An approximate fixed-configuration method for collective rotational bands in the Hartree-Fock-Bogoliubov theory

6 April 1995 PHYSICS LETTERS B Physics Lettew R 348 ( 1995) 320-324

An approximatefixed-configuration method for collective rotational bands in the Hartree-Fock-Bogoliubov theory J. Terasaki I, F. Sakata, K. Iwasawa Institute fc* Ndear Study. University

of Tokyo, Tanoshi, Tokyo 188, Japan

Received 18 August 1994; revised manuscript received 31 January 1995 Editor: G.F. Bertsch

Abstract A new constraintis proposedwhich specifiesa Hartree-Fock-Bogoliubov (HFB) subspaceto describea collective rotational band in the yrare region.The ground- and S-bandsof ‘“Er are calculatedby using the gradientmethod under this constraintas an example,and it is shown that this method works well.

Detailed spectroscopic study of rotational bands in nuclei hasprogressedfrom the,yrastline [ l] to many yrare states.Due to the large level density, it remainsdifficult to obtain accuratewave functions for the yrare states. Within mean-fieldtheories,oneof the approachesis to introduceappropriateconstraints.We emphasizethatfinding appropriateconstraintsrequiresa good physicalunderstandingof the statesin question. One of theseapproachesis to specify a rotational band by approximatelyfixing configurations,which was developedin the crankedshell model 12-61. This methodcan avoid phenomenologicallythe difficulty that the ordinary mean-field theory often gives a poor approximation in the band crossingor level crossingregions [Y-9]. When this ::ind of approachis physically meaningful,it ought to be possibleto introducea certainHFB subspace in which the variationalcalculationis performedto obtain a rotationalbandin the yrareregion.The introductionof the subspacemakesit possibleto obtain self-consistentHFB wave functionswithout any significantmodification for the conventionalnumericalmethod [ lo]. The aim of this letter is to propcsea new methodof introducinga HFB subspacespecifiedby an approximatefixed configuration,and to test its feasibility. In the constrained(C-)HFB theory [ lo], a collective rotationalstateis obtainedby searchingfor a MFB state which givesthe minimumof the expectationvalueof theHamiltonianwith givenparticlenumberandthe component of the angularmomentumalong the rotational axis perpendicularto the symmetryaxis of a nucleus.When this methodis appliedto collectiverotationalstatesof a well-deformedmedium-heavyeven-evennucleus,one obtains an yrastline whoselower angulartnomentumsolutiondescribesthe groundband,whereasthe higherone describes the S-band.A differencein the configurationof thesetwo bandscan be clearly seenin the increasein one of the ufactors,which arecalculatedby applyingtheBloch-Messiahtheorem[ 1l] * to a generalBogoliubovtransformation I~’ Present address: C.P.229,UniversM Libre de Bruxelles, Boulevard du Triomphe, 1050 Bruxelles, Belgium.

; * SeeappendixE in Ref. [ 101. ~0370-2693/95/$09.50 Q 1995Elsevier Science B.V. Ail rights reserved Sso10370-2693(95)00165-4

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from a ground state to anotherstate in the yrast line. (An examplewill be shown later.) ‘I”husit is our idea to introducea I-Il?B subspacefor eachrotational band by imposinga constrainton the rl-factorsof the Bogoliubov transformation. TO explain our procedure,the groundband in a deformedeven-evennucleusis takenas an example. Firstly, we startwith the groundstate 1+e) obtainedby the ordinaryCHFB theorywith the angularmomentumZ= 0. Secondly, we calculatethe uu-factorsof the transformationfrom 1(bO)to tinother . memberof the bandwith If 0, which is also obtainedby the conventionalCI-IFB equation.Thirdly, we introducean upper limit D-(Z) to the cl-factorn,(Z) and check whetherthe inequality u,(l) I~inaJO ,

(1) is satisfiedor not. The way to determinethe IImX(Z) is the most importantpoint in the presentmethod,and this will be discussedlater. Fourthly, if the inequality ( 1) is not satisfiedfor one of the single-particlelevels cy,we solve the variationalequationof the CHFB theory underthe additionalconstraint u,(l) = UmaAO

(2)

for the level (Y.An advantageof the presentmethodis its capability of being calculatedwithin the conventional gradientmethod [ 12,131.Detailedexpressionswill be given elsewhere.Thus the HFB subspaceis characterized by the constraint ( 1) togetherwith the usualconstraintsfor the particlenumberand the angularmomentum. u,(Z) is determinedas follows: Let us imaginea Bogoliubov transformationfrom the groundstatewith Z=O to anotherHFB statewith I#0 and by which no clear configurationchangeis induced.In order to have such a transformation,we decomposethe transformaticnfrom a quasiparticlebase[a:, a,} with Z= 0 to anotherquasiparticle base{a 5, a 2 1 with Z# 0 into the following threesteps:The first is a BCS transformationfrom (u i, a,) to nsingle-particlebase{CL, c,) with Z=O, the secondisaunitary transformationfrom {CT,,c,) withZ=O toa cranked single-particlebase (c 2, CL) , and the last is a BCS transformationfrom (~2, e:) to (a$, a: 1. Here we assume the use of only the monopolecomponentin the pairing force, althoughthe quadrupolepairing force will be used also in the numericalcalculationlater. Sincethe [J-factorof the BCS transformation

is a smoothand monotonically-decreasing function of the single-particleenergyS, measuredfrom the Fermi level for any single-particlebase,it is reasonablyexpectedthat the distribution of V( &, A)’ for the base{CL, c,] and that for the crankedbase{ cb+,CL) are similar to one another.Thus the abovethree-steptransformationcombined doesnot induceany clear configurationchange,providedthat the transformation-from(CL, c,} to { cz, c&) does not give rise to a significantmixture of the single-particlewave functionsand inversionsof il.; order of the singleparticleenergies.If this is the case,one can easily calculate.the u-factorof the three-steptransformationas follows: For later convenience,hereWCconsideronly the u-factorfor an intruderorbit. The single-particleRouthianof the intruderorbit measuredfrom the Fermi level can be written as iT(Z,j&m) s

Ejfi!,

- A(Z) -Zj$~~?i*ft.F(Z) 3

(4)

wherejf,,, and Eii’,, are the rotational-axiscomponentof the angularmomentumand the energyof the intruder orbit closeto the Fermi level, respectively,andY(Z) is the kinematicalmomentof inertia of the groundband.By substitutingEq. (4) into Eq. (3)) V-factorsappearingin the first and third BCS transformations,V( Z(TJ,jf, ), A(O)) and V( Q(Z,j&,), A(Z)), respectively,can be obtained.Under the simplificationc: =ct the o-factorof the three-steptransformationis finally given by u(Z,j&&*=

i l(

S(O,j&,)~(~,jL,d

+A(O)AV)

J [Et0,j~“,)2+A(0)21[E’(Z,j~~,)2+A(Z)21 ’

(5)

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322

Letters B 348 (199.) 320-324

A significantincreasein the occupationprobability of the intruder orbit is known to be the origin of configuration changein the yrast line. Thereforewe put i!m(z,j;tntruj

=~~(l,j;~~) .

(6)

Note that this (I,,( Z,j&,,, ) togetherwith A(I), Y (I) andA(Z) in Eq. (5) shouldbe determinedself-consistently. The startingpoint of the descriptionof the S-bandis chosento be the CHFB state 1&Z,> > which hasa minimum fluctuationof theangularmomentumof theS-bandmembers.Thischoiceindicatesthatthemean-fieldapproximation is relatively good for ] +(I,)), and thus ] &la)) is suitableas a stateto characterizethe configurationof the Sband.The u-factorof the S-bandis obtainedfor a transformationfrom ]&lo)) to anotherstatewith I+&. u,,(Z, j&, )2 usedfor the S-bandis generalizedas (7) cs(Z, I,, j&&

=E,;n,w(Zo) -A(Z)

- [ZlF.dZ)

-&!~.d&)l&,fi2,

(8)

is the single-particleRouthianwith the where ~j;,,~(&) 1sdefinedin the sameway as ejcauexceptthat E~:~~(Z~) angularvelocity of ] +(Z,,)). Y,(Z) is the kinematicalmomentof inertia of the S-band. The Hamiltonianusedhereconsistsof the modifieddeformedharmonicoscillator [ 141with 6-0.29, the doublystretchedquadrupnbforce [ 15J, and the monopoleand quadrupolepairing force. We modified to a small extent the theoreticalvalue of the strengthof the doubly-stretchedquadrupoleforce of Ref. [ 151 so as to reproducethe energyof the y-vibration in the RPA. The strengthof the monopolepairing force was determinedfrom the oddevenmassdifference[ 161. As for the strengthof the quadrupolepairingforce, we adoptedthe valueof Hamamoto [ 171 multiplied by 0.9. The parametersof the modified harmonicoscillator were referredto from the textbook of Bohr and Mottelson [ 141,For detailsof the HamiltonianseeRef. [ 161.The spaceusedin our calculationconsists of two major shellswith N,,,,= 4 and 5 for the proton and 5 and 6 for the neutron,whereN,, is the total oscillator quantumnumberdefinedin the sphericalharmonicoscillator. In Fig. 1 areshownthe calculatedandexperimentalenergiesof the ground-and S-bandsof *@ErrWe show some o,(Z)’ in Fig. 2. The uppermostpanelillustratesthosefor the positive parity neutronsobtainedfrom the ordinary CHFB theory for the yrast line. It is clearly seenthat the quasiparticlelevel is occupiedalmost completelyaround Z=20, and a configurationchangetakesplace on the yrast line. The middle panel shows u,,,&Z,if,,)’ given by Eqs. (5) and (6). In the calculationof the groundbandwe ‘mposedthe constraint(1) only on the positive parity neutrons,becausethe other nucleonsdo not contributeto the bandcrossing.Actually we usedtwo o-(Z,j&,, ) for the positiveparity neutronsin the groundband,i.e. one with j&,, = 13/2 and the other with j&, = 11/2. Otherwise, thereoccursa degeneracyin the u-factorsfor Zk 18, when one usesonly one u,,( Z,jf”,,,, ) . This causesa difficulty in the convergenceprocessfor searchingthe energyminimum underthe constraints.The u,(Z)* in the lowest panel were obtainedby our methodfor the groundband. The quasiparticlelevel correspondingto the solid lines in the uppermostand the lowestpanelsin Fig. 2 consists of mainly 16513121and t642 5121 in the asymptoticquantumnumberof the Nilsson model.As for other levels, however, it is difficult to assignan asymptoticquantumnumberclearly throughoutthe range of the calculated angularmomentum.By comparingthesethreefigures,it is seenthat in the regionof Z_<11 our result is exactly the sameas that of ordinary CHFB theory. This is simply because,in this region, any of the u,(Z)~ of the ordinary CHFB theorydoesnot exceedu-( Z,j$,,,, ) 2. For the Sband I,, turnedout !o be 22. The “o-constraint” is different for the regionsof I< 22 and I> 22 in the S-band,becausethe bandcrossingwith I> 22 is inducedby the negativeparity proton. Thus in the region Z> 22 the constraint( 1) with jf& = 11/2 was imposedon the negativeparity protons,whereasin the region I< 22 the constraintwith j&, = 1312was imposedon the positiveparity neutrons.For ejs, (la) we averagedthe two values of the positive and negativesignaturelevels.

Fig. I The energy of the ground- and S-bands of “%r as a function of the angular momentum 1. Solid lines and open circles sholv the numerical and the experimental values [ 18,19], respectively.

Fig. 2. Some o-factors squared for the positive parity neutrons, which are associated with the transformation from I&,) to another )liFS state. The uppermost shows those obtained by the ordinary CHFB theory for the yeast line. The middle illustrates r,.(I, j&,)* of Qs. (5) and (6). and the lowest is that obtained by our method for the ground band. Solid, dashed, dot-dashed, etc., lines in the uppermost panel correspond to those in the lowest.

The kinematicalmomentof inertia of thesebandsis drawn in Fig. 3 as a function of the angularvelocity o. The lower curve and circles are thosefor the groundband,and the upperonesare for the S-bana. The behavior of the kinematical momentof inertia of the S-bandwith I= 8-10 (w
-0.5

Fig. 3. The kinematical moment of inertia of the ground- and S-bandsof ‘-Er as a function of the angulnr velocity o. Solid lines and open circlesshow numericalandexperimentalvalues,respectively.Upper andlower valuesareofthe S-bandandground band,respectively.A dashed (I,l&,, ) multiplied byafactor 1.2 for8ez1522. Theexperimentaldatawereobtainedby usingY= (21- 1)fi’/ [20] withenergyofin-bandtransitionE,

Fig. 4. Calculatedand experimentalg factorsof IHEr. Solid and dashedlines show calculatedvaluesof the ground-and S-band,respectively, and solid and open circles areexperimentaldata at I= 2 in the ground band [21] and I= 12 in the S-band [22], respectively. References [ 11 A. Bohr and B.R. Mottelscn, Suppl.J. Phys. Sot. Japan44 (1978) 157. (21 R. Bengtssonand S. Frauendorf,Nucl. Phyj. A 327 ( 1979) 139. ]3] I. Ragnarssonand S. Aberg, Phys.Ser. 24 (1981) 215. [4] T. Rengtssonand 1. Ragnarsson,Nucl. Phys. A 436 (1985) 14. [S J T. Bengtsson,Nucl. Phys.A 496 (1989) 56. 161 Y.R. Shimizu and K. Matsuyanagi,Prog.Theor. Phys. 74 (1985) 1346. 171 I. Hamamoto,Nucl. Phys.A 271 (1976) 15. [8] E.R. Marshalekand A.L. Goodman,Nucl. Phys.A 294 ( 1978) 92. ]9J R. Bengtssonand W. Nazarewicz,Z. Phys.A 334 ( 1989) 269. ] IO] P. Ring and P. Scbuck,The NuclearMany-Body Problem(Springer,New York, 1980) Ch. 7. [ I1 ] C. Bloch and A. Messiah,Nucl. Phys.39 ( 1962) 95. [ 12] H.J. Mang, B. Samdi and P. Ring, 2. Phys. A 279 ( 1976) 325. [ 131 J.L. Egido, H.J. Mar18and P. Ring, Nucl. Phys.A 334 (1980) 1. [ 141 A. Bohr and B.R. Mottelson,NuclearStructure,Vol. II (Benjamin, Reading,MA, 1975) Ch. 5. 115] H. Sakamotoand T. Kishimoto, Nucl. Phys.A 501 (1989) 205. [ 161 J. Terasaki,Prog.Theor. Phys.92 ( 1994) 535. [ 171 I. Hamamoto,Nucl. Phys.A 232 (1974) 445. [ 181 SW. Yateset al., Phys.Rev. C 21 (1980) 2366. I191 3.C. Bacelar,Contrib. 5th Nordic Conf. (Jyvaskyla, 1984). [20] M.J.A. de Voigt, J. Dudek and Z. Szymadski,Rev. Mod. Phys.55 ( 1983) 949. [21] C.M. Ledererand V.S. Shirley,Table of Isotopes(Wiley, New York, 1978). [221 Y. Nagai et al., Phys.Rev, Len. 51 (1983) 1247. ]23] K. Sugawara-Tanabe and A. Atima, Phys.Lett. B 229 (1989) 327. [241 I. Hamamotoand W. Nazarewicz,Phys.L&t. II 297 (1992) 25.