The energy density functional (EDF) method in nuclear physics is nowadays the approach of choice for large-scale nuclear structure calculation. It has the same roots as the density functional theory (DFT) [1] in atomic and molecular physics, which is based on the Coulomb interaction between electrons. In nuclear physics, the EDF approach still lacks firm microscopic derivation and well-defined rules that would allow for systematic construction of an exact (or optimal) functional. In practice, the nuclear EDF is constructed phenomenologically, based on the knowledge accumulated within modern self-consistent mean-field approaches built upon effective density-dependent two-body interaction.
These approaches, although successful in reproducing gross nuclear properties and certain generic features of collective and single-particle nuclear motion are only coarse in confrontation with precise spectroscopic data. This situation calls for improvement, which can be achieved in two directions, namely, either explicitly, by using better and/or more complicated parameterizations of the nuclear EDF, or implicitly, by going beyond the mean-field or Hartee-Fock (HF) level. At present, intense studies are being conducted in both directions, aiming at exploring the limits thereof and answering the question of whether they can be considered equivalent, complementary, or independent. In this work we will explore the second alternative, by employing the angular-momentum-projection (AMP) method of cranked HF (CHF) states.
The spontaneous symmetry breaking (SSB) mechanism is inherently built into the HF approach. In many cases, it allows not only for incorporating a significant fraction of many-body correlations into a single HF state, but also serves as a source of deep physical intuition. Emergence of nuclear deformation (breaking of rotational invariance) leads naturally to the collective rotational motion, and is one of the most spectacular manifestations of the SSB in nuclear physics.
The CHF approximation treats collective (rotational) and intrinsic degrees of freedom on the same self-consistent footing. This fact is at the base of the success of this simple approximation, because in atomic nuclei both energy scales are strongly interwoven, and dramatic structural changes may take place along rotational bands. Terminating rotational bands [2] are the best examples of such possible changes, whereupon the collective rotation is followed by the total alignment of valence particles.
Although the energies of rotational states are correctly reproduced
by the CHF approach, and the changes of shape and pairing as well
as the recoupling processes of individual nucleonic orbits are well
captured, the price paid is high. Indeed, the resulting wave packets
(deformed CHF states)
are well localized in the
angular degrees of freedom and thus they are broadly spread over many
angular momenta
, with
only the average value of the
projection of angular momentum on one the axes (the
axis in
our case) being constrained,
Apart from strongly deformed states, this feature of the cranking
approximation precludes applications of this formalism to compute
transition rates, which constitute extremely valuable source of
structural information. The demand for symmetry-restoration is
therefore well motivated. Starting from deformed CHF state
, the goal can be achieved by projecting onto the
eigenspaces of the angular momentum. Most of the calculations that
have been performed so far invoke the angular momentum method applied
to non-rotating states, see, e.g.,
Refs. [3,4,5] and the
reviews in Refs. [6,9,7,8]. This limits their
applicability to low spin states, where the influence of rotation on the
intrinsic states can be neglected, and leads to an overestimate of the
nuclear moment of inertia (MoI) as compared to the (realistic) cranking
estimate.
After the ground-breaking studies in Refs. [10,11], the
AMP of CHF states has not been performed in modern
self-consistent calculations in nuclear structure. Recently, in
Ref. [12], we presented results of such
calculations for the test case of collective rotation of a
well-deformed nucleus Gd. The AMP procedure we used has been
implemented within the code HFODD [13,14]. The
calculational scheme proposed and tested in Ref. [12],
dubbed hereafter
scheme, assumes the AMP of
spin component
of the self-consistent CHF state
, which is constrained to the same mean value of
the projection on the
axis, i.e.,
. It
combines the simplicity of the self-consistent 1D cranking approach,
and its ability to reproduce the correct MoI,
with the AMP after variation method.
In this paper, within the same formalism, we present the first
systematic calculations of rotational states along terminating bands
in the 44 mass region. The paper is organized as follows.
Methods of calculation and results are presented in Secs. 2
and 3, respectively. In particular, the AMP of cranked
states along the rotational band in
Ti is discussed in
Sec. 3.1 and the AMP of states near the band termination is
presented in Sec. 3.2. Finally, summary and discussion are
given in Sec. 4.