The Hartree-Fock-Bogolyubov mean fields

Since the Bogolyubov transformation does not preserve the particle number, we introduce two Lagrange multipliers $\lambda_N$ and $\lambda_Z$ to conserve the average neutron and proton number. The HFB equations are then obtained by writing the stationary condition $\delta \left[\mathcal{E}-\langle\lambda_N N +\lambda_Z Z\rangle\right]=0$. The dependence of $\mathcal{E}$ on the kinetic densities leads the effective mass

$$M_q = \displaystyle \frac{\hbar^2}{2m^*_q}\hfill$$ (22)

$$= \displaystyle \frac{\hbar^2}{2m}+\frac{t_1}{4} \left[\left(1+\frac{x_1}{2}\right)\rho -\left(x_1+\frac{1}{2}\right)\rho_q\right]\hfill$$

$$\hskip 1.4cm +\frac{t_2}{4}\left[\left(1+\frac{x_2}{2}\right)\rho +\left(x_1+\frac{1}{2}\right)\rho_q\right]$$ (23)

and to the abnormal effective mass

$$\tilde M_q=\frac{t_1'}{4}(1-x_1')\tilde\rho_q\,.$$ (24)

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

$$U_q = \displaystyle t_0\left[\left(1+\frac{x_0}{2}\right)\rho -\left(x_0+\frac{1}{2}\right)\rho_q\right] \hfill$$

$$+ \displaystyle \frac{t_1}{4}\left[\left(1+\frac{x_1}{2}\right) \le... ..._1+\frac{1}{2}\right)\left(\tau_q-\frac{3}{2}\Delta\rho_q\right) \right] \hfill$$

$$+ \displaystyle \frac{t_2}{4}\left[\left(1+\frac{x_2}{2}\right) \le... ..._2+\frac{1}{2}\right)\left(\tau_q+\frac{1}{2}\Delta\rho_q\right) \right] \hfill$$

$$+ \displaystyle \frac{t_3}{12}\left[\left(1+\frac{x_3}{2}\right)(2+... ...mma\rho^{\gamma-1}\sum_{q'}\rho_{q'}^2 +2\rho^\gamma\rho_q\right)\right] \hfill$$

$$+ \displaystyle \frac{t_3'}{24}(1-x_3')\gamma'\rho^{\gamma'-1} \sum_{q'}\tilde\rho_{q'}^2 \hfill$$

$$- \displaystyle \frac{W_0}{2}\left(\nabla J+\nabla J_q\right)$$ (25)

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

$$V_c({\mathbf r})=\frac{e^2}{2}\int\, d^3{\mathbf r}'\,\frac{\rho_... ...\vert} -e^2\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\rho_p^{1/3}({\mathbf r})\,.$$ (26)

The particle-particle (pairing) field is

$$\tilde U_q= \frac{t_0'}{2}(1-x_0')\tilde\rho_q +\frac{t_1'}{4}(1-... ...\Delta\tilde\rho_q\right] +\frac{t_3'}{12}(1-x_3')\rho^{\gamma'}\tilde\rho_q\,.$$ (27)

Finally, the spin-orbit fields ( $\propto\boldsymbol\ell\cdot\mathbf s$) have the following form factors:

$$B_q=-\frac{1}{8}\left(t_1x_1+t_2x_2\right)J+\frac{1}{8} (t_1-t_2)J_q+W_0\nabla(\rho+\rho_q)\,,$$ (28)

$$\tilde B_q=\left[\frac{t_2'}{2}(1+x_2')+W_0'\right]\tilde J_q\,.$$ (29)