Question on differential crossection, pair production

[malawi_glenn]

Hi!

I have encountered many differential crossections: \frac{d\sigma}{dE_+d\Omega _+ d<br /> \Omega _-}
(Pair production of electrons and positrons)

Where E+ is energy of positron. However, in all of these crossections, the energy of the electron; E- is included in the formula, e.g eq 2.1.1 (http://www.irs.inms.nrc.ca/EGSnrc/pirs701/node22.html)

So let's say I know the incident photon energy, k, and want to evaluate the probability to get a positron with energies between E+(1) and E+(2), should I replace the E- in the formula with (assuming that recoil energy of the nucleus is neglectable): E- = k - E+ , then integrating over dE+ ?

E- is not an independent variable, but I am wondering why all sources I have encountered so far do this? -> Putting E- and p- into the equations when they are dependent on E+ and k... is it just for making the formulas more nice and symmetric?

This might be a very trivial question, but input from someone else would save my day :-)

 

[humanino]

Hi,

just a general consideration : suppose you know your initial state completely, and you don't have any particular polarization measurement. For a scattering with N particles in the final state, complete reconstruction will give you N 4-vectors, that is 4N variables. You have 4 laws of conservation (energy and momentum) reducing this number of independent variables to 4(N-1). You also know the masses of the N particles in the final states, giving you N additional constraints, so you end up with 3N-4 variables (notice that, instead of starting with 4-vectors, we could also have started by saying you have N energies and 2N angles).

There is yet an additional trick. If your nucleon (or nucleus ?) was at rest (and with unknown transverse polarization), or if you had a head-on collision, you have an axis-symmetric situation in the initial state. That yet makes you loose one angle in the final state, which is arbitrary and only defines a reference plane of scattering.

So finally, you have 3N-5 independent variables.

If N=3 (electron, positron and nucleon) that makes 4 independent variables. If N=1 (for instance, you neglect the recoil of your nucleon) that would be only 1 independent variable.

Finally, it is very probable that the introduction of E+ and E- is only a matter of symmetrical, more beautiful (or less ugly) equations

 

[malawi_glenn]

Well yes I understand what you wrote my dear friend, I will at this level of accuracy neglect the target nucleis reqoil and initial configuration in phase space, so you are basically telling me that my substitution is accurate?

I am trying to develop my own MC-generator for gamma conversion in a detector using ROOT-functions. From that I will estimate the background to certain rare e+e- decay modes, such as pi0-e+e- and so on. Just as background why I am asking this ;-P