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When a positron and electron annihilate each other typically two or more quanta are created in order to balance momentum. It is widely thought that single quantun annihilation cannot take place. This is true only in a field-free environment. But if the positron annihilation takes place in the near vicinity of a heavy nucleus momentum and energy in a single quantum annihilation can be balanced.
It was empirically demonstrated in the late 1950's by Lester Sodickson at M.I.T. that such single quantum annihilations do take place. The write-up is given the Physical Review as L. Sodickson et al, "Single-Quantum Annihilation of Positrons," vol.124 num 6, (Dec., 1961), pp.1851-1861. In this article itis suggested that it is the recoil of the nucleus that may enable the balancing of momentum in a single quantum annihilation.
One quantum annihilation occurs only on the order of 0.2 of 1% because of the infrequency that a positron makes it to the innermost shell of the electrons of an atom. This is in part due to the prevalence of electrons elswhere, but also to the difficulty of a positive particle making it so near to the positively charged nucleus.
The purpose of this webpage is to examine theoretically the mechanism for single quantum positron annihilation. It turns out to be quite different from the recoil of the nearby nucleus.
First a simplified model be considered.
Consider an electron of mass m revolving around the center of mass of it with a nucleus of mass M. Let re and rN be the orbit radii of the electron and nucleus. respectively, with s be their sum. Let the charge of the nucleus be denoted as Z.
The electron and the nucleus have tangential velocities and momenta.
Centrifugal force is a familiar concept in elementary dynamics, but it is not really a force in the same sence that gravitation or the attraction between opposite charges are forces. Centrifugal Force is merely the magnitude of the force required to keep a body traveling in a circular orbit.
Unfortunately the notion of centrifugal force has produced a false intuition that if the constraint keeping a body moving in a circular orbit is released the body would move radially outward. Instead what really happens is that with the disappearance of the constraint the body moves in the direction of its tangential velocity at the instant of its release.
The circumstances most favorable to the balancing of momentum for a single quantum annihilation is if the collision is an over-taking one; i.e., the particles are moving in the same direction at sightly different velocities.
The momentum of the nucleus is unchanged by the annihilation. Its motion is just converted from circular motion to linear motion. The momentum of the electron disappears with its disappearance but may reappear in the momentum of the single quantum photon. The question is whether the magnitudes can balance. The positron also has momentum that disappears with its disappearance. The positron must have high energy and momentum to get to an electron in the vicinity of a positively charged nucleus. For a minimally over-taking collision its momentum would be essentially the same as that of the electron. However,one particle could have significantly higher velocity than the other in an over-taking collision.
The annihilation of the masses of the electron and the positron produces an energy of 2(0.511) million electron volts (MeV). This is equal to
The momentum p of a photon of energy E is given by E/c, where c is the speed of light.
Further computation however is not necessary. In the Bohr model of an atom the momentum of an electron in the lowest orbit around a nucleus of atomic number Z is given by
where α is the fine structure constant ≅1/137.
The figures are doubled for the combined annihilation of the masses and momenta of the electron and positron. The feasibility of single quantum annihilation thus hinges on the value of Zα.
For Cesium Z is 55 and for Uranium it is 92. Hence the computation indicates single quantum annihilation is not feasible for a Cesium nucleus or, for that matter any other element, but for Uranium the value of Zα is 0.67. This quantity is on the order of magnitude of 1 so it seems appropriate to say that single quantum annihilation is theoretically feasible. This for a minimally over-taking collision.
If the positron is traveling much faster than the electron in orbit then the momentum of the single quantum photon could be feasible.
It appears that any recoil of the nucleus is inconsequential in making feasible a single quantum annihilation of a positron. The essential thing seems to be that the annihilation collision have the particles moving in the same direction at sufficiently high velocities and could take place outside of the neighborhood of nuclei.
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