Model for independent particle motion

Rotation is the phenomenon which appears in all branches of physics, from galaxies down to atomic nuclei. In the latter, it is a collective phenomenon in which many nucleons define a nuclear deformation and contribute to rotational motion. Note that contrary to classical mechanics, a collective rotation along the symmetry axis of nuclear density distribution is forbidden in quantum mechanics.

The simplest way to describe the properties of rotating nuclei in a microscopic way is to use the cranking-model approximation suggested by Inglis Ing.54 ; Ing.56 . Over the years this model has been successfully applied to the description of different phenomena in rotating nuclei NilRag-book ; PhysRep-SBT ; VDS.83 ; Szybook ; MPZZ.13 . The consideration here is restricted to one-dimensional cranking in which the nuclear field is rotated externally with a constant angular velocity $\omega$ around a principal axis usually defined as the $x$-axis. This is done in order to outline the basic features of this model.

The basic idea of the cranking model is that a nucleus with angular momentum $I\neq 0$ can be described in terms of an intrinsic state $\Psi^{\omega}$ at rest in a rotating frame. In the one-dimensional cranking approximation for collective rotation, the total cranking Hamiltonian (or Routhian) for a system of independent particles is given by

where $H$ is the total Hamiltonian in the laboratory system, $I_{x}$ is the $x$-component of the total angular momentum and $h^{\omega}$ is the single-particle Hamiltonian in the rotating system

with $h$ being the single-particle Hamiltonian in the laboratory system and $j_{x}$ the $x$-component of the single-particle angular momentum. In Eq. (25), the sum extends over the occupied proton and neutron orbitals. The term $-\omega I_{x}$ in Eq. (25) is analogous to the Coriolis and centrifugal forces in classical mechanics.

Then the total energy $E_{tot}$ in the laboratory system is given as

and the total spin $I$ by

where $\psi_{i}^{\omega}$ are the single-particle eigenfunctions in the rotating system and $e^{\omega}_{i}=\langle\psi_{i}^{\omega}|h^{\omega}|\psi_{i}^{\omega}\rangle$ the corresponding eigenvalues (single-particle routhians). The approximation used in Eq. (28) is valid only in the limit of high spin ($I\gg 1$).

The evolution of the single-particle states in rotating potential is shown in Fig. 3. There are several important features. First, the time-reversal symmetry is broken in rotating potential and two-fold degeneracy of deformed single-particle states, which exist in non-rotating nuclei, is removed. As a consequence, each single-particle orbital has additionally to be characterized by signature quantum number (either $r_{i}$ or $\alpha_{i}$) NilRag-book ; PhysRep-SBT ; VDS.83 . This quantum number is a consequence of the fact that full cranking Hamiltonian is invariant with respect to a rotation through an angle $\pi$ around the cranking axis ($x$-axis)

The eigenvalues of ${\cal R}_{x}$ are $\exp(-i\pi\alpha)$, where $\alpha$ is the signature exponent quantum number. Alternatively, one can define signature quantum number as $r=\exp(-i\pi\alpha)$. Signature quantum number $\alpha_{i}$ ($r_{i}$) of a single-particle orbital could take the $\alpha_{i}=+\frac{1}{2}$ $(r_{i}=-i)$ or $\alpha_{i}=-\frac{1}{2}$ $(r_{i}=+i)$ values. Similar to parity, this classification of the single-particle states is an important tool for identifying the nucleon orbitals in the rotating nuclear potential.

Second, the coupling between the different single-particle orbitals increases with increasing rotational frequency. As illustrated in Fig.  3 this leads to the change of the dominant Nilsson components of the wavefunction. Third, the slope of the orbitals in Fig. 3 corresponds to the single-particle alignment $\left<j_{x}\right>_{i}$ NilRag-book ; VDS.83

for the case of cranking at fixed deformation. Note that single-particle routhians are always plotted at fixed deformation in phenomenological potentials. In contrast, they are given along the equilibrium deformation path in the DFT calculations. Thus, the condition (30) is only approximately satisfied in the latter case provided that deformation changes with increasing rotational frequency are modest. Note that in general $\left<j_{x}\right>_{i}$ depends on signature and this dependence is especially pronounced for the single-particle orbitals with low value of $\Omega$.

The knowledge of single-particle alignments is extremely useful for an understanding of physical situation and interpretation of experimental data. For example, high-$j$ or high-$N$ intruder orbitals (such as those with Nilsson labels $1/2[770]$ and $1/2[880]$) have large values of $\left<j_{x}\right>_{i}$ and, as a consequence, they are strongly downsloping with increasing rotational frequency (see Fig. 3). Large energy splitting between two signature partner orbitals of a given single-particle state appear for the states with $\Omega=1/2$ but no such splitting exist for the states with large value of $\Omega$. For example, this is a case for the $1/2[770]$ state for which a large signature splitting between the $\alpha=\pm 1/2$ branches leads to a formation of large $N=85$ gap at $\omega\approx 0.3$ MeV which is absent at $\omega=0$ MeV (see Fig. 3).

The advantage of the cranking model is that it provides a microscopic description of rotating nuclei, where the total angular momentum is described as a sum of single-particle angular momenta, and thus collective and ‘non-collective’ rotations can be treated on the same footing. On the other hand, there are limitations. The cranking-model approximation is semiclassical, because the rotation is imposed externally. The model also breaks the rotational invariance, since a fixed rotation axis is used. These latter limitations are not very important for very fast rotation $(I\gg 1)$. Another shortcoming of the cranking model is that the wave functions are not eigenstates of the angular momentum operator, which can lead to difficulties; for example, a proper calculation of electromagnetic transition probabilities requires the use of projection techniques RS.80 . Furthermore, the cranking model is a poor approximation in the crossing region of two weakly interacting bands, but this difficulty can be overcome by the removal of such crossings PhysRep-SBT .

The total density of the nucleus is defined as the sum of the single-particle densities of occupied states

which are given by

for the $k-$th single-particle state. The single-particle wave function $\psi_{k}(\vec{r})$ can be expanded into basis states $|\mu>$

where $\mu$ represents the set of quantum numbers defining the basis state and $c_{k\mu}$ the expansion coefficients. Of special interest are the cases when this expansion is dominated by a single basis state $n$ ($c_{kn}\approx 1$) since then the nodal structure of the wave function of the state $\psi_{k}(\vec{r})$ and consequently its single-particle density will be predominantly defined by this basis state. This takes place either at spherical shape or at extremely elongated nuclear shapes typical for rod shape structures or megadeformation (MD) AA.18 .

In the latter case, the wave function defined by asymptotic Nilsson quantum number is expanded into the basis states characterized by $\mu=[Nn_{z}\Lambda]\Omega$ NilRag-book ; AA.18 . For extremely elongated shapes in light nuclei this expansion is dominated by a single basis state AA.18 . This is illustrated in Fig. 4 which shows the single-particle densities of indicated states at the deformation $\beta_{2}\approx 1.6$ typical for the MD shapes. They are characterized by axial or nearly axially symmetric spheroidal/ellipsoidal like density clusters formed for the single-particle states with the $[NN0]1/2$ Nilsson quantum numbers, doughnut density distributions for the $[N01]\Omega$ states, multiply (two for $n_{z}=1$ and three for $n_{z}=2$) ring shapes for the $[N,N-1,1]\Omega$ Nilsson states with $N=2$ and 3. Fig. 4 clearly indicates the importance of the deformation which has two critical effects. First, it leads to the formation of density clusters with specific nodal structure and to the separation of the clusters in space. Second, it lowers the energies of the Nilsson states of the$[NN0]1/2$ type which favors the $\alpha$-clusterization and leads to the configurations in which all occupied single-particle states have this type of structure.

It is impossible to experimentally measure the single-particle density distributions in rotating nuclei. That is a reason why contrary to the single-particle energies and alignments they have not been used as fingerprints of independent particle motion. However, they are important for an understanding of the $\alpha$-clusterization and the evolution of extremely elongated shapes in light nuclei such as ellipsoidal and rod shapes and nuclear molecules AA.18 ; A.18 . The examples of total neutron densities of later two types of nuclear shapes are provided in Fig. 5. A rod-shaped nuclear configuration in ¹²C is built from a linear chain of three $\alpha$-clusters and could be well understood from the summation of the single-particle densities of occupied single-particle states AA.18 (see also Refs.  IMIO.11 ; YIM.14 ; ZIM.15 for additional discussion of rod-shaped nuclei built on linear chains of the $\alpha$ particles). Rod-shaped configurations appear also in heavier nuclei such as ⁴²Ca and ⁴⁴Ti AA.18 . However, their densities are more uniform since the $\alpha$-clusterization is suppressed by the occupation of the single-particle orbitals which have either doughnut or ring type single-particle density distributions (see Fig. 4 and Refs. AA.18 ; EKLV.18 ). Nuclear molecules are characterized by the formation of the neck between two nuclear fragments: the configuration of ³⁶Ar is an example of such structure (see Fig. 5).

A knowledge of the nodal structure of the single-particle densities of the states allows to understand the microscopic mechanism of the transition between different nuclear shapes. For example, in order to build nuclear molecules from typical ellipsoidal density distributions one has to move matter from the neck (equatorial region) into the polar one. This can be achieved by means of specific particle-hole excitations which move the particles from (preferentially) doughnut type orbitals or from the orbitals which have a density ring in an equatorial plane into the orbitals (preferentially of the $[NN0]1/2$ type) building the density mostly in the polar regions of the nucleus AA.18 ; A.18 .

Another example of important role of the single-particle density of the states is seen in so-called “bubble” structures of the ground states of spherical nuclei. This is due to a specific radial nodal structure of the wavefunctions of the single-particle states (see Fig. 6.2 in Ref. NilRag-book ): only $s$ ($l=0$) states build the density in the center of the nucleus. On the contrary, the $l>0$ states do not participate in the building of the densities in the center of the nucleus due to presence of the centrifugal barrier (see also Fig. 2 in Ref. KLRL.17 ). The impact of this feature is especially pronounced in proton subsystem of ³⁴Si KLRL.17 ; GGKNPOGV.09 . Left panel of Fig. 6 shows the evolution of the proton densities in the $N=20$ isotones. The ⁴⁰Ca, ³⁸Ar and ³⁶S nuclei in which the proton $2s_{1/2}$ orbital is occupied show similar proton density patterns with the density peak at the center. In contrast, the emptying of the proton $2s_{1/2}$ orbital in ³⁴Si leads to a considerable depletion of the proton density in the central region of nucleus. This “bubble” phenomenon has been indirectly confirmed in experiment Bubble-exp .

Similar depletion of the density in the central region of the nucleus exist also in superheavy nuclei (see Refs. AF.05-dep ; BRRMG.99 ; SNR.17 and right panel of Fig. 6). However, the mechanism of its creation is different as compared with the one active in ³⁴Si since the emptying of the $s$ states is not possible in such nuclei. It relies on the fact that the filling of low-$j$/high-$j$ orbitals builds the density in the central/surface region of the nucleus AF.05-dep ; BRRMG.99 ). Thus, starting from the flat density distribution in ²⁰⁸Pb (which is experimentally verified, see Fig. 2.4 in Ref. NilRag-book ) one can build “bubble”-type structure in the ${}^{292}120_{172}$ superheavy nucleus by occupying predominantly high-$j$ orbitals outside the ²⁰⁸Pb core AF.05-dep . It was suggested in Ref. SNR.17 that the depletion of density in central region is mostly due to Coulomb interaction. This, however, contradicts to the observation that spherical superheavy nuclei with $Z=126$ have significantly smaller depletion of the density in the central region as compared with the ${}^{292}120_{172}$ one AF.05-dep .

The state-of-the-art view on the nuclear landscape is shown in Fig. 7. It is based on the results of the calculations presented in Refs. AA.21 ; AARR.14 ; AATG.19 . The bands (shown by gray color) of spherical nuclei along proton and neutron numbers 8, 20, 28, 50 and 82 (as well as for neutron number $N=126$) are seen in this figure. They are due to large spherical shell gaps at these particle numbers which provide an extra stability of the nuclei. Note that these bands are more pronounced in neutron subsystem. Outside of these bands of spherical nuclei, the ground states of the $Z\leq 120$ nuclei are either oblate or prolate being in typical range of quadrupole deformations $|\beta_{2}|<0.4$. Similar structures and features are also obtained in non-relativistic calculations of Refs. MSIS.16 ; DGLGHPPB.10 ; Eet.12 but they are restricted to the nuclei below $Z\approx 120$. In general, existing experimental data confirms these model predictions SP.08 but there may be some differences between specific model and experiment.

With increasing proton number these classical features disappear and only toroidal shapes are calculated as the lowest in energy. This region (shown in white color between two black lines in Fig. 7) is penetrated only by three islands (shown in gray color) of potentially stable spherical hyperheavy nuclei. Their existence is due to substantial proton $Z=154$ and 186 and neutron $N=228$, 308, and 406 spherical shell gaps AA.21 and substantial fission barriers around spherical minima AA.21 ; AATG.19 . However, these states are highly excited with respect of minima corresponding to toroidal shapes. They could become the ground states if relevant toroidal minima are unstable with respect of so-called sausage deformations AA.21 ; Wong.73 .

Thus, the richness of nuclear structure seen in experimentally known part of nuclear landscape is replaced by a more uniform structure of the nuclear landscape in the region of hyperheavy ($Z\geq 126)$ nuclei dominated by toroidal nuclei AA.21 ; AATG.19 . This transition from compact ellipsoidal-like shapes to non-compact toroidal shapes is driven by the enhancement of the role of Coulomb interaction with increasing $Z$ AATG.19 . Thus, the former shapes become either unstable against fission or energetically unfavored in hyperheavy nuclei.

Single-particle degrees of freedom play also an important role in the definition of the boundaries of nuclear landscape. For example, the position of two-neutron drip line of ellipsoidal nuclei depends sensitively on the single-particle energies of high-$j$ states located in the vicinity of neutron continuum threshold AARR.15 . The transition from ellipsoidal to toroidal shapes drastically modifies the underlying single-particle structure and as a result lowers the energy of the Fermi level for protons AA.21 . As a consequence, a substantial shift of the two-proton drip line from its expected position for ellipsoildal shapes (shown by dashed orange line in Fig. 7) towards more proton-rich nuclei for toroidal shapes (shown by solid line in Fig. 7) takes place AA.21 .

With increasing proton number beyond $Z=100$ the fission barriers provided by the liquid drop become rather small and then for higher $Z$ they disappear (see Fig. 10.6 in Ref. NilRag-book ). It turns out that the existence of the heaviest nuclei with $Z>104$ is primarily determined by shell effects due to the quantum-mechanical motion of protons and neutrons inside the nucleus SGK.66 ; Meld-67 ; OU.15 ; GMNORSSSS.19 ; AALMZ.21 . In these nuclei the heights of fission barriers are entirely determined by the shell corrections and they would not exist without shell effects. These effects also play a central role for the production, stability and spectroscopy of superheavy nuclei.

Although state-of-the-art theoretical models provide a reasonable description of many aspects of the physics of superheavy nuclei, they face substantial challenges in the prediction of the location of next spherical shell closures BRRMG.99 ; SP.07 ; OU.15 ; GMNORSSSS.19 ; AALMZ.21 ; AANR.15 . These challenges are illustrated in Fig. 8 which shows the map of calculated ground state quadrupole deformations obtained with two covariant energy density functionals. The predictions of these two functionals are drastically different in the vicinity of the $Z=120$ and $N=184$ lines. The PC-PK1 functional predicts wide bands of spherical nuclei in the $(Z,N)$ chart along $Z=120$ and $N=184$. In contrast, in the calculations with DD-PC1, the band along $Z=120$ does not exist and narrower band along $N=184$ is seen only for $Z\leq 114$. These discrepancies are due to modest differences in the single-particle properties of these functionals and related differences in the competition of shell effects at spherical and deformed shapes AANR.15 . Note that the width of the band of spherical nuclei in the $(Z,N)$ chart along a specific particle number corresponding to a shell closure indicates the impact of this shell closure on the structure of neighboring nuclei.

There is a wide variety of the predictions for proton and neutron spherical shell closures in superheavy nuclei obtained in different models. These are proton numbers at $Z=114,120$ and 126 and neutron numbers at $N=172$ and 184 (see Fig. 9 and Refs. BRRMG.99 ; SP.07 ; GMNORSSSS.19 ; AANR.15 . However, in the CDFT the $N=172$ could be eliminated as a potential candidate when the deformation effects are taken into account AANR.15 and the $Z=120$ nuclei become deformed in a number of functionals when both deformation and correlations beyond mean field effects are considered SALM.19 . The challenges the models face in the prediction of spherical shell closures are due to two factors. At present, self-consistent models do not provide a spectroscopic quality of the description of the single-particle spectra DABRS.15 . In contrast, the phenomenological potentials better describe single-particle spectra in known nuclei (see Ref. SP.07 and references therein) but they fail to incorporate self-consistency effects (such as a depletion of the density in central region of nucleus) which are important in superheavy nuclei (see discussion of Fig. 6).

Fig. 8 illustrates that only for a few experimentally known $Z=116$ and 118 nuclei the CDFT predictions differ. In general, with possible exception of these high-$Z$ isotopic chains, existing experimental data on superheavy nuclei are described with comparable level of accuracy by all existing models GMNORSSSS.19 ; AALMZ.21 ; AANR.15 . This is undeniable success of independent particle model the global consequences of which (shell structure) led to the predictions of superheavy nuclei in 1960s within relatively simple models SGK.66 ; Meld-67 . Experimental data collected over these years fully confirmed these predictions and some differences in the predictions of next spherical shell closures are secondary to this success. This field or research (both experimental and theoretical) remains extremely active which is illustrated by recent reviews OU.15 ; GMNORSSSS.19 ; AALMZ.21 . However, at present available experimental data does not allow to give a preference to the predictions of one or another model and to decide where next (if any) proton and neutron spherical shell closures are located or whether there is an island of spherical superheavy nuclei.

Fig. 7 shows that the ground states of the nuclei in experimentally known part of the nuclear chart are characterized by quadrupole deformation $\beta_{2}$ which is located typically in the range $-0.35\leq\beta_{2}\leq+0.35$. Strutinsky has predicted excited minimum with $\beta_{2}\sim 0.6$ in the second (superdeformed, SD) potential well of potential energy curves of deformed nuclei in Ref. Strut67 . At low spin such minima are located at high excitation energies, but they can be brought down to the yrast line by fast rotation since it favors extremely deformed shapes. This was confirmed later in the mic+mac calculations of Ref. Ander.76 which predicted the doubly magic SD band in ¹⁵²Dy with 2:1 semi-axis ratio. The existence of the SD bands is due to the shell effects associated with large proton and neutron SD shell gaps. For example, in the $A\sim 150$ region of superdeformation these shell effects are produced by large proton $Z=66$ and neutron $N=86$ SD shell gaps which exist both in phenomenological potentials (see Fig. 3 in this paper and Refs. NWJ.89 ; Rag.93 ) and in the DFT calculations ALR.98 .

The phenomenon of superdeformation at high spin has been confirmed experimentally by the observation of the first SD band in ¹⁵²Dy in 1986 Twin . From that time, a significant amount of experimental data on the SD bands in different mass regions has been collected Firestone.2002 . The mic+mac and DFT-based cranked approaches rather well describe experimental observables such as dynamic $J^{(2)}$ and kinematic $J^{(1)}$ moments of inertia and transitional quadrupole moments $Q_{t}$ of these bands (see Refs. CRHB ; ALR.98 ; NWJ.89 ; Rag.93 and references quoted therein). The calculations also reveal a substantial dependence of these physical observables on the occupation of high-$N$ intruder orbitals; here $N$ stands for the principal quantum number of dominant component of the wave function. For example, they strongly depend on the number of occupied $N=6$ protons and $N=7$ neutrons in the $A\sim 150$ region of superdeformation. This region is also of special interest since pairing is negligible in the majority of the SD bands. As a consequence, it provides one of the best examples of independent particle motion in nuclear physics (see detailed discussion in the last section).

One of clear manifestations of the independent particle motion is the phenomenon of band termination PhysRep-SBT ; HFMSAD.72 ; WKW.80 . It definitely reveals the fact that each single-particle orbital occupied in nucleonic configuration possesses a limited and state-dependent angular momentum. Of special interest are so-called smooth terminating bands which show a continuous transition from high collectivity at low and medium spin values to a pure particle-hole (terminating) state at the maximum spin which can be built within the configuration PhysRep-SBT ; RJFSW.95 . Note that this feature of finite multi-fermion systems has so far only been observed and studied in atomic nuclei. In the terminating state, the symmetry axis coincides with the rotation axis. Since collective rotation along the symmetry axis is forbidden in quantum mechanics, no further angular momentum can be brought into the system with the same occupation of single-particle orbitals and thus this state represents the termination of the rotational band.

The phenomenon of band termination has been observed in several regions of nuclear chart (see review in Ref. PhysRep-SBT ). However, the best examples of smooth terminating bands are seen in the $A\approx 110$ region in which the nuclei have several valence particles and holes outside the ¹⁰⁰Sn core. A classical example of terminating bands in ¹⁰⁹Sb PhysRep-SBT ; RJFSW.95 is considered here in order to illustrate the major features of smooth band termination. A critical feature of these bands is the fact that their dynamic moments of inertia $J^{(2)}$ gradually decrease with increasing rotational frequency to unusually low values (near 1/3 of rigid-body value) at the highest observed frequencies. This is definite indication of significant suppression of pairing correlations PhysRep-SBT making this type of rotational structures as one of the best examples of independent particle motion.

Fig. 10 compares the experimental and calculated energies $E$ of rotational bands with respect of rigid rotor reference $E_{RLD}=AI(I+1)$, where $A$ is the moment of inertia parameter. The calculations are performed in configuration-dependent cranked Nilsson-Strutinsky approach (CNS) approach PhysRep-SBT ; BR.85 . The configurations relatively to the ¹⁰⁰Sn core are labelled using the shorthand notation $[p_{1}\,p_{2},n]^{\alpha_{tot}}\equiv[\pi(g_{9/2})^{-p_{1}}\,\,(h_{11/2})^{p_{2}}\,\,(g_{7/2}\,\,d_{5/2})^{Z-50+p_{1}-p_{2}}\,\,\otimes\nu(h_{11/2})^{n}(g_{7/2}\,\,d_{5/2})^{N-50-n}]^{\alpha_{tot}}$, where $\alpha_{tot}$ is the total signature of the configuration (only sign is shown) and $p_{1}$, $p_{2}$ and $n$ are the numbers of proton holes in the $g_{9/2}$ orbitals, of protons in the $h_{11/2}$ orbitals and of neutrons in the $h_{11/2}$ orbitals, respectively. One can see that the CNS calculations without pairing very well reproduce experimental data. Some discrepancies seen at low $I\leq 20\hbar$ spin are due to the neglect of pairing.

The CNS calculations suggest that rotational bands of interest are near prolate at low spin and thus they involve collective rotation about an axis perpendicular to the symmetry axis. With increasing spin the valence nucleons gradually align their spin vectors with the axis of rotation via the Coriolis interaction. This causes the nuclear shape to gradually trace a path through the triaxial ($\gamma$) plane, toward the non-collective oblate shape at $\gamma=+60^{\circ}$ (see left panel of Fig. 11). After the available spin is exhausted, consistent with the Pauli principle, the band terminates. This gradual change from collective near-prolate $(\gamma\sim 0^{\circ})$ to non-collective oblate ($\gamma=+60^{\circ}$) shape leads to a gradual decrease of transition quadrupole moment $Q_{t}$ which agrees with available experimental data (see right panel in Fig. 11). The termination spins, which depend on the configuration, are well defined property for terminating bands and they confirm the termination process. For example, the detailed structure of the $[21,2]^{-}$ (band 1) terminating $I=\frac{83}{2}^{-}$ state in ¹⁰⁹Sb is $\pi(g_{9/2})^{-2}_{8}(g_{7/2}d_{5/2})^{2}_{6}(h_{11/2})^{1}_{5.5}\otimes\nu(g_{7/2}d_{5/2})^{6}_{12}(h_{11/2})^{2}_{10}$. Note that fully self-consistent cranked relativistic mean field calculations confirm this physical picture VALR.05 . However, due to technical reasons it is difficult to follow smooth terminating bands up to their terminating states in such calculations.

The addition or removal of particle(s) to the nucleonic configuration modifies the total physical observables. But it also creates the polarization effects on the physical properties (both in time-even and time-odd channels) of initial configuration. The comparison of relative properties of two configurations can shed important light both on the impact of the added/removed particle(s) in specific orbital(s) on physical observable of interest and on the related polarization effects. In addition, the comparison with experimental data on such properties can provide a measure of the accuracy of the description of single-particle properties in model calculations.

Let us consider an example of rotational bands in the regime of weak pairing. The relative properties of different physical observables such as relative charge quadrupole moments

and relative effective alignments Rag.93

are defined as the differences between either charge quadrupole moments $Q$ or the spins $I$ of the bands A and B at the same rotational frequency $\omega$ with the band $A$ being a reference band.

They provide sensitive probes of the single-particle motion and polarization effects induced by the occupation of specific single-particle orbitals. In addition, the experimental values of relative charge quadrupole moments are to a large degree free from the uncertainties due to stopping powers which impact the absolute values of these moments. The $\Delta Q(\omega)$ probes the differences in time-even mean fields generated by the addition or removal of particle(s) from the configuration of the reference band ALR.98 ; SDDN.96 .

The effective alignment $i_{eff}$ depends both on the alignment properties of single-particle orbital(s) by which the two bands differ and on the polarization effects (both in time-even and time-odd channels of the DFTs) induced by the particles in these orbitals ALR.98 . It can also be used to investigate experimentally the structure of the underlying single-particle orbitals in the configurations of interest ALR.98 ; Rag.93 . Note that additivity principle for the single-particle observables has been tested for the first time on the example of effective alignments of superdeformed bands in the $A\sim 150$ mass region Rag.93 .

The comparison of experimental and calculated $\Delta Q(\omega)$ values for a selected set of superdeformed bands in the nuclei near ¹⁵²Dy is presented in Table 1 (for a more extensive set of data see Table 2 in Ref. SDDN.96 ). One can see that these quantities (column 3) are well described in model calculations (columns 4 or 5). In addition, the additivity rule of quadrupole moments SDDN.96 which states that relative quadrupole moments of two configurations $\Delta Q(\omega)$ can be well approximated by the sum of individual contributions $\Delta Q^{i}$ of $i$-th particles to the charge quadrupole moment defined with respect of core SD configuration in ¹⁵²Dy

is rather well fulfilled both in relativistic and non-relativistic DFT calculations (compare columns 4 and 5 in Table 1 and see Ref. SDDN.96 ). This means that the polarization effects due to individual particles or holes are largely independent.

Experimental and calculated $i_{eff}$ values for single-particle orbitals active in the vicinity of the SD shell closures in the $A\approx 150$ region of superdeformation are compared in Fig. 14. The calculations are carried out within the cranked relativistic mean field ALR.98 and the cranked Nilsson-Strutinsky Rag.93 approaches. The CRMF calculations reproduce in average the experimental $i_{eff}$ values better than the CNS approach. This indicates that alignment properties of single-particle orbitals and their polarization effects are correctly accounted for in the CRMF approach. The discrepancy between the CRMF calculations and experiment seen in Fig. 14a at $\Omega\leq 0.5$ MeV is most likely due to the fact that pairing correlations play some role at low rotational frequency in the ¹⁴⁹Gd(1) band.

It is necessary to point on principal differences in the description of these relative observables in the DFT and mic+mac based approaches. The relative quadrupole moments are self-consistently described in the DFT based approaches ALR.98 ; SDDN.96 ; MADLN.07 . In contrast, the effective single-particle quadrupole moments in the mic+mac method are not uniquely defined because of the lack of self-consistency between the microscopic and macroscopic contributions KRA.98 . In the mic+mac models (for example, CNS) polarization effects caused by time-odd fields are neglected and therefore $i_{eff}$ is predominantly defined by the alignment properties of the active single-particle orbitals Rag.93 . On the contrary, they play an important role in the DFT models DD.95 ; AR.00-to .

Similar studies of relative properties of rotational bands have been performed in different regions of nuclear chart AF.05 ; MADLN.07 ; KRA.98 ). For example, a statistical analysis of significant number of rotational configurations in the $A\sim 130$ region of superdeformation confirms the additivity principle for quadrupole moments and effective alignments MADLN.07 . This justifies the use of an extreme single-particle model in an unpaired regime typical of high angular momentum. Note that the basic idea behind the additivity principle for one-body operators is rooted in the independent particle model.