🚀 go-pugleaf

Hello sci.physics,

I'd like to understand electron positron annihilation. I'll just say what
I think I know, and let the net correct me or add info where appropriate.

Can choose center of mass as origin of coordinate system, choose
time = 0 when particles are co-incident, choose direction of coordinates
such that e is moving in -x, p moving in +x, choose handedness such that
p has +1/2 spin (I think you can always to this, if not then have to
consider two more cases). Both fermions have the same energy and equal
but opposite momenta. The incoming system can be described by two cases,
aligned spins and anti-aligned spins.

In the coordinate system chosen, total momentum is zero. The energy
of both fermions are equal, and add up to E> 2 *(eletron rest mass) * c^2.
The energy above the rest masses is the kinetic energy of the system, > 0.
The resultant products of the annihilation are two photons. The
conservation of momentum dictates that the photons have equal but
opposite momentums. Thus they must have equal energies. Spin
conservation has implications on the two cases of incoming spins:

aligned -> photons have spin 0 and 1
anti "" -> photons have spin 0 and 0, or 1 and -1

I see no restriction on the direction of the photons, only that they
go opposite ways. The directions could be decomposed into three basis
function directions of x, y, and z. So the total possible situations:

aligned ->	spin 0 in +x, spin 1 in -x,
		spin 0 in -x, spin 1 in +x,
		spin 0 in +y, spin 1 in -y,
		spin 0 in -y, spin 1 in +y,
		spin 0 in +z, spin 1 in -z,
		spin 0 in -z, spin 1 in +z

anti "" ->	spin 0 in +x, spin 0 in -x,
		spin 0 in +y, spin 0 in -y,
		spin 0 in +z, spin 0 in -z,
		spin 1 in +x, spin -1 in -x,
		spin 1 in -x, spin -1 in +x,
		spin 1 in +y, spin -1 in -y,
		spin 1 in -y, spin -1 in +y,
		spin 1 in +z, spin -1 in -z,
		spin 1 in -z, spin -1 in +z

What are the relative probabilities of these outcomes? (Hmm.... Maybe
need to specify the incoming distributions. e and p have identical
distributions to each other except for their different propogation
directions. The transverse distributions are equal in y and z. The
longitudinal distribution need not be the same as the transverse one.
The distributions are all gaussian. So this system is described by:
	
	the total energy E,
	the spin state of the electron S,
	the transverse spatial distribution variance Tv,
	the longitudinal spatial distribution variance Lv.

Of course other possible outcomes include:

	particles miss each other,
	elastic collision, with all possible recoil directions,
	for anti-aligned spins, collision could flip spin assignments,
	positronium is created.

By the way, I remember reading here about two forms of positronium with
two different lifetimes. I take it they are due to aligned spins or not.
Also, I take it positronium must decay into two photons as described
above.

Just asking,
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Ric Peregrino     c/o Hewlett Packard Co.        I represent only myself
ric@spd.hp.com    1501 Page Mill Rd. Bldg. 5M    Buy low, sell high
415-857-7526      Palo Alto, CA 94304            Get down with the facts
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