A model of vibrational even nuclei

I.D.2

]

Nuclear Physics 48 (1963) 500---511; (~) North-Holland Publishing Co., Amsterdam

!

Not to be reproduced by photoprint or microfilm without written permission trom the publisher

A M O D E L OF V I B R A T I O N A L EVEN N U C L E I N. MACDONALD t

Atomic Weapons Research Establishment, Alderrnaston, U.K. Received 7 June 1963 Abstract: The calculations of Raz, in a model of two particles coupled to harmonic surface vibrations, are corrected and extended. Attention is confined mainly to cases in which the second 2+ state lies at more than twice the energy of the first 2 + state, as in m a n y nuclei in which both these states have been studied by C o u l o m b excitation. It is found that the 3-phonon and 4phonon approximations give rather similar results. In the 3- p h o n o n approximation the results are not radically altered by changing the type of two-particle force assumed, nor are they sensitive to the angular m o m e n t u m j o f the single-particle states, so long as j _~ t. For two particles w i t h j = ~ there is a degenerate 2+, 4 + pair of states, and no cross-over transition from the ground state to the 2 ÷ member of this pair. The effect of increasing j is to separate these two states and to allow an appreciable cross-over transition probability, which is rather sensitive to the strength of the two-particle force. T h e model has a certain degree of success in fitting transition probabilities between lowlying states. H o w e v e r the first 4 + state occurs in general below the second 2 + state, in disagreement with experiment. Further the first 0 + excited state lies rather high, while there is experimental evidence for such a state near or even below the second 2 + state.

1. Introduction It has been k n o w n for some time 1) that many even nuclei o f intermediate mass have properties resembling those o f vibrating spherical nuclei. In a simple model of harmonic quadrupole vibrations o f a spherical nucleus 2) the states with successively larger numbers of vibrational quanta (phonons) are evenly spaced, and electric quadrupole (E2) transitions are allowed between states differing by one phonon. In particular the first excited state has spin 2, while at twice the energy of this state there is a degenerate triplet, of spins 0, 2, 4. It is conventional to denote the first and second states of spin 1 by I and I'. In the harmonic model one has strong E2 transitions 0 ---, 2, 2 --, 0', 2 --, 2' and 2 --* 4, while 0 --, 2' is forbidden. In actual even nuclei having two approximately equally spaced 2 + states, the 0 ~ 2 and 2 --, 2' E2 transitions are k n o w n 1, 2) to be enhanced over the single particle estimates, while the transition 0--, 2' is weak. However, deviations from the harmonic model are quite appreciable. The crossover transition can be detected, and the second 2 + state lies at energies up to 2.5 times the energy of the first 2 + state. Various attempts have been made to account for these features by modifying the harmonic model. It is necessary to introduce into the wave functions for each state some admixture of states with different numbers of phonons. This may be done by adding anharmonic terms to the vibrational Hamiltonian 3, 4), or by coupling nucleo~as t Present address: Department of National Philosophy, The University, Glosgow. 500

A MODEL OF VIBRATIONAL EVEN NUCLEI

501

to the vibrational system. The latter procedure was developed in particular by Raz 5), who considered a specific model of two interacting nucleons, in ft shell model states, coupled to the harmonic vibrations. He cut the vibrational spectrum off at three phonons, and diagonalised the resulting energy matrices for spins 0, 2 and 4. He conjectured that the detailed assumptions of the model with regard to the particle states and two-particle interaction would be of secondary importance. This of course is necessary if the model is to be used to describe a range of nuclei and not confined to specific cases in the I ft shell. It is also implicit in this approach that the effect of coupling different numbers of nucleons to the vibrations is assumed to resemble the effect of altering the strength with which a pair of nucleons are coupled to the vibrations in the model. The properties of the 2 + states mentioned above were studied intensively by Coulomb excitation 6). With light particles one could study the direct E2 excitation of these states. While other experimental work gave information on 0 + and 4 + states in the vicinity of the second 2 + state, it was not until heavy ion beams became available that an appreciable number of 4 + states were found in these nuclei, and that their collective character was established 7' a). With heavy ions the probability of Coulomb excitation is sufficiently large for the double E2 excitation process 0 ~ 2 ~ 4 to be observed, when the two successive transitions are enhanced. The new information on 4 + states makes it of interest to examine in more detail refinements of the harmonic model. In the course of investigating the values of the transition probability B(E2; 2 ~ 4) given by the Raz model, an error of sign was found in an energy matrix element for spin 2. This has made it necessary to recalculate the B(E2) values and energy ratios of ref. 5). Since one of our aims in this paper is to indicate in what way this correction alters the results of ref. 5) we shall present our results in most detail for the specific case studied by Raz. In sect. 2 we present the formalism of the Raz model for two nucleons in a pure shell model configuration. These are taken as interacting by the particular force employed by Raz, and are coupled to the surface vibrations. In sect. 3 we compare the experimental data with the results of this model for the (lft) 2 configuration, and with the phonon spectrum cut off at three phonons. In sect. 4 we consider the way in which certain results obtained differ from those in ref. 5). As already mentioned, it is important to examine how far the results of this model are affected by altering some of the assumptions regarding the particle states. In sect. 5 we give results which indicate that the case employed in sects. 3 and 4 is typical. Specifically we find that different choices of two-nucleon interaction make little difference to the results, and that the use of (lft) 2 and (2dt) 2 configurations gives rather similar results. The case of two particles in states of angular momentum j = ½ turns out to be rather a special one, and we discuss the similarity of our results in that case to those of Wilets and Jean 9). Also in sect. 5 we investigate the effect of cutting offthe phonon spectrum at two or four phonons. We find that the results of the three-phonon approximation employed in the rest of the paper are not greatly altered by including

502

N. MACDONALD

a fourth phonon. In the final section we discuss the extent to which the model is successful in fitting the experimental data, and how it may be modified. 2. The Raz Model

The model is described in detail in ref. 5) and only an outline is given here. Two nucleons are taken in a pure shell model configuration (nlj) 2 where n, l, j stand for principal quantum number, orbital angular momentum and total angular momentum, respectively. The basic states IJRP(I) > have these nucleons coupled to spin J, and P phonons coupled to spin R. The total spin of the nucleus in this state is I = J + R. The diagonal elements of the energy matrix in this representation are sums of the energy of P phonons and the energy of the two nucleons. In units of the energy hta of one phonon, this is written P + (JRP(I)IHI2IJRP(I)).

The two-nucleon Hamiltonian H12 is taken by Raz in the form - - 3 D ( 3 - 01 • a2) exp (-r2/r2),

(1)

where ol and ¢2 are the spins of the nucleons and r their separation. We discuss in sect. 5 the consequences of using other interactions. The physical significance of the parameter D is that it determines, in the absence of surface-particle coupling, the ratio of the energy of the first (2 +) excited two-nucleon state to the phonon energy hc0. For example with the (lfi) 2 configuration used in sect. 3, eq. (1) leads to a value D/1.34 for this ratio. From eq. (I) the following form can be derived for the diagonal matrix elements, as for example in ref. lo): P-6D(2j+l)2|

E L-k . . . .

J j ½--½

0] [j j +(2l+1)2{~ j j~} ~ l

' (100

l J} Fk(l)]

(2)

k ©yell

In this expression ( . . ) and {..} are 3j and 6j symbols, and the F~(/) are Slater integrals. For these we employ results tabulated by Thieberger 11), which refer to oscillator wave functions. The values of the F~(I) depend on the ratio of the radius parameter of the oscillator well to the range parameter r 0 in eq. (1). This was taken following ref. 5) as l/x/2, which corresponds to 2 = 1 in the notation of ref. 11). The off-diagonal elements in the energy matrix represent the coupling of the two particlcs to the quadrupole vibrations. They are zero when AP = [P-P'[ # 1, AR = I R - R ' I > 2, AJ = I J - J ' l > 2,

A MODEL OF VIBRATIONAL EVEN NUCLEI

503

and otherwise are given by

- 2x ( 7 ) ½{ + ( J'R'P'(I)IGIJRP(I)> }. In this expression G is the average over both nucleons of

G, = ~ b, Y2~(O,, tp,), where b, is the phonon annihilation operator and the angular coordinates refer to particle i. The parameter x determines the strength of the surface-particle coupling. From the requirement that AP = 1 for a non-zero off-diagonal element one can show that the sign of x is immaterial for the eigenvalues, only affecting the phases of eigenvector components. In consequence our results are independent o f the sign of x, with the exception of the static quadrupole moment. Written in full the off-diagonal elements are

1

,.,

The reduced matrix elements of the annihilation operator (PR[ IbI[P'R'> are tabulated in ref. 5) for all the cases we require. The energy matrix obtained in this way is exactly diagonalised to give the wave functions of states of spin I as

~ = ~ K~(JRP)IJRP(I)>. JRP

In calculating electric quadrupole transition probabilities between these states only the collective contributions are included. The transition probability is B(E2;i ~ f ) _

1 2I~ + 1

i12"

and evaluation of the reduced matrix element B(E2; i ~ f )

x

=

~a~'e'

gives the result

Ki(JRP)Kr(JR'P')

l-i:// [(2If+l)~(-)~'(eellbllP'e'>+(2li+l)~(-)R]. lr R'

(4)

In the constant factor here R0 is the nuclear radius and C is the surface deformation energy parameter. In general we shall be concerned with ratios of energies and ratios of B(E2) so that our results, for specific (n~/) and a particular upper limit on P, depend

504

N. MACDONALD

on the two parameters x and D and not on h¢o, Ro or C. However, in estimating the static quadrupole moments of 2 ÷ states we use the observed value of B(E2; 0 ~ 2) to fix the constant factor in eq. (4). Hence the static moment can be calculated using Q = 2(2)t<211M(E2)II2>.

3. Comparison with Experiment The vibrational nuclei most intensively studied by Coulomb excitation are in the mass regions near A = 76 and near A = 108. The relevant data, taken mainly from refs. 6 - s ) are given in figs. 1-4. In all these nuclei the ratio of the energies of the first two 2 + states lies between 2.0 and 2.5. It is convenient to plot the other expermental quantities against this ratio E2,/E2. In fig. 1 we show E4/E2 and Eo,/E2, and also in the cases of Cd 112 and Cd 114 the energy ratio Eo,,/E2, 0" being the second excited 0 ÷ state. The assignments of 0 + states in Se76 and Se 7s are preliminary 16). Experimental results for transition probability ratios are given in figs. 2-4. The Se 76 value of B(E2; 2 ~ 4)/B(E2; 0 ~ 2) is from ref. iT). We recall that in the harmonic vibrational model the ratio B(E2; 0 ~ 2')/B(E2; 0 ~ 2) of fig. 2 is identically zero. In that model the ratios of figs. 3 and 4 have the values B(E2; 2 --* 2') = 0.4, B(E2; 0 ~ 2)

B(E2; 2 ~ 4) _ 0.72. B(E2; 0 --* 2)

In this section we employ the theory outlined above, for the particular case of the (lf~) 2 configuration, and with the number of phonons limited to three. Thus we have at our disposal the two parameters x and D. The cross-over transition probability ratio, as displayed in fig. 2 against the ratio E2,/E2, is particularly sensitive to these parameters, and we use the data of fig. 2 to select a limited range o f x and D for further consideration. The particular sensitivity of this ratio to x and D can be understood by considering the limit x = 0. Then for D < 1.34 the first 2 + state is a particle state and the second is the one-phonon state, so that the ratio is infinite. For larger D the ratio is zero in the limit x = 0, the first 2 + state being the one-phonon state and the second either a particle state or a two-phonon state. From fig. 2 we see that to give reasonable agreement with the data on the cross-over transition, one requires x values around 0.5 and D values between 2 and 3. Thus to get reasonable results for this ratio, when the surface-particle couplingis turned on, one must start from a situation in which, in the absence of coupling, the two-nucleon 2 + state lies at an energy between 1.5 ho~ and 2.2 h~o. The general character of the D = 2 and D = 3 results in figs. 3 and 4 can be understood by noting that as x ~ 0 (that is as E2,/E 2 --* 1.5 for D = 2, or E2,/E2 ~ 2 for D = 3) the states 2', 0', 4 are all two-phonon states for D = 3, but 2' is the two-nucleon state for D = 2. In figs. 3 and 4 we see that the transition probability ratios are

A MODEL OF VIBRATIONAL EVEN NUCLEI

/

2,5

505

r=o

Pd II0 108 "k I

+

. '~d t + P d 7 114 •. t " / / _ Cd Se75 . . ~ ~ ~,~-,

Ru'04

cdll2 ~..~

Z.O

n: 3 I 2.0

Cd|l Z

Se76 Z5

EZ' / £Z

Fig. I. Experimental a n d theoretical results for t h e energy ratios EJEz (crosses) a n d Eo,/E~ (circles), plotted against E2,/E2. F o r C d n s a n d C d u4 Eo,,/Es is also given. T h e theoretical results refer to the case o f two nucleons in t h e (ift) 2 configuration, with the interaction o f eq. (1), coupled to at m o s t three p h o n o n s , a n d with the p a r a m e t e r D = 2 a n d 3.

0.40

I

I

0.10

s (E2~ o - 2 ' ) a c.Ez~ o

Cd

-0

II0

0=2 7..=.6

78

0.04

Se

• 76

X-.7

O~5

tOG

0.01

j41 i Z,O

/z~Cd 114 :- .45

I Z.5

£2'IEZ

Fig. 2. Experimental 6) a n d theoretical results for the E2 transition probability ratio B(E2; 0 --~ 2')/B(E2; 0 -~ 2). T h e theoretical results are o b t a i n e d as in fig. I, b u t with the addition o f the case D = 1.5.

506

N. M A C D O N A L D

0"75I Te 12;'

0.50 B(E2; Z--/) D-3

Cdil2

B(LZ~o--0

0.25 : !e74 "L!~~!;106 ~ Rul02 t Ru.O0 Se S¢ I I 2.0 2.5 E2'/£ 2

Fig. 3. Experimental e) and theoretical results for the E2 transition probability ratio B(E2; 2 --~ 2")/B(E2; 0 --~ 2). The theoretical results are obtained as in fig. 1.

'oI

I

I

Pd106

0.8 D-3

B (U;2-.-4)

Ru

0.6

Cd112

•

IOZ

Ru 104

76

c d "4

0.4 2.0

Lz./ E2

2.5

Fig. 4. Experimental 7,s) and theoretical results for the E2 transition probability ratio B(E2; 2 -~ 4)/B(E2; 0 --~ 2). The results for Pd 1°8 a n d Pd 11° are those obtained f r o m y-y coincidences assuming that any 0 + state near the 4 + state is weakly excited. We have also included an unpublished result o f Eccleshall for Sen . The curves m a r k e d " D = 2" a n d " D = 3" are obtained as in fig. 1. That marked " D = 2 (pairing)" is discussed in sect. 5.

A MODEL OF VIBRATIONAL EVEN NUCLEI

507

reduced from the results of the harmonic model, and that this is in agreement with the experimental results, although the theory does not give a sulfieient reduction in the case of the 2 --+ 2' transition. The curve marked " D = 2 (pairing)" in fig. 4 will be discussed in sect. 5. In fig. 1 we see that experimentally E 4 __> E2, , whereas the theory requires, throughout the useful range of values of E2,/E2, that E4 < E2,. In discussing modifications to the model in sect. 5 we shall be particularly concerned with the relative positions of the 2', 4 states. The 0' state is given by the theory at an energy well above that o f the 2' state. This is consistent with the cases Pd 1os and Se 7s, and with the higher 0 + states in Cd 112 and Cd 114, but not with the cases Pd 106, 8e76 or the lower 0 + states in Cd 112 and Cd 114. Advances in the techniques of Coulomb excitation experiments hold out some prospect is) of establishing the probabilities of double E2 excitation o f 0 + states in certain of these nuclei, and of detecting the second order "reorientation effeet"~2), which can give information on the static quadrupole moments of 2 + states. We therefore mention briefly the relevant results of this model. The ratio B(E2; 2 --+ 0']B(E2; 0 ~ 2) falls rather rapidly, as x increases, from the value 0.08 of the harmonic model. As mentioned above, the experimental values 6) of B(E2; 0 ~ 2) can be used to fix the magnitude of the static quadrupole moments of the 2 + states, while the signs o f these quantities depend on the sign of x. F o r x > 0 the first 2 + state has static quadrupole moment Q about 0.3 b for E2,/E2 = 2.2, and about 0.5 b for E2,/E2 = 2.5, while the second 2 + state has about 0.2 b and 0.4 b for these two cases.

4. Comparison with the Results of Raz We present here a discussion of how the results of ref. 5) are modified by the correction to the I = 2 energy matrix. The significant difference between the present resuits and those o f ref. s) lies in the sensitivity of the ratio E2,/E2 to the value of the surface-particle coupling parameter x. As we have stated; to have this ratio < 2.5 necessitates keeping x ~< 0.7. In figs. 5 and 6 of ref. 5) on the other hand, the ratio remains near 2 for values of x as large as 1.5. Because of this it was possible (see fig. 8 ofref. 5)) to get E2,/E2 ~ 2 and B(E2; 0 ~ 2')/B(E2; 0 --+ 2) < 0.05 for D as small as 1. Qualitatively m a n y results of ref. 5) still hold, if one remembers that they apply to rather small D values, that is to say to situations in which as x -+ 0 the first excited state is a two-particle state. We have seen, however, that such small D values are irrelevant to the nuclei examined, because they imply large cross-over transition probabilities. It is appropriate to refer at this point to the discussion in ref. 5) of the relation between B(E2; 0 --+ 2) and E2. Fig. 1 of ref. 5) gives some experimental results showing in some cases a sharp fall in B(E2) as E 2 increases. The resemblance is noted to the resuits o f the model (again for D = 1) given in fig. 11 ofref. 5). The result in that figure that B(E2; 0 --+ 2) = 0 when E2 = 0.744 hog, that is for x = 0, is of course a conse-

508

N. MACDONALD

quence of the use of a small D value. The variation of B(E2; 0 --, 2) with striking for D = 2, as we illustrate in table 1.

E2/hcois less

TABLE 1

B(E2; 0 ~ 2) and E , for D = 1 and D = 2 D

1

x

2

E2/hoo

B(E2)

0

0.744

0

0.125

0.693

1.33

E,/~

B(E2)

1

5

0.25

0.601

2.21

0.849

5.03

0.375

0.516

4.41

0.731

5.50

0.5

0.454

5.56

0.625

6.15

0.492

7.27

0.448

7.84

0.75 1.0

0.394

7.58

5. Modifications of the Raz Model In this section we examine how far the results obtained in the specific case employed in sect. 3 are affected if we alter some of the details of the model. We consider the following modifications: taking a different maximum number of phonons, changing the angular momentum of the single particle states employed and changing the type of two-nucleon interaction employed. 5.1. T H E P H O N O N C U T - O F F

We have carried out some calculations including up to two or up to four phonons, and present in table 2 various energy ratios and B(E2) ratios, compared with those TABLE 2

Results o f 2, 3 and 4 - p h o n o n calculations, for x -----0.5 D Pm~

Ez,/E2 EJEa Eo,/Ez B(E2; 0 -> 2') B(E~; 0 --~ 2) B(E2; 2 --~ 2') B(E2; 0 -~ 2) B(E2; 2 -~ 4) B(E2; 0 --~ 2)

1

2

3

3

4

2

3

4

2

3

4

2.84

2.80

2.48

2.32

2.27

2.52

2.17

2.11

2.46

2.39

2.55

2.25

2.16

2.55

2.15

2.07

3.53

3.24

3.09

2.61

2.43

2.77

2.31

2.18

0.21

0.17

0.10

0.055

0.049

0.018

0.014

0.20

0.22

0.18

0.29

0.31

0.27

0.34

0.36

0.59

0.64

0.49

0.63

0.68

0.54

0.65

0.69

0.015

A MODEL OF VIBRATIONAL EVEN NUCLEI

509

obtained when up to three phonons are included, as in sect. 3. These results are for x = 0.5, D = 1, 2 and 3. In considering how good an approximation it is to stop at three phonons one should note that the significant results for comparison with experiment are not in fact those for fixed x, but for fixed E2,/E2. In conjunction with figs. 1-4 the results in table 2 show that going from P < 2 to P < 3 makes quite a substantial difference, while going from P < 3 to P < 4 gives comparatively F.tle change. This justifies our use of the P < 3 approximation above. Qualitative results of increasing the number of phonons are that the ratios B(E2; 2--. 2')/B(E2; 0 - ~ 2) and B(E2; 2 --* 4)/B(E2; 0 ~ 2) are brought nearer to the results of the harmonic vibration model, at a given E2,/E2. Also the 4 + and 0 '+ states are lowered relative to the 2 '+ state. In the 4-phonon approximation one gets even more pronounced disagreement with experimental results for E4/E2, while not substantially improving the values of

Eo,/E2. 5.2. THE ANGULAR MOMENTUM OF THE SINGLE PARTICLE STATES We have obtained results, in the three-phonon approximation, for (2pt) 2 and (2dt) 2 configurations, as well as for the (lf~) 2 configuration employed in sect. 3. In these calculations the two-nucleon force is taken in the form used in sect. 3. We find that the results for (2dt) 2 and (lft) 2 are very similar, allowing for the fact that slightly higher D values are appropriate in the (lf~) 2 case than in the (2dt) 2 case. The case of nucleons in states w i t h j = ~ is rather a special one. The off-diagonal elements in the energy matrix are zero unless AJ = 2, as can be seen from the presence of the factor {~. -~J ~2} in eq. (3). In consequence the matrices are reducible and, since the E2 transitions are zero unless AJ = 0, selection rules occur. There is a degenerate 2 +, 4 + pair at or above twice the energy of the first 2 + state, and there is no cross-over transition to the ground state f r o m the 2 + member of this pair. Also to first order in hm/2c the static quadrupele moments of the 2 + states are zero. Thus the results are intermediate in character between those of the harmonic vibration model and those of our present model with higher j values. This situation is reminiscent of results obtained by Wilets and Jean 9) in whose work the harmonic vibration model is modified by adding to the vibrational Hamiltonian a term dependent only on the overall deformation parameter fl (y-unstable potential) or one depending on the asymmetry parameter y as well as on ft. In the y-unstable case the 2 +, 4 + members of the two-phonon triplet remain degenerate, and the cross-over E2 transition is forbidden, y-dependence splits the 2 +, 4 + pair and permits a cross-over E2 transition. Bayman and Silverberg 13) have pointed out that for one particle coupled to phonons the case o f j = ½ is a special one. The Hamiltonian is invariant under symplectic transformations and these authors have shown that this invariance is equivalent to y-instability. N o w for two j = ½ particles the two-nucleon part of the Hamiltonian is also invariant under symplectic transformations. Invariance implies and is implied by degeneracy of all states belonging to the same representation of the symplectic group. In this case the 0 + two-nucleon state belongs to the representation 14) (00), the 2 +,

510

N. MACDONALD

M = 2 . . . . . - 2 two-nucleon states belong to the representation (11), and these exhaust the two-nucleon states. Therefore the invariance is trivial and the resemblance of our j = ½ results to the 7-unstable model of ref. 9) is not surprising. 5.3. T H E T W O - N U C L E O N

INTERACTION

It is to be expected that the essential feature of the two-nucleon interaction, at any rate for the properties of 2 ÷ states, is the spacing it gives between the 0 + and 2 ÷ twonucleon states. To check this we have compared the results discussed in sect. 3 with those found using a pairing force, that is assuming the 2 ÷, 4 + and 6 + two-nucleon states to be degenerate. Taking diagonal matrix elements P for J = 2, 4, 6 and P-D~ 1.34 for J = 0 keeps the same 0 ÷ to 2 + spacing as before. Using this description of the particle states we find that in the three-phonon approximation the cross-over transition and 2 ~ 2' transition behave very much as before. However, when x is small and D = 2 the results for B(E2; 2 ~ 4)/B(E2; 0 --, 2) reflect the fact that the state 4 is now predominantly a two-nucleon state, whereas with the force of eq. (1) and D = 2 it was predominantly a two-phonon state for small x. The curve labelled " D = 2 (pairing)" in fig. 4 is the result of this calculation. The D = 3 result is almost identical to that of fig. 4. The 4 + state is lowered relative to the 2 '+ state when we use this description, throughout the whole range of E2,/E2.

6. Discussion We have seen in sect. 3 that the model of Raz gives qualitative agreement with the experimental ratios B(E2; 2 ~ 2')/B(E2; 0 ~ 2) and B(E2; 2 -~ 4)/B(E2; 0 ~ 2) once the parameters have been fixed from the cross-over ratio and the energy ratio of the first two states of spin 2. However, it runs into difficulties with regard to the order of the 0', 2' and 4 states. We may note that other models such as those treated in refs. 4) or 15) (the latter discussed in ref. s)) also have little success with this feature. A modification of the model which might give a significant change in these energy ratios would be to couple two nucleons to an anharmonic vibrational spectrum. This is mentioned by Belyaev and Zelevinsky 4) as a means of improving their anharmonie model. Alternatively one could include another two-nucleon configuration to allow a low-lying 0 ÷ particle state. We understand that a specific case of this is being examined by B. J. Raz. Either of these modifications inevitably adds at least one parameter to those at our disposal. They might be of interest in connection with Cd 112 or Cd 114, in which more states are known, especially if one could confirm experimentally the collective character of one of the 0 ÷ states in either of these nuclei. The author wishes to express his thanks to Dr. James Raz for his help in locating the origin of the discrepancy between the present results and those of ref. 5) and for subsequently checking some B(E2) calculations. Dr. D. Eccleshall is thanked for several discussions and for providing some unpublished results. The co-operation of Miss Joan Cadman and Mrs. Joan Knock in the computations is greatly appreciated. Discussions with Dr. A. C. Douglas were very helpful.

A MODELOF VIBRATIONALEVEN NUCLEI

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

G. Scharff-Goldhaber and J. Weneser, Phys. Rev. 98 (1955) 212 K. Alder et al., Revs. Mod. Phys. 28 (1956) 432 A. K. Kerman and C. M. Shakin, Phys. Lett. 1 (1962) 151 S. T. Belyaev and V. G. Zelevinsky, Nuclear Physics 39 (1962) 582 B. J. Raz, Phys. Rev. 114 (1959) 1116 P. H. Stelson and F. K. McGowan, Phys. Rev. 121 (1961) 209; F. K. McGowan and P. H. Stelson, Phys. Rev. 126 (1962) 257 D. Eccleshall et al., Nuclear Physics 32 (1962) 190 D. Eccleshall et aL, Nuclear Physics 37 (1962) 377 L. Wilets and M. Jean, Phys. Rev. 102 (1956) 788 J. B. French and B. J. Raz, Phys. Rev. 104 (1956) 1411 R. Thieberger, Nuclear Physics 2 (1957) 533 G. Breit et aL, Phys. Rev. 103 (1956) 727 B. Bayman and L. Silverberg, Nuclear Physics 16 (1960) 625 A. R. Edmonds and B. H. Flowers, Prec. Roy. Soc. A214 0952) 515 A. S. Davydov, Nuclear Physics 29 (1962) 682 D. Eccleshall and R. Middleton, private communication D. Eccleshall et aL, unpublished A. C. Douglas and D. Eccleshall, private communication

511