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Minimum distance for annihilation 
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#1
Jun114, 12:55 PM

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How close does does a particle and antiparticle pair have to be with each other in order to achieve annihilation?



#2
Jun114, 04:07 PM

P: 77

My best guess is that the closer they are the more likely annihilation gets. In positronium the separation is typically 1 ångström. For parallel spins and the lifetime is 0.12 ns and for antiparallel 0.14 µs. In excited states where the expected distance is larger, the lifetime increases.



#3
Jun114, 07:19 PM

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#4
Jun214, 09:06 AM

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Minimum distance for annihilation
Cross sections and reaction rates in QED are conveniently expressed in terms of a distance:
[tex]r_0 = \frac{e^2}{mc^2}[/tex] which goes by the (most unfortunate!) name of "classical electron radius." Its value happens to be 2.82 x 10^{13} cm. Also, if a particle is placed in a box of side a, its wavefunction at the origin is order of magnitude ψ(0)^{2} = 1/a^{3}. These two remarks lend intuitive support to the following answer obtained from QED: The e^{+}e^{} annihilation probability per unit time is [tex]\Gamma = \frac{r_0^2 \,c}{a^3}[/tex] where a is their average distance apart. For positronium, a is approximately the Bohr radius, 10^{8} cm, and if you put these values together you'll get the positronium lifetime that my2cts quoted, 0.1 ns. 


#5
Jun214, 11:24 AM

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#6
Jun214, 12:50 PM

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#7
Jun214, 04:44 PM

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Your Feynman diagram is for free particles. Include spatial bound state wave functions in the full calculation, and then "close enough for their wave functions to overlap" is relevant.
That is why P waves don't annihilate, but S waves do, with the rate proportional to \psi(0)^2 


#8
Jun214, 06:17 PM

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For a pwave, L = 1, this factor is basically v^{2}/c^{2}, or the ratio of the potential energy to the rest energy, 6 eV/0.5 MeV, about 10^{5}. Instead of nanoseconds, the lifetime for pwave annihilation is therefore in the microsecond range. But the radiative decay to swave via electric dipole transition takes place in 10^{8} sec. So the direct annihilation from pwave is perfectly possible, but has too small a branching ratio to be observed. 


#9
Jun314, 05:18 AM

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