2.1. Some Basic Concepts and Terminologies 69
on the type of cross section used in the calculation. For example, if scattering cross
section is used then the mean free path would correspond to just the scattering
process. However in most instances, such as when calculating the shielding required
in a radiation environment, one uses total cross section, which gives the total mean
free path.
The mean free path has a dependence on the energy distribution of the particles
relative to the medium. For particles that can be described by the Maxwellian
distribution^3 , such as thermal neutrons in a gas under standard conditions, the
mean free path can be computed from
λm=1
√
2 ρnσ. (2.1.7)
In all other cases, the mean free path should be estimated from
λm=1
ρnσ. (2.1.8)
The number densityρnin the above relations can be computed for any material
from
ρn=NAρ
A, (2.1.9)
whereAis the atomic weight of the material,NAis the Avogadro’s number, andρ
is theweightdensity of the material. Mean free path is usually quoted incm,for
which we must takeAing/mole,NAinmole−^1 ,ρing/cm^3 ,andσincm^2.
For a gas, the weight density term in the above relation can be computed from
ρ=PA
RT
, (2.1.10)
wherePandTrepresent respectively the pressure and temperature of the gas and
R=8. 314 Jmole−^1 K−^1 is the usual gas constant. Hence for a gas, the mean free
path can be computed from the following equations.
λm =RT
√
2 NAPσ(for Maxwellian distributed energy) (2.1.11)λm =RT
NAPσ(for non-Maxwellian distributed energy) (2.1.12)It is apparent from these relations that the mean free path of a particle in a gas can
be changed by changing the gas temperature and pressure (see example below).
The mean free path as defined above depends on the density of the medium.
This poses a problem not only for reporting the experimental results but also for
using the values in computer codes for systems whose density might change with
time. Therefore derivative of this term has been defined that does not depend on
the density. This is the so calledspecific mean free pathand is given by
λp=A
NAσ. (2.1.13)
(^3) The Maxwellian distribution describes how the velocities of molecules in a gas are distributedat equi-
libriumas a function of temperature.