Spinor symmetry and supersymmetry in the Ru, Rh, Pd and Ag isotopes

Volume 108B, number 1 PHYSICS LETTERS 7 January 1982

SPINOR SYMMETRY AND SUPERSYMMETRY IN THE Ru, Rh, Pd AND Ag ISOTOPES

J. VERVIER and R.V.F. JANSSENS 1 Institut de Physique Corpusculaire, B-1348 Louvain-la-Neuve, Belgium

Received 9 October 1981

Experimental data on excitation energies and B(E2) values in the-even-even Ru and Pd, and odd-A Rh and Ag isotopes suggest the existence of a Spin(3) spinor symmetry and a U(6/2) supersymmetry in these nuclei, wherein the even-even isotopes are described by the SU(5) limit of the interacting boson model and the odd proton of the odd-A isotopes is in a/" = 1/2 orbit.

The unified description of collective states in even- even nuclei by the interacting boson model (IBM) [1 ] and in odd-A nuclei by the interacting boson fermion model (IBFM) [2] has recently led to the introduction, in nuclear physics, of the two concepts of spinor symmetry and supersymmetry [3-5] . The former deals with levels in odd-A nuclei only, and is associated with (ordinary Lie) spinor groups. The latter includes the odd-A nuclei and their even-even cores in a single theoretical framework, and is characterized by (non-Lie) supergroups. Experimental evidence for these symmetries has been found in the mass region between the Os and Hg isotopes [3-7] . The even-even nuclei are then described by the O(6) limit of the IBM [8], and the odd proton of the odd-Z, even-N nuclei is in a single orbit with spin/" = 3/2. The associated spinor groups and supergroups are Spin(6) and U(6/4), respectively.

It is of interest to search for other examples of these symmetries, corresponding to other limits of the IBM and to other spin values for the orbit occupied by the odd fermion. The present paper deals with the SU(5) limit of the IBM [9] and with a ] = 1/2 orbit for the fermion. The analytical expressions for the energies and other observables of the levels for this case have recently been worked out by Iachello [10]. The spinor groups and supergroups are Spin(3) and U(6/2), respectively. We want to show that experimental evidence sug- gests the existence of these symmetries in the even-even

1 Present address: Physics Division, Argonne National Labo- ratory, Argonne, IL 60439, USA.

isotopes of Ru and Pd and odd-A isotopes of Rh and Ag, at least for those levels of the latter nuclei with a negative parity, and with a small degree of symmetry breaking. This statement will be based on the comparison between the available experimental data on excitation energies and B(E2) values (extracted from the Nuclear Data Sheets [11 ], unless otherwise specified) and the prediction of the model. This provides a second region of nuclei, in addition to the Os to Hg isotopes, wherein a dynamical supersymmetry is present in nuclear physics.

Nuclei in the Ru to Cd region have been interpreted, with a considerable success, in the framework of the vibrational (V) model for the even-even isotopes, and of the core-particle (weak) coupling (CPC) model for the odd-A Rh and Ag isotopes, at least for those levels of the latter where the odd proton is in the 2Pl / 2- orbit [12]. However, these models have met with problems, some of which are listed below: (i) the B(E2) values for the decay of the 0~, 2~, 4~ members'of the two- phonon triplet to the one-phonon 2~ singlet in the even-even nuclei are lower than the predictions of the V model which are: B(E2; 0~, + + 22,41 ~ 2~) = 2B (E2; 2~ ~ 0~); (ii) the levels of the odd-A nuclei are located at a much lower excitation energy than their associated states in the even-even cores. The CPC model, in its simplest version and with no monopole core-particle interaction, predicts that the center of gravity of the former levels should be close to the energy of the latter states [12] ; (iii) the B(E2) values for the transitions between the first 3/2 i - and 5/21- levels

0 031-9163/82/0000-0000/$ 02.75 © 1982 North-Holland 1


and the 1/2 i- ground state in the odd-A nuclei, which + +

should be equal to the B(E2) value for the 2 1 - 0 1 transition in the associated cores according to the CPC model [12], are systematically lower if the Ru and Pd isotopes are considered as the cores of the Rh and Ag isotopes, respectively. They are closer if the cores are the Pd and Cd isotopes, respectively, but then the energy discrepancy mentioned in (ii) is larger than for Ru and Pd cores; (iv) the B(E2) values for the decay of the two-phonon levels to the one-phonon states in the odd-A nuclei are lower than the predictions of the CPC model Examples illustrating problems (i)-(iv) can be found in fig. 1 and table 2 below.

The applicability of the IBM to the Ru and Pd isotopes has recently been discussed [ 13] in the framework of the so-caUed IBA-2, i.e. where a distinction is made between proton and neutron bosons [1] ; good overall agreement with experiment has been achieved, in particular for the B(E2) values, thus providing a possible answer to problem (i) above. In the present paper, we apply to the Ru and Pd isotopes a simpler version of the IBM, i.e. with no distinction between proton and neutron bosons (the so-caUed IBA-1 [1]), and one of its three limits, i.e. SU(5) [9]; this is an aDnroximation, which has already been applied to 102Ru [9], and which is justified below.

The eigenvalues of the boson-fermion hamiltonian,

for the case considered in the present paper and outlined above, are given by [10]

E(N, N,M, nd, o, L, J ) = E~ + e lndM + e2n d

+ ~omd(n d -- 1) +/3(n d -- o)(n d + o + 3)

+ ~ I L ( L + 1) + "[2J(J + 1). (1)

In this expression, N is the number of bosons, M, the number o f fermions (M = 0, 1 ,2 fo r ] = 1/2), and N = N + M. In even-even nuclei, when M = 0, N is determined from the nearest magic numbers as in the IBM [ 1 ] ; in odd-A nuclei, where M = 1, N is fixed by the condition that N is conserved in a nuclear supermultiplet according to the U(6/2) supersymmetry [ 10]. The quantum numbers nd, o and L have the same meaning and can be determined by the same rules as in the SU(5) limit o f the IBM [9]. The spin J = L for even-even nuclei, and J = L -+ 1/2 (for L 4= 0) or 1/2 (for L = 0) for odd-A nuclei. E~ only depends on N, Nand M. The quan- tities el, e2, a, ~, ~'1 and 72 are constant parameters in one odd-A nucleus for the Spin(3) spinor symmetry, and in one nuclear supermultiplet for the U(6/2) supersymmetry. Analytical expressions have also been obtained for observables other than the energy [10]. They are similar to previous IBFM results [14].

We have fitted the experimental excitation energies of various levels in the even-even isotopes of Ru and

Table 1 Description and results of various fits to the experimental excitation energies of levels in the Ru, Rh, Pd and Ag isotopes with respect to their ground states (for even-even nuclei) and lowest 1/2- levels (for odd-A nuclei), extracted from the Nuclear Data Sheets [ 11], with the theoretical expression (1). The indices to the levets refer to their order of appearance in the experimental spectrum, the ground state of even-even nuclei being denoted 0~, and the lowest negative-parity level of odd-A nuclei, 1/2 i.

Fit Nuclei Levels Parameters Range of 0 (%) label included fitted involved and a (keV)

1 1°°d°2 Ru 2~, 2~, 4~ e 2, 6, 71 + "r2 3.8-5.1% 1°4,1°6pd 0~, 3~, 4~, 6~ 96-133 keV

2 1°1,1°3,1°5Rh 3/2~, 5/2 i e 1 + e2, a 1.3-2.3% 105'107'109'111Ag 3/2~, 5/22, 7/2~, 9/2~ ~1, "r2 17-38 keV

3 l°°'l°2'l°4Ru 2~, 2~, 4~ el, e2, ~ 1.3-3.2% l°l'l°3'l°SRh 3/2 i, 5/21 ~1, "r2 19-52 keV 104'106'108'110pd 3/22, 5/2~, 7/2~, 9/2~ 105,107,109,111 Ag

4 100'102Ru 2~, 2~, 4~ el, e2, a 4.0-7.6% l°l'l°3Rh 0~, 3~, 4~, 6~ 71, ~2 82-126 keV l°4,1°6pd 3/2~, 5/21 l°S,l°9Ag 3/2~, 5/2~, 7/2~, 9/21


Pd and odd-A isotopes of Rh and Ag with expression

(1) using, as parameters, e l , e2, t~, ~/1 and ~/2 or linear combinations of them, each nucleus or nuclear super- mult iplet being described by a different set of parameters. For reasons which will be outlined later, only levels with assigned quantum numbers n d = o have been included in the fits, so that the parameter/~ is irrelevant. The quality of the fits is expressed as in refs. [4,5] by the quanti ty ¢ = ~'i I E th - Eexpl/~i Eexp in percent,

• _ ~ E'th Eexp 2 n k 1/2 o r by the quanti ty o - [ i ( i - i ) /( - - )] in keV, where n is the number of levels fi t ted and k, the number of parameters involved. Various calcula- tions have been carried out, whose description and results are synthesized in table 1. For the present paper, and in the language of the CPC model, the odd fermion of the odd-A Rh and Ag isotopes is coupled to Ru and Pd cores, respectively.

A general remark about the results of table 1 is the very good representation of the experimental data achieved by the model. The average relative deviation

varies between 1.3% and 7.6%; this compares favour- ably with ¢ ~ 13 -19% for the U(6/4) supersymmetry in the Os to Hg region [4,5]. Fi t 1 is a test of the SU(5)

description of the relevant even-even Ru and Pd isotopes in the framework of the IBM, which, for this case, is identical to the U(6/2) supersymmetry; its result shows that this approximation is sufficiently accu- rate for the purposes of the present paper, at least for what concerns the excitation energies of the levels included, i.e. those with n d ~< 3. Fi t 2 deals with +.he Spin(3) spinor symmetry applied to the states with n d = o ~< 2 in the odd-A nuclei only (experimental in- formation is too scarce on the n d = o = 3 levels); the agreement is seen to be very good, with ¢ ~< 2.3%. Fits 3 and 4 are testing the U(6/2) supersymmetry for the description of the excitat ion energies of the e v e n - even and odd-A nuclei of the same nuclear supermuiti- plet together with the same set of parameters• They in- clude corresponding levels in the odd-A nuclei and their cores with n d = o ~< 2 (fit 3), and some additional core states with n d = o = 3 (fit 4). The qualities of fits 4 and 3 are comparable to those of fits 1 and 2, respectively, both for ¢ and o; this shows that the description of the nuclei included in the fits by the U(6/2) supersymmetry is as good as the one of the even-even nuclei by the SU(5) (ordinary) symmetry (fits 4 and 1) and of the odd-

E e l

(keY)

1500

1000

500

lO2Ru lO3Rh (rid.V)

+ t. 1

=; o ;

=::--'- ~ 12.21

~t - - I1,11

[ 1 1 2 . 2 1

iot.pd (n,,.v) 2+ '

), _ _ } 1 1 . 1 )

0 - - I0.01 - - 10.0)

a;

o: i'; o; exp. co~. exp. co~• exp.

Fig. 1. Comparison between the experimental and calculated (fit 3 of table 1) spectra of l°2Ru and l°3Rh. The indices to the levels are as in table 1. The experimental data are taken from the Nuclear Data Sheets [11 ], with the exception of the 3/22 level in l°3Rh at 803.1 keV, to which we assign a spin 3/2, a proposal which is not incompatible with the experimental data, but not imposed by them. The calculated spectra have been obtained from eq. (1), with el = - 121, e 2 = 448, a = 186, ~1 = 8, "r2 = 10, all in keV; the quantum numbers n d and o are indicated near the calculated spectra. The corresponding experimental and calculated levels are con- nected by dashed lines. The experimental spectrum of l°4pd [ 11 ] is displayed for illustration purposes.

Table 2 Comparison between the experimental [ 1 1 , 1 5 - 1 9 ] and calculated reduced E2 transition probabilities B(E2) (in 10-48 e2 cm4) between levels in various Ru, Rh, Pd, Ag and Cd isotopes. The indices to the levels are as in table 1. The three models considered are: the U(6]2) supersymmetry [ 10] ; the harmonic vibrational and core-par t ic le coupling [ 12] models applied to the e v e n - e v e n and odd-A nuclei, respectively (V + CPC); the CPC model applied to the odd-A nuclei, the B(E2) values in the even -even cores being taken f rom exper iment (CPC). For V + CPC and CPC, the cores of l °3Rh, l°7,1°9Ag are l °2Ru, l°6,1°8pd, respectively. The calculated B(E2) values are normalized to those given in parentheses; in particular, for U(6/2) and V + CPC, the normalizing transit ion is ÷ + two 2 1 - 0 1 . When values are given for one transit ion, they correspond to ambiguities in the exper imental data [ 16, 18]. The B(E2) values in l°4pd, 108,1 lOCd [11 ] are given for illustration purposes.

Nucleus Transi t ion Exper iment Ref. Calculation.

U(6/2) V + CPC CPC

102Ru + + 2 1 - 0 1 0.130 ,+ 0.007 + +

N = 7 2 2 - 0 1 0.0042 ,+ 0.0004 + +

M = 0 2 2 - 2 1 0.117 -+ 0.015 4- + = 7 4 1 - 2 1 0.212 ,+ 0.023

t °3Rh 3 / 2 ~ - 1 [ 2 i 0.109 ,+ 0.008 N = 6 5 / 2 i - 1 / 2 ~ 0.118 ,+ 0.007 M = 1 3 / 2 2 - 1 / 2 i ~<0.011

= 7 5 / 2 2 - 1 / 2 i 0 .0044 -+ 0.0003 5 / 2 2 - 3 / 2 ~ 0.0768 ,+ 0.0097

0.0073 ,+ 0.0013 5 / 2 ~ - 5 / 2 ~ 0.225 ,+ 0.037

0.015 ,+ 0.004 7 / 2 i - 3 / 2 i 0.130 ,+ 0.020 9 / 2 1 - 5 / 2 i 0.181 ,+ 0.015

104pd + + 2 1 - 0 1 0.108 ,+ 0.006

lO6pd + + 2 1 - 0 1 0.142 ,+ 0.008 + +

N = 7 2 2 - 0 1 0.0036 ,+ 0.0003 + +

M = 0 2 2 - 2 1 0.136 ,+ 0.018 + +

= 7 4 1 - 2 1 0.220 ,+ 0.030

l°TAg 3 / 2 ~ - 1 / 2 1 0.101 ,+ 0.009 N = 6 5 / 2 1 - 1 / 2 i 0.096 ,+ 0.008 M = 1 3 / 2 ~ - 1 / 2 i 0 .0014 ,+ 0.0003

= 7 3 / 2 ~ - 3 / 2 1 <0 .032 5 / 2 ~ - 1 / 2 ~ 0.0068 ,+ 0.0007 5 / 2 ~ - 3 / 2 ~ 0.013 ,+ 0.003 5 / 2 2 - 5 / 2 ~ 0.029 ,+ 0.008

lOaCd + + 21--01 0.088 ,+ 0.004

108pd + + 2 1 - 0 1 0.152 ,+ 0.010 + +

N = 8 2 2 - 0 1 0.0034 ,+ 0.0003 ÷ +

M = 0 2 2 - 21 0.270 + 0.060 = 8 0.095 + 0.026

+ ÷ 41--21 0.280 ,+ 0.040

1°nAg 3 / 2 1 - 1 / 2 ~ 0.111 +- 0.009 N = 7 5 / 2 ~ - 1 / 2 i 0.107 -+ 0.009 M = 1 3 / 2 ~ - 1 / 2 ~ 0.00043 +- 0 .00010

= 8 5 / 2 2 - 1 / 2 i 0 .0058 -+ 0.0006 5 / 2 ~ - 3 / 2 i 0.018 +- 0.006

5 / 2 2 - 5 / 2 i 0.022 ,+ 0.011

[15] (0.130) (0.130) (0.130) [15] 0 0 (0.0042) [15] 0.223 ,+ 0.012 0.260 ,+ 0.014 (0.117) [15] 0.223 ,+ 0.012 0.260 ,+ 0.014 (0.212)

11] 0.111 ,+ 0.006 0.130 ,+ 0.007 0.130 -+ 0.007 11] 0.111 ,+ 0.006 0.130 ,+ 0.007 0.130 ,+ 0.007 16] 0 0 0 .0042 ,+ 0.0004 16] 0 0 0.0042 ,+ 0.0004 16] 0.037 ,+ 0.002 0.052 +- 0.003 0.023 ,+ 0.003 16] 16] 0.149 + 0.008 0.208 ,+ 0.011 0.094 + 0.012 16 17] 0.167 ,+ 0.009 0.234 ,+ 0.013 0.191 ,+ 0.021

[16] 0.186 -+ 0.010 0.260 ,+ 0.014 0.212 ,+ 0.023

[14] - - -

[18] (0.142) (0.142) (0.142) [18] 0 0 (0.0036) [18] 0.243 +- 0.014 0.284 -+ 0.016 (0.136) [18] 0.243 +- 0.014 0.284 ,+ 0.016 (0.220)

[19] 0.122 ,+ 0.007 0.142 -+ 0.008 0.142 -+ 0.008 [19] 0.122 ,+ 0.007 0.142 ,+ 0.008 0.142 +- 0.008 [19] 0 0 0.0036 -+ 0.0003 [19] 0.101 ,+ 0.009 0 .141 ' - + 0.013 0.095 ,+ 0.013 [19] 0 0 0.0036 ,+ 0.0003 [19] 0.041 ,+ 0.002 0.057 ,+ 0.003 0.027 ,+ 0.004 [19] 0.162 ,+ 0.009 0.227 ,+ 0.013 0.109 ,+ 0.014

[ 1 1 ] - - -

[ 1 8 ] (0.152) (0.152) (0.152) [18] 0 0 (0.0034) [18] 0.266 ,+ 0.018 0.304 -+ 0.020 (0.270) [181 (0.095) [18] 0.266 -+ 0.018 0.304 ,+ 0.020 (0.280)

[19] 0.133 +- 0.009 0.152 +- 0.010 0.152 -+ 0.010 [19] 0 .133-+0.009 0 .152 ,+0 .010 0.152 -+0.010 [19] 0 0 0.0034 +- 0.0003 [19] 0 0 0.0034 ,+ 0.0003 [19] 0.046 +- 0.003 0.061 ,+ 0.004 0.054 -+ 0.012

0.019 -+ 0.001 [19] 0.182 ,+ 0.012 0.243 ,+ 0.016 0.216 ,+ 0.048

0.076 ,+ 0.021

llOCd + + 2 1 - 0 1 0.086 ,+ 0.001 [11] -


A nuclei by the Spin(3) spinor symmetry (fits 3 and 2). As an illustration of the agreement obtained, the ex-

perimental and calculated (for fit 3) spectra of l°2Ru and 103Rh are compared in fig. 1. It is apparent that the order and relative spacings of the levels are well reproduced by this calculation wherein 9 states are fitted with 5 parameters, resulting in ~ = 1.5% and o = 19 keV. In particular, the lowering of the levels in 103 Rh with respect to the corresponding core states is nicely explained by the term elndMin eq. (1) which is negative: this solves problem (ii) of the CPC model mentioned above. This interaction term originates from linear Ca- simir operators of the groups U(5) and U(2) describing the bosons and the fermion, respectively [ 10]. The split- tings of the various J = L + 1/2 doublets in 103Rh are also well reproduced by the model. These remarks are typical of most fits obtained.

The situation with respect to the B(E2) values is displayed in table 2 [11,15-19]. Several remarks can be made on this table. It first illustrates problems (i), (iii) and (iv) above. It also shows that the SU(5) limit of the IBM as applied to the even-even Ru and Pd isotopes is able to solve problem (i), at least for the 4~-2~ transitions in 102Ru and l°6,108pd and, perhaps, 23 -2 ~ in l°Spd. The same remark applies to some other E2 transitions known experimentally in the even-even Ru and Pd isotopes [ 11,15,18] but not included in table 2, and is a well-known consequence of the finite number of bosons [1 ]. Table 2 clearly displays that the B(E2) values between the 3/2]- and 5/2]- levels with n d = o = 1 and the 1/2]- ground state with n d = u = 0 in the odd-A nuclei are well explained by the Spin(3) and U(6/2) symmetries: not only are they equal to each other within their respective experimental uncertainties - this is a test of the Spin(3) spinor symmetry, and is also predicted by the V + CPC and CPC models described in the caption to table 2 - but also their reduction with respect to the B(E2, 2~-0~) between the n d = o = 1 and n d = o = 0 levels in the corresponding cores is reasonably well predicted by the U(6/2) supersymmetry. This solves problem (iii) and is a consequence of the fact that, in the U(6/2) supersymmetry [10], the number ofbosons in the odd-A nucleiNod d is one unit less than the number of bosons in the corresponding even- even cores Nee, since these nuclei belong to the same nuclear supermultiplet characterized by the same quantum number ~ = N + M.

The known B(E2) values in the decay of n d = o = 2

levels in the even-even and odd-A nuclei are also included in table 2. Those between the 23 level and 0~ ground state, 3/2 7 and 5/2~- levels and 1/2]- ground state are hindered by comparable factors: this can be interpreted as a result of the An d = 0, + 1 selection rule for E2 transitions in the Spin(3) and U(6/2) symmetries, and is also predicted by the V + CPC and CPC models. The E2 transitions between the 7/2]- and 3/2]-, 9/2]- and 5/2]- levels in 103Rh are rather well accounted for by the U(6/2) supersymmetry, better than by the V + CPC and CPC models; this solves problem (iv) for these states, and is a consequence, both of the finite number of bosom (compare U(6/2) and CPC in table 2) and of the fact that Nod d = Nee - 1 (compare U(6/2) and CPC in table 2). The agreement is not so nice for the 5 /2~-3/2]- and, even worse, 5/2~--5/2]- transitions in 107,109Ag; the predictions of the V + CPC and CPC models are also at variance with experiment for these cases. We shall come back later on this breaking of the Spin(3) and U(6/2) symmetries.

The two-neutron transfer reactions (t, p) and (p, t) with an orbital angular momentum transfer L = 0 should only connect the ground states of the even- even nuclei and the 1/2]- levels in the odd-A nuclei, due to the An d = Ao = AJ = 0 selection rule in the Spin(3) and U(6/2) symmetries [10]. This is verified in the experimental data [11,20-24] where the ground- state to ground-state transitions strongly dominate the spectra. In particular, the predicted 1/2~- levels with o = n d - 2 in the odd-A nuclei, which should correspond to the known 03 states in the even-even cores, are not well localized experimentally [11,20-23]. This is one reason why only those levels with assigned n d = o have been included in the fits of table 1 and fig. 1. Another reason is the presence of "intruder" 0 ÷ states in the spectra of the even-even Ru and Pd isotopes noticed in ref. [13], for instance 03 in 102Ru, which are not described by the IBM and which probably interfere with the predicted 0 ÷, o = n d - 2 levels. Finally, the ratios of the experimental cross sections for the allowed two-nucleon transfers in the 106pd-107Ag and 108pd- 109Ag pairs, (t, p) and (p, t) [21,23,24], agree with the predictions of the U(6/2) supersymmetry [10], which are the same as those of the CPC model [23,24].

Breaking of the Spin(3) and U(6/2) symmetries in the Ru, Rh, Pd and Ag isotopes may arise from several causes: (a) the even-even nuclei are not exactly describ- able by the SU(5) limit of the IBM, and more involved


versions of this model [13], which imply symmetry breaking, are more successful. As pointed out earlier, the corrections so introduced are not very large, at least for the levels and observables considered here and with

+ + the exception o f some 2 2 - 2 1 transitions. In this con- nection, it has been shown previously [9] that symmetry- breaking terms in the framework of IBA-1 affect the

+ + B(E2) value for the 2 2 - 2 1 transition much more than for 4 ~ - 2 ~ , in agreement with experimental data (table 2); (b) the odd-fermion can occupy other orbits than 2Pl / 2-, in particular 2P3/2- and l f5/2- . This is shown by the experimental data from the (3He, d) reaction on e v e n - even targets leading to 3 / 2 - and 5 / 2 - levels in the odd- A nuclei [11,22,25]. These 2P3/2- and l f5/2- admix- tures in the levels analyzed in the present paper amounts for example to about 9 - 1 8 % in 3/2~- and 5/2~- for 107 Ag [22]. They represent the most important symmetry-breaking corrections, and are probably responsible for the discrepancies noted above for the E2 transitions between the 5/2~, 3/2~- and 5/2 i- states (table 2). They could in principle be dealt with by diagonalizing the full IBFM hamiltonian including these orbits [2], with however a large increase of the computat ional complex- i ty and o f the number of parameters; (c) the whole model, IBM and IBFM, probably fails for levels with higher and higher spins, as suggested by the results of recent multiple Coulomb excitation experiments on even-even targets [26]. This question should be inves- tigated experimentally in the odd-A nuclei. The comparison of table 2 shows that this is not yet a serious problem at spin 4 in l°2Ru, 1°6,108pd and spin 9/2 in lO3Rh.

In conclusion, the results o f the present paper show that new symmetries are present, with a reasonably high accuracy (better than 10% for the excitat ion energies), in the Ru, Rh, Pd and Ag isotopes: a spinor symmetry associated with the Lie group Spin(3) for the odd-A isotopes o f Rh and Ag, and a supersymmetry associated with the supergroup U(6/2) linking together the even-even isotopes o f Ru and Pd and the corresponding odd-A isotopes of Rh and Ag. The lat ter point is especially suggested by the excitation energies (table 1 and fig. 1) and the B(E2) values (table 2). The description of these nuclei based on these symmetries successfully solves several problems encountered by the vibrational and core-par t ic le coupling models with the same degree o f sophistication. The presence of symmetry-breaking contributions is apparent in

some of the analysed experimental data. However, as "i t is by no means essential the symmetry be exact" [27], the results of the present paper provide a general framework for the analysis of present and future experimental data in this region of the nuclear chart.

We thank Professor Dr. F. Iachello for communicat- ing to us his results prior to publication and for very useful discussions concerning the present paper.

References

[1] F. IacheUo, ed., Interacting bosons in nuclear physics (Plenum, New York, 1979), and references therein.

[2] F. Iachello, Nucl. Phys. A347 (1980) 51; F. IacheUo, ed., Interacting Bose-Fermi systems in nuclei (Plenum, New York, 1981), and references therein.

[3] F. Iachello, Phys. Rev. Lett. 44 (1980) 772. [4] A.B. Balantekin, I. Bars and F. lacheUo, Phys. Rev. Lett.

47 (1981) 19; Nucl. Phys., to be published. [5 ] F. IacheUo and S. Kuyucak, Ann. Phys. (NY), to be pub-

lished. [6] J. Vervier, Phys. Lett. 100B (1981) 383. [7] M.N. Harakeh et al., Phys. Lett. 97B (1980) 21;

Y. Iwasaki et al., Phys. Rev. C23 (1981) 1477; M. Vergnes et al., Phys. Rev. Lett. 46 (1981) 584; J.A. Cizewski et al., Phys. Rev. Lett. 46 (1981) 1264; J. Vervier et al., Phys. Lett. 105B (1981) 343.

[8] A. Arima and F. IacheUo, Ann. Phys. (NY) 123 (1979) 468.

[9] A. Arima and F. Iachello, Ann. Phys. (NY) 99 (1976) 253. [10] F. IaeheUo, private communication. [11] A = 100: D.C. Kocher, Nucl. Data Sheets 11 (1974) 279;

A = 101: B. Harmatz, Nucl. Data Sheets 28 (1979) 343; A = 102: R.L. Auble et al., Nucl. Data Sheets 19 (1976) 1 ; A = 103: B. Harmatz, Nucl. Data Sheets 28 (1979) 403; A = 109: F.E. Bertrand, Nucl. Data Sheets 23 (1978) 229; A = 110: F.E. Bertrand, Nucl. Data Sheets 22 (1977) 135; A = 111: B. Harmatz, Nucl. Data Sheets 27 (1979) 453.

[12] A. de-Shalit, Phys. Rev. 122 (1961) 1530. [13] P. van Isacker and G. Puddu, Nucl. Phys. A348 (1980) 125. [14] A. Arima and F. IacheUo, Phys. Rev. C14 (1976) 761. [15] S. Landsberger et al., Phys. Rev. C21 (1980) 588. [16] R.O. Sayer, J.K. Tempefley and D. EccleshaU, Nucl. Phys.

A179 (1972) 122. [17] S.H. Sieet al., Chalk River Phys. Div. Progr. Rep. AECL-

4147 (1972) 17. [18] R.L. Robinson et al., Nucl. Phys. A124 (1969) 553. [19] R.L. Robinson et al., Nucl. Phys. A150 (1970) 225. [20] R.M. Del Vecchio et al., Phys. Rev. C12 (1975) 69. [21] R.M. Del Vecchio, I.C. Oelrich and R.A. Naumann, Phys.

Rev. C12 (1975) 845. [22] A.W. Kuhfeld and N.M. Hintz, Nucl. Phys. A247 (1975)

152. [23] R.E. Anderson et al., Phys. Rev. C15 (1977) 123.


[24] K. Krien et al., Phys. Rev. C15 (1977) 1288. [25] R.E. Anderson et al., Nucl. Phys. A242 (1975) 75. [26] J. Idzko et al., Proc. Intern. Conf. Nuclear behaviour at

high angular momentum (Strasbourg, 1980), Contribu- tions p. 115; J. Stachel et al., GSI Sci. Rep. GSI 81-2 (1981) p. 40;

H. Ower et al., GSI Sci. Rep. GSI 81-2 (1981) p. 48; P. Fuchs et al., GSI Sci. Rep. GSI 81-2 (1981) p. 49.

[27] H. Feshbach, in: Interacting bosons in nuclear physics, ed. F. Iachello (Plenum, New York, 1979) p. 175.

Documents

Spinor symmetry and supersymmetry in the Ru, Rh, Pd and Ag isotopes