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6 Physical Applications of the Gamow Shell Model
6.2.1 Dependence of Nuclear Matrix Elements on Angular
Momentum and Continuum Coupling
When one considers the eigenstates of well-bound nuclei whose basis states belong
to a single proton or neutron shell of fixed total angular momentum, binding energies
are equal to the sum of core single-particle energies and nuclear two-body matrix
element. Consequently, two-body matrix elements of the nuclear Hamiltonian can
be directly extracted from binding energies by removing its one-body part. However,
due to the configuration mixing induced by the presence of scattering states in the
Berggren basis, this simple relation between binding energies and nuclear twobody matrix elements is no longer verified. One might argue that the previous
situation would be restored by considering a unique proton or neutron resonance
shell for unbound nuclear states. This is, however, not possible as in this case the
completeness properties of the Berggren basis are lost. Indeed, for calculated nuclear
states to bear positive widths if they are resonances, or to be real if they are bound,
it is necessary to use a complete Berggren basis for the considered partial wave.
Otherwise, while the real parts of binding energies might remain close to their
exact values if a continuum coupling is not large, their imaginary parts would be
uncontrolled and could no longer be associated with particle-emission widths. As
a consequence, one has to include the scattering states of the same partial wave as
that of the considered bound or resonance proton or neutron shell.
Clearly, calculation of nuclear Hamiltonian matrix elements in this case becomes
not only more complicated but also undefined. Indeed, as a very large number of the
nuclear matrix elements involving one-body scattering states enter the Hamiltonian
matrix, their extraction from a binding energy becomes undefined. A solution to
this problem is not to consider nuclear matrix elements per se, but to analyze
the difference between binding energies and the sum of occupied core singleparticle energies present in the Hamiltonian. This procedure allows to avoid the
previous mentioned problem generated by the presence of the scattering states in the
single-particle basis, while providing with energy differences from which nuclear
matrix elements involving only resonant one-body states can be inferred. Indeed,
the continuum coupling is in general not very large, so that the matrix elements
of nuclear Hamiltonian involving resonance basis states and the aforementioned
energy differences are very close. In order to simplify the following discussion,
and to allow a direct comparison with the well-bound case, these aforementioned
energy differences will be deemed as nuclear matrix elements even though, strictly
speaking, they are not exactly equal to the nuclear two-body matrix elements.
In order to average out local effects due to the use of different nuclei, shells,
and angular momenta, one does not consider directly nuclear matrix elements, but
renormalized nuclear matrix elements [22]:
(V 12 ) =
E n
¯
E n
(6.8)
E n = E − e a − e b
(6.9)
6 Physical Applications of the Gamow Shell Model
6.2.1 Dependence of Nuclear Matrix Elements on Angular
Momentum and Continuum Coupling
When one considers the eigenstates of well-bound nuclei whose basis states belong
to a single proton or neutron shell of fixed total angular momentum, binding energies
are equal to the sum of core single-particle energies and nuclear two-body matrix
element. Consequently, two-body matrix elements of the nuclear Hamiltonian can
be directly extracted from binding energies by removing its one-body part. However,
due to the configuration mixing induced by the presence of scattering states in the
Berggren basis, this simple relation between binding energies and nuclear twobody matrix elements is no longer verified. One might argue that the previous
situation would be restored by considering a unique proton or neutron resonance
shell for unbound nuclear states. This is, however, not possible as in this case the
completeness properties of the Berggren basis are lost. Indeed, for calculated nuclear
states to bear positive widths if they are resonances, or to be real if they are bound,
it is necessary to use a complete Berggren basis for the considered partial wave.
Otherwise, while the real parts of binding energies might remain close to their
exact values if a continuum coupling is not large, their imaginary parts would be
uncontrolled and could no longer be associated with particle-emission widths. As
a consequence, one has to include the scattering states of the same partial wave as
that of the considered bound or resonance proton or neutron shell.
Clearly, calculation of nuclear Hamiltonian matrix elements in this case becomes
not only more complicated but also undefined. Indeed, as a very large number of the
nuclear matrix elements involving one-body scattering states enter the Hamiltonian
matrix, their extraction from a binding energy becomes undefined. A solution to
this problem is not to consider nuclear matrix elements per se, but to analyze
the difference between binding energies and the sum of occupied core singleparticle energies present in the Hamiltonian. This procedure allows to avoid the
previous mentioned problem generated by the presence of the scattering states in the
single-particle basis, while providing with energy differences from which nuclear
matrix elements involving only resonant one-body states can be inferred. Indeed,
the continuum coupling is in general not very large, so that the matrix elements
of nuclear Hamiltonian involving resonance basis states and the aforementioned
energy differences are very close. In order to simplify the following discussion,
and to allow a direct comparison with the well-bound case, these aforementioned
energy differences will be deemed as nuclear matrix elements even though, strictly
speaking, they are not exactly equal to the nuclear two-body matrix elements.
In order to average out local effects due to the use of different nuclei, shells,
and angular momenta, one does not consider directly nuclear matrix elements, but
renormalized nuclear matrix elements [22]:
(V 12 ) =
E n
¯
E n
(6.8)
E n = E − e a − e b
(6.9)