The interacting boson model structure of 16O

The interacting boson model structure of 16O

Volume 45B, number 1 PHYSICS LETTERS THE INTERACFING BOSON MODEL STRUCTURE 25 June 1973 OF 160 H. FESHBACH ~ Laboratory for Nuclear Science and ...

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Volume 45B, number 1

PHYSICS LETTERS

THE INTERACFING

BOSON MODEL STRUCTURE

25 June 1973

OF 160

H. FESHBACH ~ Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Mass. 02139, USA and

F. IACHELLO Instituto di ~'~sica,Politecnico di Torino, Torino 10129, Italy

Received 12 May 1973

The recently proposed interacting boson approximation (IBA) is applied to the structure of 160. The T = 0, J = 0+, 2+, 4 +, 1-, 3 - states below 18 MeV are calculated and compared with the known experimental levels. The agreement obtained indicates the usefulness of the IBA. The limitations of the model are briefly discussed.

It is by now quite clear that it is not possible to describe the ground and excited states of even a single nucleus in terms of a single simple model. In the same nucleus some levels may conform to a vibrational type spectrum, while other may appear to form a rotational band; and still others will be intermediate in character. It would be useful to have a unified description of the various families of states. In principle, the shell model calculations of McGrory and Halbert and of Lawson and Macfarlane et al., if a sufficient number of levels and configurations are included, furnish such a description. Large computer efforts are involved. However, the inclusion of more than a few levels rapidly puts such calculations out of the reach of today's computers. We have devised an approximate model for dealing with such complicated cases. The approximation does not replace the shell model. Rather it is a procedure for obtaining results quickly but with a corresponding reduction in quantitative accuracy. The model it is hoped will provide insight where shell model calculations can be done and provide a method for extrapolating to nuclear excitations for which shell model calculations are impractical. This work is supported in part through funds provided by a contract with the U.S. Atomic Energy Commission under grant AT(11-1) 3069.

We call this model an interacting boson model (IBA), because we replace the ordinary shell model by a boson shell model in which the orbitals contain bosons. We shall report the details of the model and a discussion of the approximations involved in a longer paper [ 1]. In that paper a number of approximations will be presented of which the most drastic lead to the IBA with the consequence that it is simplest to apply. Here we present its application to the structure of 160. The problems involved in understanding 160 have been recently summarized by Irvine, Latorre and Pucknell [2]. Another successful application of the IBA has been made to the photo disintegration of 160 by Shakin and Wang [3]. Much of the work presented here is contained, although in a rougher form, in ref. [4]. In this example, 160, we construct boson levels of 1 ph states as follows. When the particle-hole interaction is diagonalized in the 1 ph subspace some levels move down in energy. These low lying states are collective in nature and are taken to form our boson levels. More complex excitations are formed by combining two bosons in all positive ways, 4 ph states from four bosons, etc. The diagonalization of the residual interaction in the proper symmetrized multiboson space gives the final spectrum. The residual interactions which we in-

Volume 45B, number 1

PHYSICS LETTERS

(ph)

(pp)

~ w~

~

(hh)

(cph)

(cph)

(mph] ~

(mph)

Fig. 1. Schematic representation of the basic interaction. Lines running upwards represent particles while lines running downwards represent holes. Each type of interaction consist of a direct and exchange term. Only for the (ph) interaction both terms are shown in the figure. The (ph), (pp), (hh) and (cph) interactions are included in the interacting boson model.

clude are: (1) the particle-particle (pp) and hole-hole (hh) interaction, (2) the interaction which couples the different p-h subspaces (cph). They are shown in fig. 1. The terms in the Hamiltonian which we neglect are: 1) the rearrangement terms in the (ph) interaction; 2) the (mph) in which an odd number of ph's are created or destroyed. The latter is absent if the low lying boson levels are constructed only from 1 h e excitations. The other limitations of the model will be discussed after the description of its application to 16 O is given. One should bear in mind an advantage this model has with respect to the RPA. The RPA makes approximations similar to those listed above (it does however include many particle-hole states of a given type) but neglects the pp and hh interaction as well. From our experience with the calculations of the levels in 160 and 40Ca, the effect of the pp and hh interaction is very large. The inclusion of these terms is thus a particularly important feature of our approximation; their omission makes the results obtained with the RPA suspect. The calculation proceeds by first performing a separate diagonalization of the pp and hh interaction in each n boson space. In the unperturbed situation, the energy of a two-boson system would be the sum of the energies of the two bosons. This is no longer the case 8

25 June 1973

in the presence of the pp and hh interaction. In fact some of the states move down a considerable distance in energy. The effect becomes larger as the number of bosons increases stopping when the blocking caused by the exclusion principle becomes important. After each n boson space is diagonalized, the coupling between these states because of the c ph interaction is taken into account. We have applied the IBA to 160 (see ref. [5] and [6] for the shell model results) by using a/5-function interaction with Soper mixture

Vii = - V o ( / r t + 0.46 ns) 8(xi-x/)

(1)

where n t and n s are spin triplet and singiet projection operators. The only liberty we have given ourselves is to choose independently the strength of the (ph) and of the residual (pp, hh) interaction. (Two parameters in all). To construct the boson levels we perform a standard particle-hole calculation in which the particles are allowed to occupy the ld5/2, 2Sl/2, ld3/2 levels and the holes are lp3/2, l p l / 2 levels. The single particle energies are taken as [7] elP3/2 = - 6 . 1 4 MeV, elPl/2 = 0.00 MeV,

elds/2 eld3/2

11.45 MeV, e2sl/2 = 12.32 MeV, = 16.53 MeV.

We fix the strength of the (ph) interaction to give the low-lying J 7r = 3 - , T = 0 state at 6.1 MeV. The lowlying J ~r = I - , T = 0 state comes next at 7.7 MeV. The required value of Vo is Vo = 4rr (1.76) 3 14.0 MeV. For details of the calculation see for instance fig. 2 of ref. [8]. Although Elliott and Flowers use a different interaction, the results are rather similar. In the same figure one can see clearly the boson levels t . Beyond "~This choice of Vo is larger than that of Shakin and Wang [2] who choose a value which places thek 3ph states in the giant resonance region. Our choice is correct within our model. The inclusion of 3ph states to obtain the final 1 - and 3 states does not modify the energies of these states substantiaUy, although the wavefunction is considerably changed; the percent of the lph state being 50% in for example the 1 - state. However it may be argued (Shakin, private communication) that many more multiparticle-hole need to be included for example via the RPA before fitting the data. However it, in our opinion, is difficult to assess the influence of these h~gher states because of the blocking effect of the Pauli principle as well as the neglect of the pp and hh interactions.

Volume 45B, number 1

PHYSICS LETTERS

O+

,5 ,,

14.81

F~. ,90

12.05 II .26

Ex

Fo~

16.41

60

Ex

15.26

14.10 260 14.00 4 8 0 0 - - ~ /

4 +

2+

Ex

15

25 June 1973

~ - -

1.5 2500

13.11

200

r~

~

16.80

525

~___

14,92 14.70 13,89

670 90

, / - -

- -

I0

11,52

I10

~

9.85

--

~ - -

11.09

I0. 35

/

30

6.91

6.05

5-

O-

0 Th

Ex

Th

l-

E,

Ex

5-

Fc~

Th

E~

F~

15.70 15.42

525 I00

15.25 13.10

26 60

Ex

17.30 17.15 15

15.40

60

~ / - - -

14.78

60

~

13.10 12.43

115 I00

---

-

~ S

I0-

9.60

11.63 830

490

le O _ _ ~

7.11

. . . . . .

6.13

5-

0

Th

Ex

Th

EX

Fig. 2. The calculated (TH) and experimental (EX) jTr = 0+, 2+, 4 +, 1 - , 3 - T = 0 energy levels. The experimental excitation energy E x (MeV) and the width for the (a, ,~) reaction r,~ (keV) are also shown. Uncertain levels are shown in broken lines. the 3 - and 1 - the next boson level is the j~r = 2 T = 0 about 2.5 MeV higher. In our calculation, the spurious c.m. 1 - state has been identified and eliminated by using the Elliott and Skyrme procedure [9]. Keeping only the 3 - and 1 - boson levels, we next construct multiboson states. We use spectroscopic notation (j t) n, (30) 4 is four-(30) states. H e r e / i s the

boson spin and t its isospin. The construction requires the preliminary calculation of the c.f.p, for identical bosons. We use the seniority scheme (o - seniority; o will only be written if required to distinguish the states). The residual interaction is computed again by using eq. (1). The strength Vo is adjusted at the value V o = 41r(1.76) 3 14.0 MeV. The results of the diagonalization in the multiboson

Volume 45B, number 1

PHYSICS LETTERS

25 June 1973

Table 1 Wave functions of the 0+ states below 18 MeV. In square brackets are the partial angular momenta and isospins. The seniority quantum number o is not required to specify the states. Although we have not computed the 6ph we believe that they will have large components into the 0~ and 0~states at 10.88 and 17.27 MeV thus increasing even more the c~width.

4ph

E x (MeV)

0

6.54

(30) (10) a [30] (30) 2 [20] (10) 2 [20] (30) a [10] (10)

0.000 0.007 0.001 0.033 0.105 0.162

(10)4 (30) 2 [00] (10) 2 [001 (30)4

2ph

(10)2 (30) 2

0ph

(0)

10.88

13.29

16.83

17.27

0.000 0.024 0.004 0.023 0.183 0.771

-0.003 -0.063 -0.022 -0.409 -0.476 0.251

-0.001 0.012 -0.004 0.162 -0.058 0.561

0.333 0.077 0.908 -0.193 0.132 0.005

0.084 -0.002 0.207 0.859 -0.345 -0.009

-0.187 -0.403

-0.016 -0.468

0.708 0.021

-0.218 0.753

0.008 0.051

0.276 -0.125

0.873

-0.387

0.188

0.197

0.013

0.011

space are shown in fig. 2, where our calculated levels (TH) below 18 MeV are compared with the known levels (EX) below that energy, taken from recent experiments [10, 11]. The wave functions of the 0 + states are shown in table 1. Those of the other states will be given in ref. [1]. The calculation has been cut at 5 ph. 5 ph states with occupation number for the 2Sl/2, lpl/2 -1 levels greater than 3.7 have been excluded (the maximum admissible is 4). It has been possible to relate the calculated T = 0 spectrum with the experimental spectrum for the lowlying states, 28 in all. The elements entering into this identification beside the usual ones of spin, parity and order in the spectrum, are (1) the a particle width as observed in the elastic and inelastic scattering of 0tarticles by 12C and (2) the electromagnetic decay rates. For a relatively large a particle width to occur it is necessary for a large overlap to exist between the wavefunction formed from 12C and an alpha particle and a state in 160. There are of course many other factors entering into the determination of Pa" We have however used this overlap criterion as a qualitative way of relating the calculated 160 levels and the observed resonances in 12C(ot, 0t)12C reaction. For the 0 + states for instance, a large experimental Pa corresponds to a large (10) 4 component. That correlation is clearly visible in table 1. This might be expected in advance between the (10) boson is mainly (2Sl/2)(lpl/2) -1 and therefore four (10) bosons correspond closely to an t~ particle outside of the 12C core. For 10

2 + and 4 + states the overlap criterion reduces to the requirement of a relatively large four boson component in the 160 wavefunction. For the 1 - and 3 states the amplitude of the 5 boson state is correlated with the value of F~. It is interesting to note that the 1~], 1~-, 1~- at 9.6, 11.2 and 13.1 MeV all have large 5 ph components and large experimental (a) width. Our calculation therefore confirms the suggestion made by Zuker, Buck and McGrory [5] and by Ellis and Engeland [12] that these states are indeed 5 ph in character. The association of individual theoretical 3 levels with the experimental ones is not as clear cut as for the other spins. The difficulty is associated with the 3~ state. Different experiments report different energies for this state ranging from 11.2 to 11.6 MeV and a P,, width which is anomalously large ( 8 0 0 1200 keV) considering the large centrifugal barrier. In our calculation we find two levels with large 5 ph amplitudes and of very similar structure at 10.40 and 11.16 MeV and a third with a reduced amplitude close by at an energy of 1156 MeV. This suggests the possibility that the reported level at 11.63 might be two or three unresolved levels. In fig. 2 we have assumed the experimental 3~ level to correspond to the two unresolved at 10.40 and 11.16 but of course alternative assignments are possible. Additional experiments are needed to clarify this situation. Also the possibility that some of the calculated states are largely spurious cannot be completely excluded, as we did not attempt to estimate the amount of spuriosity brought in by the

Volume 45B, number 1

PHYSICS LETTERS

Table 2 Comparison between calculated and experimental BE (2) values. All values in fm 4. The experimental values are taken from refs. [131 and [14]. Transition

BE(2)th

BE(2)ex

4~ 2323 2~2~ -

2~ 0~ 0~ O~ O~

111 37 5.5 0.005 0.41

117 ± 10 42 +- 10 4+ 1

25 -

o~

3.5

2~-0~

2.1 30

1~- 3~

4± 1 60 ± 20

neglect of the Pauli principle. In table 2 we compare our calculated BE(2) values with the experimental values. Although we use a rather large additional charge eeff = 0.8e for protons and neutron, it is seen that calculated values reproduce the trend of the experimental values and are of the right order of magnitude which is all that can be expected in virtue of the approximate nature o f the wave functions. Significantly the collective nature o f the levels has been clearly demonstrated. It should be noted that this model does predict a 0 + level near 6 MeV with a large 4 boson component (see table 1) as suggested b y Brown and Green [13]. In addition to the four 3 - boson c o m p o n e n t there is a substantial amplitude of the two 3 - bosons and of the unperturbed ground state. Similarly the ground state now contains substantial two- and four-particlehole states. The results presented here are highly encouraging and indicate the usefulness of our classification scheme. However there are approximations involved which limit the applicability of the model. Of particular importance are the omission o f the rearrangement terms mentioned earlier, the effects of the Pauli principle and the restriction to the 1 - and 3 - bosons. As

25 June 1973

R.L. Feinstein [ 15] has demonstrated b y direct calculation of the J = 0 states in the 2 particle - 2 hole subspace these errors tend to cancel for the low-lying levels. When included, the rearrangement terms push these levels up while the omitted intermediate-coupled configurations act to increase their binding. These effects do n o t cancel in general, requiring as a consequence a renormalization of the pp and hh potential compared to the ph potential. To fit the data related to 160 we did not need to renormalize. One of us (F.I.) is now starting an investigation o f 4°Ca to see whether the success of our and Shakin and Wang's calculation in 160 is fortuitous.

References [ 1] H. Feshbach and F. lachello, to be published. [2] J.M. Irvine, C.D. Latorre and F.E. PuckneU, Adv. in Phys. 20 (1971) 661. [3] C.M. Shakin and W.L. Wang, Phys. Rev. Lett. 26 (1971) 902; Phys. Rev. C5 (1972) 1898. [4] F. Iachello, Ph.D. Thesis, MIT (1969), unpublished; H. Feshbach, Invited talk at the Washington meeting of the APS (1971). [5] A.P. Zucker, B. Buck and J.B. McGrory, Phys. Rev. Lett. 21 (1968) 39. [6] H.U. Jager, Nucl. Phys. A136 (1969) 641. [7] G.E. Brown, Unified theory of nuclear models (NorthHolland, Amsterdam 1964). [8] J.P. EUiott and B.H. Flowers, Proc. Roy. Soc. 242 (1957) 57. [9] J.P. Elliott and T.H.R. Skyrme, Proc. Roy. Soc. A232 (1955) 561. [10] T.P. Marvin and P.P. Singh, Nucl. Phys. A180 (1972) 282. [ 11] M.S. Zisman, E.A. McClatchie and B.G. Harvey, Phys. Rev. 2C (1970) 1271. [12] P.J. Ellis and T. Engeland, Nucl. Phys. A144 (1970) 161. [13] G.E. Brown and A.M. Green, Nucl. Phys. 75 (1969) 401. [14] D.H. Wilkinson, D.E. Alburger and J. Lowe, Phys. Rev. 173 (1968) 995. J. Lowe, D.E. Alburger and D.H. Wilkinson, Phys. Rev. 163 (1967) 1060. [ 15 ] R.L. Feinstein, private communication.

11