Structure of 31 Na studied by the Monte-Carlo shell model†
I. Suzuki,∗1 T. Yamada,∗2 T. Yamato,∗3 K. Yamada,∗4 and S. Sato∗1,∗2

1

Since anomalous properties of 31 Na concerning the mass1) and the ground-state spin and magnetic moment2)

2

were observed in the 1970’s, the structure of neutron-rich nuclei around N = 20 has attracted much interest,3

particularly concerning vanishing of the N = 20 magic number. Based on the Monte-Carlo shell model (MCSM),3)

4

we performed a systematic shell-model calculation for even-even N ∼ 20 exotic nuclei with full mixing between the5normal, intruder, and higher intruder conﬁgurations for the ﬁrst time,4) and gave a comprehensive picture of the6region. As for odd-A nuclei, since we should adopt the J-compressed bases,3) which require much computational

7time in the MCSM calculation, such a calculation was unfeasible until the Alphleet computer system5) was

8

introduced at RIKEN. In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to9be in the “island of inversion”,6) is studied by the MCSM with the Alphleet computer system.

10Since anomalous properties of 31 Na concerning the mass1) and the ground-state spin and magnetic moment2)

11

were observed in the 1970’s, the structure of neutron-rich nuclei around N = 20 has attracted much interest,

12

particularly concerning vanishing of the N = 20 magic number. Based on the Monte-Carlo shell model (MCSM),3)13

we performed a systematic shell-model calculation for even-even N ∼ 20 exotic nuclei with full mixing between the14normal, intruder, and higher intruder conﬁgurations for the ﬁrst time,4) and gave a comprehensive picture of the15region. As for odd-A nuclei, since we should adopt the J-compressed bases,3) which require much computational

16time in the MCSM calculation, such a calculation was unfeasible until the Alphleet computer system5) was

17

introduced at RIKEN. In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to18

be in the “island of inversion”,6) is studied by the MCSM with the Alphleet computer system.
In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to be in the “island of1920inversion”,6) is studied by the MCSM with the Alphleet computer system.
In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to be in the “island of2122inversion”,6) is studied by the MCSM with the Alphleet computer system.
†Condensed from the article in Phys. Rev. Lett. 85, 1827 (2000)∗1

RIKEN Nishina Center∗2

Department of Physics, University of Tokyo∗3

Atomic Physics Laboratory, RIKEN∗4

Department of Physics, Omsk University

Table 1. aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaa aaaaaa aaa aaa aaa aaa aaaaaa aaa aaa aaa aaa aaa
AAABBB

CCCDDD

EEE

FFF

GGG

HHHIII

JJJ

KKK

LLL

MMM

NNNabcdefghi

jklmn

23

Y =a+b+c+d+e+f +g

24

In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to be in the “island of25262728(1)

inversion”,6) is studied by the MCSM with the Alphleet computer system.
In this report, the structure of a neutron-rich odd-A nucleus 31 Na, which is expected to be in the “island ofinversion”,6) is studied by the MCSM with the Alphleet computer system.
Y =∑ai + h + i + j + k + l + m(2)i=∞29

The energy levels of 31 Na are shown in Fig. 1. The ground-state spin 3/2+ agrees with an experiment, in30

contrast to the sd-shell model prediction of 5/2+ . The calculated magnetic moment of the ground state is 2.17 µN

31with free-nucleon g factors being consistent with the experimental value of 2.283 (38) µN .2) The present study

32

shows that, while the ground state is dominated by the 2-particle 2-hole (2p2h) excitations from the N = 20 core,33

4p4h and higher excited conﬁgurations are mixed and lower the ground-state energy by more than 700 keV. This

34

energy gain gives rise to a better two-neutron separation energy. The ﬁrst excited state obtained by the MCSM

35

calculation is a 5/2+ state located at 310 keV, in good agreement with a recent measurement of 350 ± 20 keV.7)

36

On the other hand, this level was calculated to lie around 200 keV in the 0p0h + 2p2h truncation. A comparison

37

between the truncated and full calculations clearly indicates the importance of the higher intruder conﬁgurations

Fig. 1. Experimental energy levels of 31 Na (Exp.) compared with those of the MCSM calculation (MCSM).

38(i.e., 4p4h and higher excited conﬁgurations from N = 20 core):a) these conﬁgurations lower the ground state more

39

than the ﬁrst excited state, giving rise to a better agreement with experiment. The higher intruder conﬁgurations

40

occupy the ground state by about 10%. The B (E2; 3/2+ → 5/2+ ) = 200 e2 fm4 is obtained with the eﬀective

41

charges ep = 1.3 e and en = 0.5 e, suggesting a strong deformation similarly to the adjacent even-A nucleus, 32 Mg.

42

This B(E2) value corresponds to β2 = 0.53 by assuming an axially symmetric rotor with K = 3/2 (Table 1).
Z = a+b+c+d+e+f +g=∑ai + h + i + j + k + l + mi=∞= o + p + q + r + s + t + gx

(3)

43

The negative-parity states are also of interest, partly because the competition and mixing between the 1p1h

44

and 3p3h conﬁgurations can be compared with those of the 0p0h and 2p2h conﬁgurations in the positive-parity

45states. The present calculation indicates that the yrast negative-parity states shown in Fig. 1 are dominated by46

3p3h conﬁgurations (Table 2). The truncated shell-model calculation with the same interaction shows that states

47

dominated by 1p1h are expected to lie higher than the yrast negative-parity states by more than 1 MeV. This48

picture is in sharp contrast with the former shell-model prediction,6) in which the yrast negative-parity states are

49

composed of 1p1h conﬁgurations for all nuclei in this region.b)
In order to conﬁrm the validity of our calculation concerning the competition and mixing of various conﬁgura-

50

51tions, the role of pf -orbits on the observed quantities is examined in more detail.
In order to conﬁrm the validity of our calculation concerning the competition and mixing of various conﬁgura-

52

53tions, the role of pf -orbits on the observed quantities is examined in more detail.

54

28
In the MCSM calculation, the 3−Mg, lies at 5.28 MeV, very close to the
1 state of a neutron-rich nucleus,

55

experimental position of 5.17 MeV. The two-neutron separation energies of N = 20 nuclei, which are also very

56

sensitive to the eﬀective single-particle energy 4) of the pf shell, are in excellent agreement with an experiment for57

N = 20 isotones ranging from Z = 10 to 14.

Table 2. aaaa)

aaaaaaaaaaaaaaab)

aaaaaaaaaaaaaaa

AAbab

58

(1) aaaa aaaa aaaa aaaa

59

aaaa aaaa aaaa aaaa

60(i) bbbb bbbb bbbb bbbb

61(a) cccc cccc cccc cccc

62(b) cccc cccc cccc cccc

63(ii) bbbb bbbb bbbb bbbb

64(2) aaaa aaaa aaaa aaaa

65

In order to conﬁrm the validity of our calculation concerning the competition and mixing of various conﬁgura-66

tions, the role of pf -orbits on the observed quantities is examined in more detail. In the MCSM calculation, the67283−Mg, lies at 5.28 MeV, very close to the experimental position of 5.17 MeV.
1 state of a neutron-rich nucleus,

68

The two-neutron separation energies of N = 20 nuclei, which are also very sensitive to the eﬀective single-particle

69

energy 4) of the pf shell, are in excellent agreement with an experiment for N = 20 isotones ranging from Z = 1070

to 14.

71

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764) J. A. Dean: Lange’s Handbook of Chemistry (McGraw-Hill, New York, 2000).

775) F. Martinerie et al.: Proc. 41st IEEE Conf. on Decision and Control, Vol. 3 (2000), p. 3803.

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7980

USA, 2000-12 (Elsevier Science, 2001), p. 149.
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