Quadrupole-moments in mirror nuclei and proton halo

Physics Letters B 299 (1993) 1-5 North-Holland

PHYSICS LETTERS B

Quadrupole-moments in mirror nuclei and proton halo Hisashi Kitagawa 1 Research Center for Nuclear Physics, Osaka University, 10-I Mihogaoka Ibaraki, Osaka 567, Japan and

Hiroyuki Sagawa Department of Physics, Faculty of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113, Japan

Received 7 July 1992; revised manuscript received 25 November 1992

Electric quadrupole moments in light mirror nuclei are studied by shell model calculations with the proton-neutron formalism. Our calculations describe successfullythe Q-moments of both loosely-boundand well-bound nuclei. The adopted effective charges are consistent with the theoretical predictions due to the core polarization effect. The large enhancement in SB and 17Fshows a clear evidence of the proton halos.

The study o f electric and magnetic moments provides an opportunity for detailed tests in nuclear wave functions. Especially, experimental data for quadrup o l e - ( Q - ) m o m e n t s in nuclei give a measure o f the extent to which the nuclear charge distribution deviates from spherical symmetry [ l ]. The shell model is known as it provides reasonable wave functions to describe these observables if appropriate effective operators are used [ 2 ]. The effective operators were first introduced to take into account the effect of the larger model space than that adopted in truncated shell model calculations. A well-known example is the effective charge for electric quadrupole (E2) observables. The origin of the E2 effective charge is now well established in p- and sd-shell nuclei as being caused by virtual excitations o f particles to E2 giant resonances [ 3 ]. Recently, much progress in the study o f unstable nuclei has been made through high-energy and medium-energy heavy-ion reactions with radioactive beams. One of the most spectacular findings is neutron halos in ~Li and ~tBe by the measurements o f the reaction cross sections [ 4 - 6 ]. So far, the study o f unstable nuclei has been much concentrated on the l JSPS Fellow for Japanese Junior Scientists.

neutron-rich side. On the other hand, proton-rich nuclei might also have interesting features, e.g. the effect o f the halo exists when overcoming the Coulomb barrier when both the proton number and the proton separation energy are small enough [ 7,8 ]. In order to study the loosely-bound nature o f protons, the Q-moment would be a better probe than the reaction cross section since the former is more sensitive to the radial expansion of protons because o f the effective charges. In this letter we extract the effect of loosely-bound protons from studying the Q-moments o f light mirror nuclei. Studying the mirror nuclei together, we can minimize the ambiguity o f the effective charges substantially and extract precise information o f the proton wave function. Especially interesting nuclei for our purpose are SB and 17F since they have extremely small separation energies of 0.137 and 0.601 MeV, respectively. The shell model calculations are performed by using the C o h e n - K u r a t h and Millener-Kurath [ 9,10 ] model for the p-shell nuclei with the effective interactions C K P O T and CKI. We employ the p r o t o n neutron formalism to take account o f the difference between them in the shell model wave functions [ 11 ].

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Volume 299, number 1,2

PHYSICS LETTERS B

The Q-moment is defined as a diagonal matrix element of the E2 operator: Jt'(E2,/z)= ~

i 2 Y2,u(ri) e~err(~-tz,)ri

~t

+ ~ e~efftl,~+tz,)rZY2u(~i),

.

p

(1)

The values e~fr and e~,ff are the effective charges of protons and neutrons, respectively. The Q-moment can be expressed by the one-particle spectroscopic factors

eQ=

E

x//T6n (Jl IIr2y2 ][/z 5

Jl , j 2 , 0 t C , J c , T C , I z l ,lz2

1

× 2 x / / 2 J + l ( 2 T + l ) (_)r~+l/2-T+2J f

X(JJ20[Jj)~S

J2j

Mrc ½t~ I T M r ) 2

Jc}

X ( J Till a~, IIIa c Jc Tc ) ( a c Jc Tc Ill~j~ IllJ T ) , (2) where the effective charges are defined as diS-- ! toeffa_oeff~

. , I V _ 1 /" elf

eff$

We use two kinds of single-particle wave functions in the following study. Firstly, we calculate the Q-moments by the isospin formalism with harmonic oscillator ( H O ) wave functions. There is no difference between the proton and the neutron wave functions in the isospin formalism. Secondly, we use singleparticle wave functions in the Woods-Saxon (WS) potential in order to include the difference between protons and neutrons. We take into account all core excited states of A - 1 nuclei in the calculations of Qmoments of A nuclei. For each core excited state with the excitation energy Ex, the separation energy of the valence nucleon Sv is determined by the formula Sv-Ex =B(A)-B(A-

which only the single configuration for the valence nucleon is assumed. There are five sets of data available for mirror light nuclei. We adopt three sets of data with A = 8, 11 and 17 to determine empirical effective charges. All data are recently measured or confirmed by the Minamisono group using the fl-nuclear magnetic resonance method [ 12]. We do not take the data of A = 12 nuclei into consideration since the Q-moment of ~2N was determined by a quite different method [ 13 ] from the measurements of other nuclei and should be confirmed by the same method. The calculated matrix elements for the Q-moments (2) are rewritten to be

eQ=e,eft q,+e~eft q~

e Is )

Xt~t~,,t~(tz~ I~elVr3~ltz2) ( T c

1),

where B(A) is the binding energy of the nucleus A. The valence single particle wave function is calculated by adjusting the depth of the WS potential to obtain the separation energy for each configuration. The loosely-bound nature of valence nucleons was also studied in ref. [2] by using the WS potential in

21 January 1993

(3)

and the matrix elements of q, and q~ are tabulated in table 1 in the case of the CKPOT interaction. The empirical effective charges are determined from the observed Q-moments and the calculated q, and q~ values using the HO and WS wave functions. The average values for the HO wave functions are determined as e~ff( H O ) = 1.81 + 1.11, e~ff(HO) =0.51 + 0.16 with the systematic errors. The empirical value for protons is much larger than the theoretical one e eft , = 1 +Je~°~= 1.25 due to the core polarization effects [ 3], and has also a large systematic error. The large uncertainty in the value e~ff is caused by the anomalously large value of the A = 8 system. The average effective charges are also obtained by using the single-particle wave functions in the WS potential whose central potential depth is changed to reproduce the separation energy of each nucleus. The radius parameter ro and the surface diffuseness a are taken to be standard values: ro= 1.27 and a=0.65. It is remarkable that the proton value has a very small systematic error: e~fr (WS) = 1.22 + 0.19, e~ff (WS) = 0.50 + 0.16. These values are also close to the theoretical predictions for light nuclei. The small systematic error and the agreement with the theoretical value are both caused by the decrease of the e~fr value for A = 8 in the case of the WS potential. Namely, the expectation value q, is increased by the loosely-bound nature of the proton wave function in SB and becomes two times larger than that of the HO case. Inversely, the proton effective charge decreases to be close to the values of the other nuclei. Thus, a substantial improvement in the systematic error of the

Volume 299, number 1,2

PHYSICS LETTERS B

21 January 1993

Table 1 Empirical effective charges in A = 8, 11 and 17 nuclei. The expectation values q (in fm2 units) are calculated by using both the HO and the WS wave functions. A

Nucleus

HO wavefunction q~

8

11

]Li5

WS wavefunction

q.

2.428

e~]r

e~ff

0.475

2.657

0.800

~B3

0.800

2.428

I~B6

1.317

2.887 0.645

17

~C5

2.887

1.317

1209

-6.050

0.000 0.423

IgFs average

0.000

proton effective charges is o b t a i n e d by the effect o f the p r o t o n halo in the SB nucleus. We calculated also the Q - m o m e n t s by using C K I interaction. The effective charges are o b t a i n e d to be e ~ f f ( H O ) = 1.82+ 1.17, e~err ( H O ) = 0 . 3 4 + 0 . 3 9 a n d e ~ f r ( w s ) = 1.25+ 0.14, e ~ f r ( w s ) = 0 . 4 7 + 0 . 1 3 . Thus the results do not d e p e n d on the choice o f the effective interactions. In the A = 11 system, all protons a n d all neutrons are well-bound to the core o f the nucleus, so that the results with the H O and WS wave functions are almost the same. We expect the p r o t o n halo effect also in 17F since the last p r o t o n is loosely-bound having the separation energy o f 600 keV. Certainly, the calculate p r o t o n q u a d r u p o l e matrix element q~ in 17F by the WS single-particle wave functions is enhanced by 25% c o m p a r e d with that o f the H O wave functions. The Q - m o m e n t observed in 17F is r e p r o d u c e d fairly well by the calculated result with the effect o f the halo proton as is shown in table 2. We now analyse all five sets o f d a t a in light m i r r o r nuclei by using the average effective charges with the corresponding single-particle wave functions. A m o n g the nuclei shown in table 2, the nucleus ~2N is particularly interesting because o f its small p r o t o n separation energy o f 600 keV. However, the calculated value o f the Q - m o m e n t for the W S case is smaller than that o f the H O case. This is due to the geometrical factor in eq. ( 2 ) . The increment o f the p r o t o n radius gives

q,

4.163

0.800

0.800

5.813

1.276

2.672

2.670

1.301

-6.641

-0.054

0.065

--7.357

1.112

1.653

--6.050 0.514

q~

1.807

e~~r

e~fr

0.577

1.095

0.559

1.204

0.374

1.363

0.503

1.221

a positive contribution to q~ in the case o f SB. It works, however, as a negative contribution for ~2N and decreases the q~ value by the cancellation with other contributions than that o f the halo configuration. In the case o f the A = 17 system, the configuration is assigned as one-particle outside the 160 core, so that the loosely-bound proton configuration increases the q~ value straightforwardly. In the case o f the A = 41 system, we assign the ground state configuration purely as one-particle outside the 4°Ca core, assuming that the core is spherical without any deformation, as in the case o f the A = 17 system. The calculated result of41Sc reproduces the observed d a t a well, while there is a small difference between the two values o f 4~Ca. The discrepancy between the calculated a n d the experimental results in 4~Ca suggests a softness o f the 4°Ca core for the quadrupole deformation. The proton-rich nuclei have in general small separation energies, so that one needs the WS wave functions to reproduce the empirical Q-moments. On the other hand, the neutron-rich nuclei are deeply-bound systems, a n d the effect o f spatial expansion in the wave functions is not large. It is noticed in table 2 that there remains a difference between the calculated and the experimental d a t a for the neutron-rich nuclei 170 a n d 41Ca. These discrepancies give an empirical constraint on the neutron effective charges for

Volume 299, number 1,2

PHYSICS LETTERS B

21 January 1993

Table 2 The Q-moments in A=8, 1l, 12, 17 and 41 systems in fm 2 units. We use the empirical values for the effective charges of protons and neutrons. A

Nucleus

T~

J~

HO 8

3SLi5

Reference

Q-moment (fm 2 )

WS

Experiment

1 -1

2+ 2+

2.695 4.880

3.073 7.501

3.28 +0.06 6.83 +0.21

[ 14] [14]

~ B6

3

1~C5

_~

323-

5.894 3.866

3.904 2.936

4.059 _ 0.010 3.327 + 0.024

[ 15 ] [ 16 ]

12B7

12N5

1 - 1

1+ 1+

2.612 1,119

1.620 0.516

1.321 + 0.026 2.6

[ 14] [ 17]

17

~709 '7Fs

3 -3

5+ 5+

-3.112 -10.933

-3.410 -9.016

-2.558 +0.022 -10 +2

[ 16] [14]

41

2oCa214t ~4~Sc2o

3 -3

~77_

-5.850 -20.550

-5.413 -14.095

-8.0 -15.8

[17] [14]

8B 3

11 12

the h e a v i e r systems a n d the effect o f d e f o r m a t i o n o f c o r e nuclei. N u c l e a r radii o f the p r o t o n s a n d the n e u t r o n s are listed in table 3. T h e h a l o effect c a l c u l a t e d by the W S w a v e f u n c t i o n s a p p e a r s also in these values. Since t h e A = 1 1 systems are d e e p l y - b o u n d , the radii are a l m o s t the s a m e for the h a r m o n i c oscillator a n d the W S w a v e functions. O n the o t h e r h a n d , o n t h e p r o t o n - r i c h side o f A = 8 a n d 12 systems w h i c h h a v e l o o s e l y - b o u n d p r o t o n s , the p r o t o n n u c l e a r radii i n c r e a s e substantially. T h e s e i n c r e m e n t s o f p r o t o n radii will e n h a n c e Table 3 The proton and neutron nuclear root mean square (RMS) radii in fm units for A = 8, 11 and 12 systems. The mass RMS radii are also shown. A

Nucleus

Radius (fro) proton

neutron

mass

]Li5 (WS)

2.316 3.034 2.164

2.164 2.164 2.727

2.260 2.740 2.531

11

~B6 (HO) lib 6 (WS) 1~C5 (WS)

2.396 2.385 2.506

2.434 2.473 2.379

2.417 2.434 2.449

12

~2B7 (HO) 12Bv (WS) t2N 5 (WS)

2.420 2.412 2.795

2.484 2.682 2.422

2.457 2.573 2.646

8

8B 3 ( H O ) 5SB3 (WS)

+_0.8 +_0.9

the r e a c t i o n cross sections o f the p r o t o n - r i c h projectiles, especially at m e d i u m energies Elab/A ~ 3 0 - 4 0 M e V c o m p a r e d w i t h t h o s e o f the n e u t r o n - r i c h m i r r o r nuclei. In c o n c l u s i o n , we s t u d i e d the Q - m o m e n t s in light m i r r o r nuclei by taking the effect o f l o o s e l y - b o u n d n u c l e o n s i n t o a c c o u n t . O u r calculations successfully describe the Q - m o m e n t s not only in the d e e p l y - b o u n d nuclei b u t also in the l o o s e l y - b o u n d ones. T h e large e n h a n c e m e n t o f e x p e r i m e n t a l v a l u e s o f SB a n d ~TF can be i n t e r p r e t e d as clear e v i d e n c e o f the p r o t o n halo effect. In the cases o f the systems o f A = 12 a n d 41 nuclei, h o w e v e r , we n o t i c e an a p p r e c i a b l e discrepa n c y b e t w e e n o u r c a l c u l a t e d results a n d the e m p i r i c a l values. F u r t h e r t h e o r e t i c a l a n d e m p i r i c a l efforts will be necessary to u n d e r s t a n d fully the Q - m o m e n t s o f light m i r r o r nuclei.

References [ 1 ] A. Bohr and B.R. Monelson, Nuclear structure, Vol. I (Benjamin, New York, 1969). 12] B.A. Brown et aL, Nucl. Phys. A 277 ( 1977 ) 77. [ 3 ] H. Sagawa and B.A. Brown, Nucl. Phys. A 430 (1984) 84. [4] I. Tanihata et al., Phys. Rev. Lett. 55 ( 1985 ) 2676. [5] M. Fukuda et al., Phys. Lett. B 268 (1991) 339. [6 ] S. Shimoura et al., to be published. [7] K. Riisager et al., Nucl. Phys. A 548 (1992) 393.

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PHYSICS LETTERS B

[ 8 ] I. Tanihata, Proc. first Intern. Conf. on Radio-active nuclear beams (Berkeley, CA, 1989). [9] S. Cohen and D. Kurath, Nucl. Phys. 73 (1965 ) 1. [ 10 ] D.J. Millener and D. Kurath, Nucl. Phys. A 255 ( 1975 ) 315. [11 ] H. Kitagawa and H. Sagawa, Nucl. Phys. A 551 (1993) 16. [ 12] T. Minamisono et al., Phys. Rev. Lett. 69 (1992) 2058.

21 January 1993

[ 13] G.M. Radutskii et al., Sov. J. Nucl. Phys. 31 (1980) 177. [14] T. Minamisono, Proc. IXth Intern. Conf. on Hyperfine interactions (Osaka, 1992). [ 15 ] D. Sundholm and J. Olsen, J. Chem. Phys. 94 ( 1991 ) 5051. [ 16] D. Sundholm and J. Olsen, J. Phys. Chem. 96 (1992) 627. [ 17] P. Raghavan, At. Data Nucl. Data Tables 42 (1989) 189.