Covariant density functional theory for nuclear chirality in 135 Nd
aa r X i v : . [ nu c l - t h ] S e p Covariant density functional theory for nuclear chirality in Nd J. Peng ∗ and Q. B. Chen † Department of Physics, Beijing Normal University, Beijing 100875, China Physik-Department, Technische Universit¨at M¨unchen, D-85747 Garching, Germany
The three-dimensional tilted axis cranking covariant density functional theory (3D-TAC CDFT) is used tostudy the chiral modes in
Nd. By modeling the motion of the nucleus in rotating mean field as the interplaybetween the single-particle motions of several valence particle(s) and hole(s) and the collective motion of a core-like part, a classical Routhian is extracted. This classical Routhian gives qualitative agreement with the 3D-TACCDFT result for the critical frequency corresponding to the transition from planar to aplanar rotation. Based onthis investigation a possible understanding of tilted rotation appearing in a microscopic theory is provided.
PACS numbers: 21.10.Re, 21.60.Jz, 21.10.Pc, 27.60. + j Triaxial nucleus, shaped like a kiwi with all three principalaxes unequal, has drawn considerable attention over the years.While being rare in the ground state [1], it becomes more com-mon at high spin [2]. For high spin, the orientation of angularmomentum vector relative to the triaxially deformed densitydistribution becomes a very important concept [3]. The an-gular momentum may align along one of principal axes (prin-cipal axis rotation), lie in one of the principal planes definedby two principal axes (planar rotation), and deviate from theprincipal planes (aplanar rotation). Note that the second andthe third motions are also called as tilted rotation.For tilted rotation, its physical interpretation was given inRef. [4] by studying the model system of one h / particleand one h / hole coupled to a triaxial rotor with the irrota-tional flow type of moments of inertia (MoIs). In detail, thehigh- j particle / hole tends to align along the short / long ( s - / l -)axis because its torus-like / dumbbell-like density distributionhas the maximal overlap with the triaxial core in the medium-long / medium-short ( ml - / ms -) plane. The triaxial rotor tends toalign along the medium ( m -) axis since it is of the largest MoI.As a consequence, the total angular momentum will firstly liein the sl -plane and with increasing spin be tilted towards the m -axis, i.e., showing a transition from planar to aplanar rota-tion.The occurrence of aplanar rotation manifests itself as theobservation of a pair of nearly degenerated ∆ I = ff er by chirality, i.e., a left- and a right- handedsystem. Correspondingly, the excitation modes based on theaforementioned planar and aplanar rotations are called as chi-ral vibration and chiral rotation, respectively.As is well-known, the chirality is a phenomenon existingcommonly in nature, such as, the spiral arm of the nebula,the spirals of snail shells, and the handedness of amino acids.Therefore, the propose [4] and observation [5] of chirality atnuclear high spin state caused quite a stir among nuclear struc- ∗ Electronic address: [email protected] † Electronic address: [email protected] ture physics [61], and has intrigued lots of investigations fromboth theoretical and experimental aspects. Up to now, morethan 50 chiral nucleus candidates spread in the mass regionsof A ∼
80, 100, 130, and 190 have been reported, see, e.g.,data tables [6].As mentioned above, the nuclear chirality was proposedbased on the study of angular momentum vector geometryof a phenomenological model system with a particle and ahole coupled to a triaxial rotor [4]. This model system re-veals the duality of the single-particle and collective prop-erties of atomic nuclei and has been used extensively in thedevelopment of (triaxial) particle rotor model (PRM) [7, 8].By the newly developed many-particle-many-hole PRM (e.g.,Refs. [9–19]), the understanding of the nuclear chirality isdeeply entrenched.To verify the prediction by the model system, microscopicthree-dimensional tilted axis cranking (3D-TAC) calculationsin a hybrid Woods-Saxon and Nilsson model combined withthe shell correction method were carried out [20]. In the3D-TAC calculation, with the mean field approximation forthe two body interaction, all of the nucleons are treated onthe same footing. It allows also for an arbitrary orienta-tion of the angular momentum vector in the intrinsic frame.The solution for each rotational frequency ω is obtained self-consistently by minimizing the total Routhian surface, whichis the energy surface of the rotating nucleus in the intrinsicframe, with respect to the orientation of angular velocity vec-tor ω = ω (sin θ cos ϕ, sin θ sin ϕ, cos θ ). Here, usually, θ is thepolar angle between the ω and l -axis, and ϕ is the azimuthalangle between the projection of ω onto the ms -plane and the s -axis. The authors of Ref. [20] found that when cranking a re-alistic system with triaxially deformation and high- j particle-hole configuration, θ , ± π/ ϕ = ± π/ ω , and θ , ± π/ ϕ , ± π/ ω (denoted as ω crit ).Subsequently, an analytical formula of ω crit , called as thecritical frequency, was derived in Ref. [21]. In the derivation,using the aforementioned model system, the particle and holeare further assumed to align rigidly along the s - and l - axis,respectively. Further self-consistent calculations by the 3D-TAC Skyrme-Hartree-Fock method show that ω crit depends onthe e ff ective interaction used [21, 22]. Very recently, 3D-TACcalculations based on covariant density functional theory (3D-TAC CDFT) [23] also support the existence of ω crit .In the 3D-TAC calculations, the model system that valencenucleons coupled to a triaxial rotor is not necessary to be as-sumed a priori, since all of the nucleons are treated on thesame footing in the rotating mean field. Then a straightfor-ward question comes: which part of nucleon(s) drives thetotal nuclear system to exhibit an aplanar rotation? Inspiredby the entrenched picture in model system, we might wonderwhether there exists really a hidden collective core-like partand this “core” plays a role driving the rotational axis towardto the m -axis to stimulate the aplanar rotation. In this work,we try to figure out this core-like part and based on it pro-vide a possible understanding of tilted rotation appearing inthe microscopic 3D-TAC calculations.As an example, the chiral modes in nucleus Nd will bestudied. The reasons for studying
Nd are as follows. Ex-perimentally, this nucleus is a possible candidate nucleus withmultiple chiral doublet (M χ D) phenomenon [24, 25] as twopairs of chiral doublet bands were reported. One of them(labeled as bands D5 and D6 in the current work), built onthe configuration π h / ⊗ ν h − / (labeled as config2), is thefirst reported chiral doublet bands of odd- A nuclei [26] andwith rare lifetime measurement results [27]. The other one(labeled as bands D3 and D4), built on the configuration π [ h / ( gd ) ] ⊗ ν h − / (labeled as config1), is just establishedvery recently [28]. Theoretically, Nd has drawn attentionsfrom various di ff eren kinds of models, such as the microscopic3D-TAC model based on hybrid of Woods-Saxon and Nilssonpotential combined with the shell correction method [26] andthe 2D-TAC CDFT [29], the beyond cranking mean field ran-dom phase approximation (RPA) [27], the algebraic interact-ing boson-fermion model (IBFM) [30], and the phenomeno-logical PRM [12, 28]. Therefore, the study of Nd is a mat-ter of general interest and has double meaning of experimentaland theoretical aspects. In this work, the 3D-TAC CDFT willbe further used to study the chirality in this nucleus.The covariant density functional theory (DFT) based on themean field approach has played an important role in a fullymicroscopic and universal description of a large number ofnuclear phenomena [31–35]. In order to describe nuclear ro-tation, covariant DFT has been extended with the crankingmode [23, 36–39]. Using these extended cranking covariantDFT, lots of rotational excited states has been described well,such as superdeformed bands [36], magnetic [37–41] and anti-magnetic rotations [42–45], etc. For chirality in nuclei, M χ Dphenomenon has been suggested and investigated based onconstrained triaxial covariant DFT calculations [24, 46–51].In particular, the newly developed three-dimensional tiltedaxis cranking (3D-TAC) covariant DFT has been success-fully applied for the chirality in
Rh [23],
Nd [52] and
Ag [53].Our microscopic calculations are performed using 3D-TACCDFT [23, 53] with the e ff ective point-coupling interactionPC-PK1 [54], and a three-dimensional Cartesian harmonic os-cillator basis with ten major shells are used to solve the Diracequation. The pairing correlation is neglected in the currentinvestigation, but one has to bear in mind it could have someinfluences on the critical frequency [21] and the descriptions of total angular momentum and B ( M
1) values [29].As mentioned above, in 3D-TAC calculations, the orienta-tion of the angular velocity ω with respect to the three princi-pal axes ( θ, ϕ ) is determined in a self-consistent way by min-imizing the total Routhian. In these minima with ( θ min , ϕ min ),the deformation parameters ( β , γ ) are about (0 . , ◦ ) forconfig1 and (0 . , ◦ ) for config2, respectively. They changewithin ∆ β = .
02 and ∆ γ = ◦ , exhibiting a stability with therotation. The θ min varies from 50 ◦ to 68 ◦ for config1 and from63 ◦ to 66 ◦ for config2 driven by rotation.Since the sign of ϕ can be used to characterize the chiralityof the rotating system [55], here we can only focus on howdoes the total Routhian behave with ϕ . By minimizing the to-tal Routhians with respect to θ for given ϕ , the total Routhiancurves are shown in Fig. 1 for two configurations at severaltypical rotational frequencies. One can see that all the curvesare, analogy to the schematic picture of left- and right- hand,symmetrical with the ϕ = ±| ϕ | for a given θ are identical on theenergy. -80 -60 -40 -20 0 20 40 60 80-150-100-50050100150200250 = . = . config1 config2 R ou t h i a n ( ke V ) (deg) = . = . = . = . = . Nd FIG. 1: Total Routhian curves as functions of the azimuth angle ϕ calculated by 3D-TAC CDFT for configurations π [ h / ( gd ) ] ⊗ ν h − / (config1) and π h / ⊗ ν h − / (config2) in Nd. All curves are nor-malized at ϕ =
0. The minima of the curves are denoted by the balls.
For both configurations, the Routhian curves are rathersteep at low ω , and become softer with respect to ϕ with in-creasing ω . The ϕ min = sl -plane for the yrast band. At slightly larger energy, theangular momentum J in fact could, according to the beyondTAC mean field approximation investigations [27, 55–57], ex-ecute harmonic oscillation with respect to the sl -plane and en-ter into the left- and right- sector back and forth. This motion,generating the yrare band, is called as chiral vibration [5].For config1, there appears two degenerate minima ( | ϕ min | ∼ ◦ ) on the Routhian curve at ~ ω = .
50 MeV. However, thecurve is rather flat in the region − ◦ ≤ ϕ ≤ ◦ . For higher ~ ω , the barrier of the Routhian at ϕ = ∼
150 keV at ~ ω = . . ~ .Convergent results of configuration-fixed calculations couldbe attained only up to ~ ω = .
55 MeV, corresponding to I ∼ ~ . By further increasing ω , a level crossing betweenthe neutron h / and h / orbits appears. This leads to a newconfiguration π h / ⊗ ν h / . Therefore, it might imply thatthe ω crit for config2 in the present 3D-TAC CDFT calculationis larger than ~ ω = .
55 MeV. For comparison, the ω crit is ~ ω = .
50 MeV in the 3D TAC calculation based on hybrid ofWoods-Saxon and Nilsson potential combined with the shellcorrection method in Ref. [26].The rotational motion of triaxial nuclei attains a chiral char-acter if the angular momentum has substantial projections onall three principal axes of the triaxially deformed nucleus [4].In a microscopic picture, the angular momentum comes fromthe individual nucleons in a self-consistent calculation. In or-der to check the mechanism behind the generation of the an-gular momentum, it is important to extract the contributionsof the individual nucleons to the angular momentum. The re-sults from 3D-TAC CDFT for config1 at the minima of theRouthian are presented as an example in Fig. 2. C o r e [ gd] h (a) s-axis C o r e [ gd] (b) m-axis A ngu l a r m o m e n t u m () h (d) s-axis h Spin I ( ) (e) m-axis proton (c) l-axis neutron (f) l-axis config1
FIG. 2: Contributions of the valence protons and neutrons in the h / and ( gd ) shells as well as the residual core-like part to the total protonand neutron angular momenta along the s -, m -, and l - axis for config1in Nd.
One observes that the contribution to the proton angularmomenta along the s -axis originates mainly from the high- j orbit, i.e., a proton filling at the bottom of h / orbit. Thevalence proton occupying in the ( gd ) shell orbit contributionsquite small along the three principal axes ( < ~ ). In contrast,a neutron sitting at the third h / shell from the top downwardcontributes an angular momentum of roughly 4 . ~ along the l -axis. When the rotational frequency increases, the contribu-tions of the valence protons and neutron change barely alongthe s - and l - axis, respectively.The equivalent core composed from the remaining nucle- ons now can be separated as two parts, i.e., proton and neu-tron parts. In more detail, the eight protons / neutrons in the( gd ) / h / shell contribute to the s -axis component, and theother nucleons make main contributions on the l -axis compo-nent. Both proton and neutron parts contribute together to the m -axis component after the ω crit . -2-10120 20 40 60-2-101 0 20 40 60 0 20 40 60 80 (a) =0.30 Total Core SP particle hole config1 (b) =0.50 C l ass i ca l R ou t h i a n ( M e V ) (c) =0.60 config2 (d) =0.30 (e) =0.40 (deg) (f) =0.55 FIG. 3: Classical Routhians as functions of ϕ for the total, core aswell as the valence particle(s), valence hole(s), and their summation(SP). All curves are normalized at ϕ =
0. The minimum of the totalRouthian is denoted by the ball.
Adopted the same ideas in Refs. [4, 58, 59], the modelstudy separates the nucleus into several valence nucleons anda collective core-like part. Correspondingly, the total classicalRouthian of the nucleus is calculated as E ′ Total = E ′ SP + E ′ core = X i = p,h ǫ ′ i − X k = s , m , l J k ω k , (1)in which the first term ǫ ′ p,h is the single-particle Routhian forthe valence particle(s) and hole(s), and the second term theRouthian for the core-like part. Inspired by this expression,we make an attempt on understanding the appearance of tiltedrotation shown in Fig. 1. Starting from 3D-TAC CDFT re-sults, we extract, intuitively, the values of ǫ ′ p,h in Eq. (1) fromthe obtained single-particle Routhian for those valence parti-cle(s) and hole(s) that the configuration information gives. Asa consequence, the “core” is composed by the remaining nu-cleons. Its angular momentum R core k is thus the summation ofthese remaining nucleons’ angular momentum. Correspond-ingly, its MoIs can be extracted as J k = R core k /ω k , and finallyits Routhian E ′ core is yielded by Eq. (1). The obtained resultsof these classical Routhians as functions of ϕ are shown inFig. 3.It is observed that the values of ǫ ′ p,h and E ′ core increase anddecrease with ϕ , respectively. The former indicates the va-lence particle(s) and hole(s) prefer to the sl -plane, and the lat-ter represents that the core-like part favors the m -axis. Theircompetitions determine the rotational orientation of the totalsystem. These features are consistent with previous modelstudies [4].For config1, after ~ ω = .
50 MeV, the steeper variation be-havior of E ′ core than those of E ′ SP provides stronger Coriolisforce and drives rotational axis deviating from the s - l plane tominimize the energy. The minima of total classical Routhianthus shift from zero to nonzero, which corresponds to the apla-nar solution, in line with the results shown in Fig. 1 though thedetailed value of ϕ min and the height of the barrier are di ff er-ent. Therefore, a transition from planar to aplanar rotation hasbeen displayed for config1.For config2, the increments in E ′ SP are larger than the decre-ment in E ′ core . This is mainly caused by the two aligned h / particles, which has a very large alignment along the s -axisand make their Routhians become very steep with the in-crease of ϕ . The minimum of total classical Routhian staysat ϕ min =
0, which corresponds to the planar regime. This isalso agreement with the microscopic results shown in Fig. 1.In the previous model studies [4, 12, 15, 16, 28], the tran-sition from planar to aplanar rotation is attributed to the MoIof the m -axis, with the assumption of irrotational flow type J irr k ∝ sin ( γ − k π/ J k = R core k /ω k ( k = s , m , l ) from the 3D-TAC CDFT results. We find that J k does not change much with the increase of ω and ϕ . In addi-tion, J m is indeed the largest for both configurations. In detail,e.g., at ~ ω = .
30 MeV, the ratio of J m : J s : J l at ϕ = .
00 : 0 .
65 : 0 .
50 for config1 and 1 .
00 : 0 .
78 : 0 . + states in even-even nuclei [60] and supports the assumption of J irr k in themodel study. In Ref. [21] the MoIs are extracted for the wholesystem from the 3D-TAC Skyrme-Hartree-Fock calculations,and the m -axis one is the largest as well. Furthermore, onenotes that the ratios of J s ( J l ) : J m for config2 is larger thanthat for config1. This makes, according to the formula of ω crit in Ref. [21], the planar rotation become more stable, and theaplanar rotation is not easy to be obtained as shown in Fig. 1.Therefore, we can possibly understand how dose the titledrotation occur in the 3D-TAC calculations. Though moving inthe same rotating mean field, some (unpaired) valence high- j nucleons are active with significant single-particle motion andact as valence particle(s) or hole(s), while the (paired) othersare not that active but exhibit collective behavior and form astable core-like part. This core-like part has finite MoIs alongthe three principal axis (as indication of triaxial deformation),and the m -axis one is the largest. At low ω , the collectivecore angular momentum is small and is driven to lie in the sl -plane by the strong Coriolis forces from the valence particle(s)and hole(s). With increasing ω , the gradual increasing coreangular momentum along the m -axis becomes comparable tothose of valence particle(s) and hole(s) along s - and l - axis andresults in the transition from planar to aplanar rotation.With the above studies, the comparisons between the 3D-TAC CDFT results and the available experimental data [26–28] are given in Fig. 4.As the calculations are carried out without additional ad-justable parameters and the theoretical bandhead energies ofboth configurations are shifted by the same value, the qualitiesof present reproduction are reasonable for energy spectra. In -0.50.00.51.01.52.0 E ( I )- . * I ( I + ) ( M e V ) (a) D3 (Exp)
D4 (Exp) D5 (Exp)
D6 (Exp) config1 config2 B ( M ) / B ( E ) ( N / e b ) (b) B ( M ) ( N ) (c) B ( E ) ( e b ) (d) Spin I ( )
FIG. 4: Calculated results by 3D-TAC CDFT in comparisons withthe available data [26–28] of doublet bands D3 and D4 and bands D5and D6 in
Nd: (a) energies minus a common rotor contribution, (b) B ( M / B ( E
2) ratios, (c) B ( M B ( E ff erences. addition, the larger experimental energy di ff erences betweenbands D6 and D5 than bands D4 and D3 at low spin regioncorrelate with their steeper behavior of Routhian as observedin Figs. 1 and 3.The calculated B ( M / B ( E
2) values of config1 show asteep falling behavior in the planar rotation regime, and thefalling tendency slows down after the transition to aplanar ro-tation. There is only one experimental value of B ( M / B ( E I = . ~ for band D3 [28], which is slightly smaller thanthose of config1. For band D5, one observes that the calcu-lated result with config2 shows a good agreement with thedata within the error bar after I = . ~ . However, it overes-timates the data in the lower spin region. It is caused by, asshown later, underestimating the B ( E
2) values from I = . . ~ .For B ( M jh / particle than config2, gives smaller B ( M
1) values. How-ever, it also shows a decreasing trend in the planar rotationregion. In the aplanar region, it increases a bit.The B ( E
2) value depends on the deformation parametersand the orientation angles θ min and ϕ min . As the deformationand θ min change slightly in config2, the calculated B ( E
2) val-ues are roughly constant, in line with the behavior of the ex-perimental value. However, the calculated values of B ( E B ( E
2) valuesincrease slowly for planar rotation while rapidly for aplanarrotation. This implies that, as shown in Figs. 2 and 3, the col-lective core-like part plays more and more important role withthe increase of spin. Thus further experimental e ff orts in par-ticular the lifetime measurement to extract B ( M
1) and B ( E Nd. The transition from planar to aplanar rota-tion is found for config 1 of band D3, while only the planarrotation for config 2 of band D5. By modeling the motionof the nucleus in rotating mean field as the interplay betweenthe single-particle motion of several valence particle(s) andhole(s) and the collective motion of a core-like part, a classi-cal Routhian is extracted. This classical Routhian gives quali-tative agreement with the 3D-TAC CDFT result of the criticalfrequency for the transition from to aplanar rotation. In ad-dition, the extracted MoIs of the core-like part has the largestvalue along the m -axis. Based on this investigation a pos-sible understanding of tilted rotation appearing in a micro-scopic theory is provided. Current study also indicates thee ff ects of core-like part indeed exists in the realistic rotatingnucleus and thus provides a justification for the assumptions of the phenomenological model, and more importantly rein-forces the duality of single-particle and collective propertiesin atomic nuclei. Acknowledgments
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