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Positron cross section with H,D,T

  1. Aug 12, 2013 #1
    I have a question about cross section probability for a positron (e+) to interact with the electron (e-) that would be bound to three different atomic systems (1) hydrogen atom: H(e-), (2) deuterium atom: D(e-), (3) tritium atom: T(e-).

    My hypothesis is that a positron (e+) would have the highest cross section interaction probability with H(e-) and lowest with T(e-), with D(e-) intermediate. My thinking is that the addition of neutrons in the D and T systems would serve to slightly shield the positron (e+) and (e-) wavefunctions from forming unstable positronium, thus lower cross section probability of interaction between the (e+) and (e-). Would my hypothesis be correct ?
  2. jcsd
  3. Aug 12, 2013 #2


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    I don't see why the neutrons would "shield" anything.

    The positron needs some minimal energy, as the positronium bond is weaker than the electron/proton bond.
    Due to the isotope effect, the energy levels of deuterium and tritium are a bit lower than those for hydrogen. If you get the right positron energy, the reaction might be possible with some isotopes, but not with heavier ones.
  4. Aug 12, 2013 #3
    OK, thanks, we can forget about any shielding effect.

    So, are you saying that, due to the 'isotope effect' you mention, the positronium formation cross section potential would be highest for H(e-) isotope, and lowest for tritium T(e-) isotope, because the electron energy density would be a bit lower in tritium, thus less possibility for (e-) and (e+) wavefunctions to overlap to form positronium ?
  5. Aug 12, 2013 #4


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    No, I did not say this.

    The binding energy for H+e in the ground state is -13.606 eV.
    The binding energy for D+e in the ground state is something like -13.611 eV (rough estimate).
    The binding energy for positronium in the ground state is -6.803 eV.

    The reaction ##e^+ + {}^1H \to (e^+e^-) + {}^1H^+## needs additional 6.803 eV of energy.
    The reaction ##e^+ + {}^2H \to (e^+e^-) + {}^2H^+## needs additional 6.808 eV of energy.

    If your positron has 6.805 eV, it can react with H, but not with D.
  6. Aug 13, 2013 #5
    Ok, very helpful, thank you.

    I have a few more questions.

    Your estimate in binding energy between H and D has a delta of 0.005 eV, so does that mean there would be a similar delta between D and T predicted, that is, we would predict the binding energy for a T+e in ground state to be about -13.616 eV, given the added neutron isotope effect ? So, we would predict this:

    The reaction ##e^+ + {}^3H \to (e^+e^-) + {}^3H^+## needs additional 6.813 eV of energy.

    If yes above, it seems reasonable that we predict that a positron (e+) with energy between 6.809 and 6.812 eV would react with either a H and D system, but not with T=tritium ? Would this be correct ? Are you aware if this experiment has ever been conducted to confirm the prediction ?

    One final question. Suppose a positron (e+) with an energy of 6.820 eV. Would it be predicted to have a higher cross section reaction probability with a D+e(-) isotope in ground state, or a T+e(-) isotope, or would the cross section be the same for the two ? Again, are you aware if this experiment has ever been conducted ?

    Again, thanks for help with these questions.
  7. Aug 13, 2013 #6
    I do not see why positron needs to form positronium in order to be unstable.

    Positron is itself a somewhat special isotope of hydrogen. Very special isotope, though, due to its having such a small reduced mass.

    Just like two hydrogen nuclei and one electron can form a dihydrogen cation, a positron and one hydrogen atom can form a molecular ion. Which would constantly have an overlap between positron and electron wavefunctions.
  8. Aug 13, 2013 #7


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    It does not have to, and I would guess that direct annihilation is the most probable reaction* (and it is the only possible reaction* for low-energetic positrons).

    *ignoring elastic collisions
  9. Aug 14, 2013 #8
    What do you mean low-energetic?

    Energetic positron can undergo direct annihilation, or lose energy by elastic collisions, or lose energy by inelastic collisions, or undergo direct capture into positronium - or undergo direct capture into positronium molecules.

    So what could be the further fate of positronium molecules and molecular ions?
  10. Aug 14, 2013 #9


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    Below the 6.8 eV I calculated.

    Positronium decays (unless it hits something and disintegrates before that happens), and ions usually capture an electron after a while, but that depends on your setup.
  11. Aug 14, 2013 #10
    Cannot positronium molecular ions also annihilate?
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