The p-shell hypernuclei and the Λ-N interaction

8.B

[

Nuclear Physics 64 (1965) 593--628; (~) North-Holland Publishing Co., Amsterdam

I

Not to be reproduced by photoprint or m i c r o f i l m without written permission from the publisher

THE p-SHELL HYPERNUCLEI

AND THE A-N INTERACTION

k. R. BODMERt

Department of Theoretical Physics, The University, Manchester, England and

Argonne National Laboratory, Argonne, Illinois, USA tt and J. W. MURPHY

Department of Theoretical Physics, The University, Manchester, England Received 6 July 1964

Abstract: The p shell hypernuclei have been studied in detail using two and three-body central Yukawa A-N interactions. Intermediate coupled wave functions and a range of density distributions are considered for the core nuclei. The A wave function and volume integrals are obtained by numerical solution of the appropi'iate two-body (A-core nucleus) eigenvalue problems, the results being most conveniently represented by the corresponding numerical values of the relevant Slater integrals and the A kinetic energies. The simple assumption of a spin-dependent and charge-independent two-body interaction turns Out to be adequate to account for all the known Ba. In particular, the values of the spin-averaged volume integral of the two-body interaction, which are obtained from AHeS, ABe° and AC18, agree very well in the absence of three-body forces. Further, quite small tipper limits on the permissible strength of the latter are obtained. Almost nothing can be deduced about the range of the two-body forces or about their interactions in relative p states. The implications of our results for the well depth in nuclear matter are discussed. The spin-dependent interaction energy is found to be completely masked by quite small uncertainties in the core sizes and, to a lesser extent, by uncertainties in the re-arrangement energies. Effectively nothing can at present be deduced from the p shell hypernuclei about the spin dependence which with most plausible assumptions about the core sizes and re-arrangement energies can be made to agree with the vaines obtained from the s shell hypernuclei. Conversely assuming such values for the spin dependence can make the A into a quite sensitive probe into small size differences.

1. Introduction D e t a i l e d analyses 1 - 5 ) o f the s shell hypernuclei (i.e. f o r A < 5, where A is the b a r y o n n u m b e r ) have been m a d e with the a i m o f o b t ai n i n g k n o w l e d g e o f the A - N interaction. In particular, D a l i t z a n d D o w n s have m a d e analyses for soft ( G au ssi an a nd Y u k a w a ) central a n d c h a r g e - i n d e p e n d e n t t w o - b o d y interactions. F o r an y assum ed r an g e a n d shape o f the interaction, the e x p e r i m e n t a l A separation energies Ba (in particular, those o f a H 3 a n d AHe 5) then d e t e r m i n e the singlet an d triplet Present address: Argonne National Laboratory, Argonne, Illinois, USA. ~t Work partially supported by the A.E.C. 593

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MURPHY

volume integrals Us and Ut, using the knowledge that the singlet interaction is more attractive than the triplet 6). The volume integrals obtained for soft interactions are expected to be almost shape independent. Analyses with hard-core potentials 3, 4) have also been made. There are indications that, for the same intrinsic range of the potentials the scattering length is not expected to depend sensitively on the hardcore radius and that therefore the scattering length may approximately be obtained independently of the shape from the results obtained for soft interactions 7). However, the importance of three-body forces may be comparable to that of twobody forces (see e.g. ref. 6)). Analyses including central three-body forces 5, s) and also certain types of non-central three-body forces 9) indicate that Us is still fairly well determined from aH 3. However, Ut and therefore in particular also the spindependence A = Us- Ut can no longer be uniquely determined from the Ba of the s shell hypernuclei, although this would in principle be possible from a determination of the excitation energy of an excited (J = 1) state in ,tHe4 if this state should turn out to be bound. At the other extreme of mass numbers, several authors 5,1o, lx) have made detailed studies of the well depth D felt by a A in nuclear matter. The two-body interactions in relative p states, especially if the interaction has a hard core, may now be of some importance. Also, the effect of central three-body forces is rather different in nuclear matter than in the relevant s shell hypernucleus aHe 5. However, these studies of the depth D have not so far given much further significant information about the A-N interaction. This is partly because of considerable uncertainties in the empirical value of D but also because of uncertainties in the form (particularly the range) of the interaction. In particular, even if D were well-known, nothing very significant could be deduced about the three-body forces unless rather definite assumptions are made about the form of the two-body interaction 5). It thus seems very desirable also to have detailed studies of the heavier hypernuclei with A > 5, i.e. the known hypernuclei with p nucleons; this is the aim of the present paper. Because the structure and density distributions of the p shell hypernuclei differ considerably on the one hand from those of the s shell hypernuclei and on the other from those of the much heavier hypernuclei relevant for the determination of D, one may hope that studies of the p shell hypernuclei will give further information about the A-N interaction. Thus one might hope to learn something about the strength of the central three-body forces, and hence also indirectly about A by making use of the results already obtained for the s shell hypernuclei. Also one could hope that the p shell hypernuclei will give independent information about the spin-dependence of the interaction and hence possibly some indications about the importance of non-central two-body forces. In principle, it may also be possible to deduce something about the range of the forces as well as about the importance of interactions in relative p states (or equivalently about the exchange character of the two-body force if this is assumed to be static). More generally, one should be able to establish whether or not the simple assumption of a purely central spin-dependent two-body force is

p SHELL HYPERNUCLEI

595

capable of accounting for all the presently known values of Ba. Another aim is to learn about the structure o f hypernuclei since they are o f interest in their own right. Related and following from this, an important result will be an indication of the extent to which uncertainties about the structure will limit the information obtainable about the interaction. One may plausibly hope that such indications will be rather general and largely independent o f the particular form of interactions used. A closely related question of considerable interest is then whether there are situations in which the A acts as a useful probe into the structure of the core nuclei. In this paper, we shall consider the different aspects of p shell hypernuclei just mentioned. Because of their special features, the A = 7 hypernuclei are discussed in another paper 12) with special emphasis on the A in the role of a nuclear probe into the A = 6 core nuclei. Previous discussions of the p shell hypernuclei 2,6,13-17) have used two-body forces and shell-model wave functions for which mostly eitherjj or L S coupling was assumed. However, intermediate coupling is in fact appropriate for most of the p shell nuclei. Further, most of the discussions have limited themselves to the matrix elements of the interaction and have taken the relevant Slater integrals as constants for all the p shell hypernuclei. This procedure corresponds to assuming the single-particle nucleon and A wave functions to be the same for all these hypernuclei. In particular, this implies that the sum of the A kinetic energy and the interaction energy between the A and the s shell nucleons is constant and equal to BA(aHe 5) = 3.1 MeV and that similarly the spin-averaged interaction energy with a p nucleon is constant. Making these assumptions, Dalitz and Soper 16) and Lawson and Soper 17) have in fact studied intermediate coupling for two-body forces, with special emphasis on deducing the matrix elements of the spin-dependent parts of the interaction. The latter authors also consider two-body tensor forces in addition to central spin-dependent forces. We have also used intermediate coupling but in addition have also relinquished the other restrictions. We consider both (central) two-body and three-body Yukawa interactions having a reasonable span o f ranges. Our calculations are based on a two-body model, consisting of the A and the core nucleus. For a given core wave function ~c, this model corresponds to the best product type wave function, i.e. ~ka~c if the core nucleus is spinless, where ~ka is the single-particle (s state) A wave function with respect to the centre-of-mass o f the core nucleus. The A wave function and the associated volume integral o f the A - N interaction are then obtained from the A separation energy by solving the eigenvalue problem corresponding to the effective A-core nucleus potential obtained for the assumed form of interactions and for the appropriate intermediate-coupled core wave functions. Our results are then directly comparable with those for the s shell hypernuclei and for D. We consider a range o f core density distributions and make use o f the electron-scattering data wherever possible. The basic results of our numerical calculations are most conveniently represented by the corresponding results for the relevant Slater integrals and for the A kinetic

596

A. R .

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MURPHY

energy from which the values of Ba (or, alternatively, of the volume integrals if B a is given) may be obtained for any case likely to be of interest Much of the discussion in ref. 12) of the A = 7 hypernuclei is also based on a two-body, A-core, model. Therefore, for convenience and completeness, we also present and discuss, in this paper, results for the Slater integrals for the A = 7 hypernuclei. The spinaveraged interaction (both two and three-body) is determined effectively by aHe 5, aBe 9 and aC 13 which have spinless core nuclei. The hypernucleus abe 9 requires special treatment because the isolated Be a core is not bound so that it is essential to take proper account of the dynamics of the core. This may be done by considering ,tBe 9 to be a three-body system consisting of two ct particles and the A as in refs. 18). A more detailed and exact treatment of aBe 9 using this three-body model has been presented in another paper 19). T h e hypernucleus abe 9 is important not only for the determination of the spin-averaged interaction but also, in practice, for the spindependence since ABe 9 is neighbouring to both aLi a and .4Li 9 whose cores have spin and for which Ba is reasonably well-known. The predominant part of the interaction energy arising from the spirt-averaged interaction between the A and the core nucleons is found to depend quite strongly on the core size. This on the one hand very severely limits the possibility of using the p shell hypernuclei to deduce anything about the spin dependence since the corresponding interaction energies are masked, in particular, by quite small uncertainties in the sizes. On the other hand, this size dependence makes possible the use of the A as a significant probe, especially for aBe 9 and the .4 = 7 hypernuclei (4Li 7, nile 7, aBeT). The latter are discussed in ref. 12), while ,iBe9 has been discussed in this connection in ref. 19). Although distortion of the core by the A cannot be included in a two-body approach without further assumptions about the size dependence of the energy of the core, we do allow for the possibility of core distortion and the associated re-arrangement energy where these are likely to be important. It is particularly significant for aBe 9. A basic assumption o f our approach (as of most others which use soft interactions) is that, for a given core configuration, correlations between the A and the individual nucleons are unimportant and that thus (apart from the core distortions just mentioned) the effect of the A - N interaction may, to a good approximation, be obtained to the first order in the interaction. In fact, the estimates of ref. s) for the secondorder contributions to the well depth D indicate that such contributions to the Anucleus potential are likely to be less than 1 0 ~ of the first-order contribution for Yukawa interactions with relevant ranges. It furthermore seems quite likely that the second-order contributions are similar for D and for all the hypernuclei we consider. Thus the volume integrals obtained in first order from different hypernuclei are expected to be quite closely comparable. These volume integrals should then probably be reduced by rather less than 10 ~ to obtain the actual volume integrals which would be obtained if second-order and higher order effects were taken into account.

p SHELL HYPERNUCLEI

597

2. The Interaction o f the A Particle with the p-Shell Nuclei 2.1. THE A-N INTERACTION We consider (effectively phenomenological) two-body and three-body A - N interactions which are central and charge independent and which are characterized by their ranges and corresponding vohLme integrals. Thus, following Dalitz and Downs1' 2) we use a soft spin-dependent two-body interaction

VAN(rAN) "~ -- (Vs Us at-Pt Ut)v(rAN),

(1)

where Ps and Pt are respectively the singlet and triplet spin projection operators for the A - N system and Us and Ut are the corresponding volume integrals. We use a Yukawa shape

v(r) - #3 e-Ur

(2)

4~ /~r This is normalized to unity, p - 1 being the appropriate Yukawa range. We shall mainly consider ordinary forces and the two ranges P2~ = 0.7 fm and/1~ 1 = 0.4 fro, corresponding respectively to the exchange of two pions and of a K meson. The latter mechanism in fact leads to an exchange force, the appropriate modifications being considered in sect. 3. A range of about p~ 1 is also appropriate if the force is predominantly due to the exchanges of single heavy mesons (t/, K, etc.) (see, e.g., ref. 2o)). Following ref. 5) we consider a central three-body force of the form

W(ra, 1, 2) -- W(ax" a2)(~1" g,2)V(rA1)l)(rA2),

(3)

which has an exchange character suggested by meson theory to lowest order 6, 2,). Here a and z are nucleon spin and isobaric spin operators, respectively, the indices referring to the nucleons with which the A interacts, and W is the total "volume" integral of the three-body interaction. With W > 0, the interaction will be attractive if the nucleon pair is in an even relative angular momentum state; the three-body interaction of eq. (3) will then be referred to as attractive if W > 0. The shape functions v(r) are again taken to be normalized Yukawa functions of range v-x. The values v-1 = 1.0, 1.4 and 2.0 fm are used. The intermediate range corresponds to a Yukawa interaction with a Yukawa range appropriate to the exchange of a single pion with each o f two nucleons, while the longer range is more nearly equivalent to an exponential shape with an exponential range 5) o f h/m=c. The shorter range is included so as to cover possibly important modifications due to higher order processes. 2.2. THE A-NUCLEUS POTENTIAL DUE TO TWO-BODY FORCES Our considerations are based on a two-body model consisting of the A and the core nucleus. The A in a ls state is considered to move in the potential Va(r ) generated

598

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through the two and three-body interactions with the nucleons of the core. Thus for the normalized A radial wave function one has

[Ta+ Va(r)+ Ba](aa(r ) = O,

(4)

where for Ta the appropriate reduced mass must be used, and where Ba is the separation energy of the A with respect to the core. With only two-body forces and for those hypernuclei for which the total angular momentum JN of the core nucleus is zero (namely aHe s, the mirror pair aHe 7 and aBe 7, aBe 9 and aCXa), the potential Va(r) is obtained by folding the A - N interaction into the density distribution of the core. Then, for the hypernncleus of baryon number A and with the appropriate value o f Ba, eq. (4) may be solved as an eigenvalue problem for the total volume integral Ua-1. In general, however, JN ~ 0 and spindependent effects will be present which will necessitate a more detailed description for the core nucleus. For this we shall assume a shell-model wave function. For the p shell hypernuclei, the spin-dependent effects will then come only from the coupling of the A with the p shell nucleons since the s shell nucleons form a closed shell. The exchange character assumed for the three-body force also requires such a more detailed description. We shall always consider density distributions to refer to the distribution of the mass centres of the nucleons. For the s and p nucleons the corresponding density distributions p, and pp, normalized to unity, will mostly be obtained from the harmonic-oscillator functions

u~(r)= F

4 ]'l'e_,e/2a~ '

Up(r)--ro-l/:___~ .~/~ir

e-r2/2av2,

(5) (6)

LJ~ a~,/ ap where a s and ap are the appropriate oscillator size parameters. For all but the A = 7 hypernuclei these will be taken as equal, i.e., the s and p nucleons are assumed to move in the same harmonic-oscillator well. It should be emphasized that we consider the density distributions always to be with respect to the centre-of-mass of the core nucleus. Thus for the undistorted core the values of as and ap will be those that give the same rms radii for the s and p nucleon distributions, respectively, with respect to the centre-of-mass of the core as those obtained from analysis o f the electron scattering experiments where these are appropriate. Our oscillator parameters thus differ slightly from the conventional ones which are obtained by consistently incorporating centre-of-mass corrections in the harmonic-oscillator wave functions. We always refer directly to the distributions with respect to the centre-of-mass of the core since this is appropriate to our two-body A-nucleus model. For small distortion of the core by the A, when the relevant density is that of the isolated core

p SHELL HYPERNUCLEI

599

nucleus, the value o f Ba in eq. (4) may, to a good approximation, be identified with the experimental A separation energy. If, on the other hand, distortion is important, then the isolated core sizes are no longer appropriate and also the value of B a to be used in eq. (4) must now be increased by the re-arrangement energy of the core nucleus. This is discussed in more detail subsequently and especially also in ref. 12). Whether or not core distortion is important depends of course on the particular core nucleus. With only two-body forces one has Va(r) =

F~(r)+V2p(r),

(7)

where V2s(r) = - U f darl

V2p(r) = -

Vvf

d3rl

ps(rl)v(rat),

(8)

pp(rl)v(rax).

(9)

In eq. (8) the volume integral U = Us + 3 Ut is four times the spin-averaged volume integral of the A - N interaction. For a H e 5 one has U = U4, where U4 is just the total volume integral for this hypernucleus (V2p = 0). With Yukawa interactions one obtains U4 = 1040+50 M e V . fm 3 and 790+40 M e V . fm 3 for #2~ and #K, respectivelyS) t for an rms radius R, = 1.44+0.07 fm for the matter distribution of the ~ particle 22). It should be noted that the more recent measurements of Burleson and Kendall 23) together with a value o f 0.854-0.05 fm for the rms radius of the proton, indicated by recent experiments, give almost the same value of R, = 1.45 40.065 fm and thus effectively the same values o f U with slightly reduced errors. The volume integral Up in eq. (9) is defined by Np

Up = (J] E (Ps Us+Pt Ut)IJ),

(10)

1

where the sum is over the Np p shell nucleons and where [J) denotes the ground state of the hypernucleus. Eq. (10) m a y be rewritten as

Up = ¼Np U+~A,

(11)

where A = U s - Ut measures the spin dependence of the A - N interaction and where Np

~c = - ( J I E o , . o a l J ) . i=l

(12)

* These values are about 10,2/oolarger than those obtained with Gaussian interactions having the same intrinsic range e), the corresponding values being 9254-45 MeV • fm8 and 7054-35 MeV. fm8.

600

A.R.

BODMER AND J. W. MURPHY

~In general, different states of the core nucleus will enter and hence, with [~JN, ½; J ) denoting the hypernuclear state obtained by coupling a core state of angular momentuna JN to the A spin to give J, one has

[J) = ~ a,,S,I~JN, ½; J).

(13)

~, J N

The angular momentum o f the core nucleus can only have the values JN = J-+ ½, labelling core states having the same value of JN. For each JN, only the lowest state is expected to be important and accordingly the label ~ will be unnecessary and will be dropped. The coefficients asr, are then obtained by diagonalizing the appropriate two-dimensional energy matrix having the elements (EsN6sNs,~+xsNs, NAF(2°)), where ~t~,(o) 2p is the relevant Slater integral which implicitly involves the A wave function (see below) and where, appropriate to the shell-model description, only core states corresponding to p nucleon configurations are considered. The energies EsN are just those of the parent nuclear states and the xs,~s'N are given by Np

xs~s,N = (JN, 3; JI E a," oalJ~, ½; J).

(14)

i=l

F o r j j coupling this matrix element is diagonal in JN and one has 15) K = --~ [ J ( J + 1 ) . J N ( J N + 1)--31.

(15)

For LS coupling one has = - ½[(2sN + 1)(2s

+ 1)y(-

1)

+'"+""

SN+'~

×

E

s=ISN-~I

(2S+I)W(SNSS S;½L)W(S SJ S;½L)[S(S+I)-S (S +I)-¼], (16)

where W(abcd; ef) is a Racah coefficient. For intermediate coupling, which is mostly appropriate for the p shell nuclei, an expansion of the core wave functions in terms of either jj or LS coupled wave functions then gives the required matrix elements. I f the energy difference between the core states o f different JN is large, then only the ground state o f the core nucleus is expected to be important and no diagonalization in the core states is then necessary. However, in particular for ALi8 and aBe 8 it is necessary to include both the JN = ~ - ground and JN = ½- excited core states, both of which contribute comparably to the J = 1 ground state t of the hypernucleus z4). Values o f x, given in sect. 5, have been reported by Dalitz 16) using intermediate coupled wave functions obtained by Soper. Using these wave functions we have obtained essentially the same results for tc. t That this is the ground-state spin and that thus A Dalitz ~4) from an analysis of the decay of ALis.

> 0 also in the p shell has been concluded by

p SHELL HYPERNUCLEI

601

Expanding the shape functions v(raN ) into a sum of Legendre polynomials one has

v(rAN) = Z

)P (cos oa ),

(17)

k=O

where coaN is the angle between r and rN. Averaging over the direction of r, as is appropriate for a A in an s state, eqs. (8) and (9) then become V2s(r) = -

ev(2°)(r),

(18)

(19)

v2,(,) = - ( ¼ N , U + where the potential shape function v~]) is given by

v~(r) =

;o ore u~(rl)v(~)(r,

rl)dr~ .

(20)

Here l refers to the angular momentum of the nucleon orbital and k to the appropriate Legendre polynomial. For Yukawa interactions, these integrals can be expressed in terms of the tabulated functions I-Ih,(x) (see reL 2s)) as discussed in the appendix. For ~2~'(°)(r) the relevant result has already been given in ref. 5) in terms of error functions. In practice the functions v~) were obtained by numerical integration. As already remarked, for hypernuclei with JN = 0 and with only two-body central forces, an explicit separation into s and p shell contributions is not necessary since only the total density distribution p and the spin-averaged volume integral U then enter. Thus for these hypernuclei one has

Va(r) = -- Ua-1 f d 3 r x p(rl)v(ra~),

(21)

where UA_ 1 = ¼ ( A - 1 ) U and where in terms of the s and p nucleons separately the (o)(r )+NpV~2°)(r)]/(A - 1). integral is given by [4v2~ 2.3. T H E A - N U C L E U S P O T E N T I A L D U E T O T H R E E - B O D Y F O R C E S

The spin and isobaric-spin exchange character assumed for the three-body interaction [eq. (3)] again requires that the s and p nucleons be considered separately. With a shell-model description the assumption that the ls shell is closed has then the consequence that the net contribution to the three-body A-nucleus potential coming from the interaction of the A with a p nucleon and any of the s nucleons will vanish. Thus only the contributions V3~(r) and V3p(r ) arising from the interactions of the A with nucleon pairs only in the s shell and only in the p shell separately need be considered. For Yukawa interactions the former has already been obtained in ref. 5) and is given by 4 (0) ( r ) = - lSWv~°)(r), V3~(r) = W(Xl ~ (a," aj)(,,. *j)[Z>v3~ i
(22)

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A . R . B O D M E R A N D J. W .

MURPHY

where [~> is the totally antisymmetric spin]isobaric-spin state of the closed ls shell and where the shape function is given by t"

v(°)(r) = J d3ra d3r2 p~(rl)p~(r2)v(ral)v(ra2 )

=

(o)

(r)]. 2

(23)

Here vt2°)(r) is given by eq. (20) but with a range v-1 appropriate to the threebody Yukawa interaction. The three-body interaction of the A with a single pair of p nucleons which are coupled to orbital angular momentum L, spin S and isobaric spin T gives rise to the potential V3v(r) =

Wvap(r),

(24)

r2)[2v(rat)v(ra2),

(25)

with =

dar2[ kvz(r

.

where ~ L ( r l , 1"2) is the orbital wave function obtained by coupling the single-nucleon wave functions Uv(r~)Y~(f2i) together to the total orbital angular momentum L. Only if there were no spatial correlations between the two nucleons could I~z(rl, r2)[ 2 be replaced by the product of the single-particle densities, i.e. by pp(rl)pp(r2), as may be done for the s nucleons *. Expanding v(raa ) and v(ra2 ) in terms of Legendre polynomials, as in eq. (17), and using the spherical harmonics addition theorem, the integration over the nucleon angles f21 and t22 can be performed and one obtains !

Vav(r) = 4n

C ( l k l , 000)C(lk 1, 000) -------------------~S,k'=O. 2 [ ( 2 k + 1)(2k'+ 1)3" Z

(k),, (S'~,, v2-trlv2-'trl ......

x [ ~ Oss,(ms, ink,, L)Y~'~(f2a)Ykm,k'(t2a)]. (26) ink, tak t

Here I2a denotes the polar angles of the A, C (abe, ~fl6) is a Clebsch-Gordan coefficient (using the notation of Rose 26), and Ok.k,(msrnk,,L ) is a number constructed from a sum of products of four such coefficients. Averaging over t2a, appropriate to the A in an s state, and performing the sum over m s and ms, one obtains (3/4n)W(lkL1, ll)dkS, for the square bracket in eq. (26). One has then v3,(r ) =

v~a°v)(r)+2-~W(12L1, ll)vCa~J(r),

(27)

which is now only a function of r, and in which

=

2.

(28)

t The effect on V8 of short-range pair correlations, due to a hard core in the nucleon-nucleon interaction, was investigated for nuclear matter in ref. ~) and was shown to be small.

p SHELL HYPERNUCLEI

603

If there were no spatial correlations between the two nucleons [i.e., if pv(rl)pp(r2)], then only the first term in eq. (27) would be obtained, the second term arising entirely from the angular correlations. For the three possible values of L in the lp shell, namely L = 0, 1 and 2, the coefficients of v(a2) in eq. (27) are ~-s, - ~ and a-Tr,1 respectively. However, the expectation value of the exchange factor of the three-body force is related to the orbital wave function through the exclusion principle and one finds that the correlation term is always attractive for W > 0. Thus, for the various two-nucleon states one obtains I~LI 2 --

V3,,(r) =

i,{,-r,1, (0)_L__6, ( 2 ) 1 rr L.,U3p T 2 5 U 3 p J

for 13S and 318 states,

(29a)

HTV~. ( 0 ) ~ 3 .~,~(2)1 vr l ~ U 3 p " r ' T ~ - ~ U 3 p j

for 13D and aiD states,

(29b)

WFO,,(o) 9 ,,(2)-I L~U3p -- 2 5U3p/

for a 11p state,

(29c)

W[-,~ (o)

for a 33p state.

(29d)

I .t,(2)"]

LU3p - - ~ U 3 p I

For W > 0, the attractive potential due to the angular correlations is normally smaller than the leading contribution which involves v(°). The opposite signs of this latter term for S and D states on the one hand and for P states on the other should be noted, this being a direct consequence of the exclusion principle. Thus, with positive IV, the potential for a P state is predominantly repulsive, while for S and D states it is predominantly attractive. If there are more than two p nucleons, one can obtain Vap(r ) in terms of v(a°)(r) , (2)(r ) by expanding the core wave function in terms of two-particle LS coupled and ~3p, states with the aid of the fractional-parentage coefficients of Elliott, Hope and Jahn 27). Thus, quite generally, for a hypernucleus of mass number A one can write V3p(r) =

(o) (r) + - WEaNp V3p

(2) (r)]. flNpYap

(30)

[[f]LNSNTN1}[f']ES'T', LST]2a2( S, T)

(31)

For an LS coupled nuclear state one has

eNp = ½Np(Np-1)

E [f']L'S'T" LST

and an exactly analogous expression for flNp. The values of ~2 and f12 are given by eq. (29) and [[f]LNSNTN l} [f']L'S'T', LST] denotes the relevant fractional-parentage coefficient. For an intermediate-coupled state the appropriate values of ~ p and fiN, may be obtained by making use of the expressions for LS coupling. It should be noted that the simple dependence of the three-body potential on the number of nucleon pairs, which has been included in the definitions of ~ and flN~, can be strongly modified by the detailed structure of the nuclear state, principally through the operation of the exclusion principle. This is discussed further in sect. 3 where the numerical values of ~ and flNp are also given.

(304

A. R. BODMER AND J. W. MURPHY

2.4. ENERGY EXPECTATION VALUES It is convenient to discuss the results, which are obtained with the procedures o f subsects. 2.2 and 2.3, in terms o f the potential and kinetic energies of the A corresponding to the wave functions which are obtained numerically from eq. (4). Thus, with angular brackets denoting expectation values with respect to this wave function, one has

~.~

= (V)2s+(V>2.+(V)3,+(V)3p-(T.~)

co)+ W(~u. Fao co) + fl.p F3p (2)) - ( T a ) , = UF~2°) + (¼Np U + •A)F)pco)+ 18 WF3,

(32)

where for (V).z the suffix n = 2 or 3 denotes the type of force, the suifix l = s or p refers to the nuclear shell, and the Slater integral F.~k) is given by .z F(k)

=

f O° (~2(r)v~.~)(r)dr.

(33)

The two-body integrals F2~°) and F2~°p) for aC 13, aBe 9 and the A = 7 hypernuclei, obtained by use of harmonic-oscillator functions for the core nuclei, are shown in figs. 1 and 2 as functions of the appropriate oscillator size parameters for both #2. and /~K- The decrease of these integrals as the size parameter is increased is primarily a reflection of the fact that the forces are o f much shorter range than the nuclear extensions and that, for a 6 function interaction, v(°) - (o) are just proportional 2s and U2p to the appropriate single-nucleon densities. Thus an increase in a s or ap results in a decrease of vt2°) or vt2°v), respectively, in the region where the A wave function is large. For the sizes of interest, e.g., a s = ap ~ 1.6 fm, there is still a significant dependence on the range which leads to smaller volume integrals for #K than for #2,. This range dependence of the volume integrals becomes less as the size increases. Apart from the core size and the range of the force, the integrals also depend on the A wave function and thus on the particular hypernucleus and the value of Ba. This dependence results in the relatively small differences in the values obtained for the various hypernuclei and, as expected, these differences again become less for more extended density distributions. It is a general feature that for a given a and # ~,(0) the integral .~'w)2sis more sensitive than ~ 2p to variations in q~.t(r) and hence the relative differences between the various hypernuclei are greater for F2~°) than for F2~°v). This is a consequence of the behaviour of the shape functions vt2°) and vC2 °) in the region (o) CO) where ~b,t is large: in this region v2p is slowly varying whereas v2s decreases relatively rapidly with r. Variations in Va(r) and Ba which result in a greater r.m.s, radius for the A wave function then decrease the value o f the integrals, the effect being more pronounced for F2(°). Thus for both aC 13 and abe 9, an increase in Ba of 1 MeV from the specified values gives increases of about 4.5 % and 3 ~ in ,2s jT(o) .o.,,~ . . . . ~.(o) 2p, respectively. By use of these values, the integrals for hypernuclei of intermediate mass numbers can then be obtained by interpolation. Thus a reasonable procedure is to obtain the

p SHELL HYPERNUCLEI

20

~

605

e f (HeoS;FK)

16

T 0 X ~m

0-

I

1.2

~

I

1.4

i

I

i

1.6

I

h8

,

1

2.0

as or a [fro]

Fig. l. The Slater integral F28 t°). The dashed curves are for AHe" with B A = 3.1 MeV; the u p p e r curve is for/*K and the lower curve is for/*sn. The values for the p shell hypernuclei are given by the curves a-h. F o r a-d, a c o m m o n oscillator well was used (i.e., as = ap = a) and the abscissa then refers to the c o m m o n size parameters a. For e-h, the values o f F~a{°) are s h o w n as a function o f as for ap = 2 fin, F28c°}being almost independent o f ap. The labelling is as follows: a - AC 18, B a = 10.9 MeV and PK, b - a C xs, B a = 10.9 MeV and Pan, c - A B eS, BA = 6.5 MeV and pK, d - a B e 9, B a = 6.5 MeV and #2~; e - A = 7, T = 1, B a = 5 MeV a n d / t K , f - A = 7, T = 1, B a = 5 MeV, a n d / * ~ , g A = 7, T = 1, B a = 3.5 MeV a n d / , K , h - A = 7, T = 1, B a = 3.5 MeV a n d / , z n .

10

% X o a.

~t,Y

I

1.4

I.B

2.2

2.6

3.0

ap or a(fm).-~

Fig. 2. The Slater integral F~p(°). F o r the labelling o f curves a - h see fig. 1. The curves a - d are again for a c o m m o n oscillator size parameter a, whereas curves e - h n o w s h o w F~p (0) as a function of ap for a s = 1.2 fm.

606

A. R. BODMER AND

J. W.

MURPHY

integrals for aLi a and aLi 9 from those for aBe 9 and those for AB12 from those for aC x3, with suitable adjustment of the Ba values. The variation of the two-body Slater integrals as the relative proportion of three-body to two-body force is increased was found to be insignificant since for a given Ba such an increase changes the overall potential Va(r), and hence also c,ba(r), very little. For the A = 7 hypernuclei the A wave function is not expected to be appreciably different for the T = 0 and T = 1 hypernuclei (if the range and the core sizes are assumed to be the same) because the values of Ba which are used are similar for both. However, with densities appropriate to the electron-scattering data (a s ~ 1.65 fro, ap ~ 2.0 fro) and with B a = 5.5 MeV, one finds that F (°)2sfor the T = 1 hypernuclei is about 10 70 greater than that for the T = 0 hypernucleus aLi 7, while with the same values of ap and B a but now with a s ~ 1.2 fm the integrals agree closely with each other and with the value obtained for aHe 5 (also shown in fig. 1). The difference for the larger values of a s arises from the fact that eq. (4) for the T = 1 hypernuclei was solved as an eigenvalue problem for U, whereas for aLi 7 the volume integral U ( = Ud) was considered as given and the equation was solved as an eigenvalue problem for A. Then, for large values of a s, the wave function c,ba(r ) obtained for aLi 7 is more extended than that obtained for the T = 1 hypernuclei because of the enhanced attraction between the A and the two p nucleons in aLi 7 (for further discussion see ref. 12)). The more extended wave function gives a correspondingly reduced value ofF~ °). I f for aLi 7 one had solved eq. (4) using the value of U obtained from the T = 1 hypernuclei (with the same ranges and core sizes) then the integrals would be very similar. Accordingly, only the results for the T = 1 hypernuclei are shown. The dependence of the Slater integrals on B a is indicated by the curves given for B a = 5.0 MeV and 3.5 MeV. The difference between the integrals for these values o r B a is seen to decrease as the relevant size parameter is increased, as expected. Because the A wave function is more strongly determined by as than by ap (there are only two p nucleons whose distribution is rather extended) the value of F~°) is almost independent of ap. Thus with #2, and for a s = 1.50 fm and Ba = 5.0 MeV, the change in F~°) as a v increases from 1.8 fm to 2.4 fm is less than 1 70. On the other ~ p for given values of ap, BA and/~ was found to be somewhat hand, the value ot"--(°) more dependent on a s, particularly for small values of ap; for example with #2,, ap 1.8 fm and Ba ~ 5.0 MeV one finds a 6 % decrease in ~,(o) -~ 2 p as as is increased from 1.20 fm to 1.80 fro. For ap ---- 2.4 fm the corresponding variation is negligible. The three-body Slater integrals r,~(o) 3s, ~.(o) 3 p and ~.(2) ± 3 p obtained for aCla are shown in fig. 3 for v - 1 = 1.0 fm and 2.0 fin and for the two-body range #27; the range v - 1 = 1.4 fm gives intermediate values. The relevant A wave function was mostly obtained from eq. (4) with only two-body forces considered since the slight dependence on the relative strengths of the two-body and three-body forces is again insignificant (see also subsect. 3.4). Values for ktK can be obtained from those given for P2~ by noting that the small contraction o f the A wave function relative to that for/t2~ leads to increase of 4.5 70, 3 70 and 7ol for ~.(o) respectively, for all three- P 3 s , ~.(o) ~ 3 p and ~,{o) _r:3p

607

p SHELL HYPERNUCLEI

i,'1= I fm (full curves)

r

12[-

/

~

F(2) \ / 3p \ Jo~

~ol-/\ ', I ~o)\\.

\

=,"= 2 fm (doshed curves)

q 2.o

\

I

K';",,Y,& 1

8

F]

/,X3,

1

'~

\

~'\ ~

--~l.5

/"....,e,"'-,..",,,

|,o

r'\ 81-

~: I-",.

t ~,\

OI

,

1.4

t

\\.\| ",,. \ "" 1

J 'a

".

-""

-I

"-..

/

I , I 1.6 I.B o(fm) ---~

~

/ 0 2.0

Fig. 3. T h e Slater integrals for t h e three-body interaction arc s h o w n for AC ]3 with B A = 10.9 M c V a n d for t w o - b o d y forces o f rangep2~ -1. T h e solid curves are for ~-z ~ 1 f m (left-hand scale) a n d the d a s h e d curves for ~-1 = 2 f m ( r i g h t - h a n d scale).

(AHeS; px)

IC

! >

8

J

v

--

(AHeS; FZT)

] 1.2

,

1

,

I

1.4 1.6 o or a s ( f m ) - - - ~

,

I 1.8"

,

I 2.0

Fig. 4. T h e / 1 kinetic energy. T h e c a p t i o n is the s a m e as for fig. 1 b u t for t h e c o m p u t e d values o f t h e / I kinetic energy.

608

A. R. BODMER AND ,l. W. MURPHY

body ranges. The corresponding integrals for aBe 9, with the same size and range as for aC 13, are approximately 10~o smaller than those for aC t3 for all two-body and three-body ranges considered. The values obtained for (Ta) for the various hypernuclei are shown in fig. 4. For the A = 7 hypernuclei ( T a ) is shown as a function of as for ap = 2.0 fro, the precise value of ap being unimportant since ( T a ) does not depend sensitively on %. Thus with #2~ and for a s = 1.5 fin and B a = 5.0 MeV, the change in (Ta) as ap increases from 1.8 fm to 2.4 fm is only about 1.5 Yo. An increase of 1 MeV in Ba from the specified values gives increases in ( T a ) of about 0.4 MeV and 0.6 MeV for aC 13 and aBe 9, respectively. The results presented for ( T a ) and for the various Slater integrals allow one, using eq. (32), to obtain the values of Ba for all reasonable sizes, ranges and volume integrals which are of interest for the p shell hypernuclei. 3. Hypemuclei with Spinless Core Nuclei: aC Is and ABe9 3.1. T H E AC 13 N U C L E U S W I T H T W O - B O D Y F O R C E S

The relevant hypernuclei for which B a is known 28) and whose cores are spinless are aC 13, aBe 9 and the mirror pair sHe 7 and aBe 7. Of these, only the first two will be considered in this paper. Because of their special features the A = 7 hypernuclei are discussed elsewhere 12). With JN = 0 and with only two-body forces, Ira(r) is given by eq. (21). The volume integrals ½U8 and ½U12 obtained for aBe 9 and aC 13 should then agree with each other and with the value of U, obtained from sHe 5 if two-body forces alone are to give a satisfactory description of these three hypernuclei. We shall mainly consider aC 13 since the C 12 core is stable and its density distribution is known from electron-scattering experiments, neither of which is true for the Be s core of aBe 9. Also, the effects of compression of the C 12 core by the A are expected to be quite small since such effects ordinarily are roughly proportional to (A-l) -1, since there are (A-l), nucleons but only one A. Thus for aC 13, estimates on the lines of refs. 2, 5) indicate a decrease of < 29/0 for U12 and one of ~< 3 ~o for the core radius if reasonable values (~> 100 MeV) are used for the compressibility coefficient K *. Such changes are ordinarily less than errors due to uncertainties in Ba and are considerably less than errors due to uncertainties in the density distribution of the undistorted core. This justifies the use of the latter for calculating Va(r ) in order to obtain Ba. For aC 13 we have considered both an oscillator and a Fermi density distribution, some results for the former having already been given by Dalitz 6). With the same t T h u s for the increase o f B a arising f r o m compression o f the core, one obtains the expression ~B,t = ½ a ~ ( O B a / 2 a ) o 2 / K ( A - 1) ~ -- ( O B a / a U ) ~ tSU, where ~U is the change in U = ½Uzl needed to compensate the change (SB,t and where a is the value o f the oscillator size parameter for the undistorted core. The corresponding change in the core size is given by tSa = a ~ ( a B 4 / O a ) t i / K ( A - 1). F o r A = 13, U = 1000 M e V - f m a, a = 1.65 fm and for the calculated values ( a B a ] a a ) v ~ - - 2 0 M e V . f m -1 and ( a B A / ~ U ) a ~ 0.02 f m -a, one obtains t S U / U ~ - - 2 . 2 / K and tSa/a ,m -- 2.75/K.

p SHELL HYPERNUCLEI

609

oscillator size parameter a for both the s and p nucleons, the oscillator distribution for the total density is given by

(

Psho(r) _ A - - 1 1 + 3n~a 3

Z-2r2~ 5

e-'2/"~.

(34)

~-i]

The electron-scattering data for C lz (Z = 6) is well fitted with a = 1.64 ___0.05 fm (refs. 29, 30)). The Fermi distribution is given by

prCr) = po Il +exp ( ~ )

]

-1

3 ( A - l ) ( 1 + rr2d2~ - I

with po - ~ eric

c2 1 .

(35)

To investigate the effect of the shape o f p on U~2 we have considered, in particular, "equivalent" Fermi distributions. These have the same half-density radius c and the same 9 0 - 1 0 ~ surface thickness s = 4.40 d as the oscillator distributions we have used. Thus, for example, with a = 1.64 fm one has for the equivalent PF the values c = 2.30 fm and s = 1.82 fm. The parameters c and s seem to be effectively the only ones determined, so far, by electron-scattering experiments t. One might expect that these parameters will also determine U~2 almost uniquely and independent of the precise shape o f p. The results for ½U12, to be compared with Ua, are shown in table 1 for Psho and for the most recent z~) value of Ba = 10.9+0.3 MeV. Our results for ordinary forces are roughly in agreement with those given by Dalitz tt. The values o f U~2 for the "equivalent" Fermi distributions are, for both/tzn and/tic, consistently smaller by about 4 ~ than for the corresponding oscillator distributions for all values of a considered. Thus uncertainties due to the shape of p are rather smaller than those due to errors in the size and are considerably smaller than the differences in U 4 which are obtained with Gaussian and Yukawa interactions of the same intrinsic range ttt. For #2n one obtains for the dependence on c and s the values ½(8U12/Oc), = 430 M e V - f m 2 and ½(8U12/dS)c = 255 M e V . fm 2. Thus for the values c = 2.24 fm and s = 2.2 fm obtained by Meyer-Berkhout et al. ao) for a Fermi charge distribution, one obtains :}O-12 = 1060 MeV • fin 3. Folding out the proton charge distribution would reduce this; in fact, Meyer-Berkhout et al. claim a somewhat more satisfactory fit with an oscillator than with a Fermi distribution. On the basis of our results we thus obtain ~U12 = 990--+60 M e V . f m a and 815_+50 M e V . f m a as reasonable values for ordinary Yukawa interactions of ranges #2~ and p~ 1, respectively, most of the error being due to size uncertainties. t In fact, direct fits o f a F e r m i distribution to t h e electron-scattering results for AC is give values for c a n d s which are in reasonable a g r e e m e n t with those o b t a i n e d by fitting a n oscillator distributiona°). tt T h u s in ref. e) the values ~U12 = 9504-150 M e V " f m 8 a n d 7404-130 M e V - f m 3 are q u o t e d for/z2n and/~K, respectively. T h e s e values are for G a u s s i a n interactions with the s a m e intrinsic r a n g e as the c o r r e s p o n d i n g Y u k a w a interactions. T h e r e is a n inconsistency in these values since t h e relative difference for/tan and/~K is as large as t h a t for ,tHe s, whereas it s h o u l d be considerably less because o f the larger radius o f C 12. t t t T h e c o r r e s p o n d i n g difference for ½U12 is expected to be rather less t h a n for U4.

610

A. R. BODMER AND J. W. MURPHY

It is seen that, within the uncertainties, the values of kU12 and U 4 are consistent with each other for both kt2, and #K, although an intermediate range is marginally favoured. To draw any firm conclusions about the range of the interaction from the (still rather light) p shell hypernuclei would thus require, above all, a very accurate knowledge of the relevant core sizes. TABLE 1 Results for AC18 with two-body forces for Ba = 10.94-0.5 MeV

/,z~

~o-~

/,zK Ordinary

Exchange

½U~

~U~

C

-:-~U~,F~

(MeV)

(MeV)

c

~-2 U~,F,p '°'

(MeV)

(MeV)

9034-15

--0.544-0.08

1.434-0.03

7504-20

7904-20

--0.364-0.15

1.414-0.0:

9984-20

--0.134-0.08

1.384-0.04

8244-20

8604-20

--0.024-0.16

1.374-0.0,

10934-25

0.314-0.16

1.334-0.04

9154-20

9604-20

0.404-0.18

1.324-0.0,

(MeV. fm 8)

(MeV. fm 8) (MeV. fm 8)

The indicated errors are only those due to the error in Ba For the K meson range, the exchange nature o f the force should also be taken into account. This was done by using the expression obtained by Dalitz and D o w n s 2) for the ratio r / = (V)(exch)/(V) ~erd) of the A potential energies for ordinary and exchange A - N interactions o f Gaussian form and for a Gaussian A wave function. Since Gaussian and Yukawa interactions with the same intrinsic range are approximately equivalent in binding the A, the corresponding ratios m a y be expected to be in excellent agreement with each other. For the size parameter o f the Gaussian A wave function, we use aa = 1.80 f m and 1.75 f m for #2~ and PK, respectively. These values give the same r.m.s, radii for q~2 as are obtained f r o m the eigensolutions of eq. (4) for a = 1.65 fm. The results do not in fact depend sensitively on the precise value of a a. For t/l = (V)[~x~h)/(V)~°rd), where l = s or p for the interactions of the A with the s or p nucleons, respectively, one finds r/,0~2~) = 0.99 and q,(/t K) = 1.00 and thus effectively no difference between ordinary and exchange forces, while r/p(#2,) = 0.75 and qp~K) = 0.91. The values for the total ratio q are then q(/z2,) = 0.86 and q(PK) = 0.95. Thus, as expected, the values of z}U12 obtained with exchange forces will be correspondingly larger than for ordinary forces. The results for the physically interesting range px~1 are given in table 1. One n o w has ½U~Xch)(pK) = 850+--50 MeV" fm 3, where the error includes uncertainties due to the size. For comparison, the corresponding values for nuclear matter (p = 0.168 fm -3) which are obtained from ref. 5) are r/~2, ) = 0.7 and ~/(/~K) = 0.86; exchange effects are thus considerably greater in nuclear matter than in aC 13. The ratio of the interaction energy in relative p states to the total interaction energy is given, to a very good approximation, by P = ½((V)¢°~d)--(V)('x°h))/ ( V ) <°rd) = ½(1--r/), since interactions in relative A - N states with l > 1 are expect-

p SHELL HYPERNUCLEI

611

ed to be negligible. Thus for ,tC la with a = 1.65 fm, one obtains P12Q.12z) = 7~o and P12(#K) = 2.5 ~ . As expected, P 1 2 is smaller for the shorter range. For nuclear matter the corresponding values are P~(/~2~) = 15~o and P~(#K) = 7~o. Interactions in relative A - N angular m o m e n t u m states with l >__ 1 are thus considerably less effective in ,~C~a than in nuclear matter. Thus in this respect, conditions in the two are still substantially different. This is, presumably, mainly a reflection o f the smaller average density of C ~2 because o f its large surface. If, as an extreme case o f a force of range #2~ for which the interaction in relative p states is weakened compared with that in relative s states, we consider a force with only interactions in relative s states (i.e., effectively a Serber force), then one obtains for a = 1.65 fm and B A = 10.9 MeV the value ~U12 = 1070 M e V - fm a which corresponds to a 10 ~ increase relative to the value for an ordinary static force. It seems clear that, at least for soft forces with the relevant short ranges, the BA of the p shell hypernuclei do not enable one to say anything significantly about the interactions in relative p states t TABLE 2 Potentials (in MeV) for .4Cla for a = 1.64 f m and B, t ----- 10.9 MeV Oscillator density distribution

-I.

Fermi density distribution

-I-

0.163

40.2

33.5

33.8

33.2

0.178

42.25

33.7

35.6

3

0.167

41.2

33.1

34.6

33.0

0.177

42.0

33.2

35.4

3

0.169

41.7

31.8

35.0

32.2

0.173

41.0

31.8

34.6

3:

0.163

40.2

29.1

33.8

29.9

0.167

39.6

29.2

33.4

2!

0.109

26.9

20.0

22.6

20.1

0.120

28.5

20.3

24.0

2q

Paho the volume integrals are }Ull = 988 M e V . f m s and 830 M e V . f m s for/~gn and/ZK, respectively; tl for PF are ½U12 = 950 MeV • f m 8 a n d 800 MeV • flu 8.

The small values of the quantity C = ~,~12-1rr w(o)_
612

A. R. BODMER A N D J. W. MURPHY

nuclei. The much smaller values of C for the heavier p shell hypernuclei are mainly due to their larger s shell sizes ~r'~(°)2sdecreases as the size increases). The assumption that C is constant and equal to 3.1 MeV throughout the p shell, made in other investigations 1 , - 17), implies that the expectation value of the spin-averaged interaction energy of a p nucleon is also constant and given by ¼UF(2°) ,~ 1.0 MeV. This is therefore an underestimate for the heavier p shell nuclei. As seen from tables 1 and 3 the value of ¼UFt2°v) is approximately 1.4 MeV for both ,iBe 9 and aC t3 and does not depend too much on the size parameter a. Results for Va~r) are shown in table 2. The results for/~2~ a n d / t K and also for Psho and the "equivalent" PF are seen to agree closely. For P~ho the m a x i m u m density of 0.169 fm -3 occurs at r = 0.8 fro. However, Va(r ) does not show a corresponding dip at r = 0. This is because the finite range of the A - N interaction smoothes out the effects o f density variations over distances of the order of the range # - 1. (See also the discussion in the appendix.) 3.2. T H E W E L L D E P T H F O R N U C L E A R M A T T E R A N D B A F O R H E A V Y H Y P E R N U C L E I

Also shown in table 2 are the relevant values of D (p) = ¼Up (1-¢U 12 P in the table) which is the first-order well depth for nuclear matter of density p for ordinary A - N forces. Second-order contributions to the well depth are small 5) and probably less than about 109/o. Again, because of the effects of the finite range and of the finite size of the core nucleus, the values of - Va(r) for the longer range P2~ are substantially smaller than the corresponding values of D, while f o r / t z the differences are much less (for zero-range forces, one has V(r) = D(p(r)). Thus, especially for ~u2~, surface effects in aC 13 are sufficiently important that the central depth, which has the value - V 4 ( 0 ) = 33.5+2 MeV for both #2~ and /tK, is not a reliable indication of the corresponding well depth for nuclear matter, which for/~2~ and for p = 0.168 fro-3 is D = 42 MeV for U = 1000 M e V . fm 3. This is in spite of the fact that the central density of C 12 is quite close to that of nuclear matter. For exchange forces of range #~ t one finds D (e~ch) = rid (°rd) = 29.5 MeV for U = 800 M e V - fm 3 and r/ = 0.88, instead o f the value D ~°rd) = 33.6 MeV which would be obtained for ordinary forces of range #~ 1. Experimental results 32) for heavy nuclei with 60 < A < 100 (corresponding to the heavy nuclei in emulsions) indicate a value o f Ba in the region of 25 MeV and not much in excess o f 30 MeV. Even for these nuclei Ba is still substantially less than D both because the A kinetic energy is still appreciable (thus for A = 60 one has ( T a ) = 6.5 and 7.5 MeV for Ba = 25 and 35 MeV, respectively) and also because an appreciable fraction of the nucleons is still contained in the nuclear surface. Thus using a Fermi density distribution appropriate for heavier nuclei (i.e., with p o = 0.168 f m - 3, s = 2.49 fm and c = 1.08 A ~ fro), one obtains for BCaa)(D) the following values (in MeV): B(A12°)(40) = 32.3, B(at2°)(30) = 22.9, B(6°)(40) = 28.2, B(a6°)(30) = 19.5 for P2~; and B(at2°)(40) = 34.3, B(a12°)(30) = 24.5, B(A6°)(40) = 30.2, Bta6°)(30) = 21 for #K-- the dependence of Ba on D being linear for all cases. Even for A = 200 there

p SHELL HYPERNUCLEI

613

is still a difference of about 5 MeV between D and Bet. The effect of the finite range is, however, very slight for these large values of A. Using a mean value A ,~ 80 appropriate for the heavier emulsion nuclei, one obtains for ordinary forces the values Bet ~ 32___2 MeV for #2~ with U = U4 = 1050___50 M e V . fm 3, and Bet ,~ 25___2 MeV for/~K with U = U, = 790_ 50 M e V . fm a. With the latter value of U and for an exchange force of range p~ 1, one would have the value Bet ~ 22 4-2 MeV. None o f these values seems inconsistent with the experimental results and it seems doubtful whether significance should be attached, at present, to the slightly large predicted values for # 2 ~ - especially if uncertainties in U of about 10 ~ , or possibly even more, are accepted t. Such uncertainties may arise not only from errors in the values o f the volume integrals U, and U12 but also from possible uncertainties due to a lack of knowledge of the shape and velocity dependence o f the A - N interaction as well as from possible differences in higher order effects o f the interaction in heavy and light hypernuclei. Our value D ~ 43 MeV for #2~ obtained with Yukawa interactions agrees well with that of Ram and Downs a3) who used a hard core (of radius 0.4 fro) together with an attractive well which in the asymptotic region possesses an exponential behaviour appropriate to the two-pion-exchange mechanism, although for such a force the relative p state contributions are considerably enhanced compared with those for soft forces such as we use. 3.3. THE H Y P E R N U C L E U S etBe°

Results for ½U8 obtained with oscillator density distributions are shown in table 3 for Bet = 6.5___0.25 MeV. Since the isolated Be a core nucleus is unstable, its size and energy when the A is present are not known. The re-arrangement energy of the TABLE 3 Results for .iBe9 with two-body forces for Bet = 6.54-0.25 MeV /t~t ½Us (MeV" fm s) 9604-15 b

C (MeV)

,UK ~'U, F~p~°) (MeV)

½Us

C

"~UsF~p

(MeV" fm 8)

(MeV)

(MeV

0.624-0.09

1.484-0.03

7854-15

0.68-4-0.11

1.454-0.(

1044-4-18

0.814-0.09

1.434-0.02

8654-15

0.924-0.10

1.404-0.1

11264-20

1.014-0.19

1.384-0.04

9504-15

1.244-0.13

1.344-0.1

The indicated errors are only those due to the uncertainty in Bet.

core (i.e., the difference between the total energy of the core with the A present and that o f the isolated core) must be positive. Thus the value of ½(-/8 obtained from our two-body model calculations for a given Bet will be a lower limit to the value which t The predicted value for/*2~ would in fact be reduced to about 28 MeV if there was an experimental bias towards the smaller values of A in the range 60 ~ A =< 100.

614

A. R. BODMER AND J. W . MURPHY

would be obtained for some value o f the core size (assumed to correspond to the actual size) if the corresponding rearrangement energy was included t. Bearing this in mind, inspection of table 3 then indicates that a size parameter in the range 1.5 fm < a < 1.6 fm gives results which are consistent with those for U4 and ½U12 as well as with reasonable rearrangement energy of < 1 MeV. To obtain more definite conclusions, we need a more dynamical approach which includes the degrees of freedom of the Be s core. A reasonable approach is to consider aBe 9 to be a three-body system consisting of two ~ particles and the A as has been done by Wilhelmsson and Zielinski 18) and more quantitatively by Suh 18). This three-body model has been investigated in more detail by Bodmer and Ali 19) using more realistic ~-~ interactions as well as a better trial wave function than used by Suh. With ct-~ interactions which give s wave phase shifts in agreement with the experimental values, one then obtains for Ba = 6.5_+0.15 MeV the values ½Us = 1026+60 M e V . fm a and 793-+50 MeV" fm a for /t2= and /tz, respectively, for an rms radius R~ = 1.44-+0.07 fm. In fact the values are very close to those obtained f r o m s H e s with the same R, and thus are also in good agreement with the values obtained from AC13. The size of the Be 8 core which is obtained from the three-body calculations may be expressed in terms of an equivalent oscillator size parameter. This has the value 1.65-t-0.03 fm, while the re-arrangement energy turns out to be somewhat less than one MeV. For such values the results of table 3 are thus reasonably consistent with those of the three-body model calculations. As discussed in ref. 19) the (not too appreciable) discrepancy between the two sets of results, for a given value of a and with allowance for the rearrangement energy in the three-body calculations, is most probably due to the fact that correlations between the A and each of the ~ particles are allowed for in the three-body model whereas for the two-body model calculations the A can see the Be 8 core only as a whole. 3.4. TWO-BODY AND THREE-BODY INTERACTIONS For investigating the strength o f the three-body forces we shall again consider mainly s H e s and aC is in view of the inherent uncertainty of a two-body model analysis of aBe 9. Although with only two-body forces there is already good agreement between s H e s and aC 13, one cannot necessarily conclude from this that three-body forces are negligible. Thus if three-body forces are included one will obtain for each hypernucleus a relation between U and W; and if this relation should be similar for both hypernuclei, then consistency could be obtained for a large range (including zero) of three-body strength. On the other hand, if the two relations are sufficiently different, then together they will give significant limits on the strength of the threebody force. Results for aC is including three-body forces have been obtained by use o f the procedure discussed in sect. 2. As for the case of s H e s previously reported in ref. s) and for * A more detailed discussion is given in ref. is) in connection with the A = 7 hypernuclei.

p SHELL HYPERNUCLEI

615

the same reasons as discussed there the relation between U and W for aC ~3 was found to be linear within the computational accuracy. Thus one has U = U 4 - Z 4 W for a H e s, U = ~U12-Zj.

2 W for

(36)

aC x3,

(37)

where the values obtained for Z a_ x (given in table 5 below) depend on # and v and where U 4 and O"12 are the volume integrals obtained with two-body forces only. F r o m eq. (32), the expressions for Z a_ z in terms o f the relevant Slater integrals (whose values depend, of course, on the particular hypernucleus considered) are Z , - 18F~a°~) F(O) ' 2s 1~,(o)~_~, ~ ( o ) ± R

(38) ~.(2)

ZI 2 = ~'-2~ ~ 8 ~ 3 p ~ ' 8 - 3 p F(O) ~_o ~,(o) 2S T z"~t 2p

(39)

TABLE 4 The coefficients 0CNp and -BNp •.rnucleus

4= 7 tCz8 i B12

Coupling ~Np(Np--I) scheme LS {LS int.

, jj I.LS

/ int.

3 84 84 84 63 63

°CNp

flNp

3 36 28.9 20 27 21.8

0.24 2.30 2.15 2.08 1.73 1.42

Hypernucleus

Coupling scheme ~Np(Np-- 1)

[LS ALi s a)

{int./

LS aLP

int.

ILS "tBe9

~int.

9 9 18 18 18 18

~Np 9 8.720.4 10 9.84 18 17.55

fl/v 0.4 0.5 0.5 0.8 0.8

The m i x i n g o f the two core states h a v i n g JN = ½- a n d ~-, w h i c h is i m p o r t a n t for the calculation o f x, is i rtant here since the three-body interaction is effectively a scalar in the space o f the n u c l e o n variables if tht is in a n s state.

The coefficients ~Np and flNp, which have been discussed in sect. 2, are given in table 4 for various hypernuclei and coupling schemes. The values for intermediate coupling have been obtained by use of Soper's wave functions. The results do not in fact depend sensitively on the coupling scheme. It should be noted that if for the p nucleons one were to assume that only S and D two-particle antisymmetric states occur (i.e., that the fractional-parentage coefficients relating to two-particle P states were zero) then one would get ~Np = 3 x ½ N p ( N p - I). This is not usually true because of the antisymmetry of the many-particle wave function. The requirement of only S and D states together with the requirement that flNp = 0 (which is obtained by neglecting the angular correlations between the p nucleons), then gives the same dependence on the number o f nucleon pairs as is obtained for the s nucleons and effectively gives an upper limit for the value of ~ s . Thus for example for aC 13 in

616

A. R. BODMER AND J. W . MURPHY

L S coupling, one obtains ~t8 --- 36, whereas ~ N p ( N p - 1) = 84. This latter value would be obtained for a core state having the permutational symmetry [8] for the spatial coordinates, which of course is not possible because of the exclusion principle. The maximum possible symmetry is [44] and corresponds to a 3 . 6 ~ and a 32 ~ admixture o f xlp and asp two-particle states, respectively. For aBe 9, on the other hand, the value of ~4 in L S coupling is just the maximum possible value since the symmetry [4] is now allowed (and corresponds to a p shell "~ particle"). The much smaller value of ~t4 for .tti 9, which has the same number of nucleons as aBe 9, should also be noted. It is due to the lower spatial symmetry ([31 ] in L S coupling) of Li s as compared with Be s. As expected, the inhibiting effect of the exclusion principle in reducing CtNpbelow the value ~:Np(Np- 1) becomes progressively more important for larger A. This is quite in keeping with the situation found for nuclear matter 5) for which also the exchange character of the three-body force very much reduces its effect. F r o m table 5 it is seen that Z 4 is approximately twice as large as Z12, its greater value corresponding to a greater effectiveness of the three-body forces in s H e 5 than in AC13. This can be understood as due to the large s nucleon density o f He 4 on the one hand (the three-body and two-body contributions 'being approximately proportional to p2 and p, respectively) while on the other hand the effect of the rather large number of p nucleon pairs in AC13 is very much reduced by the exclusion principle as discussed above. In comparison, for aBe 9 in intermediate coupling and for B a = 6.5 MeV and a = 1.60 fm, one has the values Z s = 0.125 fm - a and 0.105 fm -a for #2~ and PK, respectively, if v -1 = 1.0 fm and Z s = 0.030 fm -a and 0.025 fro- 3 if v - 1 = 2.0 fm. These values are quite close to the corresponding values of Z12 and the three-body force contributions due to the s and p nucleons together are thus quite similar for aC 13 and aBe 9. In fact, for intermediate coupling one has Z12/Z 8 < 1, for all reasonable sizes of C 12, and this ratio is smaller than the corresponding ratio o f the number of p nucleons. The similar values o f Z s and Z12 are a consequence of the shell-model description and the use of electron scattering sizes. It seems clear that with these values of Z,t_ 1 no useful limits can be placed on the strength o f the three-body force from a comparison of aBe 9 with aC 13, while a comparison of ,iBe 9 with s H e s would not be expected to give anything significantly different from compairing s h e s with AC13. On the other hand, with an ~-~-A model for a b e 9 the value of Za is most probably very close to that of Z4. In this case, a comparison of .tBe 9 and s h e 5 (see ref. 19)) gives virtually no information about the three-body forces, while comparing either aBe 9 or s H e 5 with ,tC 13 would give the same information. Similar remarks would apply to aC 13 if one believed in an ~particle rather than a shell-model description of C 12. Combining our results for s H e 5 and aC 13 then gives the values of U shown in table 5. The results for v - 1 = 1.4 fm have not been given but are intermediate between those for v - 1 = 1.0 fm and 2.0 fm. Also shown are the corresponding ratios of the three-body to the total potential energy. These ratios are 44 = ( U 4 - U ) / U , , and ~12 = (½U12-U)/:]U12 for s H e s and aC la, respectively. The corresponding ratio

TABLE 5

Results for two-body and three-body forces /z~n; U4 = 10404-60 MeV. fm s

a (fm)

v- x = 1 fm Z4 = 0.228 fm -8

½Ull Zi2 (fm-3)

U (MeV. fm s)

9034-15

0.122

1.65 9984-20 1.65 (LS) 9984-20 1.65 (ii) 9984-20 1.75 1093-t-25

0.110 0.119 0.099 0.099

1.55

(MeV" fm 3)

v- t = 2 f m Z4 = 0.062 fm -3 Z12 (fm_S)

U (MeV. fm8)

~lZ H 100

~4X 100

22.04-9

0.029

7824-60

13.34-6.5

24.74-10

25.54-10.5

6.74-10 6.7+10 5.04-7.5 --6.04-4

0.027 0.029 0.023 0.025

9664-60 9614-70 9734-60 11294-70

3.24-5.5 3.74-6 2.54-4 --3.3+3.5

7.24-10 7.64-11 6.44-9 --8.54-12

5.74- 8.5 6.54-10 5.14-7.5 --7.54-9.5

~e~2x 100

~e~x 100

~oo x 100

7454-80

17.54-8

28.34-11

9594-70 9524-80 9664-60 11344-65

3.94-6 4.64-8 3.24-5 --3.74-4

7.84-11 8.44-12 7.14-10 --9.04-7

~eooX 100

/*K; U4 = 7864-50 MeV" fm 3

a (fm)

v- 1 = 1 fm Z4 = 0.172 fm -3

½Ui8 ( M e V . f m a)

Z12 U (fm-3) (MeV" fmO

1.55 750-t-20 1.65 8244-20 1.65 (LS) 8244-20

0.102 0.091 0.098

6984-70 8674-80 8744-85

1.65 (jj) 1.75

0.082 0.084

8594-60 10334-65

8244-20 9154-20

v- 1 = 2 . 0 f m Z4 = 0.047 fm -3

~i2xlO0

~:4X 100

~e°°× 100

7.04-10.5 --5.24- 7.5 --6.14- 9

11.34-15.5 --10.34-14.5 --11.24-16

8.54-12 --9.44-13 --9.54-13.5

--4.24- 6 --13.54- 6

--9.24-13 --31.14-14.5

--7.64-11 --19.84- 9

Zla U (fm-a) (MeV" fm a) ~12x 100

~4xlO0

~x

100

0.024 0.022 0.024

7124-70 8574-60 8644-75

5.04-8 --4.14-6 --4.84-7

9.4!13 --9.14-17.5 --9.94-19

10.04-15 --11.34-20 --12.14-21

0.019 0.021

8504-45 10194-50

--3.14-4.5 --11.44-5

--8.14-12 --29.74-13.5

--9.54-14 --26.34-12

The errors shown for U and ~.4-t are due to the errors in Ba(AC TM) = 10.94-0.5 MeV and in/-/4. For U both contributions to the error are mostly comparable whereas the errors in ~e4_ 1 are dominated by the error in U4.

t" t"

618

A. R . B O D M E R A N D

J. W.

MURPHY

Coo = ( D 2 - D)/D z for nuclear matter has been obtained by use of the results of ref. 5) for the well depths D and D2 of the total and two-body force, respectively. For a given size of C 12, the values ~4, ~12 and ~o are comparable, the values of ~4 being on the whole somewhat larger than those of ~12 and 4oo for the reasons already discussed. The precise numerical values are seen to depend in a rather involved way on the particular ranges and densities considered. For a given range of the two-body force and a given size of C 12, the ratios ~a-1 which are obtained do not depend much on the range of the three-body force, while for given # and v the values of ~A-1 decrease as the size increases. The values of ~A-1 correspond to attractive threebody forces for small values of a (i.e., somewhat less than electron-scattering sizes) and repulsive forces for larger values of a, the dividing line depending on the range of the two-body force. In particular it is an important result that for the actual size expected for the C 12 core (i.e., a = 1.64-t-0.05 fro) the magnitude deduced for the three-body force is quite small, namely 1¢121 ~ 0+0.15 for both #2~ and /zz. The most probable values correspond to weak and attractive three-body forces for/t2~ and weak but repulsive forces for #z. These small strengths which are obtained for the three-body central forces seem consistent with meson-theoretical calculations 21) _ those of Spitzer give attractive three-body forces while the others give repulsive forces. 4. Hypernuclei with Core Nuclei Having Spin

Apart from aLi 7, which is discussed in ref. 12), the p shell hypernuclei whose core nuclei have non-zero spin JN ~ 0 and for which also the values of B,t are reasonably well known are aB 12, aLi 9 and the mirror pair aLi 8 and ABes. If these hypernuclei were considered individually and a calculation ofB a for each was attempted as has mainly been done so far in this paper, then the spin-dependent contribution to Ba would be masked by even fairly small uncertainties in U, as is shown in detail in ref. 12) for the most favourable case of aLi 7. This is because the spin-dependent contribution is roughly of order of magnitude 1/.4 relative to the dominant spin-averaged contribution which depends on U( and W). This uncertaintydueto U may be largely avoided by considering, in a manner similar to that in ref. 6), the differences 6Ba for the following pairs: ABe9 and aLi 8, ABe9 and aLi 9, .4C13 and aB 12. From eq. (32) one then has 6B~ =

(o) -xAF2p (o) + (aBA~ 6a+l-6~F
(40)

where 6B~ may, strictly, only be identified with the experimental value if the rearrangement energies are the same for both members of a pair. Thus B~ denotes the A separation energy which is obtained for a given core size if the re-arrangement energy of the core is neglected. For the pair aBe 9 - a L i 9 the first term on the righthand side of eq. (40) is not present since A is the same for both members of the pair.

p SHELL HYPERNUCLEI

619

In eq. (40) the term depending on U(W) is now comparable with the other terms since it arises from the interaction of the A with only one nucleon. In fact we take U ( W = 0) = 1000 MeV" f m 3 f o r / z 2 , and 760 MeV" fm a for Pz (consistent with the results for the .IN = 0 hypernuclei), the precise value of U now being unimportant. The third term is due to any difference between the core sizes of the two members o f a pair. To a good approximation the quantity denoted by (OBa/Oa)v is given by the rate of increases of Ba with respect to the oscillator size parameter, with U kept constant, for the JN = 0 partner (i.e., for a b e 9 and aC 13) and calculated assuming only two-body forces. This quantity m a y be obtained directly from the results of tables 1 and 3 or alternatively from the results for the relevant Slater integrals given in subsect. 2.4. In fact (OBa/Oa)v is given to a fair approximation by just considering the interaction energy of the A with the p nucleons only, since the remaining part of the energy, namely C = UFt2° ) - ( T a ) , is small and does not change very much with a. One always has that (OBa/Oa)v < 0 since,t-(o) 2p increases as the size parameter decreases. The last term in eq. (40) is the difference in the potential energy of the threebody forces for which the relevant values of 6~ and 6/3 m a y be obtained from table 4. Instead of W it is of more immediate physical significance, especially with reference to the results of subsect. 3.4, to use ~t2, i.e., to use the aC x3 ratio of the potential energy due to the three-body forces to the total potential energy. Correspondingly one then has in eq. (40) that W = U ( W = 0)~a2/Z12 and U(W) = ( 1 - ~ 1 2 ) U ( W = 0). Alternatively, o f course, ~4 or ~® could have been used. The results obtained from eq. (40), with the appropriate values for the various pairs, are shown in table 6 for P2~ and/z z and for the ranges v-1 = 1.0 fm and 2.0 fm of the three-body force. The results for v-1 = 1.4 fm are intermediate to those shown for these two values of v. The changes of A with respect to a and ~x2 are insensitive to the precise value used for a(JN = 0), while the value o f A for 6a = 0 and ~12 = 0 depends somewhat more on a, although not at all critically. The values used for x are those given by Dalitz 6). We have obtained effectively the same values for x. Considering first the results for only two-body forces, namely for ~x2 = 0, one notes that for the experimental values of 6Ba the values of A which are obtained for 6a = 0 (i.e., on the assumption of equal core sizes) are very considerably larger than the values (see page 625) obtained from the s shell hypernuclei. This seems true even when one allows for the substantial uncertainties due to the errors in 6Ba. This conclusion is in agreement with the results given for the matrix elements by Dalitz 6). However, it will be seen that the values of A depend very sensitively on the size differences 6a. This is because the size dependence of the dominant spin-averaged interaction energy is involved, with the result that even for quite small values of [6al the corresponding change in energy (OBa[Oa)v6a can easily become comparable to or larger than the energy difference due to the first two terms of eq. (40) which represent the contribution due to roughly only a single nucleon. A reflection of this is the increase o f (OBa/Oa)v with increasing A.

TABLE 6

T h e spin dependence A a n d related quantities for the p shell hypernuclei with A > 7 A for ~a = 0,~12 = 0 6A for ~ a = -t- 0.1 f m hA for ~12 = q- 0.1, a n d ~x~ = 0 ha = 0 ( M e V . flu ~) ( M e V . f m 3) ( M e V " f m a) ~,-1= 1.0 fln ~,-i = 2.0 f m

--0.154-0.3 ABeg-ALi8

0.6

8.83

0.44 0"854-0"3 a)

ABeg-ALi~

ACt3-AB TM

0.7

0.3

7.71

7.1

0.34

0.73

1.5 1.6 1.5 1.6

--1.5 -4-0.3

1.5 1.6 1.5

--0.5 q-0.3 a)

1.6

0.4 -4-0.4

1.6

a) Including 1 M e V re-arrangement energy for aBeL

P2~

PK

P~

PK

--13

465q z 470q z 200q z 170q z

80 80 80 90

350q: 355:]: 130q z lO0q:

65 75 65 75

qz qz q: qz

345 395 345 395

q:285 qz330 :F285 :]:330

q-30 -4-30 -4-30 4-30

4-27 q-25 4-27 :k25

-I-27 4-30 4-27 q-30

-4-19 q-20 -4-19 -4-20

--13

340q: 390q: ll0q: 130q:

70 80 70 80

280q: 327q: 90q: ll0q:

60 65 60 65

:q:: ~ ~ q:

345 395 345 395

:]:285 q:330 q:285 :q:330

q-55 +50 ±55 ±50

-4-45 ±40 q-45 -4-40

-4-60 -4-50 -4-60 -4-50

q-50 q-40 -4-50 -4-40

--19.5

610T225

:q:ll00

q:930

±55

-4-49

-4-50 -4-36

440:t:200

.>

p SHELL HYPERNUCLEI

621

For the pair a C l a - a B 1 2 one probably has positive 6a ,~ 0.I+0.1 fm, corresponding to the values a(C t2) = 1.64+0.05 fm and a(B 11) = 1.55+0.1 fm which are obtained from an analysis of the electron-scattering results 30). The re-arrangement energies are expected to be quite small and furthermore also quite similar for both hypernuclei. Any uncertainties due to re-arrangement effects are therefore expected to be much less than the uncertainty due to the experimental error in 6Ba. It is then immediately clear from table 6 that for very reasonable positive values o f 6a the value which is obtained for A is reduced from that for 3a = 0 to values quite consistent with those obtained from the s shell hypernuclei. Thus for/t2~ one gets A = 604-225 MeV" fm 3 for 6a = 0.05 fm and A = -5004-225 M e V - f m 3 for ~Sa = 0.1 fro, the errors being due to that in ~SBa. Thus, to obtain any significant information about the spin dependence A, it is necessary not only to have quite accurate experimental values for ~SBa but also very accurate knowledge of the core sizes. In fact one can reasonably reverse the procedure and supposing that A has a value about that obtained from the s shell hypernuclei (i.e., assuming that the A-N interaction is approximately known), one can consider the A as a fairly sensitive probe into size differences. In this way one obtains the value ~a = a ( C 1 2 ) - a ( B tl) 0.04_+0.02 fm independently of the range of the A - N interaction. The situation is quite similar for the other two pairs, although for these there is a further complication due to the re-arrangement effects which are almost certainly quite substantial for aBe 9. Thus an ~--~-A model of aBe 9 gives a re-arrangement energy of about 1 MeV and an effective oscillator size parameter of about 1.65 fm 19). It seems not unreasonable to expect the re-arrangement energies for a l l 8 and aLi 9 to be considerably less, since for these hypernuclei the distorting effect of the A is expected to affect all the nucleons o f the core more or less equally and thus to be a fairly small 1/A effect. (Thus the procedure discussed in the first footnote o f subsect. 3.1 gives a re-arrangement energy of < 0.2 MeV for reasonable compressibility coefficients.) This is in contrast to aBe 9 where, within the framework of an ~-~-A model, the distorting effect of the A on each of the tightly bound ~ particles will be very small but where there will be a quite large effect on the relative motion of the two ~ particles since Be a is not even bound. Assuming, then, zero re-arrangement energies for aLia and ALl9 and 1 MeV for aBe 9, one gets the (increased) effective values: fiB~ = 0.854-0.3 MeV and -0.54-0.3 MeV for a B e 9 - a L i a and a B e 9 - a L i 9, respectively. The actual values are likely to be somewhat, but not too much, less than these. The effect of the re-arrangement energy is then to reduce A to the values shown in table 6. These are consistent with the values obtained from the s shell hypernuclei even when the core sizes are assumed to be the same. In fact, for values o f 3B~ which are about those just considered, this then implies that the core sizes (in the hypernuclei) are very nearly the same. It is again clear that, because of uncertainties in the re-arrangement energies as well as in the core sizes, very little can be said about A except that there is certainly no inconsistency with the values obtained

622

A.R.

BODMER AND

J. W.

MURPHY

from the s shell hypernuclei. Also for the A = 7 hypernuclei, as shown in ref. 12) essentially nothing can be deduced about A from considering the difference tSBa between the T = 0 (aLi 7) and T = 1 (,tHe 7, aBe 7) hypernuclei. This is again because of uncertainties in the core sizes, re-arrangement energies and also in the experimental value of tSBa. Because of the increasing relative importance o f d~fferential size effects as A increases, the corresponding uncertainties in the values obtained for A will also become progressively larger for heavier hypernuclei, other things being equal. It seems that a fairly reliable value for the spin dependence is not likely to be obtained from the p shell hypernuclei unless the excitation energy of the spin flip excited state corresponding to the ground state of some hypernucleus can be determined reasonably accurately (since the core sizes can then be expected to be very nearly equal for both states). Three-body forces affect the value obtained for A by the amount shown in table 6 for 6 a = 0. This is seen to be small, although not entirely negligible, for strengths of the three-body forces which are consistent with those obtained in subsect. 3.4. The effect of three-body forces will thus be completely masked by even very small size differences of the order of 0.01 fm. It should be noted that the difference between the three-body potential energies of the members of a pair, and correspondingly the effect on A, is about the same in magnitude for the equal-mass-number pair ,tBe 9 - aLi 9 as for the other pairs whose members differ by one nucleon. Furthermore, it is interesting to note that for all the three ranges v-1 considered the difference in the three-body potential energy has the same sign and comparable magnitude for all three pairs. Consistency with the value of A obtained from the s shell hypernuclei could then be approximately obtained for 5a = 0 (and also neglecting any re-arrangement energies) with a value ¢12 ~ - 0 . 5 . This corresponds to a very strong repulsive three-body force which is unacceptable in view of our previous results. Other contributions to Ba for hypernuclei with JN ~ 0, and titus to the differences 6Ba for the pairs considered, may arise from non-central A - N forces. Thus, assuming equal sizes, Lawson and Soper 17) have obtained good agreement for the Ba of sHe s and of all the p shell hypernuclei by use o f a two-body tensor force in addition to a spin-dependent central force. For JN ~ 0, noncentral three-body forces may possibly also make an appreciable contribution in view of the fact that there are indications from meson-theoretical calculations 21) that such forces may be quite strong. It may be remarked that quite similar results, e.g., for 6a = 0 and ~12 = 0, can also be obtained by calculating the relevant Slater integral Jt~.~o) 2 p in a more approximate way. This may be done with the aid of a Gaussian A wave function having the same size parameter as the nucleon distribution and with Gaussian interactions that have the same intrinsic range as the Yukawa interactions we have used. Thus with a s = ap = aa = 1.6 fm, one obtains for the pairs a B e g - a L i s, a B e 9 - a L i 9 and aC 13-.tB 12 the values d = 450+72, 3105-62 and 6365-194 MeV- fm a for/~2z and d = 3505-66, 2905-58 and 4525-178 M e V - f m 3 /tK. For the pair a B e g - a L i 9 this

p SHELL HYPERNUCLEI

623

procedure somewhat underestimates A as compared with the values in table 6. This is because the approximate A wave function has a smaller extension than that obtained from eq. (4) and, correspondingly, x~ o2 p) is larger. For the other t w o pairs of hypernuclei this effect is masked by the presence of the term depending on U. 5. Conclusion Our studies of the p shell hypernuclei show that the simple assumption of a soft, central, spin-dependent and charge-independent, two-body A - N interaction is at present adequate to account for all the known values of B a including those of the s shell hypernuclei as well as those of the heavier hypernuclei relevant for a determination of the well depth D. In particular, if there are assumed to be no three-body forces, the values of the spin-averaged volume integral of the two-body interaction that are obtained from the hypernuclei aHe s, aBe 9 and aC la agreeverywellwith each other (for abe 9 see ref. 19)). As a reasonable average value for these hypernuclei one has U ~ 1020__+55 M e V . f m 3 and 800+45 M e V . fm 3 for ordinary Yukawa interactions of ranges #2~ and g~ 1, respectively. Further, we have been able to show that not only are central three-body forces not necessary but that any that may exist must in fact be weak; they cannot contribute more than about 20 ~o of the total interaction energy of aHe 5. The most probable sign depends on the range assumed for the two-body force. Thus for #2, the threebody interaction turns out to be most probably attractive while for #K it is most probably repulsive. These results for the three-body interaction depend on the validity of a shell-model description for C 12 with the values of the size parameters which have been determined by electron-scattering experiments. Uncertainties in the core sizes account for most o f the errors in the values of U as well as for the upper limits which it is at present possible to place on the strength o f the three-body forces. These size uncertainties furthermore preclude the possibility of deciding between the ranges spanned by #2~t and by p~ 1 (although an intermediate range is marginally favoured and a range as long as #~-1 is most probably excluded). A more accurate determination o f the relevant density distributions, especially that of C 12, would thus be very desirable. The uncertainty about the range, quite apart from other possible uncertainties in the form and/or due to higher order effects of the interaction, implies a correspondingly large uncertainty in the value predicted for the well depth D felt by a A in nuclear matter since this is proportional to U in first order. Thus for ordinary forces and with p = 0.168 fm -3, one has D(#2~ ) = 43___2.5 MeV and D(/~z) = 34.5+2 MeV; for exchange forces of range/t~ 1, the latter value would be reduced by a factor 0.88. The corresponding values of B a for A ~ 80, corresponding to the heavier emulsion nuclei, are B a(#2~) = 31 ___2 MeV and B a(/~z) = 25___2 MeV, the latter being reduced to 22__+2 MeV for exchange forces. None of these values of B a seems inconsistent with the experimental evidence.

624

A. R. BODMER AND

J. W.

MURPHY

It is to be noted that the effective A-nucleus potential Va(r) at or near the centre of C x2 depends only slightly on the range of the interaction and on the details of the shape of the density distribution of the core, but that I Va(0)l can differ very substantially from the corresponding value of D unless the range of the interaction is quite short. Thus for #2~ the value of IVa(0)l is considerably less than the corresponding value of D whereas for #K there is only a small difference. Thus, only if one believes the range of the A - N interaction to be quite short, as would be the case if the interaction is dominated by single exchanges of heavy mesons (~/, K, etc.), would I Va(0)l also be a reasonable guide to the value of D. The difference between ordinary and exchange interactions is even smaller for the p shell hypernuclei than for a A in nuclear matter, the proportion of the interaction energy in relative p states being quite small for soft forces even for the longer range #;~. Again, at least for soft forces, it does not seem possible to say anything about interactions in relative p states from the B a of the p shell hypernuclei. This may perhaps be possible for forces with a hard core, however, since for such forces interactions in relative p states are expected to be enhanced as compared with soft forces. The spin dependence A obtained from the p shell hypernuclei seems unfortunately to be at the mercy of quite small uncertainties in the core sizes and to a lesser extent also of uncertainties in the re-arrangement energies. Assuming no differences in the sizes and re-arrangement energies of the members of each of the three pairs of hypernuclei (aBe 9 - a L i s, aBe 9 - a L i 9, aC x3 - a B 12) for which the Ba are reasonably well known, one obtains very considerably larger values of A than are obtained from the s shell hypernuclei. However, for very reasonable and quite small size differences and for reasonable re-arrangement energies, there is not the slightest difficulty in obtaining values of A that are completely consistent with the s shell ones. Indeed the B a of just the p shell hypernuclei are quite consistent with a spin-independent interaction (and indeed also with one having a considerably larger spin-dependence than required for the s shell hypernuclei). It seems quite probable that an appreciable part of the fluctuations in the Ba of the p shell hypernuclei about the average trend is attributable merely to variations in the spin-averaged interaction energy, which arise from quite small size differences and from differences in the re-arrangement energies. In particular the re-arrangement energy for aBe 9 is expected to be quite appreciable (somewhat less than 1 MeV). Small differences in the spin-averaged interaction energy can be comparable with the whole of the spin-dependent part of the interaction energy because the latter is (roughly) only of order 1/A times the spin-averaged interaction energy. Central three-body forces have been shown not to contribute appreciably to the differences in Ba. Of course non-central forces (both two-body and three-body) could in principle contribute appreciably - but because of the uncertainties mentioned it seems most unlikely that anything could be deduced about such non-central forces. In fact, assuming the effect of these to be reasonably small and the effective spindependence thus to be about that for the s shell hypernuclei implies that the A is a

p SHELL HYPERNUCLEI

625

quite sensitive probe into small size differences for neighbouring core nuclei having similar or small re-arrangement energies. Quite generally it seems clear that the information obtainable about the A - N interaction is at present limited at least as much by uncertainties in the core sizes, and to a lesser extent by uncertainties in the rearrangement energies, as it is by uncertainties in Ba. In view of these conclusions, the only meaningful determination o f A (and hence of Ut) must at present therefore still be from the B a of the s shell hypernuclei. I f three-body forces are neglected and one also uses the results for a H 3 (ref. 1)), then one obtains A ~, 170+35 M e V - f m s and 25-t-20 MeV" fm 3 for /22~ and /~t, respectively, with Ba(aH 3) = 0.31_+0.15 MeV 28). The corresponding values o f U, are 3854-18 MeV" fm 3 and 2184-8 M e V . fm s for/~2~ and #K, respectively. If the very tentative indications of a range shorter than/~2~ are taken seriously, then this implies that the spin dependence A may be correspondingly smaller than for/z2~. If the results obtained for three-body forces in this paper (table 5) and also in ref. 5) for ,tH 3 are taken into account, one obtains somewhat larger values of A for/z2~ (corresponding to attractive three-body forces), namely A ,~, 2004-40 MeV" fm a, and slightly smaller values for/2K (corresponding to repulsive three-body-forces), namely A ~ 5~aSo MeV" fm 3. It is to be noted that if second-order and higher-order effects (which would decrease the value of U) are not negligible, then the values of A must be correspondingly increased (since there is no corresponding decrease in the volume integral U2 = ½Us+½Ut obtained from a H 3 as this system has been analysed 1) using an excellent variational wave function which includes pairwise correlations quite accurately). The spin-dependence could be determined more directly from the excitation energy of a state for which the A has flipped its spin with respect to (say) the ground state. This energy could be obtained from measurement o f the energy of the ),-ray of the relevant transition. For such states, which differ only in that the A has flipped its spin, it is reasonable to expect that the core wave functions will be quite similar and that therefore uncertainties due to differences in the core size and re-arrangement energy will be quite small. A favourable case seems to be ,tLi 7 for which the excitation energy of the first excited state (½+) relative to the ground state (½+) is E = ½AF(z°)(ap).The assumptions on which this value of E is obtained are that the p nucleons can be represented by oscillator wave functions and also, of course, that the wave functions of the p and the s nucleons are the same for both states. Then with A = 1504-50 MeV. fms one gets f o r / t 2 , the values E = 0.75_+0.25 MeV and 1.5__.0.5 MeV for ap = 2 fm and 1.6 fro, respectively. For/~K the values o f E will be correspondingly smaller. It is to be noted that especially for the larger values o f ap the value o f F2t°)(ap) is not very dependent on the range #-1. Clearly the accuracy with which A can be determined from a measurement of the excitation energy E will be limited by the accuracy with which one knows the extension of the p nucleons in ,tLi 7. Finally it should be remarked that no information about the A-N interaction

626

A. R. BODMER AND J. W . MURPHY

(and in particular about A) can be deduced from the Ba of the A = 7 hypernuclei with the two-body approach of this paper because of uncertainties in the core sizes, re-arrangement energies and also in the Ba. However, as shown in ref. 12) if (most plausibly) the effective A - N interaction in these hypernuclei is assumed to be about the same as that obtained from the s shell hypernuclei (and therefore for U also as obtained from aBe 9 and aCla), then the A becomes a quite effective probe into the A = 6 core nuclei. In particular, it is found that a structure consisting of an ~ particle plus two nucleons is strongly indicated for both the T = 0(Li 6) and the T = 1 (He 6, Be 6) core nuclei. We are grateful to Dr. R. D. Lawson for discussions and for communication of his work with Dr. J. M. Soper, whom we would like to thank for the use of his intermediate-coupled wave functions. We are indebted to the Manchester University Computing Laboratory for the use of their Ferranti"Mercury" and"Atlas" Computers. One of us (J.W.M.) would like to thank the Department of Scientific and Industrial Research for financial support during part of this work. Finally, we are grateful to Dr. F. E. Throw for a careful reading of the manuscript.

Appendix THE POTENTIAL SHAPE FUNCTIONS FOR TWO-BODY FORCES The functions v~)(r) defined by eq. (20) can be written in the form vt2k'(r) -- ½(2k+ 1 ) f o t 2 ~ ( O d t f ~

\ rI /

where t = r 1 - r and where z = r . t/rt. For normalized Yukawa interactions and with harmonic-oscillator functions, one can express v(2k)(r) in terms of the integrals

1

~o

Hh,(x) = -- [ y" exp [ - ½ ( x + y)2]dy, n!do

whose values are tabulated in ref. 25). Thus with xl = I~ajx/2 and zl = ~/2r/at, where l = s or p, one obtains

#2

vt2°)(r) = 4x/27t~--~ { e - ~ ' H h o ( x , - z~) - eZ'Hh o(X. + z~)} exp (¼x2), v~°p)(r) =

#2

{e-"r[nh2(xp-Zp)-zpHhl(xp-Zp)+½(2+Z2p)Hho(xp-zv)]

6V/2n~}r -

-

eU'[Hh2(xp + zp) + z v Hht(x v + zp) + ½(2 + z~)Hho(x v + Zp)]} exp (½x~).

627

p~ SHELL HYPERNUCLEI

For r = 0 the dependence on the range o f the interaction and on the nuclear size can be readily exhibited. For this case one has

l aq v~o)(0) =

3epp(ap).;/top{#ap~

'

where

~ ( x ) = x2nhl(x) a~'p(X) =

exp (½x2),

xenh3(x) exp (½x2).

The function ~ ( x ) is such that o~(f~(x) ~ 0 as x ~ 0 and ~ s ( x ) ~ 1 as x ~ oo. Thus in the limit o f very short-range forces (or o f very extended density distributions) the value o f v(2°)(0) is, as expected, just the central density. On the other hand, ~ p ( x ) ~ 0 for both x ~ 0 and x ~ oo. The short-range limit n o w reflects the fact that for a 6 function interaction the p shell nucleons will not contribute at r = 0.

References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10)

11) 12) 13) 14) 15) 16) 17) 18) 19) 20)

B. W. Downs and R. H. Dalitz, Phys. Rev. 114 (1959) 593 R. H. Dalitz and B. W. Downs, Phys. Rev. 111 (1958) 967 K. Dietrich, H. J. Mang and R. Folk, Nuclear Physics 50 (1964) 177 B. W. Downs, D. R. Smith and T. Truong, Phys. Rev. 129 (1963) 2730; D. R. Smith and B. W. Downs, Phys. Rev. 133 (1964) B461 A. R. Bodmer and S. Sampanthar, Nuclear Physics 31 (1962) 251 R. H. Dalitz, in Proc. Rutherford Int. Conf. (Heyword, London, 1961); Enrico Fermi Institute for Nuclear Stu~es, University of Chicago, Rept. EFINS-62-9 (March 1962) J. J. de Swart and C. Dullemond, Ann. of Phys. 19 (1962) 458 R. H. Dalitz, N/nth Int. Ann. Conf. on High Energy Physics, Vol. 1 (Academy of Sciences, USSR, Moscow, 1960) J. D. Chalk III and B. W. Downs, Phys. Rev. 132 (1963) 2727 B. W. Downs and W. E. Ware, Phys. Rev. 133 (1964) B 133; B. W. Downs, in Proc. Int. Conf. on Hypcrfragments, St. Cergue, Switzerland (1963) ed. by W. O. Lock, C E R N 64-1 (1964) J. D. Walecka, Nuovo Cim. 16 (1960) 342 A. R. Bodmer and J. W. Murphy, to be published L. M. Brown and M. Peshkin, Phys. Rev. 107 (1957) 272 S. Iwao, Nuovo Cim. 17 (1960) 491 R. D. Lawson and M. Rotenberg, Nuovo Cim. 17 (1960) 449 R. H. Dalitz, in Proc. Int. Conf. on Hyperfragrnents, St. Cergue, Switzerland (1963) ed. by W. O. Lock, C E R N 64-1 (1964) R. D. Lawson and J. M. Soper, private communication from R. D. Lawson, unpublished M. Wilhelmson and P. Zielinski, Nuclear Physics 6 (1958) 219; K, S. Suh, Phys. Rev. 111 (1958) 941 A. R. Bodmer and Shamsher Ali, Nuclear Physics 56 (1964) 657 B. W. Downs and R. J. N. Phillips, to be published (1964)

628

A. R. BODMER A N D J. W. MURPHY

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