Lithium isotopes: $^{6}$Li and $^{8}$Li

The Slater determinants, which we selected for the NCCI calculations in these two very light nuclei, are listed in Table 3. For the sake of simplicity, the states are labeled by spherical quantum numbers $p_{1/2}$ or $p_{3/2}$ that dominate in the s.p. wave functions of the odd-proton and odd-neutron states. It turns out that such a labeling constitutes an intuitive and relatively unambiguous way to describe the configurations, even in cases of large deformations where the Nilsson picture formally prevails. The strategy behind selecting the reference configurations is to cover basic combinations of neutron or proton particle-hole (p-h) excitations having all possible alignments predicted by a simple $K$-scheme.

Table 3: Properties of the reference Slater determinants in $^{6}$Li and $^{8}$Li, numbered by index $i$ and labeled by spherical quantum numbers of particle states above $^4$He. Listed are: the HF energies E$_{\rm HF}$ in MeV, quadrupole deformations $\beta _2$, triaxiality parameters $\gamma$, and neutron and proton s.p. alignments, $j_\nu$ and $j_\pi$, together with their orientations $k$ in the intrinsic frame.

$i$	$\vert^{6}$Li$;i\rangle$	E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$
1	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$25.972	0.008	0$^\circ$	$-$0.50	1.50	Z
2	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$26.787	0.330	0$^\circ$	0.50	0.50	Z
3	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$26.510	0.216	60$^\circ$	$-$1.50	1.50	Y
4	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$27.244	0.207	60$^\circ$	1.50	1.50	Y
5	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$26.846	0.090	60$^\circ$	1.50	0.50	Y
$i$	$\vert^{8}$Li$;i\rangle$	E$_{\rm HF}$	$\beta _2$	$\gamma$	$j_\nu$	$j_\pi$	$k$
1	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$39.081	0.381	0$^\circ$	$-$1.50	0.50	Z
2	$\nu p_{1/2}\otimes \pi p_{3/2}$	$-$34.041	0.361	0$^\circ$	0.50	0.50	Z
3	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$39.025	0.356	0$^\circ$	1.50	0.50	Z
4	$\nu p_{3/2}\otimes \pi p_{3/2}$	$-$35.680	0.027	0$^\circ$	$-$1.50	1.50	Z
5	$\nu p_{1/2}\otimes \pi p_{3/2}$	$-$33.443	0.352	0$^\circ$	$-$0.50	0.50	Z

Results of our calculations are shown in Fig. 3. In the case of $^{6}$Li, theory clearly disagrees with data, with respect to both the ordering and values of energies. Let us first discuss the $T=0$ multiplet, composed of the $1^+$ and $3^+$ states. The ground state of $^6$Li has quantum numbers $I=1^+, T=0$ and the experimental total energy of this state is $-31.995$MeV. In calculations, the lowest $I=1^+$ state is placed above the lowest $I=0^+,T=1$ and $I=3^+,T=0$ solutions, and its energy of $-27.037$MeV is almost $5$MeV higher than in experiment. For comparison, the calculated energy of the $I=3^+,T=0$ member of the isoscalar multiplet is only $1.8$MeV higher than in experiment. Hence, it is quite evident that the model lacks the isoscalar pairing $I=1,T=0$ correlations, cf. Ref. [37]. In the $N=Z$ nuclei, the model, or the underlying mean-field, seems to favor the maximally aligned $T=0$ configurations. In Sec. 4.3 we demonstrate that the results obtained for $^{42}$Sc corroborate these conclusions.

Figure 3: Comparison between experimental and theoretical energy spectra of $^6$Li and $^8$Li.

It is worth recalling here that in the context of searching for possible fingerprints of collective isoscalar $pn$-pairing phase in $N\approx Z$ nuclei, the isoscalar pairing, or deuteron-like correlations, were intensely discussed in the literature, see Refs. [38,39,40,41,42] and references cited therein. In particular, the isoscalar $pn$-pairing was considered to be the source of an additional binding energy that could offer a microscopic explanation of the so-called Wigner energy [43] - an extra binding energy along the $N=Z$ line, which is absent in the self-consistent MF mass models. In spite of numerous recent works following these early developments attempting to explain the isoscalar $pn$-pairing correlations and the Wigner energy, see Refs. [44,45,46,47,48,49] and refs. cited therein, the problem still lacks a satisfactory solution.

There are at least two major reasons for that: (i) an incompleteness of the HFB (HF) approaches used so far, which consider the $pn$ mixing only in the particle-particle channel, see discussion in Ref. [21], and (ii) difficulties in evaluating the role of beyond-mean-field correlations. Recently, within the RPA including $pn$ correlations, the latter problem was addressed in Ref. [46]. Their systematic study of the isoscalar and isovector multiplets in magic and semi-magic nuclei rather clearly indicated a missing relatively strong $T=0$ pairing. This seems to be in line with our NCCI model findings concerning description of $T=0,I=1$ states, but seems to contradict the conclusions of Ref. [45,47].

Concerning the $T=1$ multiplet consisting of the $0^+$ and $2^+$ states, the theory tends to overbind the $0^+$ state by $0.8$MeV and underbind the $2^+$ state by $0.4$MeV. This level of agreement is much better than the one obtained for the isoscalar multiplet. It should be rated as fair, but not fully satisfactory. It is, therefore, interesting and quite surprising to see that the addition of two neutrons in $^8$Li seems to change the situation quite radically. Indeed, in this nucleus, for both the binding energies and distribution of levels below 5MeV, the overall agreement between theory and experiment is very satisfactory, even if the calculated $1_1^+$ and $3_1^+$ states are interchanged, see Fig. 3. The largest disagreement is obtained for the $1_2^+$ state, where the theory underbinds experiment by almost 3MeV. The states $0_1^+$, $2_2^+$, and $4_1^+$ are predicted at the excitation energies of 5.3, 4.7, and 6.2MeV, respectively, in fair agreement with the data.