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Study of halo nuclei with phenomenological relativistic mean field approach
26 May 1994 PHYSICS LETTERS B
ELSEVIER
Physics Letters B 328 (1994) 1-4
Study of halo nuclei with phenomenological relativistic mean field approach * Z.Y. Zhu a, W.Q. Shen u, Y.H. Cai b, Y.G. Ma b, a Institute of Nuclear Research Academia Sinica, Shanghai 201800, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 12 January 1994; revised manuscript received 24 March 1994 Editor: C. Mahaux
Abstract
The properties of halo nuclei are studied in the framework of the relativistic mean field approach. Single particle wave functions for last neutron configurations are obtained by using renormalized mean field potentials which reproduce experimental rms radii of halo nuclei. The extracted experimental density distributions and separation energies of loosely-bound neutrons of halo nuclei nLi and UBe are well described by the calculated results.
The recent experimental development and theoretical studies of exotic neutron-rich nuclei have opened a new field in nuclear physics. The experimentally finding o f neutron halo phenomena in llLi and llBe are the most spectacular results [ 1,2]. These nuclei have larger rms radii, which could be related to the small neutron separation energies in such nuclei. As the non-relativistic mean field calculations failed to describe quantitatively the separation energies, Bertsch et al. [ 3 ] introduced an arbitrary renormalization factor in the mean field potential based on the HartreeFock (HF) mean field calculations with Skyrme interaction to reproduce the extracted experimental separation energies of 11Li. Further, Sagawa [4] improved the calculated separation energies and studied the effect of the centrifugal force on the density profile at large radius by using the same method. They found that the separation energy of loosely bound neutrons plays an important role in the study of halo nuclei. RelWork supported by the National N ~ Chir~
*
Science Foundation of
ativistic mean field (RMF) theories have been applied with considerable success for the quantitative description of nuclear properties over the whole periodic in recent years [ 5-8 ]. The ground state properties of Li isotopes have been investigated in the non-linear relativistic mean field theory [9]. The results showed that the relativistic effects are important and that the trend of the binding energies can he satisfactorily obtained in the RMF theory. The RMF approximation, however, cannot reproduce the experimental data quantitatively. In its spirit and in its results this model is very similar to the conventional density dependent HartreeFock theory with Skyrme forces [ 10,11 ]. But the relativistic description has the advantage of treating the spin-degrees of freedom on a more fundamental basis in the Dirac equation and to be technically somewhat simpler. In this letter, we definitively consider the present relativistic mean field theories as phenomenological approach and describe the properties of unstable neutron-rich nuclei in this framework. Starting from the Lagrangian
0370-2693/94/$07.00 © 1994 Elsevier ScienceB.V. All rights reserved SSD10370-2 69 3 ( 94 ) 00406-W
Z.Y. Zhu et al. / Physics Letters B 328 (1994) l ~ I
2
£ = ~(iy"O,, - M)O + ½(O'=o-Ogo"- mo-20"2) -
1 bo.3 -
~1c o - 4 -
!/Wxvg?
-- 4--
lo/xv
I m 2 /z
-=#v -t- ~
O P
O Ptz
+
1
__ ! p ~ v p
- e½(1 + r 3 ) ~ y ~ a ~ o
4--
2
10°
10_1
" IXV
(1)
where the Dirac spinors ~0 represent the nucleons with the mass M. A scalar meson o-, and three vector particles, the isoscalar w-meson, the isovecmr p-meson and the photon A, are also included in Eq. ( 1 ). In addition we take into account a non-linear coupling of the o--mesons amongst themselves, which has turned out to be crucial for a quantitative understanding of many nuclear quantities, and the usual phenomenological constant gap approximations as discussed in Ref. [8] in order to consider the pairing correlations in open shell nuclei. The equations for the baryon wave functions in the static case contain two potentials: the repulsive potential
V(r)
= V~o_p(r)
+ Vcoulomb(r)
(2)
with Vo,-p ( r) = go,wo ( r) + gpr3Po( r) and the attractive potential:
S(r) = g e ~ ( r ) .
I--
#
(3)
In this work, the potentials V~_o(r) and S(r) in Eqs. ( 2 ) , (3) are multiplied by the same constant normalization factor f for the last loosely-bound neutron configurations, i.e., for the neutron orbits outside the core we set f # 1 and otherwise f = 1. In the numerical procedure, the factor f is regarded as input data. The coupled differential equations derived from the Lagrangian are solved self-consistently in the spherical cases with the parameter set NL1 [7]. We adjust the factor f until a calculated mass rms radii is obtained that is very close to the extracted experimental data. The calculated density distributions in l lLi are plotted in Fig. 1. It shows that the calculated density distributions are in excellent agreement with the extracted experimental density distributions. In the present approach these results are obtained in a natural and reasonable way. For the calculations of J lLi, in which no modification is made, the mass rms radii is 2.879 fm
I
~--~'-
10-2
10 . 3
10-5
, , ~ ~ , ,, 0
2
, 4
6 r
,"" g
10
12
(fro)
Fig. 1. Calculated and extracted experimental density distributions in nLi. The solid curve shows the calculated results; the long-dashed curves give the extracted experimental data [12]. While the short-dashed curve is for the RMF calculations with no modifications. and there is no obvious halo tail in the density distributions (short-dashed curve in Fig. 1). As the multiplicative factor to V o ) _ p ( r ) and S(r) for the neutron orbits outside the core was introduced, the single particle energy level occupied by the halo neutrons is very close to the Fermi level. In the case o f 11Li it is 0.54 MeV. Meanwhile, the potential barrier becomes lower in the region beyond r = 5 fm and the halo neutrons feel a more loosely bound as the nucleons in the core. As we adjust the factor f to get the mass rms radii rrms = 3.17, which is very close to the experimental mass rms of 11Li 3.16 ± 0.01, the wave function should characterize more properly the distributions of nucleons in the nucleus, and then the halo tail occurs in the density distributions. The tail of the density distributions is dominated by the halo neutron orbits, which is shown very clearly in Fig. 2. Fig. 3 shows the density distributions in l lBe. When compared with the extracted experimental data, our calculations give very satisfactory results. It should be pointed out that the ground state of l l B e has an abnormal parity J~ --= 1+ [ 13 ]. As the predictions of the naive shell model, the present calculations can only give a negative parity ground state J~ = 1 - of ~lBe, while the 2sw2configuration has a probability of about 10%. In the above procedures, the calculations are sensitive to the
/
Z.Y. Zhu et al. Physics Letters B 328 (1994) 1-4
Table 1 Neutron separation energies and root-mean-square radii of halo nuclei HLi and HBe Nucleus
llLi l Be
Sn
S~nxp
82 n
S~xP 2n
rlTnS
exp rl'ms
(MeV)
(MeV)
(MeV)
(MeV)
(fm)
(fm)
1.00 0.53
0.824-0.30 0.5 !
0.31 8.64
0.334-0.09 7.32
3.17 2.81
3.164-0.01 2.864-0.04
The data of l]Li are taken from Refs. [1,4,14]. Those of 11Be are from Refs. [1,4,15]. multiplicative factor, the optimal value of this factor is close to 0.7 for llLi and llBe. We found also that this renormalization influences only the relative position of levels, especially for the levels nearby the Fermi level, and does not change the order o f the levels. The separation energies are very small in the cases o f 11Li and l t B e and play an important role in the study of these halo nuclei [3,4]. To quantitatively describe the separation energies of halo nuclei, we use the above phenomenological method to calculate the binding energies of halo nuclei llLi and l l B e at the mass rms radii which correspond to the density distributions in Figs. 1 and 2. The results are given in Table 1. The calculated one neutron separation energies S, and two neutron separation energies $2, of llLi and t t B e reproduce very well the observed experimental separation energies S~nxp and f22,~p. In our
calculations, the errors between the calculated binding energies and experimental binding energies are about 5% for nuclei in the same isotope chain. This errors could be canceled through the calculating of separation energies S, and $2,. For example, the calculated binding energy o f 9Li and l lLi are EB (9Li)--- 4 7 . 1 1 3 MeV and E~ (11Li) = - 4 7 . 4 1 9 MeV, respectively, while the experimental binding energies of 9Li and ttLi are - 4 5 . 3 4 1 M e V and - 4 5 . 5 4 0 M e V [ 15]. The difference between the present results and experimental data are roughly 1.8 M e V for both nuclei. In conclusion, we have studied the density distributions, separation energies and mass rms radii of the halo nuclei in the phenomenological relativistic mean field approach. The calculated density distributions of halo nuclei show a nice agreement with the extracted experimental data. The experimental separation energies and the observed mass rms radii are well repro-
10 o
... ~ ~
10-a
101 I "~
llLi
I0 ° I
':'-
'.,
I I
\k'~ 10-2
.. ,"
10-3
10-1
~, -,.,
•
~, ",, ,~, ..~,
,
~
I
10 -2
x
,,.
"..2".
i0-3
-
.
10 -4
""',,,,,,
lO-S 0
2
4
6
8
10
1;~
r (fm) Fig. 2. Calculated and extracted experimental density distributions in 11Li. The solid curve shows the calculated results; The long-dashed curves give the extracted experimental data [ 12]. The density of core 9Li is plotted with short-dashed curve. The dotted curve presents the density of halo neutrons.
10 -6
.................... 0
2
4
6
"-
"-..
~"~...... 8
I0
12
r (tin)
Fig. 3. Calculated and empirical density distributions in 11Be. The empirical data is taken from Ref. [2]. For details see the caption of Fig. 1.
Z.Y. Zhu et al. / Physics Letters B 328 (1994) 1-4
d u c e d in t h e p r e s e n t a p p r o a c h . N e v e r t h e l e s s , a sati s f a c t o r y e x p l a n a t i o n o f t h e r e n o r m a l i z a t i o n f a c t o r is lack. O t h e r w i s e , t h e r e c o u l d b e o t h e r w a y s to h a n d l e t h e n e u t r o n - r i c h n u c l e i in a r e l a t i v i s t i c m o d e l , for exa m p l e s , a refit o f t h e p a r a m e t e r set a n d a i n c l u s i o n o f a tensor coupling of the vector mesons. Further works in t h o s e d i r e c t i o n s are u n d e r c o n s i d e r i n g .
References 11] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoskikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55 (1985) 2676. 12] M. Fukuda, T. lchihara, N. Inabe, T. Kubo, H. Kumagai, T. Nakagawa, I. Tanihata, Y. Yano, M. Adachi, M. Kouguchi, M. Ishihara, H. Sagawa and S. Shimoura, Phys. Lett. B 268 (1991) 339. 131 G.E Bertsch, B.A. Brown and H. Sagawa, Phys. Rev. C 39 (1989) 1154.
14] 151 [6] 171 [8] [91 [10] [111 1121
[13] [141
[151
H. Sagawa, Phys. Lett. B 286 (1992) 7. B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. P.G. Reinhard, Rep. Prog. Phys. 52 (1989) 439. RG. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A 323 (1986) 13. Y.K. Gambhir, P. Ring, A. Thimet, Ann. Phys. (NY) 511 (1990) 129. W. Koepf, Y.K. Gambhir, P. Ring and M.M. Sharma, Z. Phys. A 340 (1991) 119. D. Vautherin, D.M. Brink, Phys. Rev. C 5 (1972) 626. D. Vautherin, Phys. Rev. C 7 (1973) 296. S. Shimoura, T. Nakamura, H. Okamura, H. Sakai, M. Ishihara, N. lnabe, T. Kubo, T. Nakagawa, H. Kumagai and 1. Tanihata, preprint (1992). I. Tahni and I. Unna, Phys. Rev. Lett. 4 (1960) 469. I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda and H. Sato, Phys. Lett. B 206 (1988) 592. A.H. Wapstra and G. Audi, Nucl. Phys. A 432 (1985) 1.
ELSEVIER
Physics Letters B 328 (1994) 1-4
Study of halo nuclei with phenomenological relativistic mean field approach * Z.Y. Zhu a, W.Q. Shen u, Y.H. Cai b, Y.G. Ma b, a Institute of Nuclear Research Academia Sinica, Shanghai 201800, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China
Received 12 January 1994; revised manuscript received 24 March 1994 Editor: C. Mahaux
Abstract
The properties of halo nuclei are studied in the framework of the relativistic mean field approach. Single particle wave functions for last neutron configurations are obtained by using renormalized mean field potentials which reproduce experimental rms radii of halo nuclei. The extracted experimental density distributions and separation energies of loosely-bound neutrons of halo nuclei nLi and UBe are well described by the calculated results.
The recent experimental development and theoretical studies of exotic neutron-rich nuclei have opened a new field in nuclear physics. The experimentally finding o f neutron halo phenomena in llLi and llBe are the most spectacular results [ 1,2]. These nuclei have larger rms radii, which could be related to the small neutron separation energies in such nuclei. As the non-relativistic mean field calculations failed to describe quantitatively the separation energies, Bertsch et al. [ 3 ] introduced an arbitrary renormalization factor in the mean field potential based on the HartreeFock (HF) mean field calculations with Skyrme interaction to reproduce the extracted experimental separation energies of 11Li. Further, Sagawa [4] improved the calculated separation energies and studied the effect of the centrifugal force on the density profile at large radius by using the same method. They found that the separation energy of loosely bound neutrons plays an important role in the study of halo nuclei. RelWork supported by the National N ~ Chir~
*
Science Foundation of
ativistic mean field (RMF) theories have been applied with considerable success for the quantitative description of nuclear properties over the whole periodic in recent years [ 5-8 ]. The ground state properties of Li isotopes have been investigated in the non-linear relativistic mean field theory [9]. The results showed that the relativistic effects are important and that the trend of the binding energies can he satisfactorily obtained in the RMF theory. The RMF approximation, however, cannot reproduce the experimental data quantitatively. In its spirit and in its results this model is very similar to the conventional density dependent HartreeFock theory with Skyrme forces [ 10,11 ]. But the relativistic description has the advantage of treating the spin-degrees of freedom on a more fundamental basis in the Dirac equation and to be technically somewhat simpler. In this letter, we definitively consider the present relativistic mean field theories as phenomenological approach and describe the properties of unstable neutron-rich nuclei in this framework. Starting from the Lagrangian
0370-2693/94/$07.00 © 1994 Elsevier ScienceB.V. All rights reserved SSD10370-2 69 3 ( 94 ) 00406-W
Z.Y. Zhu et al. / Physics Letters B 328 (1994) l ~ I
2
£ = ~(iy"O,, - M)O + ½(O'=o-Ogo"- mo-20"2) -
1 bo.3 -
~1c o - 4 -
!/Wxvg?
-- 4--
lo/xv
I m 2 /z
-=#v -t- ~
O P
O Ptz
+
1
__ ! p ~ v p
- e½(1 + r 3 ) ~ y ~ a ~ o
4--
2
10°
10_1
" IXV
(1)
where the Dirac spinors ~0 represent the nucleons with the mass M. A scalar meson o-, and three vector particles, the isoscalar w-meson, the isovecmr p-meson and the photon A, are also included in Eq. ( 1 ). In addition we take into account a non-linear coupling of the o--mesons amongst themselves, which has turned out to be crucial for a quantitative understanding of many nuclear quantities, and the usual phenomenological constant gap approximations as discussed in Ref. [8] in order to consider the pairing correlations in open shell nuclei. The equations for the baryon wave functions in the static case contain two potentials: the repulsive potential
V(r)
= V~o_p(r)
+ Vcoulomb(r)
(2)
with Vo,-p ( r) = go,wo ( r) + gpr3Po( r) and the attractive potential:
S(r) = g e ~ ( r ) .
I--
#
(3)
In this work, the potentials V~_o(r) and S(r) in Eqs. ( 2 ) , (3) are multiplied by the same constant normalization factor f for the last loosely-bound neutron configurations, i.e., for the neutron orbits outside the core we set f # 1 and otherwise f = 1. In the numerical procedure, the factor f is regarded as input data. The coupled differential equations derived from the Lagrangian are solved self-consistently in the spherical cases with the parameter set NL1 [7]. We adjust the factor f until a calculated mass rms radii is obtained that is very close to the extracted experimental data. The calculated density distributions in l lLi are plotted in Fig. 1. It shows that the calculated density distributions are in excellent agreement with the extracted experimental density distributions. In the present approach these results are obtained in a natural and reasonable way. For the calculations of J lLi, in which no modification is made, the mass rms radii is 2.879 fm
I
~--~'-
10-2
10 . 3
10-5
, , ~ ~ , ,, 0
2
, 4
6 r
,"" g
10
12
(fro)
Fig. 1. Calculated and extracted experimental density distributions in nLi. The solid curve shows the calculated results; the long-dashed curves give the extracted experimental data [12]. While the short-dashed curve is for the RMF calculations with no modifications. and there is no obvious halo tail in the density distributions (short-dashed curve in Fig. 1). As the multiplicative factor to V o ) _ p ( r ) and S(r) for the neutron orbits outside the core was introduced, the single particle energy level occupied by the halo neutrons is very close to the Fermi level. In the case o f 11Li it is 0.54 MeV. Meanwhile, the potential barrier becomes lower in the region beyond r = 5 fm and the halo neutrons feel a more loosely bound as the nucleons in the core. As we adjust the factor f to get the mass rms radii rrms = 3.17, which is very close to the experimental mass rms of 11Li 3.16 ± 0.01, the wave function should characterize more properly the distributions of nucleons in the nucleus, and then the halo tail occurs in the density distributions. The tail of the density distributions is dominated by the halo neutron orbits, which is shown very clearly in Fig. 2. Fig. 3 shows the density distributions in l lBe. When compared with the extracted experimental data, our calculations give very satisfactory results. It should be pointed out that the ground state of l l B e has an abnormal parity J~ --= 1+ [ 13 ]. As the predictions of the naive shell model, the present calculations can only give a negative parity ground state J~ = 1 - of ~lBe, while the 2sw2configuration has a probability of about 10%. In the above procedures, the calculations are sensitive to the
/
Z.Y. Zhu et al. Physics Letters B 328 (1994) 1-4
Table 1 Neutron separation energies and root-mean-square radii of halo nuclei HLi and HBe Nucleus
llLi l Be
Sn
S~nxp
82 n
S~xP 2n
rlTnS
exp rl'ms
(MeV)
(MeV)
(MeV)
(MeV)
(fm)
(fm)
1.00 0.53
0.824-0.30 0.5 !
0.31 8.64
0.334-0.09 7.32
3.17 2.81
3.164-0.01 2.864-0.04
The data of l]Li are taken from Refs. [1,4,14]. Those of 11Be are from Refs. [1,4,15]. multiplicative factor, the optimal value of this factor is close to 0.7 for llLi and llBe. We found also that this renormalization influences only the relative position of levels, especially for the levels nearby the Fermi level, and does not change the order o f the levels. The separation energies are very small in the cases o f 11Li and l t B e and play an important role in the study of these halo nuclei [3,4]. To quantitatively describe the separation energies of halo nuclei, we use the above phenomenological method to calculate the binding energies of halo nuclei llLi and l l B e at the mass rms radii which correspond to the density distributions in Figs. 1 and 2. The results are given in Table 1. The calculated one neutron separation energies S, and two neutron separation energies $2, of llLi and t t B e reproduce very well the observed experimental separation energies S~nxp and f22,~p. In our
calculations, the errors between the calculated binding energies and experimental binding energies are about 5% for nuclei in the same isotope chain. This errors could be canceled through the calculating of separation energies S, and $2,. For example, the calculated binding energy o f 9Li and l lLi are EB (9Li)--- 4 7 . 1 1 3 MeV and E~ (11Li) = - 4 7 . 4 1 9 MeV, respectively, while the experimental binding energies of 9Li and ttLi are - 4 5 . 3 4 1 M e V and - 4 5 . 5 4 0 M e V [ 15]. The difference between the present results and experimental data are roughly 1.8 M e V for both nuclei. In conclusion, we have studied the density distributions, separation energies and mass rms radii of the halo nuclei in the phenomenological relativistic mean field approach. The calculated density distributions of halo nuclei show a nice agreement with the extracted experimental data. The experimental separation energies and the observed mass rms radii are well repro-
10 o
... ~ ~
10-a
101 I "~
llLi
I0 ° I
':'-
'.,
I I
\k'~ 10-2
.. ,"
10-3
10-1
~, -,.,
•
~, ",, ,~, ..~,
,
~
I
10 -2
x
,,.
"..2".
i0-3
-
.
10 -4
""',,,,,,
lO-S 0
2
4
6
8
10
1;~
r (fm) Fig. 2. Calculated and extracted experimental density distributions in 11Li. The solid curve shows the calculated results; The long-dashed curves give the extracted experimental data [ 12]. The density of core 9Li is plotted with short-dashed curve. The dotted curve presents the density of halo neutrons.
10 -6
.................... 0
2
4
6
"-
"-..
~"~...... 8
I0
12
r (tin)
Fig. 3. Calculated and empirical density distributions in 11Be. The empirical data is taken from Ref. [2]. For details see the caption of Fig. 1.
Z.Y. Zhu et al. / Physics Letters B 328 (1994) 1-4
d u c e d in t h e p r e s e n t a p p r o a c h . N e v e r t h e l e s s , a sati s f a c t o r y e x p l a n a t i o n o f t h e r e n o r m a l i z a t i o n f a c t o r is lack. O t h e r w i s e , t h e r e c o u l d b e o t h e r w a y s to h a n d l e t h e n e u t r o n - r i c h n u c l e i in a r e l a t i v i s t i c m o d e l , for exa m p l e s , a refit o f t h e p a r a m e t e r set a n d a i n c l u s i o n o f a tensor coupling of the vector mesons. Further works in t h o s e d i r e c t i o n s are u n d e r c o n s i d e r i n g .
References 11] I. Tanihata, H. Hamagaki, O. Hashimoto, Y. Shida, N. Yoskikawa, K. Sugimoto, O. Yamakawa, T. Kobayashi and N. Takahashi, Phys. Rev. Lett. 55 (1985) 2676. 12] M. Fukuda, T. lchihara, N. Inabe, T. Kubo, H. Kumagai, T. Nakagawa, I. Tanihata, Y. Yano, M. Adachi, M. Kouguchi, M. Ishihara, H. Sagawa and S. Shimoura, Phys. Lett. B 268 (1991) 339. 131 G.E Bertsch, B.A. Brown and H. Sagawa, Phys. Rev. C 39 (1989) 1154.
14] 151 [6] 171 [8] [91 [10] [111 1121
[13] [141
[151
H. Sagawa, Phys. Lett. B 286 (1992) 7. B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. P.G. Reinhard, Rep. Prog. Phys. 52 (1989) 439. RG. Reinhard, M. Rufa, J. Maruhn, W. Greiner and J. Friedrich, Z. Phys. A 323 (1986) 13. Y.K. Gambhir, P. Ring, A. Thimet, Ann. Phys. (NY) 511 (1990) 129. W. Koepf, Y.K. Gambhir, P. Ring and M.M. Sharma, Z. Phys. A 340 (1991) 119. D. Vautherin, D.M. Brink, Phys. Rev. C 5 (1972) 626. D. Vautherin, Phys. Rev. C 7 (1973) 296. S. Shimoura, T. Nakamura, H. Okamura, H. Sakai, M. Ishihara, N. lnabe, T. Kubo, T. Nakagawa, H. Kumagai and 1. Tanihata, preprint (1992). I. Tahni and I. Unna, Phys. Rev. Lett. 4 (1960) 469. I. Tanihata, T. Kobayashi, O. Yamakawa, S. Shimoura, K. Ekuni, K. Sugimoto, N. Takahashi, T. Shimoda and H. Sato, Phys. Lett. B 206 (1988) 592. A.H. Wapstra and G. Audi, Nucl. Phys. A 432 (1985) 1.