The 6.05 MeV 0+ state in 16O: A shape isomer

Nuclear Physics A211 (1973) 565--572; ( ~ North.Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE 6.05 MeV 0 ÷ STATE IN 160: A SHAPE I S O M E R BENT SORENSEN

The Niels Bohr Institute, Bleydamsve] 17, University o/ Copenhagen, Denmark Received 30 May 1973 Abstract: A variational calculation predicts the existence of a quadrupole deformed isomeric state in

160, which may be associated with the observed rotational band starting at 6.05 MeV. Calculated B(E2) value and strength in the 180(p, t) reaction support this interpretation.

1. Introduction

The low excitation energy of the first excited 0 + state in 160 was a puzzle until it was realized that the state might be shape deformed 1). This idea was used in calculations that either employed mixed 0p-0h, 2p-2h and 4p-4h configurations, some of which were deformed orbitals of a prescribed quadrupole deformation (but all taken with respect to a spherical vacuum [refs. 2, 3)] or, alternatively, attempted to calculate the energy of such states as function of deformation, in order to find a possible minimum 4, 6). The reason for constructing the excited 0 ÷ state as mainly a 4p-4h configuration 2-4) must be the belief that no shape isomeric states exist, in other words that the potential energy as function of deformation exhibits no minima except the spherical one. Support for this point of view seems to come from the self-consistency calculations 4.6), yet we shall in this communication direct the attention to a different point of view. The apparent disagreement with the earlier calculations will be shown to rest on the neglect of (or underestimate of) pairing type forces in ref. 4) and, in comparison with ref. 6) which does include a pairing force on the fact that in this reference a method 5) is used which only includes the part of the pairing force which causes vibrational excitations, not the part which contributes to the self-consistent field. In our calculation the increase of deformation, which causes some N = 2 and N = 3 orbits to come closer to each other, eventually leads to a phase transition into a superfiuid phase with gain of binding energy, thereby creating an isomeric state at an equilibrium deformation which is larger the smaller a pairing force we apply. In addition to a comparison of electromagnetic transitions we shall use the 180(p, t) reaction as a sensitive test of the properties of the calculated isomeric state. 565

566

B. SORENSEN

2. Variational calculation of equilibrium shapes The first step in our calculation consists in calculating, as a function of the quadrupole detormation parameter f12, the single particle bound state spectrum of x60 and 180 in a deformed Woods-Saxon potential 7), the geometry of which was determined separately for the two nuclei by a smooth formula 8) adjusted to electron scattering data. From the bound state energies ev separate proton (3 = 1) and neutron (3 = - 1 ) BCS ground states were constructed under influence of a pairing force Gr = - 3 5 A - l ( l + O . 7 5 r ( N - Z ) A - 1 ) . In the absence of superfluidity (A~ = 0) the ground states reduce to the ordinary shell model states. The expectation value of the total Hamiltonian, including a quadrupole residual interaction of strength Zz, in the BCS vacuum is now given, as a function of f12, by the formula 9) Eo(/~2) =

Z 3=--1,1

{2 Z ~:~,-(A,IG,)}+~JZz2. z z z

(1)

v

Although generally successful in predicting equilibrium deformations 9), this expression is poor for estimating absolute deformation energies. An alternative method 1o), which only calculates energy deviations from a smooth energy that is subsequently replaced by a liquid drop estimate, has proven superior in calculating deformation (and fission) energies in heavy nuclei. However, the small number of bound configurations in oxygen make it doubtful whether the liquid drop energy can be simply related to a smoothened version of the actual level distribution or not, and one is afraid that the difference between the shell correction calculated from the actual levels and the liquid drop energy may have significant deformation dependence. We therefore use the simple expression (1), the deformation dependence of which is most reliable for a light nucleus. The total energy Eo(~2 = 0) obtained for the Woods-Saxon well 8) in this way is still off by about a factor two, and the single particle levels of this potential are not in full agreement with the spectra of adjacent odd nuclei. In particular the particle-hole gap seems to be underestimated. However, a direct comparison between the 160 Woods-Saxon results and experiment is not justified in view of the likely deformation of the odd neighbouring nuclei. The 160 and 180 energies are shown in fig. 1 for various choices of Z2. The selfconsistent value of Zz is about 0.008 MeV -1 [ref. 1~)], implying a renormalized strength between 0.016 and 0.020 MeV -~. In all cases the equilibrium deformation of 180 is fi2 = 0.05 and ~60 exhibits two minima, one at flz = 0 and one at ~2 = 0.25. The depth of the second valley is between one and two MeV, and the difference between the ground state expectation values in the second and first minimum is around 5 MeV as compared to the experimental value 6.05 MeV. One should remember that the theoretical energy difference will be somewhat increased by the non-orthogonality of the two states. Since (1) is not a semi-classical potential energy, there is no question of adding energy of zero-point motion. On the contrary the f12 = 0 state will receive a small additional energy depression due to the residual

6.05 MeV 0 + STATE IN t60

567

TABLE 1 Optical parameters used in both coupled channels and DWBA calculations (MeV and fm)

p t

V

W

14'D

Vs.o.

--38.9

--4.14

--3.94

--6.2

0

0

--157

--20

r

r'

rs.o.

rc

a

a"

1 . 1 4 2 1 . 2 6 8 1.114 1 . 1 2 2 0 . 7 2 6 0.676 1.16

1.5

1.122

0.75

as o.

0.585

0.82

Proton parameters are from ref. 19), triton parameters from ref. 17) except for 14".

interaction, whereas the f12 ~--- 0.25 state will receive energy contributions of both signs due to a residual interaction which includes the effect of working in a nonorthogonal basis. The f12 of the isomeric state decreases if the pairing strength is increased from the present value of Go = 35 MeV and increases if Go is decreased.

-230 MeV

0

Eo(132)

X 2 ~

16 ~

- 240

=0.003 0.007 0.010 0,020 O.OL0

-245

0.080 -250 -255

-26q

L 180

-28'

~2=0.003

-290

~

-295 -300

0

0.007

0.010 0.020 O.OLO

.I I t I I i I I I I 0.1 0.2 0,3 O.& 0.5

Fig. 1. Energy of the lowest state in 160 and 1sO as function of the deformation parameter f12 multiplying the 1"20term in a multipole expansion of the nuclear surface is shown for selected values of the quadrupole interaction strength Za (1).

568

B. SORENSEN

The value 35 MeV is obtained by extrapolation of the pairing strengths needed to fit pairing phenomena in heavier nuclei 13). The reason for the new decrease in Eo at f12 = 0.20 is that the ½+[220] and ½-[101] configurations have approached each other so much that the pairing force is now capable of making both the protons and the neutrons superfluid (Ap = 3.64 MeV, An = 3.37 MeV). It would appear that our f12 = 0.25 BCS vacuum is a very different state from the 4p-4h configuration previously 2, 3) used to describe the 6.05 MeV K = 0 band. However, the models are not exclusive since the expansion of a BCS vacuum (zero quasiparticles) on real particle and real hole configurations will contain substantial 2p-2h and 4p-4h components.

3. Electromagnetic transitions Calculating the B(E2) value for the transition from the f12 = 0.25 isomeric 0 + state to the 2 + member of its rotational band 13) gives 170 e 2 • fm 4 as compared to the experimental value 200_+ 75 e 2. fm 4 [ref. ~4)]. The neglect of continuum states makes it necessary to use effective charges. In the rare earth region a polarization charge of 1.2 e fitted the intra-band transitions when an 8 MeV wide configuration span was used 13). In the present calculation all bound configurations are included, but of these only three lie outside an 8 MeV interval around the Fermi surface. These three are the ½+ [000], ½-[110] and 32-[101] configurations, of which the first two do not contribute to the intra-band E2 transitions. We therefore used a polarization charge 1.0 e, not much smaller than in the rare earth calculation. Using the overlap between the/~2 = 0 and the/~2 = 0.25 state estimated below, the inter-band transition B(E2)( 0+, /~2 = 0 ~ 2 +, /~2 = 0.25) = 18 e 2 • fro*. The experimental value is 22 e z • f m 4.

4. Non-orthogonality The bases for evaluating the BCS vacua described above have sharply defined deformations f12 = 0 and/~2 = 0.25. The energy surfaces shown in fig. 1 show that the actual shape fluctuations must be substantial. As mentioned above, this will lead to a non-zero overlap between the calculated 0 + states, which again will imply intra-band electromagnetic transitions and modified reaction cross section. We have estimated the overlap between the 1 6 0 f12 = 0 and f12 = 0.25 states by solving the Bohr Hamiltonian 22) using the energy surface of fig. 1 as potential energy (expanded in a power series in an L = 2 harmonic oscillator basis of up to 25 quanta). For simplicity the kinetic energy was kept quadratic. The resulting overlap is 0.3. Several spherical phonons are important for the f12 = 0.25 state. The largest single component has three phonons and N > 5 phonons are indispensable. The effect of the nonorthogonality on the excitation energy of the isomeric state is to raise its energy by about 250 keV.

6.05 MeV 0 ÷ S T A T E I N 160

569

5. Two-neutron transfer reaction The 6.05 MeV 0 + state and its rotational 2 + state at 6.92 MeV have been observed in the 180(p, t) reaction at Ep = 50 MeV [ref. is)]. F r o m our BCS intrinsic wave functions for the flz = 0.05 K = 0 g r o u n d b a n d in 180 and the f12 = 0 0 + state and f12 = 0.25 K = 0 b a n d in 16 0 we construct f o r m factors for the (p, t) reaction with a given L-transfer 16). Since at least one of the bands is substantially deformed, TABLE 2 The GnL coeltlcients for the t 8 0 ( p , t) re a c t i on

Present calculation a)

D ~ n a u et al. 2t)

N

o~-+0:

2:~o~

0~0L

0;-~2~

0~0:

0 1 2 3 4 5 6 7 8 9 10

0.0041 0.0462 0.4372 --0.1655 O. i 060 -- 0.0573 0.0268 --0.0121 0.0053 -- 0.0017 0.0003

--0.0042 --0.0063 0.0010 --0.0012 0.0009 -- 0.0003 0.0001 0 0 0 0

0.0584 --0.3010 0.2967 --0.0999 0.0495 -- 0.0240 0.0099 --0.0042 0.0017 -- 0.0005 0.0001

--0.0650 --0.2102 0.0782 --0.0314 0.0167 -- 0.0071 0.0031 --0.0013 0.0004 -- 0.000 l 0

0.0289 --0.1784 0.4019 0 0 0 0 0 0 0 0

0:~0L 0.0319 --0.1685 --0.1027 0 0 0 0 0 0 0 0

a) Before m o d i f i c a t i o n s due to n o n - o r t h o g o n a l i t y .

it is necessary to calculate the reaction cross section by a coupled channels m e t h o d that includes inelastic transitions within the rotational b a n d 17). Calculated and experimental results are c o m p a r e d in fig. 2. Since the 6.05 MeV 0 + state is n o t resolved f r o m the lowest 3 - state at 6.14 MeV, we have calculated a spherical 3 wave function using an octupole force that reproduces the k n o w n B(E3) value 630 e 2 . f m 6 [ref. 20)]. This approximately gives the cross section shown in fig. 2; however, there is no ambiguity in adding the calculated 0 + and 3 - cross sections, since the 3 - cross section goes to zero at f o r w a r d angles where the 0 + cross section is maximum. The sum fits the observed angular distribution, so does the calculated cross section, magnitude as well as angular distribution, for the 2 + rotational state based on the f12 = 0.25 0 + state. The figure also shows a D W B A calculation using the same f o r m factors. Here the relative magnitude o f the 2 + to 0 + deformed states disagree with experiment, indicating the importance of the indirect routes involving inelastic scattering 0 + --* 2 + or 2 + ~ 0 + in the final nucleus. To the left in fig. 2 the g r o u n d state cross section is shown together with a 2 + state at 9.85 MeV, believed to be spherical. The f o r m factor for this 2 + state was calculated

570

B. SORENSEN 10

do" dw

I

1

I

I

r

I

_~0

(mblsr)

~

-

+

~k\ ,

'J

+,6.05 MeV and 3. 6.14 MeV

O+OMeV

.

l

0, DEF, CC ptus 3; SPH, DW

~

_ 3SSPH,

~I"-""~

~

//

cc,F-- o.7~k O+DEF, DW, F = 1

10-1

/

o+, OEF, / CC,F = 1

2+9.85 MeV --

10-1

1

..-•,•,

2+,DEF,

\/ow,

2";SPH,DW F arbitrary 10-2

~

•

\] ~ N ~

-~

2

130 ( p, t ) Ep = 50MeV I

I 20

I

OCM

F-l

2 + 6.92 MeV I 40

I

I 20

I

I 40

I

®CM.

Fig. 2. Differential cross sections for the 180(p, t) reaction leading to some spherical (SPI-[) and. deformed (DEF) states in 160. A coupled channels (CC) or DWBA (DW) calculation has been performed using the optical parameters given in table 1. The absolute cross sectiort scale is approximate and has been established by comparison with a 43.5 MeV (p, t) experiment 18). However, the relative magnitudes of 50 MeV experimental cross sections 15) are correct. The normalization factor of the calculated cross sections relative to the deformed 0 + state is indicated as F.

assuming a pure (p½p~)x=2(d{t)z= -1 2 o configuration, the main purpose being to show the slightly different angular distributions o f a rotational and a spherical 2 + state. The calculated ratio o f cross sections for the f12 = 0.25 0 + and the f12 = 0 0 + would be a factor of 2.1 too big if the approximate treatment 16) of the difference in target and final state deformation were used. Correcting for the non-zero overlap calculated in sect. 4, we obtain a g r o u n d state cross section which is 0.7 times the experimental one, normalized at the f12 = 0.25 state. It is interesting to c o m p a r e this ratio with the one obtained when the 6.05 M e V 0 ÷ state is described as a combination o f 0p-0h, 2p-2h and 4p-4h configurations, at the same time as the 1 s o g r o u n d state is a mixture o f 2p and 4p-2h configurations. The (p, t) L = 0 form factors corresponding to the

6.05 MeV 0 + STATE IN 160

571

admixtures of refs. 2, 3) have been calculated earlier 2t), and we obtain in DWBA a ratio a(O+, 6.05)/a(0 +, O) that is 3.1 times smaller than the experimental one (at forward angles where the 3- is negligible). The DWBA angular distributions are similar to the ones obtained with our form factor (using the same optical parameters) although not identical. The reason for the difference is clear from table 2, where we show the expansion coefficients, G~L, of the form factors on harmonic oscillator states un~.(vR2) with v = 0.3854 fro-2. The Woods-Saxon form factors of the present calculation contain components of quite high N-values in contrast to the harmonic oscillator form factors of ref. 21). It is also conceptually clear why the description of the deformed 0 + state in terms of mainly 4p-4h configurations leads to a low value of the two-nucleon transfer strength, at least when the 4p-2h component in the 180 ground state is small. 6. Conclusions

We have shown that a consistent treatment of the pairing plus quadrupole interaction leads to the possible existence of a shape isomeric state in 160, which may be associated with the observed 6.05 MeV level. The estimate of the deformed equilibrium deformation and the excitation energy of the excited state can only be regarded as qualitative, in the first case because of the uncertainty in estimating the strength of the pairing force relative to the spacing of single particle energies [that are underestimated by the potential s)], in the second case because of the inadequacy of the expression (1) as well as of the Strutinsky prescription for calculating energy differences in light nuclei. If an accurate potential energy surface could be calculated, the phonon method could be used to obtain the excitation spectrum. The present calculation can therefore only be regarded as representative for a possibly existing shape isomeric state. It is believed, however, that when the force is fixed, then the expression (1) correctly determines the equilibrium deformation 9, 1o). Since the shape change between the ground state and the isomer is found to be rather small relative to the isomeric shape changes in heavy, fissionable nuclei, for which the energy expression (1) has been shown to fail xo), then one would not expect the calculated excitation energy, which is 5.25 MeV including the overlap correction, to be strongly modified by replacing (1) by a better prescription, still assuming that the forces are appropriate. Estimated intra-band B(E2) value and two-nucleon transfer strength are in better overall agreement with experiment than the corresponding quantities calculated in a previously proposed model that mixes the spherical ground state with deformed np-nh excitations. We have calculated the energy surfaces for different parameters of the Woods-Saxon potential, some of which led to more consistent single particle and pair binding energies. However, the second minimum in some cases is pushed to much larger deformations, usually with both ~2 and B4 non-zero, or the minimum is replaced by

572

B. S¢~RENSEN

a n a n o m a l y in the energy function. We therefore emphasize that the present calculat i o n only points to the possible i m p o r t a n c e of superfluidity in light nuclei b u t does n o t claim to have given a p r o o f of such. W e have performed a similar calculation for 4°Ca, where we find two anomalies in the energy surface, at f12 = 0.3 a n d 0.5. This would indicate a more complex type o f coexistence in this nucleus. The a u t h o r wishes to t h a n k Dr. G n e u s s for c o m m u n i c a t i n g his p h o n o n code.

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