In this Appendix, we discuss elementary rotational properties of
s.p. eigenstates of a mean-field Hamiltonian, which
is symmetric with respect to the group.
The group
comprises the time-reversal operation,
,
three signature operations,
,
,
,
which are rotations through 180
about the three Cartesian
axes, and products of the time-reversal and signature operations,
which are called
-signatures, cf. Ref. [45] for more
information about this group in the context of mean-field
calculations.
In most cranking solutions corresponding to quadrupole
deformation, the group
is a symmetry group of the s.p. Hamiltonian,
, of Eq. (3).
For a single Kramers
pair in a fixed potential, we investigate the response of the s.p. angular momenta to a cranking frequency applied in an arbitrary
direction. Our conclusions are based only on symmetry arguments, and
are thus independent on the particular implementation of the mean
field.
Irrespective of spatial symmetries, whenever the s.p. Hamiltonian is invariant under the time reversal, its spectrum
exhibits the two-fold Kramers degeneracy. We consider a single
Kramers pair, whose states are denoted as
and
, where
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(29) |
All information about the matrix elements of the angular-momentum
operator,
, between the states
and
can be represented in a convenient way in terms of
the real alignment vector,
, and the complex
decoupling vector,
, of the state
.
They are defined as
![]() |
(30) |
![]() |
(31) |
If the s.p. Hamiltonian is symmetric with respect to the
group, it is possible to chose the states forming the Kramers pair
as eigenstates of any of the three signature operators,
, where
, but only one at a time, because the signature operators do
not commute among themselves, i.e., for
We choose states
so that they correspond to eigenvalues
under the action of
, while the eigenvalues of
are
.
Multiplication rules (32) allow to easily express eigenstates
,
,
, and
through linear combinations of eigenstates
and
.
By fixing the relative phase
between states
and
we obtain the following expressions:
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![]() |
(37) |
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![]() |
(38) |
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![]() |
(39) |
![]() |
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![]() |
(40) |
The fact that
and
are eigenstates
of
, together with the transformation rules of the
components,
, of the angular momentum operator under the
three signatures,
![]() |
(41) |
From these relations and for the eigenstates defined as in Eqs. (33)-(36), it follows that all the quantities
and
can be expressed through the three
``diagonal'' components,
, of the alignment vector,
The symmetry group itself does not impose any conditions on
the``diagonal'' components,
. However, these
values can be further restricted if some other symmetry is present.
For example,
if
is axially symmetric, say with respect to the
axis,
then the states
,
are eigenstates
of
, which leads to quantization of
. In
fact,
, because
, while in the adopted convention
. For states
and
, defined by (33) and (35),
one easily finds
![]() |
(49) |
We now consider the TAC for a single Kramers pair
in a -symmetric potential. We make two simplifying
assumptions. First, that the Hamiltonian
in the Routhian
of Eq. (3) does not change with rotational
frequency (non-selfconsistent cranking). Second, that the considered
Kramers pair has no coupling to other eigenstates of
through the angular-momentum operator (isolated pair). The TAC under such
conditions becomes a two-dimensional diagonalization problem, that
can be solved analytically.
We use the basis of states
and
.
For a degenerate Kramers pair,
reduces to its eigenvalue,
. Matrix elements of the angular-momentum operator are defined by
Eq. (46). Altogether, matrix of the s.p. Routhian
(3) takes the form
Values of alignments (53) are undefined if, and only
if, all the products
vanish - in particular
when
. In such a case, the Routhian (52) is
proportional to unity, and the mean angular momenta of its
eigenstates depend on their (arbitrary) unitary mixing.
Two extreme cases of the dependence (53) deserve particular attention.
In axial nuclei, precisely one two-fold degenerate substate of each
deformation-split -shell has
and
, which represents the soft alignment.
According to Eq. (50), all other necessarily have a
vanishing decoupling parameter, and are thus stiffly aligned with the
symmetry axis. For prolate shapes, the lowest-energy substate has
, and is soft, while for oblate
shapes it is the highest substate. In triaxial nuclei, values of the
parameters
, where
corresponds to the short,
medium, and long principal axes, are equal to the s.p. alignments
obtained from the one-dimensional cranking about the three axes.
Indeed, for cranking about the axis
, the s.p. states are
eigenstates of
. For example, from the results of
Section 3.4, one can see that for the lowest
substates of a triaxial nucleus only
is non-zero, while
for the highest substates only
does not vanish. These
alignments are thus stiff. Note that there are no states with stiff
alignment on the medium axis. The response to rotation of the middle
substates is soft, because all the three
parameters,
,
,
, are
non-zero.
In realistic cranking calculations the symmetry arguments discussed
here interplay with the fact that there is angular-momentum coupling
between different Kramers pairs and that the mean field does change
with rotational frequency. In the results of the present paper,
however, the change of deformation induced by rotation is negligible.
The angular-momentum coupling of the lowest and highest
substates to other s.p. states is rather weak, what can be seen from
the small curvature of their one-dimensional Routhians in
Fig. 2. The stiff character of their alignments is
fully confirmed in our self-consistent calculations, as discussed in
Section 3.4 and further in the paper. Investigation on how
suitable the notion of soft and stiff alignment is in other physical
cases remains a subject for further research.