Pairing correlations and odd-even staggering in reaction cross sections of weakly bound nuclei

We investigate the odd-even staggering (OES) in reaction cross sections of weakly bound nuclei with Glauber theory, taking into account the pairing correlation with the Hartree-Fock-Bogoliubov (HFB) method. We first discuss the pairing gap in extremely weakly bound nuclei and show that the pairing gap persists in the zero separation energy limit even for single-particle orbits with the orbital angular momenta $l=0$ and $l=1$. We then introduce the OES parameter, defined as the second derivative of reaction cross sections with respect to the mass number, and clarify the relation between the magnitude of OES and the neutron separation energy. We find that the OES parameter increases considerably in the zero separation energy limit for $l=0$ and $l=1$ single-particle states, while no increase is found for higher angular momentum orbits with, for example, $l=3$. We point out that the increase of OES parameter is also seen in the experimental reaction cross sections for Ne isotopes, which is well accounted for by our calculations.

Figure 1

The mean square radii and the paring gap for ${}^{76}$Cr nucleus as a function of the single-particle energy ${\varepsilon }_{\mathrm{WS}}$ for the 3$s$${}_{1/2}$ orbit in a Woods-Saxon potential. Panel (a) shows the mean square radius of the 3$s$${}_{1/2}$ wave function obtained with (the solid line) and without (the dashed line) the pairing correlation. Panel (b) shows the rms radii for ${}^{74}$Cr (the dotted line), ${}^{75}$Cr (the dashed line), and ${}^{76}$Cr (the solid line). These are obtained with the Hartree-Fock (${}^{74}$Cr and ${}^{75}$Cr) and the Hartree-Fock-Bogoliubov (${}^{76}$Cr) methods. Panel (c) shows the effective pairing gap for the 3$s$${}_{1/2}$ state. The solid line is the pairing gap defined with the canonical basis, that is, the expectation value of the pair potential with respect to the canonical-basis wave function for the 3$s$${}_{1/2}$ state, while the dot-dashed line shows the lowest quasiparticle energy for the $s$${}_{1/2}$ states.

Figure 2

Same as Fig. 1, but with the simplified HFB model, in which the pair potential is assumed to be proportional to the derivative of the Fermi function. Only the Fermi energy $\lambda$ is determined self-consistently in this model. The dot-dashed lines in (a) and (b) denote the results of the self-consistent calculations shown in Fig. 1.

Figure 3

The Fermi energy $\lambda$ (a) and the neutron number (b) calculated with the simplified HFB model. These are plotted as a function of the single-particle energy ${\epsilon }_{\mathrm{WS}}$ for the 3$s$${}_{1/2}$ orbit in a Woods-Saxon potential. The solid lines are obtained by adjusting the Fermi energy so that the neutron number is $N=52$, while the dashed lines are obtained by setting $\lambda ={\epsilon }_{\mathrm{WS}}$.

Figure 4

Same as Fig. 2, but by fixing the Fermi energy to be the same as the single-particle energy for the 3$s$${}_{1/2}$ state, $\lambda ={\epsilon }_{\mathrm{WS}}\left(3s_{1/2}\right)$.

Figure 5

(a) Reaction cross sections for ${}^{74,75,76}$Cr $+$ ${}^{12}$C reactions at $E=240$ MeV/nucleon, obtained with the modified optical limit approximation of the Glauber theory, Eq. (22). The density distributions for ${}^{74,75,76}$Cr are constructed from the HFB calculations shown in Fig. 1. (b) The staggering parameter defined by Eq. (23).

Figure 6

The staggering parameter ${\gamma }_3$ defined by Eq. (23) as a function of the neutron separation energy S${}_n$ for the odd-mass nuclei. The solid and the dotted lines correspond to the reaction cross sections for ${}^{22,23,24}$O $+$ ${}^{12}$C and ${}^{74,75,76}$Cr $+$ ${}^{12}$C at $E=240$ MeV/nucleon, respectively. The dashed and the dot-dashed lines show the results for ${}^{30,31,32}$Ne $+$ ${}^{12}$C at $E=240$ MeV/nucleon, in which the valence neutron in ${}^{31}$Ne occupies the 2$p$${}_{3/2}$ or 1$f$${}_{7/2}$ orbits, respectively.

Figure 7

The experimental staggering parameter ${\gamma }_3$ of reaction cross sections defined by Eq. (23) for the Ne isotopes with the ${}^{12}$C target at $E=240$ MeV/nucleon. This is plotted as a function of the neutron separation energy $S$${}_n$ of the odd-$A$ nuclei. The experimental data for the reaction cross sections are taken from Ref. [9], while the empirical separation energies are taken from Refs. [41] and [42]. The dashed line is the calculated staggering parameter for the ${}^{30,31,32}$Ne isotopes, assuming that the valence neutron in of ${}^{31}$Ne occupies the 2$p$${}_3/2$ orbit.