multiplied by the probability of the particle being stopped within the slab:

d I = − I n σ d x . {\displaystyle dI=-In\sigma \,dx.}

This is an ordinary differential equation:

d I d x = − I n σ = def − I ℓ , {\displaystyle {\frac {dI}{dx}}=-In\sigma {\overset {\text{def}}{=}}-{\frac {I}{\ell }},}

whose solution is known as the Beer–Lambert law and has the form I = I 0 e − x / ℓ {\displaystyle I=I_{0}e^{-x/\ell }} , where x is the distance traveled by the beam through the target, and I0 is the beam intensity before it entered the target; ℓ is called the mean free path because it equals the mean distance traveled by a beam particle before being stopped. To see this, note that the probability that a particle is absorbed between x and x + dx is given by

d P ( x ) = I ( x ) − I ( x + d x ) I 0 = 1 ℓ e − x / ℓ d x . {\displaystyle d{\mathcal {P}}(x)={\frac {I(x)-I(x+dx)}{I_{0}}}={\frac {1}{\ell }}e^{-x/\ell }dx.}

Thus the expectation value (or average, or simply mean) of x is

⟨ x ⟩ = def ∫ 0 ∞ x d P ( x ) = ∫ 0 ∞ x ℓ e − x / ℓ d x = ℓ . {\displaystyle \langle x\rangle {\overset {\text{def}}{=}}\int _{0}^{\infty }xd{\mathcal {P}}(x)=\int _{0}^{\infty }{\frac {x}{\ell }}e^{-x/\ell }\,dx=\ell .}

The fraction of particles that are not stopped (attenuated) by the slab is called the transmission T = I / I 0 = e − x / ℓ {\displaystyle T=I/I_{0}=e^{-x/\ell }} , where x is equal to the thickness of the slab.

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