Observables and other characteristic quantities of the system

The rms radii for protons and neutrons are defined as ($q=n$ or $p$)

$$\langle r_q^2\rangle=\int_0^{R_{\mathrm{box}}}\!\! \mathbf r^2\rho_q^2(\mathbf r)d^3\mathbf r\,.$$ (55)

The charge radius is obtained from the proton radius by taking into account the proton charge distribution in an approximate way, i.e.,

$$\langle r_{\mathrm{ch}}^2\rangle=\langle r_p^2\rangle+\langle r\rangle_P^2\,.$$ (56)

with $\langle r\rangle_P=0.8$ fm. The mean gaps are the average values of the pairing fields

$$\langle\Delta_q\rangle=-\frac{\mathrm{Tr}\,{\tilde h}_q\rho}{A}\,.$$ (57)

The fluctuations of the particle numbers are defined as $\langle \hat N_q^2-\langle\hat N_q\rangle^2\rangle$ and given by $2\mathrm{Tr}\left[\rho_q^2-\rho_q\right]$.

Finally, the rearrangement energy, which comes from the density dependence of the force, and which shows how much the force is modified by the medium effects, is given by

$$E_{\mathrm{rear}} = \displaystyle \frac{t_3}{24}\gamma\rho^\gamma\left[\left( 1+\frac{x_3}{2}\right)\rho^2-\left(x_3+\frac{1}{2}\right) \sum_q\rho_q^2\right]\hfill$$

$$+ \displaystyle \frac{t_3'}{48}\gamma'(1-x_3')\rho^{\gamma'}\sum_q ... ...ho_q^2\,-\,\frac{e^2}{4}\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\rho_p^{4/3}\,.$$ (58)