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5.1.5. Proton Life Time

First we study the decay kinematics.

Combining Eq. (5.3), Eq. (5.14) and Eq. (5.26) we find the proton-neutron energy difference as a function of the applied magnetic field,

Equation 5.27        (5.27)
B14 is the strength of the magnetic field in units of 1014 T (1 T = 104 G). The neutron becomes stable for B > 1.5 × 1014 T and the proton becomes unstable to beta decay for B > 2.7 × 1014 T. We shall now turn to a calculation of the life time of the proton for fields satisfying the last inequality.

With the wave functions of the various particles in the magnetic fields we may define field operators for these particles. For the proton and electron we shall restrict the summation over states to the lowest Landau levels with spin up, down respectively; for magnetic fields of interest the other states will not contribute to the calculation of decay properties. For the same reason, the neutron field will be restricted to spin down only. The proton and neutron kinematics will be taken as non-relativistic.

Equation 5.28        (5.28)

with phim(x, y) defined in Eq. (5.5) and the energy, Em(pz) in Eq. (5.8). am(pz) is the annihilation operator for a proton with momentum pzz and angular momentum m; bm(pz) is the same for the negative energy states. For the neutron the field is

Equation 5.29        (5.29)

with an obvious definition of the annihilation operators. For the electron we use fully relativistic kinematics and the field is

Equation 5.30        (5.30)

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