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In a recent article:
Physical Review A 83, 032903 (2011), A.Yu. Voronin, P.Froelich, V.V. Nesvizhevsky, Antihydrogen Gravitational Quantum States

the authors claim that anti-hydrogen has a lifetime of 0.1 seconds when placed in a container with metal walls. They say that the atom is kept from annihilating with the walls because of the Casimir-Polder effect. This raises in my mind the question of how it is that the atoms annihilate. My first guess is that the positron would annihilate against an electron, and then the proton would be attracted into the metal by the image charge effect. But this would seem to be in contradiction to the Casimir-Polder effect mentioned in the paper.

So how does slow anti-hydrogen decay in the presence of normal matter?

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It may be useful to you - antiprotons in helium –  voix Mar 14 '11 at 18:59

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Dear Carl, the correct paper to derive the 0.1-second lifetime of the anti-Hydrogen atom in the gravitational field is described after the very sentence you quoted.

There is a "[20]" symbol which means that the sentence is justified in the reference number 20 in the list of literature at the end of the paper you quoted. So the correct paper that answers your question is

Quantum reflection of ultracold antihydrogen from a solid surface
A Yu Voronin and P Froelich 2005 J. Phys. B: At. Mol. Opt. Phys. 38 L301, doi: 10.1088/0953-4075/38/18/L02

http://iopscience.iop.org/0953-4075/38/18/L02
http://iopscience.iop.org/0953-4075/38/18/L02/pdf/0953-4075_38_18_L02.pdf

which is fully available online - click the last link for the PDF file. At distances longer than 15 nanometers from the metal, the Casimir-Polder potential has the form $-74/z^4$ in atomic units. At shorter distances between 1 Bohr radius or so to 15 nanometers, the potential becomes $-0.25/z^3$ in atomic units - a van der Waals form.

Only at distances shorter than 1 Bohr radius or so, the anti-Hydrogen behaves differently than the Hydrogen, and is annihilated. This portion of the potential is not probed in the 0.1-second case: note that to get annihilation, the positron must approach the electron at a shorter distance than the Bohr radius - comparable to the Compton wavelength of the electron (and even shorter, nuclear distances if we want to annihilate the hadrons). The authors calculate the reflection probability as a function of the kinetic energy of the anti-atom. The lower energy/temperature they have, the more likely they will bounce off. For low enough kinetic energy, the reflection probability is as high as 0.99987 (a table).

In those cases, one doesn't probe the short-distance portion of the potential. So the answer to your question why there's no annihilation is the same as the answer to the question why you don't get fusion if you fill a tank with the Hydrogen gas.

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As usual your comment is useful, and as usual I'll upvote it when I get my votes back in 4 hours (I always use them up). But what I'm asking is "how does anti-hydrogen annihilate". Is it the case that the positron goes first and then the proton is attracted into the metal and eventually hits a nucleus? Does that mean that the anti-hydrogen can escape the metal container with a fairly high probability (if it misses the nuclei)? –  Carl Brannen Mar 14 '11 at 19:03
    
Dear Carl, thanks for your niceties. The antiproton is negatively charged so it is electrically attracted to the normal nuclei. As it accelerates, it emits and loses energy, and eventually falls to one of those normal nuclei and they annihilate. The electrons and positrons play the role of the shield that protects the hadrons from close encounter - additional potential energy - but the shield only works when the distances are atomic or longer. I don't know who annihilates "first" (proton or electron) and whether the question makes sense at all. e- e+ will probably form a positronium for a time –  Luboš Motl Mar 20 '11 at 7:21

Let me say that antihydrogen is never " slow" in the process of annihilation, when it comes to matter it is already accelerated by Casimir-Polder potential to the energies of few eV, which corresponds to temperature of dozens of thousands of degrees. The attraction forces between particles and antiparticles tear antihydrogen into antiproton, which is captured by matter atom, and positron, which most likely forms pair with electron. Antiproton falls down on nuclei, ejecting electrons from the atom. Then it Usually destroys nuclei and annihilates. This process usually takes time of order of 10^-9 s or so, electron-positron in best case could survive for 10^-7 s.

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