7.3 Asymptotic Normalization Coefficient in Mirror Nuclei
333
far from the region of nuclear interaction. β = β j (S
(a)
c ) is the single-particle
asymptotic normalization coefficient of Eq. (7.1), where β is directly related to the
many-body asymptotic normalization coefficient [82, 83]:
C = S
1/2
β .
(7.37)
The relations (7.35–7.37) are satisfied also for many-body resonances. Indeed, it
was demonstrated in Gamow shell model [27] that the overlap function I a
bc;j is well
approximated by a product of the spectroscopic amplitude S j and the single-particle
resonance wave function which reproduces the Q-value of the reaction studied. Let
us also remind that the astrophysical factor S bc in the limit of zero center-of-mass
energy, E cm = 0, is proportional to β 2
[84]:
S bc (0) ∼ β
2
≡ β
2
j (η) .
(7.38)
Let us now calculate the norm of bound state in the external region where the
nuclear part of the potential is practically zero. In the region (r ≥ R f ), where R f
(R f R) is the radius in the external region where the nuclear part of the potential
vanishes, the single-particle wave function u j is described by Eq. (7.1), i.e. is given
by W −η,,+1/2 (2κr). For the normalized bound state wave function: N int +N ext = 1,
where N int is the norm of the internal part of the wave function, and
N ext = β
2
+∞
R f
|W −η,,+1/2 (2κr)|
2 dr
(7.39)
is the norm of the external part. On the condition that the energy dependence of N int
is weak, which is a reasonable assumption even if the separation energy is close to
zero, β j should be strongly depending on the value of N ext .
An extensive analysis of proton and neutron asymptotic normalization coefficients for light mirror nuclei has been performed using standard shell model, cluster
model, and various effective NN interactions [85–90]. Calculations of the overlap
functions I a
bc;j (r) in the Gamow shell model has been for the p-shell nuclei: 7 Li/ 8 Li
and 7 Be/ 8 B [55]. The scattering contours in these calculations have been discretized
utilizing a Gauss-Legendre quadrature. The number of discretization points have
been chosen in order to maintain a correct asymptotic behavior of I
a
bc;j (r) up to at
least ∼8 fm. Gamow Hartree–Fock ensemble [91] corresponding to a parent nucleus
has been taken for the single-particle basis.
One will firstly discuss the asymptotic normalization coefficients corresponding
to single-nucleon capture reactions involving only bound states. The asymptotic
normalization coefficients in Gamow shell model can be directly extracted from the
calculated radial overlap integrals by fitting their tail to Whittaker functions at large
The radial overlap integrals are calculated using Eq. (7.33), where the sum over B
states runs over the complete Berggren ensemble; hence, the result is independent
333
far from the region of nuclear interaction. β = β j (S
(a)
c ) is the single-particle
asymptotic normalization coefficient of Eq. (7.1), where β is directly related to the
many-body asymptotic normalization coefficient [82, 83]:
C = S
1/2
β .
(7.37)
The relations (7.35–7.37) are satisfied also for many-body resonances. Indeed, it
was demonstrated in Gamow shell model [27] that the overlap function I a
bc;j is well
approximated by a product of the spectroscopic amplitude S j and the single-particle
resonance wave function which reproduces the Q-value of the reaction studied. Let
us also remind that the astrophysical factor S bc in the limit of zero center-of-mass
energy, E cm = 0, is proportional to β 2
[84]:
S bc (0) ∼ β
2
≡ β
2
j (η) .
(7.38)
Let us now calculate the norm of bound state in the external region where the
nuclear part of the potential is practically zero. In the region (r ≥ R f ), where R f
(R f R) is the radius in the external region where the nuclear part of the potential
vanishes, the single-particle wave function u j is described by Eq. (7.1), i.e. is given
by W −η,,+1/2 (2κr). For the normalized bound state wave function: N int +N ext = 1,
where N int is the norm of the internal part of the wave function, and
N ext = β
2
+∞
R f
|W −η,,+1/2 (2κr)|
2 dr
(7.39)
is the norm of the external part. On the condition that the energy dependence of N int
is weak, which is a reasonable assumption even if the separation energy is close to
zero, β j should be strongly depending on the value of N ext .
An extensive analysis of proton and neutron asymptotic normalization coefficients for light mirror nuclei has been performed using standard shell model, cluster
model, and various effective NN interactions [85–90]. Calculations of the overlap
functions I a
bc;j (r) in the Gamow shell model has been for the p-shell nuclei: 7 Li/ 8 Li
and 7 Be/ 8 B [55]. The scattering contours in these calculations have been discretized
utilizing a Gauss-Legendre quadrature. The number of discretization points have
been chosen in order to maintain a correct asymptotic behavior of I
a
bc;j (r) up to at
least ∼8 fm. Gamow Hartree–Fock ensemble [91] corresponding to a parent nucleus
has been taken for the single-particle basis.
One will firstly discuss the asymptotic normalization coefficients corresponding
to single-nucleon capture reactions involving only bound states. The asymptotic
normalization coefficients in Gamow shell model can be directly extracted from the
calculated radial overlap integrals by fitting their tail to Whittaker functions at large
The radial overlap integrals are calculated using Eq. (7.33), where the sum over B
states runs over the complete Berggren ensemble; hence, the result is independent