288
6 Physical Applications of the Gamow Shell Model
in the single-particle spectra with at least one resonance. As a rule, the relative error
for the imaginary part of the total energy is larger than the corresponding error of
the real part.
The Richardson-like solution for the generalized pairing problem including
single-particle states of the discretized continuum becomes less accurate when the
occupation of nonresonant continuum states is important, as in the case of strong
pairing strength. In the weak pairing case, on the contrary, the relative error of the
solutions of generalized Richardson equations remains small.
6.5.1.1 Physical Applications of the Generalized Richardson Equations
The generalized Richardson equations can find an application in the problem of
ultra-small superconducting grains to understand the influence of continuum on
pairing properties, in particular in the transitional region of the weak coupling limit
[135–137]. To illustrate another possible application of the generalized Richardson
equations, we will discuss the energy spectrum of 20 C as an example.
The Hamiltonian (6.44) is parametrized using experimental energies of the
spectrum of 13 C and the binding energy of 14 C. 12 C is used as an inert core and
the energies of all states in 14−20 C are calculated with respect to the energy of the
12 C core.
The Berggren basis consists of the pole single-particle states 0p 1/2 , 1s 1/2 ,
0d 5/2 , 0d 3/2 , 0f 7/2 , and of the two nonresonant continua {d 3/2 }, {f 7/2 }. The
single-particle energies of bound states 0p 1/2 , 1s 1/2 , 0d 5/2 are given by the
experimental energies of 1/2
−
1 , 1/2
+
1 and 5/2
+
1 states in 13 C: 0p 1/2 = −4.946 MeV,
1s 1/2 = −1.857 MeV, and 0d 5/2 = −1.093 MeV. The energy of resonances 0d 3/2
and 0f 7/2 are taken from Ref.[125]: 0d 3/2 = (2.267 MeV; −0.416 MeV) and
0f 7/2 = (9.288 MeV; −3.040 MeV). The complex contours {d 3/2 } and {f 7/2 },
associated with the single-particle resonances 0d 3/2 and 0f 7/2 , are discretized with
ten points per segment, so that one has thirty points per contour. The pairing strength
is given by G = χ/A, where χ = −11.13 MeV, and A is the number of nucleons.
The constant χ is adjusted to reproduce the experimental binding energy of the
ground state of 14 C with respect to the 12 C core.
To evaluate the role of the continuum in the spectra of carbon isotopes, the
results of the generalized Richardson equations (6.39) are compared to results of
the standard Richardson calculations (see Eq. (6.39)) without continuum couplings
and with real single-particle energies. In the latter case, the single-particle energies
of the bound states: 0p 1/2 , 1s 1/2 , 0d 5/2 , are the same as in the previous paragraph,
and the energies of 0d 3/2 and 0f 7/2 resonances are real: 0d 3/2 = 2.267 MeV and
0f 7/2 = 9.288 MeV. One uses the value χ = −15.064 MeV in order to reproduce
the experimental binding energy of 14 C.
The spectrum of 20 C, calculated using either the generalized Richardson equations with the continuum, or the standard Richardson equations without the
continuum, is presented in Fig. 6.17. In both cases, the pairing strength G is fitted
to reproduce the experimental energy of the ground state of 14 C with respect to 12 C.
One can see significant relative shifts of energy eigenvalues which are calculated in
6 Physical Applications of the Gamow Shell Model
in the single-particle spectra with at least one resonance. As a rule, the relative error
for the imaginary part of the total energy is larger than the corresponding error of
the real part.
The Richardson-like solution for the generalized pairing problem including
single-particle states of the discretized continuum becomes less accurate when the
occupation of nonresonant continuum states is important, as in the case of strong
pairing strength. In the weak pairing case, on the contrary, the relative error of the
solutions of generalized Richardson equations remains small.
6.5.1.1 Physical Applications of the Generalized Richardson Equations
The generalized Richardson equations can find an application in the problem of
ultra-small superconducting grains to understand the influence of continuum on
pairing properties, in particular in the transitional region of the weak coupling limit
[135–137]. To illustrate another possible application of the generalized Richardson
equations, we will discuss the energy spectrum of 20 C as an example.
The Hamiltonian (6.44) is parametrized using experimental energies of the
spectrum of 13 C and the binding energy of 14 C. 12 C is used as an inert core and
the energies of all states in 14−20 C are calculated with respect to the energy of the
12 C core.
The Berggren basis consists of the pole single-particle states 0p 1/2 , 1s 1/2 ,
0d 5/2 , 0d 3/2 , 0f 7/2 , and of the two nonresonant continua {d 3/2 }, {f 7/2 }. The
single-particle energies of bound states 0p 1/2 , 1s 1/2 , 0d 5/2 are given by the
experimental energies of 1/2
−
1 , 1/2
+
1 and 5/2
+
1 states in 13 C: 0p 1/2 = −4.946 MeV,
1s 1/2 = −1.857 MeV, and 0d 5/2 = −1.093 MeV. The energy of resonances 0d 3/2
and 0f 7/2 are taken from Ref.[125]: 0d 3/2 = (2.267 MeV; −0.416 MeV) and
0f 7/2 = (9.288 MeV; −3.040 MeV). The complex contours {d 3/2 } and {f 7/2 },
associated with the single-particle resonances 0d 3/2 and 0f 7/2 , are discretized with
ten points per segment, so that one has thirty points per contour. The pairing strength
is given by G = χ/A, where χ = −11.13 MeV, and A is the number of nucleons.
The constant χ is adjusted to reproduce the experimental binding energy of the
ground state of 14 C with respect to the 12 C core.
To evaluate the role of the continuum in the spectra of carbon isotopes, the
results of the generalized Richardson equations (6.39) are compared to results of
the standard Richardson calculations (see Eq. (6.39)) without continuum couplings
and with real single-particle energies. In the latter case, the single-particle energies
of the bound states: 0p 1/2 , 1s 1/2 , 0d 5/2 , are the same as in the previous paragraph,
and the energies of 0d 3/2 and 0f 7/2 resonances are real: 0d 3/2 = 2.267 MeV and
0f 7/2 = 9.288 MeV. One uses the value χ = −15.064 MeV in order to reproduce
the experimental binding energy of 14 C.
The spectrum of 20 C, calculated using either the generalized Richardson equations with the continuum, or the standard Richardson equations without the
continuum, is presented in Fig. 6.17. In both cases, the pairing strength G is fitted
to reproduce the experimental energy of the ground state of 14 C with respect to 12 C.
One can see significant relative shifts of energy eigenvalues which are calculated in