Odd-even and two-particle mass staggering

In order to analyze the odd-even mass differences, the three-point pairing indicators [74] (staggering parameters),

$$\Delta_{p}^{(3)}=\frac{1}{2}\Big(B(Z+1,N)+B(Z-1,N)-2B(Z,N)\Big)$$ (6)

with odd $Z$ and even $N$, and

$$\Delta_{n}^{(3)}=\frac{1}{2}\Big(B(Z,N+1)+B(Z,N-1)-2B(Z,N)\Big)$$ (7)

with even $Z$ and odd $N$, where $B(Z,N)$ is the (positive) binding energy of the nucleus, have been plotted in Figs. 9 and 10, respectively. In our HFB and RHB calculations, increased sizes of the shell gaps can, in principle, be seen in the odd-even mass staggering parameters (6) and (7), because the density of single-particle states has an immediate bearing on the calculated intensity of pairing correlations. This is under assumption that the differences obtained by blocking different orbitals along the isotopic or isotonic chains have a lesser impact on the odd-even mass staggering. Irrespective of the detailed reproduction of the experimental values of these parameters, calculated results may thus illustrate the shell structure corresponding to the EDFs studied here.

Figure 9: Three-point proton odd-even mass staggering, Eq. (6), shown for the odd-$Z$ and even-$N$ nuclei in the nobelium region. Experimental values are based on the AME2012 atomic mass evaluation [75].

FigSHE09.eps

Experimental results, shown in panels (e) of Figs. 9 and 10, indicate that in the nobelium region, values of the staggering parameters are within the range of 500-700keV. On closer inspection, we see several trends in the mass dependence of these parameters, which may indicate variations of the shell structure due to the influence of the level density on pairing correlations, or due to other fine structural effects. In particular, values of $\Delta_{p}^{(3)}$ seem to have a small dip for the $N=146$ isotones at $Z=95$, are fairly constant in the $N=148$ isotones, and gradually decrease with mass in the $N=150$-154 isotones. None of these values indicate a particularly significant shell-gap opening near $Z=100$. Similarly, small dips in $\Delta_{n}^{(3)}$, which show up in the $Z=96$-98 isotopes at $N=149$ and in the $Z=100$ isotopes at $N=153$, do not point to a particularly large shell gap at $N=152$.

This lack of large variations in odd-even mass staggering is at variance with the analysis of two-particle mass staggering given by quantities

$$\delta_{2p}^{(3)}=2B(Z,N)-B(Z+2,N)-B(Z-2,N) = S_{2p}(Z,N)-S_{2p}(Z+2,N)$$ (8)

and

$$\delta_{2n}^{(3)}=2B(Z,N)-B(Z,N+2)-B(Z,N-2) = S_{2n}(Z,N)-S_{2n}(Z,N+2) ,$$ (9)

which were typically used to identify two-nucleon shell gaps in experiment and in calculations for spherical [62,76] and deformed [77,8] shell closures in the predictions of Skyrme and covariant EDFs. As discussed in Ref. [2], experimental values of $\delta_{2n}^{(3)}$ show clear maxima for the $Z=96$-102 isotopes at $N=152$ and those of $\delta_{2p}^{(3)}$ exhibit maxima for the $N=148$-150 isotones at $Z=98$ and for the $N=152$-154 isotones at $Z=100$, see also Figs. 11 and 12.

Figure 10: Same as in Fig. 9 but for the neutron odd-even mass staggering, Eq. (7), shown for the odd-$N$ and even-$Z$ nuclei.

FigSHE10.eps

Figure 11: Two-proton mass staggering, Eq. (8), shown for the even-$Z$ and even-$N$ nuclei in the nobelium region. Experimental values are based on the AME2012 atomic mass evaluation [75].

FigSHE11.eps

Figure 12: Same as in Fig. 11 but for the two-neutron mass staggering, Eq. (9).

FigSHE12.eps

When looking at the most pronounced features of the calculated odd-even mass staggering shown in Figs. 9 and 10, we see that minima of $\Delta_{p}^{(3)}$ can be seen at $Z=95$ (NL1 and NL3* EDFs) and $Z=97$ (D1M, SLy4, and UNEDF2 EDFs). For the Gogny and Skyrme EDFs, these minima disappear at higher neutron numbers and rather monotonic trends are then obtained. Similarly, minima of $\Delta_{n}^{(3)}$ appear at $N=149$ (NL1 and NL3* EDFs) or $N=151$ (D1M, UNEDF2, and SLy4 EDFs); in the latter case, in lighter isotopes they tend to shift to $N=149$. For the calculated two-proton-staggering indicators (8), covariant EDFs, NL1 and NL3*, exhibit very strong maxima at $Z=96$, at variance with the data, whereas the non-relativitic EDFs, D1M, SLy4, and UNEDF2, reproduce experimental maxima at $Z=98$ in the $N=146$-150 isotones but fail to shift these maxima to $Z=100$ in heavier isotones. This conspicuous experimental fearure thus remains unsolved. The calculated two-neutron-staggering indicators (9), do not reproduce experimental maxima occurring at $N=152$. These results illustrate the fact that none of the studied EDFs reproduces the experimental trends in shell gaps extracted from the two-particle indicators (8) and (9). We note here that the inclusion of the LN method into the calculations renders pairing correlations much less sensitive to the shell structure. Therefore, one then obtains fairly structureless trends of $\Delta_{p}^{(3)}$ and $\Delta_{n}^{(3)}$ [21], although for covariant EDFs, one at the same time obtains a significant improvement of the overall agreement with experimental values [20].