Question on differential crossection, pair production

In summary, you should replace the E- in the equation for the probability of a positron with energies between E+(1) and E+(2) with (assuming that recoil energy of the nucleus is neglectable). This might be a very trivial question, but input from someone else would save my day :-)
  • #1
malawi_glenn
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Hi!

I have encountered many differential crossections: [tex]\frac{d\sigma}{dE_+d\Omega _+ d
\Omega _-}[/tex]
(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 :-)
 
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  • #2
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 :smile:
 
  • #3
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
 

1. What is differential cross section?

Differential cross section is a measure of the probability of a particle interaction occurring in a specific direction or range of directions. It is commonly used in particle physics to describe the scattering of particles.

2. How is differential cross section calculated?

Differential cross section is calculated by dividing the number of particles scattered in a particular direction by the flux of incident particles and the target area. This results in a unit of area per solid angle.

3. What is pair production?

Pair production is a process in which a high-energy photon interacts with a nucleus or another photon to produce an electron-positron pair. This process is commonly observed in high-energy particle collisions.

4. What is the significance of differential cross section in pair production?

In pair production, differential cross section is used to describe the angular distribution of the produced particles. It provides information about the scattering process and can be used to test theories and models of particle interactions.

5. How does pair production relate to the study of high-energy physics?

Pair production is a fundamental process in high-energy physics and is used to study the properties of particles and their interactions. It also has important applications in fields such as medical imaging and radiation therapy.

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