Pseudospin symmetry and octupole correlations for multiple chiral doublets in 131Ba
aa r X i v : . [ nu c l - t h ] A ug Pseudospin symmetry and octupole correlations for M χ D in Ba Y. P. Wang, Y. Y. Wang, and J. Meng
1, 21
State Key Laboratory of Nuclear Physics and Technology,School of Physics, Peking University, Beijing 100871, China Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan (Dated: August 7, 2020)
Abstract
A Reflection-Asymmetric Triaxial Particle Rotor Model (RAT-PRM) with three quasiparticlesand a reflection-asymmetric triaxial rotor is developed, and applied to investigate the observedmultiple chiral doublets (M χ D) candidates with octupole correlations in
Ba, i.e., two pairs ofpositive-parity bands D3-D4 and D5-D6, as well as one pair of negative-parity bands D7-D8. Theenergy spectra, the energy staggering parameters, the B ( M /B ( E
2) ratios, and the B ( E /B ( E χ D candidates are examined by the azimuthal plots , and the evolution of chiral geometry with spin is clearly demonstrated. Theintrinsic structure for the positive-parity bands is analyzed and the possible pseudospin-chiralquartet bands are suggested. . INTRODUCTION Chirality is a phenomenon of general interest in chemistry, biology and particle physics.Since the concept of chirality in atomic nuclei was first proposed by Frauendorf and Meng [1],many efforts have been made to understand chiral symmetry and its spontaneous breakingin atomic nuclei, see e.g., reviews [2–6].Based on the adiabatic and configuration fixed constrained triaxial covariant densityfunctional theory (CDFT) [7], a new phenomenon named multiple chiral doublets (M χ D) ispredicted [8]. The chiral doublet bands are a pair of nearly degenerate ∆ I = 1 bands withthe same parity. M χ D suggest that more than one pair of chiral doublet bands can exist inone single nucleus. The first experimental evidence for M χ D is reported in
Ce [9], andfollowed by
Rh [10], Br [11],
Nd [12] and
Tl [13], etc. More than 60 chiral doubletcandidates in around 50 nuclei (including 10 nuclei with M χ D) have been reported in the A ∼
80, 100, 130 and 190 mass regions [13–16].Theoretically, nuclear chirality has been investigated with many approaches, including thetriaxial particle rotor model (PRM) [1, 17–21], the tilted axis cranking (TAC) model [1, 22–25], the TAC approach with the random phase approximation (TAC+RPA) [26, 27] and thecollective Hamiltonian (TAC+CH) [28–30], the interacting boson-fermion-fermion model(IBFFM) [31], the generalized coherent state model [32] and the projected shell model(PSM) [33–37].The PRM is a quantal model coupling the collective rotation and the single-particle mo-tions in the laboratory reference frame, and describes directly the quantum tunneling andenergy splitting between the doublet bands. Various versions of PRM have been developedfor the nuclear chirality. For odd-odd nuclei, the triaxial PRM with 1-particle-1-hole config-uration is mainly used [1, 17, 18, 38]. To simulate the effects of many valence nucleons, thetriaxial PRM with 2-qusiparticle configuration was developed with pairing correlations takeninto account by the Bardeen-Cooper-Schrieffer (BCS) approximation [19, 39–44]. For odd-Aand even-even nuclei, the many-particle-many-hole versions of triaxial PRM with nucleonsin 2 j shells [20, 45], 3 j shells [9, 10, 46, 47], and 4 j shells [21], have been developed. Therecently observed octupole correlations between the M χ D candidates in Br [11] stimulatethe development of the reflection-asymmetric triaxial PRM (RAT-PRM) with both triaxialand octupole degrees of freedom [48]. 2he RAT-PRM in Ref. [48] is based on a reflection-asymmetric triaxial rotor coupled withtwo quasiparticles. In Ref. [16], the M χ D candidates with octupole correlations, i.e., twopairs of positive-parity bands D3-D4 and D5-D6, as well as one pair of negative-parity bandsD7-D8 are observed in nucleus
Ba. The positive-parity bands, D3-D4 and D5-D6 with πh / ( g / , d / ) ⊗ νh / , and the negative-parity bands D7-D8 with πh / ⊗ νh / , are builton 3-quasiparticle configurations. Furthermore, the four nearly degenerate positive-paritybands are suggested to be the pseudospin-chiral quartet bands [16].In this work, a RAT-PRM with three quasiparticles and a reflection-asymmetric triaxialrotor is developed, and applied to investigate the observed M χ D with octupole correlationsin
Ba. The model is introduced in Sec. II, and the numerical details are presented inSec. III. The energy spectra, the electromagnetic transitions, and the angular momentumorientations are calculated and compared with the data available in Sec. IV, and a summaryis given in Sec. V.
II. FORMALISM
The total RAT-PRM Hamiltonian can be expressed asˆ H = ˆ H core + ˆ H π intr . + ˆ H ν intr . , (1)where ˆ H core is the Hamiltonian of a reflection-asymmetric triaxial rotor and ˆ H π ( ν )intr . is theintrinsic Hamiltonian for valence protons (neutrons) in a reflection-asymmetric triaxiallydeformed potential.Similar as Ref. [48], the core Hamiltonian generalized straightforwardly from the reflection-asymmetric axial rotor [49] reads,ˆ H core = X k =1 ˆ R k J k + 12 E (cid:0) − (cid:1) (1 − ˆ P c ) , (2)with ˆ R k = ˆ I k − ˆ J k , the indices k = 1 , , R k , ˆ I k , and ˆ J k denote the angular momentum operators for the core, thenucleus and the valence nucleons, respectively. The moments of inertia for irrotational flow, J k = J sin ( γ − kπ/ E (0 − ) is viewed as theexcitation energy of the virtual 0 − state [49]. The product of the core parity operator ˆ P c p π ( ν ) gives the total parity operatorˆ P . The intrinsic Hamiltonian for valence nucleons is [48]ˆ H π ( ν )intr . = ˆ H π ( ν )s . p . + ˆ H pair = X τ> (cid:16) ε τ π ( ν ) − λ π ( ν ) (cid:17) (cid:16) a † τ π ( ν ) a τ π ( ν ) + a † τ π ( ν ) a τ π ( ν ) (cid:17) − ∆2 X τ> (cid:16) a † τ π ( ν ) a † τ π ( ν ) + a τ π ( ν ) a τ π ( ν ) (cid:17) , (3)where λ π ( ν ) denotes the Fermi energy of proton (neutron), ∆ is the pairing gap parameter,and the single-particle energy ε τ π ( ν ) is obtained by diagonalizing the reflection-asymmetrictriaxial Nilsson Hamiltonian ˆ H π ( ν ) s.p. [48]. The indices τ π ( ν ) and τ π ( ν ) represent the proton(neutron) single-particle state and its corresponding time reversal state, respectively.Taking into account the pairing correlations by the BCS approach [19, 48, 50, 51], theintrinsic Hamiltonian becomesˆ H π ( ν )intr . = X τ π ( ν ) > ε ′ τ π ( ν ) (cid:16) α † τ π ( ν ) α τ π ( ν ) + α † τ π ( ν ) α τ π ( ν ) (cid:17) , (4)with the quasiparticle energies ε ′ τ π ( ν ) = r(cid:16) ε τ π ( ν ) − λ π ( ν ) (cid:17) + ∆ .In reference [48], the RAT-PRM with a quasi-proton and a quasi-neutron coupled with areflection-asymmetric triaxial rotor is developed. In order to describe the M χ D bands withmore than two quasiparticles, the intrinsic wavefunction for a system with Z valence protonsand N valence neutrons in RAT-PRM is generalized as, χ = Z Y i =1 α † τ πi ! N Y j =1 α † τ νj ! | i . (5)For Ba, Z = 2 and N = 1.The total Hamiltonian ˆ H is invariant under the operation ˆ S = ˆ P c e iπ ˆ R , which is thereflection with respect to the plane perpendicular to 2-axis. It can be diagonalized in thesymmetrized strong-coupled basis with good parity and angular momentum [48], | Ψ IMK ± i = 12 p δ K δ χ, ¯ χ (cid:16) S (cid:17) | IM K i ψ ± , (6)where | IM K i = q I +18 π D I ∗ MK is the Wigner function, ¯ χ = ˆ S χ , and ψ ± are the intrinsicwavefunctions with good parity, ψ + = (1 + ˆ P ) χ Φ a = (1 + ˆ P c ˆ p π ˆ p ν ) χ Φ a , (7)4 − = (1 − ˆ P ) χ Φ a = (1 − ˆ P c ˆ p π ˆ p ν ) χ Φ a . (8)Here, Φ a describes the same orientation in space between the core and the intrinsic single-particle potential, and χ Φ a is the core-coupled intrinsic wavefunction.The diagonalization of the total Hamiltonian ˆ H gives the eigenvalues and eigenfuctions.Then one can calculate the reduced electromagnetic transition probabilities [48], angularmomentum components [48], and the probability profile for the orientation of the angularmomentum, i.e., azimuthal plot [35, 52], etc. III. NUMERICAL DETAILS
For the M χ D candidates with octupole correlations observed in nuclei
Ba, twopairs of positive-parity bands D3-D4 and D5-D6 are suggested to have the configuration πh / ( g / , d / ) ⊗ νh / [16]. For this configuration, the microscopic configuration-fixedtriaxial covariant density functional theory [8] with PC-PK1 [53] gives the quadrupole de-formation β = 0 .
22 and γ = 27 . ◦ . An octupole deformation β = 0 .
02 is adopted toinvestigate the octupole correlations in the present RAT-PRM calculations.With the deformation parameters above, the reflection-asymmetric triaxial Nilsson Hamil-tonian is solved by expanding the wavefunction by harmonic oscillator basis [54]. Accordingto the configuration suggested in Ref. [16], the Fermi energies of proton and neutron arechosen as λ π = 43 .
86 MeV and λ ν = 48 .
76 MeV, respectively. The single-particle spaceincludes seven above and six below the Fermi level. The pairing correlation is taken intoaccount by the empirical formula ∆ = 12 / √ A MeV.The moment of inertia J = 19 ~ / MeV and the core parity splitting parameter E (0 − ) =2 . g R = Z/A for the core and g π ( ν ) = g l + ( g s − g l ) / (2 l + 1) for the proton (neutron) [55, 56]. For the calculations of the electric transitions,the empirical intrinsic dipole moment Q = π R Zeβ and quadrupole moment Q = √ π R Zeβ are taken with R = 1 . A / fm [48].5 S i ng l e P a r t i c l e E n e r g y [ M e V ] g [deg] b b (a) (b) (c) |1æ|2æ|3æ|4æ|5æ|6æ|7æ|8æ |9æ|10æ |11æ|12æ|13æ Proton l p FIG. 1. The proton single-particle levels, obtained by diagonalizing the reflection-asymmetrictriaxial Nilsson Hamiltonian in Eq. (3), as functions of the deformation parameters (a) β ( γ = 0 ◦ , β = 0), (b) γ ( β = 0 . β = 0), and (c) β ( β = 0 . γ = 27 . ◦ ). The red and blue representthe positive and negative parity. The solid lines represent the chosen proton single-particle spacein the RAT-PRM calculation. The green dash dot line represents the proton Fermi energy. IV. RESULTS AND DISCUSSION
By diagonalizing the reflection-asymmetric triaxial Nilsson Hamiltonian in Eq. (3), onecan obtain the proton (neutron) single-particle levels. As an example, Fig. 1 shows theproton single-particle levels as functions of the deformation parameters. The proton single-particle space, represented by the solid lines in Fig. 1, includes seven above and six belowthe Fermi level. The quasiparticle states are obtained from the single-particle states atthe deformation β = 0 . , γ = 27 . ◦ and β = 0 .
02 by the BCS approximation. Fromthese quasiparticle states, the intrinsic wavefunctions with good parity can be constructedfrom Eqs. (5), (7) and (8). Then the intrinsic wavefunctions with good parity are usedto generate the symmetrized strong-coupled basis in Eq. (6). By diagonalizing the total6 E ( I ) - E [ M e V ] (a) (b) (c)(d) (e) (f)(h) (i) (j) S ( I ) [ ke V / h ] B ( M ) / B ( E ) [ m N / e b ] D3 (exp.) D3 (exp.)
D3 (cal.) D4 (cal.)
Spin I [ h ] D5 (exp.) D6 (exp.)
D5 (cal.) D6 (cal.) D7 (exp.) D8 (exp.)
D7 (cal.) D8 (cal.)
FIG. 2. The energies E ( I ), the energy staggering parameters S ( I ) = [ E ( I ) − E ( I − / I , andthe B ( M /B ( E
2) ratios for the two pairs of positive-parity bands D3-D4 (left panels) and D5-D6(middle panels) as well as for the negative-parity doublet bands D7-D8 (right panels) in
Ba byRAT-PRM (lines) in comparison with the experimental data (symbols) [16]. The bandhead energyof band D3 at I = 23 / ~ is taken as a reference. RAT-PRM Hamiltonian on the strong-coupled basis, the energies and eigenfunctions at eachspin I are obtained.In Fig. 2, the energies E ( I ), the energy staggering parameters S ( I ) = [ E ( I ) − E ( I − / I ,and the B ( M /B ( E
2) ratios calculated by the RAT-PRM for the two pairs of positive-parity doublet bands D3-D4 and D5-D6 as well as for the negative-parity doublet bands D7-D8 are shown in comparison with the data available [16]. The bandhead of band D3 is taken7 -4 -3 -2 -1 B ( E ) / B ( E ) [ - f m - ] Spin I [ h ] exp. cal. FIG. 3. The calculated B ( E /B ( E
2) ratios between the interband E → D3) and the intraband E as reference. The experimental spectra, the energy staggering parameters S ( I ), and the B ( M /B ( E
2) ratios for bands D3-D8 are reproduced well by the RAT-PRM calculations.For each pair of the doublet bands, the spectra are nearly degenerate and the correspoding B ( M /B ( E
2) ratios are similar, which are the characteristics for the chiral rotation [40].Since the octupole degree of freedom is included in the present RAT-PRM calculations,the electric dipole transition probabilities B ( E
1) between the positive- and negative-paritybands can be calculated. In Fig. 3, the calculated B ( E /B ( E
2) ratios between the in-terband E → D3) and the intraband E B ( E /B ( E
2) ratios arein general determined by that of the B ( E
1) values. The experimental data are available atspin I = 29 / ~ and 31 / ~ [16] and can be well reproduced. Here the value of β is chosen toreproduce the experimental B ( E /B ( E
2) data. Changing the β by 10%, the influences onthe energies, the staggering parameters, the B ( M /B ( E
2) ratios, and the chiral geometry8 f [ d e g ] I=29/2 h I=35/2 h I=41/2 h
30 60 90 q [deg]
D3 D4
FIG. 4. The azimuthal plots , i.e., probability distribution profiles for the orientation of the angularmomentum on the ( θ, φ ) plane, calculated for bands D3 and D4 at I = 29 / ~ , 35 / ~ , and 41 / ~ .The black star represents the position of a local maximum. are negligible, while the B ( E
1) values are changed by about 20%.In order to investigate the chiral geometry for the three chiral doublet candidates, the azimuthal plots [35], i.e., the probability distribution profiles for the orientation of the an-gular momentum on the ( θ, φ ) plane, are shown in Figs. 4 - 6. The polar angle θ is the anglebetween the angular momentum and the intrinsic 3-axis, and the azimuthal angle φ is theangle between the projection of the angular momentum on the intrinsic 1-2 plane and theintrinsic 1-axis. For γ = 27 . ◦ , the 1, 2, and 3 axes are respectively the intermediate ( i ),short ( s ), and long ( l ) axes.The azimuthal plots for bands D3-D4 are shown in Fig. 4. At spin I = 29 / ~ , the azimuthal plot for band D3 has a single peak at (65 ◦ , ◦ ), which suggests that the angularmomentum stays within the s - l plane, in accordance with the expectation for a 0-phononstate. The azimuthal plot for band D4 shows a node at (65 ◦ , ◦ ), and two peaks at (70 ◦ , ◦ )and (70 ◦ , ◦ ). The existence of the node and the two peaks supports that the state in bandD4 can be interpreted as a 1-phonon chiral vibration across the s - l plane on the state in9 f [ d e g ] I=29/2 h I=31/2 h I=33/2 h q [deg] D5 D6
30 60 90
FIG. 5. Same as Fig. 4, but calculated for bands D5 and D6 at 29 / ~ , 31 / ~ , and 33 / ~ . band D3.At spin I = 35 / ~ , the azimuthal plots for bands D3 and D4 are similar, with two peaksat (80 ◦ , ◦ ) and (80 ◦ , ◦ ) for band D3, and (70 ◦ , ◦ ) and (70 ◦ , ◦ ) for band D4. Thisshows the feature of static chirality.At spin I = 41 / ~ , the peaks of the azimuthal plot for band D3 are at (90 ◦ , ◦ ) and(90 ◦ , ◦ ), namely along the i axis. The peaks of the azimuthal plot for band D4 locate(85 ◦ , ◦ ) and (85 ◦ , ◦ ), which are similar to those at I = 35 / ~ but approaching the i -axis. The tendency from chiral rotation to principle axis rotation at high spins is shown.In Fig. 5, the azimuthal plots for bands D5-D6 are shown. Similar as the evolution of thechiral geometry for bands D3-D4 in Fig. 4, bands D5-D6 show the chiral vibration along the φ direction at I = 29 / ~ , static chirality at I = 31 / ~ , and a tendency to a princpal axisrotation around the i axis at I = 33 / ~ .In Fig. 6, the azimuthal plots for negative-parity doublet bands D7-D8 are shown. At I = 25 / ~ , the peak of the azimuthal plot for band D7 is at (70 ◦ , ◦ ), which is a planarrotation within the s - l plane. The angular momentum is mainly contributed by the valenceprotons and the neutron hole. For band D8, the peaks are at (60 ◦ , ◦ ) and (60 ◦ , ◦ ), which10 f [ d e g ] D7 I=25/2 h I=31/2 h I=35/2 h I=39/2 h D8
30 60 90 q [deg]
30 60 90 30 60 90
FIG. 6. Same as Fig. 4, but calculated for bands D7 and D8 at I = 25 / ~ , 31 / ~ , 35 / ~ and39 / ~ . corresponds to an i - l planar rotation. The angular momentum is mainly contributed by thecore and the valence neutron hole. The contribution of the valence protons are negligible.For I ≥ / ~ , similar as the evolution of the chiral geometry for bands D3-D4 in Fig. 4,bands D7-D8 show the chiral vibration along the φ direction at I = 31 / ~ , static chiralityat I = 35 / ~ , and a tendency to a princpal axis rotation about the i axis at I = 39 / ~ .The positive-parity M χ D candidates observed in nuclei
Ba, bands D3-D4 and D5-D6, are suggested to have the configuration πh / ( g / , d / ) ⊗ νh / [16]. One possibleM χ D mechanism is that the “yrast” and “excited” chiral doublets are built on the sameconfiguration, similar as
Rh in Ref. [10]. Another possible mechanism is the appearanceof the pseudospin-chiral quartet bands due to the pseudospin partners ( g / , d / ).To pin down the mechanism for the positive-parity M χ D candidates in
Ba, the mainoccupation of the valence protons are investigated. In Fig. 7, the occupation of the single-particle level | c τ | is shown with τ = 5 , , , I = 31 / ~ for bands D3-D6, which iscalculated from the corresponding strong-coupled basis including | τ i in its intrinsic wave-function. The occupation patterns are similar for each chiral doublet bands. Examiningmore carefully, the top components for bands D3-D4 are levels | i and | i , while for bandsD5-D6 are levels | i and | i .As shown in Fig. 1, the levels | i and | i stem from 1 g / , the level | i stems from11 .00.10.20.30.40.50.60.70.80.9 | c t | D3 D4 D5 D6
I=31/2 h |5æ |6æ|7æ|8æ FIG. 7. The occupation of the single-particle level | c τ | contributed from the corresponding strong-coupled basis which includes | τ i in its intrinsic wavefunction with τ = 5 , , , I = 31 / ~ forbands D3-D6. h / , and the level | i stems from 2 d / . Figure 8 shows the main components N lj
Ω forthe levels | i , | i , | i , and | i in the present calculation with β = 0 . , γ = 27 . ◦ , β =0 .
02. The dominent components for the levels | i and | i are respectively 1 g / and 1 h / .For the levels | i and | i , there is a strong mixture of 1 g / and 2 d / . Therefore thedominent component of the intrinsic wavefunctions is πh / g / ⊗ νh / for bands D3-D4,in comparison with πh / ( g / , d / ) ⊗ νh / for bands D5-D6.Thus the positive-parity M χ D candidates involve the pseudospin partners ( g / , d / ) andare suggested to be pseudospin-chiral quartet bands. V. SUMMARY
A Reflection-Asymmetric Triaxial Particle Rotor Model (RAT-PRM) for three quasi-particles coupled with a reflection-asymmetric triaxial rotor is developed, and applied toinvestigate the observed multiple chiral doublets (M χ D) candidates with octupole correla-tions in
Ba, i.e., two pairs of positive-parity bands D3-D4 and D5-D6, as well as one pairof negative-parity bands D7-D8.The calculated energy spectra, energy staggering parameters S ( I ) = [ E ( I ) − E ( I − / I , B ( M /B ( E
2) ratios and B ( E /B ( E
2) ratios reproduce well the data available [16]. The12 .00.10.20.30.40.50.60.70.8 | c N lj W | |5æ |6æ |7æ |8æ b =0.22, g=27.1 (cid:176) , b =0.02 Proton
FIG. 8. The main components
N lj
Ω of the proton single-particle levels | τ i with τ = 5 , , , chiral geometries for the three pairs of chiral doublet bands are clearly illustrated by the azimuthal plots . They show the chiral vibration at low spins, static chirality at intermediatespins, and a tendency to a princpal axis rotation around the i axis at high spins. For bandD8, an i - l planar rotation occurs before the chiral vibration. The possible pseudospin-chiralquartet bands are suggested for the positive-parity M χ D candidates D3-D6 based on theirintrinsic structure.
ACKNOWLEDGMENTS
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