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Coulomb term in Potential of ions in a Penning Trap,Green's Function

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How do you write out the potential of an ion in a Penning trap? Specifically, how does the inter-ionic Coulomb interaction come to be written as a Green's function? I found it here : http://journals.aps.org/rmp/pdf/10.1...vModPhys.71.87

They do say something about interactions with image charges in the electrodes, but I can not see why a simple expression of all Coulomb pairs is not good enough. They seem to use it well enough in some of the other papers I read.

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