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Microscopic three-cluster study of 20 O, 20 Mg and 19 N, 19 Mg exotic nuclei
1 October 1998
Physics Letters B 437 Ž1998. 7–11
Microscopic three-cluster study of O, 20 Mg and 19 N, 19 Mg exotic nuclei 20
P. Descouvemont
1
Physique Nucleaire Theorique et Physique Mathematique, C.P. 229 UniÕersite´ Libre de Bruxelles, B-1050 Bruxelles, Belgium ´ ´ ´ Received 13 February 1998; revised 29 May 1998 Editor: J.-P. Blaizot
Abstract The 20 O and 19 N neutron-rich nuclei are studied in a 16 O q 2n q 2n and 15 N q 2n q 2n three-cluster model respectively. The mirror systems are also analyzed without approximation on the Coulomb force. Spectroscopic properties of 20 O and 20 Mg are in reasonable agreement with experiment, and do not show evidence for an exotic structure. The 19 Mg ground state is predicted to be close to particle stability. q 1998 Elsevier Science B.V. All rights reserved.
Nuclear spectroscopy near the neutron and proton drip lines is one of the most successful applications of radioactive ion beams w1x. For example, measurements of r.m.s. radii in neutron-rich nuclei gave evidence for a new phenomenon, the halo structure w2x. The 11 Li nucleus was the first observed halo nucleus, but several other candidates have been recently investigated, such as 6 He or 14 Be. Even though proton halos should be strongly hindered by the Coulomb force, such a proton halo has also been suggested in some nuclei, such as 8 B w3x. More recently, Chulkov et al. w4x analyzed interaction cross sections of A s 20 isobars, ranging from 20 N to 20 Mg, on carbon targets. The r.m.s. radii were then derived from a theoretical analysis within the Glauber model, and show an irregular dependence on isospin projection. For example the difference be-
1
ˆ de Recherches FNRS. Maitre
tween the 20 Mg and 20 O r.m.s. radii is found to be 0.2 fm. Chulkov et al. also suggest the existence of a neutron skin in 20 N and of a proton skin in 20 Mg. In the present letter, we aim at investigating spectroscopic properties of A s 20 and A s 19 exotic nuclei in a microscopic cluster model w5x. In the Generator Coordinate Method ŽGCM., the wave functions of the system are defined from internal wave functions of clusters. When this cluster structure and the nucleon-nucleon interaction are chosen, the model is parameter free. The GCM has been successfully applied to several halo systems, such as 11 Li w6x or 17 B w7x. Here we shall consider 20 O and 20 Mg by an 16 O q 2n q 2n Žor mirror. cluster structure, where notation ‘‘2n’’ stands for a pointlike dineutron, i.e. a Ž0 s1r2. 2 configuration with the same oscillator parameter as for the core. This approximation has been shown to give fairly good results for the 17 B nucleus in a 13 B q 2n q 2n threecluster model w7x. In parallel, the 19 N Žs15 N q 2n q 2n. and 19 Mg systems will be analyzed. Very few
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 9 4 5 - 9
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
8
information is known about 19 Mg, which is expected to be particle unstable w8x. Let us briefly define the three-cluster microscopic model. Technical details are given in Ref. w7x and are not repeated here. If the core wave function with spin I1 Ž16 O, 15 N or 15 O in the present case. is denoted as f I1 K 1 and the dineutron wave function is denoted as f 2 n , a three-cluster GCM state reads
F I1 K 1Ž S1 ,S2 ,S3 . s Af I1 K 1Ž S1 . f 2 n Ž S2 . f 2 n Ž S3 .
Ž 1. where S1 , S2 , and S3 are the locations of the clusters. Of course, extension to the mirror nucleus is trivial. Within shell model description of clusters with identical oscillator parameters, functions Ž1. are Slater determinants. In practice, the distance between the core and the center-of-mass of the dineutrons is taken as the generator coordinate R 1 , the distance between dineutrons is chosen as R 2 ; the angle between both directions is denoted as a . After projection on total spin and parity, we have p F IJM Ž R1 , R 2 , a . s 12 1K1, K
J) MK
HD
Ž V . RJŽ V .
= Ž 1 q p P . F I1 K 1Ž S1 ,S2 ,S3 . d V
Ž 2.
where R J Ž V . is the rotation operator involving the Euler angles V , DMJ )K Ž V . is the Wigner function, and P the parity operator; K is the spin projection over the intrinsic z axis, chosen along R 1 Žwe select K s 0 for 20 O, and K s 1r2,3r2 for 19 N.. Matrix elements of the Hamiltonian between projected states Ž2. yield the total wave functions of the system. These matrix elements are obtained from numerical integration of matrix elements between unprojected states Ž1.. Let us point out that the Coulomb term is treated exactly; predictions on proton-rich nuclei can therefore be reliably done from the study of their mirrors. The 16 O wave function is defined by a closed p-shell structure. For 15 N and 15 O, 11 particles in the p shell yield the I1 s 1r2y ground state, and the I1 s 3r2y excited state, which are both included in the GCM basis Ž2.. The oscillator parameter is taken as b s 1.8 fm, which approximately reproduces the 16 O and 15 N experimental radii. The generator coordinates are taken as R 1 s 1 fm to 6 fm Žwith a step of 1 fm., R 2 s 0.5 to 4.5 fm Žwith a step of 1 fm.,
and a s Ž08,458,908., which are expected to cover reasonably well the configuration space. The calculations are performed with two nucleon-nucleon interactions: the Volkov V2 force w9x and the Minnesota force w10x Žhereafter referred to as MN.. This will allow us to evaluate the sensitivity of the results with respect to the force. The exact Coulomb interaction, and a zero-range spinorbit force w11x are included in the Hamiltonian. The amplitude of the spin-orbit potential is taken as S0 s 50 MeV.fm5, which approximately reproduces the 3r2y excitation energy in 15 N and 15 O with b s 1.8 fm. Parameters m Žin the V2 force. and u Žin the MN force. are determined on the 20 O– 16 O binding energy Žy23.75 MeV. for A s 20, and on the 19 N– 15 N binding energy Žy16.53 MeV. for A s 19. This yields m s 0.577 and 0.580, and u s 1.029 and 1.031 for 20 O and 19 N respectively. Proton-rich mirror nuclei are studied without any fitting procedure. Notice that, within the Ž0 s1r2. 2 description, the Coulomb energy of the diproton is 2rp e 2rb. Before using the full GCM basis mentioned above, let us consider the energy surfaces displayed in Fig. 1. The energy surfaces w7x correspond to the energy of the system with a single Ž R 1 , R 2 . value; diagonalization is performed with respect to the a angle. Energy surfaces give a qualitative picture of the
'
Fig. 1. Energy surfaces of 20 O and 19 N states.
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
9
Fig. 2. Spectra of 20 O and 20 Mg with the V2 and MN forces. Experimental data are taken from Ref. w12x.
cluster structure. From Fig. 1, it appears that the structures of 20 O and 19 N ground states are rather similar. Both present a minimum near R 1 s 1.5 fm and R 2 s 2.0 fm. The variation with respect to R 2 is
rather weak, which suggests a low correlation between the dineutrons. Energy surfaces for the mirror nuclei have similar shapes and are therefore not shown.
Fig. 3. Spectra of 19 N with the V2 and MN forces Žthe levels are labeled by 2 J .. Experimental data are taken from Ref. w14x. The r.h.s. of the figure shows the predicted energy of the 19 Mg nucleus, along with p, 2p and 3p experimental thresholds.
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
10
Table 1 R.m.s. radii Žin fm., quadrupole moments Žin e.fm 2 . and B ŽE2. Žin e 2 .fm 4 . in 20
2: m
(
'² r
2: n
Dr QŽ2q . q. B Ž E2,2q 1 ™ 01 q. B Ž E2,2q ™ 0 2 1 q. B Ž E2,4q 1 ™ 21 a
Interaction radii.
b
MN
exp. w12x
20
Mg
Mg
V2
exp. w12x
MN
2.78
2.64 " 0.03 a
2.87
2.80
2.86 " 0.06 a
2.66
2.64
b
3.01
2.90
3.05 " 0.10 b
2.94 0.28 -2.6 3.5 0.14 2.1
2.87 0.23 -1.4 2.4 0.12 1.1
2.64 0.37 -10.5 41.8 2.9 28.2
2.64 0.26 -3.8 29.6 1.9 11.6
2.83
² r 2 :p
O and 20
O
V2
'² r
20
2.61 " 0.06
0.05 " 0.06 b 4.03 " 0.16 c
Extrapolated from neighboring nuclei.
c
0.5 "0.3 b
From Ref. w12x.
Energy spectra of A s 20 and A s 19 nuclei are shown in Figs. 2 and 3 respectively. Let us repeat that only the ground state of 20 O and 19 N are fitted by the nucleon-nucleon interaction. For 20 O, we obtain a reasonable agreement with the experimental data w12x. For both interactions the 2q excitation energy is well reproduced. A second 0q state is found at E x s 4.7 MeV ŽV2. and 5.2 MeV ŽMN., close to the experimental energy Ž4.46 MeV.. The 4q states are however predicted at too low energies, which might arise from the lack of spin-orbit force in the present description of 20 O. With respect to the shell-model theory w13x, the GCM improves the 0q– 2q excitation energy, which is underestimated in Ref. w13x. In agreement with experiment, no negative-parity state is found in the low-energy spectrum. The first theoretical state has a 1y assignment, and is located at E x s 11.89 MeV for V2, and 10.37 MeV for MN. For 20 Mg, only the ground state has been observed. As expected, the GCM spectra are similar to those of 20 O. The Coulomb displacement energy is underestimated by 1 MeV with the V2 force and by less than 0.5 MeV with the MN force. In Fig. 3, we compare the GCM 19 N spectra with the experimental data of Catford et al. w14x. As expected from shell-model approximations, the ground state is found to be 1r2y. The first excited state is 5r2y with V2 and MN; from its excitation energy, this state might correspond to the experimental first excited state at E x s 1.11 MeV. The ground state of 19 Mg is shown on the r.h.s. of Fig. 3, with different experimental thresholds. The V2 force pre-
dicts the ground state to be slightly below the lowest Ne q 2p threshold, whereas the MN force gives a positive value. Of course, a microscopic model is not able to give results with such a high accuracy, but we may conjecture that the 19 Mg should be close to the particle stability. Spectroscopic properties of 20 O and 20 Mg are displayed in Table 1. The difference between matter radii of 20 Mg and 20 O is calculated as 0.04 fm, smaller than the experimental value 0.2 fm w4x. However, experimental radii are actually interaction radii, which are known to be partly model dependent w15x. The 20 Mg matter radius is well reproduced by the GCM but the 20 O radius is overestimated. The same behaviour appears in the relativistic mean-field model w4,16,17x and in Hartree–Fock models w18,19x which yields matter radii quite similar to our’s. The proton radii are in fair agreement with the extrapolations suggested by Chulkov et al. From these estimates, 17
Table 2 R.m.s. radii Žin fm. and B ŽE2. Žin e 2 .fm 4 . in
'² r
2: m
(
² r 2 :p
'
² r 2 :n
Dr B Ž E2,3r2y ™1r2y . B Ž E2,5r2y ™1r2y . B Ž E2,3r2y ™ 5r2y .
19
N
V2
MN
2.86
2.80
2.64
2.63
2.98 0.34 4.2 4.0 0.13
2.89 0.26 2.3 2.7 0.23
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
Chulkov et al. determine neutron radii and conclude on an exotic structure of 20 Mg, with a proton skin of 0.5 " 0.3 fm. Our result Ž0.37 fm for V2, 0.26 fm for MN. is not in contradiction with this experimental value. However, the theoretical ratio ² r 2 :p r
(
(² r
2: n
s 1.14 ŽV2. or 1.10 ŽMN. is fairly close to the ratio expected from the shell model Ž1.09.. We therefore conclude that the difference between proton and neutron radii is a normal effect, and does not need any assumption concerning a ‘‘halo’’ structure. This conclusion is also supported by the energy surfaces ŽFig. 1. which present their minima for small generator coordinates, contrary to what is expected for halo nuclei w6x. Spectroscopic properties of 20 O and 20 Mg are complemented by the E2 transition probabilities. For 20 O, an effective charge d ere s 0.3 is included; this rather small effective charge w20x aims at simulating core polarization effects, and yields a B Ž E2,2q 1 ,™ . 0q value is in fair agreement with experiment. The 1 results are however sensitive to the choice of the nucleon-nucleon interaction. The large transition q. q. probabilities B Ž E2,2q and B Ž E2,4q 1 ,™ 0 1 1 ,™ 2 1 suggest a rotational band in the low-energy spectrum. Spectroscopic properties of 19 N are presented in Table 2. Here also, the ratio between neutron and proton radii can be explained by the shell-model value ² r 2 :p r ² r 2 :n s 1.10, and does not require introducing a halo effect. The GCM radii with the V2 interaction are close to those of Ren et al. w21x in the relativistic mean field approach. In conclusion, the model gives a satisfactory accuracy, as for the first excited level of 20 O and 19 N. Of course, the GCM basis assumes a core plus two dineutrons Žor diprotons. structure, although other configurations Žfor example 18 O q n q n in 20 O. have a lower threshold. However, antisymmetrization makes these configurations nearly equivalent in bound states. This approximation has been shown to be quite justified in the 17 B nucleus, defined as a 13 B q 2n q 2n three-cluster structure w7x. It is also supported by the recent work of Chulkov et al. w22x who conclude on a strong correlation between the
(
(
11
valence protons in 20 Mg. The present model is consistent with the radius measurement of Chulkov et al. w4x, but does not support a ‘‘halo’’ structure in 20 Mg. From the analysis of the 19 N and 19 Mg systems, we find that the 19 Mg nucleus should be close to particle stability.
Acknowledgements This text presents research results of the Belgian program on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs.
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x
I. Tanihata, J. Phys. G 22 Ž1996. 157. P.G. Hansen, B. Jonson, Europhys. Lett. 4 Ž1987. 409. T. Minamisono et al., Phys. Rev. Lett. 69 Ž1992. 2058. L. Chulkov et al., Nucl. Phys. A 603 Ž1996. 219. Y.C. Tang, in: Topics in Nuclear Physics II, Lecture Notes in Physics, vol. 145, Springer, Berlin, 1981, p. 572. P. Descouvemont, Nucl. Phys. A 626 Ž1997. 647. P. Descouvemont, Nucl. Phys. A 581 Ž1995. 61. G. Audi, A.H. Wapstra, Nucl. Phys. A 565 Ž1993. 1. A.B. Volkov, Nucl. Phys. 74 Ž1965. 33. D.R. Thompson, M. LeMere, Y.C. Tang, Nucl. Phys. A 286 Ž1977. 53. D. Baye, N. Pecher, Bull. Cl. Sc. Acad. Roy. Belg. 67 Ž1981. 835. F. Ajzenberg-Selove, Nucl. Phys. A 475 Ž1987. 1. J.B. McGrory, B.H. Wildenthal, Phys. Rev. C 7 Ž1973. 974. W.N. Catford, L.K. Fifield, N.A. Orr, C.L. Woods, Nucl. Phys. A 503 Ž1989. 263. J.S. Al-Khalili, J.A. Tostevin, Phys. Rev. Lett. 76 Ž1996. 3903. Z. Ren, W. Mittig, B. Chen, Z. Ma, G. Auger, Z. Phys. A 353 Ž1996. 363. Z. Ren, Z.Y. Zhu, Y.H. Cai, G. Xu, Phys. Lett. B 380 Ž1996. 241. N. Fukunishi, T. Otsuka, I. Tanihata, Phys. Rev. C 48 Ž1993. 1648. H. Kitagawa, N. Tajima, H. Sagawa, Z. Phys. A 358 Ž1997. 381. E.K. Warburton, J. Weneser, in: D.H. Wilkinson ŽEd.., Isospin in Nuclear Physics, North-Holland, 1969, p. 173. Z. Ren, B. Chen, Z. Ma, G. Xu, J. Phys. G 22 Ž1996. L1. L. Chulkov, E. Roeckl, G. Kraus, Z. Phys. A 353 Ž1996. 351.
Physics Letters B 437 Ž1998. 7–11
Microscopic three-cluster study of O, 20 Mg and 19 N, 19 Mg exotic nuclei 20
P. Descouvemont
1
Physique Nucleaire Theorique et Physique Mathematique, C.P. 229 UniÕersite´ Libre de Bruxelles, B-1050 Bruxelles, Belgium ´ ´ ´ Received 13 February 1998; revised 29 May 1998 Editor: J.-P. Blaizot
Abstract The 20 O and 19 N neutron-rich nuclei are studied in a 16 O q 2n q 2n and 15 N q 2n q 2n three-cluster model respectively. The mirror systems are also analyzed without approximation on the Coulomb force. Spectroscopic properties of 20 O and 20 Mg are in reasonable agreement with experiment, and do not show evidence for an exotic structure. The 19 Mg ground state is predicted to be close to particle stability. q 1998 Elsevier Science B.V. All rights reserved.
Nuclear spectroscopy near the neutron and proton drip lines is one of the most successful applications of radioactive ion beams w1x. For example, measurements of r.m.s. radii in neutron-rich nuclei gave evidence for a new phenomenon, the halo structure w2x. The 11 Li nucleus was the first observed halo nucleus, but several other candidates have been recently investigated, such as 6 He or 14 Be. Even though proton halos should be strongly hindered by the Coulomb force, such a proton halo has also been suggested in some nuclei, such as 8 B w3x. More recently, Chulkov et al. w4x analyzed interaction cross sections of A s 20 isobars, ranging from 20 N to 20 Mg, on carbon targets. The r.m.s. radii were then derived from a theoretical analysis within the Glauber model, and show an irregular dependence on isospin projection. For example the difference be-
1
ˆ de Recherches FNRS. Maitre
tween the 20 Mg and 20 O r.m.s. radii is found to be 0.2 fm. Chulkov et al. also suggest the existence of a neutron skin in 20 N and of a proton skin in 20 Mg. In the present letter, we aim at investigating spectroscopic properties of A s 20 and A s 19 exotic nuclei in a microscopic cluster model w5x. In the Generator Coordinate Method ŽGCM., the wave functions of the system are defined from internal wave functions of clusters. When this cluster structure and the nucleon-nucleon interaction are chosen, the model is parameter free. The GCM has been successfully applied to several halo systems, such as 11 Li w6x or 17 B w7x. Here we shall consider 20 O and 20 Mg by an 16 O q 2n q 2n Žor mirror. cluster structure, where notation ‘‘2n’’ stands for a pointlike dineutron, i.e. a Ž0 s1r2. 2 configuration with the same oscillator parameter as for the core. This approximation has been shown to give fairly good results for the 17 B nucleus in a 13 B q 2n q 2n threecluster model w7x. In parallel, the 19 N Žs15 N q 2n q 2n. and 19 Mg systems will be analyzed. Very few
0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 0 9 4 5 - 9
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
8
information is known about 19 Mg, which is expected to be particle unstable w8x. Let us briefly define the three-cluster microscopic model. Technical details are given in Ref. w7x and are not repeated here. If the core wave function with spin I1 Ž16 O, 15 N or 15 O in the present case. is denoted as f I1 K 1 and the dineutron wave function is denoted as f 2 n , a three-cluster GCM state reads
F I1 K 1Ž S1 ,S2 ,S3 . s Af I1 K 1Ž S1 . f 2 n Ž S2 . f 2 n Ž S3 .
Ž 1. where S1 , S2 , and S3 are the locations of the clusters. Of course, extension to the mirror nucleus is trivial. Within shell model description of clusters with identical oscillator parameters, functions Ž1. are Slater determinants. In practice, the distance between the core and the center-of-mass of the dineutrons is taken as the generator coordinate R 1 , the distance between dineutrons is chosen as R 2 ; the angle between both directions is denoted as a . After projection on total spin and parity, we have p F IJM Ž R1 , R 2 , a . s 12 1K1, K
J) MK
HD
Ž V . RJŽ V .
= Ž 1 q p P . F I1 K 1Ž S1 ,S2 ,S3 . d V
Ž 2.
where R J Ž V . is the rotation operator involving the Euler angles V , DMJ )K Ž V . is the Wigner function, and P the parity operator; K is the spin projection over the intrinsic z axis, chosen along R 1 Žwe select K s 0 for 20 O, and K s 1r2,3r2 for 19 N.. Matrix elements of the Hamiltonian between projected states Ž2. yield the total wave functions of the system. These matrix elements are obtained from numerical integration of matrix elements between unprojected states Ž1.. Let us point out that the Coulomb term is treated exactly; predictions on proton-rich nuclei can therefore be reliably done from the study of their mirrors. The 16 O wave function is defined by a closed p-shell structure. For 15 N and 15 O, 11 particles in the p shell yield the I1 s 1r2y ground state, and the I1 s 3r2y excited state, which are both included in the GCM basis Ž2.. The oscillator parameter is taken as b s 1.8 fm, which approximately reproduces the 16 O and 15 N experimental radii. The generator coordinates are taken as R 1 s 1 fm to 6 fm Žwith a step of 1 fm., R 2 s 0.5 to 4.5 fm Žwith a step of 1 fm.,
and a s Ž08,458,908., which are expected to cover reasonably well the configuration space. The calculations are performed with two nucleon-nucleon interactions: the Volkov V2 force w9x and the Minnesota force w10x Žhereafter referred to as MN.. This will allow us to evaluate the sensitivity of the results with respect to the force. The exact Coulomb interaction, and a zero-range spinorbit force w11x are included in the Hamiltonian. The amplitude of the spin-orbit potential is taken as S0 s 50 MeV.fm5, which approximately reproduces the 3r2y excitation energy in 15 N and 15 O with b s 1.8 fm. Parameters m Žin the V2 force. and u Žin the MN force. are determined on the 20 O– 16 O binding energy Žy23.75 MeV. for A s 20, and on the 19 N– 15 N binding energy Žy16.53 MeV. for A s 19. This yields m s 0.577 and 0.580, and u s 1.029 and 1.031 for 20 O and 19 N respectively. Proton-rich mirror nuclei are studied without any fitting procedure. Notice that, within the Ž0 s1r2. 2 description, the Coulomb energy of the diproton is 2rp e 2rb. Before using the full GCM basis mentioned above, let us consider the energy surfaces displayed in Fig. 1. The energy surfaces w7x correspond to the energy of the system with a single Ž R 1 , R 2 . value; diagonalization is performed with respect to the a angle. Energy surfaces give a qualitative picture of the
'
Fig. 1. Energy surfaces of 20 O and 19 N states.
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
9
Fig. 2. Spectra of 20 O and 20 Mg with the V2 and MN forces. Experimental data are taken from Ref. w12x.
cluster structure. From Fig. 1, it appears that the structures of 20 O and 19 N ground states are rather similar. Both present a minimum near R 1 s 1.5 fm and R 2 s 2.0 fm. The variation with respect to R 2 is
rather weak, which suggests a low correlation between the dineutrons. Energy surfaces for the mirror nuclei have similar shapes and are therefore not shown.
Fig. 3. Spectra of 19 N with the V2 and MN forces Žthe levels are labeled by 2 J .. Experimental data are taken from Ref. w14x. The r.h.s. of the figure shows the predicted energy of the 19 Mg nucleus, along with p, 2p and 3p experimental thresholds.
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
10
Table 1 R.m.s. radii Žin fm., quadrupole moments Žin e.fm 2 . and B ŽE2. Žin e 2 .fm 4 . in 20
2: m
(
'² r
2: n
Dr QŽ2q . q. B Ž E2,2q 1 ™ 01 q. B Ž E2,2q ™ 0 2 1 q. B Ž E2,4q 1 ™ 21 a
Interaction radii.
b
MN
exp. w12x
20
Mg
Mg
V2
exp. w12x
MN
2.78
2.64 " 0.03 a
2.87
2.80
2.86 " 0.06 a
2.66
2.64
b
3.01
2.90
3.05 " 0.10 b
2.94 0.28 -2.6 3.5 0.14 2.1
2.87 0.23 -1.4 2.4 0.12 1.1
2.64 0.37 -10.5 41.8 2.9 28.2
2.64 0.26 -3.8 29.6 1.9 11.6
2.83
² r 2 :p
O and 20
O
V2
'² r
20
2.61 " 0.06
0.05 " 0.06 b 4.03 " 0.16 c
Extrapolated from neighboring nuclei.
c
0.5 "0.3 b
From Ref. w12x.
Energy spectra of A s 20 and A s 19 nuclei are shown in Figs. 2 and 3 respectively. Let us repeat that only the ground state of 20 O and 19 N are fitted by the nucleon-nucleon interaction. For 20 O, we obtain a reasonable agreement with the experimental data w12x. For both interactions the 2q excitation energy is well reproduced. A second 0q state is found at E x s 4.7 MeV ŽV2. and 5.2 MeV ŽMN., close to the experimental energy Ž4.46 MeV.. The 4q states are however predicted at too low energies, which might arise from the lack of spin-orbit force in the present description of 20 O. With respect to the shell-model theory w13x, the GCM improves the 0q– 2q excitation energy, which is underestimated in Ref. w13x. In agreement with experiment, no negative-parity state is found in the low-energy spectrum. The first theoretical state has a 1y assignment, and is located at E x s 11.89 MeV for V2, and 10.37 MeV for MN. For 20 Mg, only the ground state has been observed. As expected, the GCM spectra are similar to those of 20 O. The Coulomb displacement energy is underestimated by 1 MeV with the V2 force and by less than 0.5 MeV with the MN force. In Fig. 3, we compare the GCM 19 N spectra with the experimental data of Catford et al. w14x. As expected from shell-model approximations, the ground state is found to be 1r2y. The first excited state is 5r2y with V2 and MN; from its excitation energy, this state might correspond to the experimental first excited state at E x s 1.11 MeV. The ground state of 19 Mg is shown on the r.h.s. of Fig. 3, with different experimental thresholds. The V2 force pre-
dicts the ground state to be slightly below the lowest Ne q 2p threshold, whereas the MN force gives a positive value. Of course, a microscopic model is not able to give results with such a high accuracy, but we may conjecture that the 19 Mg should be close to the particle stability. Spectroscopic properties of 20 O and 20 Mg are displayed in Table 1. The difference between matter radii of 20 Mg and 20 O is calculated as 0.04 fm, smaller than the experimental value 0.2 fm w4x. However, experimental radii are actually interaction radii, which are known to be partly model dependent w15x. The 20 Mg matter radius is well reproduced by the GCM but the 20 O radius is overestimated. The same behaviour appears in the relativistic mean-field model w4,16,17x and in Hartree–Fock models w18,19x which yields matter radii quite similar to our’s. The proton radii are in fair agreement with the extrapolations suggested by Chulkov et al. From these estimates, 17
Table 2 R.m.s. radii Žin fm. and B ŽE2. Žin e 2 .fm 4 . in
'² r
2: m
(
² r 2 :p
'
² r 2 :n
Dr B Ž E2,3r2y ™1r2y . B Ž E2,5r2y ™1r2y . B Ž E2,3r2y ™ 5r2y .
19
N
V2
MN
2.86
2.80
2.64
2.63
2.98 0.34 4.2 4.0 0.13
2.89 0.26 2.3 2.7 0.23
P. DescouÕemontr Physics Letters B 437 (1998) 7–11
Chulkov et al. determine neutron radii and conclude on an exotic structure of 20 Mg, with a proton skin of 0.5 " 0.3 fm. Our result Ž0.37 fm for V2, 0.26 fm for MN. is not in contradiction with this experimental value. However, the theoretical ratio ² r 2 :p r
(
(² r
2: n
s 1.14 ŽV2. or 1.10 ŽMN. is fairly close to the ratio expected from the shell model Ž1.09.. We therefore conclude that the difference between proton and neutron radii is a normal effect, and does not need any assumption concerning a ‘‘halo’’ structure. This conclusion is also supported by the energy surfaces ŽFig. 1. which present their minima for small generator coordinates, contrary to what is expected for halo nuclei w6x. Spectroscopic properties of 20 O and 20 Mg are complemented by the E2 transition probabilities. For 20 O, an effective charge d ere s 0.3 is included; this rather small effective charge w20x aims at simulating core polarization effects, and yields a B Ž E2,2q 1 ,™ . 0q value is in fair agreement with experiment. The 1 results are however sensitive to the choice of the nucleon-nucleon interaction. The large transition q. q. probabilities B Ž E2,2q and B Ž E2,4q 1 ,™ 0 1 1 ,™ 2 1 suggest a rotational band in the low-energy spectrum. Spectroscopic properties of 19 N are presented in Table 2. Here also, the ratio between neutron and proton radii can be explained by the shell-model value ² r 2 :p r ² r 2 :n s 1.10, and does not require introducing a halo effect. The GCM radii with the V2 interaction are close to those of Ren et al. w21x in the relativistic mean field approach. In conclusion, the model gives a satisfactory accuracy, as for the first excited level of 20 O and 19 N. Of course, the GCM basis assumes a core plus two dineutrons Žor diprotons. structure, although other configurations Žfor example 18 O q n q n in 20 O. have a lower threshold. However, antisymmetrization makes these configurations nearly equivalent in bound states. This approximation has been shown to be quite justified in the 17 B nucleus, defined as a 13 B q 2n q 2n three-cluster structure w7x. It is also supported by the recent work of Chulkov et al. w22x who conclude on a strong correlation between the
(
(
11
valence protons in 20 Mg. The present model is consistent with the radius measurement of Chulkov et al. w4x, but does not support a ‘‘halo’’ structure in 20 Mg. From the analysis of the 19 N and 19 Mg systems, we find that the 19 Mg nucleus should be close to particle stability.
Acknowledgements This text presents research results of the Belgian program on interuniversity attraction poles initiated by the Belgian-state Federal Services for Scientific, Technical and Cultural Affairs.
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