A shell-model analysis of Λ binding energies for the p-shell hypernuclei II. Numerical Fitting, Interpretation, and Hypernuclear Predictions

ANNALS

OF

PHYSICS:

72, 445-488 (1972)

A Shell-Model Analysis of A Binding Energies for the p-Shell Hypernuclei. II. Numerical

Fitting,

Interpretation,

and Hypernuclear

Predictions

oj’ Jerusalem,

Israel

A. GAL Racah

Institute

of Physics,

The Hebrew

Unir;ersity

Jerusalem,

AND

J. M. SOPER Theoretical

Physics

Division,

Atomic

Energy

Research

Establishment,

Harwell,

England

AND

R. H. DALITZ Department

of Theoretical

Physics,

Oxford

University,

England

Received July 29, 1971

A phenomenological analysis of the (1 binding energies known for the p-shell hyper nuclei is given in terms of their representation by the shell-model configuration {(l~)~~(lp)~-“(ls)~~, using the formalism set up in Part I. The dN interactions are described quite generally by five free parameters, and the inclusion of ANN interactions with the forms arising through the double-OPE mechanism requires five further parameters. Various constraints motivated by theoretical arguments concerning the nature of dN and llNN interactions are imposed on these parameters, and best fits are then obtained for them following a minimization procedure. A good and economical fit to these data is obtained for a parameter set in good accord with our theoretical notions about these interactions (although the fit otherwise most attractive does require dN spinorbit forces with sign opposite those known for the NN system). On the basis of these fits to the BA data, a discussion is given of the energy-level spectra for the p-shell hypernuclei, of isomeric states for jHe* and jLi*, and of the resulting predictions for the properties of the A hypernuclei corresponding to core nuclei assigned to the pllZ shell.

1.

INTRODUCTION

In this paper, we describe a phenomenological analysis of the (1 binding energies

BA for the A hypernuclei whose core nuclei belong to the nuclear lp shell. This analysis is based on the assumption that, for mass number A, these /I-hyper-

445 Copyright All rights

0 1972 by Academic Press, Inc. of reproduction in any form reserved.

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nuclear states are well described within the shell-model configuration {(l~),~(lp)~~~(ls)~}, appropriate for the simplest intermediate-coupling model for them. The basic formulas for the elements of the energy matrix for the n hypernuclear states with mass number A, isospin T, and spin-parity (J, P), have been derived and discussed in a previous paper1 [I]. The basic states formed from the {(l~),~(lp)$-~} nuclear configuration were taken from the phenomenological intermediate-coupling calculations made by Soper [2, 31 and others [4, 51, in which the parameters of a simple NN potential are adjusted to give a good overall fit to a variety of data available on the properties of the ground state and low-lying states of the p-shell nuclei, as discussed briefly in Chapter 2 below. The wavefunctions thus obtained for the low-lying states are then used as the nuclear base states for the /I-hypernucleus calculation, the energy of excitation for each nuclear state being taken to have its empirical value rather than the value obtained from the nuclear intermediate-coupling calculations. In the shell-model configurations for the (1 hypernucleus of mass number A, the contribution of the dN forces to the B,, values depends quite generally on just six parameters, B(5) which measures the fl interactions with the closed shell (1s) N4, taken together with the kinetic energy of the LI particle in the state (1~)~ , and the five possible two-body matrix elements between the /I particle and the p-shell nucleons. It is convenient to discuss the problem in terms of a particular set of parameters, namely B(5), V, d, S+ , S- , T, as discussed in Chapter 2 of Part I. The latter five parameters correspond in turn to the spin-independent energy, the spin-spin energy, the symmetric (+) and antisymmetric (-) spin-orbit energies, and the noncentral spin-spin energy arising from clN interactions of tensor form. Their use is purely phenomenological in spirit and does not necessarily correspond to any particular assumptions about the existence of static /IN potentials. However, for assumed clN potentials, the calculation of the corresponding values for these parameters 7, A, S+ , S- and T has been given in Appendix B of Part I. As discussed in the Introduction to Part I, there are powerful theoretical and experimental reasons for believing that three-body (1NN forces may play a significant role for a (1 particle in interaction with a nucleus. The contribution of such /INN forces to the BA values for (1 hypernuclei would depend on a rather large number of disposable parameters, if they were treated in the completely phenomenological spirit adopted for the AN forces; indeed, there would be considerably more parameters needed to specify the ANN contributions to BA than there are known A hypernuclear species at present. In this situation, we have decided to confine attention to the particular ANN potential forms which arise 1 This previous paper will henceforth be referred to as Part I. Equations contained in it will be specified in this paper by adding the prefix I to the equation number; thus, for example, Eq. (I 2.2) will indicate Eq. (2.2) of Part I, and Eq. (I A.4) will indicate Eq. (A.4) contained in Appendix A to Part I.

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for the double-OPE2 processes between the fl particle and two nucleons. One great simplification which results from this assumption is the quite general feature that the (1NN forces are then independent of the n-spin on . Even with this simplification, there are still five terms possible; the corresponding energy matrix elements were discussed generally for this case in Chapter 4 of Part I, and were specified there by five parameters denoted by the symbols Qfm, where (Mm) take the values (000), (022), (122), (202) = (220), and (222). With these /INN forces and the general specification for the /IN forces, there are thus 11 parameters in all, whereas the established /I hypernuclear species listed in Table I of Part I are also 11 in number. In view of the relatively large uncertainties given for some of the BA values listed, there is no hope of determining all these parameters empirically in any meaningful way. With this situation, it is really necessary to limit the number of disposable parameters, by adopting some specific hypotheses which are motivated by other physical considerations bearing on the nature of LlN and /INN forces. In Part I, we discussed the various AN-force terms which result from the OBE3 processes possible, and we may be guided in our choice of parameters by the dominant features found for these terms, as we shall discuss below. As for the (1NN terms, the simplest reasonable hypothesis is that their spin and orbital dependence is that given by the double-OPE term, after the singular short-range terms contained within its general expression have been eliminated (since this elimination will occur in the physical situation as a result of the strongly repulsive short-range core known to occur in the NN forces and expected to occur for the /lN forces also). The dominance of this particular term appears reasonable, both because of the long range associated with the OPE process and because of the great strength known for the z-NN and nLY’l interactions. With this term alone, the Qfm have a definite ratio, calculated in Part I and given in Table V there. To specify the Qfm , it is then sufficient to give the value of one of these parameters and we shall choose this to be Qz, . Wherever the symbol Q* appears in Fig. 1, this indicates, by convention, that all five QFm are contributing in the following ratio:

= 0.0259 : 1 : -0.4823

: -0.0446

: 0.2131,

(1.1)

the value of Qi, then being the value given there for Q*. It is convenient to relate this convention with the parameter C, introduced by Bhaduri et al. [6] to charac* OPE denotestheterm“one-pion-exchange.” The double-OPE processes are those corresponding to Fig. 3 of Part I. 8 OBE denotes “one-boson-exchange.” 595/7+-Io

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terize the overall strength of these (1NN forces. Using the last column of Table V of Part I, and the definition C = 3C,/(4+$) following Eq. (I-4.4), we have

,

Qi2 = -0.73C,

MeV.

(1.2)

A theoretical expression has been given for this parameter C, by Bhaduri et al. [6] on the basis of the static theory of the pion-baryon interactions, and has been reproduced in Eq. (I-3.37), from which they obtain the theoretical estimate C, = 1.43 MeV. Here, however, we shall regard C, as a parameter to be determined phenomenologically. Most of the analysis to be described here was made with the 1968 3,, data, as listed in Table I of Part I. This data includes the (I-hypernuclear species ,6He. During the course of the analysis the 1969 data listed in Table I of Part I became available; in particular, this includes improved BA values for ‘jB, ‘jB, and ‘jC. Since it appeared undesirable to have to recommence the analysis whenever the input data was changed, the analysis was completed using the 1968 data. Subsequently, some analysis has also been made using the 1969 data, but excluding the case of jHe; this case will be referred to as “New Data” and its analysis will be discussed only briefly in the closing Chapter 5 below, since the results were found to be in qualitative accord with those from the previous analysis. The exclusion of the species ,6He from the New Data was made on the grounds that the shell model may be expected to be an especially poor approximation for this case. The value B,(6) = 4.28 * 0.15 MeV is given relative to the lowest level of 5He*, which occurs at 0.95 MeV relative to (N + 4He). Since B,(5) = 3.08 MeV, the p-shell neutron of ,6He is bound by only B, = 0.25 f 0.15 MeV, and its wavefunction will extend out to greater distances than does the p-shell nucleon wavefunction appropriate to our shell model. In fact, these shell-model formulas do have difficulty in fitting both jHe and ,6He, generally giving too high a value for B,(6), an error in the direction expected from this argument. Calculations specific to the case of ,6He have been made by a number of authors, especially by Lovitch and Rosati [7] who treated it as a three-particle system (4He + N + (1), and by Ananthanarayanan [8], who used the Bethe-Goldstone procedure for two particles N and /I moving in the presence of a closed shell of nucleons, within a harmonic-oscillator framework. The former calculation aimed directly to predict (or interpret) B,(6), whereas the latter took into account particularly well the N/l spatial correlations and is more reliable as concerns the energy-level spectrum of jHe than for the total separation energy B,(6). However, the inclusion or exclusion of ,6He has a relatively weak effect on our analysis, as we have found by trial, since it is rather generally a poor fit when it is included. In Part I, we gave some theoretical discussion of the possible OBE origins for the various f’IN potential terms. Here we collect together the theoretical estimates

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for the phenomenological parameters, for orientation concerning their order of magnitude. The value estimated for r depends particularly on the assumptions made, for example on the shape and intrinsic range assumed for the spin-independent part of the /lN potential. If we assume the (1-4He interaction to be characteristic of F, and assume the spin-independent flN potential to have Gaussian shape with intrinsic range b, then, knowing the size and shape of 4He from electronscattering data, we can deduce a corresponding estimate for F, following the discussion in Appendix B of Part I, we have b = 1.52

2.0

v = 1.55 + 0.08

1.76 * 0.09

2.35 1.87 * 0.09

fm MeV.

(1.3)

The intrinsic range parameter b is not well known for the clN interaction. By fitting the energy dependence of the total cross section u(dP), and assuming that the s-wave flP interaction has no spin dependence, Londergan and Dalitz [9] have deduced the best fit b = 2.05 + 0.3 fm, for a Gaussian shape. This estimate leads to V = 1.78 & 0.13 MeV. However, if we adopt the viewpoint that p for jHe is strongly affected by (1NN forces, then we can only estimate it from the cross sections o(flP). To do this requires additional assumptions. One simplifying assumption which is compatible with the data at present is that of equality for the singlet and triplet /lP s-state interactions. This assumption leads to a relatively large value for I? With the intrinsic range b = 2.05 & 0.3 fm for a /lP interaction of Gaussian shape, the scattering data requires a flP potential strength [notation as in Eq. (I-B.5)] given by zl = (27 & 7) MeV4 [9], which corresponds to the estimate V = (2.5 * 0.2) MeV, from the expression (I-B.6). However, as discussed in Part I, all the llNN interactions which involve one of the s-shell nucleons also contribute to E with the double-OPE (INN interaction, this contribution is given by (-XnS), where X,, is given by Eq. (I-D. 18b). The net estimate for V is then V = (2.5 & 0.2) - 0.325 C, .

(1.4)

Next, we consider the spin-dependent terms. The spin-spin difference A is not well known. There are now some theoretical reasons (dominance of 2~-- or o-exchange in the flN interaction, cf. Refs. [IO--121) and some experimental indications (from the analysis of rlP scattering data [13]), which favor the assumption that A should be small. However, this conclusion is not firm, and the most direct way to determine A would be by the observation and measurement of the 4 Note that this error is strongly correlated with the error in b, and that these two uncertainties tend to cancel in the estimation of r.

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y-ray* emitted from the excited states jHe* and jH*. The absence of any clear evidence for this y-ray may be due either to a low efficiency for the formation of these states in the K--capture reactions studied, or to the smallness of A, since the y-ray would be particularly difficult to detect if the excitation energy were less than 0.3 MeV. The AN tensor forces are expected to be relatively ineffective since their dominant contributions arise from the exchange of K-mesons and heavier mesons and so have correspondingly short range. The estimates given in Part I lead to T = f0.05 MeV; although such a value would be relatively unimportant for the lower p-shell A hypernuclei, we recall from Table III of Part I that T contributes with a large coefficient for p-shell core nuclei heavier than 12C, which implies a corresponding sensitivity to values of T of even this low magnitude for the BA predictions for these heavier p-shell A hypernuclei. The spin-orbit terms may generally be expected to be quite significant, on the basis of the magnitudes known for the vector-meson couplings to baryons. Quite comparable estimates have been obtained [14] for a variety of sets of coupling parameters which have been proposed in the literature. Here again, we recall that all the (double-OPE) ANN interactions which involve one of the s-shell nucleons contribute (+X,,J to S- , where X,,, is defined by Eq. (I-D.19) and has the to s, 2 and (-X,,,) numerical value (-0.13 C,) MeV. From Part I, the net estimates are S, = (-0.3

- 0.13 C,) MeV,

(1.5a)

S- = (f0.1

+ 0.13 C,) MeV.

(1.5b)

We note that this theoretical estimate for AN spin-orbit matrix-element S, is comparable, in magnitude and sign, with the corresponding estimate5 of (-0.45) MeV for the matrix element of the NN spin-orbit interaction between an s-shell and a p-shell nucleon. To summarize, our present theoretical views about the origin of the AN interaction would incline us to expect rather small values for A and T, and appreciable values for both S+ and S- , together with a significant contribution from ANN three-body interactions. * Note added in proof: Recently, Bamberger et al. (34) have reported the observation of y-rays of energy 1.09 and 1.42 MeV emitted following the nuclear capture of K- measons coming to rest in BLi and ‘Li. These authors attribute the 1.09 MeV line as definitely due to y-decay following the formation of either A4H* or A&He*, and the 1.42 MeV line as possibly due to this origin.

5 This rough estimate is obtained from the N-“He spin-orbit potential deduced by Sack et al. [15] and the P-We spin-orbit potential deduced by van der Spuy [16], from the corresponding differential and polarization angular distributions. The estimate is made by calculating onequarter of the expectation value for this empirical spin-orbit potential (WQ * L exp(-A?)) for a p-shell nucleon with the harmonic oscillator wavefunction (I 1.3b), which gives the result (w/~)(v/(v f A))5/Z. The roughness of this estimate for S+(NN) is already indicated by the fact that the parameter values of Sack et al. give -0.55 MeV, whereas the parameter values of van der Spuy give -0.35 MeV.

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Our procedure will be to find the parameters describing the clN and (1NN interactions which give the best fit to the observed ground-state BA values, by the least-squares procedure of minimizing x2, for various constraints on these parameters. In order to assess the relative importance of the various parameters for obtaining a good fit, these constraints will be relaxed gradually, allowing more and more parameters to adopt their optimum values. The detailed procedure by which this minimization is carried out will be described in Chapter 2. The pattern of best fits obtained in this way is summarized in diagrammatic form in Fig. 1, and the outstanding features of this pattern will be discussed in detail in Chapter 3. A small number of best-fit solutions are selected for their goodness-of-fit and/or their general attractiveness in terms of the theoretical notions summarized above. These solutions will then be discussed in particular detail in Chapter 4, giving some features of their energy matrices, the form obtained for the fl-hypernuclear wavefunctions for some hypernuclear species of particular interest, and the pattern of excited states predicted for these species. Since some of these solutions are almost diametrically opposed in some qualitative features, it is of interest to compare their predictions, especially for cl-hypernuclear ground-state spins, for their spectra of excited states, and for the unknown species for which the nuclear core is in the plia shell. The final chapter will include a discussion of the effects on this analysis of replacing the 1968 data by the New Data, and of some of the uncertainties involved in using these analyses for extrapolating into the realm of prediction. 2. THE CALCULATIONAL PROCEDURE In Chapters 2 and 4 of Part I, we derived formulas for the dN and (1NN interactions in the nuclear lp shell. These have been written down in terms of an L-S coupling basis for the nuclear wavefunctions. On the nuclear-structure side, it is known that neither a simple L-S or a simple j-j coupling scheme is adequate to describe the nuclei of the lp shell. An intermediate coupling scheme is necessary, reflecting the comparable importance of both central forces (which alone would give L-S coupling) and spin-orbit forces (which would lead to j-j coupling on their own). Much work has been expended on this scheme [2-5, 17-201, and we shall assume that the low-lying states of all the nuclei in this region can be described by a simple version of it. The central potential used in the nuclear calculations can be described by five parameters, in terms of which the two-body (diagonal) nuclear matrix elements may be written V(T&L,)

= f(T&(L

+ a(L,)K).

(2.1)

The f(r,&) are four parameters describing the exchange character of the potential, while L and K are (direct and exchange) radial integrals reflecting its depth,

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range and shape. The quantity a(L,) has a geometrical origin; its values are 2, -3 and -1 for L, = 0, 1 and 2. Analysis of the Li isotopes with this model [2-51 gives for f( T&J, f(0, 0) = -0.50;

f(0, 1) = 1.00 (normalization);

f(1, 0) =

f(1, 1) = 0.15.

0.46;

(2.2)

Here as elsewhere in this paper we adopt the convention that a positive sign indicates attraction. The radial integrals L and K and the strength parameter of the spin-orbit potential are then determined by fitting the spectra and the reaction and electromagnetic properties of individual nuclei throughout the shell. Generally speaking, good fits are obtained and we feel that the uncertainties in the nuclear wavefunctions that we use are probably small compared with the other uncertainties of the problem. Within the intermediate-coupling scheme itself, the simplest assumption that we could make would be that the ./I was simply coupled on to the various nuclear states without perturbing them in any way. The result would be a series of doublets (for JN # 0) whose energies would be given by the diagonal matrix elements of the A-nucleus interaction. While it may prove ultimately that this is a valid model, it would be dangerous to incorporate this assumption (that the parent nuclear states are unperturbed) from the beginning. The /I-nucleus forces are strong, and a given hypernuclear state of spin J(#O) is coupled by these forces to all the nuclear states of both nuclear spins JN = J rt +. We have no a priori knowledge of the strength of this coupling, since we shall be varying the flN forces freely in order to fit the experimental results. While it might be true (for example) that the lowest J = 1 state in .Li8 may be well represented as a (1 particle coupled to the JN = $ ground state of Li7 for some values of the /IN interaction parameters, it is certainly not true in general as we vary the parameters. In many regions of parameter space the JN = 4 excited state of Li7 at 477 KeV will be a strong component of this state. We therefore do not make this “unique parent” approximation, but allow for mixing in the hypernuclear state of as many of the low-lying states of the core nucleus as we can. Physically, of course, this corresponds to allowing the clN interaction to polarize the core nucleus within the chosen space. The computational procedure appropriate to the above ideas is the following one: (i) The expressions for the matrix element of our ten rlN and /INN potentials in an L-S coupling basis are combined to form ten matrices for every possible spin for every hypernuclear species (known or not) that lies in the p-shell region and is likely to be of any interest. This was done once-for-all and the results stored on magnetic tapes.

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(ii) These matrices are then transformed from their L-S coupling basis to the intermediate coupling basis, using the calculated eigenvectors of our nuclear states. Once we have chosen a set of intermediate-coupling wavefunctions this step can also be done once-for-all. It has two advantages; firstly, the nucZear interactions are diagonal in this scheme, and we are able to correct our nuclear model by inserting observed values of nuclear excitation energies in place of the ones given by the model. Secondly (and more important), we can truncate our problem by retaining only the low-lying nuclear states lying within 10 MeV of the ground state; these are the only ones likely to be important for the low-lying hypernuclear states. (iii) We now have the following situation: We have a number of parameters [our two-body matrix elements, F, A, S, , S- and T, the three-body integrals Ql”m , and B(5), which represents the kinetic energy of the A and its interaction with the nucleons in the 1s shell]. We call them x, , and we choose “suitable” starting values for them. Since the x, specify the problem completely we can at once calculate and examine the whole spectrum of low-lying states of every hypernuclear species corresponding to our initial choice of x, . In particular we can determine the starting values of the spins of the hypernuclear ground states. The energy-fitting procedure to be described next assumes the ground states to have these spins and uses the corresponding energy matrices. We check for consistency later. (iv) We wish to fit a set of observed hypernuclear binding energies {(B&}. The theoretical values are the lowest eigenvalues & for each corresponding energy matrix,

where the Mel are the matrices described under (ii). Let the corresponding vector be V, . Then Ek = V&,V,

eigen-

(scalar product) (2.4)

= 1 x,Cka (say), 01 where c,a = V&f,“V, We want to minimize

(scalar product).

(2.5)

the quantity

x2= c ~d@J,- JW, k

(2.6)

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where the W, are a set of weight factors that we may choose as we like. We have ‘3 and then, for the minimum ax,=

(2.7)

x2, -2 c w,

cka = 0.

k

cw

Define D@ by Duo = c C&“CkB.

(2.9)

k

We then have C DsBxB= Au,

(2.10)

B

where Acr = c

Ck”

Wk@A)k

(2.11)

k

These equations have a solution X,

= C (D-l)mBA6 B

(2.12)

which should normally give a better fit to the data than the original X, . If the problem were strictly linear this would in fact be the exact minimum; this is not the case since the vk change whenever the x, do. However, we can usually iterate quite quickly to a minimum. It sometimes happens that the indicated change in x, gives an actual increasein x2; in this case we keep the indicated direction but halve the step length until we obtain a decrease. (v) We have now fitted the lowest hypernuclear states with the starting values of J to the observed values of Bn . In doing so we have altered the x, and these states may no longer be the calculated ground states of their particular species. We therefore examine again the spectrum of every hypernucleus used in the fit. If its calculated ground-state spin is not the same as its starting value, we repeat the iterative procedure (iv) and again examine the indicated ground-state spins, and so on. In this way we almost always reach a stable state where the final J values are the same as the starting ones. We call this a consistent solution. We shall return briefly at the end of the chapter to the case where we do not find a consistent solution [section (viii)].

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(vi) We have discussed the calculation so far in terms of independent parameters x, . In fact the X, are eleven in number, and an unrestricted fit to the eleven known BA values would be practically meaningless. We shall wish to constrain the x, in various ways; we saw in Chapter 4 of Part I that we could estimate the ratios of the Q% for a particular model, and in the next chapter we shall want to investigate the effects of fixing the values of various terms in the two-body /1N potential. To achieve this we allow up to nine general linear constraints between the possible eleven parameters, in each of which some linear combination of them is set equal to some constant. These constraints are taken by the computer, orthogonalized to each other (by a Schmidt process) and normalized. The Schmidt process is then used to construct a further set of othonormal vectors to fill out the eleven-dimensional parameter space. The energy matrices are then transformed using this orthogonal matrix, and the transformed matrices corresponding to the jixed combinations are combined to form a single matrix. The minimizing procedure described above is then carried out, varying the remaining independent parameters in this transformed parameter space, and everywhere adding the constant energy matrix corresponding to the fixed parameters. At the end of the process we transform back to our original parameters. (vii) When we have reached a minimum in x2 for any particular set of constraints and checked that it is stable, we then print details of all the low-lying levels of all the possible hypernuclei of interest. These details include the wavefunction (i.e., the amplitude with which each of the parent nuclear states enters the hypernuclear state, which tells us the degree of polarization of the core nucleus), the energy (as a number), and finally the energy expressed as a linear combination of the eleven potential parameters; this latter is a very useful aid in discussing the influence of various parameter changes on different binding energies and in gauging the importance of the various contributions to these energies. (viii) One final use of the linear constraint facility may be mentioned in this connection. If we do not obtain a consistent set of J values as described in (v), we must find ourselves in a recurring cycle of sets of J values. If the number of independent parameters is allowed to grow too large, quite long cycles can occur, but in the calculations described in the next chapter the only cycles which occurred were of length two. These situations we call b&able. A typical case would involve .Li7 (the usual culprit) where J = 4 and J = g compete to be the ground state. We would find that a fit of the lowest J = 4 to the observed BA would give parameters that made the lowest J = g come a few hundred kilovolts below it, and vice versa. We observe that neither of these states is a solution to our problem. The “nearest” solution is some intermediate set of parameters which will make the J = Q and J = g states degenerate. We can easily achieve this by applying a further

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constraint to the problem. We have the energy of these two states expressed as linear combinations of the parameters; equating them gives the necessary constraint. Normally it is a very gentZe constraint, in the sense that it is almost satisfied by the parameters of each of the bistable states. In theory we should have to iterate to obtain degeneracy, but in practice this is not necessary as the first approximation is very close. We should remark here that the x2 value given by this “nearest” solution is necessarily larger than the x2 values given by the two minima between which the bistable iteration oscillates. (ix) In the great majority of cases we find two solutions (each either consistent or “stabilized bistable”) for any given set of constraints. The prototype is the familiar situation where we try to fit the data with central two-body forces alone, and find a solution both for positive A and for negative d. Such a multiplicity of solutions will occur in any nonlinear situation of this kind (note that the important nonlinearity is not the relatively small one discussed in (iv), but the sharp changes in the x2 surface that occur when levels cross and the ground-state quantum numbers change as we vary the parameters). There is no general way of ensuring that we find all solutions, and a systematic search through the whole parameter space for every set of constraints we use would be prohibitive. We have searched extensively in some cases but found only one new solution (discussed in Chapter 3 below); we often iterate through a large range of parameter values to reach the solutions we have, without finding others. In a few cases, however, our iterative procedure has led us to local x2 minima which correspond to x2 values lying far above the x2 minima we shall discuss in this paper; in these cases the physically interesting solutions with low x2 minima were reached very quickly by modifying the starting values x, for the iterative procedure. We feel it is reasonable to suppose that we have found all the important minima, even though we cannot guarantee it.

3. THE NUMERICAL

FITTING

OF THE B,, VALUES

For the numerical analysis of the empirical BA values to be described here, the goodness-of-fit for the calculated values E(k) was measured by the expression (2.6) with weight function WV = (4W2,

(3.1)

where o(k) denotes the empirical uncertainty in the BA value determined for the A-hypernuclear species labelled k. Thus, the function minimized was the conventional “chi-square,” given by x2 = ; Wk)

- B&)“l(4kN2.

(3.2)

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Some of the a(k) values listed in Table I of Part I are rather small, especially for :He(0.02 MeV), iBe(0.04 MeV), jLi(O.04 MeV) and ;Li(O.O5 MeV). Indeed, for these species, the BA value is known empirically to an accuracy far greater than the simple shell-model calculations described in this work could possibly be expected to achieve. It could therefore be argued with some force that the function (3.2) gives undue weight to the fitting of the BA values for these particular species. On the other hand, it is far from clear what would be the most appropriate choice for the W(k). One possibility among many would be the use of a weight function such as J+‘(k) = M(~W

+ c+,Y,

(3.3)

chosen to avoid such undue weighting for the well-established species, the parameter u0 being chosen to give some measure of the accuracy with which these theoretical calculations for BA(A, T) can be carried out. To avoid the additional freedom such a choice as (3.3) would introduce into the BA analysis, we decided to retain the conventional choice (3.1) for W(k), but it then follows that it would be unreasonable for us to expect that even the physically correct values for the AN and ANN parameters should give a value for x2 lying near to the number of degrees of freedom, s = H - P, where H = 11 denotes the number of hypernuclear species included and P denotes the number of independent parameters determined by the B,-fitting procedure. The flow pattern of the minima found is shown in Fig. 1. In each box, all of the parameters which are not held zero are specified [except for the parameter B(5), which always has value very close to the BA value for jHe], and the corresponding ($)min is given in the lower part of the box. When the number of parameters is small, say B(5) and V only, the minima are deep and the parameter values rather sharply determined; however, the (x2) min value is then correspondingly large [628 for the fit with only B(5) and v] and the fit obtained is correspondingly poor [with B(5) and V on 1y, we are attempting to fit the data on Fig. 1 of Part I by a single straight line, whereas the purpose of the present work is to attempt to explain the observed deviations from such a mean line]. Figure 1 shows how the solutions move as one after another of the parameters are allowed to vary freely from zero. Generally speaking, the arrows shown on Fig. 1 indicate that the minimum specified in the box at the tip of the arrow may be reached by our minimization procedure, starting from the parameter set given in the box at the other end of the arrow and taking the parameters now allowed free in this step to have starting values zero. The only exceptions to this remark are the pairs of arrows leading from the box (628) for which only B(5) and V are free, to the boxes for which in addition either A or (S, , X) are allowed free. Clearly, a definite set of starting values, with a definite set of parameters released, can only lead to one definite minimum; the explanation is that the two minima, with d 2 0 for

458

GAL,

SOPER

AND v=,

-

”

=

= l-04

iv,

,v

go=-0 32 148 .

v

DALITZ

t _ ii=170 A=-004 a$ -0 31

= I3O--,V S+=O28 s-=0-43

s+=o27

o&=-O-IS v = I71 (J*=+oo

33

= I-12/ A =0.30

s-=051 I

cg+l’2 30-O

I-31

A =-0. I3

FIG. 1. The AN and ANN potential parameters (and the x2 corresponding to them), as obtained by minimizing the expression x2 for the A-hypernuclear separation energies given as “1968 Data” in Table I of Part I, under various sets of constraints on these parameters. In each box, the parameters whose value is not specified [excepting B(5)] have been constrained to remain zero during the minimization of x2 with respect to the parameters whose optimum values are specified; the only exceptions to this remark are the entries Q*, for which cases all five Qt;, were included but with the constraint that they should have a definite ratio (l.l), so that they are all specified by the value Q* given for Qi, and by these fixed ratios. For each minimum, the x2 value is given in the bottom section of the box. The notation (b) after (x2)min signifies that the minimization procedure leads to a bistable oscillation in the iterative sequence; the significance of the parameters then given in this box is discussed in Chapters 2 and 4.

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

II

459

S* = 0 [or (S, + S-) 2 0 for d = 01, both exist but are both reached only when both positive and negative starting values with a substantial magnitude (&l MeV, for example) are adopted for d (or for &). The minima corresponding to a large number of free parameters can generally be reached through a number of paths starting from the primitive solution for which only B(5) and Vare nonzero, which is reassuring. The only exception to this remark has been displayed explicitly in Fig. 1. The two minima in the bottom right-hand corner, for which x2 has values 13.3 adn 22.9, have qualitatively similar values for B(5), v, d, Si , T and Q*; the parent minima, that with B(5), 7, d, S* , and T free and (x2)min = 13.4 and that with B(5), 7, d, S* , and Q* free and (x~)~I~ = 33.7, have a qualitative similarity, but when Q* is released for the first minimum, x2 falls from 13.4 to 13.3, whereas when T is released for the second minimum, (x2)min falls from 33.7 to 22.9, thus reaching a different minimum for B(5), v, d, S+ , T and Q* free, a local minimum with (~~)~u, appreciably larger than that for the qualitatively similar minimum just mentioned. It must be emphasized that very many searches were also made starting from sets of parameter values chosen randomly. Almost all of these searches led back to minima already known from the paths starting from the primitive solution and shown on Fig. 1. However, one disconnected set of solutions was found in this way, displayed in the sequence of boxes on the far left of Fig. 1, the solution found having nonzero parameters B(5), v, d, Sh and Q* with (~~)~n, = 19.3. By starting from these parameter values, but with Q* constrained to value zero, the search procedure led back to a new minimum for B(5), Jz, d and S+ free, with (x7 mm = 21.2, which had not been reached from any of the minima flowing from the primitive solution nor found as a result of our random searches with B(5), r, d and S+ free. By relaxing the parameters T and Q* from value zero, these disconnected minima led to the further disconnected minima shown in Fig. 1, reaching the disconnected minimum with (x2)min = 14.1 for B(5), F, d, S* , T and Q* free. We shall comment on the characteristics of these solutions towards the end of this chapter. For the present, the interesting point is that none of the parameter sets connected with the primitive solution through the network depicted in Fig. 1 leads to any of these disconnected minima. Four bistable solutions were found in these searches. These are indicated by the notation (b) placed to the right of the (x2)mrn value given in each box of Fig. 1. Their characteristics are as follows: (i) The tninittum (x2)min = 64.6. This minimization was found to oscillate between two configurations which differed in the spin values found for the ground states of the species(A, T) = (7,O) and (8, Q). One configuration had .7 = =&for (7,O) and J = 1 for (8, $), with (~~)~u, = 48.7 and d negative; starting from this configuration, the minimization led to the other configuration, with J = $ for

460

GAL,

SOPER

AND

DALITZ

(7,O) and J = 0 for (8, $) with (X2)mtn = 62.3 and L3 positive; and vice versa. The minimization procedure was stabilized by requiring that the J = ; and # states of (7,O) be degenerate. This stabilized minimum has a higher (X’)mtn , of course. It has d small and negative, and happens to give J = 1 for (8, -A), as is observed empirically [21, 221; however, this last point is not significant since the J = 0 state for (8, &) then lies very low in excitation energy (~0.06 MeV), rather close to the ground state. (ii) The minimum (X2)mrn = 22.9. This minimization was found to oscillate between two configurations which differed in the spin values found for the ground states of the species (A, T) = (6, Q), (7,O) and (12, 3). One configuration had spin values J = 2, #, and 2 for these species, in turn, with (X2)min = 20.4 and d = 0.39 MeV; starting from this configuration, the minimization led to the other configuration, with spins J = 1, 4, and 1 for these species, in turn, with undergoing much (x2>mm = 20.8 and A = 0.55 MeV (the other parameters smaller changes); and vice versa. The minimization procedure was stabilized by requiring that the J = 4 and 3 states of (7,O) be degenerate. The stabilized minimum has A = 0.47 MeV and happens to give J = 1 for both (6, &) and (12, &), although the J = 2 states lie rather low in excitation energy, at 0.12 MeV for (6, 3) and at 0.07 MeV for (12, $)); for (8, $), the ground state has BA = 6.70 MeV and J = 0, the J = 1 level lying at excitation energy 0.65 MeV. (iii) The two minima with (x2)min = 11.2. These minimizations were found to oscillate between two configurations differing in the spin value found for the ground state of the species (7,O). One configuration had J = 4 for (7, 0), with (X2)min = 5.8 and Y = 0.57 MeV, A = 0.60 MeV and T = -1.30 MeV; starting from this configuration, the minimization led to the other configuration, which had J = # for (7,0), (x2)min = 10.2 and r = 0.70 MeV, A = 0.89 MeV and T = -1.02 MeV (the other parameters undergoing smaller changes); and vice versa. This minimization was stabilized by requiring that the J = + and $ states of (7, 0) be degenerate. The stabilized minima are given in Fig. 1; they give J = 1 for the ground state of the species (8, $), the J = 2 level lying at excitation 0.17 MeV. More complicated minimization situations, which involved a repetitive cycling through a number of minima differing in the spin values found for some number of hypernuclear species, were found in the course of this work but, fortunately, they connected local minima which lay rather high in x2 relative to the minima arrayed in Fig. 1. Apart from the four bistable situations just discussed, the minima found which were of physical interest were all simple. We shall first discuss a series of special solutions included among those shown in Fig. 1. With purely central AN forces, the most general situation has only the parameter A, beyond the primitive solution. There are two minima found for

2.95(i)

4.52(l)

6.04(i)

4.89&

6.60(l)

6.78(;)

7.76(s)

8.25(i)

9.48(S)

10.12(i)

10.65($)

(6.;)

(7,O)

(7,l)

(8,$

(9,o)

(Y,l)

iO,$

11 ,O)

12,;)

13,0)

B*(J)

(A+;26l)

(5~0)

(A,T)

lpecies

I

0.1

68.5

34.4

1.2

7.5

60.8

6.6

1.9

70.1

3.7

6.6

x2

7.9

4.67(2)

0.2

8.11(S)

11.38($

lCL78(2)

9.73#

2.9

5.2

16.5

0.9

IL.5

6.62($)

8.32(2)

30.9

29.6

6.45(2)

5.91($

37.7

2.2

2.97(s)

5.88(3)

x2

B*(J)

10.25(i)

0.3

20.3

0.7

0.18@

0.53(l)

0.6

0.0

8.430)

8.20($)

6.3

1.4

6.67(l)

6.57(s)

8.4

10.6

5.68(i)

5.52(i)

15.1

2.5

x2

4.85(i)

2.97(i)

B,,(J)

(5:&*;66.2)

0.1

0.5

1.1 0.0

0.45(3)

6.2

0.1

12.4

0.3

0.2

0.3

lo.gr(2)

0.65(g)

8.63(Z)

7.64(s)

6.4?(i)

6.71(l)

5.00($

5.50(5)

14.3

2.99($) 4.83(2)

X2

B,,(J)

(A+S-;35.5)

0.67(&)

3.85(l)

0.55(5)

3.65(l)

8.02@)

6.52($)

6.66(l)

4.95&

5.45($)

5.05(l)

2.99&

B*(J)

0.1

2.6

3.1

0.1

1.0

1.3

1.8

0.9

0.0

25.8

0.1

X2

0.26(;)

0.69(z)

0.61($

8.54(2)

8.19($)

6.53&J

6.67(2)

5.37($)

5.40(Z)

4.95(Z)

2.99($

B,,(J)

S-&+;42.8)

0.3

9.3

5.0

0.3

0.0

7.1

I .5

3.6

~ 11 I 1 0.4&

0.78(l)

0.45(S)

8.54(l)

8.20(q)

6.52($)

6.69(l)

5.42(i)

5.38(;)

4.98(l)

0.4

2.99(;)

0.4

B*(J)

20.0

x2

(S+Q*;36.1)

0.0

5.3

1 .I

0.3

0.0

I.&

0.5

4.7

0.9

21.5

0.4

X2

10.37(i)

10.85(2)

10.63($

8.59(2)

7.82(5)

6.50$

5.69(l)

0.1

2.7

5.6

0.2

5.5

0.5

0.8

1.6 0.0

5.54(i)

15.6

0.3

X2

5.09($)

4.86(Z)

2.49(i)

B*(J)

(A+S-Q*;32*)

10.50($

10.81(l)

10.51(s)

8.59(l)

8.12($

6.51($)

6.63(l)

5.20(i)

5.43(5)

5.02(l)

2.99(;)

B*(J)

0.0

4.1

2.2

0.2

0.1

1.2

1.1

0.5

0.1

24.0

0.3

X2

(A+S+Q!:337) I f

i

0.0 0.0 0.3

1 i.O4(1) 1D.P,O(~)

0.3

3.6

0.1

0.1

0.4

0.2

ti.2

0.2

X2

10.27(s)

8.55(l)

7.99(s)

6.1+8($

6.71(i)

5.18($)

5.48(Z)

4.83(l)

7.99($)

B*(J)

(A-S+Q*;i9.3)

The Bn and spin values calculated for the ground states of the p-shell n hypernuclei known empirically are tabulated for the dN and &IN parameters appropriate to seven minima selected from those shown in Fig. 1 (i.e., for the 1968 data), as function of mass number A and isospin Z’. For each of these minima, the contribution to (~&i~ from the fit obtained to each hypernuclear species individually is also

TABLE

462

GAL,

SOPER

AND

DALITZ

this case, shown on the line of boxes immediately below the Y box. Both provided poor fits to the Bn data, as was already well known from earlier attempts [23] to fit the B., data with central LIN forces alone. Their points of failure can readily be seen by looking at the contributions to x2 from the individual species (A, T). These are given in Table I for the solution with positive d, which gives J = 1 for the species (8, 4) and (12, a). There are three large x2 contributions, for the species (7, 0), (9, 0) and (12, +); at the same time, this value of d gives much too great a separation between the Bn values for (7,O) and (7.1). This is in accord with the earlier observation [23] that the p-shell BA values could only be accounted for with central forces if d could for some reason increase progressively through the p shell. The central solution with negative d fails in the same way. Another solution of interest is that with the parameters B(5), v, d, and Q$ only. It has been pointed out by Gal [24] that a central three-body /INN potential with spin-isospin dependence given by ((I~ * 02~l * ~~~ would give especially low BA values to the (1 hypernuclei with a 4n nuclear core, in the p shell. He suggested that this could account for the ((9, l)-(9,0)) BA difference, and for the relative values of Bn for (12, $) and (13,0). In fact, Table I shows that the solution with B(5), v, d, and Q& nonzero does give quite a good fit to the data on these points. Actually, the optimum value of d for this solution (0 = -0.04 MeV) is not significantly different from zero, so that we have not ascribed a definite sign to Ll in specifying this solution; even small positive values for d [giving J = 1 for the ground state (8, &)I would give equally acceptable x2 values. As shown in Table I, the large x2 for this solution arises primarily from the species (7,0), (7, 1) and (8, +), the calculated BA values being too large for the A = 7 species and too low for the A = 8 species. A minimization was also made allowing all five QFm to vary freely, together with B(5) and p, as shown in Fig. 1. This 7-parameter fit has no special merit; the x2 value is not especially low among 7-parameter fits. Also, there is no simplicity in the optimum parameters, for at least three of the QL contribute significantly and in a way which is very far from that given by Eq. (1.1). Minimizations were also made allowing just two of the Qfm to vary from zero, and including S+ together with B(5) and I? By far the most favorable of this set of 5-parameter fits was that obtained with only Qi, and Qi, nonzero, giving (~~)~i~ = 18.1; the parameter values were r = 1.55 MeV, S, = 0.76 MeV, Qg2 = -2.84 MeV, and Qi, = 1.66 MeV. The interesting point is that this optimum fit has /1N parameters not unlike those calculated for the double-OPE process, which gives Qi, and Qi, dominant and opposite in sign, as indicated in Eq. (1.1) above. Another minimization giving a low x2 for a small number of parameters is that shown with (x2)min = 66.2 on the first row below the primitive solution in Fig. 1, the parameters being S, and Ql”, in the fixed ratios (1.1) in addition to B(5) and I? This solution reproduces the B, difference for the species (9, 1) and (9,O) satis-

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

II

463

factorily, the difference being (0.33 S, - 0.55 Q*), therefore due primarily to the QFm ; the major difficulty is that Bn for (12, 8) is given significantly low, relative to that for (13,0). However, the introduction of S, did lead to a significant improvement, (~~)~u, then falling from I39 to 66.2. This improvement is due to the A-spin dependence of the S, interaction, which allowed the B,, values calculated for (8, 4) and (9, 1) to increase towards their empirical values; without S, , the hypernuclear states (JN i +) for core nucleus spin JN + 0 are degenerate, since neither V nor the QTm in the ratios (1.1) have any dependence on the A spin. The introduction of the further parameter S- then reduces (x2)min to 36.1 and gives a very satisfactory solution, as we shall discuss below. From this point on, we shall discuss only minimizations which include both S+ and S- , and for which the AN parameters QI”,, bear the ratios (1.1). Returning to Eqs. (2.9) and (2.14) of Part I, the spin-orbit terms take the form <# I c {G(s,, + SN> . ZN + ff(s,, - SN) . M I #i N

= <# I (G -

ff) c&v

* SN) + (G + ff) S,I . L I ti),

(3.4)

N

where the first term does not depend on the A spin (and therefore has the same value for the hypernuclear states JN f A), and the second term couples the A spin with the total orbital angular momentum L in the A-hypernuclear state 4. From the discussion in Chapter 2 of Part I, it follows that the first term contributes to BA with coefficient proportional to (S, - S-), and the second term contributes with coefficient proportional to (S, + S-). In that discussion, we remarked [Eq. (2.17) of Part I] that, with j-j coupling, the first term would contribute -n(S+ - S-)/2 to B, for n nucleons in the p3,2 shell; if this were precisely the situation, the parameter (S, - S-) could not be distinguished from v, since the B, data available at present is confined to nuclei which would be assigned to the pZi2 shell, and the parameter combination (S, - S) would therefore be redundant. In Table II, we have given the coefficients of (S, & S-) and of Q* [= pi, , the other Q,,li bearing the ratios of Eq. (1.1) to it] in the approximation that the core nucleus wavefunction is simply that for the ground state in the intermediatecoupling approximation, as function of the hypernuclear species (A, T). From this table, we see that the coefficient of (S, - S-) is by no means proportional to IZ through the p3,2 shell, not a surprising conclusion in that it is well known that the coupling is more nearly L-S than j-;j at the beginning of the p shell; j-j coupling is a fair approximation in the p shell only above about r”B. We note also that the coefficient of (S, + SJ is zero for spinless core nuclei, but that it has appreciable values between these cases, always with the same sign all through the p shell. The coefficient of Q* rises through the p 312 shell, rather faster than linearly; in the upper part of this shell, Q*/n has mean value about 0.2 MeV. Hence Table II 595/72/2-I

1

II

Q*

n(S+- s-)

0, + s-1

(A, T) =

0.260

-0.121

(-0.012)

0.023

(730) 0.825

(8, i)

0.198

-0.410 0.612

-0.170

0.000 (-0.495)

(7,l)

-0.3475 0.540

1.104

(-0.555)

0.832

(9,l)

-0.1855

0.000

(930)

1.111

-0.280

(-0.498)

0.914

(lO,B,

0.931

-0.421

(-1.0805)

1.441

U1,O)

1.323

-0.406

(-0.598)

0.997

(l&g

2.017

-0.370

o.ooo

(13,O)

2.236

-0.296

(-0.335)

1.004

(1434)

3.079

-0.191

(-0.686)

1.371

(15,O)

-

4.544

-0.091

(-0.333)

1.ooo

(16,&

The coefficients with which (S+ + S-), n(S, - SJ, and Q* contribute to the net BA value are tabulated as function of the hypernuclear species (A, T) in the approximation that the A particle is coupled only to the ground state of the nucleus (A-l, T), this ground state being described by the intermediate coupling shell-model calculations referred to in Chapters 1 and 2 of this paper. Here n = (A-5) denotes the number of p-shell nucleons. For (S, + S-), the upper value given is for the spin state J = JN - 3, (JN # 0), the lower value in brackets being for the spin state I = (JN + 3).

TABLE

I.

c x

P

5;1 g u

E

9 r

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

11

465

tells us that the slope of the B, curve in the upper half of the p3/2 shell may be roughly approximated by (V - 0.4(S+ - S-) + 0.2Q*) MeV/(unit The outstanding follows:

step in A).

(3.5)

features of the pattern of solutions given in Fig. 1 are as

(1) The optimum S, and S- almost always have the same sign. The only exceptions are the three parameter sets with (x2jmi, -= 13.4, 13.3, and 22.9, given in the right-hand bottom corner of Fig. 1. However, S, is essentially zero in the first two cases, and achange of S, to a small positive value would not affect their (X2hn . In the third case, S, has a larger negative value, but (S+ + S-) is still positive, as for all the other minima in the bottom right-hand corner of Fig. 1; this minimum is not an important one [because its (x~)~I~ lies so much above 13.31 and it will not be referred to again in this paper. In order to identify a solution we need therefore only specify the sign of S = (S, $- S-). (2) The minima occur in pairs with respect to A. For every solution with a positive A, there occurs a corresponding solution with a negative A, all the remaining parameters having the same signs and roughly the same magnitudes for the pair of solutions. The negative values for d generally have magnitude of order twice that for the corresponding positive value of d. Generally speaking, the spin assignments are (JN + +) for d < 0, and j (JN - $) 1 for A > 0, and the two minima then correspond to the choice of one or other of these two sets of spin values. (3) The minima occur in pairs with respect to S = (S, , SJ. Here again, this corresponds roughly to the two sets of spin values, corresponding to the two orientations of the coupling sA * L, whose coefficient is (S, + SJ; if S, and Sdiffer appreciably, the values of r will be significantly different for the two minima in order to keep the same mean slope (3.5), although this relationship between V and (S, - S-) may also change in consequence of a simultaneous change in the magnitude of Q*. (4) For the few cases in which T # 0 is permitted, the same as that for (S, + S-).

the sign of T is generally

(5) The sign of Q* is generally negative, corresponding to net repulsion. [In Fig. 1, there are two exceptions to this observation, namely the minima given by the two boxes in the central and right-hand positions in the bottom row; however, in both cases, the value of Q* is small and its introduction has produced little effect on (x2)mtn-for the right-hand box, the introduction of Q* has reduced (x2)min only from 13.4 for Q* = 0 to 13.3 for Q* = +0.17-the effect of Q* is

466

GAL,

SOPER

AND

DALITZ

far from dominant for these cases.] This results from two effects: (a) the sign and magnitude of the Bzl difference between the species (9.0) and (9.1), since this is accounted for dominantly by the Q* term, and (b) the concavity downwards for the B-, curve (excluding ,iBe) as a function of A, since the coefficients of Q* show a concavity upwards, tabulated as function of (A, T) in Table II. (6) When the number of parameters [including B(5)] allowed to be nonzero exceeds five, there are no minima for which both d < 0 and (S, + X) < 0. Starts from the five-parameter minimum x2 = 50.3, for which d < 0 and (S+ + S-) < 0, lead to the minima x 2 = 32.8, for which Q* # 0, A > 0, and (S, + S-) < 0, and x2 = 13.4, for which T f 0, A > 0, and (S, + S-) > 0, as shown in Fig. 1. Minimizations were also made starting from many other sets of parameters for which A < 0 and (S, + S-) < 0, but none of these led to a minimum with both A < 0 and (S, + SJ < 0. In the discussion below, we shall adopt the following notation to specify a particular minimum. In brackets, we shall write each of A, S, T and Q* only when the parameter is not held zero, followed by a semicolon; the symbols B(5) and V will not be written, since they always vary freely. To the symbols A and S, will be added, according to the sign of the correa superscript “+” or “-” sponding parameter [(S+ + S-) in the case of S] for this minimum; this superscript is not necessary for Q*, which is always negative when it is significant. Following the semicolon, we write the (X2)min value inside the bracket, thus

(A*S*TQ*;

(x2)min>

specifies uniquely the depth and nature of the minimum considered. Although the inclusion of (x2)min in this notation is to some extent redundant, it is a quantity which allows one to pick out rather quickly which minimum is under consideration. We now discuss briefly the properties of the solutions corresponding to the various minima shown in the flow pattern of Fig. 1. For a small number of free parameters, the minima are deep and well defined; however, the (x2)min values are then usually unreasonably large. When the number of free parameters is large, the (~~)~r~ values are frequently quite reasonable in magnitude, but the minima are then poorly determined; the x2 surface is then rather flat, with small local hollows here and there. For example, the fit given by the minimum (kS+TQ*; 14.1) is barely more significant than that given by (A-SfT; 16.1) yet the parameter values change appreciably when Q* is introduced, r from 0.70 to 0.90 MeV, A from - 1.64 to - 1.20 MeV, and Q* from zero to -0.62 MeV. Many of the parameter sets given by these minima are quite unreasonable. For example, the minimum (A-S+; 21.2) has a much lower (x2)mrn than any other 5-parameter fit, yet it is not to be preferred relative to them since the physical

SHELL MODEL FOR P-SHELL HYPERNUCLEI.

II

467

considerations summarized in Chapter 1 here make it unreasonable to accept the parameters B = $0.77 MeV, d = - 1.70 MeV at their face value, since we have no doubt that the clN interaction is strongly attractive in the lS, state, from the evidence of the B‘, value for :H. This remark applies to all of the minima with A :O. The most acceptable minimum is (S&Q*; 36.1). It is a 5-parameter fit which has a (X”)min which is quite low among 5-parameter fits. The value P = 1.44 MeV has a reasonable magnitude, as do S, and S- . The value for Q* corresponds to C,, = 2.72 MeV, only about twice the theoretical estimate given by Bhaduri et al. [6], hence quite an acceptable value. We note from Table I that 21.5 is contributed by ;He to (~~)~r~ in this solution; if this specieswere omitted in this analysis,Gfor the reasonsdiscussedin the Introduction here, this minimum would give (~~)~i~ = 14.6, which would appear reasonably acceptable for (10 - 5) = 5 degreesof freedom, even on an absolute scale. This minimum also gives the spin 1 to the ground states of :Li and YB, in fact generally the value J = I(JN - +)[ for the ground-state hypernucleus. If the parameter A is included, the resulting minimum remains an attractive solution; V falls to 1.15 MeV, the optimum A is not unreasonably large, and the Q* required falls by a factor 2. For this minimum (A+S+Q*; 33.7), the x2 obtained after excluding >He is only 9.6, now for 4 degrees of freedom, so that this set of parameters also appears quite reasonable, on all grounds, and they will both be discussedfurther in the later chapters. For completeness and contrast, we shall also discussthe two minima, (S-Q*; 42.8) and (A+S-Q*; 32.8), for which S, and S- have reversed sign relative to the two minima just discussed. For the minimum (S-Q*; 42.8), the exclusion of :He reduces (x2)minfrom 42.8 to 22.8; this minimum gives J = (JN + 4.)for all the ground-state hypernuclear spins in the pal2 shell. The spin situation for the minimum (A+S-Q*; 32.8) is less transparent, since the positive sign for A favors the spin state I(JN - $)I and the negative sign for S favors the spin state (JN + 4). As shown in Table I, the net effect is that the hypernuclear spin is (JN + 4) for 9 < A < 13 (and also for A = 6), and I(& - +))I for 7 < A < 9. For this minimum, (x2)mi, drops to 17.2 for 4 degreesof freedom, when :He is omitted from the analysis. The minimum (A-S+Q*; 19.3) is also an opposite to (A+S-Q*; 32.8) as regards reversal of sign for A and S. It has a much lower (x2)min , and this falls from 19.3 to 5.1 for 4 degreesof freedom when ;He is omitted. It also gives J = I(& - $)I for the ground-state hypernuclear spins, except for ,?,Li. It has a large negative A, sufficient for this parameter set to be excluded on physical grounds; however, we shall include it for contrast in our later discussion, especially because of its 6 We note that the x2 values given here and below, obtained by subtracting out the x2 contribution obtained from the fit to ,BHe, are upper limits on the value &z),i, for the true minimum corresponding to the optimum fit for the set of A hypernuclei excluding ,GHe.

468

GAL, SOPER AND DALITZ

peculiarities as a member of a disconnected branch found in random searches and as an especially deep minimum among those parameter sets including Q*. As we shall find that all parameter sets including Q,“, or Q* have some common characteristics as the hypernuclear mass number moves up into the pl,z shell, we shall also consider the minimum (d+S+; 36.8) and its S-reversed counterpart (d+S-; 35.5). When ,6He is excluded from the analysis, these (X’)mrn values fall from 36.8 to 11.0, and from 35.5 to 21.2, respectively, for 5 degrees of freedom. The minimum (A+&‘+; 36.8) gives J = I(& - $)I for the (I-hypernuclear spins. Again, A+ and S- favor opposing tendencies for the ground-state spins for the minimum (A+S-; 35.5), and the situation is not transparent; the result is (JN + 4) for 9 < A < 13, and also for A = 6, and otherwise I(& - +)I, thus giving the physically observed spin value for jLi.

4. DISCUSSION OF SOME SELECTED MINIMA In this chapter, we will discuss the hypernuclear states in greater detail for the parameter sets corresponding to the seven minima discussed in the closing sections of the last chapter. From a theoretical standpoint, the most interesting parameter sets are those corresponding to the related minima (S+Q*; 36.1) and (A+S+Q*; 33.7), and to a lesser extent, the minimum (A+S+; 36.8). These minima give acceptable x2 when the species ,6He is omitted, and they require J = 1 for jLi (as is observed), and also for ‘:B . The other four of these seven minima represent opposites to these three minima, in some sense, and they are included in the discussion primarily for contrast. Table III is a counterpart to Tables VI and VII of Part I. It gives the coefficient of A, S+ , S- , T and the Qfm for the ground state (spin J = j(JN - &)I) for the same series of p-shell hypernuclear species, for the minimum (A+S+Q*; 33.7). These coefficients show a good qualitative similarity to those given in the tables of Part I, which were calculated using the nuclear wavefunctions obtained from the intermediate-coupling studies of nuclear properties, without allowing for the possibility of distortion of the nucleus by the presence of the il particle. For the large coefficients, the change is generally less than 10 %; for the smaller coefficients, the change is rarely more than a factor of two (signs are all in agreement). Table IV displays the wavefunctions for all levels of :Li below 6 MeV excitation energy, according to the calculations for the selected minima. This case is of particular interest, not only because the ground-state spin has been determined but also because its lowest J = I state involves a strong admixture of the JN = + first-excited state of 7Li. In the L-S coupling approximation, the JN = 2 and JN = 4 states of ‘Li both contribute to the J = 1 jLi wavefunction, the coupling amplitude of the JN = $ state being (- I/x~). For the three favored solutions,

Qb’, QZ, Qi, Q,“* Qi, (Q*)

T

S+ s-

A

(A, T) = .I=

2

(7$)

0.9615 -0.0152 -0.2780 -0.8637 0.8956 0.3551 0.0256 -0.0490 1.4658 1.3557 0.3158 0.2678 0.2797 0.2536 -0.0326 0.1321 0.0122 0.0989 0.2355 0.1832

(7,O) if

(9,O) i

0.3008 0.0021 0.3766 -0.8387 I .4692 0.8431 -1.0808 -0.0003 5.0338 9.7790 0.5473 1.0957 0.1723 0.6474 0.1229 0.1263 0.0645 0.1975 0.6028 1.0732

(8, $) 1 1

W,$) (14,# 0

(15;O) 2

-0.0353 -0.3342 -0.4295 -3.2930 -1.7034 -0.499 3.3650 3.8719 3.358 -0.0823 5.1715 7.231 15.7311 17.8811 21.856 2.7392 3.1813 3.795 3.4889 3.5345 2.883 1.6976 1.2476 1.015 2.1191 1.2185 0.354 1.8398 2.1438 3.001

(13,O) 4

-0.250 0.000 2.000 6.000 25.981 4.648 1.610 0.000 o.ooo 4.544

0

5 B 5 o pK =

8 ; $ E

B F

Ul,O) g

U6,+)

(10, & I

E p P

to the net B4 value are tabulated as function of the hypernuclear species wavefunctions calculated here for the AN and (1NN parameters correthe coefficient given is that appropriate to the hypernuclear ground state, $)I in all cases for this parameter set.

III

0.5291 0.2746 0.1884 0.4968 -0.5041 -0.4960 -1.0584 -1.7632 2.4142 2.4406 4.0554 3.8805 -0.5397 -0.8400 -1.9271 -1.3801 5.1255 9.8237 10.7163 12.9249 0.6555 1.2415 1.3922 2.0440 0.6488 1.0182 1.7369 2.5943 0.2649 0.2060 1.1248 1.4292 0.2858 0.3822 0.9789 1.4166 0.5244 1.0771 0.9769 1.3792

(9, 1) -g

The coefficients with which d, S+ , ,S’- , Tand the Q’E;,contribute (A, T), for the intermediate-coupling shell-model hypernuclear sponding to the minimum (d+S+Q*; 33.7). For each species, which has spin J = / (JN -

TABLE

470

GAL,

SOPER

AND

TABLE

DALI-l-2

IV

For the hypernucleus ,RLi, and for each of the seven selected parameter sets, the excitation energy E*, the spin value J, and the wavefunctions are given for each state lying below E* = 6 MeV, together with the BA value calculated for ground-state (IRLi. Hyp. ext. energy E* (MeV)

(Minimum) BA MeV @l+s-; 35.5)

Parent nuclear level-(&)&*(MeV) Spin JP

($)O

(#0.48

@6.6

($)*8.20(*)*8.40

og.;sb 3.41 4.02

l2lO-

0.488 1.000 0.868

0.873 -0.487 0.991

& 1.68 1.89 6.00

lo2l2-

0.999 0.999 0.039 -0.004

-0.039 1.000

g.s. 0.19 1.81 2.02

2llo-

1.ooo 0.373 0.926

-

0.928 -0.372 -1.000

El 0.98 1.23 5.55

lo2I2-

0.963 0.999 0.268 -0.002

-0.268 -1.000 0.963 -

32.8)

g.s. 0.14 2.84 3.36

l2Io-

0.477 I .ooo 0.875 0.997

0.879 -0.476 -

(d+S+Q*; 33.7)

g.s. 0.56 1.31 1.50

lo2l-

0.991 1.000 0.132

-0.132 -1.000 0.991

0.008 -

-0.002 -

0.031 0.039 0.005

-0.005 0.006 0.002

K; 1.20 2.67 5.64

lo2l2-

0.867 0.997 0.497 -0.014

-0.496 -1.000 0.868 -

0.041 0.458

-0.010 0.886

0.016 0.066 0.008 0.071

-0.033 0.020 0.013 -

6.71 (d+S+; 36.8)

6.66 (S-Q*; 42.8) 6.67 (S+Q*; 36.1)

6.69 (d+S-Q*; 6.69

6.68 (A-S+Q*;

6.71

19.3)

0.999 -

-0.027 -

($)*7X

0.003 0.996 -0.014 0.013 0.372 -0.025 -

0.008 -

0.017 0.000 0.073 -

0.007 0.059 0.136

-0.001 0.085

0.031 0.034 0.001 0.019

-0.002 0.016 0.020 -

0.004 -

0.020 0.010 0.047 -

-0.021 0.032 0.002

0.030 -0.004 0.046 0.012 0.928 0.010

-0.005 0.025 -0.015 -

0.016 0.008 0.007 0.067 -

0.000 0.053 0.079

-

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

TABLE

471

II

V

For the hypernucleus ZB, and for each of the seven selected parameter sets, the excitation energy E*, the spin value J, and the wavefunctions are given for each state lying below E* = 6.5 MeV, together with the En value calculated for ground-state 1,2B. P arameter

set

( g.s.B*(MeV))

T

Hypernuc1ear

E*(MeV)

-L-

( A+S-;35.5) (10.91)

9.8. 1.86 4.18

( (10.69)

(10.78)

( A+S-&';32.a) (10.85)

JP

$)O.O() (,

zI-

0.977 0.680 0.652

l-

g.s. 1.71

l-

1 .ooo

2-

0.999

3.06 4.01

Ol-

0.014

4.57

2-

.0.012

g.s.

2-

1.73

lI-

0.993 0.922 .0.366

3.14 4.60 5.32

(:S+Q*;36.1)

(JN)~:

0.125

6.44

(10.85)

-

T

2O-

5.17

( A+s+;36.8)

state

g.s. 1.09

O2l-

0.995 0.9~6

3.70 4.89

2-

0.063 .0.055

g.s.

2-

0.902

1.82

l-

3.83

l2-

0.746 ,0.606 0.118

2.78

5.24 5.86

-

O-

1 .ooo

0.264 -

0.947 -

0.014

0.013

0.009 0.024

-

-0.020

1 .ooo 1.000

1 .ooo

0.972

0.994 0.990

0.624

-

1.000

0.266 -

5 - 0.040

0.998! 0.998

(Ic.aj)

2O-

3.84 4.75

l2-

0.039 .0.034

0.996

g.s. 1.26

I2Ol-

0.970 0.969

0.183

2-

0.099

0.998

0.129

-0.084 -

0.778

l-

4.37 5.00

0.965

-

0.997 0.910

-

-0.064

-0.141

0.085

-0.186

0.145 -0.075

0.033

-

0.006

0.003 0.013 0.002 -0.007

-0.050

-0.065 0.004 -0.087 -0.132 -0.113 0.156

0.063

0.214

-0.147 0.109

-0.042 0.950

-0.049

.0.119

-0.103 O.O@O

-0.031

0.040

0.031 -

0.012 -

0.991

-0.054 -0.067

0.107

-

-0.114 0.081

0.177

0.133

k&.30

-0.020

-0.067

0.026

-

(ij7.56

0.015

-0.082 0.066

0.203

0.045

,

-0.110

0.006

-

1.40 2.91

2.50

-0.045 -

state8

-0.131

-0.020

0.997

0.066

"B*

0.023

0.928

g.s.

(11.04)

-0.122 0.177 0.078

(’ 6+S+Q*;33.7)

(’ A-S+G*;IT.3)

-0.096 -

0.359

parent

I:# 6.6C

0.692 0.716

0.086

2O,-

for

-

-0.391

0.909

-0.362

0.092

0.089 -0.117 -0.037 -

472

GAL,

SOPER

AND

DALITZ

those for which neither S nor d is negative, the coupling still has some resemblance to this extreme approximation, but the amplitude of the coupling of the JN = + state is considerably smaller, ranging from -0.039 tot -0.268. The three minima with S < 0 give dominant coupling to the JN = + state of ‘Li for the J = 1 state of ,SLi, with amplitude from 0.873 to 0.928, even though two of them still predict this to be the ground state; the minimum (A-S+Q*; 19.3) gives J = 1 for the ground state, with a coupling quite close to that for the L-S coupling limit. The lowest J = 2 state of jLi has a rather pure coupling to the ground state in all cases, owing to the high excitation energy (6.6 MeV) for the first excited state (JN = $) with which the A particle can couple to give J = 2; more generally, these higher ‘Li states contribute only very weakly to the lowest three states of jLi. E”( Mev)

\

\ .I-

+ /

\

\-

/

\ \

.

‘,

7‘

\

-\

Bn(MeVl

= IO.91

IO 69

IO 85

FIG. 2. The energy level spectrum for ZB is shown for each of the seven selected parameter sets. The central axis gives the scale for the excitation energy E*, in MeV; for comparison, the energy-level spectrum of the parent nucleus I18 is shown by the black spots on this scale.

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

11

473

The relative positions of the energy levels vary quite widely even as we go from one favored solution to another. For these solutions, the first excited state of :Li generally has J = 0, with the excitation energies calculated lying between 0.24 and 0.95 MeV. Another species of particular interest is “B n (see Table V), for which there is some hope for a spin determination in the near future, from the study of the angular correlations in the decay sequence ‘jB + 7~~ + Y*, lzC* + 301, passing through the 12.71 MeV level of 12C. All of these minima with S > 0 give J = 1 for YB. Except for the case (d-S+Q*; 19.3), the ?B ground-state wavefunction is completely dominated by the ground state of the parent llB nucleus; even for this exceptional case, the ljB amplitude for coupling to the first-excited state of llB is still only -0.183. For the other three minima, J = 2 holds for the ‘jB ground state; in this case, there is a rather pure coupling to the llB ground state, since the lowest excited state of llB which can couple with the II particle to give J = 2 lies at 4.46 MeV. The spectrum for each of these seven minima is given in Fig. 2. We see that the four 5’ > 0 solutions give qualitatively similar spectra, the sequence of levels being the same and their excitation energies quite comparable. The species :He is of special interest because of the suggestion by Pniewski and Danysz [25] that ,7He may have an excited state which is isomeric, preferring to decay by hypernuclear modes rather than by y-emission to its ground state. Our calculations for this system are summarized in Table VI. For all of these parameter sets, there is a strong splitting between the J = 4 and J = + states derived from the JN = 2 6He* parent state with excitation energy 1.71 MeV. For the minimum (d+S-; 35.5), the spin-orbit coupling favors the state (JN + $) and is especially strong, so much so that it drives this J = Q state below the J = 4 state of ,7He normally expected to be the ground state. Otherwise, J = 4 does hold for the jHe ground state but the first-excited state IHe* then lies rather low in energy, with excitation ranging from 0.06 MeV to 0.52 MeV. For all parameter sets with S > 0, the low-lying iHe* state has J = $ and can in principle decay to ground state :He by Ml y-emission through the JN = 1 6He* (5.0 MeV) admixture in both of these states, a mechanism discussed by Dalitz and Gal [26]. However, these admixtures are large enough to allow this J = -$ Ml decay to compete successfully with the weak hypernuclear decay for this state, only for the case (d-S+Q*; 19.3); in the other three cases, especially for (d+S+; 36.8), the J = g level will show up as isomeric, at least to some extent. For all parameter sets with S < 0, the low-lying iHe* state has J = g and is isomeric; in these cases, the J = $ state lies high (2.5-3.0 MeV) and will decay rapidly by Ml y-emission to the J = s state. In all cases where the fit displayed leads to an isomer iHe*, the excitation energy for this isomer does not exceed about 0.5 MeV, certainly far from the value of about 1.7 MeV obtained assuming a weak spin-dependence for the /IN interaction. The sensitivity of the position of

474

GAL,

SOPER

4ND

DALITZ

TABLE VI For the hypernucleus JHe, and for each of the seven selected parameter energy E*, the spin value J, and the wavefunctions are given for each state lying together with the Bn value calculated for ground-state iHe. Also tabulated with which S, , S- , and T contribute to the separation energy BA* = (BA -

(~.s.BA(MeV))

'Z*(MeV)

JP

(0)O.O

(A+E- ;35.5)

g.s.

22+

-

0.08

$+

0.987

1.85

J+

-

g.s.

;+

1 .ooo

0.42

6

-

2.58

52+

8.s.

$+

0.996

0.33

22+

-

2.68

12+

-

(5.00)

(n+s+;36.8)

(4.95)

(S-Q*;42.8)

(5.37)

2

sets, the excitation below E* = 6 MeV, are the coefficients E*) for each state.

(2)1.71

1-0.930 0.644

1

0.998: 0.596

-0.oy1

' 0.897

0.243

L

1

'I.347

C.72

-3.19

I

(5.42)

(A+S-Q*;32.8)

(5.09)

(A+S+Q*;33.7)

(5.20)

(A-S+Q*;lY.3)

(5.18)

B.S.

$+

0.?97

0.52

1;

--

-

0.995: 0.997

1.96

C.OR3

I

0.048

-0.09

-0 .Olci

-1 .m

0.071

-0.133

kT.8.

;+

0.06

2+ 5.

2.71

12+

~.s.

$+

0.9985

0.46

&

2.25

s+

g.s.

;+

0.908

0.43

&

-

0.983

a.143

2.24

2+

-

0.959

0.283

0.9895

-

+I.39

I

I

(S+Q+;36.1)

j

-C.&b

1 +0.92

0.63

+2.06

1 .Ol

-1 .bY

+0.06

0.18

-0.94

+O.EO

-

-0.983

0.185

-

0.83

-1.51

-0.33

-

0.695

0.446

0.565

0.48

-0.45

+0.51

0.027

-0.05

-0.86

+0.90

0.994:

.0.017

-0.027

-1.63

0.70

+2.09

0.997

0.076

-

-

j

I .Ol

0.133

-

4.119

/ -0.29 I /-2.05

-

/

0.30

-1.69

+0.07

-0.80

+O.VE

0.76 -1.69

12.02 +0.29

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

475

II

the jHe levels to changes in S, and S_ , and to the introduction of a tensor interaction T, are indicated by giving the coefficients for the contribution of each of these terms to BA . These coefficients are each calculated as the expectation value of the corresponding interaction operator for the optimum wavefunction calculated for that minimum; thus, these coefficients give the derivatives aB,jaT, aB,/ZS+ , and BB.,IBS_ for T = 0 and for S., , S- , taking the values appropriate to that minimum. The tensor interaction would have to be unexpectedly large if it were to modify this iHe* situation appreciably. The possibility of an isomeric state in :Li has also been discussed in the literature [27]. This requires that the hypernuclear forces should bring either the ($) level or the (S) level of JLi* derived from the JN -= 3 level of 6Li* at 2.18 MeV down sufficiently close to the ,?,Li ground state to slow down the E2 transition rate between the lower of the (3) and (f) levels and the (4) and (8) levels of ,7Li, derived from the JN = 1 ground state of ‘jLi, to the point where hypernuclear decay competes favorably with y decay. This turns out to be the case for every one of the parameter sets discussed here, although there are many differences E*(MeVI

B,(MeVl=

5.50

5.40

5.54

5.45

5.38

5.43

5.48

FIG. 3. The energy-level spectrumfor JLi is shown for each of the seven selected parameter sets.The centralaxisgivesthe scale for the excitationenergyE*, in MeV; for comparison, the energy-levelspectrumof the parentnucleusBLi is shown by the black spotson this scale.

476

GAL,

SOPER

AND

DALITZ

between the level spectra given by each of these minima. This range of variation results from the differing effects of d and S. For parent level JN , positive d here puts the j(JN - +)I spin lowest, and vice versa, and the splitting produced does not depend strongly on JN ; positive (S, + S-) also puts the i(JN - i)i spin lowest, and vice versa, but the splitting increases strongly with LN , the orbital angular momentum in the nuclear parent state, being zero for LN = 0. Hence the (s)-(g) splitting is small (due to LN # 0 admixtures in the JN = 1 ground state wavefunction) if L3 = 0, and the order of these two levels is determined primarily by the sign of d when it is nonzero. When (S+ + S) # 0, the (g)-(g) splitting is determined dominantly by S and the order of these levels reflects mainly the sign of S. These remarks are sufficient to explain the variations found in the level patterns shown in Fig. 3 in terms of the varying signs and magnitudes of d and S. The sensitivity of the energy-level pattern to small changes in S, , S- and T from their optimum values is indicated in Table VII by giving the derivatives awas, , BB,/BS- and aB,jaT at these optimum parameter values. Fast Ml transitions (/l spin-flip) will occur within the split doublet in each case, unless the splitting is less than about 0.1 MeV, in which case the two levels coincide, for all practical purposes at present. For minima with S > 0, the ($+) level is brought low in excitation energy. Accepting the decay rate (0.85 f 0.1) x 1Ol2 set-l for the E2 transition (3) + (1) in 6Li, which has been determined by Bernheim and Bishop l-281, we follow the estimate made by Pniewski et al. [27] to conclude that the ($+) state of jLi* will be isomeric if it lies lower than the (z+) state and less than about 0.8 MeV above the ground state. [The Ml transition (g+) -+ ($+) is very slow in view of the purity of the ($+) state in its coupling to the ground state of the parent nucleus “Li. Thus it may compete with the E2 transition (g+) + (3+) only for transition energies much smaller than 0.8 MeV, in which case the discussion is essentially academic.] We note that this condition is essentially satisfied for all S > 0 minima except (O-S+Q*; 19.3) so that isomerism may show up, to some extent. In this last case, the ILi ground state has (Q+) because d < 0 and the first excited state (at 1.35 MeV) has ($+) because (S, + S-) is positive and large, and the condition for isomerism is then correspondingly weaker [not by more than a factor 3, since for (g+) -+ (#+) there is another factor of + multiplying the factor of Qfound in Ref. [27] for the transition (g+) + (Q+)]; even here, the hypernuclear decays will be competitive with y-emission for the first-excited lLi* state. For minima with S < 0, the (z+) level lies low in excitation energy, and sufficiently low to be isomeric for each of the parameter sets under consideration here; the condition for isomerism is now that the (z+) state lies lower than the (++) state and less than about 0.8 MeV above (or lower than) the ($+) state [note that the electromagnetic transition (z+) -+ ($+) can proceed only through A43 or E4 multipoles, which can be neglected here].

TABLE VII For the hypernucleus JLi, and for each of the seven selected parameter sets, the excitation energy E*, the spin value J, and the wavefunctions are given for each state lying below E* = 6 MeV, together with the Bn value calculated for ground-state JLi. Also tabulated are the coefficients with which S, , S_ , and r contribute to the separation energy BA* = (BA - E*) for each state. aramet.eT

set

Bypernuclear state

(J&(W)

for

parent

6Li*

)2.18

1 .ooo

(2)4.52

-

0.549

S-&*;42.8)

(5.40)

s+c*;3fi.1:

(5.38)

A+S-c.*;32.8)

A+S+,';j3.7)

(1 .!.i>

A-s+;*;l9.3)

(5.43)

B.S.

$+

0.94

3

1 .I1

12+

0.999

3.88

;+

-

4.28

($+T -0.003

1)9.70

0.027

.0.021

-0.078

0.836

-

0.444

0.068

0.999 -

(1)5.5c

-

-0.033

(5.45)

C

eparati

(3

A+S+;36.8)

I-oefft

levels

-0.032 -1 .ooo

0.018

0.087 0.034

-

0.002 1.000

0.029

XI.026

T

-0.037

.o .004

0.21

-2.00

1.20

0.00

-0.37

0.05

-0.70

0.26

0.01 -0.27

-0.32

1.80

-0.66 0.13

0.29

1.20

0.00

2.19

0.80

0.03

0.25

1*+

0.995 0.997

0.80

$t

-

1.5io

-

7.55

z+

-

0.572

0.820

-

3.48

cg+,* -0.078

0.379

0.919

0.075

-0.14

-0.67

6.S.

;*

0.995

0.054

0.082

-0.26

-0.14

0.04

2+

0.997

0.056

0.062

-3.29

0.97

5+

-

2.hC

$+

-

3.24

g+,*

0.020

P.S.

$+

o.sgc,

C.88

2+ L

0.999

1.04

g+

-

2.42

s+

-

3.47

($+,*

o.rr4

E.C.

;+

o.yge

0.57

&

O.YS9

0.95

s+

-

-O.?%

3.23

;+

-

I.000

3.74

is+,*

O.OO?

B.S.

s+

1 .coo

1.35

3

-

1.81

;+

0.933

2.91

;+

3.147

~,J+i’

-0.006 -0.994 1 .LJ@

-0.21

-0.219 0.037

-0.70

0.29

-3.316 I

.0.025

1.99

0.00

2.20

0.94 0.23 0.30

-2.00

1.20

-0.70

0.28

-0.04

0.006

0.057

d.28

3.003

0.015

O.O!r7

-0.31

0.057

-

0.40

-

-2.00 .0.013 0.027

1 .YO -0.99 0.64

-0.124

0.093

-0.396

-0.048

0.917.

2.43

1 .:o

0.01

0.386

-

0.76

-0.34

0.894

0.005

0.35

0.049

0.441

o.co9

-1.76

0.00

-

0.310

-2.01

0.00 -0.72

-0.19

-

-0.119

0.11

0.27

,0.002

0.823

O.YY3

1 .@OO

1.25

0.46

-

-0.15

-2.00

-7.00

-5.009

-0.9'11

-

/

-0.29

0.040

C'.975

1 .%;I 0.568

-0.505

0.111

0.073

0.35 2.35

$+

0.067

-1 .a0

-0.10

0.06

0.074

0.31

-0.80

-1.67

g.s.

0.010

S-

0.07

-2.00

0.999

BA(MeV)

-0.18

0.36 0.030

of contributjon c>n energy

-0.76 0.03 -0.01

0.00 -0.74 -1.79 0.36 0.23

-1 .31

2.39

1.20

0.00

2.26

0.89

-0.10 -7.68

0.26 2.35

-0.37

0.33

0.56

-2.CO

1.20

0.00

2.05

2.33

0.88

to

478

GAL, SOPER AND DALITZ

We should note here that, for all these minima, the ground-state wavefunction for :Li is remarkably pure in its coupling to the ground state of the parent nucleus ‘jLi; this is simply a consequence of the fact that the first-excited state of ‘jLi* which can couple with the A particle to give the :Li ground-state spin is in no case lower than 4.52 MeV above the 6Li ground state. Finally, we consider the extrapolations to hypernuclei in the pljz shell given by our calculations for these parameter sets. The B, values calculated for the species with 14 < A < 17 are given in Table VIII for the two spin states I(& i- $)I for TABLE

VIII

For each of the seven selected parameter sets, the BA value and the coefficient with tensor interaction T contributes to this value are tabulated for both spin states J = as function of (A, T) for all the p,,,-shell hypernuclei. The BA values given here with represent the separation energies for the ground state of the hypernuclear species (A, priate for each parameter set under consideration. 1 (A,T)

(14,;)

=

(15,C)

(1-j (A+S-;35.5) T coefft

(A+S+;36.8)

c;+-+, ($+)

9.c4

u

9.20

Q&3

5.172

-0.917

4.861

-2.881

u

11.08

($+I

I

/

/

(16,;)

(17,o)

(0-l 1 (1-I

11.98

9.9%

Mu

0.460

6.000

-1.531

a.71

10.j6

6.0ao

-1.831

~

(;+I -

12.74

T coeff:

5.336

-1

(S-Q+;42.8)

9.96

r1.78

9.52

12.21

T coefft.

5.463

-1.327

7.312

-3.272

T coeff:

4.456

-1.998

(A+S-Q*;32.8)

9.21

it369

(15,l)

which the I(& * +)I underlines T), appro-

7.568 I

(S+Q*;J6.1)

u

Il.14

12.19

6.978

11.08 0.207

10.76

__ 11.01

-3.699

-I).136

10.40 6.000

9.51

u 7.60

-2.115

-

-2.082

-

-2.436

-

the seven parameter sets under consideration. There is a marked difference between the A dependence predicted for minima with Q* = 0 and for minima

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

II

479

with Q* # 0. For the minima with Q* = 0, the Bad values calculated for the JN = 0 core nuclei continue to increase with A as we go through the pliZ shell, reaching values of order 13 MeV for 20; the ground-state BA values calculated for JN # 0 core nuclei all lie well above this line. With Q* allowed to vary freely, all the minima have Q* < 0, with 1 Q* 1 quite large. The dominant terms are Qi, and Q& , which contribute with opposite signs, a negative Q!& corresponding to are also significant and contribute to a decrease in BA . The Qi, contributions decrease BA if Qi,, is negative. As A increases through the pi/a shell, the coefficients of the terms Qi,, and Qi, are positive and increase monotonically, whereas the coefficient of the opposite term Qi, is positive but falls monotonically, reaching zero at TO. The coefficients of Qi, and Q& vanish for both A = 16 and A = 17, but these are smaller terms which do not modify the qualitative picture appreciably, in any case. Since Qi, is large and negative, Q!& and Q& give an increasingly strong repulsion, whereas the attraction provided by Qi, falls away, so that these LINN forces lead to a strong repulsion in YO and therefore predict rather low B, values for ‘j0, the minima (S-Q*; 36.1) and (S+Q*; 42.8) both giving BA about 8 MeV. This effect is particularly marked for the closed-shell core nucleus 160. It will occur even if the LlNN interaction has only a term Qi, , for its coefficient. increases with A faster than linear for the pl,Z shell nuclei; however, the effect is more marked for the double-OPE (INN forces, because of the degree of cancellation between the Qi, and Qi, terms in the p 8,,2shell. For the minima which have nonzero d, S and Q*, the effect is less marked because the value of Q*, needed to follow the variations in the BA values observed for the p3,2 shell hypernuclei, is then about a factor of two smaller, in consequence of the additional flexibility introduced by the parameter d. The BA values predicted for 20 by the (dSQ*; x”) minima are in the region of 10-l 1 MeV, whereas the (AS; x”) minima give about 13 MeV. It is clear that the measurement of B, values for any hypernuclei near the upper end of the p1,3 shell would be very illuminating concerning the nature of the fl-hypernuclear forces. We note that the spin values predicted for the ground states of the p,,,-shell hypernuclei are correlated with the sign of S. It is clear from Table VI of Part I, and from Table II here, that the coefficient of (S, + S-) in the difference between the BA values for the states (JN + 3) and [(JN - +)I is much larger than the coefficient of d. With d alone, a positive d would give (JN + 4) for the hypernuclear ground state, and a negative d would give I(JN - i)i. When S, and S- have the same sign, as is the case for essentially all of our parameter sets, then the S term will generally dominate the d term in this BA difference, and it is the sign of S which will then decide which of these two states lies lowest, rather than that for d; (S, + SJ > 0 leads to J = I(JN - +)I, and vice versa. We have already referred to the sensitivity of the Bz, values for p,,,-shell hypernuclei to the value of the tensor interaction parameter T. To make these remarks .595/742-12

480

GAL,

SOPER

AND

DALITZ

quantitative, we have given in Table VIII the coefficient for the contribution of T to the BA value for the ground state of each prjZ -shell hypernucleus, for all seven parameter sets under discussion here. These coefficients are frequently quite large. For example, for the species (l&O), and for the minimum (O+S+Q*; 33.7), the tensor term contributes +7.2T to BA for the J = $- state, and -3.8T for the J = $ state; the two states are separated by 1.43 MeV for this parameter set, without T. A value for T as small as -0.13 MeV would be sufficient to invert the order of these two states, and to make the J = 3 the ground state of the hypernucleus; this value for T would also invert the order of the J = 0 and J = 1 states of the other two p,,,-shell species, (14, 3) and (16, 4). In fact, however, when the trial was made of allowing T to vary freely from the fixed value of zero which it had for the minimum (d+S+Q*; 33.7), T moved in the positive direction to the new minimum (Li+S+T+Q*; 22.9) for which T takes the rather large value +0.58, with quite large changes in all of the other parameters also. We should note here that none of the parameters A, S, , S- and T, for the spin-dependent interactions makes any contribution to Bn for ‘JO, a consequence of the fact that its nuclear core 160 closes the p shell. For the 4n nuclei in the p shell, we have the following BA expressions (in the no-polarization approximation for the core nuclei) for the Ll-hypernuclei (a) iHe, (b) jBe, (c) YC, (d) 20, in turn: (4.la)

B(5),

9.89Q:o + l.lOQ& + 0.57Qi2 + 0.21Q& + O.lSQ;, , (4.lb)

B(5) + 4v-0.74(S+-S-)

+

B(5) + 8 r-

+ 16.77Q:, + 2.8OQ;z + 3.04Q;, + 2.72QE2 + 1.75Qi2, (4.lc)

B(5) + 128

2.96(S+ -S-j

+ 31.18Q;, + 5.58Q;, .

(4.ld)

If we neglect three-body forces for the /I-nucleus interaction, we can use these expressions to determine B(5), Y and (S+ - SJ from the BA values. With the 1968 data, we have V = 0.80(*0.06) MeV, (S, - S-) = -0.385(FO.3) MeV, which then lead to the extrapolated value, BA = 12.55(&0.7) MeV for 20 [the New Data leads to BA = 12.0(*0.4) MeV]. This is not a satisfactory value for r, according to our discussion in the last chapter. However, when three-body forces are included, it is clear from the expressions (4.1) that BA for ‘JO may have any value, even when the three-body forces are parameterized by a single number [e.g., by taking some definite ratio for the Q& such as that given by (l.l)].

SHELL MODEL FOR P-SHELL HYPERNUCLEI.

II

481

5. DISCUSSION AND CONCLUSION

First we will discussbriefly the repetition of this analysis with the “New Data.” These fits have been carried out less extensively than those with the Old Data; the flow pattern for the minima is shown in Fig. 4. The structure and qualitative properties of the pattern are precisely the same as before. The (X’)min values are generally rather higher than in Fig. 1 because the experimental error is now smaller for many of the species,as discussedin the Introduction. From a theoretical

FIG. 4. The AN and ANN potential parameters (and the x3 corresponding to them), as obtained by minimizing the expression x2 for the -4-hypernuclear separation energies given as “1969 Data” (“New Data”) in Table I of Part I, under various sets of constraints on these parameters. In each box, the parameters whose value is not specified [excepting B(5)] have been constrained to remain zero during the minimization of x2 with respect to the parameters whose optimum values are specified; the only exceptions to this remark are the entries Q*, for which cases all five Qt;, were included but with the constraint that they should bear the definite ratio (l.l), so that they are all specified by the value Q* given for Q:, and by these fixed ratios. For each minimum, the x3 value is given in the bottom section of the box. The notation (b) after WUli, signifies that the minimization procedure leads to a bistable oscillation in the iterative sequence; the significance of the parameters then given in this box is discussed in Chapters 2 and 5.

I

6.80

6.63

8.25

8.G

10.19 11.06

11.32

CR,;)

(9,o)

(9,1)

(1%;)

(11 ,O) (12 $1

03,O)

0.15

I

ro.e14&)

Il.C4!2)

'.I;

.-.i;(;)

IL~i(p)

j

1 7.6:-(i)

r. a?(;)

6.73(l)

d.lC

c .2"

0.13

0.04

0.04

'I

lJ.3

I> , i

!7.1

\ .I

.O

1.9

3.c

0.0

?.f,cl($ i

O.!:_

5.09

(7,i)

n.0 4.3

5.68(k)

0.06

3.08($

x2

3.08

0.02

(o+s-;57.5)E

E!*(J)

I

(Q L:eV)

Dsta

5.56

BA(MeV)

Input

(7,O)

I

(590)

(A,'f)

SpSCi.SS

IX

;

I

I

10.9?($

'10.$4(l)

ic.y(<)

J,. 7 3 ( 1 1

i S.OB($

6.65.($)

.-.73(1:

5.05(g)

5.50(t)

3.08($)

B*(J)

(A+S+;31.7),

9.3

1 .I,

iL.2

: . :a

I .6

I

13.66(;)

lO.Bh(2)

ilO.62(;)

'3 . 6 J ( 2 :

P.:ii($)

/ 6.74(;)

j G.Cf'(>)

2.6

5.2

5.',;'(f)

3.07(i)

>.52if)

j

B*(J)

0.0

0..

0.0

x2

1 (S-u*;69.Y)x

I

19.2

3.9

in.5

c . :I

11.0

7.3

7.1

72.0

0.;

0.i

2

".52&

r..47($)

3.07@

E*(J)

(S+Q*;he.6)N

I

'

iC.73($)

10.93(l)

lO.:l($)

8.65(j)

3.:5(s)

6.71(i)

1 6.74(j)

!

I

li.L+

1 .E

1. .I

0.0

0.c

3.9

1.9

12.9

2.2

0.1

x2

I

I

($1

lC.7$)

11 .C1(?)

,c.<,,(;)

r .+-6c:)

7.7

F&C($)

6.77(?)

5.16($

5.69($)

3.07(3)

BiJ,

(A+;-~*;

jr.7

1" . j

:-.:

5. 0

1'1.7

I.

3.6

0.1,

5.1)

0.1

x2

56.3) N

I

I

i

T.'l(j)

r,-.cl.(,,

:-.<7&

".71,(1‘

?.erf;l

6. .'mf,y, 1,

(.73(i)

lr.ec(:)

CL'+)

3.C'1($)

B,,(J)

x2

".(

',i

:L.I

,..I;

:.j

: .!l

3.?

0.i

>.3

c.0

( i\+s+ C*;31.3)N

i .r.

'-4.

Ii".?'

LT

,I

-:

,::

7::

..'.-'._

1 m'.-.

I

4)

i, . 7 <, f + :

:.~:(Y>

..'?Cs!

3.i:

En(J)

(A-s+a*;14.")N

I,<,.

I

.c

2 . ,,

1:

.-

-.I

..:

i.1

0.9

0.3

c . i'

2

The E4 and spin values calculated for the ground states of the p-shell A-hypernuclei known empirically are tabulated for the AN and ANN parameters appropriate to seven minima selected from those shown in Fig. 4 (i.e., for the New Data), as function of mass number A and isospin T. For each of these minima, the contribution to (x2)~in from the fit obtained to each hypernuclear species individually is also given.

TABLE

I

I

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

II

483

standpoint, the most reasonable minima are (S+Q*; 48.6), and (S-Q*; 69.9), . The first of these has the lower x2, and has smaller (positive) values for S. However, the minimum with negative S is quite acceptable. All minima which allow d to vary freely require a rather low value for F, besides a large value for d. For example, when d is introduced, the first minimum moves to (d+S+Q*; 31.3), , for which V drops to 0.73 MeV, d rises to 0.85 MeV, and Q* changes markedly, from -1.86 MeV to 0.31 MeV; these central parameters require V,/V, M 0.4, an unreasonably low value. Again, there is a disconnected branch, starting with the minimum (d-S+; 14.6)N , giving by far the lowest (x2) among the five-parameter fits; however, for this branch, d is very large and negative, so that these parameter setsare physically unreasonable. Among the six-parameter fits, given in the bottom row, the minima which allow T to vary and require Q* = 0 give the best x2 values. The fit with d and S positive gives x2 = 2.7 for 4 degreesof freedom, but has a large value (+0.43 MeV) for T; this unusually low x2 value must be regarded as a matter of chance. The introduction of T into the minimum (Lil+S--;57.5), gives a remarkable change in x2, down to (~~)~r~ = 12.5, for quite small changes in the parameter values, apart from that for T, which moves from T = 0 to T = -0.96; qualitatively, this is the same behavior as for the corresponding minimum with the Old Data (where x2 drops from 35.5 to 11.2), but the drop is much more abrupt. This illustrates well how the minima become deeper and more well defined, when the experimental errors on the data points are reduced. The (x~)~I~ values for the most reasonable of these parameter sets receive their major contributions from the species (9, 1) [the calculated difference (BJ9, 1) - B,(9,0)) being too small] and/or from the species(11,O) [the calculated B;,( 11,O) being too high] and to a lesserextent from the species(13,0) [the calculated value Bn(13, 0) being generally too small]. The ground-state hypernuclear wavefunctions resulting from these parameter sets are qualitatively similar to those found using the Old Data. Apart from that for the species (8, &), these wavefunctions are generally dominated by the coupling to the ground state of the core nucleus; the small admixtures vary in magnitude from those found before, but they remain small. For (8, &), the minimum (d+S-; 35.5) found with the Old Data corresponded to a wavefunction whose amplitude was 0.488 for the 7Li ground state and 0.873 for the excited state 7Li* (0.48 MeV); these wavefunction amplitudes for the corresponding “New Data” minimum (d+S-; 57.5)N are 0.522 and 0.853, respectively. The stability shown by these minima with respect to this improvement in the experimental data is very reassuring. With the Old Data, there were two particular minima which fitted rather well with our theoretical notions about the origin and nature of the /lN and /INN interactions, namely, (S+Q*; 36.1)

and

(S-Q*; 42.8),

(5.la)

484 their counterparts

GAL,

SOPER

AND

DALITZ

in the analysis with the New Data being (S+Q*; 48.6)N

and

(S-Q*;

69.9,jN.

(5.lb)

The value of Q* is about - 1.9 MeV for all these minima, and corresponds to C, = 2.72 MeV, using Eq. (1.2). This is about twice the estimate given by Bhaduri et al. [6]. From Eq. (1.4) the contribution to the effective r from the sum of all the ANN forces involving one s-shell and one p-shell nucleon is (-0.325C,), which now takes the value -0.88 MeV. From the value of r, about 1.46 MeV for all these minima, we then obtain the following value for the contribution from two-body AN forces: v/l* = 2.34 MeV.

(5.2)

The estimate given by these minima is therefore in satisfactory agreement with the phenomenological estimate obtained from the analysis of AP scattering cross sections at low energy, as discussed in the Introduction. However, it is quite possible that this agreement is simply fortuitous. When A is allowed to vary, the minima (5.1) move to (A+S+Q*;

33.7)

and

(d+S-Q*;

32.8)

(5.3a)

31.3)N

and

(d+S-Q*;

56.3)N

(5.3b)

for the Old Data, and to (A+S+Q*;

for the New Data. For these minima, the value of Q* has dropped significantly, by a factor of 2 to 3 for cases (5.3a) and by a factor of about 6 for the cases (5.3b); the value of V is also lower for these lower minima, and there is no longer any agreement between the value of V -AN obtained in this way and the independent estimate from the AP scattering data which was discussed in the Introduction. If we accept the minima (5.1) as reasonable, then phenomenological arguments would incline us to prefer the minimum of type (S+Q*; x2), because this gives J = 1 for the ,SLi ground state, as has been deduced from the qualitative properties of jLi decay. On the other hand, our theoretical arguments (cf. Part I) about the origin of the AN spin-orbit forces suggest that they should have the same negative sign as is known to hold for the NN spin-orbit force, which would incline us to prefer the minimum of type (S-Q*; x2). We should note that, even though J(;Li) = 2 holds for the minima of type (S-Q*; x2), these minima also indicate a low-lying state for ,8Li* with spin 1, at excitation energy 0.19 MeV for the “Old Data” parameter set. For small changes of the parameters, this excitation energy is given by E*(J = 1) = (0.19 + 0.704 - 0.28(65’+ - SS) + 0.69T - O.O88Q*) MeV.

(5.4)

SHELL

MODEL

FOR

P-SHELL

HYPERNUCLEI.

II

485

A small negative value for d (or a relatively large negative value for 7’) would be sufficient to reduce this excitation energy to the point where the J = 1 state of :Li would be isomeric. In actual fact, when d is allowed to vary freely, the minimum moves to a point with positive d. Near the (S-Q*; x2) minimum, x2 is quadratic in &S+, 8X and SQ* but linear in d (and in T) so that x2 will be increasedappreciably by the addition of a small negative d to this parameter set. For changes in the other parameters S, , S- and Q*, x2 will increase much less,but they produce much less shift in E*(J = l), since their coefficients in expression (5.4) are so much smaller. We conclude that, at present, the minimum with (S+Q*; x2) appears the more natural fit, although contrary to our expectation for the /lN spin-orbit forces. Bn data on p,,,-shell hypernuclei will provide very informative evidence, in due course. The most interesting species would be 20, but this system is not readily available for study with the present techniques, since nuclear emulsion contains only the heavy nuclei Ag and Br, beyond 160. The parameter sets for minima of the type (SQ*; x2) require Q* to have a large negative value, and this would imply a dramatic fall in BA as we approach the upper end of the p shell, for the reasons discussed in Chapter 4. For example, the parameter set for (S-Q*; 42.8) would predict the BA values 11.1 MeV for the isospin triplet systems ‘:O and ‘;C, 10.4 MeV for the doublet systems (l,6N, ‘,“O), and 8.0 MeV for ‘:O; the parameter set for (S+Q*; 36.1) gives essentially the samepredictions, the main difference being that the former set predicts J = 1 for the ground states of (YN, ?O) while the latter set predicts J = 0. It is conceivable that YO and YN might be detected and measured in nuclear emulsion experiments in due course, but the chancesfor this fortunate situation to be realized are far from high. A more likely possibility is that the hypernuclear species :N, (5C, 20) and (‘,“C, 1,4N) may become established in nuclear emulsion work; their Bn values would be sufficient to indicate whether or not this turn-down to lower BA values predicted for the end of the p shell does correspond to the physical situation. The parameter sets of type (OSQ*; x2) have more flexibility and can fit the outstanding variations in BA through the pS12shellwith a smaller magnitude for Q *, namely, 1Q* / w 1 MeV. With this value, the turn-down of BA values in the p1,2 shell is lessdramatic; BA for 20 then takes values in the range lo-11 MeV, but the intermediate species(‘,“C, y N) and (?C, 20) would have larger BA values than those suggestedjust above, for parameter sets of the type (SQ*; x2). This contrasts with the expectations based on a simple extrapolation using a cl-nucleus potential well with constant depth U, the value of U being deduced from B, for YC, which leads to Bn = 15 MeV for 20. A criticism of such an extrapolation, based on the possibility of f’lNN three-body forces, has been put forward by Bhaduri et al. [29]. Theseauthors also point out that the basically repulsive doubleOPE (INN force considerably reduces (1 binding for closed-shell core nuclei; in

486

GAL,

SOPER

AND

DALITZ

particular, they have made an estimate for BZ1(17, $), leading to values in the range 9-12 MeV depending on the cut off they imposed on the singular part of the (INN force. Even without llNN interactions, our shell model gives a lower B,, value for YO than the value ~15 MeV given by the uniform potential-well model. In order to reproduce the variations of Bn through the p3,2 shell in the absence of ANN interactions, we need to invoke strong spin-dependent noncentral flN potentials. However, none of these noncentral terms contribute to BA for 20, although (S, - S-) does contribute to B* for ‘;C, for example, as Table II shows. If we fit expressions (4.1) with Qt = 0 to the /iHe, :Be and YC BA values, the three equations give V = 0.80 MeV and (S+ - S-) = -0.385 MeV, which correspond to BA = 12.5 MeV for “:O [for the New Data, the corresponding numbers are F = 0.75 MeV, (S+ - SJ = -0.77 MeV and B,(17,+) = 12.0 MeV]. A less extensive analysis of the B* values for p-shell hypernuclei has recently been published by Lee et al. [33]. This analysis includes data only for the species A > 8, including a Bn value for 13B n , and assumes that the ground-state spins are all I(JN - &)I. Their one solution corresponds to parameter values F= 0.68 MeV, A = -2.00 MeV, S, = -0.75 MeV, and T = -0.96 MeV, which are not physically reasonable, according to the criteria we have discussed above. These parameter values also give unreasonable B, values for the A = 6 and 7 hypernuclei. For the pl,z shell, these parameters lead to the same bending down of the B* value with increasing A as we have found in Table VIII here, and give BA = 11.1 MeV for the species 20. Lee et al. [33] also considered roughly the effect of including the /INN parameter Qt,, , and found that it produced only a small improvement to the fit given just above; this is consistent with the result shown in Fig. 1 here, that Qz, has an important effect when S* = T = 0 is imposed, but that the inclusion of Qi,, is relatively unimportant when S, and T are already allowed to vary freely. To conclude, we must ask whether it is reasonable to expect our uniform shell model to be capable of accounting satisfactorily for these BA data. This has been questioned especially by Bodmer and Murphy [30]. Bodmer [31] has written the BA difference (SB,) between two neighboring nuclei (mass number difference 8A), one of which has spin zero, in the form BA = S + @A) vi, - (&z)(~B,/&z)~

- cleBrr + cCOIr+ (INN

contributions,

(5.5)

where 6 is the energy due to spin-dependent terms, vt, is the spin-averaged energy for clN forces for a nucleon in the p shell, a denotes the radius of the nuclear core, and cream , +Orr denote the rearrangement energy arising from distortion of the core nucleus and the additional energy due to correlations between the A particle and individual nucleons, respectively. In our shell-model calculations the rearrangement energy is included insofar as the nuclear distortions can be described

SHELL MODEL FOR P-SHELL HYPERNUCLEI.

II

487

by states within the lp shell, and the correlation energy is included insofar as it can be represented simply by changesin the two-body clN interaction parameters T, d, & , and T, since these are treated as phenomenological parameters within the p shell. A correction proportional to 6a has not been included. Bodmer and Murphy emphasize that (as,/&), is large, with values in the range 13-20 MeV . fm-l, and that the radii measured for the p-shell nuclei have substantial uncertainties, and possibly irregular variations with increasing mass number A. For example, the core nuclei 5He and sBe are not bound; 6Li and gBe are only lightly bound and are known to have anomalously large radii. However, in the hypernuclear ground states,the binding of a nucleon is considerably larger than it is in the corresponding free core nucleus. For example, JBe is bound by 3.4 MeV relative to its lowest threshold (4He + jHe) whereas 8Be is energetically unstable, and the nucleons in ILi are bound by 2.5 MeV more than their counterparts in 6Li. In our New Data analysis, the species:He has been omitted becauseof the low nucleon binding, as was discussedearlier. It is true that there is appreciable empirical uncertainty in the nuclear radii but a uniform shell model for nuclei doesgive quite an adequate fit to the shapeand size of the p-shell nuclei A 3 7 [32]. For a core nucleus which is lightly bound in the free state, the nuclear system bound in the il hypernucleus is in a much more tightly bound state; the hypernuclear core nucleus will then not have the radius observed for the free nucleus. It appears a reasonable first approximation to treat these nuclei within (1 hypernuclei according to the same uniform shell model as has been successfulin accounting for the properties of free nuclei, especially as we recall that this also accounts quite well for the energy-level spectra and the electromagnetic and beta-decay transition rates, and this even for the lightly bound nuclei.

6. ACKNOWLEDGMENTS The first author (A. G.) wishes to acknowledge the hospitality of the Rutherford High-Energy Laboratory, during a visit when a part of the calculations described in this paper were carried out. The second author (J. M, S.) wishes to acknowledge fruitful discussions with Dr. R. D. Lawson, especially concerning the calculations with two-body AN parameters alone. The third author (R. H. D.) wishes to,acknowledge the hospitality of the Center for Theoretical Physics of M. I. T. during two periods when several parts of the calculations described here were carried out. REFERENCES 1. 2. 3. 4.

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