6.3 Effective Nucleon–Nucleon Interactions for Gamow Shell Model. . .
263
The helium chain has been studied in Ref. [62] by reducing as much as possible
the number of parameters needed to reproduce its main features. It was shown in
this analysis that it is sufficient to adjust a single parameter, the spin-singlet central
parameter denoted as V 01
C in Table 6.7 to reproduce the experimental energies and
widths of 5−8 He up to a few tens of keV. A parity inversion of narrow resonances
in 9 He has also been predicted. The calculation of the ground state of 10 He [62]
provided with a wave function dominated by s-waves decaying predominantly by
the two-neutron emission. The calculation of the ground state of 10 He [62] provided
with a wave function dominated by s-waves decaying predominantly by the twoneutron emission. For a complete description of the two-neutron emission of 10 He
it is necessary, however, to separate different emission channels in the Gamow shell
model calculation. For this, one has to rely on the formulation of Gamow shell
model in the representation of coupled-channels (see Sect. 9).
As we have discussed above, quite a few terms of the Furutani–Horiuchi–
Tamagaki interaction are poorly constrained. In fact, only the central V 00
C , V 10
C , V 01
C ,
and tensor V 10
T terms of this interaction are well fitted. This suggests another
strategy which was put forward in Ref. [63]. In this strategy, one restricts the
number of terms to those which are well constrained by the fitting procedure and the
statistical analysis of the Furutani–Horiuchi–Tamagaki interaction. The reduction of
the interaction terms can also be motived by arguments issued from the effective
field theory [64–68], as the terms proportional to V 10
C , V 01
C , and V 10
T appear at
the leading order in the effective field theory expansion of the Furutani–Horiuchi–
Tamagaki interaction [63]. Inclusion of the central term V 00
C , of higher order in this
expansion, is motivated empirically as it improves the overall fit of energies. This
simplified effective interaction was used to describe the chain of Li, Be isotopes
[63] in a smaller model space including only the s 1/2 , p 3/2 , and p 1/2 partial waves.
It was then possible to calculate the ground states of Li isotopes up to the neutron
dripline, ending with the 11 Li two-neutron halo state, as well as the isobaric analog
states of the Li ground states.
Figure 6.8 shows the energies calculated in Gamow shell model for the ground
states and selected excited states of N = 3 isotones from 7 Be to 11 O. As one can see,
the devised interaction allows for a good reproduction of experimental energies. It is
to be noted that the results for higher excited states, not included in the fit, are very
satisfactory as well. The 5/2 − and 7/2 − excited states in 7 Be are slightly above the
corresponding experimental values, whereas the position of the resonant 3 + states
in 8 B and 5/2 − state in 9 C are well reproduced, as well as the weakly bound ground
states of 8 B and 9 C.
The spectrum of an unbound nucleus 10 N is not experimentally known with
certainty. Figure 6.8 shows the tentative level assignments given in http://www.nndc.
bnl.gov/ensdf. According to Refs. [71, 72], the ground state of 10 N is most likely a
1 − state with an excitation energy with respect to the core E = (1.81 − 1.94) MeV.
In a more recent work [73], two low-lying negative-parity states were observed but
the spin assignment was not possible. The Gamow shell model calculation predicts
1 − ground state of 10 N which is a resonance (E, Γ ) = (−8.93, 0.9) MeV that
263
The helium chain has been studied in Ref. [62] by reducing as much as possible
the number of parameters needed to reproduce its main features. It was shown in
this analysis that it is sufficient to adjust a single parameter, the spin-singlet central
parameter denoted as V 01
C in Table 6.7 to reproduce the experimental energies and
widths of 5−8 He up to a few tens of keV. A parity inversion of narrow resonances
in 9 He has also been predicted. The calculation of the ground state of 10 He [62]
provided with a wave function dominated by s-waves decaying predominantly by
the two-neutron emission. The calculation of the ground state of 10 He [62] provided
with a wave function dominated by s-waves decaying predominantly by the twoneutron emission. For a complete description of the two-neutron emission of 10 He
it is necessary, however, to separate different emission channels in the Gamow shell
model calculation. For this, one has to rely on the formulation of Gamow shell
model in the representation of coupled-channels (see Sect. 9).
As we have discussed above, quite a few terms of the Furutani–Horiuchi–
Tamagaki interaction are poorly constrained. In fact, only the central V 00
C , V 10
C , V 01
C ,
and tensor V 10
T terms of this interaction are well fitted. This suggests another
strategy which was put forward in Ref. [63]. In this strategy, one restricts the
number of terms to those which are well constrained by the fitting procedure and the
statistical analysis of the Furutani–Horiuchi–Tamagaki interaction. The reduction of
the interaction terms can also be motived by arguments issued from the effective
field theory [64–68], as the terms proportional to V 10
C , V 01
C , and V 10
T appear at
the leading order in the effective field theory expansion of the Furutani–Horiuchi–
Tamagaki interaction [63]. Inclusion of the central term V 00
C , of higher order in this
expansion, is motivated empirically as it improves the overall fit of energies. This
simplified effective interaction was used to describe the chain of Li, Be isotopes
[63] in a smaller model space including only the s 1/2 , p 3/2 , and p 1/2 partial waves.
It was then possible to calculate the ground states of Li isotopes up to the neutron
dripline, ending with the 11 Li two-neutron halo state, as well as the isobaric analog
states of the Li ground states.
Figure 6.8 shows the energies calculated in Gamow shell model for the ground
states and selected excited states of N = 3 isotones from 7 Be to 11 O. As one can see,
the devised interaction allows for a good reproduction of experimental energies. It is
to be noted that the results for higher excited states, not included in the fit, are very
satisfactory as well. The 5/2 − and 7/2 − excited states in 7 Be are slightly above the
corresponding experimental values, whereas the position of the resonant 3 + states
in 8 B and 5/2 − state in 9 C are well reproduced, as well as the weakly bound ground
states of 8 B and 9 C.
The spectrum of an unbound nucleus 10 N is not experimentally known with
certainty. Figure 6.8 shows the tentative level assignments given in http://www.nndc.
bnl.gov/ensdf. According to Refs. [71, 72], the ground state of 10 N is most likely a
1 − state with an excitation energy with respect to the core E = (1.81 − 1.94) MeV.
In a more recent work [73], two low-lying negative-parity states were observed but
the spin assignment was not possible. The Gamow shell model calculation predicts
1 − ground state of 10 N which is a resonance (E, Γ ) = (−8.93, 0.9) MeV that