Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Nuclear fusion cross sections

  1. Jul 14, 2011 #1

    I am inquiring as to what the theorem function is for the mean product of cross section and velocity for stellar fusion reactions? [tex]\langle \sigma v \rangle[/tex]

    300px-Fusion_rxnrate.svg.png
    Mean product of nuclear fusion cross section and velocity. [tex]\langle \sigma v \rangle[/tex]

    360px-MaxwellBoltzmann-en.svg.png
    Maxwell–Boltzmann probability density function:
    [tex]f(v) = \sqrt{\frac{2}{\pi}\left(\frac{m}{kT}\right)^3}\, v^2 \exp \left(- \frac{mv^2}{2kT}\right)[/tex]

    The mean speed is the mathematical average of the speed distribution:
    [tex]\langle v \rangle = \int_0^{\infty} v \, f(v) \, dv = \sqrt{\frac{8kT}{\pi m}}[/tex]

    For a mono-energy beam striking a stationary target, the cross section probability is:
    [tex]P = n_2 \sigma_2 = n_2 \pi r_2^2[/tex]

    And the reaction rate is:
    [tex]f = n_1 n_2 \sigma_2 v_1[/tex]
    Reactant number densities:
    [tex]n_1, n_2[/tex]
    Target total cross section:
    [tex]\sigma_2 = \sigma_\text{A} + \sigma_\text{S} + \sigma_\text{L} = \pi r_2^2[/tex]
    Mono-energy beam velocity:
    [tex]v_1[/tex]
    Aggregate area circle radius:
    [tex]r_2[/tex]

    Stellar nuclear fusion reaction rate (fusions per volume per time):
    [tex]f = n_1 n_2 \langle \sigma v \rangle[/tex]

    What is the theorem and solution for the mean cross section in stellar nuclear fusion? [tex]\langle \sigma \rangle[/tex]

    Is the mean cross section the mathematical average of the cross section distribution?:
    [tex]\langle \sigma \rangle = \int_0^{\infty} \sigma \, f(\sigma) \, d\sigma = \, \text{???}[/tex]

    Reference:
    http://en.wikipedia.org/wiki/Cross_section_(physics)#Nuclear_physics"
    http://en.wikipedia.org/wiki/Maxwell-Boltzmann_distribution" [Broken]
    http://en.wikipedia.org/wiki/Nuclear_fusion#Requirements"
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Nuclear fusion cross sections
  1. Cross section (Replies: 0)

  2. Fusion cross sections (Replies: 13)

Loading...