# 5. Scattering in classical and in quantum mechanics¶

Scattering experiments are perhaps the most important tool for obtaining detailed information on the structure of matter, in particular the interaction between particles. Examples of scattering techniques include neutron and X-ray scattering for liquids, atoms scattering from crystal surfaces, elementary particle collisions in accelerators. In most of these scattering experiments, a beam of incident particles hits a target which also consists of many particles. The distribution of scattering particles over the different directions is then measured, for different energies of the incident particles. This distribution is the result of many individual scattering events. Quantum mechanics enables us, in principle, to evaluate for an individual event the probabilities for the incident particles to be scattered off in different directions; and this probability is identified with the measured distribution.

Suppose we have an idea of what the potential between the particles
involved in the scattering process might look like, for example from
quantum mechanical energy calculations (programs for this purpose will
be discussed in the next few chapters). We can then *parametrise* the
interaction potential, i.e. we write it as an analytic expression
involving a set of constants: the parameters. If we evaluate the
scattering probability as a function of the scattering angles for
different values of these parameters, and compare the results with
experimental scattering data, we can find those parameter values for
which the agreement between theory and experiment is optimal. Of course,
it would be nice if we could evaluate the scattering potential directly
from the scattering data (this is called the *inverse problem*), but
this is unfortunately very difficult (if not impossible) as many
different interaction potentials can have similar scattering properties
as we shall see below.

Many different motivations for obtaining accurate interaction potentials can be given. One is that we might use the interaction potential to make predictions about the behaviour of a system consisting of many interacting particles, such as a dense gas or a liquid.

Scattering might be *elastic* or *inelastic*. In the former case the
energy is conserved, in the latter energy disappears. This means that
energy transfer takes place from the scattered particles to degrees of
freedom which are not included explicitly in the system (inclusion of
these degrees of freedom would cause the energy to be conserved). In
this chapter we shall consider elastic scattering.

## 5.1. Classical analysis of scattering¶

A well known problem in classical mechanics is that of the motion of two
bodies attracting each other by a gravitational force whose value decays
with increasing separation as . This analysis is also correct
for opposite charges which feel an attractive force of the same form
(Coulomb's law). When the force is repulsive, the solution remains the
same -- we only have to change the sign of the parameter which
defines the interaction potential according to . One of the
key experiments in physics which led to the notion that atoms consist of
small but heavy kernels, surrounded by a cloud of light electrons, is
*Rutherford scattering*. In this experiment, a thin gold sheet was
bombarded with -particles (i.e. helium-4 nuclei) and the
scattering of the latter was analysed using detectors behind the gold
film. In this section, we shall first formulate some new quantities for
describing scattering processes and then calculate those quantities for
the case of Rutherford scattering.

Rutherford scattering is chosen as an example here -- scattering problems can be studied more generally; see Griffiths, chapter 11, section 11.1.1 for a nice description of classical scattering.

We consider scattering of particles incident on a so-called 'scattering
centre', which may be another particle. The scattering centre is
supposed to be at rest. This might not always justified in a real
experiment, but in a standard approach in classical mechanics, the full
two-body problem is reduced to a one-body problem with with a reduced
mass, which is the present case (in
problem 5.1 we will perform the same procedure for a
quantum two-body system). The incident particles interact with the
scattering centre located at by the usual scalar two-point potential which satisfies
the requirements of Newton's third law. Suppose we have a beam of
incident particles parallel to the -axis. The beam has a homogeneous
density close to that axis, and we can define a *flux*, which is the
number of particles passing a unit area perpendicular to the beam, per
unit time. Usually, particles close to the -axis will be scattered
more strongly than particles far from the -axis, as the interaction
potential between the incident particles and scattering centre falls off
with their separation . An experimentalist cannot analyse the
detailed orbits of the individual particles -- instead a detector is
placed at a large distance from the scattering centre and this detector
counts the number of particles arriving at each position. You may think
of this detector as a photographic plate which changes colour to an
extent related to the number of particles hitting it. The theorist wants
to predict what the experimentalist measures, starting from the
interaction potential which governs the interaction process.

In figure 5.1, the geometry of the process is shown. In addition
a small cone, spanned by the spherical polar angles and
, is displayed. It is assumed here that the scattering takes
place in a small neighbourhood of the scattering centre, and for the
detector the orbits of the scattered particles all seem to be directed
radially outward from the scattering centre. The surface of the
intersection of the cone with a sphere of radius around the
scattering centre is given by
. The quantity
is called *spatial angle* and is
usually denoted by . This defines a cone like the one
shown in figure 5.1. Now consider the number of particles which will
hit the detector within this small area per unit time. This number,
divided by the total incident flux (see above) is called the
*differential scattering cross section*, :
The differential cross section
has the dimension of area (length).

First we realise ourselves that the problem is symmetric with respect to
rotations around the -axis, so the differential scattering cross
section only depends on . The only two relevant parameters of
the incoming particle then are its velocity and its distance from
the -axis. This distance is called the *impact parameter* -- it is
also shown in figure 5.1.

We first calculate the scattering angle as a function of the
impact parameter . We perform this calculation for the example of
Rutherford scattering, for which we have the standard Kepler solution
which is now a hyperbola (see your classical mechanics lecture course).
The potential for the Kepler problem is . The orbits are
given by specifying , . However, for the Kepler
problem it is more convenient to focus on the *shape* of the orbitals,
which is given as
with
this parameter is
called *eccentricity* -- for a hyperbola, we have . Here,
is the angular momentum and the reduced mass. The
integration constant reappears in the cosine because we have not
chosen at the perihelion -- the closest approach occurs
when the particle crosses the dashed line in
figure 5.1 which bisects the in- and outgoing particle
direction.

We know that for the incoming particles, for which , , we have Because of the fact that cosine is even we can infer that the other value of for which goes to infinity, and which corresponds to the outgoing direction occurs when the argument of the cosine is , so that we find or . The subscript indicates that this value corresponds to . From the last two equations we find the following relation between the scattering angle and :

We want to know as a function of rather than however. To this end we note that the angular momentum is given as where 'inc' stands for 'incident', and the total energy as so that the impact parameter can be found as Using Eq. \eqref{Eq:KepHelp} and the fact that , we can finally write \eqref{Eq:ThetaEps} in the form:

From the relation between and we can find the
differential scattering cross section. The particles scattered with
angle between and , must have
approached the scattering centre with impact parameters between
particular boundaries and . The number of particles flowing
per unit area through the ring segment with radius and width is
given as , where is the incident flux. We consider a
segment of this ring. Hence:
Relation
\eqref{Eq:BTheta} can be used to express the right hand side in
terms of :
This can be worked out straightforwardly to
yield: This is known as the famous *Rutherford
formula*.