Particle-number conserving analysis of rotational bands in 247,249Cm and 249Cf

0.33 MB·English

by  Zhen-Hua Zhang

Particle-number conserving analysis of rotational bands in 247,249Cm and 249Cf
Zhen-Hua Zhang,1 Jin-Yan Zeng,2 En-Guang Zhao,1,3,2 and Shan-Gui Zhou1,3,∗
1Key Laboratory of Frontiers in Theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China
2School of Physics, Peking University, Beijing 100871, China
3Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China
(Dated: January 20, 2011)
The recentlyobserved high-spinrotational bandsin odd-Anuclei247,249Cm and249Cf[Tandel et
al.,Phys. Rev. C82(2010)041301R]areinvestigatedbyusingthecrankedshellmodel(CSM)with
thepairingcorrelationstreatedbyaparticle-numberconserving(PNC)methodinwhichtheblocking
effectsaretakenintoaccountexactly. Theexperimentalmomentsofinertiaandalignmentsandtheir
variations with therotational frequency ω are reproduced verywell bythePNC-CSMcalculations.
1 By examining the ω-dependenceof the occupation probability of each cranked Nilsson orbital near
1 theFermi surface and thecontributions of valenceorbitals to theangular momentum alignment in
0 eachmajorshell,thelevelcrossingandupbendingmechanismineachnucleusisunderstoodclearly.
2

n PACSnumbers: 21.60.-n;21.60.Cs;23.20.Lv;27.90.+b
a
J
Therotationalspectraoftheheaviestnucleicanreveal symmetric nucleus in the rotating frame is
9
1 detailedinformationonthesingleparticleconfigurations,
H =H +H =H −ωJ +H , (1)
the shell structure, the stability against rotation, etc., CSM 0 P Nil x P

] thusprovidingabenchmarkfornuclearmodels. Inrecent whereHNil istheNilssonHamiltonian,−ωJx istheCori-
h
years, the in-beam spectroscopy of nuclei with Z ≈ 100 olis interaction with the cranking frequency ω about the
t
- has been one of the most important frontiers on nuclear x axis (perpendicular to the nuclear symmetry z axis).
l
c structure physics [1]. Besides even-even nuclei [2–4], ex- HP =HP(0)+HP(2) is the pairing interaction,
u
perimentaleffortshavebeenalsofocusedonthe studyof
n H (0)=−G a+a+a a , (2)
high-spin states of odd-A nuclei, such as 253No [5] and P 0X ξ ξ¯ η¯ η
[
251Md [6]. Quite recently, the rotationalbands of odd-A ξη

1 247,249Cm and 249Cf were observedup to veryhigh spins H (2)=−G q (ξ)q (η)a+a+a a , (3)
v (≈ 28~) and appropriate single particle configurations P 2X 2 2 ξ ξ¯ η¯ η
ξη
7
have been assigned to these bands [7]. It is worthwhile
0 to mention that the neutron ν1/2+[620] band in 249Cm where ξ¯ (η¯) labels the time-reversed state of a Nilsson
36 isthehighest-lyingneutronconfigurationinvestigatedup state ξ (η), q2(ξ) = p16π/5hξ|r2Y20|ξi is the diago-
. to very high spins. Although the cranking Woods-Saxon nal element of the stretched quadrupole operator, and
1 calculations reproduced well some of the observed prop- G0 and G2 are the effective strengths of monopole and
0
erties,thisexperiment,togetherwithsomepreviousones, quadrupole pairing interactions, respectively.
1
1 stillchallengenuclearstructuremodels, e.g., theabsence Instead of the usual single-particle level truncation
: ofthealignmentofj neutronsinseveralnucleiinthis in common shell-model calculations, a cranked many-
v mass region needs a1c5o/n2sistent explanation [7]. particle configuration (CMPC) truncation (Fock space
i
X In this paper, the cranked shell model (CSM) with truncation)is adoptedwhichiscrucialto makethe PNC
calculations for low-lying excited states both workable
r pairing correlations treated by a particle-number con-
a and sufficiently accurate [15]. An eigenstate of H
serving(PNC)method[8]isusedtoinvestigatetherota- CSM
tional bands in 247,249Cm and 249Cf observedin Ref. [7]. can be written as
In contrary to the conventional BCS approach, in the
|Ψi= C |ii , (C real), (4)
PNC method, the particle-number is conserved and the X i i
i
Pauliblockingeffectsaretakenintoaccountexactly. The
where |ii is a CMPC (an eigenstate of the one-body op-
PNC-CSM treatment has been used to describe success-
fullythenormallydeformedandsuperdeformedhighspin erator H0). By diagonalizing the HCSM in a sufficiently
largeCMPCspace,sufficientlyaccuratesolutionsforlow-
rotationalbandsofnucleiwithA≈160,190,and250[9–
14]. lying excited eigenstates of HCSM are obtained.
The kinematic moment of inertia (MOI) for the state
The details of the PNC-CSM treatment can be found
|Ψi is
in Refs. [9, 10]. For convenience,here we briefly give the
related formalism. The CSM Hamiltonian of an axially 1
J(1) = hΨ|J |Ψi
x
ω
1  
= C2hi|J |ii+2 C C hi|J |ji .(5)
∗ [email protected];http://www.itp.ac.cn/˜sgzhou ω Xi i x Xi<j i j x 
2
Considering J to be a one-body operator, the matrix and G = 0.04 MeV for protons, G = 0.30 MeV and
x 2p 0n
element hi|J |ji for i 6= j is nonzero only when |ii and G =0.02 MeV for neutrons. The stability of the PNC
x 2n
|ji differ by one particle occupation [9, 10]. After a cer- calculationresultsagainstthechangeofthedimensionof
tainpermutationofcreationoperators,|iiand|jicanbe theCMPCspacehasbeeninvestigatedinRefs.[9,11,15].
recastinto |ii=(−1)Miµ|µ···iand|ji=(−1)Mjν|ν···i In the present calculations, almost all the CMPC’s with
where the ellipsis “···” stands for the same particle oc- weight > 0.1% are taken into account, so the solutions
cupation and (−1)Miµ(ν) = ±1 according to whether the to the low-lying excited states are accurate enough. A
permutation is even or odd. Therefore, the angular mo- larger CMPC space with renormalized pairing strengths
mentum alignment of |Ψi can be expressed as gives essentially the same results.
Figure 1 shows the calculated cranked Nilsson levels
hΨ|Jx|Ψi=Xjx(µ)+Xjx(µν) . (6) near the Fermi surface of 247Cm. The positive (nega-
µ µ<ν tive) parity levels are denoted by blue (red) lines. The
signature α = +1/2 (α = −1/2) levels are denoted by
The diagonal contribution jx(µ) = hµ|jx|µinµ where solid (dotted) lines. For both protons and neutrons, the
nµ = Pi|Ci|2Piµ is the occupation probability of the sequence of single-particle levels near the Fermi surface
crankedNilssonorbital|µiandPiµ =1(0)if|µiisoccu- is the same as the experimental data taken from 247Cm
pied (empty). The off-diagonal (interference) contribu- and 247Bk [1] with the only exception of the ν5/2+[622]
tion jx(µν)=2hµ|jx|νiPi<j(−1)Miµ+MjνCiCj. orbital. Many theoretical models predict that the first
excited state in N = 151 isotones should be ν7/2+[624]
(see, e.g., Ref. [23]). This is not consistent with exper-
TABLE I. Nilsson parameters κ and µ proposed for nuclei
imental results, i.e., the first excited state in N = 151
with A≈ 250 [18].
isotones is ν5/2+[622]. The low excitation energy of the
ν5/2+[622] state in the N = 151 isotones have been in-
N l κp µp N l κn µn terpretedas a consequenceof the presenceof a low-lying
4 0,2,4 0.0670 0.654 Kπ =2−octupolephononstate[24]. Figure1showsthat
5 1 0.0250 0.710 6 0 0.1600 0.320
there exist a proton gap at Z = 96 and a neutron gap
3 0.0570 0.800 2 0.0640 0.200
atN =152,whichis consistentwiththe experimentand
5 0.0570 0.710 4,6 0.0680 0.260
6 0,2,4,6 0.0570 0.654 7 1,3,5,7 0.0634 0.318 the calculation by using a Woods-Saxon potential [25].
The cranked Nilsson levels of 249Cm and 249Cf are quite
similar to that of 247Cm and not shown here.
The Nilssonparameters(κ,µ) systematics[16, 17] can Figure 2 shows the experimental and calculated MOIs
not reproduce well the order of single particle levels for and alignments (see the caption of Fig. 2) of the ground
the very heavy nuclei with A≈ 250 (see, e.g., Ref. [14]). state bands (gsb’s) in 247,249Cm and 249Cf. The ex-
Recently we have proposed a new set of (κ,µ) which perimental MOIs and alignments are denoted by solid
is given in Table I and deformation parameters for nu- squares (signature α = +1/2) and open squares (sig-
clei with A ≈ 250 by fitting the observed single parti- nature α = −1/2), respectively. The calculated MOIs
cle spectra of all known odd-A nuclei in this mass re- and alignments are denoted by solid lines (signature
gion[18]. NotethatthereadjustmentofNilssonparame- α = +1/2) and dotted lines (signature α = −1/2), re-
tersisalsonecessaryinsomeotherregionsofthenuclear spectively. TheexperimentalMOIsandalignmentsofall
chart [19, 20]. The deformation parameters (ε ,ε ) are these three 1-quasiparticle bands are well reproducedby
2 4
(0.242, 0.002) for 247Cm and (0.248, 0.008) for 249Cm the PNC-CSM calculations, which in turn strongly sup-
and 249Cf. There are no experimental values of the de- port the configuration assignments for these high-spin
formation parameters for these nuclei. The values used rotational bands adopted in Ref. [7]. Moreover, the sig-
in various calculations or predicted by different models naturesplittingintheν1/2+[620]bandisalsowellrepro-
are different. What we adopt here are larger than those duced by our calculation, which is understandable from
usedinRef.[7]orpredictedinRef.[21],andsmallerthan thebehaviorofthecrankedNilssonorbitalν1/2+[620]in
the interpolated values from Ref. [22] where, e.g., the ǫ2 Fig. 1. The upbending frequency ~ωc ∼0.25 for the gsb
values for 248Cf and 250Cf are 0.260 and 0.265 respec- in 249Cf is a little larger than that of the Cm isotopes
tively. (~ωc ∼ 0.20 MeV). These results agree well with the ex-
Theeffectivepairingstrengths,inprinciple,canbede- perimentandthecrankingWoods-Saxoncalculations[7].
terminedbytheodd-evendifferencesinbindingenergies, One of the advantages of the PNC method is that the
and are connected with the dimension of the truncated total particle number N = n is exactly conserved,
Pµ µ
CMPC space. The CMPC space for the very heavy nu- whereas the occupation probability n for each orbital
µ
cleiisconstructedintheprotonN =4,5,6shellsandthe varies with rotational frequency ~ω. By examining the
neutron N = 6,7 shells. The dimensions of the CMPC ω-dependence of the orbitals close to the Fermi surface,
space are about 1000 for both protons and neutrons in one can learn more about how the Nilsson levels evolve
our calculation. The corresponding effective monopole with rotation and get some insights on the upbending
and quadrupole pairing strengths are G = 0.50 MeV mechanism. Figure 3 shows the occupation probability
0p
3

) (a) proton (b) neutron
0
6.8 [512]5/2 7.8 [615]9/2
[725]11/2
s ( 6.7 [761]1/2
evel [624]9/2 7.7 [[661232]]73//22
on l 6.6 [[551241]]71//22 7.6 [620]1/2 N=152
s [734]9/2
s 6.5 [633]7/2
Nil [521]3/2 [624]7/2
ed 6.4 [642]5/2 Z=96 7.5 [[663212]]15//22
rank 6.3 [[[454020230]]]351///222 7.4 [[670463]]173/2/2
C [651]3/2
6.2 [5[60650]1]11//22 7.3 [[765321]]53//22
-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. 1. (Color online) The cranked Nilsson levels near the Fermi surface of 247Cm for protons (a) and for neutrons (b). 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.

247 - 249 + 249 -
1 )120 Cm gsb 9/2[734] Cm gsb 1/2[620] Cf gsb 9/2[734]
- V
e
M
80
2
Exp = -1/2
(
1) 40 Exp =+1/2
( J Cal = -1/2
Cal =+1/ 2 (a) (b) (c)
0
247 - 249 + 249 -
Cm gsb 9/2[734] Cm gsb 1/2[620] Cf gsb 9/2[734]
8
)
( 6
i
4
2
(d) (e) (f)
0
0.0 0.1 0.2 0.3 0.1 0.2 0.3 0.1 0.2 0.3

(MeV)
FIG. 2. (Color online) The experimental and calculated MOIs J(1) and alignments (or called alignments difference) of the
ground state bands (gsb’s) in 247,249Cm and 249Cf. The alignments difference i is defined as i = hJxi−ωJ0 −ω3J1 and
the Harris parameters J = 65 ~2MeV−1 and J = 200 ~4MeV−3 are taken from Ref. [7]. The experimental MOIs and
0 1
alignments differenceare denoted bysolid squares (signature α=+1/2) and open squares (signature α=−1/2), respectively.
The calculated MOIs and alignments difference are denoted by solid lines (signature α = +1/2) and dotted lines (signature
α=−1/2), respectively.
n ofeachorbitalµneartheFermisurfaceforthegsb’sin son levels shown in Fig. 1(a). The π7/2+[633] is slightly
µ
249Cm and 249Cf. The top and bottom rows are for the above the Fermi surface at ~ω = 0. Due to the pair-
protons and neutrons respectively. The positive (nega- ing correlations, this orbital is partly occupied. With
tive) parity levels are denoted by blue solid (red dotted) increasing ~ω, this orbital leave farther above the Fermi
lines. We can see from Fig. 3(a) that the occupation surface. So after the band-crossing frequency, the occu-
probability of π7/2+[633] (i ) drops down gradually pationprobabilityofthisorbitalbecomessmallerwithin-
13/2
from 0.5 to nearly zero with the cranking frequency ~ω creasing ~ω. Meanwhile, the occupation probabilities of
increasing from about 0.20 MeV to 0.30 MeV, while the those orbitals which approach near to the Fermi surface
occupation probabilities of some other orbitals slightly become larger with increasing ~ω. This phenomenon is
increase. This can be understoodfrom the crankedNils- evenmoreclearinFig.3(b),buttheband-crossingoccurs
4

(proton) 112...050 (5a/)2 2-[459C23m] 5/2+[642] 3/32/-2[5+[2615]1] (b5 )/ 2274+9[/C624+f[26]33] 5/23-[/522-[35]21] (proton) 114826 (a) 249CN=m6 diagonal N=6 N=5 (b ) 249CfN=6 diagonal N=6
n 0.5 7/2+[633] 1/2-[521] Jx N=4 N=4N=5
0.0 7/2-[514] 7/2-[514] -04 N=6 off-dia gonal N=6 o ff-diagonal
n (neutron) 0112....5050 (9c/)2 2-[479C34m]7/277+//[226+-1[[7 6542]34]] 1/2+[620] (d7 )/ 224+9[C62f14/2]+[620 ] 7/29-/[27-4[733]4] J (neutron)x 1482 (c) 249Cm N =N7= d6iNag=o7nal (d ) 249Cf NN==67 diagonal N=7
0.0 3/2+[622] 0 N=7 off-diagonal N=7 off-diagonal
0.1 0.2 0.3 0.1 0.2 0.3
0.0 0.1 0.2 0.3 0.1 0.2 0.3

(MeV) (MeV)
FIG. 3. (Color online) Occupation probability nµ of each FIG. 4. (Color online) Contribution of each proton and neu-
orbitalµ(includingbothα=±1/2)neartheFermisurfacefor tron major shell to the angular momentum alignment hJxi
the gsb’s in 249Cm and 249Cf. The top and bottom rows are forthegsb’sin249Cmand249Cf. ThediagonalPµjx(µ)and
forprotonsandneutronsrespectively. Thepositive(negative) off-diagonal parts Pµ<νjx(µν) in Eq. (6) from the proton
paritylevelsaredenotedbybluesolid(reddotted)lines. The N =6 and neutron N =7 shells are shown bydotted lines.
Nilsson levels far above the Fermi surface (nµ ∼ 0) and far
below (nµ ∼ 2) are not shown. For the ν9/2−[734] band in
247Cm,nµofproton(neutron)orbitalsarenotshownbecause
they are nearly thesame as those of 249Cm (249Cf). tonorbitalsi13/2 (toprow)andintruderneutronorbitals
j (bottom row) to the angular momentum align-
15/2
ments hJ i are presented in Fig. 5. The diagonal (off-
x
at ~ωc ∼ 0.25 MeV, a little larger than that of 249Cm. diagonal) part jx(µ) [jx(µν)] in Eq. (6) is denoted by
So the band-crossings in both cases are mainly caused blue solid (red dotted) lines. Near the proton Fermi
by the πi13/2 orbitals. In Fig. 3(c), with increasing ~ω surfaces of Cm and Cf isotopes, the proton i13/2 or-
theoccupationprobabilityofν1/2+[620]decreasesslowly bitalsareπ3/2+[651],π5/2+[642]andπ7/2+[633]. Other
andthatofthe high-Ω(deformationaligned)ν9/2−[734] orbitals of πi13/2 parentage are either fully occupied
orbital (j ) increases slowly. Thus only a small con- or fully empty (cf. Fig. 3) and have no contribution
15/2
tribution is expected from neutrons to the upbending to the upbending. In Fig. 5(a), the PNC calculation
for the gsb in 249Cm. In Fig. 3(d), the neutron orbital shows that after the upbending (~ω ≥ 0.20 MeV) the
ν9/2−[734] of j15/2 parentage is totally blocked by an off-diagonal part jx(π5/2+[642]π7/2+[633]) changes a
odd neutron, so it has no contribution to the upbending lot. The alignment gain after the upbending mainly
for the gsb in 249Cf. comes from this interference term. The off-diagonal
The contribution of each proton and neutron major part jx(π3/2+[651]π5/2+[642]) and the diagonal part
shell to the angular momentum alignment hJxi for the jx(π7/2+[633]) also contribute a little to the upbend-
gsb’s in 249Cm and 249Cf are shown in Fig. 4. The di- ingin249Cm. FromFig.5(b)onefindsthatfor249Cfthe
agonal j (µ) and off-diagonal parts j (µν) in main contribution to the alignment gain after the up-
Eq. (6)Pfroµmxthe proton N = 6 and thePneµu<tνronx N = 7 bending comes from the diagonal part jx(π7/2+[633])
shells are shown by dotted lines. Note that in this fig- and the off-diagonal part jx(π5/2+[642]π7/2+[633]).
ure, the smoothly increasing part of the alignment rep- Again this tells us that the upbending in both cases is
resented by the Harris formula (ωJ0+ω3J1) is not sub- mainly causedby the πi13/2 orbitals. The absence ofthe
tracted (cf. the caption of Fig. 2). It can be seen clearly alignment of j15/2 neutrons in nuclei in this mass region
that the upbendings for the gsb’s in 249Cm at ~ω ∼ can be understood from the contribution of the intruder
c
0.20 MeV and in 249Cf at ~ω ∼ 0.25 MeV mainly come neutron orbitals (N = 7) to hJxi. For the nuclei with
c
fromthecontributionoftheprotonN =6shell. Further- N ≈150,among the neutron orbitals of j15/2 parentage,
−
more,theupbendingforthegsbin249Cmismainlyfrom only the high-Ω (deformation aligned) ν7/2 [743] and
−
the off-diagonal part of the proton N = 6 shell, while ν9/2 [734] are close to the Fermi surface. The diagonal
both the diagonal and off-diagonal parts of the proton partsofthesetwoorbitalscontributenoalignmenttothe
N = 6 shell contribute to the upbending for the gsb in upbending,onlytheinterferencetermscontributealittle
249Cf. if the neutron j15/2 orbital is not blocked [cf. Fig. 5(c)].
In order to have a more clear understanding of the Insummary,therecentlyobservedhigh-spinrotational
upbending mechanism, the contribution of intruder pro- bands in odd-A nuclei 247,249Cm and 249Cf [7] are inves-
5
tigatedusingthePNC-CSM.InthePNCmethodforthe
pairingcorrelations,theparticle-numberisconservedand
the blocking effects are taken into account exactly. The
12 (a) 249Cm (b) 249Cf 1/2+[660] experimental ω variations of MOIs and alignments are
(proton) 48 3/2+[651]51//22++[[664620]] 3+/2+[651 ] 35//22++[[6654127]]/72/+2[6+[3633]3 9] /2+[624] raitenyparlooyfdzieunacgcehdthcverearωny-kdweedeplleNnbidylsestnohcneePoorfNbtCihta-eCloSncMecaurpcaatlhtcieuonlFatepirormonbis.asbuBirly--
Jx 0 7/2[633] faceandthecontributionsofvalenceorbitalsineachma-
jor shell to the angular momentum alignment, the level
-4 5/2+[642] 5/ 2+[642]7/2+[633] 5/2+[64 2] 7/2+[633] crossingandupbendingmechanismineachnucleusisun-
n) 12 (c) 249Cm 1/2-[770] (d) 249Cf 1/2-[770] dtaetrisotnoaoldbcalnedarslyin. Tthheeseunpubcelnediiinsgmianintlhyecgaruosuenddbsytathteerino--
neutro 48 7/2-[743]9/2-[734]93//22--[[773641]] 3/2-[761] 9/2-[734] tarbusdeenrceproofttohne(aNlig=nm6)enπti1o3f/2j1o5r/b2itnaelust.rTonhseirsedasisocnusosfetdh.e
(

Jx 0
-4 5/2-[752] 7/2-[743] 5/2-[752] 7/2-[743]
-8
0.0 0.1 0.2 0.3 0.1 0.2 0.3

(MeV)
This work has been supported by NSFC (Grant
Nos. 10875157 and 10979066), MOST (973 Project
FIG. 5. (Color online) Contribution of each proton orbital 2007CB815000),andCAS(GrantNos. KJCX2-EW-N01
in the N =6 major shell (top row) and each neutron orbital and KJCX2-YW-N32). The computation of this work
in the N = 7 major shell (bottom row) to the angular mo- wassupportedbySupercomputingCenter,CNICofCAS.
mentum alignments hJxi for the gsb’s in 249Cm and 249Cf. Helpful discussions with G. G. Adamian, N. V. Anto-
The diagonal (off-diagonal) part jx(µ) [jx(µν)] in Eq. (6) is nenko, X. T. He, and F. Sakata are gratefully acknowl-
denoted by bluesolid (red dotted) lines.
edged.
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