Reaction-matrix calculations of the Λ-particle binding in nuclear matter for central ΛN potentials

l.C

I

Nuclear Physics A201 (1973) 145--178; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

REACTION-MATRIX

CALCULATIONS

OF T H E A - P A R T I C L E B I N D I N G I N N U C L E A R M A T T E R FOR CENTRAL

AN POTENTIALS

A. R. BODMER

Physics Division, Argonne National Laboratory, Argonne, Illinois 60439 and University of Illinois at Chicago Circle, Chicago, Illinois 60680 and D. M. ROTE t

Center for Environmental Studies, Argonne National Laboratory, Argonne, lllinois 60439 tt Received 12 June 1972 (Revised 18 September 1972) Abstract: Extensive results, obtained in the G-matrix approximation for a wide range of central A N Yukawa potentials both with hard cores (I-IC) and soft repulsive cores, are presented for the A well depth (binding energy of a A-particle in nuclear matter) and related quantities. The HC results, in particular, span a wide range of scattering lengths a, effective ranges ro and hard-core radii e, sufficient to include any potentials likely to be proposed. The s-state well depth Ds is to a good approximation determined by a, ro and the s-state correlation volume (wound integral) KsNMwhich characterizes the core in an effectively shape-independent way. The depth Ds decreases with KsNMfor given a, ro. A successful search was made for a "scattering" parameter Kssc which also characterizes the core in a shape-independent way but which is determined only by (lowenergy) s-wave scattering calculations; ,qsc is the (zero-energy) scattering correlation volume up to the separation distance d. The parameterization o f D~ in terms of a, ro and Kssc is then effectively universal for local potentials, permitting one to obtain Ds for any reasonable local potential by use only of s-wave scattering calculations. The p-wave depth Dp is generally quite large with only a moderate dependence on c for given a and ro but a strong dependence on ro. The p-wave correlation volume is not a useful shape-independent parameter for Dp. The d-wave depth is quite small although not negligible. Limited results are obtained for the dependence on density. The implications o f a comparison of the calculated and the phenomenological well depths for the A N interaction are discussed.

1. Introduction E x t e n s i v e r e a c t i o n - m a t r i x s t u d i e s o f t h e A - p a r t i c l e b i n d i n g in n u c l e a r m a t t e r - t h e s o - c a l l e d A well d e p t h D - h a v e r e c e n t l y b e e n m a d e in t h e G - m a t r i x a p p r o x i m a t i o n [refs. l - s ) ] .

This approach

is s u i t a b l e f o r p o t e n t i a l s w i t h s t r o n g l y r e p u l s i v e c o r e s .

t Work done while at the University of Illinois, Chicago Circle, Chicago, Illinois. ** Work supported in part by the US Atomic Energy Commission, by the National Science Foundation under grant G.P.9229, and by the U K Science Research Council while the first author was with the Nuclear Physics Laboratory, Oxford, U K wl',ere part of this work was done. 145

146

A.R. BODMER AND D. M. ROTE

Another approach also suitable for such potentials is the variational approach using Jastrow-type trial functions. It has also been extensively used to calculate D[ref. 9)]. In implementing the reaction-matrix approach, the G-matrix has been calculated by two distinct methods. One is the independent-pair approximation (IPA)3, 10) which uses the Bethe-Goldstone equation, and the other is the reference-spectrum method (RSM)1, 2, 4, 5, 7, 11). The two seem to give very similar results for D in practice, although the assumptions involved are somewhat different. The RSM, with the assumption of the pure kinetic energies for the single-particle unoccupied states, has been used to obtain D by Dabrowski and Hassan ~' 4) and by Rote and Bodmer 2, 5, 7). The former use the integral-equation method of Brueckner and Gammel 12) to obtain the G-matrix, while the latter use the Kallio-Day integrodifferential equation ~3). For the same assumptions about the single-particle spectra, the two methods are effectively equivalent, and indeed the results agree very well for the same potentials. Reaction-matrix calculations have also been made for A N interactions with a tensor force 4-6) and with coupling to the 27N channel 7, 8). In the present paper, as in ref. 2) (referred to as I), we consider central AN potentials and calculate D in the G-matrix approximation with the Kallio-Day procedure [ref. 13)]. The aims of the present paper are twofold. First, we wish to obtain a better understanding of those properties of central AN potentials that determine D. The potentials considered in refs. 1, 2) were restricted to a quite limited set of phenomenological central hard-core potentials related to the Ap scattering 14) and to the binding energies of the s-shell hypernuclei 15); in particular the potentials obtained by Tang and Herndon 16. 17) were used. The results strongly indicated that for a given scattering length a and effective range ro, the sstate well depth D s depends quite strongly on the hard-core radius c, becoming larger for smaller c. In the present paper, we extend and generalize these results by considering a large variety both of hard-core and soft-repulsive-core potentials. In particular, we confirm the important result that, for the local central potentials we consider, the short-range effects of the potential on Ds are effectively completely determined by the s-wave correlation volume x~ rd - i.e., the "wound integral" - defined by eq. (3) with L = 0. Then for given nuclear-matter conditions, D~ is completely determined by three effectively shape-independent parameters a, r 0 and x~ M. (For hard-core potentials x~ M is of course uniquely related to c for given a, r 0.) Especially, in view of the results obtained by various authors for non-local potentials is), we may conjecture very plausibly that for any potentials, including non-local ones, D s is in effect completely determined by a, r 0 and xp M. We may then regard the original parameterization of our extensive hard-core results (in terms of c, and the strength and range of the attractive part) merely as a scaffolding which we may finally discard in favor of a "universal" parameterization in terms of a, r 0 and x~ M applicable to any of a large class of potentials.

A-BINDING IN NUCLEAR MATTER

147

Of the higher partial waves, only L = 1 in general makes a substantial contribution. For the presentation of our results, we thus use also the ratio p/s of the p-state to s-state potential strengths on the assumption that the potential shapes are the same. Our p-state results are not in fact restricted to this assumption but may be used for p-state potentials with any shape in the range of those used for our s-state potentials. The d-state well depths, although small, are non-negligible and results for these are also presented. Our second main purpose is to provide an extensive set of results parameterized - if possible - such that D for any local central A N potential that is likely to be suggested can be obtained without having to make nuclear-matter calculations. Thus our hardcore potentials cover a wide range of a, ro, c and p/s. Since the detailed shape of the attractive part turns out to be unimportant for hard-core potentials, our results are also immediately applicable to any hard-core potential with an attractive tail of exponential or Yukawa type. However, the "universal" parameterization for D~ in terms of a, to, and x~NM is not very useful since x sNM is obtainable only from a nuclear-matter G-matrix calculation. We have therefore looked for a third parameter, in addition to a and ro, that is effectively shape-independent for local potentials and that makes our results for D~ "conveniently" available for any central local potential. By "convenient" is m e a n t that no nuclear-matter calculation is necessary and that the parameter can be obtained from only scattering calculations. The latter requirement implies that only integrations of the s-wave Schr/Sdinger equation are involved. These are in any case necessary to obtain a and r o. We considered several candidates for such a third parameter, and our search is described in sect. 5. This was successful and yielded as an acceptable parameter a suitably defined scattering "correlation" volume x SC s which is closely related to x~NM. The parameterization of D~ in terms of a, ro and Ksc is then "universal" to a fair approximation (for local potentials) and allows one to obtain Ds for any local AN potential without having to perform a nuclear-matter calculation. This result seems significant and of wider interest, since the characteristics of the AN potentials which determine D are quite similar to those of the nucleon-nucleon interaction which determine the binding energy of nuclear matter. In the final section we compare the calculated and phenomenological well depths and review the resulting conclusions about the AN interaction.

2. The A N potentials

We consider two types of central local potentials, namely hard-core (HC) potentials and soft-repulsive-core (SRC) potentials. The potentials all have a "reasonable" shape and correspond in general to an inner repulsive part and an outer attractive part. The potential parameters and associated scattering parameters are shown in table 1 for both H C and SRC potentials.

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

148

TABLE 1 Hard-core (HC) a n d soft-repulsive-core ( S R C ) potentials a n d

Scattering quantities

Potential ") V (MeV)

# ( f m - l)

--a (fro)

ro (fm)

1.0 1.5 2.0 3.0 1.0 1.5 2.0 3.0 1.0 1.5 2.0 3.0

0.75

7.51 3.64 2.33 1.27 4,90 2.73 1.80 1.04 3.82 2.21 1.51 0.90

2.0 3.0 4.0 1.0 2.0 3.0 4.0 1.0 1.5 2.0 3.0 4.0

0.75

2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0 1.5 2.0 3.0 4.0 5.0

0.75

E(t$ = 0) (MeV)

b (fm)

non-selfd (fro)

10Kssc

Ds (MeV)

Dp (MeV)

Dp(½) (MeV)

31.8 35.4 36.0 35,5 45,3 47,9 47.7 45.6 58.5 59.6 58.0 54.2

12.0 7.4 4.6 2.0 16,9 9.8 5.9 2.4 21.6 12.0 6.9 2.7

6.0 3.7 2.3 1.0 8,4 4.9 2.9 1.2 10.7 5.9 3.4 1.3

c = Ofm

20.50 60.60 126.76 346.42 28.80 79.94 160.95 415.90 36.70 97.O0 188.70 467.30

1.25

2.00

2.00 1.36 1.03 0.69 2,00 1.36 1.03 0.69 2.00 1.36 1.03 0.69

c = o.3fm

430.97 1810.7 5459.3 57.396 512.69 2078.0 6122.0 70.123 237.28 588.11 2308.2 6666.8 c =

1.25

2.0

5.68 3.66 2.82 8.77 3.92 2.67 2.12 6.31 4.08 3.07 2.17 1.77

172.7 271.5 362.5 104.5 233.2 350.2 454.3 140.4 217.2 289.3 417.8 529.3

1.86 1.47 1.27 2.94 1.86 1.47 1.27 2.94 2.23 1.86 1.47 1.27

1.05 0.85 0.75 1.39 0.96 0.79 0.70 1.25 1.02 0.88 0.75 0.67

0.50 0.42 0.38 0.65 0.47 0.40 0.37 0.60 0.50 0.45 0.39 0.36

28.2 32.1 33.4 28.3 43.4 46.1 46,4 44.8 54.5 58.0 59.0 58.1

14.6 9.3 6.7 32.7 17.7 11.1 7.9 40.3 28.5 20.7 12.6 8.9

6.7 4.0 2.6 15.8 8.1 4.7 3.1 19.5 13.5 9.5 5.4 3.5

7.57 5.02 3.93 3.35 5.00 3.50 2.84 2.47 4.96 3.80 2.77 2.30 2.04

100.3 153.7 201.4 243.4 135.7 199.6 254.7 302.5 128.3 168.9 238.9 299.8 351.2

2.17 1.76 1.55 1.42 2.17 1.76 1.55 1.42 2.56 2.17 1.76 1.55 1.42

1.33 1.09 0.96 0.88 1.21 1.01 0.91 0.84 1.29 1.13 0.96 0.87 0.81

1.30 1.11 1.01 0.94 1.23 1.07 0.98 0.92 1.31 1.18 1.03 0.96 0.90

15.9 23.1 26,0 27,5 32,6 38.6 40.5 41.2 42,3 48.9 53.2 53,9 53,9

20.4 14.1 10.8 8.9 24.3 16.6 12.6 10.3 37.0 28.4 18.9 14.2 11.6

8.5 5.2 3.4 2.3 10.4 6.2 4.1 2.8 16.7 12.2 7.2 4.7 3.2

0.429fm

666.53 3246.9 11311 32813 776.10 3660.8 12488 35695 327.57 880.0 4030.3 13504 38138

1.25

2.0

A-BINDING IN NUCLEAR MATTER

149

the associated scattering and nuclear-matter quantities Nuclear-matter quantities ") consistent (,<1a = 30 MeV) Dd (MeV)

self-consistent

--10D's --100D'p--10D"

10KsNM

100Kp

Ds (MeV)

10t¢sNM

D (MeV)

10x NM D(1.35) (MeV)

D(½) (MeV)

3.5 1.3 0.5 0.1 5.0 1.7 0.6 0.1 6.3 2.0 0.7 0.1

0.04 0.09 0.15 0.26 0.08 0.18 0.28 0.44 0.14 0.29 0.43 0.64

0.03 0.03 0.03 0.02 0.06 0.06 0.05 0.02 0.10 0.09 0.07 0.03

0.04 0.10 0.16 0.26 0.09 0.18 0.28 0.44 0.15 0.30 0.43 0.64

0.03 0.08 0.14 0.25 0.07 0.16 0.26 0.43 0.12 0.26 0.40 0.63

0.03 0.03 0.03 0.03 0.06 0.06 0.06 0.03 0.07 0.09 0.07 0.03

31.8 35.3 35.9 35.3 45.2 47.6 47.2 44.9 58.1 58.8 56.9 52.7

0.03 0.08 0.13 0.24 0.06 0.14 0.22 0.38 0.09 0.19 0.31 0.51

47.3 43.9 40.9 37.4 66.9 58.9 53.5 47.3 85.6 72.4 64.2 55.4

0.03 0.08 0.13 0.24 0.05 0.12 0.21 0.37 0.06 0.17 0.29 0.49

42.4 39.4 36.7 33.5 60.0 52.8 48.0 42.4 76.8 64.9 57.6 49.7

41.3 40.2 38.7 36.4 58.4 54.0 50.6 46.2 74.9 66.5 60.8 54.1

1.7 0.6 0.3 9.9 2.0 0.7 0.3 12.1 5.0 2.3 0.7 0.3

0.49 0.45 0.44 0.58 0.49 0.49 0.52 0.56 0.52 0.52 0.56 0.61

0.23 0.25 0.23 0.20 0.35 0.35 0.33 0.30 0.43 0.48 0.47 0.42

0.51 0.48 0.47 0.60 0.53 0.53 0.55 0.59 0.56 0.57 0.61 0.66

0.49 0.45 0.44 0.58 0.49 0.48 0.50 0.56 0.52 0.50 0.54 0.59

0.21 0.24 0.23 0.16 0.31 0.34 0.33 0.22 0.36 0.43 0.45 0.42

28.3 32.1 33.2 28.4 42.8 45.4 45.6 44.0 53.3 56.6 57.4 56.5

0.49 0.45 0.44 0.58 0.48 0.47 0.48 0.55 0.50 0.48 0.50 0.53

43.8 41.5 39.9 68.7 61.5 56.4 53.3 93.5 84.9 78.3 69.9 65.0

0.50 0.47 0.45 0.57 0.50 0.49 0.50 0.54 0.51 0.50 0.51 0.55

39.3 37.2 35.8 61.6 55.1 50.6 47.8 83.8 76.1 70.2 62.7 58.3

36.3 36.3 36.0 52.6 42.4 50.4 48.7 73.9 70.7 67.7 63.1 59.9

2.6 1.0 0.5 0.3 3.1 1.1 0.6 0.4 6.9 3.5 1.3 0.6 0.4

1.23 1.10 1.03 0.99 1.20 1.10 1.06 1.04 1.27 1.19 1.13 1.11 1.I1

0.45 0.50 0.50 0.48 0.60 0.66 0.65 0.63 0.67 0.79 0.85 0.83 0.78

1.28 1.15 1.08 1.04 1.26 1.17 1.12 1.10 1.34 1.27 1.21 1.20 1.20

1.23 1.10 1.03 0.99 1.20 l.lO 1.06 1.04 1.27 1.19 1.12 1.11 1.11

0.40 0.49 0.49 0.48 0.55 0.64 0.64 0.63 0.58 0.70 0.82 0.82 0.78

17.4 23.8 26.4 27.7 32.3 37.7 39.5 40.2 40.9 46.9 50.8 51.6 51.5

1.24 1.10 1.03 0.99 1.20 1.10 1.05 1.03 1.26 1.18 1.10 1.08 1.07

37.9 37.3 36.6 36.1 56.7 53.5 51.3 49.7 79.6 75.1 68.6 64.7 62.0

1.26 1.14 1.07 1.03 1.23 1.14 1.09 1.07 1.26 1.20 1.14 1.12 1.11

34.0 33.4 32.8 32.4 50.8 48.0 46.0 44.6 71.4 67.3 61.5 58.0 55.6

27.3 29.3 29.9 30.1 44.2 44.3 43.6 43.0 61.7 60.7 58.2 56.1 54.6

150

A . R . BODMER AND

D. M. ROTE TABLE 1

Potential ") V (MeV)

Scattering quantities

non-self-

/~ (fm -1)

--a (fm)

ro (fm)

E(6 = 0) (MeV)

b (fm)

d (fm)

10r~sc

D~ Dp D=(½) (MeV) (MeV) (MeV)

2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0 2.0 3.0 4.0 5.0 7.0

0.75

10.71 7.26 5.83 5.05 6.70 4.79 3.96 3.50 4.90 3.66 3.10 2.78 2.42

58.9 87.8 111.9 132.2 80.3 115.4 144.5 168.7 100.6 140.2 172.4 199.1 240.6

2.58 2.15 1.93 1.79 2.58 2.15 1.93 1.79 2.58 2.15 1.93 1.79 1.63

1.67 1.37 1.22 1.13 1.53 1.29 1.16 1.08 1.43 1.23 1.11 1.04 0.95

3.22 2.78 2.55 2.40 3.07 2.69 2.49 2.36 2.96 2.63 2.44 2.32 2.17

--13.5 0.1 6.3 9.6 5.4 17.8 23.1 25.8 24.2 34.9 39.0 40.9 42.4

28.7 21.4 17.2 14.7 34.2 25.1 20.1 17.2 39.7 28.6 22.9 19.5 15.7

9.8 5.7 3.2 1.6 12.3 7.2 4.3 2.5 14.8 8.7 5.3 3.2 0.7

2.0 2.5 3.0 3.5 2.0 2.5 3.0 3.5 1.75 2.0 2.5 3.0 3.5 4.5

0.75

9.74 7.37 5.93 4.92 6.14 4.85 3.98 3.37 5.13 4.50 3.65 3.08 2.65 1.96

69.0 94.0 122.4 156.7 95.7 127.7 165.4 210.9 104.7 122.5 160.9 205.5 260.6 451.6

2.68 2.34 2.09 1.88 2.61 2.28 2.03 1.83 2.78 2.56 2.23 1.98 1.78 1.43

1.61 1.40 1.25 1.12 1.46 1.28 1.14 1.02 1.44 1.35 1.19 1.07 0.96 0.76

1.62 1.22 0.92 0.67 1.43 1.07 0.80 0.58 1.49 1.28 0.96 0.72 0.52 0.21

7.9 17.3 23.4 27.7 26.4 34.7 40.3 43.7 39.4 44.8 51.8 55.9 58.4 60.4

26.0 20.9 16.9 13.6 30.7 24.4 19.6 15.7 40.3 35.5 27.8 22.0 17.5 10.5

12.3 9.8 7.9 6.4 14.6 I 1.5 9.2 7.3 19.3 16.9 13.1 10.3 8.1 4.9

c = 0.6fm 1147.9 6754.5 28268 98265 1306.2 7473.0 30743 105600 1460.7 8138.2 32948 112000 958890

1.25

2.0

SRC 1039.4 2512 5290. l 10072 1172.1 2772.0 5746.8 10780 798.33 1300.0 3015.2 6144.5 11379 31268

1.25

2.0

a) The parameters V and ~ refer to the attractive part of the potential for both HC and SRC potentials b) The nuclear-matter quantities are all for kF = 1.4 fro- 1 and p/s = 1 (i.e., for equal p- and s-state matter parameters are AN = 85.4 MeV, M*/MN = 0.638 for kv = 1.4 fm-1 (80 MeV and 0.653 for In addition to the low-energy s-wave scattering parameters, namely the scattering l e n g t h a n d effective r a n g e r o , w e h a v e a l s o o b t a i n e d a n d t a b u l a t e d : (i) t h e A l a b e n e r g y

E ( 6 = 0 ) a t w h i c h t h e s - w a v e p h a s e s h i f t 6 is z e r o , (ii) t h e i n t r i n s i c r a n g e b, (iii) t h e M o s z k o w s k i - S c o t t s e p a r a t i o n d i s t a n c e d f o r s c a t t e r i n g , a n d (iv) a z e r o - e n e r g y s c a t t e r i n g c o r r e l a t i o n v o l u m e x sc. T h e s e " s c a t t e r i n g " q u a n t i t i e s a n d t h e i r i m p l i c a t i o n s f o r D s a r e d i s c u s s e d i n sect. 5. A p a r t f r o m a a n d r o, o n l y E ( 6 = 0) is a p u r e l y scattering parameter determined only by the on-energy-shell scattering amplitude.

A-BINDING IN NUCLEAR MATTER

151

(continued) Nuclear-matter quantities b) consistent (A a = 30 MeV) Dd (MeV)

self-consistent

- - 1 0 D ' s - - 1 0 0 D ' p - - 1 0 D " 10Xs NM

100Kp

Ds (MeV)

10KsNM

D (MeV)

10KNM D (1.35) (MeV)

D(½) (MeV)

4.4 2.0 1.1 0.8 5.1 2.2 1.3 0.9 5.7 2.4 1.4 0.9 0.6

3.06 2.75 2.56 2.44 2.99 2.70 2.54 2.44 2.93 2.68 2.55 2.47 2.38

1.15 1.22 1.23 1.22 1.33 1.44 1.47 1.47 1.56 1.72 1.75 1.74 1.68

3.18 2.88 2.69 2.56 3.13 2.85 2.69 2.59 3.10 2.86 2.73 2.64 2.55

2.96 2.71 2.54 2.42 2.92 2.68 2.53 2.43 2.89 2.67 2.55 2.48 2.40

1.09 1.18 1.21 1.20 1.24 1.38 1.44 1.44 1.49 1.63 1.71 1.71 1.68

--3.3 6.5 11.1 13.6 11.0 20.4 24.5 26.6 25.5 33.9 37.2 38.8 40.0

3.03 2.74 2.55 2.44 2.95 2.69 2.54 2.44 2.90 2.67 2.54 2.46 2.38

22.2 24.9 25.8 26.1 41.2 41.7 41.4 41.0 60.3 58.0 56.1 54.8 52.8

3.09 2.84 2.67 2.55 3.02 2.80 2.66 2.57 2.96 2.78 2.66 2.59 2.50

19.9 22.3 23.1 23.4 36.9 37.4 37.1 36.8 54.1 52.0 50.3 49.1 47.3

7.8 12.7 14.7 15.7 24.5 27.9 29.0 29.3 41.3 42.5 42.3 41.9 41.0

3.9 2.3 1.4 0.9 4.5 2.6 1.6 1.0 6.8 5.0 2.8 1.7 1.1 0.4

1.29 1.03 0.80 0.61 1.20 0.95 0.75 0.58 1.27 1.13 0.90 0.72 0.58 0.40

0.44 0.43 0.40 0.35 0.58 0.58 0.55 0.49 0.73 0.76 0.77 0.72 0.64 0.42

1.34 1.07 0.84 0.65 1.26 1.01 0.80 0.63 1.35 1.21 0.98 0.80 0.65 0.44

1.28 1.03 0.81 0.62 1.20 0.96 0.75 0.58 1.27 1.14 0.91 0.72 0.57 0.37

0.40 0.40 0.37 0.34 0.48 0.51 0.51 0.46 0.61 0.63 0.66 0.66 0.61 0.42

10.5 18.5 23.9 27.8 26.8 34.3 39.6 42.9 38.3 43.3 50.0 54.2 56.8 59.2

1.33 1.05 0.82 0.62 1.21 0.95 0.74 0.57 1.25 1.11 0.88 0.69 0.54 0.32

36.9 39.5 40.8 41.5 58.1 58.8 59.1 58.6 79.8 79.3 77.8 7.60 74.1 69.6

1.30 1.05 0.83 0.64 1.18 0.96 0.76 0.59 1.20 1.08 0.88 0.71 0.56 0.33

33.1 35.4 36.6 37.2 52.1 52.7 53.0 52.5 7t.6 71.1 69.8 68.1 66.4 62.4

24.8 29.5 32.5 34.7 43.8 47.1 49.5 50.7 61.3 62.7 64.3 65.1 65.3 64.2

defined by eqs. (1) and (2), respectively. potentials) except that D (1.35) is for kF = 1.35 f m - 1 and Dp(½) and D(½) are for p/s = ½. The nuclearkF = 1.35 f m - 1). 2.1. H A R D - C O R E (HC) P O T E N T I A L S

To establish that D is independent of the shape of the attractive part of HC potentials, we have considered some of Herndon and Tang's potentials whose attractive parts have a pushed-out exponential shape. However, our basic and most extensive results are for HC Yukawa potentials V(r) =

~o e - • ' / l t r

r > c.

152

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

For each partial wave L, the potential is then defined by the three parameters c, Vo and ~t. Although the original parameterization is in terms of c, Vo and #, a more significant one is in terms of c, a and ro since a and ro are the primary experimental scattering data. There is (assuming no bound state) a one-to-one correspondence between the two sets of parameters. A wide range of values of c was considered, namely c = 0, 0.3, 0.429 and 0.6 fro. For a given c, the values o f # and Vo are then chosen to give the three values a = - 0 . 7 5 , - 1 . 2 5 and - 2 fm as well as reasonable values o f r o . These ranges of values of a and r o (together with possible extrapolations) cover any likely phenomenological value. For the calculation of D it suffices to know only the spinaverage potential (as explained in I). It is therefore not neccessary to distinguish specifically between the singlet and triplet potentials. 2.2. SOFT-REPULSIVE-CORE (SRC) POTENTIALS These potentials are a superposition of two Yukawa potentials, namely an (inner) repulsive one and an (outer) attractive part, V(r) = VR e--aRr/pR

r--

V A e-/~Ar/~/A r.

(2)

For our potentials the repulsive part was kept fixed with VR = 82000 MeV and /~R = 6.0 fm-1. Although our span of values of VA and #A corresponded to a span for a and r 0 comparable to that for the HC potentials, nevertheless the restriction to only one repulsive shape means that the span of our SCR potentials is considerably less than that of our H C potentials. Again there is no need to distinguish explicitly between singlet and triplet potentials. 3. Calculation of the well depth We calculate D in the G-matrix approximation with the assumptions and procedures described in I. The G-matrix approximation 11) corresponds to the leading term, proportional (explicitly) to the density p, in an expansion in powers of p. Insofar as this expansion actually exists and there is indeed a (small) expansion parameter, this parameter is believed to be closely related to /£NM, the correlation volume relative to the volume per nucleon (the latter is proportional to p - l ) . The parameter x NM will play an important role in our considerations, especially for the s-state results. For a relative angular momentum L and relative AN momentum k, the correlation volume is defined by o

x~ M = 41t(2L+ l)p

z2dr,

(3)

where )~L = r j L ( k r ) - - u L ( r ; k)

(4)

is the deficit wave function and r j L ( k r ) i s the unperturbed wave function. Asymptotically one has u L ,,, rjL(kr), i.e., the correlated wave function "heals". The total correla-

A-BINDING IN NUCLEAR MATTER

153

NM = E X~M.

(5)

tion volume is L

The superscript N M denotes that x NM is the value appropriate to nuclear matter, and is used to distinguish x NM from the related "scattering" quantity x sc (defined in subsect. 5.4). In terms of the L-state well depths DL the total well depth is given by

D = 2 DL. L

(6)

The G-matrix is obtained using the Kallio-Day version t3) of the reference-spectrum method. The free kinetic energies are assumed for the unoccupied single-particle nucleon and A states. These are separated from the occupied states by energy gaps AN and Aa, respectively. The single-particle spectra are then completely specified, in the effective-mass approximation, by these gaps and by the nucleon effective mass M* for the occupied states. Since, for D, the A-momentum is zero, the only relevant parameter of the A-spectrum is Aa; the A effective mass M* for the occupied states is then not relevant. The G-matrix (and hence D) then depend, in particular, on the total gap A = AN+ Aa between the occupied and unoccupied states. The A-gap is identified with the well depth [as discussed in refs. 1- 2)]. This gives the self-consistency condition D(A) = Aa,

A = AN+A a.

(7)

If the singlet and triplet potentials 1 V and 3 V have different strengths (but the same shape), then in principle one should make separate calculations for the singlet and triplet well d e p t h s - denoted by 3D and 1D, respectively (with the suffix L o m i t t e d ) - a n d thus obtain D from D = ¼(1D+33D). However, it was shown in I that it is an excellent approximation to calculate D for only the spin-average interaction ¼(1 V-I- 3 3 V). This is of course immediately obtainable if both 1 V and 3 V are known. If i V and 3 V are also different in shape, then separate calculations of 1D and 3D are necessary. However, our results may still be used for this more general situation by interpreting the potentials in the tables as ~V and 3 V for 1D and 3D, respectively, and remembering that the self-consistency condition is for the total well depth D. The angle-average approximation is used for the exclusion principle as discussed in detail in I. This approximation is exact for purely s-state interactions (i.e., potentials that are non-zero only for L = 0). More generally, the approximation was shown in I to be excellent. Our basic results for DL and related quantities are shown in table 1 for HC and SRC potentials, respectively. The primary results are not self-consistent but are given for Aa = 30 MeV and for a Fermi momentum k F = 1.4 f m - I corresponding to which we use AN = 85.4 MeV and M * / M N = 0.64 [ref. 19)].

154

A. R. BODMER AND D. M. ROTE

The dependence o l D L on A was found to be linear over a large range for all cases examined andis therefore determined by D [ = dDL/dA (for the above values of the parameters). Thus for other values of Aa and/or AN one can obtain DL by using the tabulated results. In particular, with the use of eq. (7), the self-consistent values shown in the tables may be obtained from the primary results. The dependence on M * / M N was shown in I to be very slight. Most of the results in table 1 are for k v = 1.4 fm -1 (p = 0.185 fm-3). However, we have also obtained some results for k r = 1.35 fm -1 (p = 0.167 fm-3). The dependence on k v, or equivalently on p, is discussed in sect. 8. The dependence of the s-wave correlation volume x sNM on k is very slight for the range of nucleon m o m e n t a corresponding to the occupied states (the A-momentum is zero). In fact, to the accuracy to which x sNM is given in the tables, this is equal to both the average value (XsNM(k)) over the nucleon momenta as well as to the value ~c~M(/~) appropriate to the average m o m e n t u m ~ = ~/~ k v . The quantities D~ and Ks are both shown in table 1. It is seen that the relation D'L = dDL/dA =

-- KL

(8)

is very well satisfied for L = 0. In contrast to K~u the correlation volumes x~ M for L > 0 depend quite strongly on k because of the centrifugal barrier; in particular for L > 0 one expects that XL ~ 0 and hence ~:~M ~ 0 as k ~ 0. This is confirmed by our numerical results for L = 1 and 2. The values of the p-wave correlation volume and of D'p are also shown in table 1. The values of ~:p = (%NU(k)) and of ~CpNM(/C)are quite close, the latter being generally 3-4 ~o less than the former. Eq. (8) is reasonably well satisfied for L = 1 (and also for L = 2 for which we have not shown the relevant results). The term following the G-matrix term in the hole-line expansion for D is proportional to p2 and leads to a three-body (A + 2N) problem involving equations of the Bethe-Faddeev type. However, in addition to the contributions associated with these equations, there is the so-called hole-rearrangement term calculated by Dabrowski and K6hler 20). Its contribution is ~ --K(NN)D where K7(NN) ( ~ 0. l) is the nucleonnucleon correlation volume. It thus reduces the G-matrix values of D by about 10 ~ . [See also ref. x) for a discussion.]

4. The s-state well depth and related results 4.1. DEPENDENCE OF Ds ON LOW-ENERGY SCATTERING PARAMETERS; LONG-RANGE EFFECTS OF POTENTIALS The basic results for Ds and related quantities are shown in table 1. The dependence on the scattering length a is large; D s increases with [al, the dependence becoming stronger for smaller c. This is illustrated by figs. 1 and 2. This strong dependence on lal is expected since lal is closely related to the overall strength of the potential.

=

I

60•

I

'

I

'

-a (fm)

f

~ ~

; G) =E

'

c=O

-

2

X

1.25

40-0.75

20

,

I 2

~

I 4

~

I 6

i

ro (fro)

Fig. 1. T h e self-consistent s-state well d e p t h Ds for H C potentials, plotted as a function o f the effective r a n g e r0 for v a r i o u s scattering lengths a a n d for zero hard-core radius. T h e n u c l e a r - m a t t e r p a r a m e t e r s u s e d for this a n d s u b s e q u e n t figures s h o w i n g n u c l e a r - m a t t e r results are those o f table 1, n a m e l y AN = 85.4 MeV, MN*/MN = 0.64 a n d kF = 1.4 f r o - 1. '

I

40-

'

I

'

I

'

I

c = 0 . 6 fm

~

Q=-

-

f

20

25 frn

0 =- 0 7. 5 f r o 0

-

1

i

I 2.0

0

I

I I 4.0 r o {fro)

I 6.0

i

I 8.0

Fig. 2. Same as fig. I except that the hard-core radius used is c = 0.6 fm. I

I

I

I

I

6C 5C

;

-a(fml ro(fml

4o

3

"~ 3 o

5 4

2G 4

I0

q

I 0.05

I 0.10

I 0.15 Ks

I 0.20

I 0.25

Fig. 3. The self-consistent well depth D,, plotted as a function o f correlation volume Ks f o r various values o f a and to. The lines, full and dashed, are f o r H C potentials. The full lines correspond to use o f the nuclear-matter correlation volume K, - - K,T M defined by eq. (3), and the dashed lines to use ot the scattering correlation volume Ks ~ K,sc defined by cq. (9). Where only the full line is shown, the dashed line coincides with it. The crosses arc the results f o r SRC potentials with the values o f a and ro equal to those appropriate to the nearest line.

156

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

The dependence on the effective range r o , also shown in figs. 1 and 2, is more complicated than that on lal, especially for small c. For these, D~ is not even a monotonic function of r o . For large c, D~ decreases with r o . (This is presumably in part because for larger r o the attractive part of the potential is of longer range and correspondingly shallower, and consequently gives smaller "higher-order" contributions.) Fig. 3 shows a similar characteristic dependence of D~ on a and r o when the correlation volume ~:~M is used instead of c. Since (as will be shown in subsect. 4.3) x~ M is efTABLE 2

Comparison of self-consistent s-state well depths Ds and nuclear-matter correlation volumes ,qNM for HC potentials (Yukawa tail) and Tang and Herndon's potentials (pushed-out exponential tail) for the same values of a, ro and c Potential b)

c (fro)

- - a (fm)

ro (fm)

D~ (MeV)

/¢sNMa)

C' HC

0.45 0.45

0.99 0.99

2.99 2.99

33.0 32.9

0.115 0.113

FI HC

0.6 0.6

2. i 2.1

3.42 3.42

35.9 36.0

0.267 0.262

A" HC

0.0 0.0

0.75 0.75

3.34 3.34

36.2 35.5

0.006 0.008

a) The nuclear-matter parameters are those o f table 1. b) Tang and H e r n d o n ' s potentials are designated by the letter used in ref. ~7); the bar denotes that the results are for the spin average o f their singlet and triplet potentials.

fectively a shape-independent parameter characterizing the short-range repulsion, the use of tc~M instead of c allows the dependence on a and ro to be depicted more generally. Although a and r o are the primary scattering parameters, they are not primary from the point of view of the binding energy. For the latter, one expects the part V (lOrlg), defined in subsect. 4.3, to be the significant quantity in determining the long-range effects of the potential. 4.2. S H A P E I N D E P E N D E N C E

O F Ds F O R H A R D - C O R E P O T E N T I A L S

We consider the shape independence of Ds for H C potentials. This is demonstrated in table 2 which shows results for pairs of H C potentials with different shapes, namely our Yukawa potentials (1) and some of the pushed-out exponential potentials of Tang and Herndon 16, tT). For both members of a pair, the values of a, r o and c are the same. The two members of a pair then give values of Ds and x~M that are also almost equal. Thus for D~ the three quantities a, r o and c are expected to give a complete and effectively shape-independent parameterization for a large class of H C potentials with attractive tails which fall off rapidly (e.g., Yukawa or exponential shapes). However, it is not clear that this shape independence would still hold for shapes with a sharper edge, such as Gaussian or square-well shapes.

A - B I N D I N G IN N U C L E A R M A T T E R

I

I

'

~

I

157

'

r o =3 fm

60

A

4C

2C ~

0 L

I

0"75 -

l

i

0.2

o

I

t

I

0.4 c (fro)

0.6

Fig. 4. Self-consistent well depth Ds for HC potentials, plotted as a function of the hard-core radius c for various values of a and for ro = 3 fm.

60

'

I

'

I -

A

40

m),ro(fm)

_ 3

a =

5

20 .75

,

I

,

4

I - 0 ( f r o ) , ro(f m) 2

0.3

5

0.2 0.1

0.1

0.2 cSlfml

5

Fig,-5. Self-consistent values of Ds (upper part) and of the nuclear-matter correlation volume KlNM (lower part), both plotted as functions of c 3 for the indicated values of a and ro.

158

A.R. BODMER

A N D D. M. R O T E

TABLE 3 Self-consistent s-state well depths a n d correlation v o l u m e s a) for s o m e H C a n d S R C potentials -- a (fm)

ro (fm)

c (fm)

10 KsT M

2

2

0.2 SRC 0.3 0.429 0.6 0.2 0.3 SRC 0.429 0.6 0.2 0.3 SRC 0.429 0.6 0.2 0.3 SRC 0.429 0.6

0.27 0.34 0.51 1.06 2.27 0.21 0.49 0.66 1.11 2.51 0.2 0.50 0.97 1.19 2.73 0.20 0.53 1.22 1.26 2.91

3

4

5

l 0 r~sc

0.13 0.23 0.38 0.89 1.97 0.16 0.44 0.69 1.07 2.40 0.18 0.49 1.09 1.20 2.72 0.20 0.54 1.45 1.31 2.98

Ds (MeV) 60.0 59.1 57.2 51.4 41.0 59.5 56.7 54.7 50.1 37.7 57.5 53.6 47.4 46.0 31.6 54.2 49.7 38.9 40.6 24.8

-- a ro (fro) (fro)

c (fm)

10 KsT M

1.25

SRC 0.3 0.429 0.6 0.3 SRC 0.429 0.6 0.3 SRC 0.429 0.6 SRC 0.3 0.429 0.6 0.3 SRC 0.429 0.6

0.45 0.47 1.05 2.30 0.48 0.75 1.12 2.55 0.50 0.98 1.19 2.72 0.036 0.45 1.04 2.3 0.48 0.635 1.09 2.43

3

4

5

0.75

4

5

I 0 Ks$C

Ds

(MeV) 0.43 0.42 1.00 2.19 0.47 0.80 1.12 2.50 0.52 1.10 1.23 2.73 0.038 0.43 1.01 2.26 0.47 0.69 1.09 2.40

44.8 45.0 39.1 28.8 42.6 39.4 35.9 24.3 38.8 33.5 32.3 19.5 31.3 31.6 26.2 16.5 29.8 27.5 23.8 13.7

a) T h e n u c l e a r - m a t t e r p a r a m e t e r s are t h o s e o f table 1. TABL~ 4 C o m p a r i s o n o f self-consistent well depths for H C a n d S R C potentials with the s a m e values o f a, ro a n d KsN~ for the u p p e r p a r t o f the table =) a n d with a, ro a n d K,sc for the lower part --a

rO

(fm) 2 2 2 2 1.25 1.25 1.25 0.75

10KsNM

(fm) 2.0 3.0 4.0 5.0 3.0 4.0 5.0 5.0

0.34 0.66 0.97 1.22 0.45 0.75 0.98 0.635

¢ (fm)

Ds (SRC) (MeV)

0.235 0.345 0.392 0.423 0.29 0.362 0.393 0.338

59.1 54.7 47.4 38.9 44.8 39.4 33.5 27.5

D~ ( H C ) (MeV) 59.3 54.7 48.3 41.0 45.4 39.6 34.4 28.1

10xssc SRC

HC

0.23 0.69 1.09 1.45 0.43 0.80 1.10 0.69

0.62 0.92 1.23 0.4 0.75 1.0 0.625

1 0 rs sc

2 2 2 2 1.25 1.25 1.25 0.75

2.0 3.0 4.0 5.0 3.0 4.0 5.0 5.0

0.23 0.69 1.09 1.45 0.43 0.80 1.10 0.69

1 0 x~ NM

0.245 0.359 0.42 0.45 0.30 0.382 0.41 0.354

59.1 54.7 47.4 38.9 44.8 39.4 33.5 27.5

a) T h e nuclear-matter p a r a m e t e r s are those o f table 1.

59.1 54.0 46.5 38.6 45.0 38.5 33.4 27.4

SRC

HC

0.34 0.66 0.97 1.22 0.45 0.75 0.98 0.635

0.35 0.72 1.15 1.44 0.48 0.85 1.09 0.71

A-BINDING IN NUCLEAR MATTER

159

4.3 DEPENDENCE OF Ds ON THE SHORT-RANGE PART OF THE POTENTIAL The dependence on c is shown in table 1 and in fig. 4 (and implicitly also in fig. 3). In particular the rate of decrease of Ds with c is not too strongly dependent on a or r o. The effect of a hard core in reducing Ds was already established in refs. 1.2) for some of Tang and Herndon's potentials, but the present results are much more complete and systematic. The reasons for this reduction were to some extent discussed in I and are further considered below. The correlation volume tc~M turns out to be an effectively shape-independent parameter characterizing the effect of the short-range part of the potential on Ds. For the HC potentials, the dependence of Ds and x~ M on c a is shown in figs. 5a and b, respectively. As expected, both vary linearly with c a for not too large values of c. It will be noted that ~¢~Mdepends predominantly on c and only slightly on a and r o . The dependence of D~ on x~ Mis explicitly shown in table 3 and fig. 3. The full lines are for HC potentials whereas the crosses are for "equivalent" SRC potentials which have the same a, r 0 and x~M as the HC potential corresponding to the nearest full line. It is seen that the values of D~ are very nearly equal for the "equivalent" HC and SRC potential. This result is further demonstrated in table 4 which shows D~ for pairs of H C and SRC potentials with the same a, r 0 and x~ M. A quite large range of values of K~M ( < 0.12) is spanned by our results and even for the largest values available for comparison ( ~ 0.12) the difference in D~ for "equivalent" potentials is only a few per cent. Thus x~ra is an effectively shape-independent parameter which characterizes the contribution to D~ from the short-range part of the potential. We recall that ~:NM is also closely related to the significant expansion parameter in the density (hole-line number) expansion. For KNM~ 0.12 (corresponding to c ~ 0.4 fm) the convergence for the nuclear-matter case seems quite reasonable and may similarly be expected to be quite good for D. For large values of x ~> 0.2 (appropriate to c ~> 0.5), the convergence (if any) is presumably much slower, and it becomes correspondingly more important to evaluate the higher-order terms. (For the hypernuclear problem, it should be noted that both the nucleon-nucleon and Anucleon correlations will enter.) However, even if for larger values of K, the higher-order terms do contribute appreciably, the equivalence of potentials with the same a, r o and ~c~ may still remain approximately valid, since the higher-order contributions may well be determined mainly by just ~. The dependence of D~ on the repulsive core through x~ M may be qualitatively understood as follows. For increasingly repulsive cores the value of Z~ and hence of ~c~ra becomes correspondingly larger. Since the effect of A as well as of the exclusion principle (i.e., of Q) occur via just the deficit part of the correlated wave function u s, the dependence on A and on Q is then also expected to increase with x~ M and hence with the size of the repulsive core. The effect of A and Q is to increase the curvature of the wave function, i.e., to make this "heal" faster, and correspondingly to reduce D; thus the self-consistent values of D~ decrease as x~ M increases.

160

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

Some further understanding may be obtained through the Moszkowski-Scott separation method 21). The separation distance d is shown in table 1 (and also in fig. 6 as a function of ro for a = - 2 fm and for various values of c). It is the distance at which just the part of the potential V (s~'°~t) for r < d gives no s-wave scattering at some energy which for convenience we take as zero. Thus V(r) = V(sh°rt)+ V (t°"g), I a = -2

'

I

I

fm

1.6

1.4

c = 0.6 fm ~

~

SRC

E 1.2

fm

'ID

1.0

//

0.8

/ ~ ' /

0.6

i

0

I

2

/

-c=0.429fm = "

,

I

4

~

r o (fm)

I

6

g

I

,

8

Fig. 6. The ro dependence of the separation distance d for a = --2 fm. Results for HC potentials are labeled by the appropriate hard-core radius c, those for SRC potentials by SRC.

where

V (sh°rt) and V 0°"g) are non-zero only for r < d and r > d, respectively. In

particular, V(sh°a) is such that the repulsive core is balanced by just enough of the attractive part of the potential to give a = 0. The scattering length a is then entirely reproduced by just V0°"g). As expected, d decreases (for given a and to) as the repulsion becomes less; in particular, one has d = 0 for purely attractive potentials. The potential V(l°"g) is sufficiently weak, especially for longer ranges, that Zs and n Nsr ~ are quite small for r > d and that therefore the well depth D~~°"g) due to V 0n"g) is to a good approximation just the first-order result (V0°"g))s. Hence D~1°"~) is quite insensitive to A and to Q since these enter only through the small higher-order contributions (corresponding to the small ;(s for r > d). This was explicitly confirmed by the detailed results obtained in I. In particular, for purely attractive potentials (c = d = 0) x sNM is very small, appropriate to very small ;(s For appreciable repulsive cores, on the other hand, the correlated wave function for r < d will differ appreciably from the unperturbed one and thus Zs and K~M will be large. However, if in spite of a large value of x~ M, the wave function Us for r < d were in fact the same for nuclear matter as for scattering, then one would have G (sh"rt) = 0, and consequently D~sh°rt) = 0 and hence D ~ D 0°"g). In fact, the effects of A and Q give rise to important modifications (increased curvature) of us which correspond to a repulsive contribution, as manifested by negative values of/)(short) --s The effect of A and Q can, however, occur only via Zs which is predominantly determined by Vtsh°rt). Thus x~ M, which is a measure of the importance of ;(s, characterizes

IN N U C L E A R M A T T E R

A-BINDING

161

the effects of g (sh°rt) and in particular the decrease, for given a and r 0' of D s as the core becomes increasingly repulsive.

5. Search for a shape-independent "scattering" parameter characterizing the shortrange (s-state) effects of the potential For given a and r 0 , the correlation volume tc~M has been seen to effectively characterize the effects on Ds of the short-range part of the potential in a shape-independent manner. Thus a, r 0 and x~ M effectively constitute a universal parameterization. However to make use of this for some given local potential involves the determination I

I

.I

0=-2 fm

I

1

I

.c=0.6 fm

|.0~

///~,SRC

0.9-

T . 0.80.7 T

0.6

0.5 ,% I o 0.4-o 0.3-0.2-Od-0 0

I 2

I 3

I 4 ro(fm)

I 5

I 6

I 7

Fig. 7. T h e inverse o f the A laboratory energy E(6 = 0) at which the s-wave p h a s e shift 6 vanishes, plotted as a function o f ro. R e s u l t s for H C potentials are labeled by the appropriate hardcore radius c, t h o s e for S R C core potentials by SRC.

of ~c~M and thus implies a nuclear-matter calculation. We therefore look for a single conveniently obtainable parameter that characterizes the short-range effects in a shape-independent way but that depends only on the potential. By "convenient" is meant that only integrations of the radial Schr6dinger equation are involved since these are in any case needed to obtain a and r o. We attempt to find just a single parameter in view of the result that only the one nuclear-matter quantity x~NM suffices.

162

A.R. BODMER AND D. M. ROTE

If such a "scattering" p a r a m e t e r can be found, then our results for D s can be universally parameterized (at least for local potentials) by a, r o and this parameter, all three being obtainable f r o m scattering calculations and therefore depending only on the potential. The quantities we have considered as candidates (sect. 2) are E(6 = 0), b, d, and rsc. We again emphasize that of these only E(t5 = 0) is a purely scattering (phaseshift) quantity. TABLE 5

Comparison of self-consistent s-state well depths a) for HC and SRC potentials with the same values of a and ro and of the A lab energy E(6 = 0) energy at which the s-wave phase shift is zero --a (fm)

ro (fm)

100 [E(6 = 0)]- ~ (MeV- 1)

c (fro)

D, (SRC) (MeV)

Ds (HC) (MeV)

2 2 2

1.88 2.86 4.0

0.202 0.434 0.7

0.3 0.429 0.51

59.4 55.5 47.4

57.0 50.5 39.5

a) The nuclear-matter parameters are those of table 1. The quantities b, d and xsc, like a, r o and E(6 = 0), are completely determined by the s-state potential t h r o u g h low-energy scattering calculations involving numerical integrations of the Schr6dinger equation. H o w e v e r they depend on off-energy-shell properties (i.e., they depend at least in p a r t on the wave function in the potential region), in contrast to a, r o and E(~ = 0). 5.1. THE ENERGY E(~ = 0) The energy E ( 6 = 0) is the A lab energy for which the s-wave phase shift is zero and seems the single most pertinent phase-shift quantity characteristic of the repulsive core. Thus [E(6 = 0)] -1 increases as the repulsion increases, as shown particularly clearly for the H C potentials in fig. 7. F u r t h e r m o r e a, r o and E(3 = 0) determine as a function of E quite well up to a b o u t E(3 = 0). Fig. 7, which also includes some S R C results, shows that E(3 = 0) is quite a good discriminator o f the repulsive p a r t of the potential for a given shape. Thus for the H C potentials, [E(c5 = 0)]-1 depends quite strongly on c for given a and r o. This ability to discriminate seems a necessary condition, although in no way a sufficient one, required of an acceptable parameter. The suitability of E(di = 0) as a shape-independent p a r a m e t e r can be assessed f r o m table 5, which lists the values of D s for several paired potentials whose m e m b e r s are "equivalent" H C and S R C potentials with the same a, ro and E ( 6 = 0). F o r fairly small [E(6 = 0)]-1, corresponding to fairly small c < 0.3 fm, the " e q u i v a l e n t " potentials give fairly similar values of Ds. H o w e v e r for larger [E(6 = 0 ) ] - 1 , corresponding to larger repulsive cores, the differences in Ds become quite large. Thus E(~ = 0) is not a useful shape-independent p a r a m e t e r except perhaps for quite small repulsive cores.

A-BINDING IN N U C L E A R MATTER F o r e q u i v a l e n t p o t e n t i a l s ( f o r w h i c h t h e p h a s e shifts u p to

163

E(6

= 0) a r e a p p r o x i -

m a t e l y t h e s a m e ) o n e has D~ ( S R C ) > D s ( H C ) . T h i s agrees w i t h t h e a n a l o g o u s r e s u l t f o r t h e b i n d i n g e n e r g y o f n u c l e a r matter*. 5.2. THE I N T R I N S I C R A N G E b T h e i n t r i n s i c r a n g e is t h e effective r a n g e w h e n t h e s t r e n g t h o f t h e p o t e n t i a l is s u c h as t o j u s t give a (single) b o u n d state (or e q u i v a l e n t l y a = - o o )

at z e r o energy. T h e

s t r e n g t h is v a r i e d w i t h t h e s h a p e k e p t fixed. T a b l e 1 s h o w s t h a t b is a q u i t e p o o r disc r i m i n a t o r o f t h e r e p u l s i v e c o r e , since f o r g i v e n a a n d r 0 t h e intrinsic r a n g e b is q u i t e i n s e n s i t i v e to c o r m o r e g e n e r a l l y to t h e r e p u l s i v e core. T h u s b is q u i t e u n s u i t a b l e as a third parameter. TABLE 6

Comparison of self-consistent s-state well depths a) for HC and SRC potentials with the same values of a, ro and of the separation distance d --a (fro)

ro (fro)

d (fm)

c (fm)

Ds (SRC) (MeV)

Ds (HC) (MeV)

2 2 2

2.35 3.0 4.0

0.88 1.05 1.26

0.429 0.492 0.548

58 54.7 47.4

51.5 45.9 36.2

a) The nuclear-matter parameters are those of table 1.

5.3. SEPARATION DISTANCE d T h e s e p a r a t i o n d i s t a n c e was d e f i n e d in subsect. 4.3. Its d e p e n d e n c e o n r o a n d c, in p a r t i c u l a r , is i l l u s t r a t e d in fig. 6. A s e x p e c t e d , d increases as the c o r e b e c o m e s m o r e repulsive. F u r t h e r m o r e , d is a q u i t e r e a s o n a b l e d i s c r i m i n a t o r o f t h e r e p u l s i o n f o r a g i v e n shape. C o m p a r i s o n o f t h e results f o r E ( 3 = 0) a n d d s h o w t h a t b o t h d e p e n d in a s i m i l a r w a y o n r 0 a n d c f o r H C p o t e n t i a l s , a n d t h a t also the S R C results s t r a d d l e t h e H C results in a s i m i l a r m a n n e r . T a b l e 6 s h o w s D s f o r s o m e e q u i v a l e n t H C a n d S R C p o t e n t i a l s w i t h t h e s a m e a, r o a n d d. T h e v a l u e o f Ds differs q u i t e a p p r e c i a b l y - s o m e w h a t m o r e so t h a n for p o * That Ds(SRC)--Ds(HC) is positive and increases as the repulsion becomes stronger may plausibly be understood as follows. The characteristic energy for nuclear matter is the Fermi energy EF ~ 40 MeV. Consider the case in which E(6 = 0) is much greater than EF, as for a weakly repulsive core (small c). Then for SRC potentials, the part of the core that mainly determines 6 for E ~ E(6 = 0) is also the part important for nuclear matter. Thus the HC and SRC potentials that give the same up to E(6 = 0) are also expected to be approximately equivalent for nuclear matter and thus to give about the same value of Ds. If E(6 = 0) is comparable to Ev, on the other hand, then also the inner part of the core becomes important for nuclear matter. This part will correspond to negative phase shifts which are no longer effectively determined by a, ro and E(6 -- 0). Since for the SRC potentials, the inner part of the core is on the average less repulsive than the outer part, the net effect of the core for nuclear matter corresponds to less repulsion, and hence larger Ds, than for the "equivalent" HC potential.

164

A.R. BODMER AND D. M. ROTE

tentials equivalent with respect to E(5 = 0). The difference is again in the same direction, i.e., D s (SRC) > D s (HC). Thus d also is not a suitable parameter.

5.4. SCATTERING CORRELATION VOLUME Kssc The conclusions for XsNM naturally suggest the use of some related measure of the correlation but one which is obtainable from scattering calculations. To sensibly define such a quantity, we have to take account of the fact that the scattering wave function does not heal. We therefore define a (zero-energy) s-state correlation volume by

sc = 4np

zSC(r; k = 0 ) ] 2 d r ,

(9)

where

k) = rjo(kr)-u,(r; k)

(10)

is the scattering deficit wave function for a relative m o m e n t u m k, and where d is the separation distance. The density p is introduced merely to make x sc s non-dimensional and comparable with x~ u (the ratio of correlation volume to volume per nucleon in nuclear matter). Both k = 0) and d are obtained by integration of the Schr6dinger equation for k = 0. This is the value of k appropriate to the scattering length and is chosen for convenience. F r o m the definition of d one has

us(r;

~dr l°g us(r,

k)lr=d,k=O =

I~r l°g

rj°(kr)lr=d,k=O"

Furthermore the magnitude o f us is chosen to be equal to the unperturbed wave function with the normalization appropriate for nuclear matter. Because for scattering Us does not heal, the upper limit in the integral in eq. (9) cannot be oo (since then the integral would diverge). In fact d is a natural upper limit in view of the Moszkowski-Scott considerations. These suggest that for us is close to the unperturbed wave function for r > d (i.e., in the region of V t~°ng)) and that the dominant contributions to x sNM therefore come mostly from r < d. However, the region r > d gives of course some contribution to x sNM, but - by definition - not to Xssc . This contribution will therefore tend to increase tc~Mrelative to tcSsC . On the other hand, the effect of Q and A depends on kF and is to enhance healing and hence to decrease tcNr~ s relative to ~ssc • Even though these two effects are in opposite directions, we cannot expect exact cancellation and can only expect x~ M and x sc s to agree approximately for some particular range of values of k F. It must be emphasized that even approximate agreement is by no means necessary for x sc to be an acceptable parameter. For this it is only necessary that x.sc together with a and r o, determine S ' Ds, i.e., that effectively there be a unique shape-independent relation between r~ M and rssc. In fact our results, in particular those of fig. 8 and table 3, show that rsc and ~:~Mare quite close for k F ~ 1.4 f m - t, especially for not too small ro. This near equali-

nuclearmatter

A-BI'NDING IN N U C L E A R MATTER

165

ty must be considered as largely accidental and a result of the near cancellation of the two opposing effects just mentioned for the particular values of k F considered. Table 4 and fig. 3 clearly demonstrate that for equivalent H C and SRC potentials with the same a, ro and Ks~c, the values of Ds are to a good approximation equal, the agreement being about equally good as for potentials that are equivalent with respect to KNM s Of course this conclusion is, strictly speaking, limited to the range of repulsive cores and values of a and r o considered. However it seems plausible that the equivalence should hold for any "reasonable" local potentials, although it would clearly be worthwhile to investigate an even wider range of potentials than we have dono. .

I

I

I

to=3 fro, O = - 2 f m 0.3 -KS

0.2 -

NM

K~

SC

~ . . ~ ~ j - - K

s

0.1 0.0

0

, -'"i'0.2

/

I 0.4 c (fro)

,

I 0.6

Fig. 8. The correlation v o l u m e ,q for H C potentials, plotted as a function o f the hard-core radius c for a = - - 2 fm, ro = 3 fro. The full line is for the self-consistent nuclear-matter correlation v o l u m e ,¢s --=- KsNM and the dashed line for the scattering correlation v o l u m e ,q ~ Kssc.

Thus, with the qualifications just made, we have found a "scattering" parameter that characterizes the effects on D s of the short-range part of a local potential in an effectively shape-independent way. The parameterization of D~ in terms of a, ro and KSsc is then effectively universal for local potentials. In particular, this allows our H C results for Ds, when parameterized in terms of a, r o and Ksc (as presented in table 1 and in figs. 1-4), to be used for any local potential. Such use then involves only scattering calculations which are in any case necessary for a and ro. For any H C potential, our results can of course be used directly in terms of a, r o and c. If Ksc is used for the parameterization, then the self-consistent values (or those for other values of AN and Aa than the ones used in table 1) may be obtained by use of eqs. (7) and (8) together with K~ ~ x sc.

6. T h e p - s t a t e w e l l d e p t h

Dp

For each potential in table 1, we give two values of Dp corresponding to two values 1 and 0.5 of the ratio p/s of the L = 1 to the L = 0 potential strengths (1 is appropriate to the potential strength actually given in the table). Also shown for p/s = 1 are D'p and the p-wave correlation volume Xp (for a precise definition and discussion see sect. 3). Only the nuclear-matter value is considered and

166

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

I

'

a =-2

I

'

I

fm

c(fm)

=18}< x

0

~

12-

0

~

60

.

6

SRC 0.429 0,3

. , ~ . ~

I

r

2

4 ro(fm)

I

I

I

6

Fig. 9. The p-state nuclear-matter correlation volume % for p/s = 1, plotted as a function of ro for a = --2 fm and Azt = 30 MeV. Results for H C potentials are labeled by the appropriate hardcore radius c, those for SRC potentials by SRC.

)

I

'

I

~

I

'

I

e(fm) 40 _ A

~

0.6 SRC

t

0,6

~

Q=-2fm

0.429

30 -

0.429

O=-1.25fm

1

¢= 20 I0 ,

O(

)

2

,

I

,

4

)

,

6

I

8

ro (fm) Fig. I0. The p-state well depth Dp (for p/s = 1 and ,dA = 30 MeV), plotted as a function of ro. The full lines are for H C potentials and the dashed line for SRC potentials.

i

I

'

I

ro(fm)

o=-2fm 40

---------4

30 :E

/

2O

o. E3

IC ,

I

0.2

~

I

0.4 e(fm)

~

I

0.6

Fig. 11. The p-state well depth Dp (for p/s = 1 and A4 = 30 MeV), plotted as a function of the hard-core radius c for various values o f ro and for a = --2 fro.

A-BINDING IN NUCLEAR MATTER

167

the superscript N M is omitted. As shown in fig. 9, ~:p is an order of magnitude smaller than xs ; but again Xp increases with the size of the repulsive core. Table 1 and also figs. 10 and 11 show that Dp n o t only increases strongly with a, as expected, but also with r o. Especially for p/s ,,~ 1, Dp can be quite large (typically about half of Ds), especially so for the longer ranges for which Dp can be comparable with Ds. For given a and ro, the value of Dp for H C potentials (fig. 11) is almost independent of ¢ for small r o and increases only moderately with c for larger values of r o. The dependence of Dp on p/s is approximately linear for p/s ~- I if the repulsive core is fairly small and if the range/~- 1 of the attractive potential is not too small. Thus with

Np = [Dp(p/s = l ) - 2 D p ( p / s = 0.5)]/2Dp(p/s = 0.5) as a measure of the nonlinearity, one has for a = - 2 fm and r o ~ 3.9 fm the values Np ~ 0.01(1), 0.055(1.5), 0.164(2) and 0.664(3.3) for c = 0, 0.3, 0.43 and 0.6 fm, respectively where the numbers in parentheses are the approximate values of # in f m - a ; for a = 0.75 fm and ro ~ 5 fm, the corresponding values are Np ~ 0(1), 0.09(2), 0.36(3) and 3.6(5). The non-linearity thus becomes rapidly larger for larger repulsive cores and shorter ranges, the significant measure of range being/2-1. Thus Np is already quite appreciable for ¢ ~ 0.4 fm and #-~ ~ 0.5 fm, and linear scaling o f Dp with p/s is reasonable only for rather small repulsive cores corresponding to c ;~ 0.3 fm. In all cases one has Np > 0, i.e., the increase ofDp withp/s is more rapid than linear. The above characteristic features of Dp are primarily accounted for by the moderately strong p-wave centrifugal barrier. On the one hand, as a result of the Fermi motion of the nucleons, the average kinetic energy of a AN pair is large enough for the correlated wave function Up to penetrate the centrifugal barrier at least into the outer and mainly attractive region of the potential where Up is mostly close to the unperturbed wave function. (The equivalent impact-parameter argument implies that the product of k F and the range of the potential becomes ~ 1 for appreciable penetration, i.e., kv p - 1 > 1.) This penetration accounts for the large values of Dp. On the other hand, the centrifugal barrier is large enough to quite strongly damp up in the inner region of the potential, in particular in the region of the repulsive core. Thus unless the core is quite large or the attractive potential very strong, the deficit wave function Xp is mostly quite small since Up can differ appreciably from the unperturbed wave function only in the inner region where it is, however, strongly damped by the centrifugal barrier. Hence Xp is mostly quite small even though Dp may be quite large. To understand the dependence of Dp on the repulsive core, one must be careful to distinguish between the case in which c is varied with a and ro fixed and that in which c is varied with a and the attractive shape (i.e., the range # - 1) kept fixed. The result of the latter variation which is illustrated in table 7, is to greatly enhance the effect of changing

168

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

c as compared to the variation with r o fixed. Thus if a and r o are kept fixed as c is increased, then # - t must be decreased to keep ro the same while at the same time VA must be increased to keep a the same. The two changes affect Dp in opposite directions such that they largely compensate each other, especially for small r o. (Thus Dp decreases if/z-1 is decreased, whereas it increases if VA is increased.) On the other hand, if a and p-~ are kept fixed, then VA must be TABLE 7 The p-state well depth Dp and correlation volume Kp as functions of the hard-core radius c for a : --2 fm and a fixed range parameter tt = 2 fm -1

c

ro

b

Dp

(fm)

(fro)

(MeV)

0

1.51

1.03

6.9

3.4

0.3

3.07

1.86

20.7

9.5

0.14

0.429

3.80

2.17

28.4

12.2

0.23

0.6

4.90

2.58

39.7

14.8

0.47

(fm)

Dp (p/s

=

½)

lOOxp ")

(MeV) 0.02

a) The nuclear-matter parameters are those o f table 1.

increased as c is increased in order to keep a the same. This implies, as illustrated by table 7, that the intrinsic range b and therefore, for a given a, also r o increase strongly with c, with a consequent strong increase also in Dp. In fig. 11 this variation then corresponds to going diagonally up to the right across the curves of constant r o . The dependence of Dp on p/s may be understood as follows. The dependence will be approximately linear if Xp is small since Dp is then given to a good approximation by the perturbation-theory expression which is linear in the potential strength. Ifp/s is appreciable, however, then for large repulsive cores and for short ranges/~- 1, Up may differ appreciably from rjl. Such appreciable deficit wave functions imply correspondingly large non-linearities through the "higher-order" contributions. These increase the binding, consistent with the sign of the non-linear terms. The situation regarding the values of Dp for "equivalent" HC and SRC potential (i.e., potentials with the same a, ro and Xp) is illustrated by a joint consideration of figs. 9 and 10. For small to, the values of Dp for equivalent potentials are quite close; but for such values of ro, Dp depends in any case only slightly on the repulsive core (and hence on xp). For larger ro (e.g., for r o > 4 fm, a = - 2 fm), the SRC results for D v are in fact closer to the HC results for c -- 0.6 fm than to those for c = 0.4 fm although the values of x v for the latter are closer to SRC values of rv. However, even for these larger values of r o the dependence on the repulsive core is fairly small. Thus although "equivalent" potentials generally give quite similar values of Dp, this approximate equality is not very useful or significant and is in fact merely a reflection of the weak dependence of Dp on the repulsive core. In fact a more useful equivalence is for potentials with the same values of a, r o and E(t5 = 0). For such potentials the corresponding values of Dp are quite close.

A-BINDING IN NUCLEAR MATTER

169

This is because, unlike the s-wave, the p-wave does n o t p e n e t r a t e into the inner core region.

7. Well-depth results for L > 2 The d-state well d e p t h s D a a n d the c o r r e s p o n d i n g n u c l e a r - m a t t e r c o r r e l a t i o n v o l u m e s x a are shown in table 1 for d-state potentials equal to the ones listed. A s expected, D a increases with the range; b u t except for extremely long a n d unrealistic ranges, D a is fairly small a l t h o u g h n o t negligible. T h e large L = 2 centrifugal b a r r i e r accentuates the shielding effects a n d r e l a t e d features discussed for Dp. I n p a r t i c u l a r ~:d is very small a n d typically o f the o r d e r o f 1 0 - 4 ( < 0.5 × 10-3). The c o n t r i b u t i o n s D L from L > 2 are generally quite negligible. TABLE 8 C o m p a r i s o n o f well depths a) for kv ~1~ = 1.35 fro-~ and kr ~2) = 1.4 f m - ~ /~ (fm -1)

1.5 2.0 3.0 4.0 2.0 3.0

c (fm)

0 0.3 0.429 0.6 SRC SRC

ro (fm)

2.21 3.07 2.77 3.10 4.50 3.08

10xs D~l)

0.26 0.50 1.12 2.55 1.14 0.72

A a = 30 MeV DI(1) Dd(t)

Self-consistent D~) D O)

Os ~2)

Op(2)

Oa(2~

Os ~2)

O t2)

0.91 0.92 0.92 0.95 0.95 0.93

0.86 0.86 0.85 0.85 0.86 0.85

0.83 0.82 0.77 0.79 0.80 0.82

0.92 0.92 0.93 0.96 0.96 0.93

0.90 0.90 0.91 0.93 0.91 0.91

R 2

R a

R6

R 2

R3

0.93

0.865

0.804

0.93

0.897

") The appropriate nuclear-matter parameters are AN = 80 and 85.4 MeV and M N * / M N = 0.653 and 0.638, respectively. The results are all for a = --2 fm. The appropriate powers of R = kvta)/kF ~2) = 0.964 are also shown (R 3 = ptl)/pc2)).

T h e small values o f D L for L > 2 are a result o f the large centrifugal barriers. These are n o w generally so large t h a t for the ranges considered a n d for the nucleon m o m e n t a characteristic o f nuclear matter, the A N wave f u n c t i o n is a l m o s t c o m p l e t e l y excluded f r o m the entire p o t e n t i a l region ( a n d n o t merely f r o m the inner region as for L = 1 ). T h e t o t a l (self-consistent) well d e p t h [eq. (6)] is also given in table 1 for an L i n d e p e n d e n t p o t e n t i a l (V L the same for all L), a n d a p o t e n t i a l with p / s = 0.5 b u t which is the same for L = 0 a n d L = 2. These results s h o w clearly the generally large L = 1 c o n t r i b u t i o n . A s s o c i a t e d with the c o n t r i b u t i o n s to D f r o m L > 0 is a slight r e d u c t i o n which arises p r e d o m i n a n t l y f r o m D s because o f the self-consistency c o n d i t i o n (EL> o D L increases d ) .

170

A.R. BODMER AND D. M. ROTE

8. Dependence of well depths on kr The results of table 1 are mostly for k F = 1.4 fm-~. However, we have obtained results also for k r = 1.35 fm -~. Some of these are shown in table 8. These indicate that for k F not too different from 1.4 f m - t the dependence on k F is to a reasonable approximation given by

DL

L+2.

(11)

This proportionality may then be used to obtain D L from the results of table 1 for values of k F not too different from k F = 1.4 fm -1. Eq. ( l l ) is consistent with the dependence expected from first-order perturbation theory if furthermore the small k-dependence of.jL(kr) is a reasonable approximation. For the case of an L-independent potential one finds, to an excellent approximation, that D oc p. This is precisely the dependence predicted by first-order perturbation theory 22). (This gives D = pU, where U is the spin-average volume integral.) It may however be expected that if k F differs more appreciably from 1.4 fm-x, then the above simple dependence of k F will become inadequate. The deviations from eq. (1 l) are expected to be greatest for D s in view of its strong dependence on the repulsive core via Xs, and for hard-core potentials with a very short-range attraction (i.e. large #) since higher-order effects are then relatively larger. The results of table 8 in fact give some indications for this.

9. Discussion and limitations of well-depth results Although our investigation of the adequacy of ~csc s and KNM s as s-state shape-independent parameters for the core covered a fairly wide range of HC and SRC potentials, it would be desirable to consider an even wider r a n g e - in particular of values of a and r o and especially of SRC potentials. Since the properties of the G-matrix elements for the A in nuclear matter are very similar to those for nuclear matter, one may expect that a, r o and x.NM s or KSC s constitute a shape-independent parameterization also for the s-wave contribution to the binding energy of nuclear matter. The present work was limited to local HC and SRC potentials which cannot be phase-shift equivalent at all energies - although they may be approximately so if a, r o and E(5 = 0) are the same. An important conclusion of our work is that two potentials need not be completely phase-shift equivalent in order to give approximately the same D s ; the only requirement, in addition to equality of ~cs, is that the two potentials should have the same a and r o. In view of the results obtained by various authors is) for phase-shift equivalent potentials (at least one of which must be nonlocal), we expect equivalence with respect to a, r o and x NM s to hold also for non-local potentials. The parameters a and ro seem effectively to give a good shape-independent representation of the long-range part of the potential (for a given value of xs). However, although a and r o are the primary low-energy scattering parameters, they are not

A-BINDING IN NUCLEAR MATTER

171

particularly fundamental for the nuclear-matter problem. For this a more fundamental quantity representing the long-range part of the potential is expected to be V 0°"g), i.e., the potential outside the Moszkowski-Scott separation distance. Thus it may well be that the single quantity (V°°ng))s and Ks would suffice to completely characterize the s-state effect of the potential for nuclear matter. Our results are limited to central potentials for which (for a given spin state) the s-wave effect of the core, more precisely of V(~h°r', is determined by ~s alone. The (triplet) s-wave effect of a tensor force occurs through admixture of a d-wave component, and an interesting question is whether the corresponding (d-wave) correlation volume x~o characterizes the s-wave effect of a tensor force for given low-energy scattering. Even further one may ask whether the single quantity x~+ t%a characterizes the total s-state effect. That x~d is the relevant shape-independent parameter for tensor forces is perhaps plausible since the "suppression" of a tensor force in nuclear matter (i.e., the difference between the effect appropriate for nuclear matter and for scattering) occurs through the exclusion principle and the spectral gaps via the d-wave admixture. This then plays the role of the deficit wave function. The question of suppression for the A case was discussed in ref. s). In particular, the suppression was found to depend strongly on the range of the tensor force, being less for shorter ranges. This result could be consistent with ~sa as a shape-independent parameter since a short-range tensor force may well give a smaller value of ~sd than would one of longer range for the same s-wave scatterin~ effect. We have assumed zero single-particle potentials for the unoccupied (A and nucleon) states, i.e., the use of the free kinetic energies. This is the most neutral assumption and corresponds to no single-particle potential insertions for the unoccupied states, and thus to no implicit assumptions about the cancellation of terms which correspond to such insertions with higher-order G-matrix contributions. Any non-zero assumption for the single-particle potentials of the unoccupied states can only be justified a posteriori by actual calculation of higher-order terms, in particular those proportional to p2. This requires the solution of a Bethe-Fadd6ev type three-body (ANN) problem. If the p2 terms are dominated by the rearrangement-energy terms (which are not included in the Bethe-Fadd6ev contributions) then to the G-matrix value of D one should add a correction ~ -IcNND, where ~NN is the nucleon-nucleon correlation volume, defined in analogy to eqs. (3) and (5). For ~cNN ~ 0.1, one then has a reduction of about I0 % in the G-matrix values of D for hard-core AN potentials. However it would clearly be most desirable to calculate the Bethe-Fadd6ev contributions for a A in nuclear matter. 10. Implications of well-depth results for the AN interaction A comparison of the calculated and phenomenological well depths can give information about the AN interaction. A reasonable value for the phenomenological well depth is Dph ~ 30_+3 MeV, with an upper limit of about 35 MeV. The uncertainties arise not only from errors in the experimental values of Ba but even more

172

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

from uncertainties in the extrapolation of the binding energies to A = oo. For a review and discussion see I. The information about the AN interaction is of course much more limited than that about the N N interaction. Nevertheless, the available hyperon-nucleon scattering data (in particular the low-energy Ap scattering data, especially when analyzed in conjunction with the binding energies of the s-shell hypernuclei) place fairly strong constraints on the AN interaction. The Ap scattering data are reviewed in refs. 14.23), in particular by Alexander. For the low-energy Ap scattering parameters, the Rehovoth-Heidelberg ( R H ) group 14) which has obtained the most extensive data gives as g - 1 . 8 fm, r s ~ 2.8 fm, a t ~ - 1 . 8 fm and r t m 3.3 fm without constraints and as = - 1 . 5 fm, r s = 3.3 fm, a t = - 1 . 7 fm and r t = 3.1 fm on the assumption of equal singlet and triplet intrinsic ranges; the Maryland-IIT (M) group 14, z4) gives as ~ - 2 . 0 fro, r s ~ 5.0 fro, a t ~ - 2 . 2 fm and r t ,.~ 3.5 fm. (The errors are large but strongly correlated since it is difficult to extract four parameters from the low-energy Ap scattering data.) The corresponding spin-averaged values, more directly relevant for D, are ~ ~ - 1 . 8 fm, Po ~ 3.2 fm ( R H without constraints), ~ = - 1 . 6 5 fro, Po = 3.15 ( R H with constraints) and ~ ~ 2.15 fm, ~o ~ 3.9 fm (M)*. The binding energies of the s-shell hypernuclei have been analyzed in conjunction with the Ap scattering data, especially by Tang and Herndon a 6. t 7) who allowed for lack of charge symmetry. [See also ref. 23) for a recent review and ref. 25) for a review of earlier analyses. ] F r o m their analysis of the A = 3 and 4 hypernuclei and with only two-body AN forces, they obtain potentials that also give results consistent with the scattering data. The parameters for their best (interpolated H C ) potential are c = 0.45 fm, ~ = - 1 . 8 fm and ~o = 3.1 fm for one data set and c = 0.45 fm, ~ = - 1 . 9 fm and ?o = 3.2 fm for another. For the latter data set, their potential E gives c = 0.45 fm, ~ = - 1 . 9 fm and ?o = 3.1 fm. The scattering parameters are seen to be quite close to the R H values. Several coupled-channel ( A N - Z N ) analyses of the low-energy hyperon-nucleon scattering data have recently been made 2a, z6, 27) which use meson-theory potentials with the SU(3) relations for the coupling constants. The Nijmegen group 23"26) allow also for lack of charge symmetry and use repulsive cores which are adjustable parameters. Their results give ~ g - 1 . 4 5 fm, ?o ~ 2.3 fm and c ~ 0.35 fm (c is in fact different for S = 0 and 1 but the difference is rather small for our purposes). They also predict a 3S 1 AN resonance just below the Z N threshold (due to a " b o u n d " Z N state in the A N continuum in the absence of A N - Z N coupling). For our discussion we use two sets of values: (A) ~ = - 1 . 8 fm, ?0 = 3.1 fm and c = 0.45 fm and (B) ~ = - 1.45 fm, ~0 = 2.3 fm and c = 0.35 fm; these correspond to Tang and Herndon's and the R H values and to the Nijmegen results, respectively. t M o r e correctly one s h o u l d use the values appropriate to the spin-averaged potentials. However, the differences between the two averages is small if the singlet a n d triplet p a r a m e t e r s are n o t very different, as is in fact the case. T h u s even for the N i j m e g e n parameters, o n which o u r p a r a m e t e r s B are based a n d for which the singlet a n d triplet scattering lengths differ appreciably, the difference between the two averages is generally less t h a n 1%.

A-BINDING IN NUCLEAR MATTER

173

The differences between (A) and (B) give some indication of the uncertainties involved; both sets are consistent with the scattering data. The hard-core radii in particular must be considered rather uncertain since the evidence for these is indirect; we may thus consider low-energy scattering parameters corresponding to sets (A) and (B) but associated with rather different hard-core radii than those given above. It should be remarked that Tang and Herndon find that the two Ap parameters as = - 1.75+0.1 fm and r t 3.0+__0.25 fm are quite insensitive to the potential shape assumed; it will be noted that this value of r t is considerably larger than that used for (B). It is unfortunate that for the well depth the value of a t is considerably more important than that of as. However, the situation is probably considerably better if the scattering data are also included*. The scattering parameters quoted suggest errors of roughly _+0.3 fm for both ~ and ~o for any particular set of scattering parameters. It is well known that central two-body AN potentials consistent with the scattering data (and also with B a for A = 3, 4) overbind .~He. Thus the Tang and Herndon pgtentials referred to above give Ba(~He ) ~ 6.5 MeV instead of the experimental value of 3.08 MeV. This strongly suggests that the A-nucleon interaction is weakened by many-body effects arising from the close proximity of the other nucleons. Two mechanisms proposed are (repulsive) three-body A N N forces 25,29,30,33,34) and suppression of the two-body interaction mainly through suppression of the AN-XN coupling 7, s, 25, 28-32). The two mechanisms are partially equivalent 29). However, in particular, for the A = 4 and 5 systems the details of the core structure may be so important as to largely obscure this equivalence 28, 32)**. F r o m their analysis of 3He and aSHe, with the idea that suppression effects for A = 3 and 4 are much smaller than for aSHe, Tang and Herndon 16, 17) have also obtained some (primed) central AN potentials that allow purely phenomenologically for A,~ suppression in SaHe. These primed potentials have an appropriately smaller strength than the corresponding unprimed potentials with the same shape and will of course no longer give a fit to the scattering data. In principle, the scattering data at somewhat higher energies give information about the p-state interaction. In fact, however, this seems rather poorly determined at present. The Ap differential cross section has been studied at higher energies by K a d y k et al. 14,35). They find that up to about 100 MeV c.m. energy the data are well described by an s-wave interaction only and that for such energies there is no evidence for any non-zero p-state interaction. The analysis of the low-energy data (for 10-20 MeV) by Tang and Herndon leads to p / s ,~ 0.5_+0.2. It is interesting that the p-state phase shifts for the Nijmegen interactions (with -----

t Thus if one accepts that as is quite well determined from the binding-energy analyses--mainly from BA(,taH)--and notes that the scattering data strongly emphasize the triplet parameters, then it seems plausible that these data probably determine at reasonably well [see e.g. ref. 23)]. tt With inclusion of three-body forces Tang and Herndon cannot obtain consistency for A = 3, 4 and 5 with the forms of three-body potentials which they use. See also ref. a4) for a sharpening of this conclusion. In view of the partial equivalence of suppression effects and three-body forces, the significance of this result is not clear.

174

A. R. BODMER AND D. M. ROTE

the repulsive cores a n d c o u p l i n g constants d e t e r m i n e d from their s-wave Y N fits) are very small even to quite high energies 36). I n particular the p-wave phase shift averaged over spin a n d total a n g u l a r m o m e n t u m is quite small ( a n d even corresponds to a slight net repulsion). TABLE 9

Well depths (MeV) for assumptions (i) to (iv) of sect. 10 about the AN interaction a)

(i)

(ii)[(ii')]

kr = 1.4fm -1 k F = 1.4fm -1 (A)

(B)

(A)

Ds D(Vp = 0 ,

56

51

46

47

Vs = Va) D(V~ = Vp

58

52

48

= Vn)

76

64

67 18.5 9 2

58 12 6 1

D (pls = ½, V, = Va)

Dp Dp(p/s = ½) Da

(B)

(iii)

(iv)

k p = 1.35fm -1 k s = 1 . 3 5 f m - ~ k r = 1.35fm -x (A)

(B)

(A)

(B)

(A)

(B)

43 [40]

44 [40]

36

36

23(32) 24(33)

47.5

44.5141]

44.5141]

37

37

24(33) 24(33)

65

57

58 [53]

51 [46]

49

42

37(45) 30(39)

55 20 8 2

52 10.5 4.3 0.6

51 [46.5] 48 [44] 17 9 7 3.7 1.6 0.5

43

44

31(40) 27(36)

(i) Central potentials with c = 0. (ii) [(ii')] Hard-core central potentials [including the rearrangement energy ER in parentheses]. Off) Hard-core potentials with tensor force and ER. (iv) Hard-core potentials with ER, tensor force, and A S suppression of 15 MeV and (in parentheses) of 5 MeV. a) For (A): a = --1.8 fm, r = 3.1 fm and unless otherwise indicated c = 0.45 fm, similarly for (B): a = --1.45 fm, r = 2.3 fm and c = 0.35 fm. I n c o n n e c t i o n with the p-state interaction, it should be noted that Vp ~ 0 will give a n effective A mass M * ~ 0.7 M a , whereas for a n L - i n d e p e n d e n t potential (Vs = Vp . . . . ) one has M * ,.~ M A (as shown in I). The mass M * is appropriate for a A of m o d e r a t e kinetic energy p r o p a g a t i n g t h r o u g h nuclear matter. F o r a finite hypernucleus the effect of M * / M a < 1 is then to increase the effective kinetic energy which, as discussed in I, leads to a larger value o f Dph t h a n for M * / M A = 1. T h u s for M * / M A ~ 0.7 one o b t a i n s a n increase of a b o u t 3 MeV a n d correspondingly Dph ~ 34-35 MeV. Since the calculated values are generally too large, this increase of Dph helps somewhat to improve the agreement. I n the following c o m p a r i s o n of the calculated a n d the p h e n o m e n o l o g i c a l well depths, we consider progressively more complex t w o - b o d y A N interactions, proceeding from purely central potentials with n o repulsive cores (i) to interactions that include tensor a n d AN-27N couplings (iv). All the interactions are consistent with the scattering parameters of (A) or (B). The c o r r e s p o n d i n g self-consistent values of D are s h o w n in table 9; D s corresponds to a purely s-state interaction. Results are given for k r = 1.4 f m - 1 a n d 1.35 f m - 1. The former is p r o b a b l y somewhat too large since it is equivalent to a density c o r r e s p o n d i n g to a n average radius per n u c l e o n o f

A-BINDING IN N U C L E A R MATTER

175

1.05 fm whereas 1.35 fm -1 corresponds to 1.12 fm, which is probably more realistic. The change from 1.4 to 1.35 fm -1 reduces D by 7-10 ~ (and in general brings it closer to Dph). For different interactions consistent with (A) and (B), inclusive of the appropriate hard core, the main difference in D is in the value of Dp. The values of D, are then very similar for (A) and (B) because D s increases with lal but decreases with c. Thus both [al and c are smaller for the (B) than for the (A) parameters. If c were the same for (A) and (B), then the values of D~ for (A) would be about 5 MeV larger than for (B). On the other hand, Dp depends only slightly on c but increases rapidly with r o which is considerably larger for (A) than for (B). Of course, the s-state parameters may be quite inappropriate for Vp, and comparison of the results for (A) and (B) inclusive of the corresponding hard-core radii may thus be considered as illustrating the use of different assumptions for Vp. (i) Purely attractive central A N potentials without repulsive cores and which are the same for all L are seen from table 9 to give much too large values of D. This remains true even if the potential acts only in s-states. This is the well-known result of the overbinding of D. To obtain agreement with Dph for these simplest forms of potential would require a strongly repulsive p-state potential somewhat larger in magnitude than Vs for (A) and more than twice as large for (B). Such strongly repulsive potentials Vp, which give appreciable contributions to D and hence must have a fairly long range, seem inconsistent with the evidence about the p-state interaction indicated by the scattering data. It should be noted that for purely attractive potentials the rearrangement energy is expected to be negligible because there is no repulsive core.

(ii) For central potentials with repulsive cores, both (A) and (B) (with their appropriate hard cores) give, if the potential is the same for all L, the values D s ~ 47 MeV and 44 MeV for k v = 1.4 and 1.35 fm -1, respectively. To obtain agreement with Dph then again implies quite repulsive Vp, especially for (B) and k v = 1.4 fm-1. To obtain agreement while assuming Vp ~ 0 but allowing the hard cores to vary, the hard-core radii must be very large (thus for (B) and kr = 1.35 f m - 1 one needs c > 0.6 fm in order that Ds ~ 35 MeV). Such large values of c seem implausible and are inconsistent with the Nijmegen analysis, but they are probably not ruled out by a purely phenomenological analysis of just the Ap and hypernuclear data. (ii ') The rearrangement energy E R and the related uncertainties due to higher-order contributions have been briefly discussed in sects. 3 and 9. The value of ER depends on x NN. We use E R ~ - 0 . 1 D as a representative value. With this reduction and for k v = 1.35 f m - 1, the corresponding values of D may be considered to be close to the minimum ones obtainable for purely central potentials consistent with (A) or (B). Even now one needs a moderately repulsive lip to obtain agreement with Dph, although with somewhat larger values of c one is close to agreement (assuming Vp 0) for (B). Thus with Vp ~ 0 and c ~ 0.45 fm one obtains Ds ~ 36 MeV. (iii) Reduction o l D due to suppression of a possible A N tensor component is most

176

A.R. BODMER AND D. M. ROTE

probably slight (less than about 4 MeV). This has been discussed in particular in refs. 4-6). The small reduction is mainly a result of the short range of the AN tensor forces expected according to OBE models 5). For the appropriate results shown in table 9, a reduction of 4 MeV due to tensor-force suppression has been used and furthermore also a 10 % reduction due to E R has been included. Tensor-force suppression helps marginally to improve agreement with Dph. For k F = 1.35 f m - 1 , the value D s ~ 36 MeV is now very close to the upper limit of Dph. With allowance for errors in a and r o and for uncertainties in c, the values of D for Vp ~ 0 are now consistent with Dph. Thus with a larger hard-core radius c ~ 0.45 fm for parameters (B), one has Ds(k F = 1.35 f m - 1 ) m 32 MeV and thus consistency with Dph. Furthermore, it should be recalled that, as discussed above, the value of Dph is somewhat larger for Vp ~ 0 than for Vp ~ I s , and this also helps to further agreement. Thus allowing for uncertainties in the scattering parameters and hard-core radii enables one to achieve consistency with Dph for central plus (reasonable) A N tensor potentials which are consistent with the scattering and the A = 3, 4 hypernuclear data and which also have a reasonable repulsive core and at most a very weakly attractive Vp.

However such forces will still overbind 5AHe since for this effectioely only the s-state interaction is important. As already mentioned, repulsive A N N forces and suppression of the AZ coupling have been proposed to deal with this difficulty. Here we shall consider only the latter mechanism. (iv) Suppression of the coupling between the AN and (virtual) ZN channel in nuclear matter ("A27 suppression") might be quite strong and could give a very substantial reduction fiDA~ of perhaps 15 MeV or even more 7, s), with the important qualification that attractive higher-order contributions are not dominant 30). In the (leading) G-matrix approximation, the AZ suppression has been discussed in detail in ref. 7) for OBE potentials. In particular, suppression of the strong long-range OPE tensor coupling as well as of the central couplings due to p-meson exchange could give large reductions, particularly for a net attractive Z N potential strong enough to give a (S = 1) ZN bound state as for the Nijmegen interactions. Also for given A2~meson coupling constants 6Da~ is larger for smaller hard-core radii*. For 6DAz = 15 MeV plus a 4 MeV reduction due to AN tensor-force suppression plus a 10 % rearrangement-energy reduction, D is now easily consistent with Dph for k F = 1.35 fm - t and for a reasonable strong attractive p-state interaction Vp ~ ½Vs. Clearly, with smaller 6DA~ consistency can be obtained only for less attractive Vp or larger hard-core radii. It is interesting to consider the (suppressed but central) primed Tang and Herndon potentials that reproduce, purely phenomenologically, the experimental value of Ba(ASHe). These give a reduction of only about 5 MeV relative to the value of D ob* Thus the results of ref. 7) give d(tSDA~)/dc~ --30 MeV fm -1 for both AY.zr and ASp couplings. For OPE coupling and with fa~n ~ 0.2 [a reasonable value of the PV coupling constant a6)], one then obtains ~SDa~m 18.5 MeV for c = 0.35 fm, as compared to 16 MeV for c = 0.43 fm obtained in ref. 7).

A - B I N D I N G IN N U C L E A R MATTER

177

tained with the corresponding unprimed potentials. (See I for the appropriate results.) Consistency with Dph can then be obtained with a moderately strong attractive p-state potential Vp < 0.4 Vs, especially if allowance is made for uncertainties in a or c. Since at least some weakening of the A-nuclear interactions seems to be required for aSHe, a smaller purely phenomenological value 6Da~ ~ 5 MeV based on Tang and Herndon's primed potentials might perhaps be regarded as a rather conservative estimate for the A27 suppression in nuclear matter. It is clear from the above discussion that, with allowance for uncertainties in the scattering parameters and repulsive core, it is possible to obtain agreement with just Dph for very weak p-state interactions (Vp ~ 0) and reasonable hard-core radii (c ~ 0.4 fm) by invoking only a rather small reduction due to suppression which could all be attributed to a AN tensor force; appreciable A2~ suppression is not needed if the interactions are not also required to fit Ba(~He ). If AE suppression is invoked to account for Ba(SaHe) and if a corresponding suppression of about 5 MeV is considered reasonable for a A in nuclear matter, then for reasonable hard-core radii a moderately attractive Vp would be required. One hopes that more accurate Ap scattering data will in future give more reliable information about Vp. This should then allow more definite conclusions about the magnitude of the A2~ suppression that is required in order to fit the well depth. Of course the above conclusions depend on the theoretical adequacy of the G-matrix approximation for the well depth. We are grateful to Prof. J. J. de Swart and to Dr. M. M. Nagels for discussions about their interactions and to Dr. Nagels for obtaining and sending us the p-wave phase shifts appropriate for these. We would like to thank Dr. F. E. Throw for editing the manuscript. References 1) J. Dabrowski and M. Y. M. Hassan, Phys. Rev. C1 (1970) 1883 2) D. M. Rote and A. R. Bodmer, Nucl. Phys. A148 (1970) 97 3) J. D. Walecka, Nuovo Cim. 16 (1960) 342; B. W. Downs and W. E. Ware, Phys. Rev. 133 (1964) B134; B. Ram and B. W. Downs, Phys. Rev. 133 (1964) B420; G. Ranft, Nuovo Cim. 45B (1966) 138; J. R. James and B. Ram, Nucl. Phys. A140 (1970) 688; B. Ram and W. Williams, Phys. Lett. 31B (1970) 49; Phys. Rev. C3 (1971) 45 4) J. Dabrowski and M. Y. M. Hassan, Phys. Lett. 31B (1970) 103; Binding energy of a A-particle in nuclear matter with noncentral A N forces (Institute of Nuclear Research, Warsaw, 1970) 5) A. R. Bodmer, D. M. Rote and A. L. Mazza, Phys. Rev. C2 (1970) 1623 6) G. F. Goodfellow and Y. Nogami, Nucl. Phys. B18 (1970) 182 7) A. R. Bodmer and. D. M. Rote, Proc. Int. Conf. on hypemuclear physics, Argonne National Laboratory, May 1969, ed. A. R. Bodmer and L. G. Hyman (Argonne National Laboratory, Argonne, Illinois, 1969) p. 521; Nucl. Phys. A169 (1971) 1 8) Y. Nogami and E. Satoh, Nucl. Phys. B19 (1970) 93; Phys. Lett. 32B (1970) 243; E. Satoh, Effect of A2~ conversion in hypernuclear matter, preprint, Tokyo Institute of Technology, 1972; J. Law, Nucl. Phys. B17 (1970) 614, B21 (1970) 332

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