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Selfconsistent calculation for Sn
As a first example we discuss here the results obtained for
the semi magic isotope Sn ( and )
using the effective force SLy4
.
This
nucleus is an interesting example since it has a rather small
neutron Fermi energy (
MeV).
Small neutron separation energy allows us to discuss
possible effects of the integration box size as well as
the choice of the boundary condition.
If not explicitly stated otherwise, in all calculations the
integration step is fm
for the box size of
fm, i.e., the number of
integration points is 150+1.
The global properties
of the nucleus for Dirichlet or Neumann boundary
condition are shown in table 5.
Table 5:
Total binding energy, mean pairing gap, Fermi energy (all in MeV)
and neutron rms radius (in fm) in Sn, calculated for the force
SLy4
and two different
boundary conditions: Dirichlet (vanishing wave functions
at
) or Neumann (vanishing derivative of
the wave functions at
).
Boundary condition 




Dirichlet 
1131.863 
1.431 
1.067 
5.264 
Neumann 
1131.862 
1.431 
1.067 
5.264 

In addition to the small neutron separation energy,
one finds several single particle states with a significant
occupation around
with different values of and indeed different asymptotic
bahaviour. The single particle neutron states in the vicinity
of the Fermi energy are shown in table 6. It is
gratifying to observe that despite the fact that the quasiparticle
states are a little bit influenced by the change of the boundary
conditions, the canonical states remain unchanged.
Table:
Single particle neutron levels in Sn using the
effective force SLy4
. Calculations
have been made in a 30fm box with an integration step of
0.2fm. and are the quasiparticle energies
and norms of the lower components while
is
the canonical energy (54) and is the occupation
factor of the corresponding canonical state (55).
Boundary condition 





Dirichlet 
(MeV) 
0.926 
0.937 
1.330 
1.556 


0.221 
0.577 
0.254 
0.438 

(MeV) 
1.169 
1.051 
1.445 
1.572 


0.251 
0.619 
0.264 
0.438 
Neumann 
(MeV) 
0.919 
0.932 
1.328 
1.556 


0.211 
0.562 
0.251 
0.438 

(MeV) 
1.168 
1.051 
1.445 
1.572 


0.251 
0.619 
0.263 
0.438 

When a rather exotic nucleus is studied, it is important
to check how well the results are converged with respect to
the various approximations which are made in the numerical
treatment of the HFB problem. First of all, we limit the
calculation to a maximum number of partial wave given
by
, which has to by consistent with the
mass of the nucleus.
The convergence of the kinetic, pairing,
and total energies are shown in the left part of Fig. 2.
Between
and 37 the total energy
difference is 1.3keV.
The changes in the kinetic and pairing energies are a little
bit bigger (7keV and 6keV, respectively); since these
quantities are not at their extrema,
they react more rapidly to a small change of the wave function.
Figure 2:
Kinetic (top panels), pairing (middle panels) and
total energies (lower panels) for Sn as functions
of
in a 30 fm box with an integration step of
0.2 fm (left) and
of the pairingfield cutoff radius
defined
in Eq. (77) (right).

As discussed in section 5.3, the introduction of
the cutoff radius
in the pairing channel,
Eq. (77), can improve
the numerical stability of the iterative procedure when
solving the HFB equation. The right part of Figure 2
displays the evolution of the kinetic, pairing, and total
energies with the same parameters as in the left part of the
Figure, except for the value of
which is fixed at 31. It can
be clearly seen that beyond a rather small distance, the total energy
and its components are not affected by the introduction of this
cutoff. Specifically, for
fm the change in
the total energy is less than 0.001keV.
Figure 3:
Difference between the total
energy of Sn (solid line) and its average asymptotic value
(see text) as a function
of the box size
, and the analogous difference for
the twoneutron separation energy (dashed line).

Figure 3 shows the convergence of the total energy
and twoneutron separation energy as functions of the
size of the box. Calculations have been made for
ranging
from 10 fm to 60 fm with a step of 0.4 fm. The cutoff equivalent energy of
=60MeV and the diffuseness of 1MeV have been used.
In order to clearly exhibit the asymptotic trend of these
quantities we have estimated their average asymptotic values
and
.
Since for
fm
the results
do not show any significant evolution, these values are
estimated by taking averages
for 35 fm
fm, which gives
MeV and
MeV.
Here, the last significant digit corresponds to 0.001keV, which is the
accuracy required to stop the iterations when we solve the HFB
equations for any given value of
.
The fluctuations around the average values
result from the individual continuum states entering into
the cutoff window with increasing
.
We see that for relatively small boxes ( fm) the
twoneutron separation energy is an order of magnitude more
stable than the total binding energy of the nucleus.
For fm, the two quantities reach their asymptotic
values with a random dispersion of about 0.5 keV.
The value of
fm is obviously unnecessarily
large but it represents a good test of the stability of the
integration procedure. From these results we conclude that
the solution of the HFB equations with the box boundary conditions
and the energy cutoff is precise up to about 1keV.
A typical asymptotic behaviour of the particle and pairing densities
for neutrons
is shown in Fig. 4 for two different choices of the boundary condition.
The impact of the boundary conditions only shows
up within about 5fm near the box edge. It is
small enough to have no significant effect on the calculated
observables.
Figure 4:
Neutron particle and pairing densities in Sn
for Dirichlet (solid line) and Neumann (dashed line)
boundary conditions. The inset represents the same
quantities on a linear scale, in that case the impact of
the boundary condition cannot be seen.

Neutron particle and pairing densities for different sizes of the box
are shown in Fig. 5.
The box radius of 10fm is obviously too small and it has only been
included
to show the effect of a very small box on the densities.
For
fm, no differences can be seen in the linear
scale. An interesting fact is that despite the cut that
has been applied for the pairing field at
fm,
see Sec. 5.3,
no consequences can be observed on the densities in the asymptotic region.
Figure:
Neutron particle (left) and pairing (right) densities for different box sizes.
In the insets the dashed lines correspond to
fm
while the solid lines correspond to the other
values of
that cannot be separated in the linear scale.

Next: Regularization of the pairing
Up: Examples
Previous: Examples
Jacek Dobaczewski
20050123