Nucleon localization function in rotating nuclei
Tong Li, Mengzhi Chen, Chunli Zhang, Witold Nazarewicz, Markus Kortelainen
NNucleon localization function in rotating nuclei
T. Li ( 李 通 ),
1, 2
M. Z. Chen ( 陈 孟 之 ),
1, 2
C. L. Zhang ( 张 春 莉 ),
1, 2
W. Nazarewicz,
2, 3 and M. Kortelainen
4, 5 National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan 48824, USA Department of Physics, PO Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland (Dated: August 10, 2020)
Background:
An electron localization function was originally introduced to visualize in positional space bondstructures in molecules. It became a useful tool to describe electron configurations in atoms, molecules, and solids.In nuclear physics, a nucleon localization function (NLF) has been used to characterize cluster structures in lightnuclei, formation of fragments in fission, and pasta phases appearing in the inner crust of neutron stars.
Purpose:
We use the NLF to study the nuclear response to fast rotation.
Methods:
We generalize the NLF to the case of nuclear rotation. The extended expressions involve both time-even and time-odd local particle and spin densities and currents. Since the current density and density gradientcontribute to the NLF primarily at the surface, we propose a simpler spatial measure given by the kinetic energydensity. Illustrative calculations for the superdeformed yrast band of
Dy were carried out using the crankedSkyrme-Hartree-Fock method. We also employed the cranked harmonic oscillator model to gain insights intospatial patterns revealed by the NLF at high angular momentum.
Results:
In the case of deformed rotating nucleus, several NLFs can be introduced, depending on the definitionof the spin quantization axis, direction of the total angular momentum, and self-consistent symmetries of thesystem. Contributions to the NLF from the current density, spin-current tensor density, and density gradientterms are negligible in the nuclear interior. The oscillating pattern of the simplified NLF can be explained in termsof a constructive interference between kinetic-energy and particle densities. The characteristic nodal pattern seenin the NLF in the direction of major axis of a rotating nucleus comes from single-particle orbits carrying largealigned angular momentum. The variation of the NLF along the minor axis of the nucleus can be traced back todeformation-aligned orbits.
Conclusions:
The NLF allows a simple interpretation of the shell structure evolution in the rotating nucleus interms of the angular momentum alignment of individual nucleons. We expect that the NLF will be very usefulfor the characterization and visualization of other collective modes in nuclei and time-dependent processes.
I. INTRODUCTION
Nuclear collective motion, such as rotations and vi-brations, provides rich information about nuclear struc-ture and nuclear response to external fields. When dis-cussing nuclear collective motion, one is often makinganalogies to molecules and their collective modes. Onehas to bear in mind, however, that the A -body nuclearwave function cannot, in general, be expressed in termsof slow and fast components because the time separationbetween single-particle (s.p.) and collective nuclear mo-tion is poor. Consequently, deviations from the perfectrotational and vibrational patterns are abundant. Suchdeviations indicate that the nuclear collective modes re-sult from coherent superpositions of individual nucleonicexcitations.The observation of rotational bands in atomic nucleihas provided us with many insights on nuclear defor-mations and the underlying shell structure [1–4]. The-oretically, high-spin states can be described in a fullyself-consistent way by the nuclear energy density func-tional (EDF) method [5], which is closely related to den-sity functional theory [6, 7]. Although rotation is essen-tially a time-dependent problem, the introduction of arotating intrinsic frame through the cranking approxima- tion transforms the time-dependent problem into a time-independent one [8]. The cranking term added to thenuclear Hamiltonian can be interpreted as a constrainton the angular momentum, with the rotational frequencyplaying the role of the Lagrange multiplier.The spatial electron localization function (ELF) wasoriginally introduced in the context of electronic Hartree-Fock (HF) studies to characterize shell structure in atomsand chemical bonds in molecules [9–14]. In nuclear struc-ture research, the nucleon localization function (NLF)turned out to be a useful tool for the identification of clus-ters in light nuclei [15–17] and nuclear reactions [18]; for-mation of fragments in fission [19–23]; and nuclear pastaphases in the inner crust of neutron stars [16]. Comparedto nucleonic distributions that are fairly constant in thenuclear interior, the NLF more effectively quantifies nu-clear configurations through its characteristic oscillatingpattern due to shell effects. Consequently, it is expectedto be a good indicator of the competition between s.p.motion and collective nuclear modes.In this work, we use the NLF to study the nuclearresponse to rotation. We consider the case of superde-formed (SD) Dy, a quintessential nuclear rotor thathas been investigated in a number of self-consistent works[24–26].This paper is organized as follows. Our theoretical a r X i v : . [ nu c l - t h ] A ug framework is described in Sec. II, which contains a com-prehensive discussion of the NFL and its extension to thecase of rotation. Section. III contains the results of HFcalculations for Dy that are supplemented by crankedharmonic oscillator model results that illuminate essen-tial points. Finally, Sec. IV presents conclusions and per-spectives for future studies.
II. THEORETICAL FRAMEWORKA. Density matrices
The starting point in the derivation of the spatial lo-calization function is the one-body HF density matrix inthe coordinate representation: ρ ( r sq, r (cid:48) s (cid:48) q (cid:48) ) ≡ (cid:104) Ψ | a † r (cid:48) s (cid:48) q (cid:48) a r sq | Ψ (cid:105) , (1)where a † r sq and a r sq create and annihilate, respectively,a nucleon q (= n or p ) at point r with spin s = ± , and | Ψ (cid:105) is the HF independent-particle state. In what follows,we shall consider pure proton and neutron HF states, i.e., q (cid:48) = q , and we define ρ q ( r s, r (cid:48) s (cid:48) ) = ρ ( r sq, r (cid:48) s (cid:48) q ).Expressed in terms of spin components, the nonlocalHF density matrices can be written as [5, 27, 28]: ρ q ( r s, r (cid:48) s (cid:48) ) = 12 [ ρ q ( r , r (cid:48) ) δ ss (cid:48) + ( s | σ | s (cid:48) ) s q ( r , r (cid:48) )] , (2)where ρ q ( r , r (cid:48) ) = (cid:88) s ρ q ( r s, r (cid:48) s ) , (3a) s q ( r , r (cid:48) ) = (cid:88) ss (cid:48) ρ q ( r s, r (cid:48) s (cid:48) ) (cid:104) s (cid:48) | σ | s (cid:105) . (3b)In the EDF method with the zero-range Skyrme in-teraction, the energy functional depends only on localdensities and currents. Following the standard defini-tions [5, 27], in the present study we employ the followingdensities: ρ q ( r ) = ρ q ( r , r ) , (4a) s q ( r ) = s q ( r , r ) , (4b) τ q ( r ) = (cid:2) ∇ · ∇ (cid:48) ρ q ( r , r (cid:48) ) (cid:3) r = r (cid:48) , (4c) j q ( r ) = 12 i (cid:2)(cid:0) ∇ − ∇ (cid:48) (cid:1) ρ q ( r , r (cid:48) ) (cid:3) r = r (cid:48) , (4d) J q ( r ) = 12 i (cid:2)(cid:0) ∇ − ∇ (cid:48) (cid:1) ⊗ s q ( r , r (cid:48) ) (cid:3) r = r (cid:48) , (4e) T q ( r ) = (cid:2) ( ∇ · ∇ (cid:48) ) s q ( r , r (cid:48) ) (cid:3) r = r (cid:48) , (4f)where ⊗ stands for the tensor product of vectors in thephysical space. B. Nucleon localization function
Let us first consider the probability of finding two nu-cleons of a given isospin q and spin s at spatial locations r and r (cid:48) : P qs ( r , r (cid:48) ) = (cid:104) Ψ | a † r sq a † r (cid:48) sq a r (cid:48) sq a r sq | Ψ (cid:105) . (5)For the HF product state | Ψ (cid:105) this probability can bewritten as P qs ( r , r (cid:48) ) = ρ q ( r s, r s ) ρ q ( r (cid:48) s, r (cid:48) s ) − | ρ q ( r s, r (cid:48) s ) | . (6)Because of the Pauli exclusion principle P qs ( r , r ) = 0. Ifa nucleon with spin s and isospin q is located with cer-tainty at position r , the conditional probability of findinga second nucleon with the same spin and isospin at posi-tion r (cid:48) is R qs ( r , r (cid:48) ) = P qs ( r , r (cid:48) ) ρ q ( r s, r s ) . (7)To study the local (short-range) behavior of R qs , oneassumes that the second nucleon is located within a shellof small radius δ around r . The conditional probability(7) that r (cid:48) = r + δ can be written as: R qs ( r , r + δ ) = e δ · ∇ (cid:48) R qs ( r , r (cid:48) ) (cid:12)(cid:12)(cid:12) r = r (cid:48) . (8)After performing an angular averaging over the δ -shelland carrying out Taylor expansion in δ , one obtains: (cid:68) e δ · ∇ (cid:48) (cid:69) = 14 π (cid:90) e δ · ∇ (cid:48) d Ω= 1 + 13! δ ∇ (cid:48) + 15! δ ∇ (cid:48) + · · · . (9)The resulting local probability becomes R qs ( r , δ ) = 16 δ ∇ (cid:48) R qs ( r , r (cid:48) ) (cid:12)(cid:12) r = r (cid:48) + O ( δ ) . (10)By introducing a localization measure D qs ( r ) throughthe relationship R qs ( r , δ ) = 13 D qs ( r ) δ + O ( δ ) , (11)one can capture the short-range limit of the conditionallike-spin pair probability.For a rotationally-invariant and spin-unpolarized sys-tem, D qs ( r ) is independent of the choice of the spin quan-tization axis. However, for the deformed and rotatingnuclei considered in this study, one has to consider threedifferent measures D qs µ ( r ) with µ = x, y, z .If one chooses µ -axis as the spin quantization axis, onecan define three spin-dependent local densities: ρ qs µ ( r ) = 12 ρ q ( r ) + 12 σ µ s qµ ( r ) , (12a) τ qs µ ( r ) = 12 τ q ( r ) + 12 σ µ T qµ ( r ) , (12b) j qs µ ( r ) = 12 j q ( r ) + 12 σ µ J q ( r ) · e µ , (12c)where σ µ = 2 s µ = ± e µ is the unit vector in the di-rection of the µ -axis. After straightforward algebraic ma-nipulations based on the density-matrix expansion tech-nique [29, 30], the measure D qs µ ( r ) can be expressedthrough the local densities (12): D qs µ = τ qs µ − (cid:12)(cid:12) ∇ ρ qs µ (cid:12)(cid:12) ρ qs µ − (cid:12)(cid:12)(cid:12) j qs µ (cid:12)(cid:12)(cid:12) ρ qs µ . (13)Following Ref. [9], a dimensionless and normalized NLFcan now be defined as C qs µ ( r ) = (cid:32) D qs µ ( r ) τ TF qs µ ( r ) (cid:33) − , (14)where the normalization τ TF qs µ ( r ) = (cid:0) π (cid:1) / ρ / qs µ ( r ) isthe Thomas-Fermi kinetic energy density.It should be noted that the densities (12) constitutingthe NLF contain both time-even and time-odd compo-nents. Indeed, the particle density ρ q ( r ), kinetic density τ q ( r ), and spin-current tensor density J q ( r ) are all time-even, while the spin vector density s qµ ( r ), spin-kineticvector density T qµ ( r ), and current vector density j q ( r )are time-odd. If time-reversal symmetry is conserved, s qµ ( r ) = 0, T qµ ( r ) = 0, and j q ( r ) = 0. Consequently, fora system that conserves time-reversal symmetry and isgoverned by spin-independent interactions, one obtains: D q ± = D q = 12 τ q − | ∇ ρ q | ρ q , (15)which is the familiar atomic physics expression [9].In general, the tensor density J q ( r ) does not vanisheven if the time-reversal symmetry is conserved [31]. Itcan be decomposed into trace, antisymmetric, and sym-metric parts [28]. In many practical applications, thespin-current tensor is approximated by its antisymmetric(spin-orbit current) part [32]. However, all componentsof J q are important to characterize nuclear spin-orbit andtensor interactions [33–36] and the resulting spin polar-ization, which is sensitive to spin saturation of nucleonicshells. Consequently, the current j qs µ ( r ) does not vanisheven in ground-state configurations of even-even nuclei.While its contribution to the NLF was ignored in severalprevious calculations [15, 18, 19], the current density con-tribution to the NLF practically vanishes in the nuclearinterior, see discussion in Sec. III A. Consequently, onecan safely neglect this term when the goal is to use theNLF as a configuration-characterization tool.The choice of the normalization function in Eq. (14)is somehow arbitrary. In atomic physics applicationsand for time-reversal invariant nuclear configurations, thedensity ρ qs µ does not depend on spin. In general case,however, nucleonic densities depend on the spin polariza-tion. In this work, in order to emphasize the rotation-induced effects, we decided to stick to the normalization function τ TF qs µ , which is different for spin-up and spin-down subsystems.As discussed in Refs. [10, 13], the localization functioncan also be interpreted in terms of the Pauli exclusionprinciple. Let us consider a situation, in which an iso-lated fermion of given spin s and isospin q , is located insome region of space. The wave function of this particlecan be written as ψ qs ( r ) = √ ρ qs e iχ ( r ) , (16)where χ ( r ) is a position-dependent phase factor relatedto the current density via j qs = ρ qs ∇ χ. (17)The corresponding s.p. kinetic energy density is the sumof last two terms in D qs (13): τ s . p .qs = | ∇ ψ qs | = 14 | ∇ ρ qs | ρ qs + (cid:12)(cid:12) j qs (cid:12)(cid:12) ρ qs , (18)where the first term is the von Weizsacker kinetic energydensity [37]. Therefore, D qs can be interpreted as a mea-sure of the excess of kinetic energy density due to thePauli exclusion principle: D qs = τ qs − τ s . p .qs . (19)This interpretation of the NLF is more flexible as it doesnot involve the notion of the conditional probability (7),which is not straightforwardly generalized to the case ofpoint-group symmetries of the nuclear mean field. C. Cranked Hartree-Fock calculations
Superdeformed nuclei around
Dy can be viewed asunique laboratories of extreme single-particle behavior[25, 38]. The nucleus
Dy plays a role of superdeformeddouble-magic core due to large shell closures at Z = 66and N = 86. Because of this, Dy has been a sub-ject of many studies of self-consistent nuclear responseto collective rotation, see, e.g., Ref. [24, 26, 39, 40]. Be-cause of large deformed gaps and rapid rotation, pairingcorrelations are weak in SD
Dy [41, 42]. Indeed witha reasonable pairing strength, adjusted to experimentalodd-even mass difference in
Sn as done in Ref. [43],the static pairing vanishes in the SD yrast band of
Dyin Hartree-Fock-Bogoliubov (HFB) calculations.The intrinsic configurations of SD bands in the A =150 mass region are well characterized by nucleons inthe intruder orbitals carrying large principal harmonicoscillator (HO) numbers N , namely the proton N = 6and neutron N = 7 states [44, 45]. Because of theirlarge intrinsic angular momenta, these orbitals stronglyrespond to nuclear rotation; hence, their occupations andalignment patterns well characterize SD bands.To study the impact of rotation on shell structurethrough the nucleon localizations, we carry out unpairedcranked HF (CHF) calculations for superdeformed Dyusing the HF solver hfodd [46]. Following Ref. [24],s.p. wave functions have been expanded in a stretcheddeformed HO basis with frequencies (cid:126) ω z = 6 .
246 MeVand (cid:126) ω ⊥ = 11 .
200 MeV along the directions parallel andperpendicular to the symmetry axis, respectively. Thetotal number of basis states is 1013 with HO quantanot exceeding 15 in each direction. We employed theSkyrme energy density functional parametrization SkM*[47], with its generic time-odd terms [24, 48].The angular momentum has been generated by meansof a cranking term − ω ˆ J y , where ˆ J y is the y -componentof the total angular momentum operator and ω repre-sents the angular velocity of rotation. In the presenceof the cranking term, parity ( ˆ P ), y -signature ( ˆ R y =exp( − iπ ˆ J y )), and y -simplex ( ˆ R y = ˆ P ˆ R y ) symmetriesare preserved while time-reversal and axial symmetriesare broken, see Refs. [49–51] for more discussion. Sincethe time-reversal operator commutes with the signatureand simplex operators, the time-reversed s.p. CHF states(Routhians) belong to opposite signature and simplexeigenvalues.Every CHF configuration can be labeled using thestandard notation in terms of parity-signature blocks[ N + , + i , N + , − i , N − , + i , N − , − i ], where N πr y are the num-bers of occupied s.p. orbitals having parity π ind y -signature r y . As discussed in Ref. [24], the yrast configu-ration of SD Dy is [22 , , , n ⊗ [16 , , , p .The relative variation of the quadrupole moment Q within this state is much less than 1% in the frequencyrange (cid:126) ω = 0 . ∼ . Q = 42 b to eliminate its possible impact on thecomputed localizations.Single-particle Routhians obtained in the CHF+SkM*calculations for the SD yrast band of Dy are shown inFig. 1. The large deformed shell closures at Z = 66 and N = 86 are clearly seen. The lowest N = 7 neutron and N = 6 proton Routhians indicated in the figure are rota-tion aligned, i.e., they are strongly impacted by the Cori-olis coupling and their s.p. aligned angular momenta arelarge at high rotational frequencies. Many other statesaround the Fermi level are weakly impacted by rotation.Such states are usually referred to as deformation-aligned(strongly coupled) [1, 52, 53]. D. Cranked Harmonic Oscillator calculations
In the previous study of the NLF, the harmonic oscil-lator model was used to provide an illustrative guidance[16]. In this work, we study the NLF patterns of the SDcranked harmonic oscillator (CHO) model with frequen-cies ω ⊥ = ω x = ω y = 2 ω z . Since the HO potential is spin-independent, every s.p. HO level is doubly-degenerate.As in the CHF calculations, we assume that the rota-tion takes place around the y -axis. The s.p. Routhiansand wave functions of the CHO can be obtained analyt-ically [2, 54, 55]. We wish to emphasize that our CHO N = 86 D ¯ hω (MeV) Z = 66 E [541]1 / [ ] / 6 L Q J O H S D U W L F O H 5 R X W K L D Q V 0 H 9 FIG. 1. Single-particle neutron (a) and proton (b) Routhiansas functions of ω , obtained in the CHF+SkM* calculations forthe SD yrast band of Dy. The ( πr y ) combinations are indi-cated by solid lines (+ , + i ), dotted lines (+ , − i ), dot-dashedlines ( − , + i ), and dashed lines ( − , − i ). The Routhians origi-nating from the lowest neutron N = 7 and proton N = 6 and[541]1/2 levels are marked by thicker lines. results were obtained without imposing the consistencyrelation between mean-field ellipsoidal deformation andthe average density distribution [1, 55]To relate the CHO analysis to the CHF results forSD Dy, we study a SD HO potential filled with 60particles, which corresponds to a closed SD supershell N shell ≡ n + n ) + n = 6 [1, 53, 56, 57]. The corre-sponding s.p. Routhians are shown in Fig. 2 as functionsof ω . A supershell of a SD HO consists of degeneratepositive- and negative-parity states. This degeneracy islifted by rotation: the orbits with no CHO quanta alongthe rotation axis ( n = 0) and the largest possible valueof the difference ( n − n ) carry the largest s.p. angularmomentum. In Fig. 2 those are the [0,0,7] ( N = 7) and[0,0,6] ( N = 6) Routhians. E. Nucleon localization function at high spins
Since parity, y -signature r y , and y -simplex r y are self-consistent symmetries in our cranking calculations, inorder to see the angular momentum alignment effectscaused by different orbits, it is convenient to study the ω/ω E / ¯ h ω z [0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , N shell = 6 N shell = 7 FIG. 2. Single-particle Routhians of the SD CHO model be-longing to the supershells N shell = 6 and 7. The CHO quan-tum numbers [ n , n , n ] are given in brackets. Positive-parityand negative-parity states are indicated by solid and dashedlines, respectively. The rotational frequency ω is expressedin units of ω = (cid:112) ω z ω ⊥ while the Routhians E in units of (cid:126) ω z . Each level is doubly degenerate due to the two possiblespin orientations. The crossing between the lowest N = 7Routhian [0,0,7] and the [3,0,0] Routhian at ω/ω ≈ . NLFs of a given r y or r y . This can be done by expressinglocal densities and currents in terms of their symmetry-conserving components. In practice, this can be done bysumming up the contributions from HF s.p. wave func-tions belonging to a given symmetry block [49–51]. Forinstance, if y -simplex is conserved, ρ q ( r ) = ρ q ˘ σ y =+1 ( r ) + ρ q ˘ σ y = − ( r ) , (20)where ˘ σ y ≡ r y /i = ±
1. A similar decomposition holdsfor τ q ( r ) and j q ( r ).By decomposing these densities into time-even andtime-odd parts, they can be expressed in a form simi-lar to Eq. (12): ρ q ˘ σ y ( r ) = 12 ρ q ( r ) + 12 ˘ σ y s (cid:48) q ( r ) , (21a) τ q ˘ σ y ( r ) = 12 τ q ( r ) + 12 ˘ σ y T (cid:48) q ( r ) , (21b) j q ˘ σ y ( r ) = 12 j q ( r ) + 12 ˘ σ y J (cid:48) q ( r ) , (21c) where s (cid:48) q ( r ) = ρ q ˘ σ y =+1 ( r ) − ρ q ˘ σ y = − ( r ) , (22a) T (cid:48) q ( r ) = τ q ˘ σ y =+1 ( r ) − τ q ˘ σ y = − ( r ) , (22b) J (cid:48) q ( r ) = j q ˘ σ y =+1 ( r ) − j q ˘ σ y = − ( r ) . (22c)The fields s (cid:48) and T (cid:48) are time-odd and J (cid:48) is time-even. III. RESULTS AND DISCUSSIONA. General considerations
In a rotating system, the current density j character-izes the collective rotational behavior [26, 58–65]. Fig-ure 3 shows how the current density builds up in theCHO model. As rotational frequency increases, a pat-tern of the vector field j resembling a rigid-body ro-tation gradually develops. At ω = 0 . ω , the lowest N = 7 Routhian [0,0,7] becomes occupied and the [3,0,0]level becomes empty, see Fig. 2. As the orbital [0,0,7] isstrongly prolate-driving and it carries large s.p. angularmomentum, and the Routhian [3,0,0] has large negativequadrupole moment (oblate), the associated configura-tion change (band crossing) results in a large increase inthe angular momentum alignment and intrinsic deforma-tion, see in Fig. 3. This effect is also present in CHOcalculations which consider the potential-density consis-tency relation [57]. -5 0 5-10010 ~j ω = 0 . -5 0 5 ω = 0 . -5 0 5 ω = 0 . -5 0 5 ω = 0 . × -2 x (fm) z (f m ) FIG. 3. Current density j in the x - z ( y = 0) plane, calculatedin the CHO model with 60 particles in a SD HO well forfour values of rotational frequency ω (in units of ω ). Themagnitude | j | (in fm − ) is shown by color and line thickness. When it comes to the realistic description, Fig. 4 showsthe neutron and proton current densities of
Dy calcu-lated in the CHF method at four rotational frequenciesup to (cid:126) ω = 0 . I y ≈ (cid:126) ). Theleftmost column in Fig. 4 shows the result of the bench-mark FAM-QRPA calculation [66], which corresponds tothe ω → Dy.In addition to the current j , two other time-odd vec-tor densities enter the expression for the NLF: spin den- -10010 ~j n FAM ω = 0 . ω = 0 . ω = 0 . ω = 0 . -5 0 5-10010 ~j p -5 0 5 -5 0 5 -5 0 5 -5 0 5 04.59.0 × -3 × -3 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 4. Current density j in the x - z ( y = 0) plane forneutrons (top) and protons (bottom) in the SD yrast band of Dy obtained in the CHF calculations, as a function of ω (inunits of MeV/ (cid:126) ). The magnitude | j | (in fm − ) is shown bycolor and line thickness. The FAM-QRPA result is presentedin the first column with a different color range. ~s n ) $ 0 ω = 0 . ω = 0 . ω = 0 . ω = 0 . ~T n × × x I P y I P FIG. 5. Spin density s (top) and spin-kinetic density T (bottom) in the x - y ( z = 0) plane for neutrons in the SD yrastband of Dy obtained in the CHF calculations, as functionsof ω (in units of MeV/ (cid:126) ). The magnitudes, | s | (in fm − ) and | T | (in fm − ), are shown by color and line thickness. TheFAM-QRPA results are presented in the first column with adifferent color range. sity s and spin-kinetic density T . They are displayed inFig. 5 for several values of ω . Both spin fields are polar-ized along the direction of the total angular momentum(here: y -axis). It is interesting to see that the distribu-tion themselves hardly change with rotational frequency;what is changing are the magnitudes | s | and | T | thatgradually increase with rotation. This is also seen in theFAM-QRPA results that produce flow patterns close tothose obtained in the CHF calculations.To complete discussion of spin fields, the spin-currenttensor density J · e y is shown in Fig. 6. As compared tothe current density j shown in Fig. 4, J · e y changes veryweakly with ω . This field has a surface character, i.e.,it practically vanishes within the nuclear volume. Since J q · e y is time-even, its contribution to C qs µ does notvanish at ω = 0. z I P J n · ~e y ω = 0 . ω = 0 . x I P ω = 0 . ω = 0 . ω = 0 . × FIG. 6. Spin-current tensor density J · e y in the x - z ( y = 0)plane for neutrons in the SD yrast band of Dy, as a functionof ω (in units of MeV/ (cid:126) ). Its magnitude (in fm − ) is shownby color and line thickness. B. Simplified nucleon localization function
An important consequence of the rigid-body flow isthat the current density only contributes significantly tothe NLF at the surface. This observation should be validin most cases even if an irrotational flow exists (see ex-amples in Refs. [61, 66]). The same argument is also validfor the contribution to the NLF from the density gradi-ent term | ∇ ρ qs | , which has a surface character. Conse-quently, we define a simplified localization measure as: C τqs µ ( r ) = (cid:32) τ qs µ ( r ) τ TF qs µ ( r ) (cid:33) − , (23)which does not include contributions from the currentdensity and density gradient. Figure 7 shows C , C τ , andtheir difference obtained in the CHO model; we indeedsee that C τ exhibits the same pattern as C inside thenuclear volume. A similar behavior is present in the CHFcalculation for the SD yrast band of Dy. Figure 8shows C and C τ for neutrons with ˘ σ y = − y -simplex r y = − i ) at (cid:126) ω = 0 . C qσ → C qσ ρ qσ / [max ρ qσ ] (with σ being either spin, sig-nature, or simplex) to avoid large values in the regions ofsmall particle density. However, as shown in Figs. 7 and8, replacing C with C τ mitigates this unwanted behaviorand leaves the internal pattern unaffected, thus eliminat-ing the need for this additional normalization. Comingback to the interpretation of D qs as a measure of thePauli repulsion, it is not surprising to see that | ∇ ρ qs | and (cid:12)(cid:12) j qs (cid:12)(cid:12) are significant only at the surface where onlya limited number of s.p. orbits are available and thusbecome ‘localized’. Therefore, the simplified localizationfunction C τ is a useful tool to characterize intrinsic con-figurations in most cases, except perhaps for dynamicprocesses and high-energy modes where the current den-sity and density gradient can become appreciable insidethe nucleus. -10010 C ω = 0 . ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 C τ -5 0 5-10010 C − C τ -5 0 5 -5 0 5 -5 0 5 -5 0 5 00.30.60.900.30.60.900.30.60.90.0 0.2 0.4 0.6 0.8 1.0 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 7. C (top), C τ (middle), and their difference (bottom)in the x - z ( y = 0) plane, calculated in the CHO model with60 particles in a SD HO well for five values of rotational fre-quency ω (in units of ω ). -5 0 5-10010 z (f m ) C -5 0 5 x (fm) C τ -5 0 5 C − C τ FIG. 8. C (left), C τ (middle), and their difference (right) inthe x - z ( y = 0) plane for neutrons with ˘ σ y = − y -simplex r y = − Dy at (cid:126) ω = 0 . C. Dependence of nucleon localizations on thechoice of spin quantization axis
As discussed in Sec. II B, in the general case of de-formed nuclei, nucleon localization functions C qs µ (14)and C τqs µ (23) depend on the choice of the spin quan-tization direction µ . This directional dependence is il-lustrated in Figs. 9 and 10 for the SD Dy at (cid:126) ω =0 . µ , especially in the case of the y - z cross section.More importantly, C τqs µ ≈ C qs µ in the nuclear interior,independently of µ . -10010 n ↑ C x C τx C y C τy C z C τz -10010 n ↓ -10010 p ↑ -5 0 5-10010 p ↓ -5 0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5 00.30.600.30.600.30.600.30.60.0 0.2 0.4 0.6 0.8 1.0 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 9. Nucleon localizations functions C qs µ (14) and C τqs µ (23) in the x - z ( y = 0) plane for three spin quantization di-rections µ = x, y, z , obtained in the CHF calculation for theSD yrast configuration of Dy at (cid:126) ω = 0 . -10010 n ↑ C x C τx C y C τy C z C τz -10010 n ↓ -10010 p ↑ -5 0 5-10010 p ↓ -5 0 5 -5 0 5 -5 0 5 -5 0 5 -5 0 5 00.30.600.30.600.30.600.30.60.0 0.2 0.4 0.6 0.8 1.0 y (fm)0.00.20.40.60.81.0 z (f m ) FIG. 10. Similar as in Fig. 9 but shown in the y - z ( x = 0)plane. D. Angular momentum alignment: CHO analysis
In this section, we use the CHO model to illustratesome general features of NLFs and densities, which willhelp us understand the CHF results. First, to show theusefulness of C τ when it comes to the visualization ofnucleonic shell structure and angular momentum align-ment, we come back to Fig. 7. A characteristic regularpattern seen at ω = 0 gradually gets blurred with ω . At ω = 0 . ω , where the band crossing occurs, C τ rapidlychanges. Namely, the number of maxima along the z -axis increases as the [0,0,7] orbit becomes occupies, andthe number of maxima along the x -axis decreases as the[3,0,0] state gets emptied.To clearly see the evolution of C τ with ω , we considerthe indicator∆ C τ ( r ; ω ) ≡ C τ ( r ; ω ) − C τ ( r ; ω = 0) . (24)This quantity is shown in Fig. 11 together with the cor-responding variations ∆ τ and ∆ τ TF relative to the non-rotating case. -10010 C τ ω = 0 . C τ ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 τ ∆ τ -5 0 5-10010 τ TF -5 0 5 ∆ τ TF -5 0 5 -5 0 5 -5 0 500.30.6 -0.20.00.200.030.06 -5.005.0 × -3 × -3 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 11. C τ (top), τ (in fm − , middle) and τ TF (in fm − ,bottom) in the x - z ( y = 0) plane, calculated in the CHOmodel with 60 particles in a SD HO well. The first columnshows the reference plots at ω = 0 while the other columsshow the rotational dependence relative to the ω = 0 referenceas a function of ω (in units of ω ). One can notice that there is a clear correspondencebetween the peaks of ∆ C τ and valleys (peaks) of ∆ τ (∆ τ TF ), which is consistent with Eq. (23). This observa-tion suggests that ∆ τ and ∆ τ TF are in antiphase, whichresults in a constructive interference when consideringtheir ratio.To analyze this pattern in more detail, Fig. 12(a) dis-plays τ , τ TF and C τ for 60 particles in the nonrotatingSD HO along z axis ( x = y = 0), together with the den-sity profile of the [0,0,6] state. One can see that valleys(peaks) of τ , τ TF , and C τ roughly coincide with maximaof the [0,0,6] density, while the contributions from otherstates contribute to a smooth background.This effect is even more pronounced in the one-dimensional HO model, as shown in Fig. 12(b) where HOorbits with quantum number N ≤ τ and τ TF is expectedsince τ is related to the gradients of s.p. wave functionswhile τ TF depends on s.p. wave functions alone. The ad-vantage of C τ is that it amplifies the characteristic nodalstructure of aligned high- N s.p. orbitals thanks to theconstructive interference between τ and τ TF . D ττ TF C τ | ψ | z I P E τ τ TF C τ | ψ | ' H Q V L W L H V G L P H Q V L R Q O H V V R U 1 / ) τ FIG. 12. C τ (thick solid line), τ (solid line), and τ TF (dashedline) for the nonrotating HO model plotted along z axis ( x = y = 0). (a) 3D SD HO case for 60 particles. The densityprofile of the [0,0,6] orbit is marked by a dotted line; (b) 1Dcase. HO orbits with principal quantum number N ≤ N = 6 orbit is marked bya dotted line; here τ TF = π ρ /
3. Some quantities are scaledfor a better visualization.
As discussed above, the kinetic energy density τ is sen-sitive to the nodal structure of s.p. wave functions. Itcan thus be utilized for the visualization of the align-ment process seen in the pattern of ∆ τ in Fig. 11. (Fordiscussion of quasi-molecular states in light nuclei basedon the nodal structure of the s.p. densities and cur-rents, see Ref. [65].) The cranking operator ω ˆ L y in-duces the particle-hole (p-h) excitations across the Fermilevel. The low-energy excitations correspond to ∆ N = 0(∆ n = ± , ∆ n = 0 , ∆ n = ∓
1) transitions.Figure 13 shows the variation of τ at ω = 0 induced bysix such p-h excitations across the N = 60 gap from theoccupied superhell shell N shell = 6 to the empty super-shell N shell = 7, see Fig. 2. The [0,0,6] → [1,0,5] excitationcan be associated with that between [660]1/2 (6 , ) and[651]3/2 (6 , ) Nilsson levels. Both are rotation-aligned,prolate driving orbits, and the corresponding ∆ τ plot ex-hibits a nodal pattern along the symmetry axis. On theother extreme, the [2,0,2] → [3,0,1] excitation correspondsto a [420]1/2 ([422]3/2) → [411]3/2 ([413]5/2) transition,which involves deformation-aligned orbits. The related∆ τ plot exhibits a nodal pattern along the minor axis.By summing up all six contributions, one arrives at apattern in the last panel of Fig. 13, which is indicativeof a change in τ due to rotation. Interestingly, this pat-tern is quite similar to that of Fig. 11 at ω = 0 . ω .We can thus conclude that for a system that is stronglyelongated along z axis, rotation aligned s.p. states withlarge n leave a strong imprint on ∆ τ and ∆ C τ . ∆ τ [0 , , → [1 , , [0 , , → [1 , , [1 , , → [2 , , x I P [0 , , → [1 , , [1 , , → [2 , , [2 , , → [3 , , W R W D O ! z I P FIG. 13. Changes in the kinetic energy density τ due to p-h excitations (at ω = 0) from the SD shell N shell = 6 to the nextsupershell N shell = 7 in Fig. 2. These excitations are induced in the CHO description of a 60-particle system by the crankingterm. The rightmost panel show the uniform average of individual p-h contributions. E. Angular momentum alignment: CHF analysis -10010 C τ n ↑ ω = 0 . ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 C τ n ↓ -10010 C τ p ↑ -5 0 5-10010 C τ p ↓ -5 0 5 -5 0 5 -5 0 5 -5 0 5 00.30.600.30.600.30.600.30.60.0 0.2 0.4 0.6 0.8 1.0 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 14. C τq ˘ σ y in the x - z ( y = 0) plane as a function of ω (inunits of MeV/ (cid:126) ), obtained in the CHF calculation for the SDyrast band of Dy. The symbols ↑ and ↓ represent ˘ σ y = +1and − y -simplex r y = + i and − i ), respectively. In this section, we study the localization patterns ob-tained in the CHF calculations for the SD yrast band in
Dy. Figure 14 shows the simplified NLF C τq ˘ σ y in the y = 0 plane for different values of ω . The first columncorresponds to the nonrotating case, where we see NLFpatterns characteristic of a deformed nucleus, similar tothose for Zr,
Th and
Pu discussed in Ref. [19].As ω increases, new patterns gradually emerge inside thenucleus, with C τq ↑ (cid:54) = C τq ↓ due to the time-reversal symme-try breaking terms in (21).For a better visualization of rotational dependence, we -10010 ∆ C τ n ↑ ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 ∆ C τ n ↓ -10010 ∆ C τ p ↑ -5 0 5-10010 ∆ C τ p ↓ -5 0 5 -5 0 5 -5 0 5 -0.100.1-0.100.1-0.100.1-0.100.10.0 0.2 0.4 0.6 0.8 1.0 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 15. Similar as in Fig. 14 but for ∆ C τq ˘ σ y . The referencevalue of C τ at ω = 0 is shown in the first column of Fig. 14. will be using relative indicators, cf. Eq. (24). Figure 15presents the relative indicator ∆ C τq ˘ σ y in the y = 0 plane.As the local densities and currents can be decomposedinto time-even and time-odd parts, see Eqs. (12) and(21), their relative indicators can also be decomposedinto time-even and time-odd components. For instance,∆ τ q ˘ σ y ( r ; ω ) = 12 ∆ τ q ( r ; ω ) + 12 ˘ σ y T (cid:48) q ( r ; ω ) , (25)where the quantity∆ τ q ( r ; ω ) = τ q ( r ; ω ) − τ q ( r ; ω = 0) (26)does not depend on simplex and provides a backgroundthat is an even function of ω . The simplex-dependent0term in Eq. (25) is ω -odd; together with the time-oddcomponent of ρ q ˘ σ y ( r ; ω ) is responsible for the differencebetween the values of C τq ˘ σ y of different simplex. (Thesame argument also holds for signature.) This differenceis clearly shown in Fig. 15.Also, to illustrate the directional dependence of ∆ C τq ˘ σ y ,we show it in the x = 0 plane in Fig. 16. A differentpattern along the y direction results from the breakingof axial symmetry by the cranking term. -10010 ∆ C τ n ↑ ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 ∆ C τ n ↓ -10010 ∆ C τ p ↑ -5 0 5-10010 ∆ C τ p ↓ -5 0 5 -5 0 5 -5 0 5 -0.100.1-0.100.1-0.100.1-0.100.10.0 0.2 0.4 0.6 0.8 1.0 y (fm)0.00.20.40.60.81.0 z (f m ) FIG. 16. Similar as in Fig. 15 but for ∆ C τq ˘ σ y in the y - z ( x = 0)plane Figures 17 and 18 show the variations of ∆ τ and ∆ τ TF with ω . Similar to the CHO case discussed in Sec. III D,∆ τ and ∆ τ TF are in antiphase that results in a construc-tive interference when it comes to ∆ C τ . Furthermore,∆ τ , ∆ τ TF , and ∆ C q ˘ σ y of opposite values of ˘ σ y changein opposite direction with ω . That is, a ridge in ∆ C q ↑ corresponds to a valley in ∆ C q ↓ . According to Eqs. (21),this is due to the contributions from time-odd densitieswhich change sign between different values of ˘ σ y .By investigating the behavior of the NLF from the per-spective of individual s.p. orbits, we can gain useful in-sights on the s.p. motion in the rotating nucleus. Follow-ing the CHO discussion in Sec. III D, we focus on ∆ τ .In particular, we shall study the rotational dependenceof kinetic energy densities of s.p. orbits near the Fermilevel as these orbits are expected to primarily affect thenuclear response to rotation.In the example discussed below, for the sake of simplic-ity we consider small rotational frequency (cid:126) ω = 0 . -10010 τ n ↑ ω = 0 . τ n ↑ ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 τ n ↓ ∆ τ n ↓ -10010 τ p ↑ ∆ τ p ↑ -5 0 5-10010 τ p ↓ -5 0 5 ∆ τ p ↓ -5 0 5 -5 0 5 -5 0 500.020.04 -1.501.5 × -3 × -3 × -3 × -3 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 17. Similar as in Fig. 15, but for ∆ τ q ˘ σ y (in fm − ). Thereference value of τ q ˘ σ y at ω = 0 is shown in the first column. -10010 τ TFn ↑ ω = 0 . τ TFn ↑ ω = 0 . ω = 0 . ω = 0 . ω = 0 . -10010 τ TFn ↓ ∆ τ TFn ↓ -10010 τ TFp ↑ ∆ τ TFp ↑ -5 0 5-10010 τ TFp ↓ -5 0 5 ∆ τ TFp ↓ -5 0 5 -5 0 5 -5 0 500.020.04 -3.003.0 × -3 × -3 × -3 × -3 x (fm)0.00.20.40.60.81.0 z (f m ) FIG. 18. Similar as in Fig. 17, but for ∆ τ TF q ˘ σ y (in fm − ). this discussion can be repeated by following the diabaticRouthians within each parity-signature block.Figure 19 shows ∆ τ at (cid:126) ω = 0 . τ can be un-derstood by inspecting the contributions from several in-dividual s.p. orbits close to the Fermi energy shown inFig. 20. The main contribution to ∆ τ n in the negative-parity blocks comes from the high- N orbits 7 and 7 .For the π = + neutrons, four close-lying deformation-aligned states [651]1/2, [642]5/2, [413]5/2, and [411]1/2,1 ∆ τ n (+ , + i ) (+ , − i ) ( − , + i ) ( − , − i ) ∆ τ p × × x I P z I P FIG. 19. Neutron (top) and proton (bottom) contributionsto ∆ τ (in fm − ) in the x - z ( y = 0) plane for different parity-signature blocks ( π, r y ) in Dy at (cid:126) ω = 0 . are most important. For the protons, the main contribu-tions to ∆ τ come from the N = 6 states 6 , , , and 6 (for π = +) and [541]1/2 (for π = − ). It is seen that thes.p. contributions shown in Fig. 20(a-h) explain the be-havior of ∆ τ in Fig. 19. As discussed earlier in Sec. III D,characteristic nodal structures of ∆ τ along the z -axis pri-marily come from the evolution of rotation-aligned s.p.orbits with large N and n z , below the Fermi energy. Thefeatures in the direction of minor axis can be attributedto deformation-aligned s.p. states. n (+ , + i ) D n (+ , − i ) E n ( − , + i ) F n ( − , − i ) G +6 p (+ , + i ) H +6 p (+ , − i ) I [541]1 / p ( − , + i ) J [541]1 / p ( − , − i ) K × × x I P z I P FIG. 20. Contributions to ∆ τ (in fm − ) in the x - z ( y = 0)plane for different parity-signature blocks from individual s.p.Routhians in Dy at (cid:126) ω = 0 . π = + , r = + i neutron levels [651]1/2, [642]5/2, [413]5/2, and [411]1/2 with r = + i (a) and − i (b) that appear below the N = 86 shell gapin Fig. 1 (see Fig. 1 of Ref. [45] for the asymptotic (Nilsson)quantum numbers [ N n z Λ]Ω of s.p. levels in SD
Dy); the N = 7 neutron intruder states 7 (c) and 7 (d); the N = 6proton intruder states 6 + 6 (e) and 6 + 6 (f); and the[541]1/2 (g) and [541]1/2 (h) proton states. IV. CONCLUSIONS
In this study, we extended the concept of the fermionlocalization function to anisotropic, spin-unsaturated,and spin-polarized systems. In particular, we consid-ered the case of broken time-reversal symmetry. Wedemonstrated that in the general case of rotating de-formed systems three localization measures C qs µ ( r ), with µ = x, y, z , which depend on the anisotropy of the spindistribution, can be defined.We used the NLF to interpret the results of crankedSkyrme-HF calculations for rotating nuclei, especiallyto study the interplay between collective and s.p. mo-tion. While the standard probabilistic interpretation ofthe NLF cannot be easily extended to the case of self-consistent symmetries associated with point groups, suchas signature or simplex, there are no conceptual problemswhen viewing the NLF as a measure of the excess of ki-netic energy density due to the Pauli principle.The localization function involves various local densi-ties, among which the current density j , density gradi-ent ∇ ρ , and spin-current tensor density J are appreciableonly in the surface region. If one neglects these surfaceterms, one can define a simplified localization measure C τ , which involves only the kinetic energy density τ andthe Thomas-Fermi kinetic energy density τ TF . We arguethat C τ is amplified by the out-of-phase spatial oscillationof τ and τ TF attributed to the specific nodal structure ofhigh- N s.p. states.To show the usefulness of the extended NLF, we carriedout the Skyrme-CHF analysis of the superdeformed yrastband of Dy. As the rotational frequency increases,rotationally aligned s.p. states with high- N and high- n z produce a characteristic oscillating pattern in the NLFalong the major axis of the nucleus, while the patternvariations along the minor axis come from deformation-aligned s.p. states close to the Fermi energy.Our CHF and CHO results demonstrate that C τ is anexcellent indicator of the nuclear response to collectiverotation. Many applications of the NLF to the visual-ization of nuclear rotational and vibrational modes andtime dependent processes [18, 36, 66–71] are envisioned,especially after incorporating pairing correlations via theHFB extension of the formalism. One can also considerapplying the concept of the NLF beyond the mean-fieldapproach. In particular, since the kinetic energy den-sity can be computed within realistic A -body frameworks[72], studies of many-body correlations with the help of C τ could offer new perspectives.Finally, let us note that while in the usual atomic ap-plications the current term in Eq. (13) is ignored, the con-tribution to ELF from the spin-current tensor density J isexpected to be nonzero in relativistic superheavy atoms.For instance, the spin-orbit splitting for the valence 7 p or-bital of the element Og ( Z = 118) is predicted to be verylarge, around 10 eV [14, 73]. While Og is believed to bea spin-saturated system (the whole 7 p shell is filled), thisis not the case for, e.g., Fl ( Z = 114, 7 p / shell empty)2for which J and the resulting spin-orbit current shouldbe consider when analyzing the corresponding ELF. ACKNOWLEDGMENTS
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