Volume 64B, number 2 PHYSICS LETTERS 13 September 1976

COEXISTENCE OF SPHERICAL AND DEFORMED STATES

NEAR CLOSED SHELLS

K. HEYDE, M. WAROQUIER' , H. VINCX’ and P. Van ISACKER Laboratorium voor Kernfysika, B-9000 Gent, Belgium

Received 16 April 1976

In the mass regions near single-closed shell configurations such as Tl(Z = 81), Sb(Z = 5 l), In(Z =49) and N = 81, N = 83 isotones; deformed Nilsson orbitals can give rise to minima in the total potential energy of the nucleus. Thus low-lying, deformed states can occur whereas the ground state in these nuclei is spherical. Comparison with the exist- ing experimental data is made in all cases.

In many nuclei, where one kind of nucleons nearly consists of a closed shell configuration (+ 1 nucleon), indications for possible nuclear coexistence between spherical and deformed states exist. One of the regions of interest is formed by the In isotopes (2 = 49). In these nuclei, a group of low spin positive parity states [l-8] cannot be explained at present as pure single- hole (lgG,$ or single-hole states coupled to the 2+ quadrupole vibrational mode in the Sn isotopes [9-l 11. A rotational band associated with the l/2+ [43 I] Nilsson orbital could serve as an explana- tion. Recent experiments [ 121 also indicate evidence for a AJ = 1 normal rotational band on the [SOS] 1 l/2- Nilsson orbital in 111T1131n.

Other mass regions of interest are the Tl isotopes (Z = 81) where again experimental evidence for a normal, rotational band on top of the 9/2- [5 141 orbi- tal is observed (201Tl[13], 1g1~1g3*1g5~1g7T1[14]). Also, possible indications for a low-lying (E, 2

1 MeV) 13/2+ [606] orbital are at present availa- ble [14]. In the light-odd-mass Sb (Z= 51) and I iso- topes (2 = 53), recent experiments by Gaigalas et al. [15, 161 clearly indicate a normal rotational band structure on top of the 9/2+ [404] Nilsson orbital.

Finally, in the N= 83 isotones, at an excitation energy 1.5 MeV < E, < 2.0 MeV, low-spin positive parity levels have been excited in N= 84 (d, t)N= 83 experiments [ 17, 181, indicating levels with hole-char-

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acter. However, no evidence is available if also in this case, a rotational band structure extends on top of the intrinsic excitations. All the possible candidates for the band head of rotational bands are excitations through one major oscillator shell as compared with the spherical ground state and low-lying levels. In nuclei with one nucleon less than a closed shell config- uration, such as In, Tl, the high a values from single- particle states with a large i-value one major shell higher can lower the total energy of the nucleus con- siderably in oblate deforming (lh,,,2 : 1 l/2- [505]

in In; lhg,2 : 9/2-[514] and li13i2 : 13/2+[606] in Tl). Also, in the case of In, the large negative slope of the l/2+ [43 1 ] orbital on the prolate side, can give this deforming tendency. In nuclei with one nucleon more than a closed shell configuration such as Sb, the high L-l values from single-particle orbits with large j value, can also lower the total energy of the nucleus in prolate deforming ( lgg12 : 9/2+ [404]) because one particle is lifted from a pairwise filled high a orbital in going to excited states. If in the actual nuclei, this strong deforming tendency will result into stable de- formed minima, will depend strongly on the potential energy of the underlying core nucleus, which in the case of Tl (Hg) and In (Cd) are known to be rather flat.

We made use of the macroscopic-microscopic renor

malisation procedure [ 191 to obtain the total poten- tial energy of the nucleus. The single-particle energies were calculated using a modified harmonic oscillator potential [20], with only quadrupole degrees of free- dom taken into account. The parameters K, p which determine the it l S and if term respectively, are

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Volume 64B, number 2 PHYSICS LETTERS 13 September 1976

taken for the N= 81,83, Z = 81 nuclei as interpolated between rare earth and transuranic nuclei values [ 181, whereas for the lighter ones (Z = 49, Z = 5 1) values as determined by Ragnarson have been used [2 11. The macroscopic part of the energy was obtained using the liquid drop formula with Myers-Swiatecki parame- ters [22]. The pairing strength used was G = 19.2/A f 7.4 (N-Z)/A2 (+protons, -neutrons) with fl (or a?$ twofold degenerate levels taken into account in the pairing calculation for the heavier nuclei (N= 81,83; Z = 81). For the Sb and In isotopes however,

Gp = 22.5/A, G, =22.5/A - 18(&Z)/A2

and &@ (m% twofold degenerate levels taken into account.

In the odd-mass nuclei,‘pairing calculations with

and without blocking have been performed in order to reach definite conclusions not depending on some approximations used. In the no blocking calculation, the total energy *l was calculated as

EL:; = + (EN+2 + EN) + d”) 19P’

where EN, EN+2 denote the total energy of the adjac- ent doubly-even nuclei and $ (E,v+2 t EN) t A corre- sponds, within the BCS approach (linear variation of total energy as a function of partical number) with a hypothetical “odd-even” system. The one quasi-parti- cle energies aad have been calculated for a system with odd particle number.

In the blocking calculation [24], the total energy was obtained as

where EN! denotes the energy of the nucleus with even number of protons and neutrons but with the level a blocked in the determination of A and I&, for the

odd system of nucleons *2

*r Direct calculation of the total energy for the odd-system,

by using formally the formulae for the doubly-even system,

yields about the same result as the interpolation formula

(within 5%) [23]. *2 Z’ means blocked level not taken into account, Z” means

that only pairwise occupied levels are considered in the no

pairing limit, such as to obtain a pairing correction going towards zero for vanishing pairing.

136

_ @(‘)I2 _ 6 1 _ f 1 _ (Ece)) _ (E(O)) . G(O) P v

Calculations have been performed for the 185-201 Tl isotopes, for the 113-133Sb isotopes, the 117-1271 iso-

topes, the 107-t311n isotopes and N= 81 andN= 83 isotones (Ce, Nd, Sm).

Only in the case of N= 8 1,83, no pronounced de-

formed minimum occurs and the excitations through one major oscillator shell remain as 2p-lh(2h-lp) spherical excitations. The reason for this has to be found in the much larger neutron shell correction (-6.7 MeV at ~2 = 0) associated with the N= 80,82 neutron shell configuration as compared with all other mass regions studied here (-4.0 MeV > Eshell > -4.5 MeV for In, Sb, I, Tl). This is a reflection of the large energy gap between major shells at N = 8 1,83 (5.2 MeV) combined with a small pairing strength G, = 0.13 MeV as compared with the Tl region (Egap =

3.00 MeV, G, = 0.10 MeV) and Sb, In region (Egap = 4.60 MeV, G, = 0.20 MeV). It is impossible to discuss all results extensively. Therefore, only the most impor- tant points will be given.

i) In the case of 1 171n, a clear-cut example for a deformed minimum exists with e2 = 0.17 and E, =

1.270 MeV *3, resulting from the l/2+ [43 1 ] Nilsson orbital. Also the lh,,,z orbital gives rise, on the oblate

side, to a deformed minimum associated with the 1 l/2- [SOS] orbital at e2 = -0.18 and E = 2.45 MeV.

Experimental evidence for such a level ecists at 2.236 MeV and 2.404 MeV in 111~1131n respectively [12].

ii) For the Sb isotopes, a deformed minimum asso-

ciated with the 9/2+ [404] Nilsson orbital occurs (fig.1). In table 1, the change of E, as a function of e2 is indi- cated. For the lower mass Sb nuclei (113-125Sb), the neutron shell correction works opposite to the strong proton shell correction thus making the potential energy of the underlying doubly-even core nucleus soft against quadrupole deformation. For the heavier Sb

*3 E, measured between the odd-even system at spherical

point and the deformed minimum.

Volume 64B, number 2 PHYSICS LETTERS 13 September 1976

d-even backgroud

I I I I I I I I I I I1 I1 I I I I I I I -02 -OS 0 OS 0.2 -0.3 0.2 -0.1 0 0.1 0.2 0.3

-& 2 -E 2

Fig. 1. The total potential energy surfaces for ‘iTSb64 with the odd particle placed in different Nilsson orbitals as indicated. The odd-even background is also given with a dashed line. The corresponding, relevant Nilsson scheme is given together with the Fermi

‘15 Sb 51 64

level.

Table 1 Theoretically calculated equilibrium deformation Eequi and excitation energy E,, for the prolate minimum for the 9/2+[404] orbital.

___--

Eequi E exe

l13Sb lrsSb l17Sb llgSb lzl Sb lz3Sb 12sSb lz7Sb

133Sb

0.140 1.24 0.145 1.20 0.150 1.20 0.145 1.24 0.140 1.35 0.125 1.48 0.105 1.72 0.085 1.97 0.055 2.22 0.030 2.38 0.015 2.45

isotopes, however, the neutron and proton shell cor- rections add up such as to remove the permanent de- formation and to make Ex higher in going towards 133Sb. For the Sb isotopes, also the l/2- [330] orbi- tal, originating from the 2pIj2 hole state, gives an ob- late, permanent deformation. In l15Sb this occurs at e2 = -0.21 and E, = 2.4 MeV.

iii) For the N= 81,83 isotones, the deforming tend- ency from the high L! values originating in 2f7,2 and lhg12 orbitals towards oblate shapes and from the 52 values, originating in the 2dFh and 3sib hole orbitals towards prolate shapes respectively, are not observed. This is caused by the very large negative neutron shell correction energy at e2 = 0 (-6.7 MeV for ‘iiNd82).

iv) For the odd-proton Tl isotopes, again the high GZ components from the lhg12 and li1312 orbitals i.e. 9/2 [S 141 and 13/2- [606] respectively, induce on the oblate side stable, permanent deformations (fig. 2). In lowering the neutron number, the excitation energy is

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Volume 64B, number 2 PHYSICS LETTERS 13 September 1976

Fig. 2. The total potential energy surfaces for 19’Tl

d\. 116 with the odd particle placed in different Nilsson orbitals as indicated. The

odd-even background is also given with a dashe me. The corresponding, relevant Nilsson scheme is given together with the Fermi level.

rapidly descending in going from 2u1 Tl towards lgl Tl, thereby reproducing qualitatively the experimental trend [ 141. Even the ground state is evolving towards permanent deformation due to the very soft potential energy surface of the underlying Hg-nuclei. So, one can qualitatively understand why near closed shells (+ 1 nucleon), coexistence between spherical and de- formed states often exists and is no exceptional situa- tion. In the heavier nuclei (Sb, In, Tl, N= 8 1,83), it becomes possible, by lifting the spherical degeneracy to deform the nucleus and lower its total energy con- siderably for AN= 1 excitations into high j orbitals as

compared with the spherical ground state configura- tion. This happens, especially for the high R compo- nents, mainly on the oblate side (one nucleon less than a closed shell) such as Tl, In and on the prolate side (one nucleon more than a closed shell) for Sb, I isotopes.

The authors are indebted to Professor Dr. A.J.

138

Deruytter for his interest during the course of this work. One of the authors (K.H.) is grateful to Dr. B. Fossan for communicating results on the Sb, In and I

isotopes prior to publication.

References

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Volume 64, number 2 PHYSICS LETTERS 13 September 1976

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