Potential model treatment of 3He cluster states in 17O

Volume 130B, number 5

PHYSICS LETTERS

27 October 1983

POTENTIAL MODEL TREATMENT OF 3He CLUSTER STATES IN 170 A.C. MERCHANT SERC, Daresbury Laboratory, Warrington WA4 4AD, UK

Received 1 July 1983

A potential cluster model is used to calculate the energies of states in t70 having a 14C + aHe structure. Excellent agreement with recently reported experimental data on two low-lyingpositive parity bands of states having isospins T = 1/2 and T = 3/2 is found, and some additional states are predicted.

Recently, Cunsolo et al. have studied the 14C(6Li, 3H)170 reaction at a beam energy of 34 MeV. They found that the main reaction mechanism was direct 3He transfer [1 ], and that a number of positive parity states in 170, up to an excitation energy of about 18 MeV, were heavily populated [2]. These states fell into two rotational bands, having isospins T = 1/2 and T = 3/2, and appeared to have a predominant 14C + 3He structure. They are listed in table 1. The T = 1/2 states at energies of 7.38 MeV, 9.88 MeV and 12.27 MeV stand out very strongly in the triton energy spectrum at 0LA B = 5 ° , whilst the known T = 1/2 states at 6.36 MeV and 8.89 MeV are less heavily populated. The spin-parity values of the lowest three of these levels have previously been determined [3], and Cunsolo et al. propose that the 9.88 MeV level has j r = 9•2 + and the 12.27 level J ~ = 7/2 + on the grounds of weak coupling considerations [4]. Two T = 3/2 states at 13.64 MeV and 16.24 MeV having J n = 5/2 + and 9/2 + respectively stand out in the triton spectrum, and together with the known J 7r = 1/2 +, T = 3/2 state at 12.94 MeV, they are proposed as members of a higher lying T = 3/2 band. In addition to these I have tentatively added the known J = 3/2, T = 3/2 state at 15.20 MeV [3], because of weak coupling considerations. Buck et al. have introduced a model [5] which should be ideal for calculating very simply the properties of states having such a cluster-core structure. They were originally stimulated by the observation that alpha transfer reactions onto 12C and 1 6 0 targets 0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland

strongly populate a small subset of states in 160 and 20Ne. The excitation energies of these states are approximately rotationally spaced, the electromagnetic E2 transitions between them are very strong (several Weisskopf units) and those above the alpha emission threshold have large alpha decay widths (in excess of the Wigner single-particle limit). This suggests that these states might have a simple alpha-cluster-core structure, and Buck et al. proposed a folding procedure to evaluate the effective cluster-core potential. Their model was so successful in describing the properties of these cluster states that it was soon extended to deal with cases where the cluster and/or core carry a spin. It was then used, with equal success, to describe similar alpha-cluster states in 19F [6], 18F and 180 [7] (which were heavily populated in alpha transfer reactions onto 15N, 14N and 14C targets) and also the trinucleon cluster states in 19F [6], 18F and 180 [8] (which were heavily populated in 3 H transfer reactions onto 160,150 and 15N targets, as well as 3He transfer reactions onto 15 N targets). Although this model has not so far been applied to trinucleon cluster states in 170, the recent experimental investigations of Cunsolo et al. [1,2] provide a very strong incentive to carry out such a study. It is therefore the purpose of this letter to compare the predictions of the folded potential model of Buck et al. concerning states of 1 7 0 having a 14C + 3He structure with the aforementioned experimental results, and to suggest energy regions where other similar states might be found. 241

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The philosophy of the model of Buck et al. is to produce a cluster-core potential by folding the cluster and core densities with an effective nucleon-nucleon interaction. A delta force is chosen in this latter role, so that a local effective cluster-core potential of the form

V(r)-

2nh 2 "7

~ J

f d3r ' p a (r -

r')PB(r')

(1)

results, where M is the nucleon mass, PA and PB are the cluster and core densities respectively, and f-is an adjustable real strength constant. The cluster spin-orbit force is taken as Vso(r) = -Vso(h/mTrc)2r-1 IdV(r)/drlL'a,

(2)

where (h/m~rc)2 = 2 fm 2, L is the relative orbital angular momentum of the cluster about the core, o is the Pauli spin matrix for the cluster, and Vso is another adjustable real strength constant. The cluster is viewed as a single particle with principal quantum number N, and in the present application to 3He cluster states in 170 the condition

2N+L/> 6

(3)

will be imposed. This restriction on the quantum numbers of the cluster corresponds to excluding the cluster nucleons from the two vacant p-shell orbitals and placing them in (sd) or higher shells. This avoids the extremely unsatisfactory situation of having cluster nucleons in the same major shells as the core nucleons. The problem of antisymmetrizing between cluster and core nucleons is mitigated, since the main requirements of the Pauli principle are satisfied immediately by using eq. (3). A one-particle Schr6dinger equation, involving the potentials of eqs. (1) and (2), together with a Coulomb and centrifugal potential, where appropriate, can then be solved for a state specified by

N and L. In actual fact the present cluster-core system is very similar to the 19F = 160 + 3H system studied by Buck and Pilt [6]. The energies of the triton cluster states in 19F were found to be split from almost perfect rotationally spaced centroids by the cluster spin-orbit force. So, in the spirit of the weak coupling approximation used by Cunsolo et al. [2] to discuss their data, a remarkably successful phenomenological formula for the energies of the states of given total angular momentum,J, in the lowest lying T = 1[2 and T = 3/2 bands may be obtained by inspection. 242

27 October 1983

E(J, L ) = C+ aL (L + 1) + A[J(S+ 1 ) - L ( L + 1 ) - ¼ ] .

(4)

The rotational parameter, a, is assumed to be the same for the 14C + 3He states as for the equivalent 160 + 3H states, namely 0.2 MeV. The cluster spin-orbit force is also assumed to be the same, and thus to show almost no dependence within a given band on the radial part, r -1 I(dV/dr)l, so that it may be parametrized as in eq. (4) with a constant A = -0.3 MeV. The effects of isospin are wholly contained in the different values of C, the heads of the T = 1/2 and T = 3[2 bands. The spectra obtained by setting C(T = 1/2) = 6.8 MeV and C(T = 3/2) = 13.1 MeV are compared with experiment in fig. 1, where the agreement is seen to be very impressive. The two bands exhibit identical level splittings, displaced from one another in the 170 spectrum by about 6 MeV. Buck and Pilt advocate the following parametrization of the folded potential of eq. (1)

V(r) = - V0 [ 1 +cosh(R/a 0)] / [cosh(r/a O)+cosh(R/a 0)] .

(5) Using this central potential, together with the prescription of eq. (2) for the cluster spin-orbit force and a Coulomb potential appropriate to a uniformly charged sphere of radius 3 fm, eigenvalues and eigenfunctions of the 14C + 3He cluster states in 170 were obtained. The energy levels are displayed in column B of fig. 1 and listed in table 1. The values R = 2.0 fro, a 0 = 1.3 fm and Vso = 2.5 MeV, which are the same as the parameters of the 160 + 3H effective potential, were used whilst well depths of V0 = 123.6 MeV for the T = 1/2 states and V0 = 105.1 MeV for the T = 3/2 states were chosen to place the band heads at 6.8 MeV and 13.1 MeV again. There are some minor quantitative differences from the predictions of eq. (4), but the agreement with the experimentally determined energies of the cluster states remains excellent. In addition, energies and wave functions of states in higher-lying negative parity bands were predicted. The lowest three levels of the 2N + L = 7 bands are indicated in fig. 1. They were calculated with Vso = 4.5 MeV, by analogy with the triton cluster states in 19F, and those of them predicted to lie below excitation energies in 170 of 20 MeV are listed in table 2. Although there are not yet any clear experimental

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PHYSICS LETTERS

Volume 130B, number 5

% T=~

1

_3 T-'~

T=~

1

_1 T-~

_3

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Fig. 1. A comparison between the phenomenologically, (A), and theoretically, (B), predicted and experimentally determined energies of the proposed 3He cluster states in 170.

243

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Table 1 The theoretically predicted and experimental spectra of the lowest lying positive parity bands of 3He cluster states in 170. jTr

1/2 ÷ 5/2 ÷ 3/2 + 9/2 + 7/2 + 13/2 + 11/2 +

Energiesof states of isospin T = 1/2 (MeV)

Energies of states of isospin T = 3/2 (MeV)

experiment

theory

experiment

theory

6.36 7.38 8.89 9.88 12.27

6.80 7.39 8.92 9.65 12.39 13.80 17.73

12.94 13.64 (15.20) 16.24

13.10 13.77 15.00 16.09 19.26 20.32 23.31

candidates which might correspond to these states, they should be accessible to experiments like those o f Cunsolo et al. Indeed, several peaks in their triton spectrum are apparent in the energy region where these states are predicted. Probably the best candidate among these is a heavily populated level at 18.1 MeV which may be the known J ~r = 3 / 2 - , T = 3/2 level [3]. If so, it would be in excellent agreement with the value listed in table 2. The difference between the energies of the heads o f the T = 1/2 and T = 3/2 bands indicates that there must be some isospin dependence of the cluster-core potential. This has previously been explicitly introduced in a semi-microscopic calculation of the properties of trinucleon cluster states in 18F and 180 [9], and that work can be easily adapted to estimate the difference between the T = 1/2 and T = 3/2 band head energies in 170. In ref. [9] a nucleon-trinucleon potential was summed over the core nucleons to calcu-

Table 2 The theoretically predicted energies (up to 20 MeV) of members of the lowest lying negative parity bands of 3 He cluster states in 17O. jlr

Energies of states with T = 1/2 (MeV)

Jn

Energies of states with T = 3/2 (MeV)

3/27/21/25/211/2-

13.54 14.65 14.72 17.34 17.45

3/21/27/2-

17.96 18.86 19.18

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27 October 1983

late an effective d u s t e r - c o r e potential. The most general nucleon-trinucleon potential contains a term U(x)~t" gn (involving the scalar product of cluster and nucleon isospins, xt and Xn, multiplied by a function of their separation distance, x) which gives rise to a term in the d u s t e r - c o r e potential o f the form F(r) X xt" Xc (involving the scalar product of the cluster and core isospins). The form of U(x) is not well known, but for simplicity, it was assumed to have a gaussian geometry and its width, 1/a 2 , allowed to range from 0.125 fm - 2 to 0.25 fm -2 with its corresponding strength, U0, determined by the P-state resonances of 4He lying at about 25 MeV. If the 14C core is described by two p-shell proton holes in a 160 core, which are represented by harmonic oscillator wave functions, an analytic form for F(r) can be obtained:

F(r) = 00 [a2/(a 2 + b2)] 5/2 [1 + 2b2r2/3a2(a 2 + b2)] X exp[-r2/(a 2 + b2)] ,

(6)

where b is the oscillator parameter. This form factor can now be integrated with the previously evaluated wave function for the T = I/2 band head state to find the energy, relative to it, of the T = 3/2 band head. Using the parameters of ref. [9] which gave the best fit to the trinucleon cluster state spectra of 18F and 180, the T = 3/2 band head is predicted to lie 5.8 MeV above the T = 1/2 band head, with a range of 5.4 MeV to 7.0 MeV as the width, 1/a 2, is varied (c.f. the fitted value of 6.3 MeV). To conclude, it has been shown that the folded potential model of Buck et al. gives an excellent account of the energies of the T = 1/2 and T = 3/2 states which were found heavily populated by Cunsolo et al. in their investigation of the 14 C(6 El, 3 H)17 O re action. In addition, the energies of the remaining states in these two positive parity bands, and the lowest lying members of two similar negative parity bands, have been predicted. They lie in a region where several unidentified peaks in the triton spectrum of Cunsolo et al. occur. All of the parameters in the calculation were previously determined by Buck and Pilt in their study of triton cluster states in 19F, except for the positive parity band heads (or equivalently, the depth of the central part of the cluster-core potential) and the difference between these band head energies can be accounted for in a semi-microscopic calculation. This success prorides another piece o f evidence for the efficacy of the

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PHYSICS LETTERS

cluster model o f Buck et al. in calculating the properties o f bands o f cluster states in nuclei w i t h i n the mass range 16 ~< A < 20.

References [ 1 ] A. Cunsolo et al., Phys. Rev. C24 (1981) 2127. [2] A. Cunsolo et al., Phys. Lett. 124B (1983) 439. [3] F. Ajzenberg-Setove, Nucl. Phys. A375 (1982) 1.

27 October 1983

[4] A. Arima, H. Horiuchi and T. Sebe, Phys. Lett. 24B (1967) 129. [5] B. Buck, C.B. Dover and J.P. Vary, Phys. Rev. C l l (1975) 1803. [6] B. Buck and A.A. Pilt, Nucl. Phys. A280 (1977) 133. [7] B. Buck, H. Friedrich and A.A. Pilt, Nucl. Phys. A290 (1977) 205. [8] B. Buck and A.A. Pilt, Nucl. Phys. A295 (1978) 1. [9] A.C. Merchant, J. Phys. G, to be published.

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