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Finding neutrino mean free path using cross section for interaction

  1. Apr 26, 2012 #1
    1. The problem statement, all variables and given/known data

    A supernova can produce a neutron star with typical radius 10km. Assume the neutron star matter consists of iron nuclei (A=56), produced by the electron capture reaction:

    [itex]e^{-}[/itex]+Co[itex]\rightarrow[/itex]Fe+[itex]\nu_{e}[/itex]

    The matter density is [itex]\rho[/itex]=10[itex]^{5}[/itex] tonne mm[itex]^{-3}[/itex] and the neutrino cross section for interaction with this material is [itex]\sigma[/itex]=3[itex]\times[/itex]10[itex]^{-46}[/itex]m[itex]^{2}[/itex].



    Find the neutrino mean free path

    2. Relevant equations

    w[itex]_{t}[/itex]=JσN (1)
    w[itex]_{t}[/itex]=Iσρ[itex]_{T}[/itex]t[itex]\frac{N_{A}}{A}[/itex] (2)
    I=JS (3)

    Where w[itex]_{t}[/itex] is the rate of reaction
    J is incoming flux
    I is intensity of incoming beam
    S is surface area of incoming beam
    σ is the cross section of reaction
    N[itex]_{A}[/itex] is Avogadro's number
    A is number of protons in target material
    ρ[itex]_{T}[/itex] is the density of the target material
    t is the length the beam passes through



    3. The attempt at a solution

    Ok so in my course on nuclear and particle physics there are often questions like this but I am not sure how to approach them.

    I figured that the target is essentially the neutron star, so t=20km (length through the star). I can see that I have most of the things required in equation (2), except w[itex]_{t}[/itex] and I. I figured that as w[itex]_{t}[/itex] is the rate of reaction it is essentially the number of reactions divided by the time. As we are only concerned with the reaction of 1 neutron in the star, I thought that in this case the time would be the mean free time (i.e the time before the neutrino reacts) which would be equal to the distance travelled/c (speed of neutrino). The distance travelled in this case would be the mean free path λ. This means that we can rewrite w[itex]_{t}[/itex] as:

    w[itex]_{t}[/itex]=c/λ


    I come up with a problem though. As no information about the incoming beam is given (its flux J, intensity I or area S) I have no idea how the intensity I can be calculated so that (2) can be used.

    I can see that, with the density and volume of the star, potentially the number of atoms in the target could be computed. However I cannot see how this would be useful to compute the quantity that I want.

    The question is only 5 marks so I am guessing it is fairly simple but I can't see it. As I said, for my exam there are usually several similar questions concerning the cross section of interaction so any help would be much appreciated. The answer is λ=3km, I just want to find how it is done!

    Thanks
     
  2. jcsd
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