**5.1.6. Decay Rates and Spectrum**

The part of the weak Hamiltonian responsible for the decay
*p*
*n* + *e*^{+} +
_{e} is

(5.31) |

For non-relativistic heavy particles the matrix element of this Hamiltonian
between a proton with quantum numbers *p*_{z} = 0 ,
*m* = *m*_{i}, a neutron with momentum
**p**_{n}, a neutrino with momentum
**p**_{} and an
electron in state
*m* = *m*_{f} and with *p*_{z,e} is

(5.32) |

_{} is the azimuthal angle of
the neutrino. The integral in the above
expression can be evaluated in a multipole expansion. Note that the natural
extent of the integral in the transverse direction is 1 /
sqrt[*eB*] whereas the
neutron momenta are, from Eq. (5.27), of the order of
sqrt[*eB*(0.12 + 0.04*B*_{14})]; thus setting the
exponential term in this integral equal to one
will yield a good estimate for the rate and spectrum of this decay.
The positron spectrum is given by

(5.33) |

where is defined in
Eq. (5.27). For
>> *M*_{e}
the total rate is easily obtained

(5.34) |

For *B* = 5 × 10^{14} T, the lifetime is
= 6 s.