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Nuclear mean field on and near the drip lines
PHYSICS ELSEVIER
REPORTS
Physics Reports 264 (1996) 297-310
Nuclear mean field on and near the drip lines Takaharu
OtsukaaTb*, Nobuhisa
Fukunishib
aDepartment of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113, Japan bRIKEN, Hirosawa,
Wako-shi, Saitama 351-01, Japan
Abstract
We discuss two subjects related to the structure of nuclei near the drip lines. The first is the vanishing of N = 20 magic structure in Z < N = 20 nuclei. Large-scale state-of-the-art shell-model calculations with 2sld and lower 2plf shells are shown to present a unified description of N = 20 isotones with Z = 10-20, covering both stable and unstable nuclei. The calculations demonstrate that, although the N = 20 closed-shell structure remains for Z 2 14, the N = 20 closed-shell structure vanishes naturally towards nuclei with Z I 12, giving rise to various anomalous features including those in 32Mg and 31Na. It is suggested that, in these nuclei, the deformed mean field overcomes the shell gap created by the spherical mean potential. Furthermore, the almost perfect agreement with a recent experiment is presented for the B(E2; 0: + 2:) value of 32Mg. The second part is devoted to the mean field for loosely bound neutrons. The variational shell mode1 (VSM) is explained with an application to the anomalous ground state of “Be. The VSM has been proposed recently to describe the structure of neutron-rich unstable nuclei. Contrary to the failure of spherical Hartree-Fock, the anomalous 2’ ground state and its neutron halo are reproduced with Skyrme SIII interaction. This state is bound due to dynamical coupling between the core and the loosely bound neutron which oscillates between 2s1,2 and ldS,2 orbits. The direct neutron capture is discussed briefly in its relation to the neutron halo.
1. Introduction The recent developments of radioactive nuclear beams are opening a new rich field of nuclear physics: the structure of neutron-rich unstable nuclei. The observation of the neutron halo [l] is an example. In this talk, we discuss the structure of neutron-rich unstable nuclei characterized by Z 4 N with Z(N) being the proton (neutron) number. Among the various topics in the field, we focus upon (i) the vanishing of the N = 20 magic shell and the emerging of large deformation in nuclei with Z < N = 20, and (ii) the single-particle motion of nucleons in exotic circumstances such as extremely neutron-rich nuclei. 2. Vanishing of N = 20 magic structure Since the pioneering work on exotic sodium isotopes by Klapisch et al. [2] and Thibault et al. [3], experimental studies on N = 20 isotones have been conducted in spite of their technical 0370-1573/96/$9.50 0 1996 SSDI 0370-1573(95)00044-5
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difficulties. On the other hand, a unified theoretical description, particularly covering from stable (2 - 20) to unstable (Z - 10) regions within a single framework, has been missing. We shall report the result of such a unified calculation of N = 20 isotones from 40Ca to 30Ne in terms of the shell model with a realistic effective interaction [4]. Stable sd-shell nuclei have been studied extensively in terms of the shell model with valence particles in the sd shell, as will be referred to as the sd-shell model hereafter. Although the sd-shell model, in particular the one with the effective (USD) interaction of Wildenthal [S], has been highly successful for stable sd-shell nuclei, various anomalies have been observed in exotic nuclei with Z 4 N = 20 including the anomalous ground state spin of 31Na [6] and the anomalously low first 2+ state of 32Mg [7]. The above anomalies have been studied in shell-model [8-123 and Hartree-Fock (HF) calculations [13]. We shall demonstrate in this report that the anomalies are explained, without any specific adjustment for unstable exotic nuclei, by the same Hamiltonian as the one used for stable nuclei, while large-scale shell-model calculations appear to be more crucial in unstable nuclei, because of stronger mixing between the sd and pf shells. The present calculation includes the configurations made up by the three sd-shell orbits, and furthermore allows excitation of two nucleons into If,,, and 2p 312 orbits. The excitation into the pf shell is extended later up to four nucleons for 32Mg, where the size of the calculation is quite huge with the maximum M-scheme dimension 9 10 878 for the O+ states. Such large-scale calculations have become feasible by a state-of-the-art shell-model program for the supercomputer coded by Sebe [14, 151 recently. This report thus announces, incidentally, the availability and significance of such a shell-model code. The effective nucleon-nucleon interaction used here consists of three parts, For matrix elements within the sd shell, we use the USD interaction [S], which has been obtained from the G-matrix interaction of Kuo-Brown [ 161 with slight empirical refinement. The USD interaction is the most successful effective interaction for the sd-shell model. We use the Kuo-Brown G-matrix interaction [17] for matrix elements within the pf shell and for matrix elements involving both the sd-shell orbits and the If,,, orbit. The remaining part gives only a very minor contribution to the primary results, and we use the Millener-Kurath interaction [lS]. All two-body interaction matrix elements including non-sd parts are scaled by the mass number dependence A-o.3 in the same way as the USD interaction [S]. In the USD interaction, core polarization effects are assumed to be included by renormalization of two-body matrix elements. However, we treat explicitly the If,,, and 2~3~2 orbits as active orbits in this calculation. The renormalization effects in the USD interaction due to these orbits are subtracted by using the second order perturbation so as to avoid double counting. All nucleons on the top of the N = Z = 8 closed shell are treated explicitly as valence particles. The single-particle energies (s.p.e.‘s) are assumed to be constants, independent of the nucleus as in the sd-shell model with the USD interaction, and are actually adjusted to reproduce the observed energy spectra of 41,40,3gCa. The configuration space here includes all configurations, excluding those really negligible, up to 3-particle-2-hole (3p2h) for 41Ca, 2p2h for 40Ca and 2p3h for 3gCa with respect to the N = Z = 20 closed shell. By using the experimental energy levels of s- and $for 41Ca, and those of 4’ and f’ for 39Ca [19], we adjust the s.p.e.‘s, which are defined to include the kinetic energies and the effects due to the interaction with the N = Z = 8 core nucleons but not the interaction with the sd- or pf-shell nucleons. The adjusted values are -2.97, 1.66, - 5.88,2.84 MeV for 2sr12, ld3iZ, lf7,2, 2p3,2, respectively. For Id,,,, we use the same s.p.e. as in the USD
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calculation [S], i.e., - 3.95 MeV. Since there are twelve valence neutrons and some valence protons in the following calculations, these orbits become much more bound and their ordering is changed, due to the interaction with the nucleons in the sd and pf shells. Note that the above values for the sd-shell orbits do not differ significantly from the s.p.e.‘s of the USD interaction [S]. As already mentioned, since a sufficient number of valence nucleons in the sd shell are assumed, the present set of s.p.e.‘s should work basically for nuclei which are not extremely far from 40Ca, but not necessarily for much lighter nuclei, for instance r’0. Note that the mass number difference between 40Ca and “0 is almost three times greater than that between 40Ca and 32Mg. The present parametrization, constant s.p.e.‘s and the A-o.3 scaling for the two-body interaction, should have certain limitations. A more general prescription should be studied in order to cover a wider range including 40Ca, 32Mg, r’0, etc. The spurious center-of-mass motion is removed by using the approximate method of Gloeckner and Lawson [20]. The positive-parity energy levels are shown in Fig. 1 for even-A nuclei with N = 20 and 10 I 2 I 16, in comparison with experiments [21]. As mentioned above, the calculations in Fig. 1 include up to 2p2h excitations into the pf shell. The excited states are found only above Ex = 2 MeV for 34Si and 36S, both experimentally and theoretically. The calculated first 2+ level comes down sharply to Ex = 0.98 MeV in going from 34Si to 32Mg, consistently with experiments. This dramatic change is one of the anomalies being studied, and has never been described before in a unified way covering both N - 2 stable and N $ Z unstable nuclei with mixing of the sd and pf shells. The first 2+ state of 32Mg is predicted as Ex = 1.68 MeV in the sd-shell model with the USD interaction.
= -
-4 -
2
zy -2
--
-2
-1
:I
-;
-2
2
2
=1 =;
-5
-
-4
=o
-
-3
1
=1
54z
3-
4
-
-3
-2 =4
2
-
-4
2
t
t
I.-t
0
-0
Th.
-0
-0
Exp.
-0
Th.
Exp.
-0
Th.
-0
Exp.
34Si Fig. 1. Experimental
(Exp.) and calculated
(Th.) energy levels of the positive-parity
-0
-0
Th.
Exp.
3%e states of even-2
N = 20 isotones.
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Fig. 2. (a) Occupation number of the pf shell in the ground states. The squares (crosses) indicate neutrons (protons). Lines are drawn to guide the eye. (b) Effective single-particle energy of the neutron orbits evaluated at subshell-closed nuclei (see the text). The diamonds (crosses) denote the values for If,,, (ld3i2) orbit. Lines are drawn to guide the eye.
Fig. 1 indicates also that the level density in the low-energy region increases for Z I 12. In the sd-shell model of N = 20 nuclei, only protons are active and neutrons are frozen in the N = 20 closed shell. The structure of low-lying states therefore can be interpreted in terms of the seniority of protons. This tendency remains approximately for stable nuclei in the present calculation. On the other hand, the structure of the low-lying states of Z I 12 nuclei is determined not only by valence protons in the sd shell, but also by neutrons lifted to the pf shell and by neutron holes created in the sd shell. This means that N = 20 is no longer the magic number in these nuclei. We shall present results for odd-A N = 20 isotones elsewhere because the length of this article is limited. We only mention that, within the positive-parity states up to 2p2h excitations, the present calculation produces the anomalous $‘-3’ spin sequence of 3‘Na, whereas the sd-shell model with the USD interaction puts, because of the Id,,, occupancy of the last proton, the 3’ level below $+, contrary to experiment. The situations in 35P and 33A1 are normal, for which both the sd-shell model and the present calculation work equally well. As for magnetic moments, the experimental value for the above 3’ state of 31Na is 2.28~~ [6], while the present calculation yields 2.20,~~~ with free nucleon g-factors. The occupation number of the pf shell in the ground state is shown in Fig. 2(a) for 40Ca-30Ne. For stable nuclei 4oCa-35P, the total occupation number of protons and neutrons remains small (-0.6), while the neutron occupation number increases as Z decreases below Z = 15 (““P). The small and almost constant pf-shell occupation number in N - Z nuclei is quite reasonable because
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of the non-vanishing matrix elements connecting the sd and pf shells. This small mixing of the pf-configuration changes only slightly the energy spectra of 40Ca-35P from those of the sd-shell model. The continuous growth of the neutron pf-shell occupation probability on the N 9 2 side indicates that the N = 20 closed shell is broken gradually below 2 - 15. This continuous change of the probability conflicts with the truncation used in the unmixed calculation in Ref. [12]. The proton pf-shell occupation number decreases as Z decreases, because the proton Fermi level goes away from the pf shell. The effective s.p.e. is introduced as the sum of the original s.p.e. and the expectation value of the two-body interaction with respect to the relevant proton closed subshell plus one neutron particle in lf7,2 or one neutron hole in Id,,,. The actual calculation is made for 40Ca. 36S (2s1,* closed), 34Si (Id,,, closed), and 280 in a way similar to the Hf particle or hole energy. Fig. 2 (b) shows the effective s.p.e.‘s thus obtained. The resultant effective s.p.e. approaches zero with decreasing Z, because the expectation value of the two-body interaction becomes smaller in magnitude as the number of valence protons decreases. Fig. 2(b) shows also that, as Z decreases, the difference between lf7,2 and Id,,, effective s.p.e.‘s decreases gradually, starting from a large value (- 7 MeV) in 40Ca . In 32Mg it reaches a smaller value, which happens to be similar to the one (- 1.5 MeV) adjusted in Ref. [lo]. Note that this small value causes the collapse of the closed shell in Ref. [lo]. In fact, the smaller values of this lf,,2-ld3i2 difference for lighter isotones in Fig. 2(b) imply the vanishing of the N = 20 shell gap towards Z Q N = 20. We stress that this vanishing seems to be a general feature of the G-matrix interaction as discussed for 280 in Ref. [22]. The If 7,2-1d3,2 difference in Fig. 2(b) shows a prominent correlation to the pf-shell occupation number in Fig. 2(a). We note that the effective s.p.e. is not used in the present shell-model diagonalization, since the actual calculation includes all two-body matrix elements. The effective s.p.e. is introduced for schematic demonstration. There is no effect on the effective s.p.e. from the deformation, because it is evaluated with spherical subshells. On the other hand, the smaller lf,,2-ld3,2 energy difference drives the nucleons towards larger mixing of the pf-shell configuration, giving rise to stronger deformation as discussed later. We point out that the binding of neutrons in If,,, and ld3,2 becomes extremely weak for the isotones lighter than 3oNe, e.g., 280 and 2gF. In such nuclei, due to weak binding, the single-particle wave functions of If,,, and Id,,, are probably changed from the ones given by the harmonic oscillator potential, and consequently the G-matrix prescription applied to such loosely bound wave functions produces a new set of matrix elements different from those used so far. The present shell-model approach remains valid up to 32Mg or 31Na where the shell gap begins to be narrower but those orbits are still bound deeply enough by about 8 MeV (see Fig. 2(b)). In contrast, the validity may become questionable beyond 30Ne for the aforementioned reason. This is a very important and intriguing point, and should be clarified in the future by combining a realistic nuclear force (e.g. G-matrix interaction) with loosely bound single-particle wave functions. Thus, the physical significance of the lf,,2-ld3,2 crossing seen at far right in Fig. 2(b) is not very certain from the viewpoint of reality. We now move on to E2 matrix elements, using, for simplicity, the same effective charges, 0.5e and 1.3e for neutrons and protons, respectively, as in Ref. [S]. Fig. 3 shows the B (E2; 0: -+ 2:) values. One finds an enormous increase from 34Si to 32Mg. This is in accordance with the sudden drop of the 2: level. Note that the B(E2) values for 32Mg and 30Ne are sensitive to the value of neutron effective charge, whereas those for heavier isotones are not. Much smaller B(E2)‘s are reported in
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500
2 “; -
400 i
+n300 +-
t
1
0-
“‘Ar
36S
34Si
32Mg 3%e
Fig. 3. B(E2; 0: + 2:) values of the present calculation (solid line) and the sd-shell moue1 calculation with the USD interaction (dashed line). The squares show the observed values.
Refs. [lo, 111, probably due to the more restrictive configuration space. The above B(E2)‘s have been measured for 38Ar and 36S with small values [23], which are in good agreement with the present and the sd-shell model calculations (see Fig. 3). These small values suggest, together with higher 2: excitation energies, that the 0: and 2: states are spherical in these nuclei. On the other hand, Fig. 3 shows that the 0: and 2: states are connected by stronger E2 transition in 32Mg and 30Ne indicating stronger “deformation” as discussed for 30Ne also in Ref. [12]. Such stronger defoimation occurs, as mentioned above, due to larger number of neutrons lifted to If,,, and holes created in the sd shell. Similar “deformed” states are found at higher excitations energy in heavier N = 20 isotones, while 34Si is in the intermediate situation [24]. Fig. 3 includes the result of very recent measurement of B(E2; 0: + 2:) value of 32Mg by Motobayashi et al. [25]. The observed value comes right to the present prediction. We would like to stress that the present calculation has been done much before such an experiment became feasible. Thus, the strong deformation of 32M has been confirmed in terms of E2 transitions. We shall now focus on 32Mg which is the kfy nucleus in this study because it is located at the entrance to the unstable regime. We have carried out a further extended calculation including up to 4p4h excitations to the pf shell for 0’ and 2’ states of 32Mg. Excitations not only from ld3,2 but also from Id,,, and 2sIi2 are included, while really negligible components are excluded. The effective interaction and the s.p.e.‘s are taken to be the same, for simplicity, as those used in the 2p2h calculation. The resultant pf-shell occupation number for the 0: (2:) state is 2.8 (2.9), while it was 1.8 (1.9) in the 2p2h calculation. Since the effective two-body interaction should be weaker in general in the 4p4h calculation than in the 2p2h, the present 4p4h result overestimates more or less the mixing of the pf shell. Nevertheless, the significance of the 4p4h excitation is clear. We mention that the 4p4h excitation has never been included for 32Mg in previous studies [lo, 121 in spite of the severer truncation that only the excitations from ld3,2 were included. Although the occupation is changed from the 2p2h to the 4p4h cases, the excitation energies and E2 transitions are not sensitive; the 2: level is changed from 0.98 MeV for the 2p2h to 1.17 MeV for the 4p4h case, and B(E2; 0: + 2:) is changed from 448 (e2 fm4) to 449(e2 fm4). The structure of the lowest states of
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32Mg contains an intriguing aspect; the pairing(-type) interaction creates valence particles in the pf shell and holes in the sd shell, both of which promote deformation significantly. Through this process, the pairing(-type) interaction enhances the deformation to a certain extent. This is in contrast to usual cases where the pairing works simply against deformation. Because of the importance of the pairing, the rotational band is not well established in the present case, as remarked also by Poves and Retamosa [ll], and the applicability of the deformed HF can be questioned. We comment that, if two Nilsson levels with the If,,, origin would have come down into the ground state of 32Mg, the pf-occupation number should have been near four in the 4p4h calculation, whereas this is not the case actually. We comment briefly on the spectrum of 34Si in Fig. 1. For the 2: level, a large difference is seen between theory and experiment. The possible reasons are as follows. The inclusion of the 4p4h excitation probably pushes down, through the pairing, the ground state of 34Si, more than most other lowest states. The lowest excited level of 34Si is thus expected to be raised by this mechanism. This excitation energy may be also sensitive to s.p.e.‘s due to different configurations from the ground state. The binding energies have also been calculated. The largest deviation of the 2p2h calculation from experiment is 2.5 MeV, which is about l-2% of the total binding energy. The deviation becomes larger naturally for the 4p4h calculation. Since the binding energies are more sensitive to the interaction than the wave functions, one has to refine interaction. We note that the interaction is not adjusted at all in the present calculation, but should be improved by comparing more detailed and systematic calculations to more data which are unavailable presently.
3. Variational shell model In unstable nuclei with N b Z, neutrons occupy single-particle orbits from the bottom up to weakly bound states in the mean potential for nucleons. In addition, if the number of nucleons is small, the mean potential itself may not be very static. Thus, the shell structure of light neutron-rich unstable nuclei can be different from that of stable nuclei, and provides various intriguing problems. One good example of such problems is the ground state of “Be where the ratio N/Z is nearly two. The naive picture of this state is that neutrons occupy Is,,~ and lp,,, completely, while lpliz holds just one neutron. The ground-state J” is then expected to be i-, whereas experimentally it is known to be 3’ [ 261. Although this has been pointed out by Talmi and Unna more than three decades ago [27], this anomaly has remained a challenge to theories. For instance, the HF approach, which normally gives a satisfactory description of nuclei around the ground state, has shown to fail in this case [28]. Recently, a new framework has been proposed to handle many-body systems containing loosely bound particles in general [29]. In this section, we present the outline of this framework, called hereafter variational shell model (VSM), and its first result applied to r’Be. This nucleus is known by the anomalous ground state, as stated earlier, and its neutron halo [30]. The result of the VSM with the Skyrme SIII interaction [31,32] nicely reproduces these exotic features, as shown later. In conventional shell-model calculations, the single-particle wave functions are provided by other methods such as the harmonic oscillator potential, HF calculation, etc. In other words, one obtains the single-particle wave functions from a certain mean field potential, assuming stable single-particle motion. The shell model calculation is then carried out, in order to treat the
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“residual” interaction between nucleons moving on such stable single-particle orbits. The validity of this procedure can be questioned, however, in certain neutron-rich unstable nuclei where the last neutrons (or protons) are forced to occupy loosely bound or even unbound orbits of the mean potential. We shall now formulate the VSM; One parametrizes the radial part, R, of the single-particle wave function, 4, in terms of variational parameters denoted collectively by (a}; 4i(X; (a>) E Ri(r; {ct})[Y"'
X
u](j)
3
where i stands for the index of a single-particle orbit, x denotes symbolically all relevant coordinates, and Ymeans the distance from the center of the nucleus. Here Y and u imply the spherical harmonics and spin wave function, respectively, and are coupled to the total angular momentumj. Multi-nucleon wave functions are constructed from the single-particle bases 4i’s:
Iyi; la)) K d{dh,(Xli{~})dh,(X2; (a})&,(X3; {~})..*}IO)
,
(2)
where 1Yi) is the ith Slater determinants, the k’s stand for single-particle bases forming Yi, & denotes an antisymmetrizer, and IO) means the vacuum. One then calculates matrix elements of the Hamiltonian for these multi-nucleon wave functions, and diagonalizes the matrix. The lowest eigenvalue with the angular momentum J and parity n: can be given as a function of (a}, E(J”; (a}), and its wave function is written with amplitude ci as
IJ”; {a>)= 1 Ci(J”; (a})1Yi;
{a})
.
i
One carries out the variation for this E(J”; (a}) with respect to {a>, by searching the minimum; FE(J”; {a})/6(a} = 0. Th e single-particle basis $i(r; {a>), the configuration mixing amplitude are thus determined simultaneously in the VSM. ci and the energy of the nucleus E(J";(a}) In practical VSM calculations, the effective nucleon-nucleon interaction has to be chosen so that the density in the interior region satisfies the saturation. The Skyrme interaction [31] is useful for this purpose, and the SIII interaction [32] is actually adopted in the present work. The SIII interaction contains, in its original form, a three-body repulsive force, which prevents the nucleus from collapsing. We take, in this work, a simplified but widely used version of SIII, where the three-body term is replaced by a density-dependent two-body repulsive term. This version of SIII still brings about the density saturation despite its practical simplicity. Thus, the Hamiltonian consists of the kinetic energy term and the SIII interaction. There are two methods for the parameterization of the Ri(r; {a}>'~ in Eq. (1). In the first method, a “black box” is used for generating radial wave functions. For instance, solutions of Woods-Saxon potential can be used by changing the values of radius, depth, diffuseness etc. of the potential [33]. This method is simple and useful, but appears not to be accurate enough for reproducing halo properties. We therefore use, in this work, the second method, which is the direct in Eq. (1). In this method, we start from a variational principle. At a local variation of Ri(r; {a]) minimum, any infinitesimal variation of Ridoes not change the total energy. This leads us to a set of coupled differential equations with Lagrange multiplier for keeping orthonormalities of the Ri'S. This differential equations of VSM contain terms like kinetic energy and average potential from
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5-1/2+
0 -
. . . . . . . . =1/2’
. . _-1/2-I.
-
-
1/2+
55‘
-
4
_
i
(b) VW -1/2+
(J _ . . . . . . . . -l/2_
. . -l/2_.
-
-1/2+
“Be Fig. 4. Energy levels of “Be and 13C obtained to +-.
by (a) experiment,
% (b) VSM, and (c) HF-shell.
Levels are shown relative
other nucleons in addition to other terms, for instance those shifting nucleons from one orbit to another, and hence shows certain similarities to HF equations. It is not easy, however, to solve the VSM equation because of partial occupancies and jumping of nucleons from one orbit to another. A numerical procedure for this solution has been developed. In the following, we present its first result. Details of the procedure and the results will be published elsewhere at length [34]. Fig. 4 shows the lowest 3’ and 3- energy levels of iiBe, obtained by experiment and from VSM. The 3’ level is shown relative to I-. 2 The VSM calculation is performed with SIII without any adjustment. The configuration space for i- is comprised of four (seven) nucleons in the 1s (lp) shell, whereas one nucleon is raised from the lp shell to the 2sld shell for 3’. The isospin is conserved. The same levels of i3C are included for comparison in Fig. 4. The observed ground state of “Be is i+, and VSM reproduces it correctly. Moreover, the VSM reproduces the change from “Be to 13C where the ordering of 3’ and i- becomes normal. It
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should be remarkable that the SIII interaction, which has been utilized for the description of a wide variety of stable nuclei, remains useful for an unstable nucleus such as ‘iBe. It is of particular importance that, in going from ‘iBe to 13C, the energy of the i’ level relative to i- is increased in the VSM to the about right amount. One cannot claim that the agreement between experiment and calculation should be perfect in Fig. 4 because SIII is designed so as to be simple yet useful for a global description of ground and near-ground-state properties. We shall next discuss the physical mechanism responsible for lowering the 3’ state in “Be. The major components of the lowest $’ state in “Be can be written as
13’) = 51x 0,‘12~1,2>Y + MC x ll~5,2hl(1’2) 9
(4)
where rl and iZ are amplitudes, 0,’ and 2: stand for the lowest core (i.e., “Be) state of given J”, and I)y means neutron single-particle orbits. The resulting values are il - 0.74 and c2 - 0.63. Note that the VSM (experimental) value of the 2: energy level of the core, “Be, is 3.2 (3.4) MeV, and that its B(E2; 0: + 2:) is calculated as 54 (e” fm4) with effective charges eP = 1.5e and e, = OSe [35] in agreement with the experimental value 52 & 6 (e” fm4) [36]. The coupling between the two components on the right-hand side of Eq. (4) plays an indispensable role in “Be in order to produce the bound $’ state [33]. Without this coupling, the system is not bound as discussed below in the context related to HF calculation. The physical meaning of this coupling is that the motion of the core surface (i.e., “Be) is coupled dynamically with the motion of the particle (i.e., a neutron in 2sli2 or lds,2). It does not matter presently whether the surface motion is vibrational or rotational. The total system then becomes bound, and the above mechanism can be referred to as dynamical mean field. We emphasize that the VSM includes this coupling effect in determining @iin Eq. (l), and that this coupling is precisely what was expected when the VSM was proposed [33]. On the other side, the single-particle explanation is not successful as has been pointed out by Millener and Kurath [37]. As discussed below, if the single-particle motion is restricted on either 2sl,2 or Id,,, (i.e., no orbital change), the nuclear force does not supply sufficient binding, and thereby the total system is left unbound. In this situation, a nucleon cannot complete the circle on either 2s1,2 or Id,,, in the classical picture. We note that the structure of the wave function of the anomalous ground state can be examined by g-factor or transfer reactions. The calculated value of the g-factor of the ground state is - 3.0,~~ where the free nucleon g-factors are assumed. In the usual single-particle picture, this is -3.&, i.e., free neutron spin g-factor. The mixing in Eq. (4) may be too strong due to SIII in the present calculation, and the measurement of the mixing amplitudes provides us with precious information for modifying the effective interaction. We discuss briefly the relation to the HF approach. Fig. 4 shows also the result of the so-called open-shell-HF plus shell-model (HF-shell) [38]. The HF-shell method is an extension of the usual spherical HF to open-shell nuclei; one starts from a set of single-particle wave functions, and carries out a shell-model calculation which produces the occupation number of each single-particle orbit. The spherical HF calculation is then carried out by using these occupation numbers as input, giving rise to a new set of single-particle wave functions. The shell-model calculation is repeated with the new single-particle wave functions. One iterates this process until the result is converged. The HF-shell works well for most of the stable nuclei, whereas it shows difficulties in nuclei where highest orbits occupied by neutrons turn out to be unbound in the HF-shell calculation. In such cases, the whole scheme breaks down. In order to avoid this obstacle, an infinitely high wall is
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introduced sometimes [28]. All nucleons are evidently “bound” inside the wall. We carry out such a calculation with the wall at r = 20 fm. Using the SIII interaction, the energies of 2sliZ and Id,,, are obtained for I’Be as 0.79 and 1.96 MeV, respectively. These are positive energy solutions bound by the artificial wall, and their wave functions are extremely spread. Fig. 4 includes the lowest i’ and i- energy levels of “Be and 13C calculated by the HF-shell. For 1‘Be, the HF-shell clearly fails to reproduce the anomalous ground state, while the result is not too bad for 13C. The latter case indicates that the HF-shell is reasonable for stable nuclei. We shall briefly discuss the origin of this failure, taking the usual HF with two-body interaction for simplicity. The HF equation is written in the form Tmi
+
1
( vjm,
ji
-
I/mj, ji}
=
&iJmi
Y
(5)
j=l,N
where Tmiis the kinetic term, I’ij,kl stands for matrix element of a two-body interaction, si denotes an eigenvalue, and 6ij is the Kronecker’s symbol. The two-body interaction matrix elements in Eq. (5) are limited to diagonal ones. This limitation remains in the HF-shell calculation. If si is a large negative number, single-particle wave functions are determined well by HF or HF-shell. On the other hand, if I’ij,ij - I/ij,ji is canceled almost completely by ‘I& as is the case for loosely bound orbits, Eq. (5) does not produce optimum single-particle wave functions because of other more significant contributions. This is the case for 2sl,2 and Id,,, of “Be, where the off-diagonal matrix element shifting a nucleon between 2sl12 and ld5,z plays a crucial role as seen in Eq. (4). The deformed HF may be better, but it assumes static deformation. It is of interest to see the result of angular-momentum projected deformed HF. Although a Nilsson calculation with large deformation has not shown the inversion of the 3’ and j- states [39], the present work and Ref. [39] are qualitatively consistent on the role of deformation. We now present the density profile obtained by the VSM calculation. Fig. 5 shows the matter density profile calculated by the VSM and that measured by Fukuda et al. [30]. One sees a reasonable agreement between them. Although the calculation is to be modified by corrections such as center-of-mass and nucleon-size ones, it is unlikely that such corrections change the result drastically. We point out that the neutron halo observed for “Be is clearly seen in the VSM. The calculation is made with SIII without adjustment. In fact, the present calculation reproduces the neutron halo of “Be without adjustment for the first time. The halo is primarily due to the slowly damping tail of Ri for i = 2sliZ where the radial dependence is very different from that in the harmonic oscillator potential. Wave functions of other orbits, including Id,,, , are quite similar to those in the harmonic oscillator potential. The halo formation has been pointed out by Hansen and Jonson as a consequence of small separation energy of neutron [40]. If the configuration mixing takes place, however, one cannot relate the single-particle wave function directly to the separation energy of the nucleus. In the VSM, the halo arises as a result of coherence and competition among various effects including configuration mixing. Fig. 5 shows also the density profile of 13C, where one finds again the halo for the first 3’ state. In other words, the halo structure of the 3’ state is carried over from “Be to 13C. This can be viewed as halo universality, if one includes excited states. The rms matter radius is calculated for some Be isotopes to show the anomalous radius of ‘iBe. Fig. 6 shows the result in comparison with experiment [41]. One sees that “Be has larger radius, and the VSM result is in good agreement with experiment. The neutron halo in 14Be is seen also in
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lo-‘-
---Matter
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VSM (S3)
10-z
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X .<
10-4
% 1o-6 2 10-B 1o-7 4
6
8
r(fm)
10
9
10
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12
14
A
Fig. 5. (a) Density profile of the lowest i’ state of 1‘Be. Experimental matter (hatched area), VSM matter (solid line) and proton (dotted line) densities are shown. (b) Density profile for lowest i’ (solid line) and f- (dashed line) of 13C calculated by VSM. Fig. 6. Experimental (points) and calculated (line) rms matter radius for 9- 14Be.
Fig. 6 in both experiment and VSM calculation. The El transition between the first 4’ and +states is known to be strong in r’Be [42]. The pres ent calculation produces relatively strong El matrix element, but does not reach the observed value [42]. The SIII interaction is probably not suitable for the study of this El, because this strong El transition seems to be sensitive to subtle cancellations of several matrix elements 1421. By the calculation of the center-of-mass kinetic energy, it is confirmed that the mixture of spurious center-of-mass motion is negligible for the states discussed in this work.
4. Direct neutron capture into a halo orbit The 2sliz neutron orbit of r3C has a halo structure, as stressed in Section 3 with the concept of the halo universality. This halo structure has been taken into account for describing the direct neutron capture of incoming neutrons in the keV region [43]. This 2sliz neutron halo in 13C is shown to play an indispensable role in reproducing a recent data of 12C(n, y)13C [44]. In this direct
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capture, the incoming p wave is converted into the halo 2~r,~ orbit by emitting the El y ray. This is in sharp contrast with the s-wave capture.
5. Summary
We have performed large-scale shell-model calculations for N = 20 isotones with 10 I 2 I 20 within a single framework including the lower pf shell. The N = 20 closed-shell structure then disappears for the nuclei with Z I 12, and the sd and pf shells melt into one shell. This vanishing of N = 20 magic structure makes these nuclei much softer towards dynamical deformation, producing anomalous properties. The anomalous change of B(E2; 0: + 2:) has been predicted, as an example, and has been confirmed experimentally quite recently [25]. The VSM has been proposed as a scheme to describe the structure of nuclei containing loosely bound nucleons. While the VSM is almost equivalent to HF-shell for stable nuclei, the VSM plays an indispensable role in some neutron-rich unstable nuclei. One of the implications of the VSM is the dynamical determination of the single-particle wave functions, which results in the neutron halo and anomalous ground state for “Be. We are currently working on calculations on neighboring nuclei, and also on improving the effective interaction suitable for unstable nuclei as well as stable ones. The direct neutron capture in the keV region is shown to be a good tool for clarifying the halo structure of the capturing neutron orbit.
Acknowledgements
The authors are grateful to Professor T. Sebe for providing a new computer code which enabled us to carry out the calculations presented in Section 2. They thank Professor A. Arima for inspiration at the initial stage of the work shown in Section 2. They acknowledge Professor H. Sagawa and Dr. H. Nakada for valuable discussions. They are grateful to Professor A. Gelberg for reading a part of the manuscript, and to Dr. D.J. Millener for useful comment. They thank Professor M. Ishihara for valuable inspirations and discussions, particularly, for the work shown in Section 4. This work is supported in part by Research Center for Nuclear Physics, Osaka University. Numerical calculations were carried out in part at Computer Center and Meson Science Laboratory, both of which belong to the University of Tokyo. This work is supported in part by Grant-in-Aid for General Scientific Research (Nos. 01540231 and 0484012) and by Grant-in-Aid for Scientific Research on Priority Area (No. 05243102) from the Ministry of Education, Science and Culture.
References [l] I. Tanihata, Nucl. Phys. A 552 (1991) 27%. [2] R. Klapisch et al., Phys. Rev. Lett. 31 (1973) 118. [3] C. Thibault et al., Phys. Rev. C 12 (1975) 644.
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T. Otsuka aud T. Sebe, Phys. Lett. B 296 (1992) 279. [S] B.H. Wildenthal, Prog. Part. Nucl. Phys. 11 (1984) 5; B.H. Wildenthal and B.A. Brown, Ann. Rev. Nucl. Sci. 38 (1988) 29. [6] G. Huber et al., Phys. Rev. C 18 (1978) 2342. [7] D. Guillemaud-Mueller et al., Nucl. Phys. A 426 (1984) 37. [S], B.H. Wildenthal and W. Chung, Phys. Rev. C 22 (1980) 2260. [9] A. Watt, R.P. Singhal, M.H. Storm and R.R. Whitehead, J. Phys. G 7 (1981) L 145. [lo] A. Poves and J. Retamosa, Phys. Lett. B 184 (1987) 311. [11] A. Poves and J. Retamosa, Nucl. Phys. A 571 (1994) 221. [12] E.K. Warburton, J.A. Becker and B.A. Brown, Phys. Rev. C 41 (1990) 1147. [13] X. Campi et al., Nucl. Phys. A 251 (1973) 193. [14] H. Nakada, T. Otsuka and T. Sebe, Phys. Rev. Lett. 67 (1991) 1086. [15] H. Nakada, T. Sebe and T. Otsuka, Nucl. Phys. A 571 (1994) 467. [16] T.T.S. Kuo and G.E. Brown, Nucl. Phys. 85 (1966) 40. [17] T.T.S. Kuo and G.E. Brown, Nucl. Phys. All4 (1968) 241. [18] D.J. Millener and D. Kurath, Nucl. Phys. 73 (1965) 1. [19] A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 1 (Benjamin, New York, 1969) p. 322. [20] D.H. Gloeckner and R.D. Lawson, Phys. Lett. B 53 (1974) 313. [Zl] P.M. Endt. Nucl. Phys. A 521 (1990) 1. [22] M.H. Storm, A. Watt and R.R. Whitehead, J. Phys. G 9 (1983) L 165. [23] S. Raman, At. Data Nucl. Data Tables 36 (1987) 1. [24] P. Baumann et al., Phys. Lett. B 228 (1989) 458. [25] T. Motobayashi et al., private communication. [26] C. Michael Lederer and VS. Shirley, eds., Table of Isotopes (Wiley, New York, 1978). [27] I. Talmi and I. Unna, Phys. Rev. Lett. 4 (1960) 469. [28] T. Hoshino, H. Sagawa and A. Arima, Nucl. Phys. A 506 (1990) 271. [29] T. Otsuka, N. Fukunishi and H. Sagawa, Phys. Rev. Lett. 79 (1993) 1385. [30] M. Fukuda et al., Phys. Lett. B 268 (1991) 339. [31] D. Vautherin and D.M. Brink, Phys. Rev. C 5 (1972) 626. [32] M. Beiner et al., Nucl. Phys. A 238 (1975) 29. [33] T. Otsuka, N. Fukunishi and H. Sagawa, in: Structure and Reactions of Unstable Nuclei, eds. K. Ikeda and Y. Suzuki (World Scientific, Singapore, 1991) p. 100. [34] N. Fukunishi and T. Otsuka, to be published. [35] A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. 2 (Benjamin, Reading, 1975). [36] S. Raman et al., At. Data Nucl. Data Tables 36 (1987) 1. [37] D.J. Millener and D. Kurath, Nucl. Phys. A 255 (1975) 315. [38] G.F. Bertsch, B.A. Brown and H. Sagawa, Phys. Rev. C 39 (1989) 1154. [39] I. Ragnarsson et al., Nucl. Phys. A 361 (1981) 1. [40] P.G. Hansen and B. Jonson, Europhys. Lett. 4 (1987) 409. [41] I. Tanihata et al., Phys. Lett. B 206 (1988) 592. [42] S.S. Hanna et al., Phys. Rev. C 3 (1971) 2198; D.J. Millener et al., Phys. Rev. C 28 (1983) 497. [43] T. Otsuka, M. Ishihara, N. Fukunishi, T. Nakamura and M. Yokoyama, Phys. Rev. C 49 (1994) R2289. [44] T. Ohsaki et al., Astrophys. J. 422 (1994) 912.
REPORTS
Physics Reports 264 (1996) 297-310
Nuclear mean field on and near the drip lines Takaharu
OtsukaaTb*, Nobuhisa
Fukunishib
aDepartment of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113, Japan bRIKEN, Hirosawa,
Wako-shi, Saitama 351-01, Japan
Abstract
We discuss two subjects related to the structure of nuclei near the drip lines. The first is the vanishing of N = 20 magic structure in Z < N = 20 nuclei. Large-scale state-of-the-art shell-model calculations with 2sld and lower 2plf shells are shown to present a unified description of N = 20 isotones with Z = 10-20, covering both stable and unstable nuclei. The calculations demonstrate that, although the N = 20 closed-shell structure remains for Z 2 14, the N = 20 closed-shell structure vanishes naturally towards nuclei with Z I 12, giving rise to various anomalous features including those in 32Mg and 31Na. It is suggested that, in these nuclei, the deformed mean field overcomes the shell gap created by the spherical mean potential. Furthermore, the almost perfect agreement with a recent experiment is presented for the B(E2; 0: + 2:) value of 32Mg. The second part is devoted to the mean field for loosely bound neutrons. The variational shell mode1 (VSM) is explained with an application to the anomalous ground state of “Be. The VSM has been proposed recently to describe the structure of neutron-rich unstable nuclei. Contrary to the failure of spherical Hartree-Fock, the anomalous 2’ ground state and its neutron halo are reproduced with Skyrme SIII interaction. This state is bound due to dynamical coupling between the core and the loosely bound neutron which oscillates between 2s1,2 and ldS,2 orbits. The direct neutron capture is discussed briefly in its relation to the neutron halo.
1. Introduction The recent developments of radioactive nuclear beams are opening a new rich field of nuclear physics: the structure of neutron-rich unstable nuclei. The observation of the neutron halo [l] is an example. In this talk, we discuss the structure of neutron-rich unstable nuclei characterized by Z 4 N with Z(N) being the proton (neutron) number. Among the various topics in the field, we focus upon (i) the vanishing of the N = 20 magic shell and the emerging of large deformation in nuclei with Z < N = 20, and (ii) the single-particle motion of nucleons in exotic circumstances such as extremely neutron-rich nuclei. 2. Vanishing of N = 20 magic structure Since the pioneering work on exotic sodium isotopes by Klapisch et al. [2] and Thibault et al. [3], experimental studies on N = 20 isotones have been conducted in spite of their technical 0370-1573/96/$9.50 0 1996 SSDI 0370-1573(95)00044-5
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difficulties. On the other hand, a unified theoretical description, particularly covering from stable (2 - 20) to unstable (Z - 10) regions within a single framework, has been missing. We shall report the result of such a unified calculation of N = 20 isotones from 40Ca to 30Ne in terms of the shell model with a realistic effective interaction [4]. Stable sd-shell nuclei have been studied extensively in terms of the shell model with valence particles in the sd shell, as will be referred to as the sd-shell model hereafter. Although the sd-shell model, in particular the one with the effective (USD) interaction of Wildenthal [S], has been highly successful for stable sd-shell nuclei, various anomalies have been observed in exotic nuclei with Z 4 N = 20 including the anomalous ground state spin of 31Na [6] and the anomalously low first 2+ state of 32Mg [7]. The above anomalies have been studied in shell-model [8-123 and Hartree-Fock (HF) calculations [13]. We shall demonstrate in this report that the anomalies are explained, without any specific adjustment for unstable exotic nuclei, by the same Hamiltonian as the one used for stable nuclei, while large-scale shell-model calculations appear to be more crucial in unstable nuclei, because of stronger mixing between the sd and pf shells. The present calculation includes the configurations made up by the three sd-shell orbits, and furthermore allows excitation of two nucleons into If,,, and 2p 312 orbits. The excitation into the pf shell is extended later up to four nucleons for 32Mg, where the size of the calculation is quite huge with the maximum M-scheme dimension 9 10 878 for the O+ states. Such large-scale calculations have become feasible by a state-of-the-art shell-model program for the supercomputer coded by Sebe [14, 151 recently. This report thus announces, incidentally, the availability and significance of such a shell-model code. The effective nucleon-nucleon interaction used here consists of three parts, For matrix elements within the sd shell, we use the USD interaction [S], which has been obtained from the G-matrix interaction of Kuo-Brown [ 161 with slight empirical refinement. The USD interaction is the most successful effective interaction for the sd-shell model. We use the Kuo-Brown G-matrix interaction [17] for matrix elements within the pf shell and for matrix elements involving both the sd-shell orbits and the If,,, orbit. The remaining part gives only a very minor contribution to the primary results, and we use the Millener-Kurath interaction [lS]. All two-body interaction matrix elements including non-sd parts are scaled by the mass number dependence A-o.3 in the same way as the USD interaction [S]. In the USD interaction, core polarization effects are assumed to be included by renormalization of two-body matrix elements. However, we treat explicitly the If,,, and 2~3~2 orbits as active orbits in this calculation. The renormalization effects in the USD interaction due to these orbits are subtracted by using the second order perturbation so as to avoid double counting. All nucleons on the top of the N = Z = 8 closed shell are treated explicitly as valence particles. The single-particle energies (s.p.e.‘s) are assumed to be constants, independent of the nucleus as in the sd-shell model with the USD interaction, and are actually adjusted to reproduce the observed energy spectra of 41,40,3gCa. The configuration space here includes all configurations, excluding those really negligible, up to 3-particle-2-hole (3p2h) for 41Ca, 2p2h for 40Ca and 2p3h for 3gCa with respect to the N = Z = 20 closed shell. By using the experimental energy levels of s- and $for 41Ca, and those of 4’ and f’ for 39Ca [19], we adjust the s.p.e.‘s, which are defined to include the kinetic energies and the effects due to the interaction with the N = Z = 8 core nucleons but not the interaction with the sd- or pf-shell nucleons. The adjusted values are -2.97, 1.66, - 5.88,2.84 MeV for 2sr12, ld3iZ, lf7,2, 2p3,2, respectively. For Id,,,, we use the same s.p.e. as in the USD
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calculation [S], i.e., - 3.95 MeV. Since there are twelve valence neutrons and some valence protons in the following calculations, these orbits become much more bound and their ordering is changed, due to the interaction with the nucleons in the sd and pf shells. Note that the above values for the sd-shell orbits do not differ significantly from the s.p.e.‘s of the USD interaction [S]. As already mentioned, since a sufficient number of valence nucleons in the sd shell are assumed, the present set of s.p.e.‘s should work basically for nuclei which are not extremely far from 40Ca, but not necessarily for much lighter nuclei, for instance r’0. Note that the mass number difference between 40Ca and “0 is almost three times greater than that between 40Ca and 32Mg. The present parametrization, constant s.p.e.‘s and the A-o.3 scaling for the two-body interaction, should have certain limitations. A more general prescription should be studied in order to cover a wider range including 40Ca, 32Mg, r’0, etc. The spurious center-of-mass motion is removed by using the approximate method of Gloeckner and Lawson [20]. The positive-parity energy levels are shown in Fig. 1 for even-A nuclei with N = 20 and 10 I 2 I 16, in comparison with experiments [21]. As mentioned above, the calculations in Fig. 1 include up to 2p2h excitations into the pf shell. The excited states are found only above Ex = 2 MeV for 34Si and 36S, both experimentally and theoretically. The calculated first 2+ level comes down sharply to Ex = 0.98 MeV in going from 34Si to 32Mg, consistently with experiments. This dramatic change is one of the anomalies being studied, and has never been described before in a unified way covering both N - 2 stable and N $ Z unstable nuclei with mixing of the sd and pf shells. The first 2+ state of 32Mg is predicted as Ex = 1.68 MeV in the sd-shell model with the USD interaction.
= -
-4 -
2
zy -2
--
-2
-1
:I
-;
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2
=1 =;
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-0
-0
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(Exp.) and calculated
(Th.) energy levels of the positive-parity
-0
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Exp.
3%e states of even-2
N = 20 isotones.
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Fig. 2. (a) Occupation number of the pf shell in the ground states. The squares (crosses) indicate neutrons (protons). Lines are drawn to guide the eye. (b) Effective single-particle energy of the neutron orbits evaluated at subshell-closed nuclei (see the text). The diamonds (crosses) denote the values for If,,, (ld3i2) orbit. Lines are drawn to guide the eye.
Fig. 1 indicates also that the level density in the low-energy region increases for Z I 12. In the sd-shell model of N = 20 nuclei, only protons are active and neutrons are frozen in the N = 20 closed shell. The structure of low-lying states therefore can be interpreted in terms of the seniority of protons. This tendency remains approximately for stable nuclei in the present calculation. On the other hand, the structure of the low-lying states of Z I 12 nuclei is determined not only by valence protons in the sd shell, but also by neutrons lifted to the pf shell and by neutron holes created in the sd shell. This means that N = 20 is no longer the magic number in these nuclei. We shall present results for odd-A N = 20 isotones elsewhere because the length of this article is limited. We only mention that, within the positive-parity states up to 2p2h excitations, the present calculation produces the anomalous $‘-3’ spin sequence of 3‘Na, whereas the sd-shell model with the USD interaction puts, because of the Id,,, occupancy of the last proton, the 3’ level below $+, contrary to experiment. The situations in 35P and 33A1 are normal, for which both the sd-shell model and the present calculation work equally well. As for magnetic moments, the experimental value for the above 3’ state of 31Na is 2.28~~ [6], while the present calculation yields 2.20,~~~ with free nucleon g-factors. The occupation number of the pf shell in the ground state is shown in Fig. 2(a) for 40Ca-30Ne. For stable nuclei 4oCa-35P, the total occupation number of protons and neutrons remains small (-0.6), while the neutron occupation number increases as Z decreases below Z = 15 (““P). The small and almost constant pf-shell occupation number in N - Z nuclei is quite reasonable because
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of the non-vanishing matrix elements connecting the sd and pf shells. This small mixing of the pf-configuration changes only slightly the energy spectra of 40Ca-35P from those of the sd-shell model. The continuous growth of the neutron pf-shell occupation probability on the N 9 2 side indicates that the N = 20 closed shell is broken gradually below 2 - 15. This continuous change of the probability conflicts with the truncation used in the unmixed calculation in Ref. [12]. The proton pf-shell occupation number decreases as Z decreases, because the proton Fermi level goes away from the pf shell. The effective s.p.e. is introduced as the sum of the original s.p.e. and the expectation value of the two-body interaction with respect to the relevant proton closed subshell plus one neutron particle in lf7,2 or one neutron hole in Id,,,. The actual calculation is made for 40Ca. 36S (2s1,* closed), 34Si (Id,,, closed), and 280 in a way similar to the Hf particle or hole energy. Fig. 2 (b) shows the effective s.p.e.‘s thus obtained. The resultant effective s.p.e. approaches zero with decreasing Z, because the expectation value of the two-body interaction becomes smaller in magnitude as the number of valence protons decreases. Fig. 2(b) shows also that, as Z decreases, the difference between lf7,2 and Id,,, effective s.p.e.‘s decreases gradually, starting from a large value (- 7 MeV) in 40Ca . In 32Mg it reaches a smaller value, which happens to be similar to the one (- 1.5 MeV) adjusted in Ref. [lo]. Note that this small value causes the collapse of the closed shell in Ref. [lo]. In fact, the smaller values of this lf,,2-ld3i2 difference for lighter isotones in Fig. 2(b) imply the vanishing of the N = 20 shell gap towards Z Q N = 20. We stress that this vanishing seems to be a general feature of the G-matrix interaction as discussed for 280 in Ref. [22]. The If 7,2-1d3,2 difference in Fig. 2(b) shows a prominent correlation to the pf-shell occupation number in Fig. 2(a). We note that the effective s.p.e. is not used in the present shell-model diagonalization, since the actual calculation includes all two-body matrix elements. The effective s.p.e. is introduced for schematic demonstration. There is no effect on the effective s.p.e. from the deformation, because it is evaluated with spherical subshells. On the other hand, the smaller lf,,2-ld3,2 energy difference drives the nucleons towards larger mixing of the pf-shell configuration, giving rise to stronger deformation as discussed later. We point out that the binding of neutrons in If,,, and ld3,2 becomes extremely weak for the isotones lighter than 3oNe, e.g., 280 and 2gF. In such nuclei, due to weak binding, the single-particle wave functions of If,,, and Id,,, are probably changed from the ones given by the harmonic oscillator potential, and consequently the G-matrix prescription applied to such loosely bound wave functions produces a new set of matrix elements different from those used so far. The present shell-model approach remains valid up to 32Mg or 31Na where the shell gap begins to be narrower but those orbits are still bound deeply enough by about 8 MeV (see Fig. 2(b)). In contrast, the validity may become questionable beyond 30Ne for the aforementioned reason. This is a very important and intriguing point, and should be clarified in the future by combining a realistic nuclear force (e.g. G-matrix interaction) with loosely bound single-particle wave functions. Thus, the physical significance of the lf,,2-ld3,2 crossing seen at far right in Fig. 2(b) is not very certain from the viewpoint of reality. We now move on to E2 matrix elements, using, for simplicity, the same effective charges, 0.5e and 1.3e for neutrons and protons, respectively, as in Ref. [S]. Fig. 3 shows the B (E2; 0: -+ 2:) values. One finds an enormous increase from 34Si to 32Mg. This is in accordance with the sudden drop of the 2: level. Note that the B(E2) values for 32Mg and 30Ne are sensitive to the value of neutron effective charge, whereas those for heavier isotones are not. Much smaller B(E2)‘s are reported in
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36S
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Fig. 3. B(E2; 0: + 2:) values of the present calculation (solid line) and the sd-shell moue1 calculation with the USD interaction (dashed line). The squares show the observed values.
Refs. [lo, 111, probably due to the more restrictive configuration space. The above B(E2)‘s have been measured for 38Ar and 36S with small values [23], which are in good agreement with the present and the sd-shell model calculations (see Fig. 3). These small values suggest, together with higher 2: excitation energies, that the 0: and 2: states are spherical in these nuclei. On the other hand, Fig. 3 shows that the 0: and 2: states are connected by stronger E2 transition in 32Mg and 30Ne indicating stronger “deformation” as discussed for 30Ne also in Ref. [12]. Such stronger defoimation occurs, as mentioned above, due to larger number of neutrons lifted to If,,, and holes created in the sd shell. Similar “deformed” states are found at higher excitations energy in heavier N = 20 isotones, while 34Si is in the intermediate situation [24]. Fig. 3 includes the result of very recent measurement of B(E2; 0: + 2:) value of 32Mg by Motobayashi et al. [25]. The observed value comes right to the present prediction. We would like to stress that the present calculation has been done much before such an experiment became feasible. Thus, the strong deformation of 32M has been confirmed in terms of E2 transitions. We shall now focus on 32Mg which is the kfy nucleus in this study because it is located at the entrance to the unstable regime. We have carried out a further extended calculation including up to 4p4h excitations to the pf shell for 0’ and 2’ states of 32Mg. Excitations not only from ld3,2 but also from Id,,, and 2sIi2 are included, while really negligible components are excluded. The effective interaction and the s.p.e.‘s are taken to be the same, for simplicity, as those used in the 2p2h calculation. The resultant pf-shell occupation number for the 0: (2:) state is 2.8 (2.9), while it was 1.8 (1.9) in the 2p2h calculation. Since the effective two-body interaction should be weaker in general in the 4p4h calculation than in the 2p2h, the present 4p4h result overestimates more or less the mixing of the pf shell. Nevertheless, the significance of the 4p4h excitation is clear. We mention that the 4p4h excitation has never been included for 32Mg in previous studies [lo, 121 in spite of the severer truncation that only the excitations from ld3,2 were included. Although the occupation is changed from the 2p2h to the 4p4h cases, the excitation energies and E2 transitions are not sensitive; the 2: level is changed from 0.98 MeV for the 2p2h to 1.17 MeV for the 4p4h case, and B(E2; 0: + 2:) is changed from 448 (e2 fm4) to 449(e2 fm4). The structure of the lowest states of
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32Mg contains an intriguing aspect; the pairing(-type) interaction creates valence particles in the pf shell and holes in the sd shell, both of which promote deformation significantly. Through this process, the pairing(-type) interaction enhances the deformation to a certain extent. This is in contrast to usual cases where the pairing works simply against deformation. Because of the importance of the pairing, the rotational band is not well established in the present case, as remarked also by Poves and Retamosa [ll], and the applicability of the deformed HF can be questioned. We comment that, if two Nilsson levels with the If,,, origin would have come down into the ground state of 32Mg, the pf-occupation number should have been near four in the 4p4h calculation, whereas this is not the case actually. We comment briefly on the spectrum of 34Si in Fig. 1. For the 2: level, a large difference is seen between theory and experiment. The possible reasons are as follows. The inclusion of the 4p4h excitation probably pushes down, through the pairing, the ground state of 34Si, more than most other lowest states. The lowest excited level of 34Si is thus expected to be raised by this mechanism. This excitation energy may be also sensitive to s.p.e.‘s due to different configurations from the ground state. The binding energies have also been calculated. The largest deviation of the 2p2h calculation from experiment is 2.5 MeV, which is about l-2% of the total binding energy. The deviation becomes larger naturally for the 4p4h calculation. Since the binding energies are more sensitive to the interaction than the wave functions, one has to refine interaction. We note that the interaction is not adjusted at all in the present calculation, but should be improved by comparing more detailed and systematic calculations to more data which are unavailable presently.
3. Variational shell model In unstable nuclei with N b Z, neutrons occupy single-particle orbits from the bottom up to weakly bound states in the mean potential for nucleons. In addition, if the number of nucleons is small, the mean potential itself may not be very static. Thus, the shell structure of light neutron-rich unstable nuclei can be different from that of stable nuclei, and provides various intriguing problems. One good example of such problems is the ground state of “Be where the ratio N/Z is nearly two. The naive picture of this state is that neutrons occupy Is,,~ and lp,,, completely, while lpliz holds just one neutron. The ground-state J” is then expected to be i-, whereas experimentally it is known to be 3’ [ 261. Although this has been pointed out by Talmi and Unna more than three decades ago [27], this anomaly has remained a challenge to theories. For instance, the HF approach, which normally gives a satisfactory description of nuclei around the ground state, has shown to fail in this case [28]. Recently, a new framework has been proposed to handle many-body systems containing loosely bound particles in general [29]. In this section, we present the outline of this framework, called hereafter variational shell model (VSM), and its first result applied to r’Be. This nucleus is known by the anomalous ground state, as stated earlier, and its neutron halo [30]. The result of the VSM with the Skyrme SIII interaction [31,32] nicely reproduces these exotic features, as shown later. In conventional shell-model calculations, the single-particle wave functions are provided by other methods such as the harmonic oscillator potential, HF calculation, etc. In other words, one obtains the single-particle wave functions from a certain mean field potential, assuming stable single-particle motion. The shell model calculation is then carried out, in order to treat the
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“residual” interaction between nucleons moving on such stable single-particle orbits. The validity of this procedure can be questioned, however, in certain neutron-rich unstable nuclei where the last neutrons (or protons) are forced to occupy loosely bound or even unbound orbits of the mean potential. We shall now formulate the VSM; One parametrizes the radial part, R, of the single-particle wave function, 4, in terms of variational parameters denoted collectively by (a}; 4i(X; (a>) E Ri(r; {ct})[Y"'
X
u](j)
3
where i stands for the index of a single-particle orbit, x denotes symbolically all relevant coordinates, and Ymeans the distance from the center of the nucleus. Here Y and u imply the spherical harmonics and spin wave function, respectively, and are coupled to the total angular momentumj. Multi-nucleon wave functions are constructed from the single-particle bases 4i’s:
Iyi; la)) K d{dh,(Xli{~})dh,(X2; (a})&,(X3; {~})..*}IO)
,
(2)
where 1Yi) is the ith Slater determinants, the k’s stand for single-particle bases forming Yi, & denotes an antisymmetrizer, and IO) means the vacuum. One then calculates matrix elements of the Hamiltonian for these multi-nucleon wave functions, and diagonalizes the matrix. The lowest eigenvalue with the angular momentum J and parity n: can be given as a function of (a}, E(J”; (a}), and its wave function is written with amplitude ci as
IJ”; {a>)= 1 Ci(J”; (a})1Yi;
{a})
.
i
One carries out the variation for this E(J”; (a}) with respect to {a>, by searching the minimum; FE(J”; {a})/6(a} = 0. Th e single-particle basis $i(r; {a>), the configuration mixing amplitude are thus determined simultaneously in the VSM. ci and the energy of the nucleus E(J";(a}) In practical VSM calculations, the effective nucleon-nucleon interaction has to be chosen so that the density in the interior region satisfies the saturation. The Skyrme interaction [31] is useful for this purpose, and the SIII interaction [32] is actually adopted in the present work. The SIII interaction contains, in its original form, a three-body repulsive force, which prevents the nucleus from collapsing. We take, in this work, a simplified but widely used version of SIII, where the three-body term is replaced by a density-dependent two-body repulsive term. This version of SIII still brings about the density saturation despite its practical simplicity. Thus, the Hamiltonian consists of the kinetic energy term and the SIII interaction. There are two methods for the parameterization of the Ri(r; {a}>'~ in Eq. (1). In the first method, a “black box” is used for generating radial wave functions. For instance, solutions of Woods-Saxon potential can be used by changing the values of radius, depth, diffuseness etc. of the potential [33]. This method is simple and useful, but appears not to be accurate enough for reproducing halo properties. We therefore use, in this work, the second method, which is the direct in Eq. (1). In this method, we start from a variational principle. At a local variation of Ri(r; {a]) minimum, any infinitesimal variation of Ridoes not change the total energy. This leads us to a set of coupled differential equations with Lagrange multiplier for keeping orthonormalities of the Ri'S. This differential equations of VSM contain terms like kinetic energy and average potential from
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5-1/2+
0 -
. . . . . . . . =1/2’
. . _-1/2-I.
-
-
1/2+
55‘
-
4
_
i
(b) VW -1/2+
(J _ . . . . . . . . -l/2_
. . -l/2_.
-
-1/2+
“Be Fig. 4. Energy levels of “Be and 13C obtained to +-.
by (a) experiment,
% (b) VSM, and (c) HF-shell.
Levels are shown relative
other nucleons in addition to other terms, for instance those shifting nucleons from one orbit to another, and hence shows certain similarities to HF equations. It is not easy, however, to solve the VSM equation because of partial occupancies and jumping of nucleons from one orbit to another. A numerical procedure for this solution has been developed. In the following, we present its first result. Details of the procedure and the results will be published elsewhere at length [34]. Fig. 4 shows the lowest 3’ and 3- energy levels of iiBe, obtained by experiment and from VSM. The 3’ level is shown relative to I-. 2 The VSM calculation is performed with SIII without any adjustment. The configuration space for i- is comprised of four (seven) nucleons in the 1s (lp) shell, whereas one nucleon is raised from the lp shell to the 2sld shell for 3’. The isospin is conserved. The same levels of i3C are included for comparison in Fig. 4. The observed ground state of “Be is i+, and VSM reproduces it correctly. Moreover, the VSM reproduces the change from “Be to 13C where the ordering of 3’ and i- becomes normal. It
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should be remarkable that the SIII interaction, which has been utilized for the description of a wide variety of stable nuclei, remains useful for an unstable nucleus such as ‘iBe. It is of particular importance that, in going from ‘iBe to 13C, the energy of the i’ level relative to i- is increased in the VSM to the about right amount. One cannot claim that the agreement between experiment and calculation should be perfect in Fig. 4 because SIII is designed so as to be simple yet useful for a global description of ground and near-ground-state properties. We shall next discuss the physical mechanism responsible for lowering the 3’ state in “Be. The major components of the lowest $’ state in “Be can be written as
13’) = 51x 0,‘12~1,2>Y + MC x ll~5,2hl(1’2) 9
(4)
where rl and iZ are amplitudes, 0,’ and 2: stand for the lowest core (i.e., “Be) state of given J”, and I)y means neutron single-particle orbits. The resulting values are il - 0.74 and c2 - 0.63. Note that the VSM (experimental) value of the 2: energy level of the core, “Be, is 3.2 (3.4) MeV, and that its B(E2; 0: + 2:) is calculated as 54 (e” fm4) with effective charges eP = 1.5e and e, = OSe [35] in agreement with the experimental value 52 & 6 (e” fm4) [36]. The coupling between the two components on the right-hand side of Eq. (4) plays an indispensable role in “Be in order to produce the bound $’ state [33]. Without this coupling, the system is not bound as discussed below in the context related to HF calculation. The physical meaning of this coupling is that the motion of the core surface (i.e., “Be) is coupled dynamically with the motion of the particle (i.e., a neutron in 2sli2 or lds,2). It does not matter presently whether the surface motion is vibrational or rotational. The total system then becomes bound, and the above mechanism can be referred to as dynamical mean field. We emphasize that the VSM includes this coupling effect in determining @iin Eq. (l), and that this coupling is precisely what was expected when the VSM was proposed [33]. On the other side, the single-particle explanation is not successful as has been pointed out by Millener and Kurath [37]. As discussed below, if the single-particle motion is restricted on either 2sl,2 or Id,,, (i.e., no orbital change), the nuclear force does not supply sufficient binding, and thereby the total system is left unbound. In this situation, a nucleon cannot complete the circle on either 2s1,2 or Id,,, in the classical picture. We note that the structure of the wave function of the anomalous ground state can be examined by g-factor or transfer reactions. The calculated value of the g-factor of the ground state is - 3.0,~~ where the free nucleon g-factors are assumed. In the usual single-particle picture, this is -3.&, i.e., free neutron spin g-factor. The mixing in Eq. (4) may be too strong due to SIII in the present calculation, and the measurement of the mixing amplitudes provides us with precious information for modifying the effective interaction. We discuss briefly the relation to the HF approach. Fig. 4 shows also the result of the so-called open-shell-HF plus shell-model (HF-shell) [38]. The HF-shell method is an extension of the usual spherical HF to open-shell nuclei; one starts from a set of single-particle wave functions, and carries out a shell-model calculation which produces the occupation number of each single-particle orbit. The spherical HF calculation is then carried out by using these occupation numbers as input, giving rise to a new set of single-particle wave functions. The shell-model calculation is repeated with the new single-particle wave functions. One iterates this process until the result is converged. The HF-shell works well for most of the stable nuclei, whereas it shows difficulties in nuclei where highest orbits occupied by neutrons turn out to be unbound in the HF-shell calculation. In such cases, the whole scheme breaks down. In order to avoid this obstacle, an infinitely high wall is
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introduced sometimes [28]. All nucleons are evidently “bound” inside the wall. We carry out such a calculation with the wall at r = 20 fm. Using the SIII interaction, the energies of 2sliZ and Id,,, are obtained for I’Be as 0.79 and 1.96 MeV, respectively. These are positive energy solutions bound by the artificial wall, and their wave functions are extremely spread. Fig. 4 includes the lowest i’ and i- energy levels of “Be and 13C calculated by the HF-shell. For 1‘Be, the HF-shell clearly fails to reproduce the anomalous ground state, while the result is not too bad for 13C. The latter case indicates that the HF-shell is reasonable for stable nuclei. We shall briefly discuss the origin of this failure, taking the usual HF with two-body interaction for simplicity. The HF equation is written in the form Tmi
+
1
( vjm,
ji
-
I/mj, ji}
=
&iJmi
Y
(5)
j=l,N
where Tmiis the kinetic term, I’ij,kl stands for matrix element of a two-body interaction, si denotes an eigenvalue, and 6ij is the Kronecker’s symbol. The two-body interaction matrix elements in Eq. (5) are limited to diagonal ones. This limitation remains in the HF-shell calculation. If si is a large negative number, single-particle wave functions are determined well by HF or HF-shell. On the other hand, if I’ij,ij - I/ij,ji is canceled almost completely by ‘I& as is the case for loosely bound orbits, Eq. (5) does not produce optimum single-particle wave functions because of other more significant contributions. This is the case for 2sl,2 and Id,,, of “Be, where the off-diagonal matrix element shifting a nucleon between 2sl12 and ld5,z plays a crucial role as seen in Eq. (4). The deformed HF may be better, but it assumes static deformation. It is of interest to see the result of angular-momentum projected deformed HF. Although a Nilsson calculation with large deformation has not shown the inversion of the 3’ and j- states [39], the present work and Ref. [39] are qualitatively consistent on the role of deformation. We now present the density profile obtained by the VSM calculation. Fig. 5 shows the matter density profile calculated by the VSM and that measured by Fukuda et al. [30]. One sees a reasonable agreement between them. Although the calculation is to be modified by corrections such as center-of-mass and nucleon-size ones, it is unlikely that such corrections change the result drastically. We point out that the neutron halo observed for “Be is clearly seen in the VSM. The calculation is made with SIII without adjustment. In fact, the present calculation reproduces the neutron halo of “Be without adjustment for the first time. The halo is primarily due to the slowly damping tail of Ri for i = 2sliZ where the radial dependence is very different from that in the harmonic oscillator potential. Wave functions of other orbits, including Id,,, , are quite similar to those in the harmonic oscillator potential. The halo formation has been pointed out by Hansen and Jonson as a consequence of small separation energy of neutron [40]. If the configuration mixing takes place, however, one cannot relate the single-particle wave function directly to the separation energy of the nucleus. In the VSM, the halo arises as a result of coherence and competition among various effects including configuration mixing. Fig. 5 shows also the density profile of 13C, where one finds again the halo for the first 3’ state. In other words, the halo structure of the 3’ state is carried over from “Be to 13C. This can be viewed as halo universality, if one includes excited states. The rms matter radius is calculated for some Be isotopes to show the anomalous radius of ‘iBe. Fig. 6 shows the result in comparison with experiment [41]. One sees that “Be has larger radius, and the VSM result is in good agreement with experiment. The neutron halo in 14Be is seen also in
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4
lo-‘-
---Matter
(Calc)
10W7-
.,...Proton
(Calc)
10-O
‘...,,
Be isotopes
““...,,_.,2
...“....‘..,.‘....‘..”
1
l-
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VSM (S3)
10-z
g
1o-3
X .<
10-4
% 1o-6 2 10-B 1o-7 4
6
8
r(fm)
10
9
10
11
12
14
A
Fig. 5. (a) Density profile of the lowest i’ state of 1‘Be. Experimental matter (hatched area), VSM matter (solid line) and proton (dotted line) densities are shown. (b) Density profile for lowest i’ (solid line) and f- (dashed line) of 13C calculated by VSM. Fig. 6. Experimental (points) and calculated (line) rms matter radius for 9- 14Be.
Fig. 6 in both experiment and VSM calculation. The El transition between the first 4’ and +states is known to be strong in r’Be [42]. The pres ent calculation produces relatively strong El matrix element, but does not reach the observed value [42]. The SIII interaction is probably not suitable for the study of this El, because this strong El transition seems to be sensitive to subtle cancellations of several matrix elements 1421. By the calculation of the center-of-mass kinetic energy, it is confirmed that the mixture of spurious center-of-mass motion is negligible for the states discussed in this work.
4. Direct neutron capture into a halo orbit The 2sliz neutron orbit of r3C has a halo structure, as stressed in Section 3 with the concept of the halo universality. This halo structure has been taken into account for describing the direct neutron capture of incoming neutrons in the keV region [43]. This 2sliz neutron halo in 13C is shown to play an indispensable role in reproducing a recent data of 12C(n, y)13C [44]. In this direct
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capture, the incoming p wave is converted into the halo 2~r,~ orbit by emitting the El y ray. This is in sharp contrast with the s-wave capture.
5. Summary
We have performed large-scale shell-model calculations for N = 20 isotones with 10 I 2 I 20 within a single framework including the lower pf shell. The N = 20 closed-shell structure then disappears for the nuclei with Z I 12, and the sd and pf shells melt into one shell. This vanishing of N = 20 magic structure makes these nuclei much softer towards dynamical deformation, producing anomalous properties. The anomalous change of B(E2; 0: + 2:) has been predicted, as an example, and has been confirmed experimentally quite recently [25]. The VSM has been proposed as a scheme to describe the structure of nuclei containing loosely bound nucleons. While the VSM is almost equivalent to HF-shell for stable nuclei, the VSM plays an indispensable role in some neutron-rich unstable nuclei. One of the implications of the VSM is the dynamical determination of the single-particle wave functions, which results in the neutron halo and anomalous ground state for “Be. We are currently working on calculations on neighboring nuclei, and also on improving the effective interaction suitable for unstable nuclei as well as stable ones. The direct neutron capture in the keV region is shown to be a good tool for clarifying the halo structure of the capturing neutron orbit.
Acknowledgements
The authors are grateful to Professor T. Sebe for providing a new computer code which enabled us to carry out the calculations presented in Section 2. They thank Professor A. Arima for inspiration at the initial stage of the work shown in Section 2. They acknowledge Professor H. Sagawa and Dr. H. Nakada for valuable discussions. They are grateful to Professor A. Gelberg for reading a part of the manuscript, and to Dr. D.J. Millener for useful comment. They thank Professor M. Ishihara for valuable inspirations and discussions, particularly, for the work shown in Section 4. This work is supported in part by Research Center for Nuclear Physics, Osaka University. Numerical calculations were carried out in part at Computer Center and Meson Science Laboratory, both of which belong to the University of Tokyo. This work is supported in part by Grant-in-Aid for General Scientific Research (Nos. 01540231 and 0484012) and by Grant-in-Aid for Scientific Research on Priority Area (No. 05243102) from the Ministry of Education, Science and Culture.
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