Re: [Phys-l] Electron vs. Alpha particle...

[Hugh Haskell <haskellh@verizon.net>]

At 15:38 -0700 01/12/2011, Jeff Loats wrote:
> In discussing Rutherford scattering I ask students to use the simple case of
> an alpha particle colliding head on with an electron at rest. The idea is to
> use conservation of energy and momentum to show that in a classical
> "billiard ball" model, the alpha particle can ignore electrons in its path
> to a good approximation.
>
> This term a curious student asked some great questions about what would
> happen when such a collision took place.
>
> This exceeded my knowledge a bit, so I thought I would ask here.
>
>        What would happen if an alpha particle was fired head-on at an
> electron.
>
> (I know the question is posed in an incorrect "billiard ball" fashion.)
>
> Any insights?

Of course, Rutherford scattering isn't between electrons and alpha 
particles, it's between alpha particles and gold nuclei (or other 
heavy bound nuclei, which enables one to treat the system without 
having to worry about the heavy nucleus recoiling. Since, in this 
case both particles are positively charged, it is possible to treat 
the problem, at least to first order as a classical interaction, 
using Newton and Coulomb to do the physics. Make the interacting 
particles of opposite charge, as you suggest, makes the interaction 
much more complex, and quantum mechanics cannot be avoided in the 
problem for the most part, except in a rather narrow range of 
energies where a classical gravitation-type interaction can be 
reasonably achieved. But at low energies we run into the problem that 
there is a minimum-energy bound state (the ionized helium atom), and 
at high energies the electron and alpha particle will interact at the 
nuclear level, and the individual particles lose their distinct 
identities.

In general, though, to answer your main question: it depends. 
Kinematically, it's no different from firing an electron at an alpha 
particle. Since the force between the two is attractive, The energy 
of the interacting pair is important. If it is low (about 25 eV or 
less--that is, the initial KE of the incoming electron) one of the 
options is that the electron is captured into a bound state and the 
pair become a singly ionized helium atom. But that's not guaranteed, 
even at low energies there is an elastic scattering channel that is 
also available. At higher energies the situation becomes more 
complex, with possible Raman scattering channels, high order Rydberg 
channels and even nuclear channels opening up, as well as elastic 
channels.

Electron-proton scattering has been extensively studied for many 
years, and more details should be available in intermediate level 
atomic scattering theory texts and monographs.

It's really hard to think about electron-proton or electron-alpha 
scattering in billiard ball terms, since the force between them is 
long range and attractive, so a quantum solution is probably going to 
be the only realistic possibility.

Since low energy atomic scattering theory was the subject of my 
dissertation, I can testify that the mathematics of these processes 
are daunting. My understanding is that the experiments are not 
trivial, either.

What is usually studied is electron-atom scattering, since that 
allows one to treat the forces between the electron and the atom as a 
finite-range process, in the sense that the net Coulomb forces among 
the particles (which can loosely be thought of as similar to the Van 
Der Waals forces that chemists are fond of) drop off much faster than 
an inverse square rate. That simplifies the mathematics considerably. 
I made the mistake of studying electron-ion scattering, in which the 
net force is pure Coulomb, and thus "long-range" in any practical 
sense, and quickly found that I was dealing with versions of 
Schroedinger's equation whose solutions had to be expressed in terms 
of confluent hypergeometric functions. I ended up learning a lot more 
about numerical analysis than I did about physics.

Hugh
--

Hugh Haskell
mailto:hugh@ieer.org
mailto:haskellh@verizon.net

It isn't easy being green.

		--Kermit Lagrenouille