Since the Bogolyubov transformation does not preserve the particle
number, we introduce two Lagrange multipliers and
to conserve the average neutron and proton number.
The HFB equations are then obtained by writing the stationary
condition
.
The dependence of
on the kinetic densities leads
the effective mass

(22) | |||

(23) |

and to the abnormal effective mass

(24) |

The particle-hole (Hartree-Fock) fields is given by

(25) |

In the case of protons, varying the expressions (21) and (22) leads to the following expression for the Coulomb field

(26) |

The particle-particle (pairing) field is

(27) |

Finally, the spin-orbit fields ( ) have the following form factors:

(29) |