Microscopic shell-model description of the exotic nucleus 16 C

The structure of the neutron-rich carbon nucleus 16 C is described by introducing a new microscopic shell model of no-core type. The model space is composed of the 0 s , 0 p , 1 s 0 d and 1 p 0 f shells. The eﬀective interaction is microscopically derived from the CD-Bonn potential and the Coulomb force through a unitary transformation theory. Calculated low-lying energy levels of 16 C agree well with the experimental values. The B ( E 2; 2 +1 → 0 +1 ) value is calculated with the bare charges. The anomalously hindered B ( E 2) value for 16 C, measured recently, is well reproduced.

The study of nuclear structure far from the stability line is being developed with great interests from both the experimental and theoretical sides.One of the recent exciting phenomena on these lines is the anomalously hindered E2 transition strength between the first 2 + and the ground 0 + states in 16 C [1], while this transition is rather strong in other nuclei.The dominance of neutron excitation is also suggested for the same transition [2,3].
In order to theoretically investigate such interesting properties, several calculations have been performed.For neutron-rich carbon isotopes with the mass number A ≥ 15, neutrons begin to fill the 1s 1/2 and 0d 5/2 orbits, while the protons occupy mainly the orbits up to 0p 3/2 .Low-lying energy levels of such systems can be well described by the conventional shell model using effective interactions for the 0p1s0d shell [4][5][6][7].However, the B(E2; 2 + 1 → 0 + 1 ) values calculated by the shell model using the usual effective charges are significantly larger than the experimental ones though the systematic decrease from 14 C to 16 C is described.A similar tendency of B(E2; 2 + 1 → 0 + 1 ) is seen in the calculations using the bare charges within the anti-symmetrized molecular dynamics (AMD) framework [8,9].The discrepancies between the calculations and the experiments may be due to the fact that those models do not include certain exotic features of unstable nuclei because of their bases with some assumptions or adjustments in and near stable nuclei.
Given this situation, ab initio nuclear structure calculations starting with the two-and three-body nuclear forces in free space may help to describe correctly the nuclear structure without any assumptions.For this purpose, the Green's function Monte Carlo (GFMC) [10,11] and the no-core shell model (NCSM) [12,13] are promising methods up to the mass number A ≃ 12.As the mass number becomes larger, however, those methods may not be practical because of a huge computation size.It is, therefore, still of much importance to construct a model for describing light nuclei starting with the bare nuclear force.
In this Letter, we shall show that the B(E2; 2 + 1 → 0 + 1 ) value as well as lowlying excited levels of 16 C can be well described by a new shell-model framework being introduced.The model space consists of the 0s, 0p, 1s0d, and 1p0f shells.Since the model space is of no-core type and is rather large as compared to that of the conventional shell model, we do not employ the effective charges but use the bare charges.Although, in principle, a sort of effective operator is needed for the calculation of B(E2; 2 + 1 → 0 + 1 ) [14], we use the bare operator for simplicity.The effective two-body interaction is microscopically derived from a bare nucleon-nucleon interaction through a unitary transformation [15,16] which is used in the unitary-model-operator approach (UMOA) [17,18].
Here we outline the method of deriving the effective interaction for the present shell model.Although the method is essentially the same as in the UMOA in Ref. [18], there are several differences regarding the model space.In Fig. 1, the model space P 2 and its complement Q 2 for the neutron-proton channel are illustrated.The other channels are also treated similarly.First we derive the effective interaction ṽ(1) 12 in the large model space P 1 (= P 2 + Q 2 ) which is specified by a boundary number ρ 1 through the same procedure as in Ref. [18].The number ρ 1 is defined as ρ 1 = 2n 1 +l 1 +2n 2 +l 2 with the harmonic-oscillator (h.o.) quantum numbers {n 1 , l 1 } and {n 2 , l 2 } of the two-body states.If we diagonalize the effective Hamiltonian in the P 1 space with a large number of shell-model many-body basis states, it leads to the NCSM.However, since the mass number in the present calculation is around 16, it is difficult to obtain a converging result by enlarging the model space P 1 .Hence, in order to derive an effective interaction suitable for a smaller model space, we employ the procedure stated below.
The P 1 space is divided into the small model space P 2 and the complementary space Q 2 as in Fig. 1.By using the effective interaction ṽ(1) 12 in the P 1 space where the short-range correlation is taken into account, we derive the effective interaction ṽ(2) 12 in the P 2 space for the present shell-model calculation.The P 2 space contains the orbits from the 0s up to the 1p0f shell for both the proton and neutron.The two-body effective interaction ṽ(2) 12 in the P 2 space is determined by solving the decoupling equation [19] Q 2 ṽ(2) where h 0 is the one-body part of the two-body subsystem which consists of the kinetic energy t 1 (t 2 ) and the one-body potential u (2) 2 ) as 1 stands for a mean potential for the nucleon 1 from the other nucleons which are assumed to occupy the orbits according to the normal filling scheme for 14 C.The u 1 and ṽ(2) 12 are self-consistently determined for the unperturbed ground-state wave function of 14 C by means of iteration.
This means that we derive the two-body effective interaction ṽ (2) 12 which can describe 14 C as a good closed-shell nucleus.This self-consistency is also used to determine the effective interaction in the P 1 space.
It is known that, with the usual projection operators P and Q, the general solution S 12 for the decoupling equation is written with the restrictive conditions P S 12 P = QS 12 Q = 0 as S 12 = arctanh(ω − ω † ), where the operator ω satisfies ω = QωP [16].The unitary transformation e S 12 is also expressed in terms of the operator ω by the block form concerning P and Q as which agrees with the unitary transformation by Ōkubo [15].
We note here that the model space P 2 is not completely separated from the Q 2 space in energy but in the orbits.Some two-body states in the P 2 space can be degenerate with those in the Q 2 space in energy.This degeneracy often causes difficulty with the convergence of the self-consistent calculation.This means that some matrix elements of ṽ(2) 12 in a step are significantly different from those in another step.As we enlarge the P 1 space by taking a large value of ρ 1 , this instability increases.In the present study, we take ρ 1 = 6 as the maximum value for obtaining the converged effective interaction.In this case, the model space P 2 is bounded on the middle of the diagonal line specified by ρ 1 .On the other hand, in the usual NCSM, one takes a sufficiently large value of ρ 1 for obtaining the converged result and does not perform the double unitary transformation for determining the effective interaction in the small model space P 2 as in the present work.Therefore, the present approach may be regarded as a sort of model to describe low-lying states in neutronrich carbon isotopes though that is still based on the bare nucleon-nucleon force and is thus microscopic.It is also noted that the self-consistent onebody potential u 1 is introduced only in the P 2 space for the convergence.For the one-body potential in the Q 2 space in solving the decoupling equation in Eq. ( 1), we use the fixed one which is self-consistently determined with the effective interaction in the large model space Next, we mention truncations in the actual shell-model calculation.The nucleon excitations from the hole states of 14 C are restricted to up to two nucleons, and the excitations to the 1p0f shell are also up to two nucleons.The three-or-more-body effective interactions are not taken into account for simplicity though the many-body effective interactions can be generated through the unitary transformation.As for the treatment of the spurious center-ofmass (c.m.) motion, we follow the Gloeckner-Lawson prescription [20] as has been done in many of the shell-model calculations in which a multi-hΩ model space is considered.Thus, the present shell-model Hamiltonian is composed of the transformed Hamiltonian which contains the sums of the kinetic energy and the two-body effective interaction, and the c.m. Hamiltonian which is multiplied by a large value β c.m. .
In the following calculations, the CD-Bonn potential [21] is employed as the two-body bare interaction, and the Coulomb force is also included for the proton-proton channel.Since we adopt the above-mentioned approximations, our calculated results may have an hΩ dependence.We have confirmed that the hΩ dependences of calculated energy levels of 14−18 C become weak in the range between hΩ = 14 and 16 MeV, and there appear energy minima in this region.Thus, in this study, we use the value of hΩ = 15 MeV in all the calculations.Since we introduce a minimal refinement of one-body energies in a simple way which will be explained later, we need to use a common value of hΩ.
In Fig. 2, calculated and experimental low-lying energy levels of 16 C are shown.The values are also tabulated in Table 1.The first column in Fig. 2, denoted states appear somewhat higher than the experiment.
We note here that there is a difference in the model space between the present approach and the UMOA to neutron-rich carbon isotopes.In the present shellmodel calculation for 15 C, the 1/2 + 1 state appears above the 5/2 + 1 state, contrary to the experiment.This wrong ordering may be one of the reasons for the discrepancy between the results and the experimental values in the excited states of 16 C for the "original" case in Fig. 2. If we consider a sufficiently large model space and calculate these energy levels within the UMOA framework, which is fully microscopic, using the same CD-Bonn potential, we can reproduce the correct ordering in 15 C as has been shown in Ref. [22].This type of calculation is feasible for describing single-particle levels of one-nucleon plus closed-shell nuclear systems, but becomes extremely complicated for the excited states of 16 C.In order to obtain the correct ordering in 15 C in the present shell model, we make a minimal refinement of neutron one-body energies on top of 14 C so as to reproduce the UMOA results for 15 C [22].In this way, we treat complex correlations coupled to a single-particle-like state within a rather simple framework.Thus, the present approach is a hybrid method combining a no-core type of shell model with single-particle information by the UMOA.Hereafter, we refer to the calculated results with the minimal refinement as "dressed".
The magnitude of the refinement may depend on the orbit.In fact, denoting the one-body energy for a single-particle orbit j of the neutron as ǫ n (j), we need the variations as ∆ǫ n (1s 1/2 ) = −2.18 and ∆ǫ n (0d 5/2 ) = −0.04MeV which are added to the original ǫ n (j)'s, namely, the h.o.kinetic energies.We also change the one-body energy for the 0d 3/2 orbit of the neutron by the same amount as ∆ǫ n (0d 5/2 ) for simplicity.Since the variations of ∆ǫ n (0d 5/2 ) and ∆ǫ n (0d 3/2 ) are very small, the effect of the refinement arises mainly from only ∆ǫ n (1s 1/2 ).We use these dressed one-body energies not only in the calculation for 15 C but also 14 C, 16 C, and 18 C.A similar approach may be useful, in particular, to some exotic systems for which single-particle levels are unknown experimentally.
The results of the low-lying energy levels of 16 C with the above-mentioned refinement are shown in Fig. 2 and Table 1 with "dressed".The calculated results become better by introducing the dressed one-body energies.Namely, the correct ordering of the low-lying energy levels is obtained.It is noted here that a recent NCSM calculation for 16 C with the CD-Bonn potential in a 4 hΩ space also gives the correct ordering and the calculated B(E2; 2 + 1 → 0 + 1 ) is about 1 e 2 fm 4 which is not so different form our values in Table 2 [23].
In Table 2, the calculated and experimental values of B(E2; 2 + 1 → 0 + 1 ) for 16 C are tabulated together with the values for 14 C and 18 C to investigate the isotope dependence.For 16 C, the calculated B(E2) value becomes closer to the anomalously hindered experimental value by introducing the dressed onebody energies.As we have explained before, in the present calculation, we do not employ the effective charges for the neutron and proton but use the bare charges.
For 14 C, the calculated results of B(E2; 2 + 1 → 0 + 1 ) for "original" and "dressed" are identical, and show a good agreement with the experiment.We note, however, that the calculated energy spacings between the 0 + 1 and 2 + 1 states for the cases "original" and "dressed" are identically 5.39 MeV, whereas the experimental value is 7.01 MeV.Thus, the calculated energy spacings are somewhat smaller than the experiment.The energy spacing is sensitive to the magnitude of the spin-orbit splitting for the 0p state of the proton.The calculated small energy spacing indicates that the proton spin-orbit splitting in the present calculation is not sufficient for the energy spacing.The proton spin-orbit splitting may be larger if we enlarge the model space and/or include a genuine threebody force in the present approach.
For 18 C, the calculated energy spacings between the 0 + 1 and 2 + 1 states for "original" and "dressed" are 1.44 and 1.73 MeV, respectively.Since the experimental value is 1.62 MeV, our calculated results are not so different from the experiment.As for the B(E2; 2 + 1 → 0 + 1 ), as shown in Table 2, the result for "dressed" is larger than that for "original".Furthermore, the result for 18 C for "dressed" is about three times larger than the experimental value for 16 C.A similar tendency is seen in the 0p1s0d shell-model calculations [6,7].Thus, it is of great interest that the B(E2) value for 18 C is experimentally established.
In order to investigate the B(E2) results in more detail, for the case "dressed", we have decomposed the B(E2) values with the restrictions up to the 0p or the 1s0d shell using the full wave functions.The results are tabulated in Table 3.For 16 C, about a half of the total B(E2) is obtained in the result up to the 0p shell, and a large contribution from the 1p0f shell can be observed though the magnitude itself is small.For 18 C, most of the contribution to the total B(E2) is from the orbits up to the 0p shell.For 14 C, a reduction of the B(E2) is seen by including the 1p0f shell.Note that these results come from only the proton contributions because we use the bare charges.
Here we examine the dependence of calculated results on β c.m. .As we have explained before, we employed the Gloeckner-Lawson prescription to remove the spurious c.m. motion.Unlike the usual NCSM, since the present shell-model space is composed of an incomplete space in energy, spurious c.m. components are not removed completely.Therefore, our calculated results have some β c.m. dependence.However, as for the B(E2; 2 + 1 → 0 + 1 ) for 16 C which is the main interest in the present study, the β c.m. dependence is quite small if we take a sufficiently large value of β c.m. .The tendency is illustrated in Fig. 3 for the case "dressed".In a wide range between β c.m. hΩ/A = 50 and 500 MeV, the differences are within 0.01 e 2 fm 4 .We have also confirmed that the β c.m. dependence of the calculated energy spacing between the 2 + 1 and 0 + 1 states is also quite small in the same range.The differences are within 0.02 MeV.Therefore, our results for 16 C will be reliable for the removal of the spurious c.m. motion.However, we should note that, for 14 C, the situation becomes worse.For β c.m. hΩ/A = 50 and 500 MeV, the calculated energy spacings for the case "dressed" are 5.39 and 5.11 MeV, respectively.Gradually the energy spacing becomes smaller as the β c.m. value is larger, and the β c.m. convergence is slow.Therefore, we will still have room to improve the method of the removal of the spurious c.m. motion for 14 C. Furthermore, we should include many-particle many-hole excitations higher than the 2p2h excitation for the realistic description of excited states in 14 C higher than 6 MeV excitation energy.These would apply to lighter carbon isotopes.For these reasons, we have not performed detailed calculations for 14 C and lighter carbon isotopes.It should be noted, however, that the situation is different in 16 C and 18 C. Since we describe the excited states below 6 MeV excitation energy where neutron excitations within the 1s0d shell play an important role, the approximations in the present calculation may be good for 16 C and 18 C.In the present study, we used the common value of β c.m. hΩ/A = 50 MeV in the final results for 14 C, 16 C, and 18 C.
One may be interested in the calculated ground-state energy since the present shell model is a kind of no-core calculation.For 14 C, our calculated result for the case "dressed" is −80.18MeV.If we include 4p4h excited configurations, the result becomes −90.90 MeV.(Note that the inclusion of 4p4h configurations in the calculations for 16 C and 18 C is not so easy due to a huge computation size.)We notice that since the present shell model is not fully microscopic and needs some approximations, the above value is not the exact Table 4 The occupation numbers for the 0p 3/2 , 0p 1/2 , 1s 1/2 , and 0d 5/2 orbits of the proton (p) and neutron (n) for the 0 + 1 and 2 + 1 states in 14 C and 16 C for the case "dressed".Isotopes J π Nucleon 0p 3/2 0p 1/2 1s 1/2 0d 5/2 14  one.On the other hand, we can microscopically calculate the ground-state energy of the closed-shell nucleus 14 C within the UMOA framework.In the case of a sufficiently large value of ρ 1 , the calculated result is −87.70 MeV without the three-or-more-body cluster effects.The inclusion of three-bodycluster terms gives an additional attractive energy about −3 MeV.Therefore, the present shell-model result with the 4p4h effect is comparable with the UMOA result.Since the experimental value is −105.28MeV, the remaining attractive energy would be brought by including a genuine three-body force.
In Table 4, we show calculated occupation numbers for the 0 + 1 and 2 + 1 states in 14 C and 16 C for the case "dressed".The occupation numbers of the proton for 16 C are hardly changed between the 0 + 1 and 2 + 1 states.As for the neutron, only the occupation numbers for the 1s 1/2 and 0d 5/2 orbits are slightly different between the 0 + 1 and 2 + 1 states.This result supports the decoupling of the valence two neutrons from the core part [2,3,6].In the case of 14 C, the proton occupation numbers for the 0p 3/2 and 0p 1/2 orbits are rather different between the 0 + 1 and 2 + 1 states in contrast to the case of 16 C.These results indicate that the excitation mechanism is very different between 14 C and 16 C.
Here we comment on the ratio of the neutron and proton transition matrix elements for 16 C.Large ratios have been found through experiments as M n /M p = 7.6±1.7 [2] and 6.7±1.3 [3].In our calculation for the case "dressed", the ratio is 4.68 and somewhat smaller than the experiments.Our model space P 2 may be large enough for M p because of the inclusion of the 2hΩ excited 1p0f shell from the 0p shell, while it may not be sufficient for M n due to the missing 2hΩ excited 2s1d0g shell from the 1s0d shell.Thus, M n can be enlarged to a certain extent in the calculation with a larger model space.
The large M n /M p ratio means that the transition matrix element for the neu- tron M n is considerably larger than that for the proton M p .This also indicates that the B(E2; 2 + 1 → 0 + 1 ) of 16 C is sensitive to the value of the neutron effective charge.This situation is illustrated in Fig. 4. The results for 14 C and 18 C are also shown.For 16 C, the B(E2) value is rapidly increasing as the neutron effective charge e eff n becomes larger.For e eff n = 0.5e which is the standard value in the conventional shell model, the calculated B(E2) is 9.35 e 2 fm 4 .This value is about ten times larger than the result for the bare charge, namely, e eff n = 0. Also for 18 C, a similar tendency of the strong dependence on the neutron effective charge is observed, while the neutron effective charge dependence for 14 C is very weak.
Although it seems that the use of the bare charges in the calculation of the B(E2) in 14 C and 16 C is a good approximation, the situation is different for 15 C. The calculated B(E2; 5/2 + 1 → 1/2 + 1 ) of 15 C with the bare charges is 0.28 e 2 fm 4 .If we make the same analysis of the B(E2; 5/2 + 1 → 1/2 + 1 ) in 15 C as in Table 3, results for 15 C are 0.15 and 0.19 e 2 fm 4 for the cases "up to 0p" and "up to 1s0d", respectively.It is seen that the effect of the 1p0f shell has a sizeable contribution to the total B(E2) as in the case of 16 C.The experimental B(E2) of 15 C has been determined with very high accuracy as B(E2; 5/2 + 1 → 1/2 + 1 ) = 0.97 ± 0.02 e 2 fm 4 [24].Thus, our calculated result is considerably smaller than the experiment.This may indicate that the present approach is not sufficient for describing the structure of 15 C, as we introduce the minimal refinement for the energy levels of 15 C.However, we have found that the discrepancy of the B(E2) in 15 C can be compensated by introducing the neutron effective charge as e eff n = 0.164e which is much smaller than the usual neutron effective charge in the conventional shell model.The need of the effective charge in the structure calculations for 15 C and 16 C has also been discussed in 14 C + n and 14 C + n + n models [25][26][27].
If we calculate the B(E2; 2 + 1 → 0 + 1 ) values using the neutron effective charge e eff n = 0.164e for 14,16,18 C, the results become 3.66, 2.62, and 4.98 e 2 fm 4 , respectively.Due to the strong neutron effective charge dependence of B(E2) in 16 C and 18 C as shown in Fig. 4, the results for 16 C and 18 C are significantly larger than those for the bare charges despite the rather small effective charge of the neutron.Therefore, within the present shell-model framework, we cannot describe properly the B(E2) values of 15 C and 16 C simultaneously, as long as we employ the same value of the neutron effective charge.This problem may be resolved if we enlarge the model space and/or include a genuine three-body force in the present method.
In summary, we have investigated low-lying energy levels and B(E2; 2 + 1 → 0 + 1 ) of 16 C by introducing a new shell-model method of no-core type with the model space up to the 1p0f shell.The two-body effective interaction is microscopically derived from the CD-Bonn potential and the Coulomb force through a unitary-transformation theory.The present work is the first study which can successfully describe the low-lying structure of 16 C including its anomalously hindered B(E2; 2 + 1 → 0 + 1 ) value based on the microscopic effective interaction with the bare charges.In principle, we should use an effective operator in the calculation of the transition strength [14] as well as the effective interaction for the energy.Such a study should be pursued in the future.However, we expect that the use of the bare charges in the present model space is a good approximation for 16 C.There are some indications that the effective charges should be smaller as the neutron richness increases in boron and carbon isotopes even in the smaller 0p1s0d space [6,28,29].In general, the magnitude of the effective charges for reproducing the experimental value should depend on the model-space size employed and each nucleus due to different correlations, such as different core-polarization effects between stable and halo nuclei [30].
In the present study, we do not include a genuine three-body force.The recent GFMC and NCSM results indicate that the bare three-body force mainly changes the binding energy but rather keep the excitation spectrum unchanged, except for a few cases such as beryllium and boron isotopes [11], while the three-body-force effect depends on the two-body interaction employed in the calculation.We expect that the present microscopic approach with the two-body interaction is a reasonable one for the calculations of excitation energies and transition strengths of low-lying states in nuclei around 16 C.Although the successful description of 16 C seems to support this expectation, the extension of the present method for the inclusion of the effective three-body interactions from the bare two-and three-body forces may be interesting for a more realistic description.This sort of study may also improve the difficult situation for the description of the structure of 15 C.

Fig. 1 .
Fig. 1.The model space P 2 and its complement Q 2 consisting of the two-body states.The axes represent nuclear shells.

Table 1
The calculated and experimental excitation energies of low-lying excited states in16C.All energies are in MeV.