Projected total energy surface description for axial shape asymmetry in 172 W

Transcription

1 . Article. SCIENCE CHINA Physics, Mechanics & Astronomy November 014 Vol. 57 No. 11: doi: /s Projected total energy surface description for axial shape asymmetry in 17 W TU Ya 1*, CHEN YongShou *, GAO ZaoChun, YU ShaoYing 3 & LIU Ling 1 1 College of Physics Science and Technology, ShenYang Normal University, Shenyang , China; China Institute of Atomic Energy, Beijing 10413, China; 3 School of Science, Huzhou Teachers College, Huzhou , China Received March 11, 014; accepted April 5, 014; published online August 18, 014 The projected total energy surface (PTES) approach has been developed based on the triaxial projected shell model (TPSM) hybridized with the macroscopic microscopic method. The total energy of an atomic nucleus is decomposed into macroscopic, microscopic and rotational terms. The macroscopic and microscopic components are described with the liquid drop model and Strutinsky method, respectively, and the rotational energy is given by the TPSM, the term beyond the mean field. To test theory, the PTES calculations have been carried out for the yrast states of the well deformed rare earth nucleus 17 W, and the theoretical results are in good agreement with the experimental data. By using the equilibrium quardrupole deformations (ε andγ) determined by the PTES, the calculation of the transition quardrupole moment (Q t ) in function of spin also reproduces the experimental data. A comparison between the PTES and TRS methods has been made for theoretical and application uses. energy surface, projected shell model, triaxial deformation, transition quardrupole moment PACS number(s): 1.60.Ev, 1.10.Pc, q Citation: Tu Y, Chen Y S, Gao Z C, et al. Projected total energy surface description for axial shape asymmetry in 17 W. Sci China-Phys Mech Astron, 014, 57: , doi: /s Introduction The total routhian surface (TRS) method based on the cranked shell model (CSM) has been extensively used to describe the shape of highly rotating nuclei. Despite the many success of the TRS in the description of nuclear shapes, the disadvantages are clear in the fact that the angular momentum is not an adequate quantum number, but instead the rotational frequency is employed. The rotational frequency introduced in the hamiltonian permits the model having the semiclassical nature. In the description of a rotating quantum nuclear system the same rotational frequency may correspond to two different angular momenta, and therefore the quantum rotational state can not be described uniquely in the CSM *Corresponding author (TU Ya, tuya sy@16.com; CHEN YongShou, yschen@ciae.ac.cn) concept as well as in the TRS. For example, the interaction between the quasi-particle (q.p.) configurations is not properly considered in the TRS and thus the detailed description of the wave functions of the rotational states in the band mixing region becomes untenable or an unclear phenomena. The assumption of the fixed rotational axis in the TRS method is unreasonable in the description of the triaxial nuclear system because the rotational axis is not fixed at all but has a complex orientation depending on both rotation and deformation. Specifically, for a nucleus with a small elongation deformation and a considerable large γ deformation the rotational axis is tilted, thus the rotation is characterized to be three dimensional. Seemingly, the energy points of the TRS at these deformations are not correctly calculated by the theory limited to the fixed axis rotation. In the present work, the projected total energy surface c Science China Press and Springer-Verlag Berlin Heidelberg 014 phys.scichina.com link.springer.com

2 Tu Y, et al. Sci China-Phys Mech Astron November (014) Vol. 57 No (PTES) approach has been developed based on the triaxial projected shell model (TPSM) hybridized with the macroscopic-microscopic method. To overcome the shortcomings of the TRS method the classical rotational term is replaced by the full quantum mechanics rotational energy in the new approach. The total energy of an atomic nucleus is decomposed into the macroscopic, microscopic and rotational terms. The macroscopic and microscopic parts are described with the liquid drop model and Strutinsky method, respectively, and the rotational energy is given by the TPSM, as the beyond mean field term. The PTES describes the total energy surface of a triaxially deformed nucleus with the good angular momentum quantum number. Furthermore, the limitation of the fixed axis rotation in the TRS has been removed automatically in the new approach because the three dimensional rotation of the triaxial deformed nuclear system is described in the full quantum mechanics by the TPSM [1]. The single particle energies as functions of the quardrupole deformation have been well generated by the Nilsson potential and reliable for the rare earth mass region, and, therefore, the rare earth nuclei have been often selected as the test ground for the new theory to limit the uncertainties from the single particle structure. Tungsten isotopes have been confirmed the existence of extensive shapes, such as oblate shape and prolate shape [ 4]. To test the validity and efficiency of the theory, the axial asymmetry of the well deformed rare earth nucleus 17 W have been studied by the PTES method. The theoretical results of the yrast states are in good agreement with the experimental data, and by using the equilibrium quardrupole deformation (ε andγ) determined by the PTES, the calculation of the transition quardrupole moment (Q t ) in function of spin again reproduces the experimental data. A comparison between the PTES and TRS methods has been made for the theoretical and application aspects to gain an understanding of the unique feature of the method. The major advantage of the PTES approach is manifested by the fact that the energy surface has a good angular momentum so that the theoretical results can be compared directly with the experimental data in the laboratory frame. Projected total energy surface method The total energy of the nuclear system includes three terms and is expressed as: E tot = E LD + E corr + E rot, (1) where E LD is the liquid-drop model energy [5]. The value of E corr is the quantum effect correction to the energy, which is given by the Strutinsky method [6,7], as the shell correction term, and sometimes includes the pairing correction. For simplicity and more clear structure of the theory the pairing correction is not considered in the present calculation. The value of E rot is the rotational energy, which can be further decomposed into the collective rotational term and the q.p. excitation induced by rotation. All the terms in eq. (1) depend on the neutron and proton numbers (N, Z), the deformation parameters,ε,γandε 4, which are not written explicitly. In the cranking shell model calculations the collective rotational energy may be calculated microscopically to be the difference between the expectation values of the hamiltonian at the rotational frequency nonzero and zero by using the cranking wave function, Ψ H Ψ ω 0 Ψ H Ψ ω=0. The q.p. excitation energy is calculated to be the sum energy of the excited q.p.s in the rotating frame, which belong to the given configuration. The calculation becomes non-reliable at the band interaction region because of the classical rotation nature as mentioned above. In the present model, the rotational energy is obtained by the TPSM calculation, which contains not only the collective rotational energy but also the q.p. excitation energy corresponding to the given configuration, and successfully describes the bandcrossing induced by collective rotation [8 11]. The Hamiltonian in the TPSM can be written as: H=H λ χ λ Q λµ Q λµ G 0 P 00 P 00 G λ= µ= λ P µ P µ, µ= () where H 0 is the spherical single particle Hamiltonian, which contains a proper spin orbit force [1]. The second term is the quadrupole-quadrupole (QQ) interaction that includes the nn, pp and np components. In the spectroscopic calculation, the QQ interaction strengthχis determined in such a way that the term has a self-consistent relation with the quadrupole deformation [1,13]. The third term is the monopole pairing, whose strength parameter G 0 (in MeV) is determined by the N Z expression G 0 = (g 1 g A )A 1, and the minus (plus) sign stands for neutrons (protons). The last term is the quadrupole pairing, whose strength parameter G is proportional to the monopole pairing strength, G = fg 0. The TPSM wave function can be written as: Ψ IM = Fκ,K I ˆP I MK Φ κ, (3) Kκ in which the projected multi-q.p. states span the shell model space. Thus, Φ κ represents the set of multi-quasiparticle states labeled byκ, and for even-even nuclei it includes the - and 4-q.p., states associated with the triaxially deformed q.p. vacuum 0, α ν 1 α ν 0, α π 1 α π 0, α ν 1 α ν α π 1 α π 0. The triaxially deformed single particle states are generated by the Nilsson Hamiltonian. In the present calculation three major shells of N=4, 5, 6 for neutrons and N=3, 4, 5 for protons are considered, and the pairing correlations are included by a subsequent BCS calculation for the Nilsson states. ˆP I MK in eq. (3) is the three-dimensional angular-momentum-projection operator [14], so that ˆP I MK = I+ 1 8π dωd I MK (Ω) ˆR(Ω). (4) The rotational energies together with the wave functions, the coefficients Fκ,K I, are obtained by solving the eigenvalue

3 056 Tu Y, et al. Sci China-Phys Mech Astron November (014) Vol. 57 No. 11 equation: Kκ F I κ,k ( Φκ HP I K K Φ κ E I Φ κ P I K K Φ κ ) = 0. (5) In the present approach, the sum of the liquid drop and the shell correction energies, E LD + E corr in eq. (1), provides the energy of the deformed BCS vacuum state, relative to the spherical liquid drop energy, and the rotational energy E rot is calculated by eq. (5), relative to the deformed q.p. vacuum. The total energy of the deformed nuclear system, E tot in eq. (1), is consequently defined in the laboratory frame, which is in function of spin and has also a good parity. It can be noted that the rotational energy determined by the TPSM includes the contribution from the q.p. excitations induced by the alignments of the high-j q.p. orbits. For example, in an even-even rare earth nucleus the neutron q.p.s are aligned in the yrast states with spins higher than the critical spin of the band crossing, and specifically the corresponding q.p. excitation energy has been included in the calculated rotational energy E rot. The nuclear equilibrium deformation can be obtained for each spin by minimizing the total energy in function of spin with respect to the deformation parameters,ε andγ. The minimization procedure can be performed through the calculation of the total energy surface in the considered deformation plan. 3 Results and discussion 3.1 Parameters Three major shells of N=4, 5, 6 for neutrons and N=3, 4, 5 for protons are included to calculate the Nilsson single particle states in the TPSM. The used Nilsson potential parameters for each main shells are taken elsewhere [1]. In the present calculation, the hexadecapole deformation is not considered as the variable but taken to beε 4 = 0 for simplicity. The monopole pairing strength parameter G 0 is calculated with the standard parameters g 1 = 0.1 MeV and g = MeV [13], which approximately reproduces the observed odd-even mass differences in the mass region. The quadrupole pairing strength parameter G is calculated from G = fg 0 with the proportional coefficient f= 0.16 as usual. 3. PTES for the ground state 0 + The projected total energy surface is calculated for a given angular momentum as well as a given parity, this may provides an opportunity to make a direct comparison between the theory and the experiment. By considering the fact that the 17 W is well deformed nucleus, in the PTES calculation the range of the elongation deformationε is taken from 0.1 to 0.3, while the range of the triaxial deformationγ is from 0 to 50. In the numerical calculation the mesh points of 0 have been taken for both theε andγ deformations. Figure 1 shows the PTES for the ground state (g.s.) I π = 0 + of 17 W, which has a local minimum representing the equilibrium deformations of (ε = 0.5,γ=15 ). The results imply that the g.s. of the nucleus is well deformed, and it is particularly noteworthy to have the axial asymmetry withγ=15. The TRS calculation has been intensively employed to study highly rotating nuclei [15], specifically, in the rare earth mass region, and thus it may be instructive to make a comparison between the PTES and TRS results. The TRS calculated at a small rotational frequency may contain the primary information of the ground state. Because of the semi-classical nature of the TRS method it is not possible to strictly describe the g.s. which has a good angular momentum I π = 0 +. The zero spin does not indicate at all the zero frequency, however, the TRS calculated at a small rotational frequency may describe the property of the g.s. as a good approximation, and thus a comparison between the PTES and TRS for the g.s. has the certain significance. Figure shows the TRS calculated with the same set of the Nilsson potential parameters for the ground band state of 17 W, at the rotational frequency of ω=0.0 ω 0, which reports a local minimum at the deformations of (ε = 0.4,γ= ). The two methods give almost the same equilibrium elongation deformation but a considerably different triaxiality for the g.s. of 17 W. The triaxiality of highly rotating nuclei has been a substantial subject of study for several decades, but much less research focus had been paid to the axial asymmetry of the nuclear ground state. Recently, the first global calculation across the nuclear chart of axial symmetry breaking was carried out by using the macroscopic-microscopic finite-range liquid-drop model (FRLDM) [16,17]. Many nuclear ground states have been predicted to be triaxially shaped or γ-soft by the FRLDM calculations and, however, some of which were previously predicted to be axial symmetric by the TRS method. The different results of the FRLDM from the TRS method come from their associated parameterizations although the two methods are belong to the same type of macroscopic-microscopic approach. For the nucleus 17 W, both the FRLDM and TRS calculations report a axial symmetry for the ground state. Also, the present PTES calculation gives the axial asymmetry of (γ=15 ) for the g.s. of 17 W, which differs from the TRS calculation and FRLDM calculation. The PTES approach treats the nuclear rotation quantum mechanically and allows three dimensional rotation through the angular momentum projection. In contrast, the usual TRS method (not tilted cranking) treats only the semi-classical rotation and requires the system to be rotating around the shortest principle axis with a rotational frequency. Theγ-deformation of 15 is not a large triaxiality but an indication of the axial symmetry breaking in the g.s. of the nucleus. In fact, the axial asymmetry of the g.s. of 17 W manifests the appearance of the characteristic γ band observed experimentally at 0.93 MeV above the g.s.. By comparison, it can be seen that the beyond mean field effects incorporated in the PTES approach may give rise to the axial asymmetry in some nuclei which have been predicted to have an axial symmetric shape by the TRS calculations and

4 Tu Y, et al. Sci China-Phys Mech Astron November (014) Vol. 57 No γ I =0 + π =0.5 ε γ = ε Figure 1 Contour plot of total energy in units of MeV for the ground state in 17 W, the minimum is marked by +. ε sin( γ +30 ) 0.40 γ= γ= ε cos( γ+30 ) Figure Contour plot of total routhian in units of MeV for 17 W, calculated at the rotational frequency ω=0.0 ω 0, the minimum is marked by +. FRLDM calculation. + The calculations of the PTES for nonzero spins, the yrast states of 17 W, have been performed in a similar way to the g.s. calculation. There exists also a minimum in each of PTES.s for spins of I π = +, 4 +, 6 +,..., 0 +, from which the equilibrium deformations are obtained for each spin and listed in Table 1. Both the elongation and triaxial deformations presents being almost unchanged from spin 0 up to 0, that is characteristic of a well deformed nucleus where the kinetic moment of inertial does not considerably change with increasing spin. Nevertheless, a slight change of theε deformation is found after spin 1 +, first from 0.5 rising to 0.6, and then decreasing to 0.4 after These deformation changes reflect the deformation driving effects induced by the q.p. excitations. Before the spin 1 +, the yrast states form the ground band in which spin gains through the collective rotation of the entire nucleus, behaving as a good rotor and having stable deformations. With increasing spins beyond the 1 + state, the pair of the i 13/ neutrons is broken and align their angular momenta along the rotational axis, and it is that the alignments of the neutron high- j q.p.s induced by the rotation that leads to the deformation change. In addition to the well deformed nature, it is also expected that the smallness of the deformation change is associated with the gradual alignments of the q.p.s because of the moderate interaction between the ground state and the q.p bands. Figure 3 presents the calculated yrast band for 17 W, from the PTES minima as listed in Table 1, compared with the experimental data [18]. The PTES for the each state is first calculated and then the minimization procedure with respect to the elongation and triaxial deformations is performed to obtain the energy of the state together with the equilibrium deformations. In this way, the yrast band has been calculated self-consistently by the PTES approach, and the results are in good agreement with the experimental data. The transition quadrupole moments Q t of the yrast states in 17 W were determined by the lifetime measurement [19] up to spin 0 +. The transition quadrupole moment is one of best physical quantities to identify the intrinsic deformed shape of the nucleus and can be calculated by the expression related to theβ andγ deformations: Q t = 6ZeA/3 (15π) 1/ r 0 β ( 1+ 7 ( 5) 1/β ) cos(30 +γ). (6) π In the calculation of Q t, we takeβ = ε /0.946 and r 0 =1.88 fm and use the obtained equilibrium deformations of the yrast states, as presented in Table 1, and the results are compared with the experimental data [19] and shown in Figure 4. The calculated values of the transition quadrupole moments may be regarded as the self-consistent results at the meaning that the underlying nuclear deformations of the yrast states are determined by minimizing the projected total energies. It is seen from Figure 4 that the calculated transition 3.3 PTES for the yrast states Excitation energy (MeV) I Figure 3 Calculated yrast band (open circle), the energy versus spin, for 17 W compared with the experimental data (solid circle).

5 058 Tu Y, et al. Sci China-Phys Mech Astron November (014) Vol. 57 No. 11 Q t (eb) W I Figure 4 Calculated transition quadrupole moment (open circle) versus spin for the yrast band of 17 W, compared with the experimental data (solid circle) quadrupole moments well reproduce the experimental data within the error bars. 3.4 Component energy surface As discussed above the axial asymmetry is originated from the beyond mean field effects generated by the angular momentum projection. To explain this point in further details the component energy surfaces have been calculated and shown in Figure 5(a) for the E LD + E shell and (b) for the E rot. The Liquid drop model-plus-shell energy surface presents a minimum at the axial symmetry with theε =0.35, and demonstrates a modestγ-softness towards theγdeformation of 15, this provides an advantageous condition for the formation of the axial asymmetry shape by adding the rotational energies on. The rotational energy surface for the spin 4 +, shown in Figure 5(b), presents the striking feature to drive theε deformation towards a larger elongation while drive the γ deformation from both the axial symmetric prolate (0 ) and oblate (60 ) shapes towards about 0. It is illustrated that the projected rotational energy lowers the total energy in the laboratory frame by about 300 kev to cause the axial symmetry breaking in the yrast states of 17 W. The E LD + E shell surface shown in Figure 5(a) applies also to the present TRS calculation so that the TRS shown in Figure can be regarded as the result by adding the rotational energy given by CSM to the liquid drop-plus-shell energy. The axial symmetric shape described by the local minimum in the E LD + E shell surface remains in the TRS, implying that the surface remains in the TRS. Also, this implies that the γ-deformation driving effects promised from adding the classical rotational energy is not sufficient to cause the formation of a local triaxial minimum in the TRS. In contrast, the quantal rotational energy surface such as one shown in Figure 5(b) presents the strongγ-deformation driving effects for the formation of the axial asymmetry shapes of the yrast states in 17 W. Although a direct comparison between the quantal and classical rotational energies is not possible but it can be seen that at approximatelyε =0.4 the energy lowering from γ=0 towards 15 is approximately 0.1 MeV for the classical rotation in the TRS at ω=0.0 ω 0, estimated from Figures and 5(a), and approximately 0.5 MeV for the quantum rotation in the PTES at I π = 4 +, estimated from Figure 5(b). The results indicate that in the nuclear energy surface calculations the inclusion of the angular momentum projection, as the beyond mean field effects, is critical in the study of nuclear symmetry and symmetry break. Table 1 Deformations of the yrast states of 17 W determined by the energy minima in PTES for spins 0 + to 0 + I π ε γ (a) E LD + E shell (b) E rot (I π = 4 + ) 40 γ γ ε ε + Figure 5 Component energy surfaces of the total energy for the yrast state of I π = 4 + in 17 W: (a) E LD + E shell and (b) E rot.

6 Tu Y, et al. Sci China-Phys Mech Astron November (014) Vol. 57 No Conclusions We have developed the PTES approach based on the TPSM hybridized with the macroscopic-microscopic method. The nuclear total energy includes the macroscopic and microscopic terms which are described by the liquid drop model and the Strutinsky method, respectively, and the rotational energy given by the TPSM, as the term beyond the mean field. The three dimensional rotation of the triaxial deformed nuclear system has been described in a full quantum mechanics way by introducing the TPSM. The PTES describes the total energy for a good angular momentum, and therefore, the theoretical results can be directly compared with the experimental data. In comparison with the standard TRS method, the classical rotational frequency and the limitation of the fixed axis rotation in the TRS have been removed in the new approach. The validity and efficiency of the theory have been tested through the PTES calculations carried out for the yrast states of the nucleus 17 W. The theoretical results of the yrast band energies are in good agreement with the experimental data. By using the equilibrium quardrupole deformations determined by the PTES, the calculated transition quardrupole moments (Q t ) for the yrast states also reproduce the experimental data. This work was supported by the National Natural Science Foundation of China (Grant Nos , , , and ), the Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. KJCX-SW-N0) and the Key Project of Science and Technology Research of Education Ministry of China (Grant No ). 1 Gao Z C, Chen Y S, Sun Y. Signature inversion - manifestation of drift of the rotational axis in triaxial nuclei. Phys Lett B, 006, 634: Jiao C F, Shi Y, XU F R, et al. Competition between collective oblate rotation and non-collective prolate K isomerism in neutron-rich tungsten isotopes. Sci China-Phys Mech Astron, 01, 55(9): Walker P M, Xu F R. Prediction and possible observation of an oblate shape isomer in 190 W. Phys Lett B, 006, 635: Sun Y, Walker P M, Xu F R, et al. Rotation-driven prolate-to-oblate shape phase transition in 190 W: A projected shell model study. Phys Lett B, 008, 659: Myers W D, Swiatecki W. Anomalies in nuclear masses. Ark Fys, 1967, 361: Strutinsky V M. Shells in deformed nuclei. Nucl Phys A, 1968, 1: Strutinsky V M. Shell effects in nuclear masses and deformation energies. Nucl Phys A, 1967, 95: Hara K, Sun Y, Mizusaki T. Backbending mechanism of 48 Cr. Phys Rev Lett, 1999, 83: Sun Y, Egido J L. Excited bands of 168 Yb in an angular momentum projected theory. Phys Rev C, 1994, 50: Dong G X, Yu S Y, Liu Y X, et al. The study of energy bands in nucleus 10 Zr. Sci China-Phys Mech Astron, 010, 53(1): Kang X Z, Shen S F, Han G B, et al. High spin states in stable nucleus 84 Sr. Sci China-Phys Mech Astron, 010, 53(10): Bengtsson T, Ragnarsson I. Rotational bands and particle-hole excitations at very high spin. Nucl Phys A, 1985, 436: Hara K, Sun Y. Projected shell model and high-spin spectroscopy. Int J Mod Phys E, 1995, 4: Ring P, Schuck P. The Nuclear Many Body Problem. New York: Springer-verlag, Tu Y, Chen Y S, Yu S Y, et al. Triaxial superdeformed bands in odd odd Lu isotopes. Nucl Phys A, 010, 848: Möller P, Bengtsson R, Gillis C B, et al. Global calculations of groundstate axial shape asymmetry of nuclei. Phys Rev Lett, 006, 97: Möller P, Sierk A J, Bengtsson R, et al. Nuclear shape isomers. Atomic Data Nucl Data Tables, 01, 98: Espino J, Garrett J D, Hagemann G B, et al. Rotational band structures in 171,17 W: Aspects on signature partnership at high spin. Nucl Phys A, 1994, 567: Mcgowan F K, Johnson N R, Lee I Y, et al. Transition quadrupole moments of high-spin states in 17 W and their implications for the interpretation of band crossings in the light isotopes of W to Pt. Nucl Phys A, 1991, 530: