It is well known that the isospin symmetry is only weakly broken in atomic nuclei and the concepts of the isospin
conservation and isospin quantum number prevail even in the presence of the Coulomb interaction. In order to
control the approximate isospin conservation we have employed, as already mentioned, the isocranking method, which corresponds
to the lowest-order isospin projection.
We parametrize the isocranking frequency as
In Fig. 1(a), we show the total energies of the IASs in
isobars calculated with and without the
Coulomb interaction.
We have used
MeV and
MeV
for the calculations without and with the Coulomb interaction, respectively.
They are determined from
the difference of the proton and neutron Fermi energies
in the standard HF solution for
S and
Cr.
When the Coulomb interaction is switched off, our EDF is invariant under the isospin rotation.
The total energy calculated without the Coulomb interaction
is independent of the direction of the
isospin, which constitutes a test of the code.
When the Coulomb interaction is switched on, the total energy depends
on the expectation value
. One can see that the total energy depends
on
almost linearly. The effect comes predominantly
from the Coulomb energy which exhibits almost the same dependence on
as that of the total energy as shown in Fig. 1(a).
Its linearity results from proportionality of the Coulomb energy to
, that
clearly enhances the linear term by a factor of
as compared to the quadratic term
for small
.
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In Fig. 2,
we show the same results as in Fig. 1,
but calculated for the states in the
isobars.
For these calculations, we have used
MeV and
MeV
for the calculations without and with the Coulomb interaction, respectively.
The values are determined from the standard HF ground state solutions in
Mg and
Ni.
Again, the total energy calculated without the Coulomb interaction is
independent on
.
However, in Fig. 2(b) one can see traces of the quadratic dependence of the Coulomb energy
on
, although the contribution from the linear term is dominant.
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In Fig. 3, we plot the expectation values of the s.p. Routhian
,
calculated for the
IASs with
.
At
, the Fermi surface appears around
MeV, below which
14 neutron and 6 proton orbitals are occupied.
The s.p. Routhians vary as functions of
, and, unlike in the case of
[4], there is no
large shell gap above the Fermi surface.
Nevertheless, with our choice of
, the level crossings are avoided.
While the s.p. states are pure proton or neutron states at
and 180
,
which means that the
states are nothing but the standard HF states without the p-n mixing,
at all other tilting angles, the s.p. states are p-n mixed.
In particular, the proton and neutron components are almost equally
mixed at
, which corresponds to
.
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Fig. 4 shows
the expectation values of
calculated for the
states in
isobars.
In case of rigorous isospin conservation one should obtain
=72.
The Coulomb interaction breaks the isospin symmetry and gives a deviation from this value.
However, even in the case without the Coulomb interaction, the calculated
deviates from the exact value 72 due to the spurious isospin mixing within
the mean-field approximation [7,8,9]. Note that around
the spurious deviation is
even larger than in the case with the Coulomb interaction.
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Fig. 5 shows the proton, neutron and total root-mean-square (rms) radii
calculated with the Coulomb interaction for the states in
isobars,
together with the total rms radius calculated without the Coulomb interaction.
The neutron (proton) rms radius increases with increasing (decreasing)
, that is, increasing the neutron (proton) components.
With the Coulomb interaction, the total rms radius increases with increasing the proton components
due to the Coulomb repulsion among protons. Without the Coulomb
interaction, it stays constant as a function of
.
In Fig. 6, we depict the quadrupole deformation
parameter
calculated for the
and
IASs in
isobars. In both of the IAS
chains, the quadrupole deformation
is nearly constant, which illustrates
the fact that the s.p. configuration for all IASs stays the same.
In the nuclei, such as the
=40 systems discussed above,
even-
states are the ground states of even-even nuclei and their IASs.
We also performed calculations for
nuclei, in which odd-
states are the ground states of even-even nuclei.
As an example of those calculations, in Fig. 7,
we depict the calculated energies of the
triplet in
isobars in comparison with the experimental data.
Here, the
states are the ground states
of
Fe and
Ni and are described by the standard HF solutions
without the p-n mixing.
On the other hand, the
IAS, the lowest
state in
Co,
is obtained by the isocranking calculation,
and it consists of the p-n mixed s.p. states.
It is gratifying to see that both the energy of the
state
as well as those of the
states are well reproduced by
the theory. It is worth stressing that the
IAS in an odd-odd nucleus
is described here by means of a single time-even Slater determinant.
This is at variance with single-reference p-n unmixed EDF models, wherein such states
do not exist at all [11].
Jacek Dobaczewski 2014-12-06