The classical scattering cross section. Rutherford scattering.

1. The “scattering cross section”
in Classical Mechanics.

2. A seminar in the Department of Physics,
Dhenkanal Autonomous College.
Audience: Physics Honors Students and Faculty.

3.  The scattering cross sections are the most
important parameters in many branches of
Physics.
 Knowledge of nuclear and particle physics
are a result of scattering experiments, the
most famous is Rutherford scattering.
 Solid state systems employ scattering
phenomena as the most important tool in
determination of the underlying physics.
 We will discuss the most basic form of
scattering in Classical Mechanics.

4.  Central forces are found everywhere. It was
first used in understanding Astronomical
observations, viz, planetary motions.
 Bohr atomic model is based on assumptions
of central force.
 Scattering of elementary particles such as
neutron or electrons can be considered as
effected by central forces according to
classical results. Although not completely
correct these results are valid to good
approximations.

5.  The scattering of particles is effected when
they are incident on a center of force. The
equivalent motion of a 2-body system is
applicable here.
 The first assumption is that the force falls off
to zero, at sufficiently large distance from
center of force.
 We define a parameter called “Intensity I or
flux density”. It’s the number of particles
crossing an unit area normal to the incident
beam in unit time.

6.  Accordingly the direction of the particle long
before it interacts with the center of force, and
long after the interaction is straight line. But the
initial direction is not same as final direction in
general.
 We define the differential scattering cross
section in a given direction: ()d  = number
of particles scattered into solid angle d per
unit time, per unit incident intensity.
 Due to central force nature there is symmetry
about angle  around incident axis. Hence solid
angle d is given by: d=2 sin  d.

7.  is the angle between the incident and scattered
direction. Its known as “scattering angle”.
s is the “impact parameter”. The particle in the range of
impact parameter s to s + ds is scattered into the angle  to
 + d.

8.  Energy and angular momentum are the
constants of motion of any trajectory.
Scattering of particle is easily described in
terms of these two parameters.
 Angular momentum is easily described in
terms of another parameter called impact
parameter which we indicated in last slide.
 Impact parameter is defined as
perpendicular distance between center of
force and incident velocity. Thus it’s the
distance an undeflected particle would make
with center of force (c.o.f).

9.  l = mv0s = ssqrt(2mE), l is angular
momentum.
 Since number of particles coming within
range s to s+ds is same as particles scattered
within solid angle d given by scattering
angle  and  +d. 2Is|ds| = 2 ()|d |
 The important thing to remember now is to
determine impact parameter s of any
problem as a function of scattering angle 
and energy E.

10.  With s = s (, E)
differential cross
section is now
given as:
 Orbit is symmetric
about direction of
periapsis.
Remember that
periapsis is
minimum distance
of trajectory from
center of force.

11.  We can thus write
the scattering angle
 as a function of
impact parameter s
by using results for
a hyperbolic orbit.
With =-2;
 With the use of
relationship
between l and s
that we established
earlier and r = 1/u;

12.  There is a repulsive
scattering of charged
particle by Coulomb’s
force. The force and
force constants are
given by:
 The E > 0 and orbit is
hyperbolic whose
eccentricity  is given
by:
 The orbit equation
can be written as :

13.  Direction of
incoming asymptote
() is given by the
condition r  ,
which leads to;
 Thus we have a
relation between
impact parameter s,
energy E and
scattering angle .

14.  All this analysis leads
to establishment of
the classical
Rutherford
differential scattering
cross section:
 The total cross
section is then the
differential amount
integrated over all
solid angle:

15.  For Coulomb interaction total cross section
blows to infinity. This is because such an
interaction is very long range and deflections
of particles howsoever small is present.
 Total cross section is finite if the effect of
nucleus is cut off somehow at longer
distances. Electrons “screen” the force of
nucleus and cancel its charge out of nucleus.
 Cross section goes to zero at both s = 0 and s
= infinity. For values of s in-between,
scattering angle  passes through maximum
m.

16.  When scattering angle  is less than m two
values of impact parameter s gives rise to
same scattering angle . This is
accommodated by taking both values of
scattering angle.
 At  = m there is infinite cross section,
because derivative of  wrt s vanishes. But
at angles above the maximum scattering
angle the cross section has to be zero, this
is known as Rainbow scattering.

17.  There are many more
interesting properties
of the cross section
formula we derived
eg for attractive
potential it gives rise
to something called
glory scattering.
 Here is how well the
classical cross section
fits with real world
problems.

18.  The cross section
predicted classically
by Rutherford,
however, departs
from reality at
higher energy.
 We need Quantum
Mechanical cross
sectional formula to
be able to deal with
such behavior.

19.  It is a good idea to stop here for brevity. One
needs to study the much more nuanced
terrain of quantum mechanical scattering
problems to be able to describe nature in its
complete glory.