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Figure 1:
Probability distributions of even angularmomentum
components in Gd.

We solve the cranked SHF equations for Gd by using
the code HFODD for
the SIII Skyrmeforce parameters[16] and spherical harmonicoscillator
basis composed of =10 shells.
Then, we calculate the Hamiltonian and overlap kernels by using
50 GaussChebyshev integration points in the and
directions and 50 GaussLegendre points in the direction.[12]
Figure 1 shows probability distributions
of even angularmomentum components
in the intrinsic crankedSHF states
constrained to
. The probability of finding the component can be
calculated from:[1]

(10) 
The curves correspond to cranking wave functions with
averaged angular momenta
, and
. One can see that for low angular momenta (e.g., for
) the maxima of the distributions do not lie near the same
values of
. Similar results have been
obtained by Islam et al.[7] and Baye et
al.[8].
Figure 2:
Probability distributions of projections in
the even angularmomentum components projected from the state with
in Gd.

In Fig. 2 we show similar probability distributions of
the projections ,

(11) 
projected from the state with
in Gd.
One can see that the component dominates for all angular
momenta, while the components increase with the increasing
angular momentum. The component is the second in magnitude
after , which illustrates the buildup of the Coriolis coupling in
a rotating intrinsic state. Note that only even values of can
be projected from the nonrotating
state,
while all appear in the cranked state.
In order to find energies of the AMP states, we solve Eq. (9)
by diagonalizing first the norm matrix:

(12) 
The nonzero eigenvalues () of are used afterwards to
built the socalled natural states

(13) 
that span the subspace called collective subspace. Final diagonalization
of the HW equation (9) is performed within the collective subspace.
In this subspace the problem reduces
to the standard hermitian eigenvalue problem.
Figure 3:
Dependence of energies of projected states on the number of
the norm eigenstates kept in the collective subspace. Angular momenta
were projected from the state having the average
projection of angular momentum
.

In practical numerical applications we use the cutoff parameter
to construct the collective subspace, by keeping only the
norm eigenstates (12) with . The test depicted
in Fig. 3 shows the stability of projected solutions with
respect to the number of the norm eigenstates kept in the collective
subspace. The analysis was conducted for angularmomentum
states projected from the deformed solution
obtained by solving the cranked SHF equations
with the constraint
. The test clearly shows that,
starting from a certain point, the obtained solutions are stable
(plateau condition). Only for very small values of
the method becomes numerically unstable.
Figure 4:
Rotational bands in Gd nucleus:
the four bands represent:
(a) experimental data,
(b) cranked SHF calculations,
(c) band projected from the state
, and
(d) band projected from the state
into ,
for
.

Figure 4 shows rotational bands (b)(d) calculated in
Gd in comparison with the experimental data (a). Band (b)
represents the average meanfield energies obtained from the cranked
SHF calculations by constraining solutions to
.
In bands (c) and (d), we show energies obtained by the AMP from the
and
, respectively. We
see, that band (c) is much higher than bands (b) and (d), which
indicates that the PAV from the
state is not
an adequate method of describing nuclear rotation. Note that the
meanfield energies (b) and the AMP energies (d) corresponding to
turn out to be very similar to one another.
This shows that the AMP from the
states
constitutes a correct symmetry restoration of the approximate VAP
solutions realized by the cranking procedure. The remaining
discrepancy with experimental data must be attributed to pairing
correlations, which are not included in our SHF solutions.
When calculating the Hamiltonian kernels within the LEDF approach,
one has to use transition density matrices between rotated states,

(14) 
which are singular whenever the rotated states are orthogonal.
In particular, the transition multipole moments,

(15) 
become singular for certain values of the Euler angles . This
is illustrated in Fig. 5, which shows absolute values of
the neutron and proton overlap kernels in Eu (left panel), and
the transition quadrupole moments
in Eu and
Gd (right panel). Calculations have been performed for axial
shapes of nuclei, for which only the rotation about the Euler angle
matters. The neutron overlap kernel corresponds to an even
number of particles (), and is always positive, although at
it becomes as small as . On the other
hand, the proton overlap kernel corresponds to an odd number of
particles (), and it three times changes the sign in the
interval of
. Consequently, the
transition quadrupole moment of Gd is a regular function,
while that of Eu has three poles.
Figure 5:
Left panel: Dependence of the absolute value of the overlap
between rotated states on the Euler angle in Eu.
Solid line shows the neutron overlap that is always positive.
Dashed and dotted lines show these segments of the proton overlap where it is
positive and negative, respectively.
Right panel: Transition quadrupole moments versus in Gd
(thick line) and Eu (thin line).

Of course, when calculating the matrix elements of multipole
operators between the rotated states, the transition matrix elements
(15) are multiplied by the overlap kernels and the poles
disappear. However, such a compensation is absent for kernels
corresponding to higher powers of densities, viz. for the
densitydependent terms of the Skyrme interactions, or for the
directCoulombenergy terms, or for the exchangeCoulombenergy terms
in the Slater approximation. How to treat such singular kernels
within the AMP methods is currently an open and unsolved problem,
similarly as is the case for the particlenumberprojection methods
recently discussed in Ref.[17].
Next: Conclusions
Up: ANGULARMOMENTUM PROJECTION OF CRANKED
Previous: AngularMomentum Projection
Jacek Dobaczewski
20061030