Systematic investigation of the rotational bands in nuclei with $Z \approx 100$ using a particle-number conserving method based on a cranked shell model

0.86 MB·English

by  Zhen-Hua Zhang

Systematic investigation of the rotational bands in nuclei with Z ≈ 100 using a
particle-number conserving method based on a cranked shell model
Zhen-Hua Zhang,1 Xiao-Tao He,2 Jin-Yan Zeng,3 En-Guang Zhao,1,4,3 and Shan-Gui Zhou1,4,∗
1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
2College of Material Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
3School of Physics, Peking University, Beijing 100871, China
4Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
(Dated: January 11, 2012)
The rotational bands in nuclei with Z ≈ 100 are investigated systematically by using a cranked
shell model (CSM) with the pairing correlations treated by a particle-number conserving (PNC)
method, in which the blocking effects are taken into account exactly. By fitting the experimental
2
single-particle spectra in these nuclei, a new set of Nilsson parameters (κ and µ) and deformation
1
parameters (ε and ε ) are proposed. The experimental kinematic moments of inertia for the
0 2 4
rotational bands in even-even, odd-A and odd-odd nuclei, and the bandhead energies of the 1-
2
quasiparticle bands in odd-A nuclei, are reproduced quite well by the PNC-CSM calculations. By

n analyzing the ω-dependenceof the occupation probability of each cranked Nilsson orbital near the
a Fermisurfaceandthecontributionsofvalenceorbitalsineachmajorshelltotheangularmomentum
J
alignment, the upbendingmechanism in this region is understood clearly.

0
1 PACSnumbers: 21.60.-n;21.60.Cs;23.20.Lv;27.90.+b

]
h
I. INTRODUCTION magic number [13–16]; predictions from non-relativistic
t
- mean field models with Skyrme forces are Z =114, 120,
l
c Since the importance of shell effects on the stabil- 124, or 126, depending on the parametrization[13]. The
u ity of superheavy nuclei (SHN) was illustrated [1] and shell structure of SHN is important not only for the lo-
n
[ the existence of an island of stability of SHN was pre- cation of the island of stability, but also for the study
dicted around Z = 114 and N = 184 [2–5], a lot of of the synthesis mechanism of SHN [17–26], particularly

2 efforts have focused on the exploration of SHN. Great for the survival probability of the excited compound nu-
v experimental progresses have been made in synthesizing clei [27, 28].
4
the superheavy elements (SHE). Up to now, SHE’s with To learn more about the shell structure of SHN, an
3
1 Z ≤ 118 have been synthesized via cold and hot fusion indirectwayistostudylighternucleiinthedeformedre-
6 reactions [6–8]. However, these SHN are all neutron de- gionwithZ ≈100andN ≈152. Thestronglydownslop-
. ficient with neutron numbers less by at least 7 than the ing orbitals originating from the spherical subshells and
0
1 predicted next neutron magic number 184. Therefore, active in the vicinity of the predicted shell closures may
1 one still can not make a definite conclusion about the comeclosetotheFermisurfaceinthesedeformednuclei.
1 location of the island of stability. Therotationalpropertiesofnucleiinthismassregionare
: The single particle shell structure is crucial for the lo- affected strongly by these spherical orbitals. For exam-
v
i cationoftheislandofstability. Forexample,whetherthe ples, the π1/2−[521] and π3/2−[521] orbitals are of par-
X nextshellclosureofprotonsappearsatZ =114or120is ticular interest since they stem from the spherical spin
r mainly determined by the splitting of the spin doublets doublets π2f5/2,7/2 orbtials.
a
π2f . Experimentally one can not investigate di- Both the in-beam spectroscopy and spectroscopy fol-
5/2,7/2
rectlythesingleparticlelevelstructureofSHNwithZ ≥ lowing the decay of isomeric states or alpha decays
110 because the production crosssections of these nuclei have been used to study nuclei with Z ≈ 100 [29–
aretinywhichmakesthespectroscopyexperimentimpos- 32]. These nuclei are well deformed; for examples, the
sible at present. Theoretically different models usually quadrupole deformation parameter β ≈ 0.28 for 250Fm
2
predict different closed shells beyond 208Pb; even within and 252,254No [33–35]. Many high-spin rotational bands
the same model there are parameter-dependent predic- in even-even nuclei (e.g., 248,250,252Cf [36], 250Fm [34]
tions [9]. For examples, the macroscopic-microscopic 252,254No [33, 35]) and odd-A nuclei (e.g., 247,249Cm and
models or the extended Thomas-Fermi-Strutinsky inte- 249Cf [37], 253No [38, 39], 251Md [40], 255Lr [41]) have
gral approach predict that the next shell closure for the been established in recent years. The study of these nu-
proton is at Z = 114 [10–12]; the relativistic mean field clei is certainly also very interesting in itself. The ro-
models predict Z = 114 or 120 to be the next proton tational spectra in these nuclei can reveal detailed in-
formation on the single-particle configurations, the shell
structure, the stability against rotation, the high-K iso-
merism, etc..
∗ [email protected] Theoretically, the deformations, the shell structure,
2
the rotational properties, and high-K isomeric states calculations for low-lying excited states both workable
havebeen studied by using the self-consistentmeanfield andsufficientlyaccurate[63,67]. AneigenstateofH
CSM
models[42–45],themacroscopic-microscopicmodels[45– can be written as
51], the projected shell model [52–54], the cranked shell
models[55–57],thequasiparticle(qp)phononmodel[58], |ψi=XCi|ii, (Ci real) , (4)
the particle-triaxial-rotor model [59], the heavy shell i
model[60],etc.. ContinuingtherecentlypublishedRapid
where |ii is a CMPC (an eigenstate of the one-body op-
Communication[57],inthiswork,wepresentresultsofa
eratorH ). BydiagonalizingH inasufficientlylarge
systematic study of the single particle structure and ro- 0 CSM
CMPCspace,sufficientlyaccuratesolutionsforlow-lying
tational properties of nuclei with Z ≈100. The cranked
excited eigenstates of H are obtained.
shell model (CSM) with pairing correlations treated by CSM
The angularmomentum alignmentof|ψi canbe sepa-
a particle-number conserving (PNC) method [61, 62] is
rated into the diagonal and the off-diagonalparts,
used. In contrary to the conventional Bardeen-Cooper-
Spcroharicehff,eirn(BthCeSP)NorCHmarettrheoe-dF,otchke-BHoagmoliyltuobnoivan(HisFBso)lvaepd- hψ|Jx|ψi=XCi2hi|Jx|ii+2XCiCjhi|Jx|ji , (5)
i i<j
directly in a truncated Fock-space [63]. So the particle-
number is conserved and the Pauli blocking effects are and the kinematic moment of inertia (MOI) of |ψi is
taken into account exactly. Noted that the PNC scheme
has been implemented both in relativistic and nonrela- 1
J(1) = hψ|J |ψi . (6)
tivistic mean field models [64, 65] in which the single- ω x
particle states are calculated from self-consistent mean
Considering J to be a one-body operator, the matrix
field potentials instead of the Nilsson potential. x
element hi|J |ji for i 6= j is nonzero only when |ii and
Thepaperisorganizedasfollows. Abriefintroduction x
|ji differ by one particle occupation [62, 68]. After a
of the PNC treatment of pairing correlations within the
certainpermutationofcreationoperators,|iiand|jican
CSM and applications of the PNC-CSM are presented
be recast into
in Sec. II. The numerical details, including a new set of
Nilsson parameters (κ,µ), the deformation parameters |ii=(−1)Miµ|µ···i , |ji=(−1)Mjν|ν···i , (7)
and pairing parameters are given in Sec. III. The PNC-
CSM calculation results for even-even, odd-A and odd- wheretheellipsisstandsforthesameparticleoccupation,
odd nuclei are presented in Sec. IV. A brief summary is and (−1)Miµ =±1, (−1)Mjν =±1 according to whether
given in Sec. V. thepermutationisevenorodd. Therefore,thekinematic
MOI of |ψi can be written as
II. THEORETICAL FRAMEWORK J(1) =Xjµ(1)+Xjµ(1ν) , (8)
µ µ<ν
The cranked Nilsson Hamiltonian of an axially sym-
where
metric nucleus in the rotating frame reads [62, 66],
n
j(1) = µhµ|j |µi ,
HCSM =H0+HP =HNil−ωJx+HP , (1) µ ω x
2
wChoreiroeliHsiNnitlerisactthioenNwilistshonthHeacmrainltkoinniganfre[q4]u,e−ncωyJωx aisbothuet jµ(1ν) = ωhµ|jx|νiX(−1)Miµ+MjνCiCj , (µ6=ν) ,
i<j
the x axis (perpendicular to the nuclear symmetry z
and
axis). H =H (0)+H (2) is the pairing interaction:
P P P
HP(0)=−G0Xa†ξa†ξ¯aη¯aη , (2) nµ =X|Ci|2Piµ , (9)
i
ξη
HP(2)=−G2Xq2(ξ)q2(η)a†ξa†ξ¯aη¯aη , (3) is the occupation probability of the cranked orbital |µi,
P =1 if |µi is occupied in |ii, and P =0 otherwise.
ξη iµ iµ
We note that because Rx(π) = e−iπJx, [Jx,Jz] 6= 0,
where ξ¯ (η¯) labels the time-reversed state of a Nilsson the signature scheme breaks the quantum number K.
state ξ (η), q2(ξ) = p16π/5hξ|r2Y20|ξi is the diago- However,it has been pointed out that [62, 69], although
nal element of the stretched quadrupole operator, and [J ,J ] 6= 0, [R (π),J2] = 0. Thus we can construct si-
x z x z
G and G are the effective strengths of monopole and multaneouseigenstatesof(R (π),J2). EachCMPC|iiin
0 2 x z
quadrupole pairing interactions, respectively. Eq.(4)ischosenasasimultaneouseigenstateof(H ,J2).
0 z
Instead of the usual single-particle level truncation It should be noted that, though the projection K of the
in common shell-model calculations, a cranked many- total angular momentum of a deformed spheroidal nu-
particle configuration (CMPC) truncation (Fock space cleus is a constant of motion, K can not keep constant
truncation)is adoptedwhichiscrucialto makethe PNC when the rotational frequency ω is non-zero due to the
3
Coriolis interaction. However, in the low-ω region, K
0.30
may be served as an useful quantum number character-
Present work
izing a low-lying excited rotational band. 0.28 FRDM Cm Cf
The PNC-CSM treatment has been used to describe MM
0.26 Exp.
successfullythehigh-spinstatesofthenormallydeformed
nuclei in the rare-earth and the actinide region, and the 0.24
superdeformed nuclei in A ≈ 190 region. The multi-
0.22
qp high-K isomer states are investigated in detail in the 2
well-deformed Lu (Z=71), Hf (Z=72) and Ta (Z=73) 0.30
isotopes [70, 71]. The backbendings in Yb (Z=70) Fm No
0.28
and Tm (Z=68) isotopes are understood clearly, espe-
cially the occurrence of sharp backbending in some nu- 0.26

clei [72, 73]. The upbending mechanisms in the actinide
0.24
nuclei 251Md and 253No are also analyzed [56]. In the
superdeformed nuclei around A≈190 region, the effects 0.22
of the quadrupole pairing on the downturn of dynamic 0.20
MOI’s are analyzed [66], the evolution of the dynamic 144 146 148 150 152 154 144 146 148 150 152 154

MOI’s and the alignment additivity, which come from N
thecontributionsoftheinterferenceterms,havebeenin- FIG.1. (Coloronline)Thequadrupoledeformations givenin
vestigated [74]. TableII(blacksquares)andthosepredictedbyamacroscopic-
Some general features in nuclear structure physics microscopic (MM) model [46, 48] (blue triangles) and the fi-
have also been explained well in the PNC-CSM scheme, niterangedropletmodel(FRDM)[11](redsolidcircles). The
experimental values for 250Fm [34] and 252,254No [35] (pink
e.g., the large fluctuations of odd-even differences in
stars) are also shown for comparison.
MOI’s [68], the nonadditivity in MOI’s [75], the mi-
croscopic mechanism of identical bands in normally de-
formed nuclei and super-deformed nuclei [76, 77], the
TABLE II. Deformation parameters ε and ε used in the
2 4
non-existence of the pairing phase transition [78], etc..
PNC-CSM calculations for even-evennuclei with Z ≈100.
In Ref. [57], the high-spin rotational bands in
247,249Cm and 249Cf established in Ref. [37] have been N 144 146 148 150 152 154
calculatedusingthePNC-CSMmethodandtheupbend- Z =96 240Cm 242Cm 244Cm 246Cm 248Cm 250Cm
ing mechanismarediscussed. This paperis anextension ε 0.225 0.230 0.235 0.240 0.245 0.240
2
of [57], providing more details and a systematic investi- ε −0.015 −0.010 −0.005 0.000 0.005 0.01
4
gation of nuclei with Z ≈100. Z =98 242Cf 244Cf 246Cf 248Cf 250Cf 252Cf
ε 0.230 0.235 0.240 0.245 0.250 0.245
2
ε −0.01 −0.005 0.000 0.005 0.010 0.015
4
III. NUMERICAL DETAILS Z =100 244Fm 246Fm 248Fm 250Fm 252Fm 254Fm
ε 0.235 0.240 0.245 0.250 0.255 0.250
2
ε −0.005 0.000 0.005 0.010 0.015 0.02
A. A new set of Nilsson parameters 4
Z =102 246No 248No 250No 252No 254No 256No
ε 0.240 0.245 0.250 0.255 0.260 0.255
2
ε 0 0.005 0.010 0.015 0.020 0.025
4
TABLEI. Nilssonparametersκandµproposedforthenuclei
with Z ≈ 100, which has been given in Ref. [57].
set of Nilsson parameters (κ, µ) which are dependent on
N l κp µp N l κn µn
the main oscillator quantum number N as well as the
4 0,2,4 0.0670 0.654
orbitalangularmomentuml [57]. Herefor the complete-
5 1 0.0250 0.710 6 0 0.1600 0.320
3 0.0570 0.800 2 0.0640 0.200 ness, we include them again in Table I. Note that the
5 0.0570 0.710 4,6 0.0680 0.260 readjustment of Nilsson parameters is also necessary in
6 0,2,4,6 0.0570 0.654 7 1,3,5,7 0.0634 0.318 someothermass regionsofthe nuclearchart[80–82]and
the l-dependence was already included in Refs. [80, 81].
The conventional Nilsson parameters (κ,µ) proposed
inRefs.[4,79]areoptimizedtoreproducetheexperimen- B. Deformation parameters
tal level schemes for the rare-earth and actinide nuclei.
However, these two sets of parameters can not describe There are not enough experimental values of the de-
well the experimental level schemes of nuclei studied in formation parameters for the very heavy nuclei with
thiswork. Byfittingtheexperimentalsingle-particlelev- Z ≈ 100. According to the data, the quadrupole de-
els in the odd-A nucleiwith Z ≈100,weobtaineda new formation parameter β ≈ 0.28 ± 0.02 for 250Fm [34],
2
4
252,254No [35]. The values used in various calcula-
tions or predicted by different models are quite differ- Exp. 248
ent. Figure. 1 shows the experimental quadrupole de- 120 Cal. Dim=1000 Cm gsb
)
formation (pink stars in Fig. 1) and those predicted in -1 V Cal. Dim=1500
e
the macroscopic-microscopic (MM) model [46, 48] (blue M sum
triangles in Fig. 1) and the finite range droplet model 80
2
(FRDM) [11] (red solid circles in Fig. 1). The deforma- (
tion parameters given in these two MM models [46, 48] 1) neutron
(
are very close to each other, hence we only show the J 40
results given in Ref. [48]. From Fig. 1 we can see that
proton
thedeformationparametersfromMMandFRDMdonot
agreewiththeexperimentalvalues,especiallythosefrom 0
the FRDM. One can find a general trend in Fig. 1, i.e., 0.0 0.1 0.2 0.3
both the experimental values and the predicted values (MeV)
indicate that the deformations reachmaximum at 254No
FIG. 2. (Color online) The calculated MOI’s of the GSB
(Z =102andN =152)partlyaccordingto whichwefix in 248Cm using different dimensions of the CMPC space
the deformation parameters used in the present study. [1000 (black solid lines) and 1500 (red dotted lines), respec-
The deformations are input parameters in the PNC- tively]. The experimental values are denoted by solid cir-
CSM calculations (black squares in Fig. 1). They are cles. The effective pairing strengths used in the calculation,
chosen to be close to existed experimental values and when the dimensions are 1500 (1000), are Gp = 0.35 MeV,
change smoothly according to the proton and the neu- G2p = 0.03 MeV, Gn = 0.27 MeV, G2n = 0.013 MeV (Gp
tron number. The deformation parameters ε and ε = 0.40 MeV, G2p = 0.035 MeV, Gn = 0.30 MeV, G2n =
2 4
0.020 MeV as given in Table III).
used in our PNC-CSM calculations for even-even nuclei
with Z ≈ 100 are listed in Table II. The deformations
of odd-A and odd-odd nuclei are taken as the averageof
the neighboring even-even nuclei. These parameters we
with weight >0.1% are included in.
choose may have some discrepancy from the empirical
values which may lead to some deviations in the single- ThestabilityofthePNCcalculationresultsagainstthe
particle levels, if the single particle levels are very close changeofthedimensionoftheCMPCspacehasbeenin-
to each other. For example, the level sequence of the 1- vestigated in Refs. [62, 67, 77]. A larger CMPC space
qp bands will change in an isotonic or an isotropic chain with renormalized pairing strengths gives essentially the
(e.g., see Figs. 6 and 11) due to the deformation stag- same results. The calculated MOI’s of the ground state
gering in the neighboring nuclei. From Fig. 1 it can be band (GSB) in 248Cm using different dimensions of the
seenthatthisstaggeringexistedintheFRDM.Somepre- CMPCspace(1000and1500,respectively)arecompared
vious calculations have shown that the octupole correla- inFig.2. Thereddottedlineistheresultcalculatedwith
tionsplayimportantrolesinthismassregion[53,83,84]. the dimensions ofCMPC spaceabout1500bothforpro-
In the present work, the octupole deformation is not in- tons and neutrons. The effective pairing strengths used
cluded yet. We note that the octupole effect may mod- in the calculationare Gp = 0.35 MeV, G2p = 0.03 MeV,
ify the single-particle level scheme. One example will Gn = 0.27 MeV, and G2n = 0.013 MeV, which are a lit-
be discussed later for the low excitation energy of the tle smaller than those used when the CMPC dimensions
ν5/2+[622] states in the N =151 isotones in Sec. IVB. are about 1000, i.e., Gp = 0.40 MeV, G2p = 0.035 MeV,
G = 0.30 MeV, and G = 0.020 MeV (see Table III).
n 2n
We can see that these two results agree well with each
otherandtheybothalsoagreewellwiththeexperiment.
C. Pairing strengthes and the CMPC space
So the solutions to the low-lying excited states are quite
satisfactory.
The effective pairing strengths G and G can be de-
0 2
termined by the odd-even differences in nuclear binding
energies. They are connected with the dimension of the
truncatedCMPCspace. The CMPCspacefortheheavy TABLEIII.EffectivepairingstrengthsusedinthePNC-CSM
calculations for thenuclei with Z ≈ 100.
nuclei studied in this work is constructed in the proton
N =4,5,6shellsandtheneutronN =6,7shells. Thedi-
even-even odd-N odd-Z odd-odd
mensionsoftheCMPCspaceforthenucleiwithZ ≈100
are about 1000 both for protons and neutrons. The cor- G0p (MeV) 0.40 0.40 0.25 0.25
G2p (MeV) 0.035 0.035 0.010 0.010
responding effective monopole and quadrupole pairing
G0n (MeV) 0.30 0.25 0.30 0.25
strengths are shown in Table III. As we are only inter- G2n (MeV) 0.020 0.015 0.020 0.015
ested in the yrast and low-lying excited states, the num-
ber of the important CMPC’s involved (weight >1%) is
notverylarge(usually<20)andalmostalltheCMPC’s
5
IV. RESULTS AND DISCUSSIONS this deviation is about 200 keV by using the Woods-
Saxon potential [50].
A. Cranked Nilsson levels Figures 5-12 show experimental and calculated band-
headenergiesofthelow-lying1-qpbandsforeven-Z and
odd-N isotones with N = 145−153 (one-quasineutron
Figure 3 shows the calculated cranked Nilsson levels
states)andodd-Z andeven-N isotopeswithZ =97−103
near the Fermi surface of 250Fm. The positive (nega-
(one-quasiproton states). In these figures, 1-qp states
tive) parity levels are denoted by blue (red) lines. The
with energies around or less than 0.8 MeV are shown
signature α = +1/2 (α = −1/2) levels are denoted by
and the positive and negative parity states are denoted
solid (dotted) lines. For protons, the sequence of single-
by black and red lines, respectively. Generally speaking,
particle levels near the Fermi surface is exactly the same
the agreement between the calculation and the experi-
asthatdeterminedfromthe experimentalinformationof
mentissatisfactory. Nextwediscussthese1-qpstatesin
249Es[31]. Thesequenceofsingle-neutronlevelsnearthe
details.
Fermi surface is also very consistent with the one deter-
mined from the experimental information of 251Fm [31], Theexperimentalandcalculatedbandheadenergiesof
with the only exception of the ν5/2+[622] orbital which low-lying one-quasineutron bands for the N = 145 iso-
tonesarecomparedinFig.5. Thedataareavailableonly
will be discussed in next subsection.
for241Cm[86]and243Cf[87]andtheyarereproducedby
From Fig. 3 it is seen that there exist a proton gap at
the theory quite well. The groundstate ofeachN =145
Z =100 and a neutron gap at N =152, which is consis-
isotonestudiedinthisworkisν1/2+[631]. Theenergyof
tent with the experiment and the calculation by using a
ν5/2+[622] steadily becomes smaller with Z increasing
Woods-Saxon potential [85]. The position of the proton
both in the experimental spectra and in the calculated
orbitals 1/2−[521] and 3/2−[521] is very important, be-
ones. The lowering of ν7/2+[624] with Z increasing is
causetheystemfromthesphericalspinpartners2f
5/2,7/2 also seen in the experimental spectra but it is not so
orbitals. The magnitude of the spin-orbital splitting of
striking from the calculation. In each of these isotones a
these spin partners determines whether the next proton
low-lying state ν7/2−[743] is predicted which is not ob-
magicnumberisZ =114or120. Notedthatthecranked
served. We note that this prediction is consistent with
relativistic Hartree-Bogoliubov theory has been used to
Ref. [50].
investigatethe spin-orbitalsplitting ofthis spin partners
Low-lying one-quasineutron levels for N = 147 iso-
in Ref. [42]. The calculated proton levels indicate that
tones 243Cm [88], 245Cf [87], and 247Fm [89] were identi-
the Z = 120 gap is large whereas the Z = 114 gap is
fied experimentally. Their ground states are ν5/2+[622],
small.
ν1/2+[631], and ν7/2+[624] respectively. However, the
Theneutron1/2+[620]orbitalisthehighest-lyingneu-
groundstateforalltheseisotonesisν5/2+[622]fromthe
tron orbital from above the N = 164 spherical subshell
PNC-CSM as seen in Fig. 6. The same is obtained re-
(2g ), based on which a high-spin rotational band has
7/2 cently by using a two-center shell model [45]. In our
been established in 249Cm [37]. This orbital also brings
calculations, the energies of these three levels are very
new information toward studying those states close to
close to each other, a small change in the deformation
the next closed shell for neutrons.
parameters would modify the order of these levels. This
might be one of the reasons for the disagreement in the
ground state configuration between the theory and the
B. 1-qp bandhead energies experiment. Similar situation happens in the Bk and
Es isotopes, in which the energies of π3/2−[521] and
Firstwechoose245Cmasanexampletoshowthecom- π7/2+[633] are very close to each other and these two
parison of theoretical 1-qp bandhead energies with the one-quasiprotonlevels cross each other at N =152.
data in Fig. 4. It can be seen that the calculated results Low-lying one-quasineutron levels for N = 149 iso-
obtainedbyusingtheconventionalNilssonparameters[4] tones 245Cm [90], 247Cf [91], 249Fm [92] and 251No [89]
deviate from the experimental values very much. With were identified experimentally. All their ground states
the modified parameters given in Table I, a remarkably are ν7/2+[624]. The comparison with the experiment is
goodagreementwiththedatacanbeachieved. Forcom- showninFig.7. Nearlyallofthese1-quasineutronband-
parison, the results from the Woods-Saxon (WS) poten- head energiesare well reproducedby the PNC-CSM cal-
tial [50] is also given (the bandhead energies are taken culations. The energy of ν1/2+[631] is much lower in
fromTableIIofRef.[50]exceptthatthatof7/2+[613]is 251No than that in 245Cm, but in our calculations this
takenfromFig. 1ofthesamearticle). Itshouldbenoted levelseems almostunchangewith the protonnumber in-
that generally speaking, for the 1-qp spectra of nuclei creasing. The calculated results are very similar with
with Z ≈ 100 the Woods-Saxon potential gives a better that in Ref. [93]. For each observed level, the energy
agreement with the data. For example, the root-mean- reachesmaximumin247Cf,whichcannotbe reproduced
square deviation of theoretical 1-qp bandhead energies by our calculations.
from the experimental values is about 270 keV for neu- Many theoretical models predict that the first excited
trons by using the modified Nilsson parameters, while stateinN =151isotonesshouldbeν7/2+[624](see,e.g.,
6

)0 6.8 (a) proton (b) neutron
[512]5/2 [615]9/2
7.8
( [725]11/2
s 6.7 [761]1/2
vel [624]9/2 7.7 [613]7/2
e [622]3/2
l 6.6 [521]1/2 [620]1/2
n
o [514]7/2 7.6 N=152
s Z=100
s 6.5 [734]9/2
Nil [[562313]]37//22 [624]7/2
7.5
ed 6.4 [400]1/2 [622]5/2
k [642]5/2 [631]1/2
n [743]7/2
a [523]5/2 7.4 [606]13/2
r 6.3 [402]3/2
C
[651]3/2
[505]11/2
7.3 [631]3/2
6.2 [660]1/2 [752]5/2
-0.1 0.0 0.1 0.2 0.3 0.4 -0.1 0.0 0.1 0.2 0.3 0.4
(MeV) (MeV)
FIG. 3. (Color online) The cranked Nilsson levels near the Fermi surface of 250Fm (a) for protons and (b) for neutrons. The
positive (negative) parity levels are denoted by blue (red) lines. The signature α = +1/2 (α = −1/2) levels are denoted by
solid (dotted) lines.
800 +
V) 1/2+[620]
7/2[613]
e -
k 7/2[743]
y ( 600
g
r
e
n
e 400 9/2-[734]
d 1/2+[631]
a
e +
h 5/2[622]
nd 200
a 245
b Cm
+
0 7/2[624]
Exp. NilLund NilNew WS
FIG. 4. (Color online) Comparison between experimental and theoretical 1-qp bandhead energies for 245Cm. The Lund
systematic Nilsson parameters are taken from [4]. The results using Woods-Saxon potential are taken from [50]. The positive
and negative parity states are denoted by black and red lines, respectively.
Ref. [50]). This is not consistent with experimental re- ment[39]observedaverysimilarlevelschemewhichwas,
sults, i.e., the first excited state in N = 151 isotones is however, assigned as the ν7/2−[734] configuration. This
ν5/2+[622]. Thelowexcitationenergyofthe ν5/2+[622] will be discussed in Sec. IV.
states in the N = 151 isotones have been interpreted as
a consequence of the presence of a low-lying Kπ = 2− Theexperimentalandcalculatedbandheadenergiesof
octupole phonon state [97]. Noted that in Ref. [42], by low-lying one-quasineutron bands for the N = 153 iso-
usingthecrankedrelativisticHartree-Bogoliubovtheory, tones are comparedin Fig. 9. The one-quasineutronlev-
the levelsequence in N =151isotones is consistentwith elsfor249Cm[98],251Cf[99],253Fm[100]and255No[101]
the experiment, but it can not reproduce the level se- were identified experimentally. The ground states for
quence in N = 149 isotones. It can be seen in Fig. 8 all these isotones are ν1/2+[620] in our calculations,
that, the level sequence in most nuclei is consistent with which is consistent with the data. But the level order
the experimental data, with the obviously exception of of ν7/2+[613] and ν3/2+[622]from our calculation is in-
the ν5/2+[622] orbital. The level ν7/2+[624] observed versed according to the data. This also happens in the
in 253No at 355 keV is from Ref. [38]. A recent experi- Ref. [50]. Another problem is that ν11/2−[725]states in
all the isotones from our calculation are systematically
7
800 800
V) V)
e e
k k
y ( 600 (a) Exp. y ( 600 (b) Cal.
g g
er er
d en 400 7/2+[624] d en 400 7/2+[624]
hea 5/2+[622] hea 5/2+[622]
nd 200 nd 200
a a -
b b 7/2[743]
+ +
0 1/2[631]241Cm 243Cf 0 1/2[631]241Cm 243Cf 245Fm 247No
FIG. 5. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasineutron bands for the
N = 145 isotones. The data are taken from [86, 87]. The positive and negative parity states are denoted by black and red
lines, respectively.
800 800
V) V)
gy (ke 600 gy (ke 600 9/2-[734]
er (a) Exp. er (b) Cal.
n n
ead e 400 ead e 400 7/2-[743]
h h
d d +
ban 200 17//22++[[663214]] ban 200 17//22+[[663214]]
+ +
0 5/2[622]243 245 247 0 5/2[6222]43 245 247 249
Cm Cf Fm Cm Cf Fm No
FIG. 6. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasineutron bands for the
N =147isotones. Thedataaretakenfrom[87–89]. Thepositiveandnegativeparitystatesaredenotedbyblackandredlines,
respectively.
800 + 800 1/2+[620]
V) 71//22+[[661230]] V) 7/2-[743]
e - e
k 7/2[743] k
y ( 600 y ( 600
g (a) Exp. g (b) Cal.
r r
e e
n - n 1/2+[631]
d e 400 19//22+[[763341]] d e 400 -
a a 9/2+[734]
e + e 5/2[622]
h 5/2[622] h
nd 200 nd 200
a + a
b 1/2[631] b
+ +
0 7/2[6242]45 247 249 251 0 7/2[624]245 247 249 251
Cm Cf Fm No Cm Cf Fm No
FIG. 7. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasineutron bands for the
N =149isotones. Thedataaretakenfrom[89–92]. Thepositiveandnegativeparitystatesaredenotedbyblackandredlines,
respectively.
8
V) 800 (a) Exp. V) 800 (b+) Cal.
e e 5/2[622]
ergy (k 600 1/2+[631] + ergy (k 600 1/2+[620]
ead en 400 17//22++[[6622047]]/2[613] ead en 400 7/2+[624]
h + h
nd 200 5/2[622] nd 200
a a
b b
- -
0 9/2[7342]47 249 251 253 0 9/2[734]247 249 251 253
Cm Cf Fm No Cm Cf Fm No
FIG. 8. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasineutron bands for the
N =151 isotones. Thedataare takenfrom [93–96]. Theconfiguration assignment 7/2+[624] in253Noare takenfrom Ref. [38]
and discussed in Sec. IV.The positive and negativeparity states are denoted byblack and red lines, respectively.
-
11/2[725]
800 800
V) V)
ke (a) Exp. ke 7/2+[624]
gy ( 600 9/2+[615] 5/2+[622] gy ( 600
d ener 400 111//22--[[772550]] 9/2-[734] d ener 400 (b) Cal.
ea ea 9/2+-[734]
h + h 7/2[613]
nd 200 3/2[622] nd 200
a a
b b
+ +
7/2+[613] 3/2+[622]
0 1/2[620]249 251 253 255 0 1/2[620]249 251 253 255
Cm Cf Fm No Cm Cf Fm No
FIG. 9. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasineutron bands for the
N = 153 isotones. The data are taken from [98–101]. The positive and negative parity states are denoted by black and red
lines, respectively.
9
higher than the experiment values. In each of these iso- Figure 13 shows the experimental and calculated
tones a low-lying state ν7/2−[743] is predicted which is MOI’softheGSB’sineven-evenCm,Cf,Fm,andNoiso-
not observed. topes. The experimental(calculated) MOI’s aredenoted
Low-lying one-quasiproton levels for Bk (Z = 97) by solid circles (solid lines). The experimental MOI’s of
isotopes 243,245Bk [102], 247Bk [103], 249Bk [104] and alltheseGSB’sarewellreproducedbythePNC-CSMcal-
251Bk [105] were identified experimentally. The compar- culations. Forsomenuclei,thereisnodata(i.e.,252Cm),
ison between the data and our calculation is shown in weonlyshowthe PNC-CSMresults. Itcanbe seenfrom
Fig. 10. The two levels π3/2−[521] and π7/2+[633] are Fig.13thatwithincreasingprotonnumberZ inanyiso-
very close to each other, usually the energy difference is tonic chain, the upbendings in each nuclei become less
lessthan50keV.Thegroundstatesareπ3/2−[521]forall pronounced. It is well known that the backbending is
Bkisotopesexcept249Bk. Inourcalculationstheground caused by the crossing of the GSB with a pair-broken
statesareallπ3/2−[521]. Thedeviationin249Bkmaybe bandbasedonhigh-j intruderorbitals[110],inthismass
due to the staggering of the deformation. Nearly all the region the πi and νj orbitals. But in several nu-
13/2 15/2
calculated 1-quasiproton energies in the Bk isotopes are clei, there is no evidence for a νj alignment. It has
15/2
a little larger than the experimental values. been pointed out in Ref. [57] that, for the nuclei with
Low-lyingone-quasineutronlevels for Es (Z =99)iso- N ≈ 150, among the neutron orbitals of j parent-
15/2
topes 245,247,249,251,253Es [106, 107] were identified ex- age, only the high-Ω (deformation aligned) ν7/2−[743]
perimentally. The comparisonbetween the data and our andν9/2−[734]areclosetothe Fermisurface. Thediag-
calculation is shown in Fig. 11. The ground states are onal parts in Eq. (5) of these two orbitals contribute no
π7/2+[633] for all Es isotopes except 251Es. In our cal- alignment to the upbending, only the off-diagonal parts
culations the ground states are all π7/2+[633]. The de- in Eq. (5) contribute a little if the neutron j orbital
15/2
viationin 251Es may be also due to the staggeringof the is not blocked.
deformation. Our calculations show that the bandhead
To see the upbending in these even-even nuclei more
energies of the negative parity states are all very small
clearly, we show the experimental (solid circles) and cal-
(less than 400 keV), which is consistent with the experi-
culated(solidlines)alignmentifortheGSB’sinN =150
ment.
isotonesinFig.14. Forotherisotonestheresultsarevery
With the proton number increasing, the data become
similar, so we do not shown them here. It can be seen
less and less. Fig. 12 shows the comparison between the
thatthe upbending frequenciesofthese GSB’s areabout
experimental values and our calculation for the Md and
0.20∼0.25 MeV.
Lr isotopes. The data are availableonly for 251Md [109],
One of the advantages of the PNC method is that the
255Md [108] and 251Lr [109] and they are reproduced by
total particle number N = n is exactly conserved,
the theory quite well. Pµ µ
whereas the occupation probability n for each orbital
From the above discussions, we see that the new Nils- µ
varies with rotational frequency ~ω. By examining the
sonparameters candescribe satisfactorilythe 1-qpspec-
ω-dependence of the orbitals close to the Fermi surface,
tra of nuclei with Z ≈ 100. But there are still some
one can learn more about how the Nilsson levels evolve
discrepancies. According to our experience, it is quite
with rotation and get some insights on the upbending
difficult to improve this situation in the framework of
mechanism. Fig. 15showsthe occupationprobabilityn
the Nilsson model. One way out might be to use the µ
of each orbital µ near the Fermi surface for the GSB’s
Woods-Saxon potential instead of the Nilsson potential.
in the N = 150 isotones. The top and bottom rows are
for the protons and neutrons, respectively. The positive
(negative) parity levels are denoted by blue solid (red
C. Even-even nuclei
dotted) lines. The orbitals in which n do not change
µ
much (i.e., contribute little to the upbending) are de-
The experimental kinematic MOI’s for each band are
noted by black lines. The Nilsson levels far above the
extracted by
Fermi surface (n ∼ 0) and far below (n ∼ 2) are not
µ µ
J(1)(I) 2I+1 shown. We can see from Fig. 15 that the occupation
= , (10) probability of π7/2+[633] (πi ) drops down gradually
~2 Eγ(I+1→I−1) from 0.5 to nearly zero with1t3h/e2 cranking frequency ~ω
separatelyforeachsignaturesequencewithinarotational increasing from about 0.20 MeV to 0.30 MeV, while the
band. The relation between the rotational frequency ω occupation probabilities of some other orbitals slightly
and the angular momentum I is increase. This can be understood from the crankedNils-
son levels shown in Fig. 3. The π7/2+[633] is slightly
~ω(I)= Eγ(I +1→I −1) , (11) above the Fermi surface at ~ω = 0. Due to the pair-
I (I +1)−I (I−1) ing correlations, this orbital is partly occupied. With
x x
increasing~ω,this orbitalleavesfartherabovethe Fermi
where Ix(I)=p(I+1/2)2−K2, K is the projectionof surface. So after the band-crossing frequency, the occu-
nuclear total angular momentum along the symmetry z pation probability of this orbital becomes smaller with
axis of an axially symmetric nuclei. increasing ~ω. Meanwhile, the occupation probabilities
10
+
800 800 1/2[400]
V) 1/2-[521] V) -
e e 1/2[521]
k k -
y ( 600 1/2-[530] y ( 600 5/2+[523]
g + g 5/2[642]
ner 15//22-[[450203]] ner -
d e 400 5/2+[642] d e 400 7/2[514]
a a
e e
h h
nd 200 (a) Exp. nd 200 (b) Cal.
a a
b + b +
7/2[633] 7/2[633]
- -
0 3/2[521] 243Bk 245Bk 247Bk 249Bk 251Bk 0 3/2[521]243Bk 245Bk 247Bk 249Bk 251Bk
FIG. 10. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasiproton bandsfor theBk
(Z =97) isotopes. The data are taken from [102–105]. The positive and negative parity bands are denoted by black and red
lines, respectively.
800 9/2+[624] 800
keV) 1/2+[400] keV) 5/2+[642] 1/2+[400]
y ( 600 y ( 600
g g
ner (a) Exp. ner (b) Cal.
d e 400 1/2-[521] d e 400 -
a a 1/2[521]
e - e
h 7/2[514] h
nd 200 nd 200 7/2-[514]
a - a
b 3/2[521] b -
3/2[521]
+ +
0 7/2[633] 245Es 247Es 249Es 251Es 253Es 0 7/2[633]245Es 247Es 249Es 251Es 253Es
FIG. 11. (Color online) (a) Experimental and (b) calculated bandhead energies of low-lying one-quasiproton bands for the Es
(Z =99) isotopes. The data are taken from [106, 107]. The positive and negative parity states are denoted by black and red
lines, respectively.
of those orbitals which approach near to the Fermi sur- The contribution of each proton (top row) and neu-
facebecomelargerwithincreasing~ω. Thisphenomenon tron (bottom row) major shell to the angular momen-
is evenmore clearin 248Cf, but the band-crossingoccurs tum alignment hJ i for the GSB’s in the N = 150 iso-
x
at ~ω ∼ 0.25 MeV, a little larger than that of 246Cm. tones is shown in Fig. 16. The diagonal j (µ) and
c Pµ x
Sotheband-crossingsinbothcasesaremainlycausedby off-diagonal parts j (µν) in Eq. (5) from the pro-
Pµ<ν x
theπi13/2 orbitals. For250Fmand252No,theoccupation ton N = 6 and the neutron N = 7 shells are shown by
probabilities of πi13/2 orbitals increase slowly with the dashed lines. Note that in this figure, the smoothly in-
cranking frequency ~ω increasing from about 0.20 MeV creasing part of the alignment represented by the Harris
to 0.30 MeV. So there is no sharpbandcrossingfromthe formula (ωJ +ω3J ) is not subtracted (cf. the caption
0 1
proton orbitals. Now we focus on the occupation proba- of Fig. 14). It can be seen clearly that the upbendings
bility nµ of the neutron orbitals (bottom row). The four for the GSB’s in 246Cm at ~ωc ∼ 0.20 MeV and in 248Cf
figures in the bottom row of Fig. 15 show a very simi- at~ω ∼0.25MeVmainlycomefromthecontributionof
c
lar pattern. It can be seen that, with ~ω increasing, the the proton N =6 shell. Furthermore, the upbending for
nµ of ν7/2+[624]orbitals increase slowly and that of the theGSBin246Cmismainlyfromtheoff-diagonalpartof
high-Ω (deformation aligned) ν9/2−[734]orbitals (j15/2) the proton N =6 shell, while both the diagonaland off-
decrease slowly. Thus only a small contribution is ex- diagonalpartsoftheprotonN =6shellcontributetothe
pectedfromneutronstotheupbendingsfortheGSB’sin upbendingfortheGSBin248Cf. Theoff-diagonalpartof
the N = 150 isotopes. The bandcrossing frequencies for the neutron N = 7 shell only contributes a little to the
neutrons are about 0.20 ∼ 0.25 MeV, very close to the upbending. It is very different from 250Fm and 252No.
proton bandcrossing frequencies. So neutrons and pro- For these two nuclei, the contribution to the upbending
tonsfromthehigh-j orbitscompetestronglyinrotation- fromtheoff-diagonalpartsoftheprotonN =6shelland
alignment as pointed out in Ref. [54]. theoff-diagonalpartofthe neutronN =7shellisnearly