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Consider the beta decay of and suppose q, , p and k are the 4-momenta of the , , and respectively. The relations below follow from momentum conservation:






The expressions for the energies and momenta of the final state particles are obtained by considering the 2-body processes below:

with final state masses , and respectively [Byc-73]. The explicit calculations in the rest frame of the decaying reveal:





The momenta follow from the energy-momentum relation


The kinematic constraints on the final state particle energies are derived as follow: from equation (gif), reaches a minimum when


To arrive at the relation (gif), one writes


where is the angle between the 3-momenta. Equation (gif) goes through a minimum at . This means that the 3-momenta of and are collinear. Suppose that the minimum is some value a to be specified. One gets:


which leads immediately to:


where and are the magnitudes of the velocities of the neutral pion and the electron respectively. Setting


the relation (gif) becomes


Taking the derivative of equation (gif), the minimum is reached at where : (gif) is proven and in addition, the neutral pion and the positron have the same velocity as can be seen from (gif). From (gif), reaches a maximum when the neutrino is born at rest. It is straightforward to establish the kinematic constraints on and . Gathering all the information together:


The above relations together with equations (gif, gif and gif) give the allowed energies of the neutral pion, the positron and the neutrino as:


Neglecting the neutrino mass, the explicit calculations show:

The maximum kinetic energy of the is about . Because of this recoil energy, the momenta of two gamma rays which originate from the decay of are not necessarily collinear. Their opening angle can be computed as follows:

The figure gif shows the distribution of the opening angle of the two gamma rays, and the energy spectrum of one of the photons is displed in figure gif.

next up previous contents
Next: Pion Beta Decay Up: Pion Beta Decay Previous: Pion Beta Decay

Bernward Krause
Mon Jan 15 14:57:06 MET 1996