358
7 Wave Functions in the Vicinity of Particle-Emission Threshold
Acting on the ground state of 6 He with an operator ˆ
T − , respectively, once and
twice, and normalizing accordingly, one obtains the T = 1 isomultiplet of this state,
which will be denoted with a T subscript. The overlap squared 6 Li| 6 Li
2
T is equal to
0.995, which signifies that | 6 Li | 6 Li T , so that isospin symmetry is almost exact
in this case. This is an example of partial dynamical symmetry, where a symmetry
is verified by some eigenstates of Hamiltonian but not by the Hamiltonian itself
[114]. The overlap squared 6 Be| 6 Be
2
T is equal to 0.951 − i0.050, so that there is
a reduction compared to the value of 1 of exact isospin symmetry. Uncertainties
being smaller for 2 + states, one can conclude that the mirror symmetry breaking
of order 5% is present in the eigenvectors calculated with the Gamow shell model.
The isospin-breaking properties of the spectroscopic factors of the pairs 13 C/ 14 N
and 13 N/ 14 O are studied as a numerical example in Exercise IV.
The explanation of isospin breaking in the framework of standard shell model
demands the use of nuclear interactions which explicitly break isospin symmetry
[115, 116]. We have seen, that the correct treatment of the continuum coupling
can also generate isospin symmetry breaking effects, even for the isospin invariant
interactions and, hence, is capable to explain at least a part of the isospin breaking
effects in atomic nucleus.
Solutions to Exercises 2
Exercise I
A. One obtains the root-mean-square radius limits (see Eqs. (7.12), (7.13)) when
κ → 0 by fitting the radii plots obtained with the attached computer code for
small values of κ. As mentioned in Sect. 7.2.1.3, the root-mean-square radius
value for = 0 depends only on κ, while that for = 1 depends on κ and on the
potential depth.
B. When ≥ 2, the root-mean-square radius converges to a finite value when κ →
0, whose limit at κ = 0 is a function of the potential depth. Note that Eq. (7.5)
does not strictly hold in the general case, as one has assumed that the Woods–
Saxon potential vanishes after R 0 + 2d. However, the root-mean-square radius
should converge to a value which is smaller or close to that given by Eq. (7.5),
so that Eq. (7.5) must be seen as a rough estimate of that limit κ → 0.
C. Negative values of m do not have to be considered because one considers axially
symmetric potentials. Axial symmetry implies that energies of eigenstates
depend only on |m|. In the spherical case, root-mean-square radius limits are
obtained similarly as in A and B. Eqs. (7.27–7.30) can be directly compared to
Eqs. (7.5), (7.12), (7.13). One has m = 1/2 with π = ±1 which corresponds,
respectively, to = 0, 1. The case m = 3/2 with π = −1 is associated with
2 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.
7 Wave Functions in the Vicinity of Particle-Emission Threshold
Acting on the ground state of 6 He with an operator ˆ
T − , respectively, once and
twice, and normalizing accordingly, one obtains the T = 1 isomultiplet of this state,
which will be denoted with a T subscript. The overlap squared 6 Li| 6 Li
2
T is equal to
0.995, which signifies that | 6 Li | 6 Li T , so that isospin symmetry is almost exact
in this case. This is an example of partial dynamical symmetry, where a symmetry
is verified by some eigenstates of Hamiltonian but not by the Hamiltonian itself
[114]. The overlap squared 6 Be| 6 Be
2
T is equal to 0.951 − i0.050, so that there is
a reduction compared to the value of 1 of exact isospin symmetry. Uncertainties
being smaller for 2 + states, one can conclude that the mirror symmetry breaking
of order 5% is present in the eigenvectors calculated with the Gamow shell model.
The isospin-breaking properties of the spectroscopic factors of the pairs 13 C/ 14 N
and 13 N/ 14 O are studied as a numerical example in Exercise IV.
The explanation of isospin breaking in the framework of standard shell model
demands the use of nuclear interactions which explicitly break isospin symmetry
[115, 116]. We have seen, that the correct treatment of the continuum coupling
can also generate isospin symmetry breaking effects, even for the isospin invariant
interactions and, hence, is capable to explain at least a part of the isospin breaking
effects in atomic nucleus.
Solutions to Exercises 2
Exercise I
A. One obtains the root-mean-square radius limits (see Eqs. (7.12), (7.13)) when
κ → 0 by fitting the radii plots obtained with the attached computer code for
small values of κ. As mentioned in Sect. 7.2.1.3, the root-mean-square radius
value for = 0 depends only on κ, while that for = 1 depends on κ and on the
potential depth.
B. When ≥ 2, the root-mean-square radius converges to a finite value when κ →
0, whose limit at κ = 0 is a function of the potential depth. Note that Eq. (7.5)
does not strictly hold in the general case, as one has assumed that the Woods–
Saxon potential vanishes after R 0 + 2d. However, the root-mean-square radius
should converge to a value which is smaller or close to that given by Eq. (7.5),
so that Eq. (7.5) must be seen as a rough estimate of that limit κ → 0.
C. Negative values of m do not have to be considered because one considers axially
symmetric potentials. Axial symmetry implies that energies of eigenstates
depend only on |m|. In the spherical case, root-mean-square radius limits are
obtained similarly as in A and B. Eqs. (7.27–7.30) can be directly compared to
Eqs. (7.5), (7.12), (7.13). One has m = 1/2 with π = ±1 which corresponds,
respectively, to = 0, 1. The case m = 3/2 with π = −1 is associated with
2 The input files, codes and code user manual associated to computer-based exercises can be found
at https://github.com/GSMUTNSR.