PHYSICAL REVIEW C VOLUME 47, NUMBER 4 APRIL 1993

Systematic study of neutron-deficient Ho isotopes in a relativistic mean field theory

Suresh Kumar Patra and Prafulla Kumar PandaInstitute ofPhysics, Bhubanestoar 751 005, India

(Received 21 August 1992)

We have calculated the binding energies, rms radii, and multipole moments of Ho isotopes(N =81—98) using a relativistic mean field theory. The quadrupole moments and the change in meansquare charge radius are compared with the experimental and other theoretical results. We observed ashape transition from oblate to prolate deformations in going from lighter to heavier isotopes.

PACS number(s): 21.60.Jz, 21.60.Cs, 21.10.Ft, 21.1D.Ky

I. INTRODUCTION

The change of shapes of the rare earth nuclei from ob-late to prolate deformation in going from the neutron-deficient to neutron-rich side is studied using a self-consistent Hartree-Fock-Bogoliubov (HFB) approach in anonrelativistic Skyrme-type interaction [1]. The oblate toprolate shape transition in going from lighter to heavierisotopes and other interesting properties of the nucleihave attracted the attention of many theoreticians [2] andexperimentalists [3—5]. The conditions needed for theobservation of shapes coexisting in such nuclei have beena matter of challenge to calculate theoretically.

In recent years, the relativistic mean field (RMF)theory [6—8] has been a successful model to reproduceexperimental observables such as the binding energies,rms radii, and multipole moments (dipole, quadrupole,and hexadecupole moments) throughout the periodictable [7—12]. The inclusion of the nonlinear self-interaction of the o field [13] improves the resultssurprisingly. The change of shape from prolate to oblatein going to the more neutron-deficient isotopes presents asuitable test of the models.

The change of shapes of the Pt isotopes was recentlystudied by Sharma and Ring [14]. The shapes of nucleisuch as Os and Pt have been studied by using the HFBformalism which are purely nonrelativistic treatments [1].In this work the multipole moments (quadrupole andhexadecupole) and other properties of Ho isotopes(%=81—98) are calculated using deformed relativisticmean field theory. The calculated quadrupole momentsare compared with experimentally observed values.

The paper is organized as follows. In Sec. II wepresent the theory and procedure of the numerical calcu-lations. The results on binding energies, quadrupole de-formation, hexadecupole moments, and rms radii bothfor prolate and oblate solutions are discussed in Sec. III.Concluding remarks are given in Sec. IV.

II. THEORY AND CALCULATION

We start with the relativistic Lagrangian density for anucleon-meson many-body system [7,8, 11]:

Electronic address: patra@iopb. ernet. inprafulla@iopb. ernet. in

'Q—t"—0 + 'm —V"V —g f y"f V ——'B" .Bpv 2 N p co l l p 4 pv

+ ,'m'—p"p„gg; y—"~g; p„,'F" +—„—1 —r„—eely" g; A„. (2.1)

The o. meson is assumed to move in a nonlinear potential[13],

V(o )=—,'m o + —,'g2a + —,'g&a (2.2)

The field for the cr meson is denoted by cr, that of the co

meson by V„, and of the isovector p meson by p„.denotes the electromagnetic field, which couples to theprotons. g; are the Dirac spinors for the nucleons, whosethird component of isospin is denoted by r3;. Here gg, g~, and e /4~= —„', are the coupling constants for o,co, and p mesons, and the photon, respectively. M is themass of the nucleon, and m, m, and m are the massesof the 0., co, and p mesons, respectively. 0", 8", andF" are the field tensors for the V", p", and the photonfields, respectively [7,8, 11].

To describe the ground-state properties, we need a stat-ic solution of the above Lagrangian. For this case themeson and electromagnetic fields are time independent,whereas the nucleon wave functions oscillate with asingle-particle energy e;. The field equations for the fer-mion and boson fields are obtained from the Lagrangianof Eq. (1) and can be found in Refs. [7,8, 11]. These arenonlinear, coupled partial differential equations, whichare solved self-consistently.

The set of partial differential equations are solved byexpanding the upper and lower components f; and g;

—of-the Dirac spinor P, and the wave functions of the bosonfields in terms of a deformed harmonic oscillator poten-tial basis, taking volume conservation into account [7].The frequencies Ace~ and A'co, can be expressed in terms ofa deformation parameter Po.

In numerical calculations the wave functions for thebosons and fermions are expanded in a deformed har-monic oscillator basis with the number of maximum os-cillator quanta, N, „=12,for both bosons and fermions.

1514 1993 The American Physical Society

47 SYSTEMATIC STUDY OF NEUTRON-DEFICIENT Ho. . . 1515

9Qo =Qo. +Qo, = AR P, (2.3)

and the hexadecupole moment is obtained from thedefinition

Q," =("~„(e))„,, (2.4)

where R = l.2 A ', Qo are the quadrupole moments, and

Q4 the hexadecupole moment.The total binding energy of the system is

E„„,=E,„,+E +E +E +Ec+E„„,+E, , (2.5)

where E „, is the sum of the single-particle energies ofthe nucleons and E, E, E, Ec, and E „,are the con-tributions of the meson fields, the Coulomb field, and thepairing energy, respectively. We have used the pairinggap as defined in Ref. 15 to take pairing into account.E, = —

—,'41 3 ' is the nonrelativistic approximationfor the center-of-mass energy correction.

We have solved the set of coupled equations for nu-cleons and bosons iteratively following the procedure ofRefs. [7,8] using the NL1 parameter set (M=938.0,m =492.25, m =795.359, and m =763.0 MeV,g = 10.138, g = 13.285, g =4.9755, g2 = —12. 172fm ', and g3 = —36.265). This set gives a good accountof various observables such as binding energies, rms radii,compressibility modulus, coefficient of asymmetry ener-

gy, and other nuclear properties, and hence we have usedthe nonlinear parameter set NL1 [11]. The initial quad-rupole deformation parameters in our calculations are

The deformation parameter P is obtained from the calcu-lated quadrupole moments for the protons and neutronsthrough

1/2

taken as Po =+0.3 to get the prolate and oblate solutions.The field corresponding to the isoscalar-scalar cr meson,which is a very broad two-pion resonance state (s wave),provides strong scalar attraction at long distance ()0.4fm). The isoscalar-vector meson co, which is a p-wave 3m.

resonance state, shows a strong repulsion at short dis-tance. Similarly, the field due to the isovector p meson(p-state 2m. resonance) is important for proton-neutronasymmetric systems. For neutron-rich or neutron-deficient nuclei, the p meson becomes important, becausethe proton-neutron isospin asymmetry is severe in thesesystems.

The three-vector part of the vector-meson fields doesnot respect time-reversal symmetry due to the odd Znumber of Ho isotopes. The contribution of the odd nu-cleon to the binding energies is nonzero. In this RMFcalculation, we take into account the nuclear current andthe resulting spatial component of the vector field due toco mesons, and we have neglected the contributions of thespatial components of the vector fields of p mesons andphotons.

III. RESULTS AND DISCUSSIONS

We have calculated the rms radii of the proton, neu-tron, and total matter radii (r~, r„,r ), binding energies,and multipole moments for Ho isotopes having neutronnumber X=81—98.

The results for the quadrupole and hexadecupole mo-ments are given in Table I for both prolate and oblatesolutions. BE(prolate) are the binding energies for theprolate solutions and BE(oblate) that of the oblate solu-tions. From the table one can note that beyond massnumber A = 149 the binding energies BE(prolate) for theprolate solution are always larger than those for the ob-

TABLE I. RMF results of the binding energies (BE) of the total quadrupole and hexadecupole moments for both prolate and ob-late solutions are given. The rms charge radius r,h of the prolate solution (r,h for ' ' Ho of the oblate solution) is given in column9. Experimental binding energy is taken from Ref. [16] and the experimental electric quadrupole moment Qo from Ref. [3]. Ener-gies are in MeV, quadrupole moments (Qo and Qo„) in b, hexadecupole moments (Q4) in b, and r,„ in fm.

148149150151152153154155156157158159160161162163164165

BEProlate

1232.4951238.8121248.7061255.9251264.5831271.5741279.8531286.7871295.2671300.5831310.2231317.0711325.3381331.1151338.3621344.734

BEOblate

1212.5311222.8851231.9501237.9671248.0281255.8821265.3151270.1301278.3351284.6051293.2731298.8281307.8771314.5221320.8391329.0651332.9241341~ 141

BEExpt.

1200.5101212.0701220.5101230.1631238.1701247.5261255.2111264.7101272.2501281.6801289.0601298.2731305.4001314.2821321.1961329.6041336.2271344.267

QoProlate

0.0610.896.388.119.41

10.5711.6412.9914.8616.6917.6318.1718.6019.1119.5519.93

QoOblate

—0.093—0.028—0.794—3.564—5.74—6.91—7.81—8.60—9.43

—10.32—11.17—11.89—12.49—12.94—13.28—13.55—13.78—13.98

Q4Prolate

—0.0020.0080.3750.4920.5490.5890.6340.7510.9611.1271.1791.1521.0991.0521.0030.947

Q4Oblate

—0.003—0.012—0.020

0.0970.1980.2490.2930.3410.4000.4720.5460.6110.6620.6930.6970.6800.6510.618

rch

5.0555.0585.0645.0755.0995.1165.1315.1585.1735.1745.1945.2155.2295.2405.2495.2605.2695.278

QopRMF

4.7695.1845.7116.4517.1807.5267.6917.8087.9568.0708.157

QopExpt.

4.2174.6096.3047.2176.8266.8266.8266.6967.609

7.478

SURESH KUMAR PATRA AND PRAFULLA KUMAR PANDA 47

late solutions. The isotopes ' Ho and ' Ho are com-pletely oblate in shape. Thus a shape transition occursfrom oblate to prolate after crossing the mass numberA =149 (see Figs. 1 and 2). For a given nucleus, themaximum binding energy correspond to the ground stateand all other solutions are the excited states. The bindingenergies for Ho isotopes indicate that all isotopes&soHo &65Ho are prolate in shape and there are excitedoblate shapes for these isotopes. In the case of ' Ho,' 'Ho, ' 'Ho, ' Ho, and ' Ho, the differences in bindingenergies between prolate and oblate solutions are 0.445,0.845, 0.678, and 0.043 MeV, respectively. This smalldi6'erence in energies shows the shape coexistence (pro-late and oblate) of these four nuclei. In our calculationsthe differences in binding energies between prolate andoblate solutions increase with an increase of neutronnumber, and the values of BE(prolate) and BE(oblate) arenot very far from each other. As the differences in bind-ing energies between these two solutions are very small,the excited oblate shapes are the low-lying excited states.

In the fifth and sixth columns of the Table I, the valuesof the total quadrupole moments are given for prolateQo(prolate and oblate solution Qo(oblate), respectively.The quadrupole moments gradually increase with the in-

crease of neutron number. For example, ' Ho has theminimum quadrupole moments [Qz(oblate) = —0.028 b]and ' Ho has the maximum quadrupole moments[Qo(prolate) = 19.93 b and Qo(oblate) = —13.98 b].With regard to the magnitude, the quadrupole momentsof the prolate solutions are much higher than the oblatesolutions.

The total hexadecupole moments of the isotopes aregiven in the seventh and eight columns of Table I. Likethe quadrupole moments, the hexadecupole moments ofprolate solutions are greater than the oblate solutions.The hexadecupole moments of the prolate [Q~(prolate)]and the oblate [Q4(oblate)] solutions are the same in

sign. Q„(prolate) and Q4(oblate) increase with an in-

crease of neutron number, and these are maximum at3 =160 and 162 for prolate and oblate solutions, respec-tively. After A =160 and 162, the values of Q4(prolate)and Q4(oblate) decrease with mass number. From TableI we further notice that the transition of hexadecupolemoments from a negative value to a positive value is ingoing from mass number 3 = 150 to 151.

In Fig. 1 we have plotted the total deformation param-eter P versus mass number. In both the prolate and ob-late solutions, the f3 values increase with the mass numberof the isotope. The deformation parameters of the pro-late solutions are relatively larger than the deformationparameters of the oblate solutions. It is minimum for

Ho. The I3 values for ' Ho and ' Ho are slightlynegative. Beyond A =149, the P values become moreand more positive.

Figure 2 shows the quadrupole moments of protonsand neutrons for both prolate and oblate solutions. Thecurves with solid dots indicate the quadrupole momentsfor the neutron and the solid and dashed curves for theproton. The quadrupole moments of prolate solutionsare comparatively larger than the oblate solutions. Fromthis figure (Fig. 2) one can see that the quadrupole mo-

0.3- ~ prola—~--obla

02-

CQ 0

-0.1

-02-

-0.3-

~~~~

~ ~

~ ~

-0.4 I

150I

1'55

I

160

Mass number

I

165 170

FIG. l. Plot of total quadrupole deformation parameter Pversus mass number of Ho isotopes using the RMF calculation.

prp(ate (neutron)Prolate &Proton)

10

Qr

OCL

i-'U

—10'l 50 155 160

~ ~ ~

I

165 170

Mass number

FIG. 2. Plot of quadrupole moments of proton and neutronversus mass number of Ho isotopes for both prolate and oblatesolutions using the RMF calculation.

ments of neutrons are greater than the quadrupole mo-ments of protons. For example, Q~„=11.78 b for theneutron and Qo =8.16 b for the proton of ' Ho for theprolate solutions and it is Qo„=—8.36 b, Qz~

= —5.63 bfor the oblate solutions.

In Fig. 3 the hexadecupole moments of proton andneutrons are given for both prolate and oblate solutions.The same as the total hexadecupole moments, the hexa-decupole moments of protons (Qi4) and that of the neu-tron (Q4) increase up to A =160 for prolate solutionsand it is 3 =162 for oblate solutions, respectively, andthen decrease with an increase of mass number. In bothsolutions the hexadecupole moments of neutrons arelarger than the hexadecupole moments of protons. Thereis also a transition of proton and neutron hexadecupolemoments from negative to positive value, while crossingthe isotope ' Ho to ' 'Ho for both prolate and oblatesolutions.

The results for the rms radii [proton (r~ ), neutron (r„),and matter rms radii (r ), respectively] versus neutronnumber are plotted in Fig. 4. From this figure it is clearthat the rms radii increase with an increase of neutronnumber. The increase of neutron radius is substantiallygreater than the increase of the proton radius. In our cal-

47 SYSTEMATIC STUDY OF NEUTRON-DEFICIENT Ho. . . 1517

0.8-

p ro late(neutron)-Oblate(neutron )

- Prolate(Proton)————0b l a te ( p r o ton)

9"

8-

RMFSKM

0.6-

0.4-

0.2-

7-COr

EO 6-Or

0CL

5'U'U

4-

—0.2150

I

155 160I

165 170155 160

Mass number

FIG. 3. Plot of hexadecupole moments of proton and neu-tron versus mass number of Ho isotopes for both prolate andoblate solutions using the RMF calculations.

culations we found quantitatively the same rms radii forboth prolate and oblate solutions. In Fig. 4 we have plot-ted the rms radii for prolate solutions. The substantialincrease of the neutron radius (r„) rather than the protonradius (r~) shows the presence of more neutrons on thesurface of the nucleus with an increase of the mass num-ber, which form a neutron skin on the nuclear surface.The thickness of the neutron skin can be calculated fromthe difference between the neutron and proton radii [i.e.,the neutron skin thickness is (r„—r ) Ir ].

The RMF and experimental electric quadrupole Qo(i.e., the quadrupole moment of the proton) moments aregiven in columns 10 and 11 of Table I, and these are plot-ted as a function of mass number in Fig. 5 included withother theoretical predictions [3]. The results of our RMFcalculations are quite close to the experimental values aswell as other values calculated theoretically.

We have also given a comparison between the changein mean square rms charge radii in Fig. 6, between RMF,Hartree-Fock (HF) solutions with the Skyrme SkM' force[3], and experimentally observed values with respect toA =165 (i.e., (b,r ),6s ~ =r, 6s r~ ). The solid cu—rve

6-

Mass number

FIG. 5. Comparison of quadrupole moments of the RMFsolutions with experimental results and other theoretical calcu-lations. The solid line from prolate solutions of the RMF calcu-lations, the dashed curve from Hartree-Fock (HF) results usingSkM force, the solid line with solid dots is the experimentalcurve (experimental error bars are not taken into account), andthe dashed curve with solid dots from the macroscopic-microscopic (MM) method of calculation.

with dots is the experimental values, the solid curve is theRMF, and the dashed curve is the Hartree-Fock solutionwith the SkM' force. The qualitative behaviors of thechange in mean square rms charge radius ( hr ),6s „ in

RMF, nonrelativistic Hartree-Fock (HF), and experimen-tally observed values are similar (Fig. 6). The onlydifference is that the magnitude of the RMF results isslightly more than the magnitude of the other two values(i.e., (b,r ) «s ~ of RMF) HF) experiment).

In RMF calculations a sudden change (rounded regionin Figs. 7 and 8) in multipole moments (quadrupole andhexadecupole moments) and hence in (Ar ),6s „ takesplace between mass numbers A = 156 and 157, agreeingwith the experimental change of nuclear deformation at

2-RMF

HF

Expt.

5.8 - — rrns C 1.2'5 "

5.6

54-

5.2-

4,8-

4.6-

rp

~O~H

~ -0~O

~ ~I

155 160I

165

u 1

A,075-

CI

0.5

0.25-

I

150 155

Mass Number

160 165

Mass number

FIG. 4. Plot of rms radii of proton (r~), neutron (r„), andmatter rms radius (r ) versus mass number of Ho isotopes forprolate solutions of the RMF calculation.

FIG. 6. Plot of change in mean square rms charge radii(Ar ) versus mass number of Ho isotopes. The dashed linewith solid dots from prolate solutions of the RMF results, thesolid line with solid dots from Hartree-Fock (HF) method withSkM' force, and the solid curve is the experimental calculations[3] of Ho isotopes.

1518 SURESH KUMAR PATRA AND PRAFULLA KUMAR PANDA 47

ps

?0

]5-

10- 0.8-

—.——p rota te——-- pbtate

/ ~~/

I

Cit

0Ct.

L'0

5-

-5

-10

-15'150

~ ~

~ ~

I

155

~ ~

~ ~ ~I I

165160

o.e-el

0 0y-

0.2

-0.1

~Y/

II

155I

160

Mass number

165

pass number

FIG. 7. Plot of total quadrupole moments versus mass num-

ber of Ho isotopes for both prolate and oblate RMF solutions.

FIG. 8. Plot of total hexadecupole moments versus massnumber of Ho isotopes for both prolate and oblate RMF calcu-lations.

A =156 and 157 [3] (Figs. 1, 2, 3, 5, 7, 8 and Table I).The sudden change of hexadecupole moments (for pro-ton, neutrons, and total hexadecupole moments) is moreprominent than the change of rms radii and quadrupolemoments (Figs. 1 —8 and Table I). Toward the neutron-deficient side of the isotopes, there is one more suddenchange (rounded region in Figs. 7 and 8) in multipole mo-ments that occurs between the mass numbers A =150,151, and 152 (Figs. 1 —3,7,8) for protons, neutrons, andtotal multipole moments.

IV. CONCLUSIONS

We found that the increase of the quadrupole moments(for protons, neutrons, and total quadrupole moments)with an increase of neutron numbers and the increase ofthe hexadecupole moments (for protons, neutrons, and

total hexadecupole moments) for a prolate solution is upto A =160 and for an oblate solution it is 3 =162. Thevalues of multipole moments for neutrons are substantial-ly larger than the values of the protons, and it is alwaysobserved that the multipole moments of prolate shapesare more than the values of oblate shapes. In our RMFcalculations, we obtain a sudden change in multipole mo-ments in the regions A =150—152 and 156—157. Thesudden change in multipole moments in the region3 =156—157 agrees well with experimental observation.We concluded from the RMF calculations that the iso-topes ' Ho and ' Ho are oblate in shape and all otherisotopes are prolate in shape. Thus a shape transitionoccurs (-148 region) in the series. For nuclei

Ho —' Ho in the series, we observed shape coex-istences and there are some low-lying oblate solutions forall other isotopes (' Ho —' Ho) in the Ho series.

[1]P. Quentin and H. Flocard, Annu. Rev. Nucl. Part. Sci.28, 523 (1978);A. Ansari, Phys. Rev. C 33, 321 (1986).

[2] M. Baranger and K. Kumar, Nucl. Phys. A110, 410(1968); K. Kumar and M. Baranger, ibid. A110, 529(1968); H. Yadav, H. Toki, and A. Fassler, Phys. Lett.76B, 144 (1978).

[3] Cr. D. Alkhazov et al. , Nucl. Phys. A504, 549 (1989).[4] C. Thibault et al. , Nucl. Phys. A637, 1 (1981).[5] I. Y. Lee et al. , Phys. Rev. Lett. 39, 682 (1974); M. V.

Hoen et al. , Phys. Rev. C 24, 1667 (1981); F. T. Baker,Nucl. Phys. A331, 39 (1979).

[6] J. D. Walecka, Ann. Phys. (N.Y.) 83, 491 (1974).[7] Y. K. Crambhir, P. Ring, and A. Thimet, Ann. Phys.

(N.Y.) 198, 132 (1990).[8] S. K. Patra and C. R. Praharaj, Phys. Rev. C 44, 2552

(1991).[9] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. 16, 1

(1986).[10]C. J. Horowitz and B. D. Serot, Nucl. Phys. A368, 503

(1981).[11]P. CJ. Reinhard, M. Rufa, J. Maruhn, W. Greiner, and J.

Freidrich, Z. Phys. A 323, 13 (1986).[12] C. E. Price and G. E. Walker, Phys. Rev. C 36, 351 (1987).[13]J. Boguta and A. R. Bodmer, Nucl. Phys. A292, 413

(1977).[14] M. M. Sharma and P. Ring, Daresbury Laboratory Report

No. DL/NUC/P329T, 1992.[15]P. Moiler et al. , At. Data Nucl. Data Tables 39, 225

(1988).[16) A. H. Wapstra and G. Audi, Nucl. Phys. A432, 1 (1985).