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 with its origin at the scattering object and with its axis aligned with the direction of the incoming beam. Since the axis of a spherical coordinate system is usually called the -axis, the horizontal coordinate will now be indicated as , not like in the one-dimensional analysis done earlier.

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 and . The far-field behavior of
the energy eigenfunctions is

The second term describes the outgoing scattered particles. The constant is called the “scattering amplitude.” The second term also contains a factor consistent with wave packets that move radially away from the target in the far field.

Finally, the second term contains a factor 1/. Therefore the magnitude of the second term decreases with the distance from the target. This happens because the probability of finding a particle in a given detection area should decrease with distance. Indeed, the total detection area is , where is the distance at which the detectors are located. That increases proportional to , and the total number of particles to detect per unit time is the same regardless of where the detectors are located. Therefore the probability of finding a particle per unit area should decrease proportional to 1/. Since the probability of finding a particle is proportional to the square of the wave function, the wave function itself must be proportional to 1/. The second term above makes it so.

Consider now the number of particles that is detected in a given small
detection area . The scattered stream of particles
moving towards the detection area has a velocity
. Therefore in a time interval ,
the detection area samples a volume of the scattered particle stream
equal to . The chances of finding
particles are proportional to the square magnitude of the wave
function times that volume. Using the asymptotic wave function above,
the number of particles detected will be

The constant includes the factor . The constant must also account for the fact that the wave function is not normalized, and that there is a continuous stream of particles to be found, rather than just one particle.

According to the above expression, the number of particles detected in
a given area is proportional to its three-dimensional angular
extent

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 occupies an angle when expressed in radians. Similarly, in three dimensions, an element of a sphere with area occupies a solid angle when expressed in “steradians.”

In those terms, the number of particles detected in an
infinitesimal solid angle is

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 is a cross sectional area element of the incoming particle beam and the number of particles passing through that area.

Physicist like to relate the scattered particle flow in a given
infinitesimal solid angle to an equivalent incoming beam
area through which the same number of particles flow.
Therefore they define the so-called “differential cross-section” as

(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 has zero angular
momentum in the -direction. Unfortunately, it does not have
definite square angular momentum. Instead, it can be written as a
linear combination of free-space energy eigenfunctions with different
values of , hence with different square angular momentum:

(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
. The required behavior of this eigenfunction in
the far field is

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 was set to 1 for
simplicity. Also, the radial part of the reflected wave function was
written using a Hankel function of the first kind

. This Hankel function produces the same
radial behavior as the second term in
(A.216), {A.6} (A.25).
However, the Hankel function has the advantage that it becomes exact
as soon as the scattering potential becomes zero. It is not just
valid at very large like the bare exponential.

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 , {A.6}, allowing various helpful results to be found in table books. If the potential energy is piecewise constant, it is even possible to solve the complete problem using Bessel and Hankel functions. These functions can be tied together at the jumps in potential in a way similar to addendum {A.27}.

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,
, is gigantic compared to the radial size
of the scattering potential. Particles of such large wave lengths do
not notice the fine details of the scattering potential at all.
Conversely, normally the scattering potential only
notices

the incoming partial wave with 0.
That is because the Bessel functions are proportional to

for small arguments. If the wave length is large compared to the typical radial size of the scattering potential, this is negligible unless 0. Now 0 corresponds to a wave function that is the same in all directions; it is proportional to the constant spherical harmonic . If only the partial wave that is the same in all directions gets scattered, then the particles get scattered equally in all directions (if they get scattered at all.)

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 is zero. But in the presence of scattering,
the final parenthetical factor will have to stay of magnitude one.
And that means that it can be written in the form

(A.227) |

If you add in the time dependent factor of the
complete unsteady wave function, you can see that indeed the waves are
shifted by a phase angle compared to the unperturbed wave
function. Without any doubt, the name of the physicist responsible
for calling the phase angle a 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 is the energy of the particle and its mass.

Born considered the case that the scattering potential is not
zero, but small. Then the wave function will still be close
to the incoming wave function, but no longer exactly the same. In
that case an approximation to the wave function can be obtained from
the so-called integral Schrödinger equation, {A.13}
(A.42):

In particular, inside the integral the true wave function can be replaced by the incoming 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 and for
large . That produces, {D.47}:

(A.230) |

One additional approximation is worth mentioning. Consider particles
of such low momentum that their quantum wave length,
, is gigantic compared to the radial size
of the scattering potential. Particles of such wave lengths do not
notice the fine details of the scattering potential at all.
Mathematically, is so small that the argument of the
exponential in the differential cross section above can be assumed
zero. Then:

Note that the integral is infinite for a Coulomb potential.