Introduction

Point-group symmetries play a very important role in nuclear mean-field theories. Two distinct aspects of this role can be singled out. Firstly, the point symmetries of a Hamiltonian provide good quantum numbers that can conveniently be used to label its eigenstates. They help to formulate selection rules for electromagnetic and/or other types of transitions, and allow for solving the stationary problems in subspaces rather than in the complete Hilbert space of the problem in question. In that respect the use of point symmetries in nuclear physics resembles their use in other branches of physics. Secondly, however, and this aspect is more specific to nuclear structure domain, the use of the self-consistent Hartree-Fock (HF) or Hartree-Fock-Bogolyubov (HFB) mean field approximations invariably leads to the problem of self-consistent symmetries and related spontaneous symmetry-breaking mechanisms[1].

In this article we aim at describing properties of nuclear one-body densities under the action of point symmetries. For the time-even densities we calculate the electric multipole moments which give information about nuclear shapes. Various point symmetries obeyed by the hamiltonian lead then to various types of allowed shapes. In addition, for the time-odd densities we calculate magnetic multipole moments which give information about current distributions in nuclei, i.e., about the "shapes" of matter flow. Again, various conserved symmetries restrict these flow patterns in different ways that are studied in this paper.

Numerous experiments indicate that a great number of nuclei are deformed in their ground states. Interpretation of the corresponding results shows that most often the shapes involved are axially-symmetric. Many realistic calculations, e.g., those based on the nuclear mean-field approximation, reproduce these experimental data. However, the same calculations suggest that the excited nuclear states often correspond to the nucleonic mass distributions that have the so-called triaxial shapes. It thus becomes clear that in a realistic description of the nuclear properties, the spontaneous symmetry breaking leading to the triaxially symmetric objects must be given attention.

The classical point group that contains three mutually perpendicular symmetry axes of the second order passing through a common point is denoted D $_{\mbox{\rm\scriptsize {2}}}$ (cf., e.g., Refs. [2,3,4,5]). Attaching to D $_{\mbox{\rm\scriptsize {2}}}$ the three mutually perpendicular symmetry planes spanned on the symmetry axes gives the D $_{\mbox{\rm\scriptsize {2h}}}$ point group which contains all the spatial symmetries of interest in the present paper.

As it is well known, the classical (single) point groups can be applied to spinless particles and/or systems of an even number of fermions, and thus to even-even nuclei. However, for odd fermion systems and, in particular, in the single-nucleon space, these have no faithful irreducible representations. There exist two methods to remedy this problem. One is to extend the notion of the group representation and to introduce projective or ray representations [2,6,7]. Another one, which is employed in the present work, is to enlarge the single groups by adjoining the rotation through angle $2\pi$ and all its products with the original group elements, and to double in this way the order of the group[5].

Physically, the need of such an extension is related to the fact that in the space of spinors the rotation through angle of $2\pi$necessarily changes the sign of the wave function of an odd-fermion system. Since within the group theory a multiplication of a group element by a number is not defined, the change of sign must be introduced as an extra group element. The point group enlarged in this way is called the double point group and usually denoted with the superscript ``D'' (cf., e.g., Refs. [4,5]), although some authors, see, e.g., [3], denote single and double point groups by the same symbols. Here we follow the former convention, and thus the double group corresponding to single group D $_{\mbox{\rm\scriptsize {2h}}}$ is denoted by D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$.

In the case of classical objects the elements of a symmetry point group are real orthogonal coordinate transformations. In quantum mechanics it is often of advantage to take into consideration both the spatial symmetries and the time-reversal operator explicitly and treat them as elements of a common ensemble of symmetry operators. Time-reversal symmetry operator (antilinear) and the space symmetries (linear) have usually been considered separately (cf., e.g., Ref. [3]). Here we follow Ref.[8] and add the time-reversal operator to the set of group elements, thus obtaining new groups denoted D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$. Hence, the D $_{\mbox{\rm\scriptsize {2h}}}$ group is a subgroup of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ composed of its linear elements, and similarly, the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {D}}}$ group is composed of the linear elements of D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$.

Gauge symmetries, which pertain to pairing correlations of nucleons, can be added independently and are not considered in the present study. In particular, neutron-proton correlations are not discussed. Therefore, the isospin degree of freedom is irrelevant in the discussion and can be disregarded to simplify the notation.

In this paper our goal is threefold: First, in Sec. 2, we present and discuss properties of the single group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {T}}}$ and double group D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ that are appropriate for a description of even and odd fermion systems, respectively. In particular, we recall the classification of representations of both groups, and classify properties of the group elements when they are represented in fermion Fock space. Second, in Sec. 3, we present explicit symmetry properties of local densities with respect to the operators in the D $_{\mbox{\rm\scriptsize {2h}}}^{\mbox{\rm\scriptsize {TD}}}$ group. This problem has been solved in particular applications [9,10]; however, it can be solved in many different ways, and it is useful to have a systematic approach which enumerates all available options. Although the local densities are most important for applications using the local density approximation (LDA), or those using the Skyrme effective interaction (see respectively Refs.[11,12] or Ref.[13] for reviews), they also define general properties of average values of any local one-body operator. Finally, in Sec. 4, we discuss symmetries of multipole moments which define the nuclear shapes and currents, and in Sec. 5 we present conclusions which can be drawn from our study. In the companion paper[14], we discuss physical aspects of the symmetry-breaking schemes pertaining to the point groups in question.