Electron scattering off 110Pd, a test of the interacting boson approximation

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28 March 1985

E L E C T R O N S C A T T E R I N G O F F ll°Pd, A T E S T OF T H E I N T E R A C T I N G B O S O N A P P R O X I M A T I O N J.B. VAN D E R LAAN, A.J.C. B U R G H A R D T , C.W. D E J A G E R and H. DE VRIES N I K H E F - K, P.O. Box 4395, 1009 AJ Amsterdam, The Netherlands" Received 15 October 1984

An electron-scattering experiment on ll°pd has been performed. Results have been compared with predictions of the interacting boson model. Quadrupole boson structure functions have been extracted from form-factor data of the first two 2 + transitions. From these a satisfactory prediction for a third 2 + excitation could be obtained.

The interacting boson approximation (IBA) has shown to yield a good description of static properties of heavy and medium-mass nuclei [1]. The model has been very successful in explaining level schemes and gamma-ray branching ratios over a broad mass region. High-resolution electron-scattering experiments now make it possible to determine not only the transition probabilities but also the spatial distribution of the densities in a model-independent way. This allows further tests of the model [2], and also a comparison with calculations on a microscopic basis, when those become available. Several earlier electron-scattering experiments have been interpreted in term of the IBA. An experiment on Sm isotopes [3] has shown that the transition charge density for the first 2 + states could be described by one set of boson quadrupole structure functions for the whole series of isotopes. Transition densities for 154Gd [4] were found to be in qualitative agreement with predictions based on those structure functions. The form-factor behaviour of the L = 2 excitations, however, could not be described properly. It was not even possible to describe the formfactor of the third 2 + excitation as an arbitrary linear combination of that of two other 2 + excitations. IBA plus configuration mixing was applied to the Ge isotopes [5]. Two different configuration spaces were allowed in the generation of a set of structure functions. In this way it appeared possible to describe satisfactorily all measured L = 2 transitions. In this let130

ter an electron-scattering experiment on 110pd is described together with the first results of a test of the IBA description. Properties of collective states are explained by the IBA-model in terms of nucleon pairs outside major closed shells, coupled to L = 0 (s) and L = 2 (d) bosons. The restriction to only two kinds of bosons limits the validity of the model to low-lying excitations (E x ~< 3 MeV) but on the other hand keeps the model relatively simple. The hamiltonian is given in a general form by [6]

H=e(ndTr+ndu)+kQ~r. Qv+ V r+ Vuu+ V w.

(1)

The first term takes into account the strong pairing interaction, assumed to be equal for protons and neutrons, in which n d are boson number operators. The second term represents the quadrupole-quadrupole interaction and the last three terms the boson-boson interactions. The parameters in the hamiltonian are assumed to vary only slowly between the shell closures. The boson E2 operator is given by [7] T(E2)=

~

e a ([dt~+st'fflK(2)+X~[ dt~'](2))-K

=TI~U

(2) In the simplest version (IBA-1) of the model no distinction is made between protons and neutrons. In this generalization the density of every L = 2 excitation is a linear combination of two-boson structure functions, which contain the information of the spa0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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tial distribution. This implies that these structure functions can be extracted from two measured densities, which then in turn can be used to predict every L = 2 transition density for which the IBA matrix elements are known. Van Isacker and Puddu have performed an IBA-2 calculation for the 44Ru and 46Pd isotopes [8]. Their calculated excitation energies and branching ratios are in many cases in good agreement with experimental values. The parameters appearing in the hamiltonian were obtained from a fit to excitation energies in both series of isotopes. Coupling constants of e,rc~zr = ev% = 0.106 eb were adapted to the reduced transition probability (B(E2)) of the first 2 + state in 102pd and then kept constant for all Pd isotopes. Predictions of the B(E2) values for the first 2 + states are close to the experimental data [9]. For the second 2 + state, however, the prediction is of the correct order of magnitude but the observed systematic decrease could not be reproduced. The experiment was performed with the NIKHEF medium-energy accelerator MEA and the high-resolution QDD spectrometer set-up [10]. Incident energies in the range 9 0 - 3 6 0 MeV were used and the scattered electrons were detected at angles between 30 ° and 80 ° to cover a momentum transfer range of 0 . 5 2.5 fm -1 . The l l ° p d target was isotopically enriched to 97% and had a thickness of 19.1 mg/cm 2. Beam currents up to 60/aA were used. The energy resolution varied from 1.5 × 10 - 4 at 90 MeV to 7 X 10 -5 at the higher energies and was always in absolute value better than 30 keV (FWHM). This enabled us to resolve the (2 +, 4 +, 0 +) triplet, where the 4 + and the 0 + are separated by only 26 keV. At the large scattering angles, data were stored on an event-by-event basis. Off-line software corrections were applied to account for kinematic broadening and spectrometer aberrations. Fluctuations in fine-channel widths were averaged out by combining 16 separate runs, where after each run the detector was shifted over one fine-channel distance. At each q-point, normalization runs were performed on a boron-nitride and a carbon target. The BN spectra used to improve the accuracy o f the energy calibration to 0.1% and the C-spectra provided the cross section normalization through a comparison with the well known carbon cross section [11]. The countrate was always kept below one event per 5/as, which implied dead-

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time corrections of at most 3%. The cross sections were determined by fitting the measured spectra with an asymmetric gaussian, folded with a theoretical radiation function. The uncertainty in the cross section was estimated to be 2%, excluding the statistical error. A subset o f spectra, covering the total momentum transfer range and measured under fixed experimental conditions, was used to evaluate in a model-independent way [12] the ground-state charge distribution. Optimum values for the number of F o u r i e r Bessel coefficients and the cut-off radius came out at 12 and 11.0 fro, respectively. This fit served as a consistency check for all other spectra. The inelastic cross section data were also analyzed in a modelindependent way [13]. Again, for each transition, the optimum number o f Fourier-Bessel coefficients and the cut-off radius were determined. The known B(E2) values [9] were used as a constraint in the fit. The resulting transition densities are presented in fig. 1. The first 2 + state exhibits the surface-peaked behaviour, frequently observed for collective excitations. The transition density of the second 2 + peaks slightly more to the interior. The transition density for the ith 2 + state is given by

Pi(r) = ~

eK[AiKo~K(r)+x B&flK(r)] ,

(3)

K = l r ,/)

0.02

"...'

,,""'

2; J

0 01

gill

g.

ooo2 0.002

5 r [fm]

10

Fig. 1. Transition charge densities of the 2~ (0.374 MeV) and 2~ (0.814 MeV) excitations, derived m odel-independently from the form-factor data.

131

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with

At, = <2~ll[dt~ + sf~],ll0+>, B/, = (2+11[df~],ll0+). These relations incorporate four boson structure functions and two coupling constants. Van Isacker and Puddu [8] have shown that the electromagnetic properties of the Pd isotopes could be reproduced quite well under the assumption of equal coupling constants for neutron and proton bosons. Saha et al. [14] extract effective charge values of e~r = 1.13 e and e v = I + 0.49e from their (rr, rr ) data on the 21 states in the Pd isotopes with the use of microscopic calculations. However, their results overestimate the B(E2)-value for the 22+ state in 110pd by a factor of two ,1 A straightforward analysis of the B(E2)-values of the 21+ and 22+ state yields indeed approximately equal values for e~rc~Trand eva v. This is also borne out by microscopic calculations of Sagawa et al. [15 ]. Therefore, we have assumed the proton and neutron coupling constants to be equal. Furthermore, because of the apparent strong polarization of the core by the neutron bosons it seems reasonable to assume the structure functions of proton and neutron bosons to be equal. The a(r) and fl(r) functions, extracted in + this way from the transition densities of the 21 and + + 22 state, are shown in fig. 2. The excitation of the 21 state is dominated by the a(r) term, while both a(r) ,1 In refs. [8] and [14] x is incorrectly defined asc~/fl instead of as file. However, in the numerical calculations the parameter was used correctly. 0.002

°°i

_ .

,"'~i!' ' . . . . 5 r[fm]

2,r,1 .... t ,

10

Fig. 2. Quadrupole boson densities e~(r) and fl(r), deduced from the transition charge densities of the 2~ and 23 state. 132

and fl(r) contribute to the excitation of other 2 + states [7]. The function a(r) has a shape characteristic for a collective surface-peaked transition density, while (3(r) resembles quite closely the derivative of a(r). An indication of such a behaviour was already observed in low-energy proton-scattering data [16]. These structure functions can how be used to predict the transition density of any other 2 + excitation with the aid o f relation (3). However, in this mass region excitations have been observed that cannot be described by any straightforward collective model [17]. This could be explained by the occurrence of two quasi-particle excitations across the Z = 50 shell. Configuration mixing calculations [ 18] have been shown to explain this problem. We considered the experimentally observed third 2 + level at 1.21 MeV to be such an intruder state, and the level at 1.47 MeV as the third 2 +, predicted by the IBA model. Hence, the matrix elements of ref. [8] for the third 2 + have been used to calculate the form factor of the level at 1.47 MeV. The results are presented in fig. 3. The transition is rather weak, so that the data points have a large uncertainty, but it is evident that the prediction is quite reasonable, in contrast to the analysis for 154Gd [4]. Dieperink [19] has suggested that the failure of

102 i 100

",/', 0

28 March 1985

i 1~"4~4"+

/

0o

i

~

, ' 110pd 1

+

,

]

1

i

i

J

2

, qe~f [fm-~l

Fig. 3. Form-factor data for electro-excitation of the 2+ states at 0.374 (2~), 0.814 (2~) and 1.470 (2;) MeV in ll°Pd. The curves show the results of Fourier-Bessel fits for the first two states and the IBA prediction for the 2~ state.

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IBA-1 in 154Gd might be attributed to quasi-particle effects. Since, however, the third 2 + excitation is relatively even weaker in l l 0 p d than in 154Gd, this explanation seems not quite plausible. Effects due to gbosons [20] play a bigger role in deformed nuclei than in vibrator-like nuclei as Pd. Furthermore the sub-shell closure observed at Z = 64 could play a role in 154Gd. Hence, the failure as found by Hersman et al. might possibly be caused b y other effects. Several attempts have been made to calculate the boson structure functions microscopically. Such calculations have been presented by Saha et al. [14] for the radial moments o f the structure functions in the Pd isotopes. The radial behaviour - a(r) and fl(r) has also been calculated for 110pd [21 ]. Core-polarization effects were taken into account in a phenomenological way b y adding a derivative of the groundstate charge distribution. Attempts to improve the calculations by extending the configuration space are in progress. Summarizing, we conclude that inelastic electron scattering allows the possibility to test dynamic properties o f heavy nuclei. The assumption that the effective coupling of proton and neutron bosons to the electromagnetic field is identical, yields a prediction for the form factor of a third 2 + state in agreement with the present data. Further investigation of the even Pd isotopes to test IBA-2 predictions systematically is in progress. The authors thank J.W. Lightbody for the loan o f the llOpd target. They are also grateful to J. Friedrich for extensive discussions on the Fourier-Bessel analysis. This work was performed as part of the research program of the National Institute for Nuclear Physics

28 March 1985

and High-Energy Physics (NIKHEF, section K), made possible by financial support from the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for the Advancement of Pure Scientific Research (ZWO). References [1] A. Arima and F. Iachello, Ann. Rev. Nucl. Part. Sci. 31 (1981) 75. [2] A.E.L. Dieperink, Nucl. Phys. A358 (1981) 189c. [3] M.A. Moinester et al., Nucl. Phys. A383 (1982) 264. [4] F.W. Hersman et al., Phys. Lett. 132B (1983) 47. [5] D, Goutte, Proc. Intern. Workshop on Interacting boson-boson and boson-fermion systems (Gull Lake, 1984). [6] B.R. Barrett, Rev. Mex. Fis. 27 (1981) 533. [7] F. Iachello, Nucl. Phys. A358 (1981) 89c. [8] P. van Isacker and G. Puddu, Nucl. Phys. A348 (1980) 125. [9] P. De Gelder et al., Nucl. Data Sheets 38 (1983) 545. [10] C. de Vries et al., Nucl. Instrum. Methods 223 (1984) 1. [11] W. Reuter et al., Phys. Rev. C26 (1982) 806. [12] B. Dreher et al., Nucl. Phys. A235 (1974) 219. [13] J. Heisenberg and H.P. Blok, Ann. Rev. Nucl. Part. Sci. 33 (1983) 569. [14] A. Saha et al., Phys. Lett. 132B (1983) 51. [15] H. Sagawa, O. Scholten and B.A. Brown, Proc. Intern. Workshop on Interaetiong boson-boson and bosonfermion systems (Gull Lake, 1984). [16] E. Cereda et al., Phys. Rev. C26 (1982) 1941; M. Pignanelli et al., Phys. Rev. C29 (1984) 434. [17] M. Sambataro, Nucl. Phys. A380 (1982) 365. [18] K. Heyde et al., Phys. Rev. C25 (1982) 3160. [19] A.E.L. Dieperink, Nuovo Cimento 76A (1983) 377. [20] K. Heyde et al., Nucl. Phys. A398 (1983) 235; P. van Isacker et al., Nucl. Phys. A380 (1982) 383. [21] O. Scholten, Proc. Intern. Workshop on Interacting boson-boson and boson-fermion systems (Gull Lake, 1984).

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