7.3 Asymptotic Normalization Coefficient in Mirror Nuclei
335
Note that as the state in A-particle system is unbound, k and η are complex [27].
In the limit of vanishing resonance width, k, I a
bc;j and H
+
,η (kr) become real and,
hence, C j given by Eq. (7.40) becomes real as well. Numerical examples showing
that the ansatz expressed Eqs. (7.32), (7.40)) are valid in practice in the asymptotic
zone are presented in Exercise II.
Exercise II
We will show in this numerical exercise that overlap functions can be effectively
fitted by Hankel or Coulomb wave functions (see Eqs. (7.32), (7.40).
While it is not strictly necessary, it is preferrable to run the many-body
Gamow shell model code using a parallel machine with at least 5–10 cores and
a total computer memory of about 30 Gb, as calculations can take several hours
otherwise.
A. Run the many-body Gamow shell model code using the Furutani–Horiuchi–
Tamagaki interaction (see Ref. [92, 93] and Eq. (6.18)) to calculate energies
and overlap functions of 8 Li/ 7 Li and 8 B/ 7 Be, as shown above. Fit the obtained
overlap functions in the asymptotic region by using the code extracting
asymptotic normalization coefficients from overlap functions.
Notice that they are well reproduced by Hankel or Coulomb wave functions
using Eqs. (7.32) and (7.40).
B. Change the interaction parameters to have slightly different binding energies
for 8 Li/ 7 Li and 8 B/ 7 Be. Show that the overlap functions obtained in this way
can be fitted with Hankel or Coulomb wave functions by changing the κ or k
parameter of Eqs. (7.32) and (7.40) accordingly.
The overlap function I a
bc;j (r) defined in Eq. (7.32) obeys a Schrödinger-like
equation, albeit inhomogeneous [66]. A source term in this equation, however,
is generated by the two-body interaction. However, separating the full interaction
into a one-body term and a two-body residual interaction, the source term can be
decomposed into a dominant homogeneous part and a residual inhomogeneous part.
Indeed, the full interaction can be written as a mean-field one-body part, to which a
small two-body residual part is added. One-body parts generate homogeneous terms,
whereas inhomogeneous terms arise from the two-body part of the interaction. The
inhomogeneous part can then be absorbed into the homogeneous part. As only
bound states or narrow resonance states are involved, one can also assume that the
potential entering the equation defining I a
bc;j (r) is real. These approximations have
been tested successfully in Ref. [27] for both bound and unbound states.
Under these assumptions, one can easily derive the relation between asymptotic
normalization coefficient and partial width [94], which reads mutatis mutandis
as Eq. (2.197). The total asymptotic normalization coefficient constant C (see
Eq. (7.34)) can then be derived by expressing the total width Γ in terms of the sum
335
Note that as the state in A-particle system is unbound, k and η are complex [27].
In the limit of vanishing resonance width, k, I a
bc;j and H
+
,η (kr) become real and,
hence, C j given by Eq. (7.40) becomes real as well. Numerical examples showing
that the ansatz expressed Eqs. (7.32), (7.40)) are valid in practice in the asymptotic
zone are presented in Exercise II.
Exercise II
We will show in this numerical exercise that overlap functions can be effectively
fitted by Hankel or Coulomb wave functions (see Eqs. (7.32), (7.40).
While it is not strictly necessary, it is preferrable to run the many-body
Gamow shell model code using a parallel machine with at least 5–10 cores and
a total computer memory of about 30 Gb, as calculations can take several hours
otherwise.
A. Run the many-body Gamow shell model code using the Furutani–Horiuchi–
Tamagaki interaction (see Ref. [92, 93] and Eq. (6.18)) to calculate energies
and overlap functions of 8 Li/ 7 Li and 8 B/ 7 Be, as shown above. Fit the obtained
overlap functions in the asymptotic region by using the code extracting
asymptotic normalization coefficients from overlap functions.
Notice that they are well reproduced by Hankel or Coulomb wave functions
using Eqs. (7.32) and (7.40).
B. Change the interaction parameters to have slightly different binding energies
for 8 Li/ 7 Li and 8 B/ 7 Be. Show that the overlap functions obtained in this way
can be fitted with Hankel or Coulomb wave functions by changing the κ or k
parameter of Eqs. (7.32) and (7.40) accordingly.
The overlap function I a
bc;j (r) defined in Eq. (7.32) obeys a Schrödinger-like
equation, albeit inhomogeneous [66]. A source term in this equation, however,
is generated by the two-body interaction. However, separating the full interaction
into a one-body term and a two-body residual interaction, the source term can be
decomposed into a dominant homogeneous part and a residual inhomogeneous part.
Indeed, the full interaction can be written as a mean-field one-body part, to which a
small two-body residual part is added. One-body parts generate homogeneous terms,
whereas inhomogeneous terms arise from the two-body part of the interaction. The
inhomogeneous part can then be absorbed into the homogeneous part. As only
bound states or narrow resonance states are involved, one can also assume that the
potential entering the equation defining I a
bc;j (r) is real. These approximations have
been tested successfully in Ref. [27] for both bound and unbound states.
Under these assumptions, one can easily derive the relation between asymptotic
normalization coefficient and partial width [94], which reads mutatis mutandis
as Eq. (2.197). The total asymptotic normalization coefficient constant C (see
Eq. (7.34)) can then be derived by expressing the total width Γ in terms of the sum