# 5.4. Partial wave analysis¶

The scattering process is described by the solutions of the
single-particle Schrödinger equation involving the (reduced) mass ,
the relative coordinate and the interaction potential
between the particle and the interaction centre:
This is a partial differential equation in three
dimensions, which could be solved using the 'brute force' discretisation
methods, but exploiting the spherical symmetry of the potential, we can
solve the problem in another, more elegant, way which, moreover, works
much faster on a computer. More specifically, in
Section 5.5 we shall establish a relation between the
*phase shift* and the scattering cross sections. In this section, we
shall restrict ourselves to a description of the concept of phase shift
and describe how it can be obtained from the solutions of the radial
Schrödinger equation.

For the potential we make the assumption that it vanishes for larger than a certain value . In case we are dealing with an asymptotically decaying potential, we neglect contributions from the potential beyond the range , which must be chosen suitably, or treat the tail in a perturbative manner, using the Born approximation.

For a spherically symmetric potential, the solution of the Schrödinger equation can always be written as where satisfies the radial Schrödinger equation:

Figure 5.2 shows the solution of the radial Schrödinger equation with for a square well potential for various well depths -- our discussion applies also to nonzero values of .

Outside the well, the solution can be written as a linear combination of the two independent solutions and , the regular and irregular spherical Bessel functions. We write this linear combination in the particular form is determined via a matching procedure at the well boundary. The motivation for writing in this form follows from the asymptotic expansion for the spherical Bessel functions:

which can be used to rewrite \eqref{Eq:LinComb} as We see that approaches a sine-wave form for large and the phase of this wave is determined by , hence the name 'phase shift' for (for is a sine wave for all ).

The phase shift as a function of energy and contains all the information about the scattering properties of the potential. In particular, the phase shift enables us to calculate the scattering cross sections and this will be done in section 5.5; here we simply quote the results. The differential cross section is given in terms of the phase shift by and for the total cross section we find

Summarising the analysis up to this point, we see that the potential determines the phase shift through the solution of the Schrödinger equation for . The phase shift acts as an intermediate object between the interaction potential and the experimental scattering cross sections, as the latter can be determined from it.

Unfortunately, the expressions \eqref{Eq:DiffCrSec} and \eqref{sigmatotdef} contain sums over an infinite number of terms -- hence they cannot be evaluated on the computer exactly. However, cutting off these sums can be motivated by a physical argument. Classically, only the waves with an angular momentum smaller than will 'feel' the potential -- particles with higher -values will pass by unaffected. Therefore we can safely cut off the sums at a somewhat higher value of -- we can always check whether the results obtained change significantly when taking more terms into account. We shall frequently encounter procedures similar to the cutting off described here. It is the art of computational physics to cleverly reduce infinite problems to ones which fit into the computer and still provide a reliable description.

How is the phase shift determined in practice? First, the Schrödinger equation must be integrated from outwards with boundary condition . At , the numerical solution must be matched onto the form \eqref{Eq:LinComb} to fix . The matching can be done either via the logarithmic derivative or using the value of the numerical solution at two different points and beyond and we will use the latter method in order to avoid calculating derivatives. From \eqref{Eq:LinComb} it follows directly that the phase shift is given by In this equation, stands for etc.

A computer program similar to the one described here was used by
Toennies *et al.* (J. Chem. Phys., **71**, p. 614, 1979). to compare the
results of scattering experiments with theory. The experiment consisted
of the bombardment of krypton atoms with hydrogen atoms.
Figure 5.3 shows the Lennard-Jones interaction potential plus
the centrifugal barrier of the radial Schrödinger equation.

For higher -values, the potential consists essentially of a hard core, a well and a barrier which is caused by the centrifugal term in the Schrödinger equation. In such a potential, quasi-bound states are possible. These are states which would be genuine bound states for a potential for which the barrier does not drop to zero for larger values of , but remains at its maximum height. You can imagine the following to happen when a particle is injected into the potential at precisely this energy: it tunnels through the barrier, remains in the well for a relatively long time, and then tunnels outward through the barrier in an arbitrary direction because it has 'forgotten' its original direction. In wave-like terms, the particle resonates in the well, and this state decays after a relatively long time. This phenomenon is called 'scattering resonance'. This means that particles injected at this energy are strongly scattered and this shows up as a peak in the total cross section.

Such peaks can be seen Figure 5.4, which shows the total cross section as a function of the energy calculated with a program as described above. The peaks are due to , and scattering, with energies increasing with .

Figure 5.5 finally shows the experimental results for the total cross section for H--Kr. We see that the agreement is excellent.

You should be able now to reproduce the data of Figure 5.4 with your program.