7.5 Mirror Nuclei and Isospin Symmetry Breaking
351
scattering continuum is present. If one uses realistic nucleon–nucleon interactions,
the continuum coupling may indeed lead to isospin breaking even in the absence of
Coulomb interaction. Indeed, the charge dependence of realistic nucleon–nucleon
interactions might lead to sizable isospin breaking. Consequently, we will study the
spectroscopic factors of isobaric analog states in mirror nuclei using a Hamiltonian
where isospin is broken solely by the Coulomb interaction. This will allow to
disentangle the isospin-breaking effects of the nuclear and Coulomb parts in the
Hamiltonian, on the one hand, and to quantify the importance of the Coulomb
interaction at particle-emission threshold, on the other hand.
Light nuclei are good candidates for that matter, as they are experimentally
accessible at both proton and neutron driplines. With this in mind, we will study
the spectroscopic factors [ 5 He(g.s.) ⊗ νp 3/2 ] 0 + , [ 5 Li(g.s.) ⊗ πp 3/2 ] 0 + , [ 5 Li(g.s.) ⊗
νp 3/2 ] 0 + and [ 5 He(g.s.) ⊗ πp 3/2 ] 0 + , Corresponding, respectively, to the 6 He/ 5 He,
6 Be/ 5 Li, 6 Li/ 5 Li and 6 Li/ 5 He pairs. In order to isolate the effect of Coulomb
interaction, the Hamiltonian parameters have been fitted so that the ground state
energies of the A = 5 mirror nuclei of a given pair have the same value.
Consequently, the differences between proton and neutron spectroscopic factors
arise solely from the different asymptotics of the corresponding wave functions.
The Hamiltonian of He and Li isotopes consists of a Woods–Saxon potential
representing the core (see Eq. (3.83)), a finite range modified surface Gaussian
interaction (see Eq. (9.174)), and the recoil term which arises from the use of cluster
orbital shell model coordinates (see Sect. 5.2). The Coulomb potential is generated
by a proton density of a Gaussian form. This form for the Coulomb potential is
convenient as it is analytical, hence its matrix elements can be calculated with a
Berggren basis numerically, as discussed in Sect. 3.7.1. The Coulomb Hamiltonian
is separated in one- and two-body parts in order to treat its infinite-range exactly (see
Eq. (5.67)). As the 6 Be nucleus has two valence protons, the two-body Coulomb
interaction, denoted as V Coul , must be taken into account. Due to the high precision
needed to calculate spectroscopic factors, it is necessary to check that the expansion
of Eq. (5.67) converges rapidly.
The convergence of the energy and width of 6 He and 6 Be ground states, as well
as that of the spectroscopic factor associated with the pairs of nuclear ground states
6 He/ 5 He and 6 Be/ 5 Li, are shown in Fig. 7.14. Even though recoil is automatically
included using cluster orbital shell model coordinates, calculations have been done
also without recoil to emphasize the infinite-range character of the recoil term.
One can see in Fig. 7.14 that the convergence is slower when recoil is taken into
account. In order to explain this fact, let us consider the point Coulomb potential
and recoil in the momentum space. The matrix elements of the Coulomb potential
in the Berggren basis bear a logarithmic divergence in ln(|k − k |) for k ∼ k ,
where k and k are the linear momenta of two scattering one-body states (see
Sect. 3.7.1). The Fourier transform of the recoil term is analytical. It consists
of the matrix elements
1 k
2 |(p 1 · p 2 )/M core |k 1 k 2 , where |k 1,2 and |k
1,2 are
states of plane waves of linear momenta k 1,2 and k
1,2 , respectively, and where
M core is the mass of the core. These recoil matrix elements are clearly equal
351
scattering continuum is present. If one uses realistic nucleon–nucleon interactions,
the continuum coupling may indeed lead to isospin breaking even in the absence of
Coulomb interaction. Indeed, the charge dependence of realistic nucleon–nucleon
interactions might lead to sizable isospin breaking. Consequently, we will study the
spectroscopic factors of isobaric analog states in mirror nuclei using a Hamiltonian
where isospin is broken solely by the Coulomb interaction. This will allow to
disentangle the isospin-breaking effects of the nuclear and Coulomb parts in the
Hamiltonian, on the one hand, and to quantify the importance of the Coulomb
interaction at particle-emission threshold, on the other hand.
Light nuclei are good candidates for that matter, as they are experimentally
accessible at both proton and neutron driplines. With this in mind, we will study
the spectroscopic factors [ 5 He(g.s.) ⊗ νp 3/2 ] 0 + , [ 5 Li(g.s.) ⊗ πp 3/2 ] 0 + , [ 5 Li(g.s.) ⊗
νp 3/2 ] 0 + and [ 5 He(g.s.) ⊗ πp 3/2 ] 0 + , Corresponding, respectively, to the 6 He/ 5 He,
6 Be/ 5 Li, 6 Li/ 5 Li and 6 Li/ 5 He pairs. In order to isolate the effect of Coulomb
interaction, the Hamiltonian parameters have been fitted so that the ground state
energies of the A = 5 mirror nuclei of a given pair have the same value.
Consequently, the differences between proton and neutron spectroscopic factors
arise solely from the different asymptotics of the corresponding wave functions.
The Hamiltonian of He and Li isotopes consists of a Woods–Saxon potential
representing the core (see Eq. (3.83)), a finite range modified surface Gaussian
interaction (see Eq. (9.174)), and the recoil term which arises from the use of cluster
orbital shell model coordinates (see Sect. 5.2). The Coulomb potential is generated
by a proton density of a Gaussian form. This form for the Coulomb potential is
convenient as it is analytical, hence its matrix elements can be calculated with a
Berggren basis numerically, as discussed in Sect. 3.7.1. The Coulomb Hamiltonian
is separated in one- and two-body parts in order to treat its infinite-range exactly (see
Eq. (5.67)). As the 6 Be nucleus has two valence protons, the two-body Coulomb
interaction, denoted as V Coul , must be taken into account. Due to the high precision
needed to calculate spectroscopic factors, it is necessary to check that the expansion
of Eq. (5.67) converges rapidly.
The convergence of the energy and width of 6 He and 6 Be ground states, as well
as that of the spectroscopic factor associated with the pairs of nuclear ground states
6 He/ 5 He and 6 Be/ 5 Li, are shown in Fig. 7.14. Even though recoil is automatically
included using cluster orbital shell model coordinates, calculations have been done
also without recoil to emphasize the infinite-range character of the recoil term.
One can see in Fig. 7.14 that the convergence is slower when recoil is taken into
account. In order to explain this fact, let us consider the point Coulomb potential
and recoil in the momentum space. The matrix elements of the Coulomb potential
in the Berggren basis bear a logarithmic divergence in ln(|k − k |) for k ∼ k ,
where k and k are the linear momenta of two scattering one-body states (see
Sect. 3.7.1). The Fourier transform of the recoil term is analytical. It consists
of the matrix elements
1 k
2 |(p 1 · p 2 )/M core |k 1 k 2 , where |k 1,2 and |k
1,2 are
states of plane waves of linear momenta k 1,2 and k
1,2 , respectively, and where
M core is the mass of the core. These recoil matrix elements are clearly equal