Deformation effects on the structures of N=7 halo nuclei

Nuclear Physics A 765 (2006) 29–38

Deformation effects on the structures of N = 7 halo nuclei J.C. Pei a , F.R. Xu a,b,c,∗ , P.D. Stevenson d a School of Physics and MOE Laboratory of Heavy Ion Physics, Peking University, Beijing 100871, China b Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China c Center for Theoretical Nuclear Physics, National Laboratory of Heavy Ion Collisions, Lanzhou 730000, China d Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom

Received 28 July 2005; received in revised form 30 September 2005; accepted 13 October 2005 Available online 21 October 2005

Abstract The structures of N = 7 isotones have been investigated using the deformed Skyrme–Hartree–Fock model. The approximate particle number projection by means of the Lipkin–Nogami method was used with the blocking effect of the odd neutron taken into account. For nuclei near the drip lines, a volume– surface mixing pairing interaction was employed, which can well reproduce odd–even differences for the long chain of carbon isotopes up to drip lines. The calculated potential-energy curve shows that the first 1 + state of the 11 Be nucleus has a very large prolate deformation. Such a large deformation can result in a 2 + good agreement between the calculated and experimental density distribution of the nucleus. The first 12 + state in 13 C is also calculated to have a large deformation, and the 12 state in 9 He could be a spherical halo

+ attached to a deformed core. The possible halo structure of the 12 neutron orbit in the N = 7 isotones is discussed.  2005 Elsevier B.V. All rights reserved.

PACS: 21.60.Jz; 21.10.Gv; 27.20.+n

1. Introduction The nuclear halo is described as being due to loosely bound nucleon(s) with a large spatial distribution of nuclear matter. The experimental observations of halo nuclei have motivated the * Corresponding author.

E-mail address: [email protected] (F.R. Xu). 0375-9474/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.10.004

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intense studies of nuclear structures of exotic nuclei that have extreme isospin values. To some extent, however, the description of the structures of halo nuclei are still challenging for nuclear structure models that were determined largely with stable nuclei. Among the observed halo nuclei, 11 Be is one of most interesting systems. The nucleus was measured to have a neutron halo structure [1]. Also of interest is that the experimental ground state and the first excited state are 1+ 1− 1− 1+ 2 and 2 states [2], respectively, rather than in the order of 2 and 2 , as expected from meanfield models. The parity-inversion phenomenon of the two lowest states has not been able to be definitely explained by mean-field calculations. Also, the observation of the strong El transition between the two states [3] attracts additional interest in the nucleus. Many theoretical works have attempted to understand the experimental observations in 11 Be. It has been mentioned that conventional mean-field models are not able to self-consistently reproduce the parity inversion. However, mean-field calculations can describe the halo structure of the nucleus [4–6]. Beyond the mean field, particle–core coupling calculations [7–12] have been suggested to explain the parity inversion. In the coupling calculations, the loosely bound neutron couples to the 10 Be core that can be partly in a vibrational excitation. In 11 Be, the importance of deformation has been noted. Earlier works [13,14] pointed out that the large deformation can sig+ nificantly lower the position of the 12 level. Indeed, recent mean-field investigation [4] showed 11 a very large deformation for Be. In particle–core coupling calculations [9–12], deformed cores were assumed. A large deformation has also been obtained in the antisymmetrized molecular dynamics model, with a large density diffuseness at the surface [15]. Experimentally, the recent coupled-channel analysis on the proton inelastic scattering indicates significant deformation effects in 10,12 Be [16]. The deformation can increase the reaction cross section and root-meansquare (rms) radius. However, the deformation in 11 Be without a halo structure is not sufficient to reproduce the large reaction cross section [1]. We used the Skyrme–Hartree–Fock (SHF) mean-field approach to investigate the structure of 11 Be. This is similar to the work of Ref. [4]. However, the calculation of Ref. [4] did not include pairing correlations between nucleons. In the present calculation, the pairing is taken into account with the inclusion of the blocking effect of the odd neutron. The configuration of − + a state can be defined by blocking the given single-particle orbit(s), e.g., the lowest 12 or 12 neutron level in 11 Be, which allows us to investigate the structure of a given state. The significant shape polarizations from blocked orbits have been seen in heavy nuclei [17]. Besides 11 Be, other N = 7 isotones have also shown very interesting structures. A resonant peak of the unbound 9 He nucleus has been observed and interpreted as a 1 + ground state that also shows the parity 2 inversion [18,19] as observed in 11 Be. The 13 C nucleus becomes to have normal parities for the − + + lowest 12 and 12 states [19]. The excited 12 state in 13 C was observed to be a neutron halo state [20]. In the present work, we investigate the structures of the N = 7 isotones. 2. The model The Skyrme–Hartree–Fock equations are solved in the coordinate space with axially symmetric shape [21]. To reduce the influence of the particle-number fluctuation, pairing correlations are treated with the Lipkin–Nogami approach [22]. For halo nuclei, due to the large density diffuse at surface, density-dependent pairing interactions should be expected to be more suitable. Dobaczewski et al. [23] have discussed the properties of pairing interactions in nuclear systems

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31

including exotic nuclei, suggesting a volume–surface mixing density-dependent delta pairing interaction,    ρ(r) 1
δ δ  δ δ(r − r ), Vmix (r, r ) = Vsurf + Vvol = V0 1 − (1) 2 2ρ0 δ = V δ(r − r ) and V δ = V [1 − ρ(r)/ρ ]δ(r − r ) are the volume and surface pairing where Vvol 0 0 0 surf interaction, respectively, with the strength V0 . The reference density ρ0 is taken to be the standard value of 0.16 fm−3 . The state-dependent pairing gaps are deduced from the local mean field,   2  2 δ (r, r )φk (r ) φi (r) d 3 r, (2) fk uk vk Vmix ∆i = k

where the additional factor fk comes from the pairing cutoff procedure [22]. It has been pointed out that the size of a halo is strongly influenced by pairing correlations, due to the self-consistent feedback between particle and pairing densities [24,25]. Hence, an appropriate form of pairing interaction including parameters is important for exotic nuclei. For an odd nucleus, the orbit occupied by the odd nucleon should be blocked, as described in Ref. [26]. The configuration of the nucleus can be defined by the spin and parity of the unpaired + nucleon. For N = 7 isotones, the K π = 12 state is obtained by blocking the lowest jz = 12 −

neutron level of the 2s1d shell. Similarly, the K π = 12 state is obtained by blocking the jz = 12 neutron level of 1p1/2 . If the k  th level is blocked, it should be excluded from the sum of Eq. (2). The density distribution for an axially deformed nucleus can be written as follows, 2  2  2  vk ψk (z, r) , (3) ρ(z, r) = ψk  (z, r) + k=k 

where cylindrical coordinates are used with the variable z for the symmetry axis and r perpendicular to the z axis. In calculations, we chose the Skyrme forces SkI4 [27] and SIII [28]. The SkI4 force was developed recently with good isospin properties [27]. The SIII force that was employed in the variational shell model can nicely reproduce the exotic feature in 11 Be [7]. The pairing strengths are estimated by fitting the neutron separation energies of carbon isotopes, with the results of Vn = −650 MeV and −550 MeV for SkI4 and SIII, respectively. We take Vp = −600 MeV for both cases. Fig. 1 shows the calculated neutron separation energies of carbon isotopes, with comparison with experiments. We can see that the odd–even differences are well reproduced for the whole chain of carbon isotopes upto the drip lines. 3. Calculations and discussions We calculated the N = 7 isotones using the model discussed above. Fig. 2 displays the SkI4 + − calculated potential-energy curves for the 12 and 12 configurations in 11 Be, 13 C and 15 O. At each given β2 deformation point, the hexadecapole deformation (β4 ) is determined by minimizing the calculated energy against β4 . Only SkI4 calculations are shown. The SIII results are very + similar. For the 12 configuration of 11 Be, the calculated energy curve (see Fig. 2) shows a very shallow minimum around the quadrupole moment of q2 = 50 fm2 (correspondingly β2 = 0.98) which is similar to the result of Ref. [4]. The β2 = 0.98 value corresponds to a ratio of axes with Rz /Rr = 2.3, where Rz and Rr are the lengths of the long and short axes of the elongated prolate − deformed nucleus, respectively. The 12 state is calculated to have a spherical shape. We see that

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Fig. 1. Neutron separation energies for the chain of carbon isotopes with the SkI4 and SIII forces. The experimental data are taken from Ref. [29].

the mean-field calculations do not solve the observed parity inversion in the 11 Be nucleus. The + − deformed 12 state is calculated to be about 4.0 MeV in energy higher than the spherical 12 state. The effects beyond the mean field could be important for the parity inversion. It has been shown that the restoration of rotational symmetry by the angular-momentum projection can lower the energy of deformed configurations [4,30]. + In 13 C, the calculated 12 state has a prolate deformation with q2 ∼ 35 fm2 (correspondingly β2 ∼ 0.5). In the chain of N = 7 isotones, the odd isotones heavier than 11 Be have no such parity + − + inversion between the 12 and 12 states. Our calculation gives that the 12 state is about 6.3 MeV in energy higher than the

1− 2

state in 13 C. The experimental data [29] show that the energy

difference is 3.09 MeV between the of the

1+ 2

1− 2

and

1− 2

states. For 15 O, the calculated energy minimum

state is very flat in the region of q2 = 0–20 fm2 . The obtained energy is 8.2 MeV higher

than that of the spherical and

1+ 2

1− 2

ground state. The experimental energy difference between the

1+ 2

states is 5.18 MeV [29]. The present calculations show such a trend that the deformations

of the first

1+ 2

states decrease with increasing the proton number in the chain of the N = 7

isotones. Our mean-field calculation gives that the with β2 ≈ 1.33. All the

1− 2

1+ 2

state of 9 He has even larger deformation

states in the N = 7 isotones are spherical in the present calculations. +

The calculated potential energies of the 12 states in the N = 7 isotones show very flat minima, indicating extreme shape softnesses. The softness could result in important deformation effects on the properties and observables of the states. The significant deformation effect has been seen

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33

+ − Fig. 2. Calculated potential energies of the 12 and 12 states for the N = 7 isotones with the SkI4 force, as a function of the quadrupole moment q2 .

experimentally in the neighbouring nuclei 10,12 Be with very large B(E2, 0+ → 2+ ) values [31]. Due to the soft deformation, the vibrational effect could be remarkable. The effect from the core vibration in 11 Be has been taken into account in the particle–core coupling calculations [7–12]. In the case of an extremely soft shape, the static deformation value of the nucleus should be less defined. In order to further check the applicability and reliability of the Skyrme mean-field calculations for light nuclei, we calculated the bulk properties of the N = 7 isotones, i.e., total binding energies and rms radii of nuclei, listed in Table 1. It can be seen that the experimental data can be reasonably reproduced with the SkI4 force. For light nuclei, the possible clustering of nucleons could play a role in the description of the detailed structures of the nuclei. Though the mean-field calculations do not solve the problem of the parity inversion, they give some useful information helping the understanding of the structures of halo nuclei. In order to make a comparison with experimental density distributions, we calculated the density distribution ρ(R) by averaging the calculated densities in the various different directions of the space

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J.C. Pei et al. / Nuclear Physics A 765 (2006) 29–38

+

Fig. 3. The SkI4 calculated matter distribution of the 12 state in 11 Be. The shadow region is the experimental matter distribution taken from [1]. ρp is the calculated proton density distribution. Table 1 calc ) of the observed ground states in the The SkI4 calculated total binding energies (B calc ) and matter rms radii (Rrms N = 7 isotones, with comparison with the corresponding experimental data Nuclei (state) 9 He( 1 + ) 2 11 Be( 1 + ) 2 13 C( 1 − ) 2 15 O( 1 − ) 2

B calc (MeV)

B expt [32] (MeV)

calc (fm) Rrms

expt

Rrms [34]

31.95

30.14 ± 0.03

5.41

62.40

65.48 ± 0.01

3.01

3.04 ± 0.04

96.27

97.11

2.55

2.42 ± 0.24

113.19

111.96

2.64

2.70 ± 0.38

of the√axially deformed nucleus (where R is the distance from the center of the nucleus, i.e., R = z2 + r 2 , here z and r are the cylindrical coordinates described in Eq. (3)). Fig. 3 shows the spatially-averaged matter distribution (i.e., the total density in the sum of the neutrons and + protons) for the deformed 12 state of 11 Be with β2 = 0.98. It can be seen that the calculated matter distribution is in good agreement with experiment [1]. With the determined deformation, the rms radius is calculated to be 3.01 fm. Experimentally, the radius ranges from 2.71 [33] to 3.04 fm [34] depending on models used in the data analyses. The matter distribution has a long tail, showing a halo structure of the state. The calculated proton density does not show any tail, hence the tail in the distribution is caused by the neutron halo. Fig. 4 displays the density + distribution of the 13 C( 12 ) state, also showing a neutron halo structure, in agreement with the experimental observation [20] and the previous theoretical calculation [7]. Correspondingly, our + calculation gives a rms radius of 2.67 fm for the 12 state of 13 C. In order to gain a more insight into the deformed structures of the neutron halos in the chain + of the N = 7 isotones, we show in Fig. 5 the density distribution of the 12 neutron orbit in +

and 13 C. It can be seen that, in 11 Be and 13 C, the 12 neutron densities have longer tails in the z axis than in the r axis. This seems to show that the spreads of the wave functions in the deformed symmetric axis provide the dominant contributions to the observed long-tailed + matter distributions in the 12 configuration of 11 Be and 13 C. The different distributions of the

9 He, 11 Be

wave functions in the z- and r-axes indicate a deformed structure of the

1+ 2

neutron orbit in 11 Be

J.C. Pei et al. / Nuclear Physics A 765 (2006) 29–38

35

+ Fig. 4. Calculated matter distribution and proton distribution (ρp ) of the 12 state in 13 C with the SkI4 force.

+

Fig. 5. The calculated density distributions of the last 12 neutron orbits in 9 He, 11 Be and 13 C with the SkI4 force. The profiles at z and r axes are denoted by solid and dashed line, respectively. +

and 13 C. For 9 He, the 12 orbit is calculated to be unbound. We obtain very similar distributions of the wave functions in the z- and r-axes, which seems to indicate a spherical structure of the 1+ 9 2 neutron orbit in He. From the wave-function and potential-energy calculations, we seem +

to come to such a picture that the 12 states in 11 Be and 13 C have deformed structures in both cores and neutron halos, while the state in 9 He could be a spherical neutron halo attached to a deformed core. It has been pointed out that the shape of a halo is determined by the structure of the weakly bound orbit(s), irrespective of the shape of the core [5]. The nuclear halo is caused by the weakly bound nucleon(s) that has a small angular momentum. A orbit with a large angular momentum should not have a long-tailed distribution in its + wave function due to the centrifugal potential. A mixing configuration for the 12 state in 11 Be has been suggested with the components of |(10 Be)0+ × 2s1/2  and |(10 Be)2+ × 1d5/2  [7]. In + order to have a better understanding of the halo structures for the 12 states in the N = 7 isotones, we estimated the orbital angular momentum of the

1+ 2

neutron orbit in our Hartree–Fock

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calculations. The expectation value of l 2  can be calculated by lˆ 2 = lˆ− lˆ+ + lˆz2 + lˆz . In the axially symmetric deformed Hartree–Fock model, lˆ+ and lˆ− can be written as [35],   ˆl± = ∓e±iϕ r ∂ − z ∂ ∓ i z ∂ , (4) ∂z ∂r r ∂ϕ where the variables z, r and ϕ are the cylindrical coordinates defined by the relations of x = + r cos ϕ, y = r sin ϕ and z = z (x, y, z are the Cartesian coordinates). If the 12 level is a pure 1d5/2 state, the l 2  value should be 6.0 (l = 2 for the d orbit). However, our calculated l 2  + values are 0.2, 2.6 and 3.7 for the 12 states in 9 He, 11 Be and 13 C, respectively. This implies a 11 Be( 1 + ), which is responsible 2 + the 12 configuration is close to

large mixture of 2s1/2 in

for the halo occurrence. For 9 He, the

zero, implying a pure 2s1/2 resonant calculated l 2  value of state, which is consistent with the spherical distribution of the wave function as shown in Fig. 5. The measurement of the mixture ratio between |(10 Be)0+ × 2s1/2  and |(10 Be)2+ × 1d5/2  in the 11 Be( 1 + ) state has attracted strong interest experimentally [36,37]. 2 It has been pointed out that the angular momentum correction can significantly lower the energy of a deformed state [4,30]. In Ref. [38], a phenomenological rotational correction has been suggested, J Erot =−

 h¯ 2
 2 J − J (J + 1) , 2

(5)

where  is the moment of inertia, J 2  is the expectation value of the total angular momentum without the angular momentum projection, and J in the last term of the formula is the spin + value of the state (J = 12 for the 12 configuration). In the present work, the moment of inertia is calculated by the Belyaev’s formula [26,39]. With the determined deformation value of the 11 Be( 1 + ) state, we obtain a rotational energy correction of −3.8 MeV for the deformed 1 + state. 2 2 For a more appropriate calculation of the angular momentum correction, sophisticated projection techniques should be used [4,30]. In the discussions above, we see that the deformation plays an + important role in lowering the energy of the 12 states, but is not enough to produce the parity inversion. This suggests that the effects beyond the mean field would be important, such as the core excitation [7] and cluster structure [15], which are out of the scope of the present work. 4. Summary In summary, we applied the deformed Skyrme–Hartree–Fock model to investigate the structures of the N = 7 isotones. Pairing correlations are calculated using the Lipkin–Nogami model with a surface–volume mixing delta-pairing interaction. For odd-A nuclei, the blocking effect of the odd nucleon was taken into account in the present calculations. The SkI4 and SIII forces were taken in the calculations with similar results. The experimental odd–even differences in the chain of carbon isotopes are well reproduced with the present model. The calculation shows + that the 12 state of 11 Be has a very large deformation. The obtained spatially-averaged density distribution agrees well with the experimental data, showing a neutron-halo structure of the state. + + Large deformations are also obtained for 13 C( 12 ) and the core of 9 He( 12 ). By calculating the +

+

average orbital angular momenta, the 11 Be( 12 ) and 13 C( 12 ) states are analysized to contain a significant component of the neutron 2s1/2 orbit that leads to the neutron halo structures of the

J.C. Pei et al. / Nuclear Physics A 765 (2006) 29–38

states. The

1+ 2

37

state of 9 He has almost a pure 2s1/2 configuration. The calculations of the poten+

+

tial energies and wave functions seem to show that 11 Be( 12 ) and 13 C( 12 ) have the structures of +

deformed halos and 9 He( 12 ) is a spherical halo attached to a deformed core. Though the meanfield calculations have not been able to solve the problem of the parity inversion, the calculations do provide some useful information about the structures of the light exotic nuclei. Acknowledgements We thank D.X. Jiang, M. Bender, R. Wyss, Z.Z. Ren and S.G. Zhou for their valuable comments. This work was supported by the Natural Science Foundation of China under Grant No. 10475002, the Doctoral Foundation of Chinese Ministry of Education under Grant No. 20030001088, the Key Grant Project of Chinese Ministry of Education under Grant No. 305001, the UK Royal Society, the UK Engineering and Physical Sciences Research Council. We also thank the Computer Center of Peking University where the calculations were done. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

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