Rotation alignment in neutron-rich Cr isotopes: A probe of deformed single-particle levels across N=40
aa r X i v : . [ nu c l - t h ] S e p Rotation alignment in neutron-rich Cr isotopes: A probe of deformed single-particlelevels across N = 40 Yingchun Yang , Yang Sun , , , Kazunari Kaneko , Munetake Hasegawa , Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Department of Physics, Kyushu Sangyo University, Fukuoka 813-8503, Japan (Dated: November 6, 2018)Recent experiments have reached the neutron-rich Cr isotope with N = 40 and confirmed en-hanced collectivity near this sub-shell. The current data focus on low-spin spectroscopy only, withlittle information on the states where high- j particles align their spins with the system rotation. Byapplying the Projected Shell Model, we show that rotation alignment occurs in neutron-rich even-even Cr nuclei as early as spin 8¯ h and, due to shell filling, the aligning particles differ in differentisotopes. It is suggested that observation of irregularities in moments of inertia is a direct probe ofthe deformed single-particle scheme in this exotic mass region. PACS numbers: 21.10.Pc, 21.10.Re, 27.40.+z, 27.50.+e
Current nuclear structure studies are devoted to thediscussion of enhanced collectivity in the neutron-rich pf -shell nuclei with neutron-number N ≈
40. One hasfound strong evidence for compressed first 2 + energy lev-els and large E2 transitions linking these and the groundstates for several isotopic chains around the proton magicnumber Z = 28, for example, in the Cr ( Z = 24) [1, 2],Fe ( Z = 26) [3–5], and Zn ( Z = 30) [6] isotopic chains.These experimental results support the early suggestionsthat near N = 40, pronounced collectivity develops cor-responding to the formation of a region of deformation[7–9].In the study of neutron-rich nuclei, an important is-sue is to understand emerging sub-shell gaps which causesubstantial modifications of the intrinsic shell structurein nuclei with a neutron excess [10]. While informa-tion on collective excitations in low-spin states is use-ful, a comprehensive knowledge for these exotic nucleirequires the study of higher-spin states in which, dueto rotation alignment, quasiparticle configurations aredominant. For an yrast band consisting of the loweststates for each angular momentum, the aligning parti-cles carry valuable information on the deformed single-particle states. Therefore, investigations of high-spinspectra can yield knowledge on the intrinsic shell struc-ture of single-particle levels.Microscopic calculations have shown that beginningfrom N ≈
30, energy minima with sizable prolate de-formations show up for the neutron-rich Cr isotopes [11].In these deformed Cr isotopes, protons occupy up to the πf / orbit whilst neutrons of the N >
28 isotopes fillin the rest of the pf -shell. With the splitting of single-particle orbits due to deformation, the proton Fermi levellies between the f / orbitals π [321]3 / − and π [312]5 / − ,and is also not far from π [300]1 / − . On the other hand,the down-sloping levels of the neutron intruder g / or-bit, ν [440]1 / + , ν [431]3 / + , and ν [422]5 / + , are foundnear the neutron Fermi levels, and therefore, neutrons can easily occupy these orbitals. Thus when nuclei ro-tate, these high- j particles (here, f / for protons and g / for neutrons) are among the first to align their ro-tation along with the rotation-axis of the system, result-ing in observable effects in the moment of inertia, whichcorrespond to the phenomenon known as rotation align-ment [12]. Thus with increasing neutron number from N = 30 towards 40 and beyond, these high- j orbits dom-inate the high-spin behavior of these nuclei. This quali-tative picture is valid also for nuclei with a soft groundstate. Angular-momentum-projected energy-surface cal-culations show [13] that as soon as the nuclei begin torotate, well-defined shapes in favor of prolate deforma-tion develop.With the experimental advances, detailed spectro-scopic measurements for neutron-rich nuclei now becomepossible. In a very recent work, Gade et al. [1] reportedtheir successful experiment for the neutron-rich isotope Cr by Be-induced inelastic scattering, obtaining thefirst spectroscopy at the N = 40 subshell for Cr isotopes.Data for some lighter Cr isotopes are presently available[2, 14, 15]. In the near future, fragmentation of a Gebeam may push the experiment to more neutron-rich re-gions [16]. On the theoretical side, large-scale shell-modelcalculations [17–19] have been successful in describing thelow-spin spectroscopy of neutron-rich nuclei. For exam-ple, the spherical shell-model calculation for Cr isotopes[19] including the g / orbit in the model space predictedthe first excited 4 + energy of Cr, which was later con-firmed by experiment [2]. There have been encourag-ing applications by beyond-mean-field approaches [11, 20]which can easily handle a large model space. Neverthe-less, models that either do not allow sufficient amountof valence particles in the spherical shell model space ordo not build excited quasiparticle configurations in thedeformed models may not be appropriate for discussionsof high-spin physics.To discuss high-spin states and to further study thedeformed single-particle structure in neutron-rich nuclei, M o m en t o f i ne r t i a ( / M e V ) PSM Exp Cr Cr Cr Cr Cr Spin ( ) Cr Cr Cr FIG. 1: (Color online) Comparison of the calculated moments of inertia (filled squares) for the yrast bands in even-even − Crwith the known experimental data (filled circles) taken from Refs. [14] ( Cr), [15] ( − Cr), [2] ( Cr), and [1] ( Cr). Notethat open circles denote those tentative data reported in these publications. we performed Projected Shell Model (PSM) [21] calcu-lations for neutron-rich, even-even Cr isotopes with neu-tron number from 30 to 44, aiming at making predictionsahead of experiment. The model has recently been ap-plied to the neutron-rich Fe isotopes [13], where large2 + state B(E2)’s were predicted for , Fe and con-firmed later by the measurement [5]. It has also beenemployed to study the yrast structure of the Ge nuclei[22]. The PSM calculation uses deformed Nilsson single-particle states [23] to build the model basis. For thepresent calculations, the quadrupole deformation param-eters for building the deformed bases are listed in Table I.These parameters are consistent with the known experi-mental trend of increasing deformation towards N = 40[15], and afterwards a slightly decreasing collectivity aspredicted by spherical shell model calculations [19]. Pair-ing correlations are incorporated into the Nilsson statesby a BCS calculation. The consequence of the Nilsson-BCS calculations defines a set of quasiparticle (qp) statescorresponding to the qp vacuum | i . The PSM wavefunc-tion is a superposition of (angular-momentum) projectedmulti-qp states that span the shell model space | Ψ σIM i = X Kκ f σIK κ ˆ P IMK | Φ κ i , (1)where | Φ κ i denotes the qp-basis, κ labels the basis statesand f σIK κ are determined by the configuration mixing im-plemented by diagonalization. ˆ P IMK is the angular mo-mentum projection operator [21] which projects an in-trinsic configuration onto states with good angular mo-mentum. As the valence space for this mass region, par-ticles in three major shells ( N = 2 , , TABLE I: Input deformation parameters ( ǫ ) used in the cal-culation.Cr 54 56 58 60 62 64 66 68 ε and protons) are activated. The multi-qp configurationsconsisting of 0-, 2-, and 4-qp states for even-even nucleiare as follows: {| i , a † ν i a † ν j | i , a † π i a † π j | i , a † ν i a † ν j a † π k a † π l | i} , (2)where a † ν and a † π are the neutron- and proton-qp creationoperators, respectively, with the subscripts i , j , k , and l denoting the Nilsson quantum numbers which run overthe orbitals close to the Fermi levels.The PSM calculation employs a quadrupole plus pair-ing Hamiltonian, with inclusion of quadrupole-pairingtermˆ H = ˆ H − χ X µ ˆ Q † µ ˆ Q µ − G M ˆ P † ˆ P − G Q X µ ˆ P † µ ˆ P µ . (3)In Eq. (3), ˆ H is the spherical single-particle Hamil-tonian which contains a proper spin-orbit force. Themonopole pairing strengths are taken to be G M =[ G ∓ G ( N − Z ) /A ] /A , where ”+” (” − ”) is for pro-tons (neutrons) with G = 18 .
72 and G = 10 .
74. Thequadrupole pairing strength G Q is assumed to be propor-tional to G M , the proportionality constant being fixed to0.30. Cr E ne r g y ( M e V ) Spin( ) ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2p ) 0-qp K=0 yrast Cr ) 4-qp K=0 (2n ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2p ) 0-qp K=0 yrast FIG. 2: (Color online) Theoreticalband diagrams for , Cr. Bandsfor some important configurations areshown and explained in Table II. Notethat, to illustrate them clearly, onlyeven-spin states are plotted to avoid astrong zigzag in curves between evenand odd-spin states.
The moment of inertia (MoI) is a characteristic quan-tity for the description of rotational behavior. In Fig.1,the calculated results (filled squares) for the yrast bandsin − Cr are presented in terms of the MoI (defined as J ( I ) = (2 I − / [ E ( I ) − E ( I − J increases nearly linearly with spin I forthe lowest spin states. However, irregularities in MoI areseen as early as I = 8, with the actual pattern differingin different isotopes. Interestingly, all the current data,with the last one or two data points assigned as tentativein several isotopes, stop at the spins where irregularitiesare predicted to occur. As we shall discuss below, theseirregularities reflect changes in the yrast structure causedby rotation alignment of nucleons from specific orbitals.Thus by studying the changes in MoI one can gain valu-able information on deformed single-particle states forthis exotic mass region.By examining the MoI patterns in Fig. 1, one findsthat with increasing spin, J of the two lightest isotopes , Cr either bends down or stops rising at spins I = 10and 12, and then shows a rapid rise. A peak is predictedto occur at I = 16 for both isotopes. Irregularities in J are especially notable in , , Cr, in which the rota-tional patterns are interrupted several times. The firstone is seen soon after I = 6 where a jump in J is pre-dicted. Later at I = 10 and 16, two peaks are predictedto appear. For the N ≥
40 isotopes , , Cr, our calcu-lation suggests an gradual increase in J (with some smallperturbations) up to high spins, until a peak is formedat I = 14 or 16.To understand what causes the variations in MoI, westudy theoretical band diagrams for these isotopes. In TABLE II: 2-qp configurations for neutrons and protons.neutron configurations protron configurations2n ν / N ν / π / N π / ν / N ν / π / N π / ν / N ν / the PSM, the energy of a calculated band κ is defined by E κ ( I ) = h Φ κ | ˆ H ˆ P IKK | Φ κ ih Φ κ | ˆ P IKK | Φ κ i , (4)which is the projected energy of a multi-quasiparticleconfiguration in (2) as a function of spin I . An ensembleof projected configurations plotted in one figure is calledband diagram [21], in which the rotational behavior ofeach configuration as well as its relative energy comparedto other configurations are easily visualized. Because ourdeformed basis states retain axial symmetry, we use K (the projection of angular momentum on the symmetryaxis of the deformed body) to classify the configurations.For even-even nuclei, the 0-qp ground band has K = 0,whereas a multi-qp band has a K given by the sum of theNilsson K quantum numbers of its constituent qp’s. Asuperposition of them imposed by configuration mixinggives the final results, with the lowest one at each spinbeing the yrast state. The study of band crossings in theyrast region can give us useful messages from which oneidentifies the most important configurations for the yraststates. E ne r g y ( M e V ) Spin( ) ) 4-qp K=0 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 0-qp K=0 yrast Cr Cr ) 4-qp K=0 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 0-qp K=0 yrast Cr ) 4-qp K=0 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 0-qp K=0 yrast FIG. 3: (Color online) Same as Fig 2, but for , , Cr.
According to the nature of band-crossings, we may di-vide the isotopes into three groups for discussion. In Figs.2-4, we plot the important configurations that are usedto discuss the MoI variations in Fig. 1. It can be seenfrom the figures that as a nucleus starts rotating, the 0-qp ground band energy increases and the band quicklyenters into the high energy region. The 2- and 4-qp bandsrise slowly at low spins, and therefore, can cross the 0-qpband. To facilitate the discussion, we term the relevant 2-qp configurations to be 2n , 2n , 2n , 2p , and 2p , withthe details listed in Table II. In the band diagrams, weuse different line styles to distinguish different qp bands,so that one can easily follow them with increasing neu-tron number. In addition, the solid circles marked as“yrast” in Figs. 2-4 are the lowest state at each spin ob-tained after diagonalization, and these are the theoreticalresults compared with the data (see Fig. 1).In the two lightest isotopes , Cr, the neutron g / orbitals are far above the Fermi level, and therefore, neu-tron 2-qp states are high in energy. Proton 2-qp states areexpected to be the first to cross the 0-qp band. In Fig. 2,we indeed find that the proton 2-qp band (marked as 2p )is the lowest 2-qp band which crosses the ground bandat I = 6. The crossing is gentle so that it causes littledisturbance in MoI. Another proton 2-qp band (markedas 2p ) exhibits a similar character and approaches the2p band near I = 10. After this spin, the two proton2-qp bands stay nearly parallel and interact with eachother. The interaction causes the first irregularity in theMoI of Cr at I ≈
10, as seen in Fig. 1. The neutron2-qp band 2n lies high in energy at low spins; it crosses however the proton 2-qp bands at I ≈
10 in Cr. Inthe spin interval I = 12 −
14, the 4-qp band consistingof two 2-qp states, 2n and 2p , sharply crosses the 2-qpbands. After I ≈
14, the yrast states are predicted tobe of a 4-qp structure. The 4-qp band crossing changesthe content of the yrast wave functions, leading to thesecond irregularity in MoI of , Cr, as seen in Fig. 1.Going toward N = 40, very irregular MoI’s are pre-dicted in Fig. 1 for , , Cr. This is because as neu-trons begin to fill the g / shell, the low- K orbitals comeclose to the Fermi level. It is known that for nucleonsof low- K , high- j orbitals it is easy to align the spinswith the rotation. In Fig. 3, one sees that althoughthe lowest 2-qp band at low spins I < , it climbs rapidly and soon becomes unimportant.The other proton 2-qp band beginning at a higher energyshows a similar character. In contrast, the neutron 2-qpband (marked as 2n ), which is coupled by two neutronsin the low- K g / orbitals, is more important becauseit crosses the ground band at I = 6, and dominates theyrast structure in the spin interval I = 6 −
14. The cross-ing leads to a jump in MoI predicted at I = 6 or 8, asseen in Fig. 1. The prediction is at variance with thecurrent tentative data [15] which seem to suggest a moreregular rise in MoI. There are two 4-qp bands in eachisotope shown in Fig. 3, among which the K = 0 4-qpband is more notable, as it crosses the neutron 2-qp band2n at spin I ≈
14, and becomes yrast after that spin.The peak in MoI at I ≈
16 in , , Cr is attributed tothe crossing of the 4-qp band.Band diagrams for the heavier isotopes , , Cr are Cr E ne r g y ( M e V ) Spin( ) ) 4-qp K=2 (2n ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 0-qp K=0yrast Cr ) 4-qp K=2 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 0-qp K=0 yrast Cr ) 4-qp K=2 (2n ) 2-qp K=1 (2n ) 2-qp K=1 (2p ) 2-qp K=1 (2n ) 0-qp K=0 yrast FIG. 4: (Color online) Same as Fig 2, but for , , Cr. plotted in Fig. 4. With increasing neutron number, the2n configuration lies far from the Fermi level. Instead,the other two neutron 2-qp bands 2n and 2n play arole. Since in the 2n and 2n configurations there arehigher- K states which are more coupled, the rotationalbehavior of 2n and 2n is different from 2n . 2-qp bandswith high- K states exhibit similar rotational behavior asthe ground band, and therefore, crossing of them is al-ways gentle. This makes an observable difference in MoI:although 2n and 2n bands cross the ground band at I ≈
6, no notable variation in MoI can been seen in Fig.1. The yrast states of high spins (
I >
14) are predictedto be strongly mixed by the K = 2 4-qp configuration.Peaks in MoI at high spins are due to the band crossingwith the 4-qp configurations.Taking a survey for the band diagrams of Figs. 2-4, weclearly see the evolution of the 2-qp and 4-qp band struc-ture along the isotopic chain imposed by shell fillings.With increasing neutron number, the role of the g / or-bit changes with occupation of different K states becausethe nucleons that first align their spin vary between thevarious isotopes of interest. For the mass region withless dense single-particle levels, a distinct behavior in ob-servable quantities such as the moment of inertia can bedetected in neighboring isotopes.The current experimental data however limit possiblecomparisons between calculations and experiment for thehigh-spin yrast states and do not allow us to draw strongconclusions about the relevance of the calculated results.This is particularly bothersome because Fig. 1 indicatesdiscrepancies between calculations and experiment at the highest spin states of the experimental bands. However,there is at least one negative-parity band in Cr thathas been observed [15] up to high spins. This band isof a 2-qp structure with one quasiparticle from the neu-tron g / orbital. In Fig. 5, we compare two rotationalbands of Cr with data taken from Ref. [15]. Theseare the lowest positive-parity (yrast) band and the low-est negative-parity band obtained from the calculation.The calculation reproduces nicely the bandhead energyas well as the rotational feature of the negative-parityband. The inserted MoI plot in Fig 5 however indicatesthat the calculation exaggerates the proton f / align-ment at I = 13 with a larger peak in MoI than whatthe data show. This suggests that one must be cautiousabout the quantitative details of the prediction in Fig. 1although we do not expect that small discrepancies canchange the alignment picture discussed in the paper.As the entire discussion in the present work dependson the single-particle states, it is important to commenton the deformed Nilsson scheme employed in the modelbasis. The Nilsson parameters used for the present dis-cussions were fitted a long time ago to the stable nuclei[23]. It has been shown that the standard Nilsson pa-rameters may need adjustments when they are appliedto proton- or neutron-rich regions [24, 25]. For neutron-rich nuclei with considerable neutron excess, the usage ofthe standard parameter set for these nuclei needs to bevalidated. It will not be surprising if the Nilsson parame-ters used here would require a modification. Thus futureexperiments for high-spin yrast spectra will be comparedwith the present predictions and serve as guidance to M o I ( / M e V ) Spin ( )
PSM Exp E ne r g y ( M e V ) _ _ _ _ _ _ Cr + PSM Exppos. parity yrast band + + + + + + + + + + PSM Exp + + + + + + + neg. parity band _ _ _ _ _ _ FIG. 5: (Color online) Comparison of the calculated energylevels with data [15] for the positive-parity yrast band andthe negative-parity band in Cr. Dashed lines denote thosetentatively assigned data. The inserted plot shows the com-parison of the calculated MoI with data for the negative-parityband. modify the deformed single-particle scheme.In conclusion, the present study has highlighted therole of the proton f / and neutron g / orbits that areinvolved in the discussion of neutron-rich nuclei. As wehave studied in detail, excitation to these orbits leads topronounced observation effects at high spins where vari-ations in the moment of inertia are explained in termsof rotation alignment of the high- j particles. As the dis-cussion is based crucially on the deformed single-particlescheme, we emphasize that any future experimental con-firmation or refutation of our predictions will be valu-able information, which can help to pin down the single-particle structure in this neutron-rich mass region.Research at SJTU was supported by the Shanghai Pu-Jiang scholarship, the National Natural Science Foun-dation of China under contract No. 10875077, the Doc-toral Program of High Education Science Foundation un-der grant No. 20090073110061, and the Chinese MajorState Basic Research Development Program through un-der grant No. 2007CB815005. [1] A. Gade et al. , Phys. Rev. C , 051304(R) (2010).[2] N. Aoi et al. , Phys. Rev. Lett. , 012502 (2009).[3] S. Lunardi et al. , Phys. Rev. C , 034303 (2007).[4] P. Adrich et al. , Phys. Rev. C , 054306 (2008).[5] J. Ljungvall et al. , Phys. Rev. C , 061301(R) (2010).[6] O. Perru et al. , Phys. Rev. Lett. , 232501 (2006).[7] R. Grzywacz et al. , Phys. Rev. Lett. , 766 (1998).[8] M. Hannawald et al. , Phys. Rev. Lett. , 1391 (1999).[9] O. Sorlin et al. , Eur. Phys. J. A , 55 (2003).[10] R. V. F. Janssens, Nature (London) , 897 (2005).[11] T. R. Rodr´ıguez and J. L. Egido, Phys. Rev. Lett. ,062501 (2007).[12] F. S. Stephens and R. S. Simon, Nucl. Phys. A , 257(1972).[13] Y. Sun et al. , Phys. Rev. C , 054306 (2009). [14] M. Devlin et al. , Phys. Rev. C , 017301 (1999).[15] S. Zhu et al. , Phys. Rev. C , 064315 (2006).[16] O. B. Tarasov et al. , Phys. Rev. Lett. , 142501 (2009).[17] E. Caurier et al. , Eur. Phys. J. A , 145 (2002).[18] M. Honma et al. , Eur. Phys. J. A , 499 (2005).[19] K. Kaneko et al. , Phys. Rev. C , 064312 (2008).[20] L. Gaudefroy et al. , Phys. Rev. C , 064313 (2009).[21] K. Hara and Y. Sun, Int. J. Mod. Phys. E , 637 (1995).[22] P. A. Dar et al. , Phys. Rev. C , 054315 (2007).[23] T. Bengtsson and I. Ragnarsson, Nucl. Phys. A , 14(1985).[24] J.-y. Zhang et al. , Phys. Rev. C , R2663 (1998).[25] Y. Sun et al. , Phys. Rev. C62