354
7 Wave Functions in the Vicinity of Particle-Emission Threshold
standard shell model considerations [12]. The Wigner cusp, characteristic of the
= 1 neutron partial wave, is visible for 6 He. The absence of the cusp in the
spectroscopic factors of the 6 Be nucleus, originates in the different asymptotic
between proton and neutron wave functions [100].
The energy dependence of spectroscopic factors changes if the A = 5 particle
system is exactly at one-nucleon threshold (E T = 0) or is unbound (E T =
0.5 MeV). In these two situations, the spectroscopic factors of bound A = 6
particle systems (S 1n or S 1p positive) slightly differ. Wigner cusp disappears in
neutron spectroscopic factor when 5 He becomes unbound (see the E T = 0.5 MeV
case in Fig. 7.15). In fact, one can only see a bump developing close to S 1n
−0.05 MeV when E T = 0.5 MeV, whose origin cannot be related to the particleemission threshold, as neutron spectroscopic factors do not change significantly at
S 1n = 0 MeV. For the experimental values of S 1n , S 2n and S 1p , S 2p in 6 He and 6 Be,
respectively, the spectroscopic factors equal: 6 He(g.s.)|[ 5 He(g.s.) ⊗ νp 3/2 ] 0 + =
0.88 − i0.386 and 6 Be(g.s.)|[ 5 Li(g.s.) ⊗ πp 3/2 ] 0 + = 1.057 − i0.181. One sees
that 6 He in the ground state is significantly more collective than its mirror partner
6 Be which can be viewed as a single proton attached to the proton-unbound core of
5 Li.
In some cases, the real part of spectroscopic factors can be larger than one
(see of the 6 He nucleus for E T = 0.5 MeV in Fig. 7.15). This non-intuitive
behavior may have several origins. Firstly, the imaginary part of the average value
of an operator acting on resonance state can be interpreted as an uncertainty
on the physical value of the associated observable induced by the possibility of
decay during the measuring process (see Sect. 5.1.2). The right column of Fig. 7.15
illustrates the uncertainty of the spectroscopic factors shown in the left column.
The uncertainty disappears when both A = 6 and A = 5 nuclei are bound. However,
one may notice in Fig. 7.15, that 1 is not totally explained by a large value
of Indeed, 1 corresponds to 0 in 6 He at E T = 0.5 MeV and
0.05 < S 1n < 0.1 MeV.
Secondly, 1 may originate from the coupling between Ψ
J A−1
A−1 ≡ Ψ
J A−1
A−1;R ,
where R stands for resonance, and the scattering states of the nonresonant continuum {Ψ
J A−1
A−1;c } of energies close to that of the resonance [113]. Let us remind
here the interference phenomenon between resonant and nonresonant continuum,
which explains the appearance of a cusp. Indeed, neglecting coupling to nonresonant
continuum destroys the unitarity of the formalism, so that no redistribution of
flux can occur. The contributions of the spectroscopic factors S c originating from
the non-resonant continuum of A − 1 particles are depicted in Fig. 7.16) (upper
and lower parts for its real and imaginary part, respectively), as well as those
corresponding to resonant states. In all the situations presented in Fig. 7.16), the
contribution of the one-body resonance state to is dominant.
The impact of the nonresonant continuum depends on E T and S 1p /S 1n . For
E T ≤ 0 and S 1p /S 1n ≥ 0, i.e. for bound ground states of A = 5 and A = 6 nucleons,
the nonresonant continuum is practically negligible. The same phenomenon is
present for 6 Be with E T ≤ 0 and S 1p < 0, hence for a bound 5 Li but for an
7 Wave Functions in the Vicinity of Particle-Emission Threshold
standard shell model considerations [12]. The Wigner cusp, characteristic of the
= 1 neutron partial wave, is visible for 6 He. The absence of the cusp in the
spectroscopic factors of the 6 Be nucleus, originates in the different asymptotic
between proton and neutron wave functions [100].
The energy dependence of spectroscopic factors changes if the A = 5 particle
system is exactly at one-nucleon threshold (E T = 0) or is unbound (E T =
0.5 MeV). In these two situations, the spectroscopic factors of bound A = 6
particle systems (S 1n or S 1p positive) slightly differ. Wigner cusp disappears in
neutron spectroscopic factor when 5 He becomes unbound (see the E T = 0.5 MeV
case in Fig. 7.15). In fact, one can only see a bump developing close to S 1n
−0.05 MeV when E T = 0.5 MeV, whose origin cannot be related to the particleemission threshold, as neutron spectroscopic factors do not change significantly at
S 1n = 0 MeV. For the experimental values of S 1n , S 2n and S 1p , S 2p in 6 He and 6 Be,
respectively, the spectroscopic factors equal: 6 He(g.s.)|[ 5 He(g.s.) ⊗ νp 3/2 ] 0 + =
0.88 − i0.386 and 6 Be(g.s.)|[ 5 Li(g.s.) ⊗ πp 3/2 ] 0 + = 1.057 − i0.181. One sees
that 6 He in the ground state is significantly more collective than its mirror partner
6 Be which can be viewed as a single proton attached to the proton-unbound core of
5 Li.
In some cases, the real part of spectroscopic factors can be larger than one
(see of the 6 He nucleus for E T = 0.5 MeV in Fig. 7.15). This non-intuitive
behavior may have several origins. Firstly, the imaginary part of the average value
of an operator acting on resonance state can be interpreted as an uncertainty
on the physical value of the associated observable induced by the possibility of
decay during the measuring process (see Sect. 5.1.2). The right column of Fig. 7.15
illustrates the uncertainty of the spectroscopic factors shown in the left column.
The uncertainty disappears when both A = 6 and A = 5 nuclei are bound. However,
one may notice in Fig. 7.15, that 1 is not totally explained by a large value
of Indeed, 1 corresponds to 0 in 6 He at E T = 0.5 MeV and
0.05 < S 1n < 0.1 MeV.
Secondly, 1 may originate from the coupling between Ψ
J A−1
A−1 ≡ Ψ
J A−1
A−1;R ,
where R stands for resonance, and the scattering states of the nonresonant continuum {Ψ
J A−1
A−1;c } of energies close to that of the resonance [113]. Let us remind
here the interference phenomenon between resonant and nonresonant continuum,
which explains the appearance of a cusp. Indeed, neglecting coupling to nonresonant
continuum destroys the unitarity of the formalism, so that no redistribution of
flux can occur. The contributions of the spectroscopic factors S c originating from
the non-resonant continuum of A − 1 particles are depicted in Fig. 7.16) (upper
and lower parts for its real and imaginary part, respectively), as well as those
corresponding to resonant states. In all the situations presented in Fig. 7.16), the
contribution of the one-body resonance state to is dominant.
The impact of the nonresonant continuum depends on E T and S 1p /S 1n . For
E T ≤ 0 and S 1p /S 1n ≥ 0, i.e. for bound ground states of A = 5 and A = 6 nucleons,
the nonresonant continuum is practically negligible. The same phenomenon is
present for 6 Be with E T ≤ 0 and S 1p < 0, hence for a bound 5 Li but for an