A.30 Three-dimensional scattering

This note introduces some of the general concepts of three-dimensional scattering, in case you run into them. For more details and actual examples, a quantum mechanics text for physicists will need to be consulted; it is a big thing for them.

The basic idea is as sketched in figure A.21. A beam of particles is sent in from the far left towards a three-dimensional target. Part of the beam hits the target and is scattered, to be picked up by surrounding detection equipment.

It will be assumed that the collision with the target is elastic, and that the particles in the beam are sufficiently light that they scatter off the target without transferring kinetic energy to it. In that case, the target can be modeled as a steady potential energy field. And if the target and/or incoming particles are electrically neutral, it can also be assumed that the potential energy decays fairly quickly to zero away from the target. (In fact, a lot of results in this note turn out not apply to a slowly decaying potential like the Coulomb one.)

It is convenient to use a spherical coordinate system

In the energy eigenfunctions, the incoming particle beam can be
represented as a one-dimensional wave. However, unlike for the one-dimensional
scattering of figure 7.22, now the wave is not just
scattered to the left and right, but in all directions, in other words
to all angles

The second term describes the outgoing scattered particles. The
constant

Finally, the second term contains a factor 1/

Consider now the number of particles that is detected in a given small
detection area

The constant includes the factor

According to the above expression, the number of particles detected in
a given area

This is the so-called “solid angle” occupied by the detection area element. It is the three-dimensional generalization of two-dimensional angles. In two dimensions, an element of a circle with arc length

In those terms, the number

As noted, the constant of proportionality depends on the rate at which
particles are sent at the target. The more particles are sent at the
target, the more will be deflected. The number of particles in the
incoming beam per unit beam cross-sectional area and per unit time is
called the “luminosity” of the beam. It is related to the square of the
wave function of the incoming beam through the relation

Here

Physicist like to relate the scattered particle flow in a given
infinitesimal solid angle

(A.217) |

Note how well chosen the term “differential
cross-section” really is. If physicists had called it
something like scattered cross-section density,

or
even simply scattered cross-section,

nonexperts would
probably have a pretty good guess what physicists were talking about.
But cross section

by itself can mean anything. There
is nothing in the term to indicate that it is a measure for how many
particles are scattered. And preceding it by
differential

is a stroke of genius because it is not a
differential, it is a differential quotient. This will confuse
mathematically literate nonexperts even more.

The differential cross section does not depend on how many particles
are sent at the target, nor on wave function normalization. Following
the expressions for the particle flows given above, the differential
cross section is simply

The total area of the incoming beam that gets scattered is called the
“total cross-section”

(A.219) |

If you remain disappointed in physicists, take some comfort in the
following term for scattering that can be described using classical
mechanics: the “impact parameter.” If you guess that it describes the local
physics of the particle impact process, it is really hilarious to
physicists. Instead, think centerline offset;

it
describes the location relative to the centerline of the incoming beam
at which the particles come in; it has no direct relation whatsoever
to what sort of impact (if any) these particles end up experiencing.

The total cross section can be found by integrating the differential
cross section over all deflection angles:

In spherical coordinates this can be written out explicitly as

A.30.1 Partial wave analysis

Jim Napolitano from RPI and Cornell notes:

Gee, what a surprise! For one, they are component waves, not partial waves. But you already componently assumed that they might be.The termPartial Wave Analysisis poorly defined and overused.

This discussion will restrict itself to spherically symmetric scattering potentials. In that case, the analysis of the energy eigenfunctions can be done much like the analysis of the hydrogen atom of chapter 4.3. However, the boundary conditions at infinity will be quite different; the objective is not to describe bound particles, but particles that come in from infinity with positive kinetic energy and are scattered back to infinity. Also, the potential will of course not normally be a Coulomb one.

But just like for the hydrogen atom, the energy eigenfunctions can be
taken to be radial functions times spherical harmonics

(A.221) |

The incoming plane wave

(A.222) |

spherical Bessel functions

Now finding the complete energy eigenfunction corresponding to the
incoming wave directly is typically awkward, especially analytically.
Often it is easier to solve the problem for each term in the above sum
separately and then add these solutions all together. That is where
the name partial wave analysis

comes from. Each term
in the sum corresponds to a partial wave, if you use sufficiently
lousy terminology.

The partial wave analysis requires that for each term in the sum, an
energy eigenfunction is found of the form

Note that the above far field behavior is quite similar to that of the
complete energy eigenfunction as given earlier in (A.216).
However, here the coefficient Hankel function of the first kind

To be sure, for a slowly decaying potential like the Coulomb one, the
Hankel function is no better than the exponential. However, the
Hankel function is very closely related to the Bessel function

In terms of the asymptotic behavior above, the differential cross
section is

(A.224) |

One special case is worth mentioning. Consider particles of such low
momentum that their typical quantum wave length,
notices

the incoming partial wave with

for small arguments. If the wave length is large compared to the typical radial size

Coincidently, equal scattering in all directions also happens in another case: scattering of classical point particles from a hard elastic sphere. That is very much the opposite case, because negligible uncertainty in position requires high, not low, energy of the particles. In any case, the similarity between the two cases is is superficial. If a beam of classical particles is directed at a hard sphere, only an area of the beam equal to the frontal area of the sphere gets scattered. But if you work out the scattering of low-energy quantum particles from a hard sphere, you get a total scattering cross section that is 4 times bigger.

A.30.2 Partial wave amplitude

This subsection gives some further odds and ends on partial wave analysis, for the incurably curious.

Recall that a partial wave has an asymptotic behavior

The first term corresponds to the wave function of the incoming particles. The second term is the effect of the scattering potential.

Physicists like to write the coefficient of the scattered wave as

(A.225) |

Now every partial wave by itself is a solution to the Hamiltonian
eigenvalue problem. That means that every partial wave must ensure
that particles cannot just simply disappear. That restricts what the
partial wave amplitude can be. It turns out that it can be written in
terms of a real number

(A.226) |

Some physicist must have got confused here, because it really is a
phase shift. To see that, consider the derivation of the above
result. First the asymptotic behavior of the partial wave is
rewritten in terms of exponentials using {A.6}
(A.24) and (A.25). That gives

The dots stand for common factors that are not important for the discussion. Physically, the first term above describes spherical wave packets that move radially inwards toward the target. The second term describes wave packets that move radially outwards away from the target.

Now particles cannot just disappear. Wave packets that go in towards the target must come out again with the same amplitude. And that means that the two terms in the asymptotic behavior above must have the same magnitude. (This may also be shown mathematically using procedures like in {A.32}.)

Obviously the two terms do have the same magnitude in the absence of
scattering, where

(A.227) |

If you add in the time dependent factor phase angle

has been
ostracized from physics. She will never be heard of again.

A.30.3 The Born approximation

The Born approximation assumes that the scattering potential is weak to derive approximate expressions for the scattering.

Consider first the case that the scattering potential is zero. In
that case, the wave function is just that of the incoming particles:

where

Born considered the case that the scattering potential

In particular, inside the integral the true wave function

It is not exact, but it is much better than just setting the integral to zero. The latter would make the wave function equal to the incoming wave. With the approximate integral, you get a valid approximation to the particle deflections.

To get the differential cross section, examine the behavior of
(A.228) at given scattering angles

(A.230) |

One additional approximation is worth mentioning. Consider particles
of such low momentum that their quantum wave length,

Note that the integral is infinite for a Coulomb potential.