IL I~UOV0 CIMENT0 VOL. L B, N. 2 11 Agosto 1967

A Note on the AA-Hypernucleus I'CA A.

S. ALI (*), L. P. KoK (**) and M. E. GRYPEOS

International Atomic Energy Agency J[~ternational Centre for Theoretical Physics - Trieste

(ricevuto il 12 Maggio 1967)

Currently a number of studies have been made of the recently discovered AA- hypernuclei lOBeAi (1) (or possibly I1BeAA ) and 6HeAA (2). Analyses of l°BeAh on a dynamical a-a-A-A model (3.4) which takes into account ra ther adequately the distortion of the unstable SBe core have been made. These analyses, agreeing essen- tially with each other, indicate tha t roughly about a th i rd of the experimentally observed addit ional binding energy A B A A ~ B A A - - 2 B A (where BAA is the AA binding energy in the AA-hypernucleus and B A is the A separation energy for the A-hyper- nucleus) arises due to core distortion. Fo r SHeAA, core distortion plays a very minor role (s) because of the high internal binding of the a-particle and the a-A-A model seems well justified (8). The strengths of the 'A-A potentials determined from the three-body analysis of SHeAA have been used by TAnG and H~RNDO~ (7) and also by ALI and BOD~ER (s) in a re-analysis of I°BeAA. Such analyses based on/~:-1 and #K 1 (the ranges corresponding to the 2-pion and K-meson exchange mechanism, respec- tively, of the A-A ° interaction) indicate a rather short range of the A-J~ interaction.

In this note we report the results of an investigation of the ye t unobserved s y s t e m 14CAA , using the A-A potent ial strengths quite reliably determined from 6HeAA. In view of the fair rigidity of the 12C core we approximate 14CAA by a three-body lzC-A-A model. The results presented here were obtained with the (( equivalent )) two-body method for three-body systems of ref. (9). This method gives the best

(*) On leave of absence f rom Atomic E n e r g y Centre, Dacca. (**) On leave of absence from the S ta te Univers i ty , Groningen. (1) ~I. ~)ANYSZ, ~ . GARBO~VSKA, J . PNIEWSKI, T. PNIEWSKI, ~. ZAKRZE~-SKI, E. R. FLETCHER,

J. LEMONNE, P. ]=~E~ARD, J. SACTON, W. To TONER, D. O'SULLIVAN, T. P. SHAH, A. THOMPSON, P. ALLEN, Sr. M. HEERAN, A. MONTWILL, J. ]~. ALLEN, M. J. BENISTON, D. H. DAVIS, D. A. GARBUTT, V. A. BULL, ~'~. C. KUMAR and 1 ~. V. MARCH: Nucl. Phys., 49, 121 (1963).

(2) D. J. PROWSE: PhyS. Rev. Lett., 17, 782 (1966). (3) ¥ . C. TANG and R. C. I~ERNDON: Phys. Rev., 138, B 637 (1965). (~) A. 1~. BODMER and S. AL~: Phys. Rev., 138, B 644 (1965). (~) Y, C. TAI~G and R. C. HERh~DON: Phys. Pev. Left., 14, 991 (1965). (6) S. ALI and A. R. BODMER: 5"UOVO Cimenfo (to be published), ICTP, Trieste, p repr in t IC/66/112. (7) y . C. TANG and R. C. t~ERNI)ON: t~rUOVO Cimenfo, 56, 517 (1966). (8) S. ALI and A. R. BODMER: Phys. Left., 24 B, 343 (1967). (~) A. R. ]~ODYrER and S. ALI: Nucl. Phys., 56, 657 (1964).

374 s. ALI, L, P. KOK a n d M. ]~. GRYPEOS

S-state var ia t iona l wave funct ion of the product form

3

~(r 1, r~, r a ) = 1- Igdr i ) , i=1

where the r~ are the in terpar t ic le separations. The two-body radial Schr6dinger w a v e f imct ion ]AA(rAA) which describes the re la t ive A-A mot ion and which corre- sponds to the appropr ia te th ree -body product wave funct ion in the in terpar t ic le separat ions is de termined by the Schr6dinger equat ion

(1) , MA (s)

] h A - - ' h ~ - [BAA q- (VAA + WAA)J/AA : 0 .

Here M A is the A-part icle mass, BAA is the separa t ion energy of the two A-part icles wi th respect to the ground-s ta te energy of the ~2C core, VAA is the A-A po ten t ia l for tke re levan t singlet s ta te and W <a>AALyA-r~ 12c, VA.~c] is t he addi t ional po ten t ia l (*) th rough which the th ree-body na ture of the system enters and is a funct ional of the re la t ive A-core funct ion g i -~c and of the A-core in te rac t ion VA_,~c. The la t te r was ob ta ined by folding a Yukawa A-J~ in terac t ion (for which we consider two ranges /x~= ~ 0.7 fm (bA~ = 1.5 fm) and ftK 1 0.4 fm (bA. v = 0.85 fm)) into the normal ized densi ty dis t r ibut ion ~ of the core. Thus

(2) VA-~c(r) = - - Gff Q~(rl)v(rA1) dr1 =

4 r~'~ + ~ ~--~ exp [ -- r~/a ~] v(rA~)dr1,

where v(rA1 ) is the normal ized shape funct ion of ti le Yukawa A - ~ ) in te rac t ion and U1, = 3 U 4, U 4 being four t imes the spin-averaged vo lume integral of the A-~N" interact ion. The electron sca t ter ing da ta for ~C are well f i t ted (~0.1~) wi th a = (1.64 ~ 0.05) fro. The numer ica l solution of the A-I~C eigenvalue problem wi th po ten t ia l ( 2 ) t o reproduce the most recent exper imenta l value of BA(13CA)= == (10.51-4-0.5) ~ e V (1~) yields for U 4 the values 970.0 MeV fm a and 819.5 MeV fm 3 for /~rt and /~K respect ively. These values are in agreement wi th the corresponding values of U 4 obta ined f rom 5HeA. Fo r gA-~c we use the t r ia l funct ion of the form

(3) gA.~c(r) = exp [-- :or] + s exp [--fir] .

This has been found to be a ve ry good tr ial wave funct ion. A two-body var ia t ional calculat ion for the A.core problem using (2) and (3) yields, for bo th /~fn and /~K,

(') See ref. (') for detailed expressions for If'¢S)AA. (1o) H . F. EI-IRENBERG, JR. HOFzTADER, U. MEYEX-BERKItgUT D. G. RAVENHALL and S. S.

SOBO'ITKA: t'hys. Rev., 113, 666 (1959). (it) U. MEYER-BERKHOUT, K. W. FORD and A. 1~. S. GREEN: A?t?t. of Phys., 8, 119 (1959). (tz) ~¢V. (AAJEWSKI, C. MAYEUR, J . SACTON, l ), VILAIN, G. WIJ,(~UEI', D. HARMSEN, 1~. LEvi

SETT[, M. RAYMOND, J . ZAKRZE~VSKI, D. STANLEY, D. H. D&VIS, E. R. FI_ETGIIER J . E. ALLEN. Y. A. BULL, A. P. CONWAY and P. V. MARCH: Nucl. Phys., B 1, 105 (1967).

A NOTE ON THE AA-HYPERNUCLEUS ~4CAA 375

values of B A which are less than 1% of the exper imenta l B A value. Thus for reasons ment ioned in ref. (9) we use (3) also for the th ree-body problem. W~A is then com- puted as a funct ion of rAA and of the var ia t iona l parameters ~, fl and s. F o r a given s t rength of the A-A potent ia l , the numer ica l solut ion of the Sehr6dinger equat ion (1) gives BAA as a func t ion of c~, fl, s arid the m a x i m u m of this funct ion gives the required value of BAA for this s t rength.

The following fo rm was chosen for the A-A potent ia l

(4) ~a) exp [--/~rAA ]

V A A ( r A A ) = ~ UAA . . . . . . , /~rAA

with # - 1 = #2-~ = 0.7 fm corresponding to the 2-pion exchange mechanism. UAA is the volume in tegra l of the A-A potent ia l . This potent ia l corresponds to an intr insic range of b = 1.5 fm (*).

The results ob ta ined for BAA aS a funct ion of UAA arc shown in Table I. Also 2 shown in the same table are the r.m.s, values of the A-A separa t ion <rAA}-, the

TABLE I. -- Results for 14CAA aS a function of the volume integral UAA of the A-A potential (4).

i i

Range ! UAA t'A'-oV !i (I~IeV fin a)

1 i

3OO i --1

I 300

J I 300

/~1 0 - - 300

~ A A (MeV)

25.01 20.91 18.79

25.20 20.90 18.64

<v) (Me'V)

38.30 29.50 26.10

39.10 29.90 26.25

2 1 <raA)~ (fm)

2.58 3.04 3.33

2.52 2.97 3.26

Op t imum parameters

(fro -I) f l ( fm -1)

1.05 1.23 1 . 0 4 1 . 2 5

1.02 1.27 i

J 1 . 0 7 1 . 2 6

1 . 0 1 1 . 3 5

[ 1 . 0 0 1 . 3 3

- - 0.92 - - (1.92 - - 0.905

- - 0.92 - -0 .87 - - 0.86

expec ta t ion value of the po ten t ia l energy <V) = <VAA-~ W~)A} and the op t imum parameters of gA.~e. I t is seen tha t the values of the op t imum parameters for dif- ferent values of UAA are more or less the same as those of the A-core var ia t ional parameters (~ = 1.086 fm 1, fl = 1.247 fm -1, s = - - 0.930) for P2~: and (a = 1.083 fm -1, f l = 1.256 fm -1, s = 0.926) for /~K" This is because the core, i.e. 12C, is ra ther heavy and hence the A-core correlat ions are quite impor tan t . In keeping ~fith this is the fact t ha t since the over-al l differences in A-~2C potent ia ls for the two ranges l*~-~ and ~.~ (not shown) are much smaller than in the ease of A-4He, one expects very lit t le difference in the three-body binding energy values BAA for these two ranges. This is indeed found to be the ease. A similar s i tuat ion was observed (s) in the ease

(*) I n ref . (~) i t was s h o w n for b ~ 1.5 h~t t h a t ~ p u r e l y a t t r a c t i v e A-A lm(CldiaI , a pheno- m e n e l ) g i c a l h a r d c~re A-A i ) ~ t e n t i a l a n d t i l e m e s o n t h e o r e t i c a l A-A 1)otent ia l a l l h a v i n g t he s ame intrt,~.qe ra.ngo arc a . lmost e q u i v M c n t fo r t h e A A - h y p c r n u e l e i in t he sense t h a t t h e b i n d i n g energy /)'~A a n d t h e s c a t t e r i n g l e n g ; h aAA t~re about, t h e s a m e for MI these potentia.Is .

376 s . ALI, L. P. KOK and ~. s. GRYPW.OS

of ~°BeAA where again the differences between the results f o r / ~ a n d / ~ i were rather small. However, it was found by TANG and HERNDON (7) and by ALl and BODMER (s) that on a four-body model of I°BeAA one obtains with #2~ too much binding for I°BeAA than with /~: and it was concluded that the experimental value of ABAA0°BeAA) favours a shorter range for the A-A" interaction. I t should be made clear here that this conclusion was based on the fact that for the experimental value of BAA (SHeAA) = = (10.8 ± 0.6) McV, the volume integrals, e.g. of ref. (s), UA A = (310+_~) MeV fm 3 and UAA = (265 ~= 25) MeV fm a for/~2= and/~g respectively, were taken to be reliably deter- mined from the three-body model calculations of 6HeAA (in fact, these calculations should be very accurate because the ~ distortion )) effects in the eases of AA-hyper- nuclei should be at their minimum in SHeAA) and since the differences in the BAA values of ~°BeAA for /~2x and /~x are very small, one obtains a greater value of BAA(X°BeAA) if one takes for UAA the value corresponding t o / ~ rather than the one corresponding to ~K (*)" In the present case, the above values of (UAA)g~ ~

and (UAA)~r¢ give BAA values of (25.2+_~-~ 6) MeV and (24.55_q_0:~) MeV respectively

for l l C A A , a s can be seen from a plot of UAA vs . BAA. A comparison of these values with the experimental one, when it becomes available, is expected to give indications about the range of the A-A ° interaction (**).

The small positive value of ABAA when VAA~ 0 arises due to the A-particle being correlated as a result of the finite mass of the core and approaches zero as the core becomes more and more heavy. In the present ease we obtain (using the varia- t ional result for B A ( = 10.42 ~/[eV) rather than the experimental value of BA) ABAA(VAA ~ 0) = 0.06 MeV for both / ~ and / ~ . This is nicely consistent with the corresponding value of 0.14MeV for XOBeAA (a). The values of (V) and ( r~A)i are in the expected direction and provide the features of the A-A correlations involved in l a C A A . As mentioned in ref. (9), the A-A correlation is treated exactly by our method.

I t is worth mentioning that although the x2C core in laCAA is fairly rigid, it is perhaps not very strictly so. Although the estimates of BODMER and M U R r ~ (l~) on the lines of ref. (~.~7) indicate a decrease of only g 3% for the core radius for reasonable values (~ 100MeV) of the compressibility coefficient K, HERI~DOI~ and TANG 0 a) find on the basis of an ~-~¢-~-A model of ~3C A that the r.m.s, value of the separation distance between two ~-par~icles in the ~C core of ~3C A becomes ~ 8 o

(*) T h e s c a t t e r i n g l eng thaAA , t h e effect ive r a n g e r 0 A A a n d t h e w e l l d e p t h p a r a m e t e r SAA for ( UAA)~2 g

and (UAA),~ are -- (1.5~:~) fro, (2.71+_g°:~) ~m, 0.59 ~0.054 and (-- 1.1 ±0.2) ~ , (3.94+°:~ °) fro, 0.51 ±0 .054 , r e spec t ive ly .

(**) A r e c e n t ana lys i s (~s) of BBeA a n d laC A b y HERNDON a n d TANG po in t s to a s h o r t r a n g e of t h e A-JV i n t e r a c t i o n . Th i s conc lus ion is h o w e v e r l i n k e d w i t h t h e r e l i ab i l i ty of t h e e x p e r i m e n t a l B A va lues for these s y s t e m s . F o r example , H e r n d o n a n d T a n g ' s ana lys i s g ives BA( 'Be A) ~ ( 6 . ~ 0 ± 0 . 2 8 ) M e V for /~2rc a n d (6.11 ~_0.26)MeV for t~K. C o m p a r i n g these w i t h the exper i - m e n t a l v a l u e of (6.24 ~0 .25 ) MeV q u o t e d b y MAyEUR et al. (~4), t h e i r a n a l y s i s seems to f a v o u r t h e s h o r t e r - r a n g e ~ 1 . H o w e v e r , if t h e m o s t r e c e n t (~) e x p e r i m e n t a l v a l u e of BA( 'Be A) = (6.66 ± 0.08) McV is t a k e n , u ~ becomes more f a v o u r a b l e . H e r n d o n a n d T a n g ' s a n a l y s i s of ~C A, however , sugges t s a s h o r t - r a n g e d A - ~ p o t e n t i a l w i t h a n i n t r i n s i c r a n g e less t h a n a b o u t 1 fro.

(is) R . C. HERNDOI~ a n d Y. C. TANG: Phys. Rev., 149, 735 (1966). (:4) C. MAyEUR, J . SAC'TON, P. VILAIN, G. WILQUET, D. STANLEY, P. ALLEN, D. H. DAVIS,

]~. R . FLETCHER, D. A. GA~BUTT, M. A. SHAUKAT, J . E. ALLEN, V. A. BULL, A o t ' . CON~VAY a n d D. V. MARCH: Nuovo Cimento, 43 A, 180 (1966).

(is) A . l~. BODY£En a n d J . W . MURPHY: Nucl. Phys., 64, 593 (1965). (~6) In. H . DALITZ a n d B. W. D o w N s : Phys. Rev., 111, 967 (1958). (~7) A. R . BODMER a n d S. SAI~IPANTHAR: NUCZ. Phys., 31, 251 (1962).

A NOTE ON THE AA-HYPERNUCLEUS 14CAA 377

smaller than the corresponding value in the free 12C core. Thus one expects that although the distortion of the core by one A-particle is not very large, distortion by two A-particles is presumably significant. The contribution to ABAA of this distor- tion effect for a variety of A-A potential shapes is being at present investigated by the authors.

On the basis of the present results for laCAA, one would predict a value BAA(~4CAA) ='(25.2 ± 0.5) MeV or (24.5 =k 0.5) Y[eV depending on whether the range p~5, is closer to -~ or #~= / ~ . Taking into account the distortion of the ~2C core is expected to widen the energy difference somewhat. We finally emphasize that since the existing results of calculations on SHeAA, being practically free from core distortion effects, provide an excellent basis for comparison, we feel that an experimental deter- mination of BAA(~4CAA ) would be il luminating on the range of the A - ~ interaction. If the experimental value of BAA(~4CAA ) favours a short range of the A-A ~interaction (corresponding to an intrinsic range b~< 1 fm) then this combined with the need (lS.~2) of a largB b (--~ 2.1 fro) to fit A-p scattering data would tend support to the existence of a repulsive core in the A-A p interaction.

$ $ $

We are very grateful to Profs. A. SALAM and P. BUDINI and the IAEA for their hospitality at the Internat ional Centre for Theoretical Physics, Trieste. One of us (L.P.K.) wishes to express his gratitude to the Foundat ion F.O.M. (Fundamentccl Onderzoek der Materie) at Utrecht, and UNESCO for granting a leave of absence and for financial support.

(x$) A. R. BODMER: Phys . Rev. , 141, 1387 (1966). (~) R. I t . DALITZ: an inv i t ed pape r presented a t the Topical Conference on the Use el Elementary

Particles in Nuclear Structure Research (Univers i ty of ]~russels, Sept. 1965), in press. (~0) S. ALI, M. E. GRYPEOS and L. P. KOK: Phys . Left. (to he published). ICTP, Trieste, prc-

p r in t 1C/67/16. ~i) ]~. C. HERNDON a nd Y. C. TANG: Lawrence I~ad. Lab. p rep r in t on Phenomenoh)yicat

A-nucleon potentials from S-sheU hyper~ucIel. - I I : Dependence on intrinsic range, Feb. 1967. (2~) G. ALEXANDER a nd U. KARSItAN: inv i t ed t a lk on Low-Energy Hyperon-Nuclcon Interaction,

presented a t the Second International Con]erence on High-Energy Physics and Nuclear Structure, Rehovoth, 27 F e b . - 3 M~r. 1967.