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 + 0.04B₁₄)]; 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 × 10¹⁴ T, the lifetime is = 6 s.