Minimum distance for annihilation

In summary: It is a bare prediction, but direct p-wave annihilation has been observed and measured to have a branching ratio matching the predictions.
  • #1
Saado
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How close does does a particle and anti-particle pair have to be with each other in order to achieve annihilation?
 
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  • #2
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.
 
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  • #3
Saado said:
How close does does a particle and anti-particle pair have to be with each other in order to achieve annihilation?

It depends on many factors, such as the relative momentum of the two particles, their masses, what particles they are etc. But I think roughly speaking you should just imagine the two particles as wave-packets, and if those wave-packets overlap, then there is some probability that the particles will annihilate. So typically they will need to be separated by a distance of the same order as their wavelength.
 
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  • #4
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/a3.

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.
 
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  • #5
Saado said:
How close does does a particle and anti-particle pair have to be with each other in order to achieve annihilation?
Close enough for their wave functions to overlap.
 
  • #6
Meir Achuz said:
Close enough for their wave functions to overlap.
Seems obvious, but actually not! Look at the Feynman diagram, there's two vertices x1 and x2. The electron arrives at point x1 and emits a photon, the positron arrives at point x2 and emits a photon. In between, a virtual particle. The amplitude is obtained by integrating over all x1, x2, but there's no requirement that they coincide.
 
  • #7
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
Meir Achuz said:
Your Feynman diagram is for free particles.
It's not "my" Feynman diagram. But I appreciate the offer! :smile:

Include spatial bound state wave functions in the full calculation, and then "close enough for their wave functions to overlap" is relevant.
"Relevant", I guess, but not accurate. The Feynman diagram approach still works. Just take the result for plane waves and integrate it over the momentum distribution for a bound state. Doesn't change the fact that there are two vertices that need not coincide.

That is why P waves don't annihilate, but S waves do, with the rate proportional to |\psi(0)|^2
P-wave states most certainly do annihilate, even though |ψ(0)|2 = 0. However the rate is suppressed by the usual factor for the centrifugal barrier, (p/mc)2L, since the particles spend more of their time farther apart.

For a p-wave, L = 1, this factor is basically v2/c2, 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 p-wave annihilation is therefore in the microsecond range. But the radiative decay to s-wave via electric dipole transition takes place in 10-8 sec. So the direct annihilation from p-wave is perfectly possible, but has too small a branching ratio to be observed.
 
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  • #9
Bill_K said:
P-wave states most certainly do annihilate, even though |ψ(0)|2 = 0. However the rate is suppressed by the usual factor for the centrifugal barrier, (p/mc)2L, since the particles spend more of their time farther apart.

For a p-wave, L = 1, this factor is basically v2/c2, 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 p-wave annihilation is therefore in the microsecond range. But the radiative decay to s-wave via electric dipole transition takes place in 10-8 sec. So the direct annihilation from p-wave is perfectly possible, but has too small a branching ratio to be observed.

Is it a bare prediction, or is direct p-wave annihilation something that can be and has been observed, and measured to have a branching ratio matching the predictions?
 

What is meant by "minimum distance for annihilation"?

Minimum distance for annihilation refers to the shortest distance at which particles or antiparticles can come into contact and annihilate each other, resulting in the production of energy in the form of photons or other particles.

Why is it important to understand the minimum distance for annihilation?

Understanding the minimum distance for annihilation is crucial in various fields of physics, such as particle physics and astrophysics. It helps us determine the conditions necessary for particles and antiparticles to annihilate, and it provides insight into the fundamental forces and interactions at play.

How is the minimum distance for annihilation calculated?

The minimum distance for annihilation is determined by the size and energy of the particles involved, as well as the strength of the forces between them. It can be calculated using mathematical models and experimental data.

Can the minimum distance for annihilation be altered?

The minimum distance for annihilation is a fundamental property of particles and cannot be altered. However, the conditions under which annihilation occurs, such as the presence of other particles or high energy collisions, can affect the distance at which it takes place.

What other phenomena involve the concept of minimum distance for annihilation?

Other phenomena related to the minimum distance for annihilation include pair production, where particles and antiparticles are created from energy, and nuclear reactions such as fusion and fission, which involve the annihilation of particles within atomic nuclei.

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