Triaxial projected shell model study of gamma-vibrational bands in even-even Er isotopes

0.25 MB·English

by  J. A. Sheikh

Triaxial projected shell model study of γ-vibrational bands in even-even Er isotopes
J. A. Sheikh1, G. H. Bhat1, Y. Sun2,3, G. B. Vakil1, and R. Palit4
1Department of Physics, University of Kashmir, Srinagar, 190 006, India
2Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
3Joint Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, Indiana 46556, USA
4Tata Institute of Fundamental Research, Colaba, Mumbai, 400 005, India
We expand the triaxial projected shell model basis to include triaxially-deformed multi-
quasiparticle states. This allows us to study the yrast and γ-vibrational bands up to high spins
for both γ-soft and well-deformed nuclei. As the first application, a systematic study of the high-
spin states in Er-isotopes is performed. The calculated yrast and γ-bands are compared with the
known experimental data, and it is shown that the agreement between theory and experiment is
quite satisfactory. The calculation leads to predictions for bands based on one- and two-γ phonon
where current data are still sparse. It is observed that γ-bands for neutron-deficient isotopes of
8
156Er and 158Er are close to the yrast band, and further these bands are predicted to be nearly
0
0 degenerate for high-spin states.
2
PACSnumbers: 21.60.Cs,21.10.Hw,21.10.Ky,27.50.+e
n
a
J
I. INTRODUCTION scheme. Itemploysonlyafewcollectivephononsandre-

1 stricts the basis to all the corresponding multi-phonon

states upto eightphonons. This approachpredicts that,
] Recentexperimentaladvancesinnuclearspectroscopic
h techniquesfollowingCoulombexcitations,in-elasticneu- for strongly collective vibrations, two phonon Kπ = 4+
t excitationsshouldappearatanenergyofabout2.6times
- tron scattering,and thermal neutron capture have made
l the energy of the one-phonon Kπ =2+ state [3, 15]. On
c it possible to carry out a detailed investigation of γ-
the other hand, the dynamic deformation model (DDM)
u vibrationalbands in atomic nuclei [1, 2, 3]. These bands
n areobservedinbothsphericalandaswellasindeformed [16], which is quite different from the models mentioned
[ above, constructs collective potential from a set of de-
nuclei. Insphericalnuclei,thevibrationalmodesarewell

formed single-particle basis states accommodating eight
1 described using the harmonic phonon model [4, 5]. Al-
major oscillator shells. This model predicts a collective
v though exact harmonic motion has never been observed,
6 there are numerous examples of nuclei exhibiting near Kπ =4+ at almost 2 MeV.
9 harmonic vibrational motion. As a matter of fact, one- All the above mentioned models (QPNM, MPM, and
2 andtwo-phononexcitationshavebeenreportedinalarge DDM)donothavetheirwavefunctionsaseigen-statesof
0
class of spherical nuclei. In deformed nuclei, vibrational angularmomentum. Strictlyspeaking,thesemethodsdo
.
1 motion is possible around the equilibrium of deformed not calculate the states of angular momentum, but the
0 shape configuration. The deformedintrinsic shape is pa- K-states (K is the projection of angular momentum on
8 rameterized in terms of β and γ deformation variables. theintrinsicsymmetryaxis). Toapplythesemodels,one
0
These parameters are related to the axial and non-axial has to assume that I K. However, since an intrinsic
: ≈
v shapes of a deformed nucleus. The one-phonon vibra- K-state can generally have its components spread over
Xi tional mode in deformed nuclei with no component of thespaceofangularmomentaofI K,thereliabilityof
≥
angular momentum along the symmetry axis (K =0) is these approaches depends critically on actual situation.
r
a called β-vibration and the vibrational mode with com- As pointed out by Soloviev [12], it is quit desirable to
ponent of angular momentum along the symmetry axis recover the good angular-momentum in the wave func-
(K = 2) is referred to as γ-vibration. The rotational tions.
bands based on the γ-vibrational state are known as γ- Some algebraic models including the extended version
bands [6, 7, 8]. One-phononγ-bandshavebeenobserved of the interacting boson (sdg-IBM) [17, 18] and pseudo-
innumerousdeformednucleiinmostoftheregionsofthe symplecticmodels[19]havealsobeenemployedtostudy
periodic table. There has also been reports on observa- the γ- excitation modes and predict high collectivity for
tion of two-phonon γ-bands [9, 10]. the double γ- vibration [20].
Severaltheoreticalmodelshavebeenproposedtostudy Recently, the triaxial projected shell model (TPSM)
γ- bands with varying degree of success. The quasipar- has been employed to describe γ- bands [21, 22]. This
ticle phonon nuclear model (QPNM) [11, 12], which re- model uses shell model diagonalization approach and in
stricts the basis to, at the most, two phonon states, has this sense, it is similar to the conventional shell model
led to the conclusion that two-phonon collective vibra- approach except that the basis states in the TPSM
tional excitations cannot exist in deformed nuclei due are triaxially deformed rather than spherical. In the
to the Pauli blocking of important quasiparticle com- present version of the model, the intrinsic deformed ba-
ponents. On the other hand, the multi-phonon method sis is constructed from the triaxial Nilsson potential.
(MPM)[13,14]embodiesanentirelydifferenttruncation The good angular momentum states are then obtained
2
throughexactthree-dimensionalangularmomentumpro- InEq. (1),thethree-dimensionalangular-momentumop-
jection technique. In the final stage, the configuration erator is [30]
mixing is performed by diagonalizing the pairing plus
2I+1
quadrupole-quadrupoleHamiltonianintheprojectedba- PˆI = dΩDI (Ω)Rˆ(Ω), (2)
sis [23, 24]. The advantage of the TPSM is that it MK 8π2 MK
Z
describes the deformed single-particle states microscopi-
with the rotational operator
callyasinQPNM,MPM,andDDM, butits totalmany-
body states are exacteigen states of angularmomentum
Rˆ(Ω)=e−ıαJˆze−ıβJˆye−ıγJˆz, (3)
operator. Correlations beyond the mean-field are intro-
duced by mixing the projected configurations.
and Φ> represents the triaxial qp vacuum state. The
|
It is to be noted that an intrinsic triaxial state in the qpbasischosenaboveareadequateto describe the high-
TPSM is a rich superposition of different K-states. For spin states upto, say I 24, and in the present analysis
∼
instance,thetriaxialdeformedvacuumstateiscomposed we shall restrict to this spin regime. The triaxially de-
of K = 0,2,4, configurations. The projected bands formed qp states are generated by the Nilsson Hamilto-
···
from these K = 0,2 and 4 intrinsic states are the dom- nian
inant components of the ground-, γ-, and 2γ-bands, re-
spectively [22]. 2 Qˆ +Qˆ
Hˆ =Hˆ ~ω ǫQˆ +ǫ′ +2 −2 . (4)
N 0 0
In the earlier TPSM analysis for even-even nuclei, the − 3 ( √2 )
shell model space was very restrictive, including only 0-
quasiparticle (qp) state [21, 22, 25, 26, 27, 28]. This Here Hˆ is the spherical single-particle Hamiltonian,
0
stronglylimitedtheapplicationofthe TPSMtothelow- which contains a proper spin-orbit force [31]. The pa-
spin and low-excitation region only. It was not possi- rameters ǫ and ǫ′ describe axial quadrupole and triax-
ble to study high-spin states because multi-qp config- ial deformations, respectively. It should be noted that
urations will usually become important for states with for the case of axial-symmetry, the qp vacuum state has
I >10inthenormallydeformedrare-earthnuclei. Inthe K = 0, whereas in the present case of triaxial deforma-
presentwork,theqp-spaceisenlargedtoincorporatethe tion, the vacuumstate Φ> is a superpositionof allthe
|
two-neutron-qp,two-proton-qpandfour-qpconfiguration possibleK-values. TheallowedvaluesoftheK-quantum
consisting of two protons plus two neutrons. This large number for a given intrinsic state are obtained through
qp space is adequate to describe the bands up to second thefollowingsymmetryconsideration. Forthesymmetry
bandcrossing [24]. The purpose of the present work is, operator, Sˆ=e−ıπJˆz, we have
as a first application of the extended model, to perform
a detailedinvestigationofthe high-spinbandstructures, PˆI Φ>=PˆI Sˆ†Sˆ Φ>=eıπ(K−κ)PˆI Φ>, (5)
MK| MK | MK|
in particular γ-bands, of Erbium isotopes ranging from
massnumberA=156to170. Inaparallelwork[29],the where Sˆ Φ>= e−ıπκ Φ>, and κ characterizes the in-
| |
TPSM analysis for odd-odd nuclei in a multi-qp space trinsicstatesinEq. (1). Fortheself-conjugatevacuumor
has been performed. 0-qpstate,κ=0and,therefore,itfollowsfromtheabove
equationthatonlyK =evenvaluesarepermittedforthis
The manuscript is organized in the following manner:
state. For2-qpstates,thepossiblevaluesforK-quantum
in the next section, a brief description of the TPSM
number are both even and odd depending on the struc-
method is presented. The results of the TPSM study
ture of the qp state. For the 2-qp state formed from the
are presented and discussed in section III. Finally, the
combination of the normal and the time-reversed states,
work is summarized in section IV.
κ = 0 and, therefore, only K = even values are permit-
ted. Forthecombinationofthetwonormalstates,κ=1
and only K = odd states are permitted.
As in the earlier projected shell model (PSM) calcu-
II. TRIAXIAL PROJECTED SHELL MODEL lations, we use the pairing plus quadrupole-quadrupole
APPROACH Hamiltonian [23]
1
Hˆ =Hˆ χ Qˆ†Qˆ G Pˆ†Pˆ G Pˆ†Pˆ . (6)
In the present work, the TPSM qp basis is extended, 0− 2 µ µ− M − Q µ µ
which consists of projected 0-qp vacuum, 2-proton (2p), Xµ Xµ
2-neutron (2n), and 4-qp states, i.e.,
The interaction strengths are taken as follows: The
QQ-force strength χ is adjusted such that the physi-
PˆI Φ>, cal quadrupole deformation ǫ is obtained as a result of
MK| the self-consistent mean-field HFB calculation [23]. The
PˆI a† a† Φ>,
PˆIMKa†p1a†p2|Φ>, (1) monopole pairing strength GM is of the standard form
MK n1 n2|
PˆI a† a† a† a† Φ> . G =[21.24 13.86(N Z)/A]/A,
MK p1 p2 n1 n2| M ∓ −
3
with “ ” for neutrons and “+” for protons, which ap- same names in the following discussion to be consistent
−
proximatelyreproducesthe observedodd–evenmass dif- withtheliterature,butstressthatinourfinalresultsob-
ferences in the rare-earth mass region. This choice of tainedafterdiagonalisation,K isnotastrictlyconserved
G is appropriatefor the single-particlespaceemployed quantum number due to configuration mixing.
M
in the PSM, where three major shells are used for each ItisevidentfromFig. 1thatthe(2,0)bandsfor156Er
typeofnucleons(N =4,5,6forneutronsandN =3,4,5 and 158Er lie very close to the (0, 0) bands. This means
for protons). The quadrupole pairing strength GQ is as- that γ- vibration has low excitation energy in these two
sumedtobeproportionaltoGM,andtheproportionality nuclei. For high-spin states, it is further noted that the
constantbeing fixedas0.18. Theseinteractionstrengths (0,0)and(2,0)bandenergiesbecomealmostdegenerate,
are consistent with those used earlier for the same mass and as a matter fact for I = 16 and above, the energy
region [21, 22, 23]. of even-spin states in the (2, 0) band is slightly lower
than the (0, 0) band. It is a well-known fact that γ-
bands become lower in energy with increasing triaxility
III. RESULTS AND DISCUSSIONS and what is also evident from Fig. 1 that they become
favored with increasing angular-momentum. As can be
The triaxial projected shell model calculations have seenfromFig. 1,the(2,0)bandsin156Erand158Eralso
been performed for Er-isotopes ranging from A = 156 depict pronounced signature splitting with the splitting
to 170. The deformation parameters (ǫ,ǫ′) used in the amplitude increasing with spin. The (4, 0) band is close
present work are same as those employed in Ref. [22]. to the (2, 0) band for 156Er and lies at a slightly higher
It has already been mentioned in section II that in the excitation energy for 158Er. The (4, 0) bands in these
presentworkthemean-fieldpotentialisconstructedwith two isotopes are also noted to have signature splitting
giveninput deformationvalues of ǫ and ǫ′. In a more re- forhigherangularmomenta,andthe splitting amplitude
alistic calculation, these deformation values for a given is nearly the same for the (2, 0) and (4, 0) bands.
system are obtained through the variational HFB calcu- In Fig. 1, several representative multi-qp bands,
lations. Thechosenvaluesofǫforthepresentcalculation namely projected 2- and 4-qp configurations, are also
arethosefromthe measuredquadrupoledeformationsof plotted. AlthoughtheK =12-qpneutron(1,2n)and2-
thenucleiasisdoneinthepreviousprojectedshellmodel qpproton(1,2p)bandsarecloseinenergyforlowspins,
analysis. Theǫ′ values usedinthe presentworkarereal- butwithincreasingspinthe2n-qpbandsarelowerinen-
istic, whichcorrectlyreproduce,for example,excitations ergythan2p-qpbandsduetolargerrotationalalignment.
of the γ band relative to the ground-state [22]. It is noted that neutrons are occupying 1i and pro-
13/2
tons are occupying 1h intruder sub-shells. For each
11/2
of the (1, 2n) and (1, 2p) bands, the projected energies
A. Band Diagrams are also shown for the corresponding γ-bands with con-
figurations(3,2n)and(3,2p). The(1,2n)bandisnoted
Banddiagramscanbringvaluableinformationregard- to cross the (2, 0) and the (0, 0) bands at I = 12. It is
ing the underlying physics [23]. These band diagrams also seen that the (3, 2n) band crosses the (0, 0) band
for the studied Er-isotopes are presented in Figs. 1 to 4 at a slightly higher spin value of I =14. It is interesting
and depict the results of the projected energies for each tonotethatafterthebandcrossing,thelowesteven-spin
intrinsic configuration. In the diagrams, the projected states originate from the (1, 2n) band, whereas the odd-
energiesareshownfor 0,2n,2pand2p+2nquasiparticle spin members are the projected states from the (3, 2n)
configurations. The qp energies for these configurations configuration. Finally, the 4-qp (4, 4) configuration lies
are given in the legend of each figure. As already men- at high excitation energies and does not become yrast,
tionedinthe lastsectionthatwiththe triaxialbasis,the at-least, up to the spin values shown in the figure.
intrinsic states do not have a well-defined K-quantum The band diagrams for 160Er and 162Er are presented
number. Each triaxial configuration in Eq.(1) is a com- inFig.2. Theenergyseparationbetweenthe(0,0)and(2,
position of several K-values and bands in Figs. 1 to 4 0)bandsislargerascomparedtothetwolighterisotopes
areobtainedbyassigningagivenK-valueintheangular- in Fig. 1. In the case of 160Er, the (2, 0) band energies
momentum projection oprator. To make the discussion do come close to the (0, 0) energies for spins I > 12.
easy,we denote a K-stateofani-configurationas(K,i), The (1, 2n) band againcrossesthe (0, 0) band atI =12
with i = 0, 2n, 2p and 4. For example, the K = 0 state for 160Er and at I = 14 for 162Er. The band diagrams
of 0-qp configuration is marked as (0,0) and K = 1 of for 164Er and 166Er shown in Fig. 3 depict larger energy
2n-qp configuration as (1, 2n). gapsamongvariousbands. Thesignaturesplittingofthe
InFigs. 1to4,theprojectedbandsassociatedwiththe (2, 0) bandhas considerably reduced. It is further noted
0-qpconfigurationareshownforK =0,2,and4,namely that2n-band-crossingisshiftedtohigherspinvalues. For
the(0,0),(2,0),and(4,0)bands. Intheliterature,these the case of 164Er, the band crossing is observed to occur
K =0,2,and4bandsarereferredtoasground-state,γ-, at I = 16 and for 166Er it occurs at I = 18. The band
and 2γ-bands. The ground-state band has κ =0 and is, diagrams for 168Er and 170Er shown in Fig. 4 indicate
therefore, comprised of only even-K values. We use the that the (2, 0) bands are quite highin excitationenergy.
4
The band crossing for these cases is further shifted to inant from I =12 to 16. For I = 18 and onwards, there
higher spin values. are many configurations with finite values contributing
to the yrast states. The band diagram of 156Er in Fig.
1 suggests that the γ-band should have the (2, 0) con-
B. Results after Configuration Mixing figuration as the dominant component. This is evident
from Fig. 7 and it is also noted that (0, 0) is significant
for the even spin states up to I = 8. The I = 10 state
In the second stage of the calculation, the projected
is mostly composed of (1, 2n) and for higher spin states
states obtained above are employed to diagonalize the
the (3, 2n) and (3, 2p) configurations are the dominant
shellmodelHamiltonianofEq. (6). Itistobementioned
components of the γ band. The 2γ band in Fig. 7 is
that for the discussion purpose, only the lowest three
composed of (4, 0) band for the low spin states. I = 8
bands from the 0-qp configurationand lowest two bands
of this band is predominantly composed of the (1, 2n)
for each other configuration have been shown in band
configuration,but the high spin states are found to have
diagram, Figs. 1 to 4. However, in the diagonalisation
quite a complex structure.
of the Hamiltonian, the basis states employed are much
more, which includes, for example, those K =1,3,5 and Theyrastwavefunctiondecompositionof164Er,shown
7 with κ=1 and K =0,2,4,6 and 8 with κ=0. inthetoppanelofFig. 8,indicatesthatthislowestband
The lowest three bands after the configuration mixing ispredominantlycomposedofthe(0,0)configurationup
are shown in Figs. 5 and 6 and are compared with the to I =12 andthere appearsto be verysmalladmixtures
experimentalenergieswhereveravailable. Althoughthey of K =2 and other configurations. After the bandcross-
are of mixed configurations in our model, we still call ing at I = 16, the yrast states are dominated by the (1,
them yrast, γ- and 2γ-bands to be consistent with the 2n) configuration. There is also a significant contribu-
literature. It is observed from these two figures that the tion of the (3, 2n) configuration after the bandcrossing.
agreement between the calculated and the experimen- The γ-band in Fig. 8 is primarily composed of the (2,
tal energies for the yrast and γ-bands is quite satisfac- 0) configuration up to I = 11 and above this spin the
tory. For156−164Er,thetheoreticalyrastlinedepictstwo states are a mixture of different configurations. There is
slopesandthese correspondtothe slopesoftwocrossing a clear distinction in the composition of the even- and
bandsshowninFigs. 1and4. Thisalsoindicatesthatthe odd-spin states above I = 11. The odd-spin states are
interactionbetween the two crossingbands is small with composed of the (3, 2n) and (2, 0) configurations, and
the result that these nuclei shall depict a back-bending theeven-spinstatesaredominatedbythe(1,2n)and(0,
effect [24]. It is also encouraging to note from Figs. 5 0) structures. The 2γ-band up to I = 7 is primarily the
and 6 that the agreement for the γ-bands is quite good, (4, 0) configuration. For I =8 and above, this band is a
except that for 164Er and 170Er, the signature splitting mixture of (1, 2n) and (3, 2n) configurations.
at the top of the bands is not reproduced properly. For The wavefunction analysis of 170Er shown in Fig. 9
the 2γ-bands, our calculations agree well with the only indicates that the yrast state, as expected for a well de-
available data in 166Er [9] and 168Er [10]. formed nuclei, is mainly comprised of the (0, 0) config-
There is another notable effect about anharmonicity uration. This contribution drops smoothly and, on the
in γ vibrations. If we regard the γ-bandhead as one γ- otherhand,the(1,2n)componentincreasessteadily. For
phonon vibrationandthe 2γ-bandheadas two γ-phonon I = 20, it is noted that the (0, 0) and (1, 2n) contribu-
vibration,itcanbeeasilyseenfromFigs. 5and6thatthe tions are almost identical and above this spin value, it
vibration is not perfectly harmonic. In fact, in the two is expected that the (1, 2n)configurationshalldominate
lightestisotopes,theγ-soft156Erand158Er,thevibration the yraststates. The γ-bandis also noted to have a well
is almost harmonic. As the neutron number increases, defined structure of (2, 0) and only for high spin states,
a clear anharmonicity is predicted from our calculation it is observed that the (1, 2n) and (3, 2n) of the 2n-
andthedegreeofanharmonicityincreaseswithincreasing alignedconfigurationbecomeimportant. The2γ-bandis
neutron number. dominated mostly by the aligning configurations above
I = 7. As is evident from the band diagram of this nu-
cleus,presented in Fig. 4, that 2n-aligned band is lower
C. Analysis of Wavefunction than the (4, 0) band for most of the spin values.
In order to probe further the structure of the bands
IV. SUMMARY AND CONCLUSIONS
presented in Figs. 5 and 6, the wavefunction decompo-
sition of the yrast, γ- and 2γ-bands are shown in Figs.
7, 8 and 9 for 156Er, 164Er and 170Er. For other nuclei, In the present work,the triaxialprojected shell model
the wavefunctionhave smiliar structure and are not pre- approach with extended basis has been employed to
sented. It is seen from Fig. 7 that the yrast band for study the high-spin band structures of the Er-isotopes
156Er is predominantly composed of the (0, 0) configu- from A = 156 to 170. In this model, the Hamiltonian
ration up to I = 10. The (0, 0) contribution suddenly employedconsistsofpairingplusquadrupole-quadrupole
dropsatI =10,and(1,2n)configurationbecomesdom- interaction. It is known that Nilsson deformed potential
5
is the mean-field of the quadrupole-quadrupole interac- 1. γ-bands are quite close to the yrast line for the
tion and this potential is directly used as the Hartree- neutron-deficient Er-isotopes, in particular, for
Fock field rather than performing the variational calcu- 156Er and 158Er. It is further evident from the
lations. It is, as a matter of fact, quite appropriate to present results that these γ-states become further
use the Nilssonstates as a starting basis because the pa- lower in energies for high-spin states. As a matter
rameters of this potential have been fitted to large body offact,for156Erand158Er,theybecomelowerthan
of experimental data. The parameters of the model are theground-statebandforI >14. Weproposethat
the deformation parameters of ǫ and ǫ′. The axial de- this is a feature of γ-soft nuclei.
formation parameter ǫ has been fixed from the observed
quadrupoledeformationofthe systemasis donein most 2. γ-bands are pushed up in energy with increasing
of the projected shell model analysis. The non-axial neutron number, and further the degree of anhar-
parameter ǫ′ was chosen to reproduce the bandhead of monicity of γ vibration also increases.
the γ band. The pairing strength parameters have been
determined to reproduce the odd-even mass differences. 3. The wavefunction decomposition of the bands
The monopole pairing interactionhas been solvedin the demonstrates that for neutron deficient Er-
BCS approximation and the qp states generated. In the isotopes,thereisasignificantmixtureoftheγ con-
present work, the qp states considered are: 0-qp, 2-qp figurationin the ground-statebandand vice-versa.
neutron, 2-qp proton, and the 4-qp state of 2-neutron The neutronrich170Ernucleus,onthe otherhand,
plus 2-proton. has the intrinsic structures as expected for a well
In the second stage of the calculations, the three- deformednucleuswiththeground-statebandcom-
dimensional angular-momentum projection is performed posed of nearly pure K =0 configuration.
to project out the good angular-momentum states from
these qp states. These projected states are then used Y.S. is supported by the Chinese Major State
as the basis to diagonalise the shell model Hamiltonian Basic Research Development Program through grant
in the third and the final stage. The salient features of 2007CB815005, and by the the U. S. National Science
results obtained in the present work are: Foundation through grant PHY-0216783.
[1] C. Fahlander, A. Baklin, L. Hasselgren, A. Kavka, V. (1988).
Mittal, L. E. Svensson, B. Varnesting, D. Cline, B. [14] M.K.JammariandR.Piepenbring,Nucl.Phys.A 487,
Kotlinski,H.Grein,E.Grosse,R.Kulessa,C.Michel,W. 77 (1988).
Spreng,H.j.Wollersheim,andJ.Stachel,Nucl.Phys.A [15] D. G. Burke and P. C. Sood, Phys. Rev. C 51, 3525
485, 327 (1988). (1994).
[2] T. Belgya, G. Molnar, and S. W. Yates, Nucl. Phys. A [16] K. Kumar, Nuclear Models and Search for unity in
607, 43 (1996). Nuclear Physics (Universitetforlaget, Bergen, Norway,
[3] H.G. Borner and J. Joli, J. Phys. G 19, 217, (1993). 1984).
[4] A. Bohr and B. R. Mottelson, Nuclear Structure, Vol. 2 [17] A. Arima and F. Iachello, Phys. Rev. Lett. 35 1069
(Benjamin Inc., New York,1975). (1975).
[5] R. F. Casten, Nuclear Structure from a Simple Perspec- [18] N. Yoshinaga, Y. Akiyama, and A. Arima, Phys. Rev.
tive, Second Edition (Oxford University Press, 2000). 56, 1116 (1986).
[6] A. Guessous, N. Schulz, W. R. Phillips, I. Ahmad, M. [19] O.Castanos,J.P.Draayer,andY.Leschber,Ann.Phys.
Bentaleb, J. L. Durell, M. A. Jones, M. Leddy, E. (New York) 180, 290 (1987).
Lubkiewicz, L. R. Morss, R. Piepenbring, G. Smith, [20] P. E. Garrett, M. Kadi, Min Li, C. A. McGrath, V.
W. Urban, and B. J. Varley, Phys. Rev. Lett. 75, 2280 Sorokin, M. Yehand, and S. W. Yates, Phys. Rev. Lett.
(1995). 78, 4545 (1996).
[7] X. Wu, A. Aprahamian, S. M. Fischer, W. Reviol, G. [21] J. A. Sheikh and K. Hara, Phys. Rev. Lett. 82, 3968
Liu, and J. X. Saladin, Phys. Rev. C 49, 1837 (1993). (1999).
[8] H. G. Borner, J. Jolie, S. J. Rabinson, B. Krusche, R. [22] Y.Sun,K.Hara,J.A.Sheikh,J.G.Hirsch,V.Velazquez,
Piepenring, R. F. Lasten, A. Aprahamian, and J. P. and M. Guidry,Phys. Rev. C 61, 064323 (2000).
Draayer, Phys.Rev.Lett. 66, 691 (1991). [23] K.Haraand Y.Sun,Int.J.Mod. Phys.E 4,637 (1995)
[9] C. Fahlander, A. Axelsson, M. Heinebrodt, T. H¨artlein, [24] K. Hara and Y. Sun,Nucl. Phys.A 529, 445 (1991)
and D. Schwalm, Phys.Lett. B 388, 475 (1996). [25] J. A. Sheikh, Y. Sun, and R. Palit, Phys. Lett. B 507,
[10] T. H¨artlein, M. Heinebrodt, D. Schwalm, C. Fahlander, 115 (2001).
Eur. Phys.J. A 2, 253 (1998). [26] Y.Sun,J.A.Sheikh,andG.-L.Long,Phys.Lett.B533,
[11] V. G. Soloviev and N. Yu. Shirikova, Z. Phys. A 301, 253 (2002).
263 (1981). [27] P.Boutachkov,A.Aprahamian,Y.Sun,J.A.Sheikh,S.
[12] V.G.Soloviev,Theory of Atomic Nuclei: Quasiparticles Frauendorf, Eur.Phys. J. A 15, 455 (2002).
and Phonons (IOP,London,1992). [28] Y. Sun, G.-L. Long, F. Al-Khudair, and J. A. Sheikh,
[13] J. Leandri and R. Piepenbring, Phys. Rev. C 37, 2779 submitted.
6
[29] Z.-C. Gao, Y.-S. Chen, and Y. Sun, Phys. Lett. B 634, S. Wycech, C. Gustafson, I. Lamm, P. Moller, and B.
195 (2006). Nilsson, Nucl. Phys.A 131, 1 (1969).
[30] P.RingandP.Schuck,TheNuclearManyBodyProblem [32] Y. Sun, P. Ring, and R. S. Nikam, Z. Phys. A 339, 51
(Springer,New York,1980). (1991).
[31] S.G.Nilsson,C.F.Tsang,A.Sobiczewski,Z.Szymanski,
7
6 156
Er
4
)
V
e
M
( 2

E
(0,0)
(2,0)
(4,0)
0
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,4)
(4,4)
6 158
Er
4
)
V
e 2
M
(
(0,0)
E
(2,0)
(4,0)
0
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,4)
(4,4)
0 2 4 6 8 10 12 14 16 18 20
Spin
FIG. 1: Band diagrams for 156−158Er isotopes. The labels (0,0), (2,0), (4,0), (1,2n), (3,2n), (1,2p), (3,2p), (2,4) and (4,4)
correspond to ground, γ, 2γ, two neutron-aligned, γ-band on this two neutron-aligned state, two proton-aligned, γ-band on
two this proton-aligned state, two-neutron plustwo-proton aligned band and γ band built on this four-quasiparticle state.
8
6
160
Er
4
)
V
e
M 2
(

E
(0,0)
(2,0)
0 (4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,4)
(4,4)
6
162
Er
4
) 2
V
e
M
( (0,0)

E
(2,0)
0 (4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,2)
(4,4)
0 2 4 6 8 10 12 14 16 18 20
Spin
FIG. 2: Band diagrams for 160−162Er isotopes. Thelabels indicate thebandsmentioned in thecaption of Fig. 1.
9
164
Er
4
)
V
e
M 2
(

E
(0,0)
(2,0)
0
(4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2
(2,4)
(4,4)
166
Er
4
2
)
V
e
M
(
(0,0)
E
(2,0)
0
(4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2
(2,4)
(4,4)
0 2 4 6 8 10 12 14 16 18 20
Spin
FIG. 3: Band diagrams for 164−166Er isotopes. Thelabels indicate thebandsmentioned in thecaption of Fig. 1.
10
6
168
Er
4
)
V
e
M
2
(

E
(0,0)
(2,0)
0 (4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,4)
(4,4)
6
170
Er
4
)
V 2
e
M
(
(0,0)
E
(2,0)
0 (4,0)
(1,2n)
(3,2n)
(1,2p)
(3,2p)
-2 (2,4)
(4,4)
0 2 4 6 8 10 12 14 16 18 20
Spin
FIG. 4: Band diagrams for 168−170Er isotopes. Thelabels indicate thebandsmentioned in thecaption of Fig. 1.