68 Chapter 2. Interaction of Radiation with Matter
Schroedinger and Heisenberg to lay the foundations of quantum mechanics. Due to
the intrinsic uncertainties in finding particle properties quantum mechanics naturally
developed into a probabilistic theory. We talk about the probability of finding a
particle at a certain position but never claim to know its exact position with absolute
certainty.
So how does quantum mechanics treat particle interactions? That’s where the
concept of cross section comes in. It tells us how likely it is for a particle to interact
with another one in a certain way. Mathematically speaking it has dimensions of
area but conceptually it represents probability of interaction.
Cross section is perhaps the most quoted parameter in the fields of particle physics
and radiation measurement because it gives a direct measure of what to expect from
a certain beam of particles when it interacts with a material. Let us now see how
this concept is defined mathematically.
Suppose we have a beam of particles with a flux Φ (number of particles per unit
area per unit time) incident on a target. After interacting with the target, some of
the particles in the incident beam get scattered. Suppose that we have a detector,
which is able to count the average number of particles per unit time (dN)thatget
scattered per unit solid angle (dΩ). This average quantity divided by the flux of
incident particles is defined as the differential cross section.
dσ
dΩ(E,Ω) =
1
Φ
dN
dΩ(2.1.4)
It is apparent from this equation that the cross sectionσhas dimension of area.
This fact has influenced some authors to define it as a quantity that represents
the area to which the incident particle is exposed. The larger this area, the more
probable it will be for the incident particle to interact with the target particle.
However, it should be noted that this explanation of cross section is not based on
any physical principle, rather devised artificially to explain a mathematical identity
using its dimensions.
The differential cross section can be integrated to evaluate the total cross section
at a certain energy (differential cross section is a function of energy of the incident
particles), that is
σ(E)=∫
dσ
dΩdΩ. (2.1.5)The conventional unit of cross section is barn (b)with1b=10−^24 cm^2.
2.1.C MeanFreePath
As particles pass through material, they undergo collisions that may change their
direction of motion. The average distance between these collisions is therefore a
measure of the probability of a particular interaction. This distance, generally known
as themean free path, is inversely proportional to the cross section and the density
of the material, that is
λm∝1
ρnσ, (2.1.6)
whereρnandσrepresent thenumberdensity of the medium and cross section of
the particle in that medium. Note that the definition of the mean free path depends