Equation of state and radii of finite nuclei in the presence of a diffuse surface layer

The definitions of nuclear surface and nuclear radii are considered within the Gibbs-Tolman-Rowlinson-Widom (GTW) approach. We demonstrate the nonmonotonic behavior of the nuclear equimolar radii, which is due to the shell effects in the chemical potential of finite nuclei. The direct variational method within the extended Thomas-Fermi approximation is used to establish the equation of state for finite nuclei. We have studied the influence of the polarization effect caused by the neutron excess on the particle density and the nuclear radii. This effect increases with the asymmetry parameter $X$ and can be responsible for the appearance of large neutron halos in nuclei well away from the $\beta$ stability line. We have performed new calculations of the $A$ dependence of the radii $R\left(A\right)$ of nucleon distribution, which are based on the use of the experimental data for the nuclear binding energy. We demonstrate the presence of the quantum shell effects in $R\left(A\right)$. We have analyzed the value of the neutron-skin thickness $\mathrm{\Delta }{r}_{np}$ in the isotopes of the Na, Sn, and Pb nuclei within the GTW approach and show the appearance of nonmonotonic behavior of $\mathrm{\Delta }{r}_{np}$ as a function of the neutron excess. We discuss the relative contributions to the neutron-skin thickness $\mathrm{\Delta }{r}_{np}$ from the skin and the halo effects.

Figure 1

Equation of state for the nucleus of ${}^{208}\mathrm{Pb}$. The calculation was performed for the $\mathrm{Sk}{\mathrm{M}}^{*}$ interaction [15]. The dashed line is the EOS for the symmetric nuclear matter $P_{\mathrm{vol}}\left({\rho }_0\right)$, solid line 1 is for $P_{\mathrm{vol}}\left({\rho }_0\right)+P_{\mathrm{capil}}\left({\rho }_0,X\right)$, solid line 2 is for $P_{\mathrm{vol}}\left({\rho }_0\right)+P_C\left({\rho }_0,X\right)$, and solid line 3 is the total pressure $P_A\left({\rho }_0,X\right)$ of Eq. (28).

Figure 2

The partial pressure $P_{A,\mathrm{sym}}$ for the nucleus of ${}^{208}\mathrm{Pb}$ calculated for different parametrizations of the Skyrme forces: KDE0v1 [19]: solid line 1; SLy230b [18]: solid line 2; and $\mathrm{Sk}{\mathrm{M}}^{*}$ [15]: solid line 3. The dotted vertical line is the mark for the spinodal instability border, and the dashed line is for the equilibrium density.

Figure 3

Dependence of equimolar radius $R_e=R_e^{*}\left(A,X^{*}\right)$ on mass number $A$ on the $\beta$-stability line in the presence of the polarization effect (the solid line). The dashed line is the equimolar radius $R_e\left(A,X=0\right)$ where the polarization effect is absent. The dotted line is obtained by elimination of the Coulomb force polarization effect, see Eq. (46). The calculations were performed for the $\mathrm{Sk}{\mathrm{M}}^{*}$ [15] interaction.

Figure 4

Specific surface particle density $F\left(R\right)=-\left({\rho }_{\mathcal{S}}\mu +{\rho }_{-,\mathcal{S}}{\mu }_{-}\right)$ versus dividing radius $R$ for nuclei with $A=208$ and $A=120$. The calculation was performed using the $\mathrm{Sk}{\mathrm{M}}^{*}$ interaction [15]. $R_e$ denotes the equimolar radius where $F\left(R\right)=0$.

Figure 5

$A$ dependence of equimolar nuclear radius $R_e\left(A\right)$. The solid points were obtained within the Gibbs-Tolman procedure where the experimental values for the nucleon chemical potential were used, and the dashed line is for the corresponding averaged values of equimolar radii $R_e$. The dotted line is for $R_e=1.13A^{1/3}\mathrm{fm}$. The $\mathrm{Sk}{\mathrm{M}}^{*}$ interaction [15] was used.

Figure 6

The rms radius of the proton distribution in the Na isotopes obtained by use of Eq. (53). The dotted line with open circles was obtained with surface layer correction $\sim b_q^2$, and the dashed line with open squares is for $b_q^2=0$. Here and below, the experimintal data were taken from Ref. [24]. The $\mathrm{Sk}{\mathrm{M}}^{*}$ interaction [15] was used.

Figure 7

The isovector shift of nuclear rms radius $\mathrm{\Delta }{r}_{np}=\sqrt{\langle r_n^2\rangle }-\sqrt{\langle r_p^2\rangle }$ in Na isotopes. The solid points are the experimental data [24], the open circles (connected by the dotted line) have been obtained using the Gibbs-Tolman approach described in the text, and the solid line is obtained using the extended Thomas-Fermi approximation with the $\mathrm{Sk}{\mathrm{M}}^{*}$ Skyrme interaction [15], see Ref. [6].

Figure 8

The same as in Fig. 7 but for Sn isotopes. The data were taken from Refs. [23, 25, 26].

Figure 9

The same as in Fig. 7 but for Pb isotopes. The experimental data were taken from Refs. [27, 28, 29].

Figure 10

The ratio $\chi =\mathrm{\Delta }{r}_{np,a}/\mathrm{\Delta }{r}_{np,a}^{*}$ versus the deviation $\mathrm{\Delta }X=X-X^{*}$ from the $\beta$-stability line for isotopes of the nuclei of Na, Sn, and Pb. The calculations have been performed using the $\mathrm{Sk}{\mathrm{M}}^{*}$ interaction [15] and Weizsecker's parameter $\beta =1/9$ [see Eq. (15)]. The crosses at the end of the lines denote the neutron drip line, which is derived by the condition ${\lambda }_n=0$.