Collective vibrations in P31

I 1.D.2: 1.E.1

[ Nuclear Physzcs 19 (1960) 303--308, © North-Holland Publzshzng Co., Amsterdam [ Not to be reproduced by photoprmt or mmrofilm without written perrmssmn from the pubhsher

COLLECTIVE VIBRATIONS IN p31 V. K. T H A N K A P P A N and S P. P A N D Y A

Physwal Research Laboratory, A hmedabad-9, Indzat Received 20 April 1960 Abstract: An a t t e m p t is m a d e to explain the energy levels and the electromagnetm tranmtlons m p31 in t e r m s of t h e collective vibrational model The results are found to be fairly satisfactory

1. Introduction The role of collective motion in explaining the properties of nuclei of mass A ~ 30 is recently being investigated with much interest, in view of the successful demonstration of the existence of the collective rotational motion of nuclei of mash A m 25, and the experimental observation of enhanced E2 transitions in some A = 29 and A = 31 nuclei1). In particular, it is found in these nuclei that the E2 cross-over transition from the second excited state of spin J = { to the ground state of spin J = ½ is about hundred times more intense than the possible M 1 transition to the first excited state of spin J = 3This and other properties of these nuclei are explained with a fair amount of success by a model which describes them in terms of a single odd particle interacting with a deformed rotating nuclear core. Analysis of nuclear stripping reaction data for these nuclei by French and Mcfarlane 2) shows evidence of considerable mixing of shell model configurations in the ground state of p3~. However, detailed calculations on the basis of nuclear shell model including mixed configurations, predicting the energy levels, magnetic moments, etc., are not yet available. It should be of interest to examine the predictions of the collective vibrational model for nuclei of A ~ 30, for several reasons. One of us has earlier described a preliminary calculation for the properties of Si 29 in terms of this model3). In view of the considerably more detailed information now available for the energy levels of p31, we present here results for this nucleus. Earlier calculations of Goldhammer4) on similar lines were based on assumptions quite different from those we adopt here; our emphasis is at present on detailed comparison of the predicted and observed energy level spectra. t Supported b y t h e D e p a r t m e n t of Atomic Energy, India a n d t h e U n i v e r s i t y Grants Commission, India 303

~04

V K THANKAPPAN AND S. P. PANDYA

2. Energy Levels W e consider the p31 nucleus as a spherical core of 14 protons and 16 neutrons, filling up the nuclear subshells u p t o ld~ and 2s~ respectively, and the last odd p r o t o n in the 2s~ or l d~ subshell. The collective properties of the core are described in terms of q u a d r u p o l e surface oscillations which are quantised. T h e single particle states of the odd p r o t o n are t h e n coupled to the 0, 1 and 2 q u a n t a states of the core. The m a t h e m a t i c a l formalism for such a model is well-known, and detailed calculations are e x a c t l y similar to those described in ref. 3). W e follow the notation described there. It is implicit in the model t h a t the collective properties of the core should be a p p r o x i m a t e l y the same as those observed in the Si 3° nucleus. We i n t e r p r e t the first excited state of Si 3° at 2.24 MeV 5) as the o n e - q u a n t u m vibrational state of the core, which gives ho) = 2.24 MeV. Of course, on the basis of such a simple model we should expect to see in SP ° a degenerate triplet of states of spins 0, 2 and 4 arising from t w o - q u a n t a exoitation at 4.5 MeV. However, the experim e n t s show only a close doublet at 3.v 1 and 3.79 MeV, and f u r t h e r excited states are not clearly known. This result should not be surprising as for excitation energies > 4 MeV, this simple model m a y not be a d e q u a t e ; in particular, effects of inter-nucleon forces and particle excitations from the core m a y have to be t a k e n into account more explicitly. I t is for this reason t h a t we confine our a t t e n t i o n to (even parity) states of p31 below 4 MeV. The other two p a r a m e t e r s of the model, viz. the separation A of the single particle states 2s~ and ld~, and the constant q indicating the strength of coupling of the odd particle to the collective oscillations of the core, are considered as free parameters, and are adjusted to obtain the best agreement of the calcula*-d and the observed energy levels. The H a m i l t o n i a n matrices for J = ½, 3, { and ~ are constructed, and are explicitly diagonalised for various values of A and q. The results for the lowest few states of each J are shown in fig. 1. The e x p e r i m e n t a l results are shown in fig. 2. We note t h a t q u a l i t a t i v e l y the order of the energy levels is correctly given b y the theory. To find the best choice of the p a r a m e t e r s A and q, we r e m a r k t t h a t the separation of the states ~ and ~*, and the splitting of the triplet {*, ½" and 9** IS quite sensitive to variation in the value of q and is relatively unaffected b y variation of A ; this enables us to choose the best value of q as 1.0. W i t h this choice for ~, the variation of the excitation energy of the state with A determines the best value for A viz., A ~ 2 0 MeV. Fig. 2 also shows the predicted energy levels for q = 1.0, A = 2 0 MeV. It m a y be n o t e d t h a t e x p e r i m e n t a l l y the spin of the 3.41 MeV state is not determined. Moreover, Simons ~) assigns spin ½ to the 3.51 MeV level, whereas Broude et al. 1) find spin t T h e n o t a t m n h e r e is t h a t t h e u n s t a r r e d , ~ t a r r f d a n d d o u b l e s t a r r e d v a l u e s r e f e r t o t h e l o w e s t , t h e first a n d t h e s e c o n d e x c i t e d s t a t e s of a g i v e n J

cOLLECTIVE

VIBRATIONS

I N I~ I

305

for the same level. Though we have shown in fig. 2 the spin sequence of the triplet as ~, ½, ~, the possibility that the 3.41 MeV state is ~ and the 3,51 MeV state is ½ cannot be ruled out from our calculations, as these levels cross in the

c --

3/2

~f.5

I

4--

~

-7/2

2

2~

J

-5t2

J ,

"----..

I

L

I

J t

IO

1.5

O'

20

t.o

(~)

1.5

20

(s)

Fig, I. V a r i a t i o n of t h e e n e r g y level s c h e m e of p a l w i t h q (fig. l(a) a n d w i t h A (fig. l(b)), T h e levels a r e n o r m a l i s e d to E = 0 for t h e g r o u n d s t a t e . ~(rlev)

4~

.3/2 7/2

S/2

l , S / 2

3/2

at2

J ,tA V~

cq0

(b)

Fig 2. Comparison of the calculated energy levels (A -----2.0 MeV, q ~ 1.0 MeV) with the observed energy levels below 4 M e V . (a) Calculated levels. (b) Experimental levels.

306

V. K. THANKAPPAN

AND S P

pANDYA

neighbourhood of q m 1.0, as m a y be seen from fig. la. The close triplet of levels ~*, ½" and ~}** is predicted about 0.5 MeV higher than the observed triplet. This discrepancy is not regarded as serious in view of the remark earlier made regarding the higher energy levels, and the availability of only two parameters in the calculation. The really serious difficulty in the predictions of this model is the presence of the J = { level between the ~* level and the higher triplet. This level is not seen experimentally. We stress this feature as important, since no reasonable variation of the parameters can avoid placing this level below ,m 4 MeV. The only comment we make is that since the observed levels in this region are very close, the probability that the 27-level is degenerate with ~* or {* should be considered. Small changes in the parameters A and q m a y easily cause this to happen. It m a y be noted that for q > 1, this ~ level would occur below 3 MeV, and would almost certainly have been detected. This is perhaps an additional argument against choosing q > 1.0.

3. Electromagnetic Transitions It is characteristic of the collective vibrational model that although E2 transitions can be considerably enhanced by introducing even a small amount of collective vibrations, the static values of the electromagnetic moments are not changed very much from the simple shell model values, p31 belongs to that grmlp of s~ nuclei which show a very large deviation of the observed magnetm moment from the Schmidt value. It is thus not surprising that as in the case of Si 29, the vibrational model fails to predict the observed value of the magnetic moment of P3k The calculated value for the magnetic moment is (for q = 1.0, A = 2 0 MeV) ~ = 2.56 n.m., which m a y be compared to the observed value /, = 1.13 n . m , and the Schmidt value/~ = 2.79 n.m. The wavefunctions of the ground state and the first two excited states are listed in table I. These show features very similar to those calculated for Si 29. The two important characteristics of the electromagnetic transitions in p31 are as follows : a) large E2 component in the decay of the first excited state, b) the possible M1 transition from the second excited state { to the first excited state ~ is less than 5 % of the crossover E2 transition to the ground state ½. Qualitatively these features are easily explained by the structure of the wavefunctions found for these states. We note that the ground state is ~ 80 % pure single particle s½ state, whereas the first excited state ~ has a large admixture of the single particle state d~, and the s~ state coupled to one vibrational quantum i he presence of the latter large component would give rise to an enhanced E2 transition to the ground state. On the other hand the second

307

COLLECTIVE VIBRATIONS IN pS1

excited state arises almost entirely from the coupling of the st state to one vibration quantum state of the core. Ficnce we should expect a strong E2 transition to the ground state, whereas the M1 transition to the first excited state is almost forbidden. We hope to report on detailed calculations of the TABLE 1 Elgenfunctions of the statesJ n

k

00½ 0 0 { I 2 ½ 20½ 22½ 24½ 2 0 ~ 22~ 2 4 .~

~ ½, ~,-~ o f p z l f o r q =

J=½

0 948 --0 308 0.056 ---

-- 0 053

J=~

0.709 -- 0 625 -- 0 247 0 079 153 0 129

1 0 , A ~--- 2 . 0 M e V

j=~

--0

0 924 175

-- 0 039

0

0 090 0 325

T h e t a b u l a t e d q u a n t i t y is t h e a m p l i t u d e of t h e s i n g l e p a r t i c l e s t a t e of s p i n I c o u p l e d t o t h e s t a t e of the nuclear core with n quanta coupled to the resultant spin k

electromagnetic transitions in such' odd-proton nuclei later in another context. It m a y be emphasised in the meanwhile that absolute measurements of the various transition probabilities in these nuclei would be very useful. We should like to add a remark on the interpretation of these electromagnetic transitions within the framework of simple shell model ideas, as we fear that this aspect has perhaps been misrepresented elsewhere. On the basis of the simple shell model, one would interpret the ground and the first excited states as pure single particle states s½ and dt, whereas the second excited state would be due to the excitation of a d I particle from the underlying closed shell to perhaps the st shell, resulting in the configuration (dt) -I (s½)2 for this state. Even on the basis of such a simple model, it is clear that the } -+ ~ transition would be absolutely forbidden (as it involves two-particle transitions), whereas } -~ ½ transition can take place as d -+ s transition. The introductio~l of collective effects would serve to enhance the E2 transitions. It is therefore unfair to infer from the large crossover transition that the shell model fails, and deformation of the core must be invoked. Perhaps a proper shell model calculation taking into account the mixing of configurations would also give quite good results. 4. Conclusions We conclude that the collective vibrational model explains satisfactorily the seven energy levels of p31 observed below 4 MeV, but predicts an unobserved

~0~

V. K. THANKAPPANAND S. P, PANDYA

level J - {. We hope that further experimental observations m a y elucidate this point. One m a y compare the predictions of this simple two-parameter model to those of the collective rotational model of Broude et aL l) for the low lying energy levels. It appears that both models are about equally successful. It is important to remark that though the rotational model predicts the ~ level above 4 MeV, the position of this level is displaced to a large extent b y the rotation-particle coupling between the bands 8 and 11. The values of the parameters A and q obtained here are also quite reasonable. In particular the small value of q and tt~e calculated wavefunctions for all the states considered here show that the amplitudes of the two-quanta excitation states of the core are small, and the neglect of more than two-quanta excitations of the core is justified. Finally, the observed features of the electromagnetic transitions in p31 are also quite easily understood, at least qualitatively, on the basis of this unified model. R~ .-rences 1) D. A. Bromley, H. E. Gore, E, B Paul. A E Lltherland and E Almqmst, Can, J Phys. 35 (1957) 1042; D. A, Bromley, H. E. Gove and A. E. Lltherland, Can. J. Phys 35 ~1957) 1057, C, Broude, L, L Green and J. C Wfllmott, Proc Phys Soc. 72 (1958) 1097, 1115, 1122 2) M. H Mcfarlane and J B. French, NYO-28~6, The Umvermty of Rochester, Rochester N Y, 3) S, P, Pandya, Prog. Theor. Phys 21 (1959) 431 4} P. Goldhammer, Phys. Rev 101 (1956) 1375 5) P, M E n d t and C. M. Braams, R~,vs. Mod Phys 29 (1957) 683 6) L. Simons, Nuclear Physics I0 (1959) 215