7.2 Halo Nuclei
325
The case of = 0 and large η in Fig. 7.4a is characteristic of a proton halo.
However, as seen in Fig. 7.4, the large values of β ,j are also expected for very
bound states, so that a large β nnj (η) is not a sufficient indicator of a proton halo.
Limiting regimes for β nnj (η) can be characterized by the complex turning point
z t (see Eq. (7.14)). The low-binding regime, which is reached for κr |z t | and
η → +∞, is characterized by the condition:
η +
η 2 + + 1)
η κ 0 R f .
(7.21)
In this regime, the proton wave function behaves asymptotically as u j (r) ∝
r 1/4 exp(−2
√
2κ 0 r), i.e. its decay is slower than exponential. The strong-binding
regime is reached at κr |z t |:
η +
η 2 + + 1)
η κ 0 R f .
(7.22)
The proton wave function in the strong-binding regime shows the expected exponential decay (7.17).
The charged particle radiative capture cross sections can be discussed in the
two limits: (i) η CM → ∞ (E CM → 0), and (ii) η → ∞ (S
(a)
c
→ 0). The
center-of-mass energy of the system + c can be varied experimentally. On the
contrary, the charged particle separation energy is fixed for any state of the anucleus. Therefore, the limiting behavior of the radiative capture cross section when
η → ∞ cannot be studied experimentally in a single physical system. In the first
limit (η CM → ∞), the radiative capture cross section σ cγ (η CM , η) for a fixed value
of η is exponentially reduced :
lim
η CM →∞
σ cγ = S bc (η CM , η)
exp(−2πη CM )
E CM
.
(7.23)
The second limit (η → ∞), the radiative capture cross section σ cγ (η CM , η) for a
fixed value of η CM diverges as:
lim
η→∞
σ cγ ∝ S bc (η CM , η) ∝ η exp(2η ln(η) − 2η) .
(7.24)
Both asymptotic behaviors of σ cγ hold for any charged particle radiative capture
reaction.
Figure 7.5 surveys the values of β ,j and η for the proton emission channels with
= 1 (p-shell) and = 2 (d-shell) in left and right panels, respectively. One can see
that the Z a -dependence of η crit is weak in both cases. All p-shell nuclei in Fig. 7.5
with the exception of the proton halo nucleus 8 B, belong to the class η < η crit . The
ground state of 12 N has η = η crit . The neutron-rich nuclei have very small η-values
in the ground state so their normalization constants β (η) are large.
325
The case of = 0 and large η in Fig. 7.4a is characteristic of a proton halo.
However, as seen in Fig. 7.4, the large values of β ,j are also expected for very
bound states, so that a large β nnj (η) is not a sufficient indicator of a proton halo.
Limiting regimes for β nnj (η) can be characterized by the complex turning point
z t (see Eq. (7.14)). The low-binding regime, which is reached for κr |z t | and
η → +∞, is characterized by the condition:
η +
η 2 + + 1)
η κ 0 R f .
(7.21)
In this regime, the proton wave function behaves asymptotically as u j (r) ∝
r 1/4 exp(−2
√
2κ 0 r), i.e. its decay is slower than exponential. The strong-binding
regime is reached at κr |z t |:
η +
η 2 + + 1)
η κ 0 R f .
(7.22)
The proton wave function in the strong-binding regime shows the expected exponential decay (7.17).
The charged particle radiative capture cross sections can be discussed in the
two limits: (i) η CM → ∞ (E CM → 0), and (ii) η → ∞ (S
(a)
c
→ 0). The
center-of-mass energy of the system + c can be varied experimentally. On the
contrary, the charged particle separation energy is fixed for any state of the anucleus. Therefore, the limiting behavior of the radiative capture cross section when
η → ∞ cannot be studied experimentally in a single physical system. In the first
limit (η CM → ∞), the radiative capture cross section σ cγ (η CM , η) for a fixed value
of η is exponentially reduced :
lim
η CM →∞
σ cγ = S bc (η CM , η)
exp(−2πη CM )
E CM
.
(7.23)
The second limit (η → ∞), the radiative capture cross section σ cγ (η CM , η) for a
fixed value of η CM diverges as:
lim
η→∞
σ cγ ∝ S bc (η CM , η) ∝ η exp(2η ln(η) − 2η) .
(7.24)
Both asymptotic behaviors of σ cγ hold for any charged particle radiative capture
reaction.
Figure 7.5 surveys the values of β ,j and η for the proton emission channels with
= 1 (p-shell) and = 2 (d-shell) in left and right panels, respectively. One can see
that the Z a -dependence of η crit is weak in both cases. All p-shell nuclei in Fig. 7.5
with the exception of the proton halo nucleus 8 B, belong to the class η < η crit . The
ground state of 12 N has η = η crit . The neutron-rich nuclei have very small η-values
in the ground state so their normalization constants β (η) are large.