Revelation of thick neutron skins in nuclei

Physics Letters B 289 (1992) 261-266 North-Holland

PHYSICS LETTERS 13

Revelation of thick neutron skins in nuclei I. Tanihata, D. Hirata ~, T. Kobayashi, S. Shirnoura 2, K. Sugimoto 3 and H. Toki 4 RIKEN, 2-1 Hirosawa, Wako, Saitama 351-01, Japan

Received 27 April 1992

Thick neutron skins ( ~ 0.9 fm) have been verified for the first time in 6He and 8He nuclei from a combined analysis of the interaction and the fragmentation cross sections of 4'6'aHeincident reactions at 800,4 MeV. A relativistic mean field (RMF) model was applied to 6He and SHe nuclei and shown to reproduce the neutron skin thicknesses very well. It is also shown that the RMF model predicts a gross linear dependence of the neutron skin thickness on the difference between the proton and the neutron Fermi energy in a wide range of nuclei. Possible observations of thick neutron skins in other nuclei, in particular in Na isotopes, are also discussed.

In spite o f detailed studies o f stable nuclei o f large neutron excess ( N - Z ) , no evidence o f a thick neutron skin was o b s e r v e d so far. F o r example, the rootm e a n - s q u a r e ( R M S ) radius o f the neutron distribution is larger than that o f the protons only b y ~ 0.2 fm in 48Ca ( N - Z = 8 ) a n d by ~ 0 . 1 5 fm for 2°8pb (N-Z=44) [1]. Recently a d e v e l o p m e n t o f high-energy radioactive nuclear b e a m s [ 2 ] enabled us to d e t e r m i n e nuclear radii o f I~-unstable neutron-rich nuclei in which one m a y expect thick neutron skins due to not only the neutron excess b u t also to the large difference between the neutron a n d the p r o t o n F e r m i energies. N u c l e a r m a t t e r radii o f light nuclei were d e t e r m i n e d by measurements o f the interaction cross sections (a~) [ 3,4 ] or b y the reaction cross section m e a s u r e m e n t s [ 5 ]. Although these m e a s u r e m e n t s p r o v i d e d the det e r m i n a t i o n s o f the m a t t e r radii, the neutron a n d the proton radii were not d e t e r m i n e d separately. F o r light unstable nuclei, N a isotopes are the only exceptions in which the charge (thus the p r o t o n ) radii were det On leave from Instituto de Estudos Avanqados - CTA, Silo Jos6 dos Campos, Brazil. 2 Also at Faculty of Science, Rikkyo University, Tokyo 171, Japan. 3 Also at Faculty of Science, Osaka University, Toyonaka 560, Japan. 4 Also at Faculty of Science, Tokyo Metropolitan University, Hachiohji 192, Japan.

t e r m i n e d by isotope-shift m e a s u r e m e n t s [6 ]. Although an experiment is p l a n n e d to measure the matter radii at GSI in G e r m a n y [ 7 ], neither the neutron nor the m a t t e r radii were known for N a isotopes. Therefore so far we h a d no i n f o r m a t i o n on the difference o f the neutron and the proton radii in I]-unstable nuclei. In this report we show the first verification o f thick neutron skins in 6He and SHe nuclei from the c o m b i n e d analysis o f the ax o f H e isotopes a n d two a n d four neutron removal cross sections o f 6He and SHe projectiles (O'_2na n d a - 4 n ) . Nuclear m a t t e r radii o f 4'6'8He d e t e r m i n e d by the interaction cross section (a~) measurements showed that the radius increases drastically from ( 1.57 + 0.05 f m ) 4He to (2.48 + 0.03 f m ) 6He [4]. Although they show quite a change in the m a t t e r radius it was not possible to d e t e r m i n e whether the protons and the neutrons have different density distributions. Recently, Ogawa, Yabana and Suzuki [8 ] d e d u c e d an i m p o r t a n t relation between the interaction cross sections a n d the particle removal cross sections by applying the G l a u b e r m o d e l to a loosely b o u n d system. According to their result, if 6He is well described by a 4He core and two neutrons, the interaction cross sections o f 6He a n d 4He and the two-neutron rem o v a l cross section o f 6He, with the same target, are related as tY_2n (6He) = o'i (6He) - o'i (4He) ,

0370-2693/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

( 1) 26 1

Volume 289, number 3,4

PHYSICS LETTERS B

where a_2~(6He) is the two-neutron removal cross sections of 6He projectile. Here they used the fact that the 5He nucleus has no particle stable state. This relation holds if the 4He core in the 6He nucleus is not modified from the free 4He nucleus, in other words if the wave function of 6He, ~/(6He), is expressed as q/(6He) = ~(4He)gt( 1,2),

(2)

where ~u(4He ), qJ( 1,2 ) are the wave functions o f " H e and of the two remaining neutrons, respectively. Although multiple scatterings between a projectile and a target nucleon are included in the Glauber model, rescattering effects such as the one from a collision between the 4He core and a knockout neutron from the 6He is not included in the derivation. However this effect is considered to be small in the present case because of an extended peripheral nature o f the reaction and the tight binding of 4He. This relation, in turn, provides a test for the persistence of a core in a nucleus because of measurements o f ai and t r 2n. Experimental values o f related cross sections are summarized in table 1. Here, all cross sections used are the ones measured at a beam energy of 800A MeV using a C target [3,9]. This energy was high enough to satisfy the assumption in the model of Ogawa, Yabana and Suzuki. Experimental values, o'_2n(He ) = 189+ 14 m b ,

a l ( 6 H e ) - a l ( 4 H e ) =219_+8 m b ,

(3)

almost satisfy eq. ( 1 ) and thus indicate that the 4He remains intact as the core in the 6He nucleus. In case of an SHe nucleus, however, the corresponding relation does not hold at all because Table 1 Interaction (a~) and neutron removal (tr_~) cross sections at 800A MeV. (aTs are from ref. [3] and tr_~'s are from ref. [9] ). Reaction

4He+C 6He+C

SHe+C

Cross section (mb) 0.1

0-_2 n a)

G _ 4 t a b)

503+5 722+ 5 817+6

189+ 14 202+ 17

95+9

a) The cross section for two neutron removal from the incident 6He or SHe nucleus. b) The cross section for four neutron removal from the incident SHe nucleus. 262

10 September 1992

tr_2n(SHe) = 2 0 2 + 17 m b , ai(SHe) - al(6He) = 9 5 + 9 m b .

(4)

Therefore the 6He wave function is strongly modified in the SHe nucleus and has thus no identity as a core. Instead, the relation a_2n (SHe) + o'_4n (SHe) = ai(SHe) - ex(4He),

(5)

where we extended eq. ( 1 ) under the same assumption for the 4He core in the SHe nucleus ~, holds very well as a_2,(SHe) + a_4n(SHe) = 2 9 7 + 19 m b , a ~ ( S H e ) - ai(4He) = 314 + 8 rob.

(6)

We note here that 5He and 7He are u n b o u n d nuclei. Therefore SHe is well described by a 4He core and four neutrons orbiting outside the core. Under the assumption that 4He forms a good core in 6He and SHe nuclei, we have deduced the pointnucleon density distribution using the optical limit calculation o f Glauber model [ 11 ] for the a~'s of 4He, 6He, and SHe incident reactions. This method is essentially the same as the one used in ref. [4 ] and provides a direct connection between nucleon distributions and the interaction cross section. In the fitting we assumed a harmonic oscillator density distribution, which gives a reasonable approximation for light nuclei, but with different size parameters for the 1s orbital and lp orbital. First, the size parameter, a, of the ls orbital was determined to fit the a~(4He), then each size parameter, b, of the lp orbital was determined by fitting the a~ (SHe) or the ai (SHe). The resultant root-mean-square ( R M S ) radii o f the proton, the neutron, and the matter density distributions are shown in table 2 and the profile o f the point density distributions are shown in fig. 1. The neutron distribution extends far beyond that o f protons. The RMS radius of neutron, R ~ s, is much larger than that of protons, Rbms, i.e. Rn~ - R ~ ms ~ 0 . 9 fm for SHe and SHe. Thus these nuclei have thick neutron skins. Although we assumed that the 4He core is not modified, the central densities of 6He and SHe are lower than that of 4He. This is because of the m o v e m e n t o f the core with respect to the center of mass of a nucleus, ~1 This equation has also been derived independently by Ogawa etal. [10].

Volume 289, number 3,4

PHYSICS LETTERS B

10 September 1992

Table 2 The fitted parameters and root-mean-square radii of a point-nucleon (protons, neutron and matter) in the He isotopes. Nucleus

4He 6He SHe

Fitted parameters a}

RMS radius (fro)

AR'ms (fro)

a (fm)

/9 (fm)

Rn

Rp

Rm

1.53 1.53 1.53

2.24 2.06

1.63 + 0.03 2.59 + 0.04 2.69 + 0.04

1.63 + 0.03 1.72 + 0.04 1.76 + 0.03

1.63_+0.03 2.33 _+0.04 2.49 +_0.04

0 0.87 + 0.06 0.93 + 0.06

a) The s-orbital size parameter a and the p-orbital size parameter/9 are defined by the density distribution formula (for a neutron) as

-,-

2

1

R 2

1N-2

R 2

R 2

where a2=a2(1-1/A ) and b2 =b 2(1-1/A ) [12].

which is taken into account as in the usual harmonic oscillator density distributions [ 12] assuming no strong correlation between nucleons. A sharp cut distinction between the neutron halo, which was observed in ~Li and ~Be, and the neutron skin is difficult. A difference "by definition", which we propose here, is the difference o f the slope factor in the density tail that is related to the separation energy. Appreciable neutron halos appear only in nuclei with an extremely small separation energy o f the last neutron(s); 0.3 MeV in case o f 11Li and 0.5 MeV for l tBe. The two-neutron separation energies are 0.97 MeV for 6He and 2.13 MeV for SHe. Therefore, the terminology "skin" is more appropriate for SHe. It is more arbitrary to call the excess neutron on the surface either as a halo or a skin for 6He. As will be shown later neutron skins appear in m a n y nuclei away from the neutron drip line. It also suggests that a considerable number o f neutrons can be included in a neutron skin. In contrast, a neutron halo is expected to include only a few neutrons in the last orbital. N o w we calculate the density distributions o f these nuclei by the relativistic mean field model ( R M F ) [ 13 ]. All the details o f the method are described in the work o f Hirata et al. [ 14 ] including the definitions o f the parameter sets used in the model. Here we used the parameter set N L I , which was determined by the Frankfurt group [ 15 ] by a least mean square fit to the masses and radii o f eight stable nuclei. This parameter set was further demonstrated to provide an extremely good fit with the nuclear properties o f not only the stable nuclei but also o f the unstable ones [ 14 ].

We show in fig. 1 the density distributions of 6He and SHe obtained by the R M F model together with the empirical ones. The empirically deduced proton and neutron density distributions are well reproduced by the R M F model except in the central region. Since the presently used experimental data are not very sensitive to the central density, this difference is not considered to be meaningful. A three-body model o f 6He by Suzuki [ 16 ] also predicts the difference o f the neutron and the proton RMS radii to be 0.8 fm and shows a good agreement with present empirical values. Stimulated by the success of the R M F predictions to the empirical 6'SHe density distributions, we extended the model calculation to other nuclei. Fig. 2 presents the results o f the calculations shown as the difference o f the neutron and the proton RMS radii (AR ~ s = R ~ s _ R ~ s ) as a function o f the difference of the neutron and the proton Fermi energies ( A E F = E n - E p ) . As illustrated in the insert o f fig. 2, neutrons and protons occupy orbitals up to almost the same Fermi energy in a stable nucleus. Then, if we assume that the nuclear potentials and the density distributions are self-consistently determined and drop the small isovector potential temporarily as a sake o f discussion, the proton and the neutron potentials are the same except for the Coulomb potential. Only the existence o f the Coulomb potential makes the radius o f protons slightly smaller. Therefore, no thick neutron skin is expected for stable nuclei even if they have large ( N - Z) values. In addition, the isovector potential, neglected hitherto, acts attractively for protons and repulsively for neutrons in neutron 263

V o l u m e 289, n u m b e r 3,4

P (0 [fm-3]

PHYSICS LETTERS B

neutron (exp.)

10-1 . ~ % ~

10 S e p t e m b e r 1992

6He

8He

]

6He

1.0 l-t

31Na , 10-2

10-3 0

,

1

0.5

208pb

0.0

I"1 /~ A ~ 4 ~ . . . . . 4 r l

3

4

r [fm] -0.5 -20

P(r) neutron (exp.)

[fm31 10-1

8H e

(RM_F)

10-3 0

,

,

1

2

, 3

4

(b)

r [fm]

Fig. 1. Proton and neutron density distributions in 6He and 8He nuclei are shown. The solid curves show the density distribution obtained from the interaction cross sections trl under the assumption that the 4He core is not modified in 6He nor in SHe. Large excesses of the neutron density at R > 1 fm are seen in both nuclei. The propagation error from the uncertainties of the cross sections is very small and is not shown in the figure. The RMF results are also shown by dashed curves that reasonably reproduce the empirically deduced excess of the neutron density. excess nuclei a n d t e n d s to m a k e the d i f f e r e n c e o f the density d i s t r i b u t i o n s e v e n s m a l l e r t h a n discussed a b o v e . It is t h e r e f o r e n o t r e l e v a n t to plot the skin thickness against the ( N - Z ) v a l u e . Instead, t h e F e r m i energy difference, AEF, is an a p p r o p r i a t e q u a n t i t y b e c a u s e it actually reflects t h e d i f f e r e n c e o f the orbital sizes o f t h e lastly filled p r o t o n (s) a n d neut r o n ( s ) . In a I]-radioactive nucleus far f r o m the stability line, A E v is large a n d t h u s a large d i f f e r e n c e bet w e e n the p r o t o n a n d t h e n e u t r o n radii are e x p e c t e d . As seen in fig. 2 the c a l c u l a t e d v a l u e s o f A R rm~ for a w i d e mass r a n g e o f nuclei lie in gross o n a u n i v e r s a l 264

24o

°

16O

,

2

~ a-d~D~ Z X

O . 20Na ~ , -10 0

~ 10

A E F (MeV)

Fig. 2. The difference between the neutron and the proton rootmean-square radii, ARrmS=R'~S-R'~S,as a function of the Fermi energy difference between the neutron and the proton, AEr:=E.-Ep,for various isotopes up to nucleon drip lines, obtained by using the RMF model. Results for 6He and SHe and other selected nuclei were indicated by arrows. Present empirical values of AR ~s of 6He and SHe nuclei are shown by the shadowed area. The horizontal lengths of the shadows have no meaning because the AEF's are not known. The insert shows the occupation of nucleons in a self-consistent potential. The dark gray area indicates the proton and the neutron occupation in a stable nucleus. The lightly shadowed area shows the extra neutrons, which contribute to a formation of neutron skin, in 13-unstable nuclei.

straight line a n d m o n o t o n i c a l l y increase w i t h AEF. A AEF was d e f i n e d as the single-particle energy difference b e t w e e n last filled p r o t o n a n d n e u t r o n orbitals. T h e c o r r e l a t i o n b e t w e e n AEF a n d A R rms is not perfect, w h i c h is c o n s i d e r e d to be due also to the o t h e r factors such as the v a l u e s o f N a n d N - Z. AR rms values larger t h a n 0.5 are seen for all nuclei w i t h AEF larger t h a n ~ 10 MeV. T h e r e f o r e , the n e u t r o n skin is c o n s i d e r e d to be a c o m m o n structure t h a t appears in n e u t r o n - r i c h ~l-unstable nuclei. A m o n g these, 6He, 8He, a n d 240 are the nuclei that are p r e d i c t e d to h a v e the thickest n e u t r o n skins. F r o m 0.1 to 0.3 f m o f z~kRrms are seen for stable nuclei (/~kEF~---0 ) in the figure t h a t is c o n s i s t e n t w i t h the o b s e r v a t i o n in 4SCa. T h e present calculation predicts a slightly larger AR ~ms (0.3 f m ) t h a n the o n e recently o b s e r v e d for 2 ° 8 P b ( ~ 0 . 1 5 f m ) [1 ]. W e c o n s i d e r t h a t the slight t u n e up o f the p a r a m e t e r in the R M F m o d e l , w h i c h we h a v e n o t m a d e in the p r e s e n t c a l c u l a t i o n because the p u r p o s e o f this p a p e r is to study the global b e h a v -

Volume 289, number 3,4

PHYSICS LETTERS B

ior of AR rms, would improve the result. A non-relativistic mean field calculation was reported by Lombard [ 17 ] for light nuclei and the present results are consistent with theirs. It is also suggestive that some nuclei show proton skins even if they have a similar number of neutrons and protons. For example, the 2°Na nucleus is expected to have a proton skin of 0.25 fro. The measurements of the matter radii of Na isotopes (2°Na-3~Na) are planned at the FRS facility of GSI in Germany [7]. As the charge radii were already known, determination of the matter radii from the interaction cross section measurement will then provide the method to separate the proton and the neutron radii for the long isotope chain for the first time. Thus the Na isotopes provide a unique opportunity for directly studying the growth of the neutron skin. We show our RMF prediction of the proton and the neutron radii for Na isotopes in fig. 3 together with the charge radii's data determined by the isotope shift measurement at CERN [6]. The calculated results on the proton radii reproduce the experimental tendency very well. Based on the deformed RMS model, Patra and Praharaj recently reported that the deformation is important for understanding the binding energies of neutron rich Na isotopes (A = 29-34) [ 19]. We also confirmed it with the de-

formed RMF calculation. However the deformed calculation does not reproduce the charge radii when it is applied to 2~-31Na [20]. Therefore we show here only the results of the spherical RMF model. It predicts a gradual increase ofAR ~ toward neutron rich isotopes. A AR ms of about 0.7 fm is expected for 3~Na. In summary, we have analyzed the interaction cross sections and two- and four-neutron removal cross sections in 4He, 6He, and SHe + C reactions at 800A MeV. In the framework of the Glauber model, we have found that 4He persists as an unmodified core in 6He or 8He nucleus. On the bases of this fact, the optical limit of the Glauber model was used to determine the point-particle density distributions of the proton and the neutron under the constraint that the 4He core persist in 6He and 8He nuclei. The deduced density distributions show thick neutron skins in 6He and SHe nuclei. The point-neutron root-mean-square radii are larger than that of protons by ~ 0.9 fm, which is the thickest neutron skin ever observed. The relativistic mean field calculation has been applied for unstable nuclei of a wide mass range. It is found that the thickness of the neutron skin is strongly correlated with the difference between the neutron and the proton Fermi energies. In agreement with the findings of Lomard it is predicted that neutron-rich 13-unstable isotopes have neutron skins in common. The authors would like to express their thanks to Professor N. Suzuki and Professor K. Yabana for useful discussion of the G l a u b e r model of nuclear fragmentation. One of the authors, D.H., is grateful to the CNPq for a fellowship that made her possible to work at RIKEN.

R;'~ [fml 3.5

10 September 1992

oRn o o o o o

3.0

o

o o

-~

o

~o

o

e°

8

6 ~ Rp

References

o

2.5

8

i

i

i

12

16

20

N

Fig. 3. The radii of the proton and the neutron distributions of Na isotopes calculated by the RMF model as a function of the neutron number, N. The measured charge radii in a long isotope chain by the isotope shift method are shown by black dots, which are normalized at 23Na to the value determined by electron scattering [ 18 ] assuming the charge RMS radii of the proton to be 0.8 fm. The RMF model reproduces the change of the charge radii and predicts a continuous increase of the neutron skin.

[ 1 ] C.A. Whitten Jr., High energy physics and nuclear structure, eds. D.F. Measday and A.W. Thomas (North-Holland, Amsterdam, 1980) p. 419; G. Igo et al., Phys. Len. B 81 (1979) 151; A. Krasznahorkay et al., Phys. Rev. Lett. 66 ( 1991 ) 1287. [2] See, for example, W.D. Myers, J.M. Nitschke and E.B. Noeman, eds., Proc. First Intern. Conf. on Radioactive nuclear beams (Berkeley, CA, USA) (World Scientific, Singapore, 1990); Th. Delbar, ed., Proc. Second Intern. Conf. on Radioactive nuclear beams (Louvain-la-Neuve, Belgium, August 1991 ) (Adam Hilger, Bristol, 1991 );

265

Volume 289, number 3,4

PHYSICS LETTERS B

I. Tanihata, Treatise on heavy-ion science, Vol. 8, ed. D.A. Bromley (Plenum, New York, 1989) p. 443. [3] I. Tanihata et al., Phys. Lett. B 160 (1985) 380; I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoshikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55 ( 1985 ) 2676. [4] I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, IC Sugimoto, N. Takahashi, T. Shimoda and H. Sato, Phys. Lett. B 206 (1988) 592. [ 5 ] M. Fukuda et al., Phys. Lett. B 268 ( 1991 ) 339; A.C. Villari et al., Phys. Lett. B 268 ( 1991 ) 345. [6] G. Hiiber et al., Phys. Rev. C 18 (1978 ) 2342. [7] I. Tanihata, G. Miinzenberg et al., Proposal to the GSI/FRS experiment ( 1991 ). [8 ] Y. Ogawa, K. Yabana and Y. Suzuki, Niigata University preprint NIIG-DP-91-3, submitted to Nucl. Phys. A. [9] T. Kobayashi, O. Yamakawa, IC Omata, K. Sugimoto, T. Shimoda, N. Takahashi and I. Tanihata, Phys. Rev. Lett. 60 (1988) 2599.

266

10 September 1992

[ 10 ] Y. Ogawa, Y. Suzuki and K. Yabana, paper submitted to Internat. Nuclear Physics Conf. (Wiesbaden, 1992); and private communications. [ 11 ] P.J. Karol, Phys. Rev. C 11 (1975) 1203. [ 12] L.R.B. Elton, Nuclear sizes (Oxford U.P., Oxford, 1961 ). 1961). [ 13] B.D. Serot and J.R. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [ 14 ] D. Hirata, H. Toki, T. Watabe, I. Tanihata and B.V. Carlson, Phys. Rev. C 44 ( 1991 ) 1467. [ 15 ] P.G. Reinhard et al., Z. Phys. A 323 (1986) 13. [ 16] Y. Suzuki, Nucl. Phys. A 528 ( 1991 ) 395. [ 17] R.J. Lombard, J. Phys. G 16 (1990) 1311. [ 18 ] H. De Vries, C.W. De Jager and C. De Vries, At. Nucl. Data Tables 36 (1987) 495. [ 19 ] S.K. Patra and C.R. Praharaj, Phys. Lett. B 273 ( 1991 ) 13. [ 20] D. H irata et al., in preparation.