Nucleon-nucleon interaction and triton binding energy

I

1.B

I i

Nuclear Physics A122 (1968) 684--688; ( ~ North-HollandPublishin# Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

NUCLEON-NUCLEON INTERACTION A N D T R I T O N B I N D I N G ENERGY L. P. K O K

Institute for Theoretical Physics of the Physical Laboratory, State University, Groninyen, The Netherlands G. E R E N S a n d R. V A N W A G E N I N G E N

Natuurkundig Laboratorium, Vrije Universiteit, Amsterdam, The Netherlands Received 25 July 1968 Abstract: T h e H u l t h 6 n a n d t h e generalized B a r g m a n n potential, which are b o t h in a certain sense equivalent to the Y a m a g u c h i separable t w o - n u c l e o n interaction, are s h o w n to give a rather different triton b i n d i n g energy. T h e implications o f this result are discussed.

In recent years various phenomenological and semi-phenomenological "realistic" nucleon-nucleon potentials which fit the available on-shell scattering data up to a few hundred MeV have been constructed. However, these potentials can differ considerably in their of~-shell behaviour, and it is important to investigate these differences. This can be done at least partially by investigating inelastic processes, e.g. e-d scattering, nucleon-nucleon bremsstrahlung, deuteron photo-disintegration, radiative capture of neutrons by protons etc. On the other hand, one may consider three-nucleon systems. In this note, we shall discuss the dependence of the ground-state energy of the triton on certain features of the nucleon-nucleon interaction. Since the treatment of realistic interactions is very complicated, we consider the simplest possible triton model, which consists of three identical spinless bosons interacting pairwise through (local or non-local) central interactions. Previous variational calculations t) with this model using different local potentials have revealed that interactions reproducing the same low-energy scattering data (scattering length a and effective range r0), and the same binding energy of the twoparticle system can give a rather different triton ground state energy ET. In this investigation, we have calculated E T for three different types of potential, namely (i) the non-local separable Yamaguchi potential 2), (ii) the Hulth6n potential and (iii) a generalized Bargmann potential constructed with the help of the recipe of Newton 3). The Hulth6n potential has not only the same two-nucleon binding energy as the Yamaguchi potential but also exactly the same two-nucleon ground-state wave function and approximately the same low-energy scattering parameters, whereas the Bargmann potential we have chosen has the same two-nucleon binding energy and identically the same S-wave phase shift (for all energies from zero to infinity) 684

NUCLEON-NUCLEON INTERACTION

685

as the Yamaguchi potential, but it has a different two-nucleon ground-state wave function. (The l > 0 phase shifts for these potentials are of course different.) The investigation is interesting because the Yamaguchi (Y) and Hulth6n (H) potential, on the one hand, and the Y and generalized Bargmann (B) potentials, on the other hand, are much more closely related to each other than any other pair of interactions compared hitherto. 400

300

200 100 r(fm)l~

V(r) MeV 0

1

5

3

i

T-IO0

7 i

9 L

H

-200 -300 -400 F i g . 1. G e n e r a l i z e d B a r g m a n n (B) a n d H u l t h ~ n ( H ) p o t e n t i a l w i t h ct = 0.10036225 fin -1 a n d fl = 1.45 f m -1 (set n u m b e r 2).

All three potentials can be expressed in terms of the two parameters ~ and fl used by Yamaguchi. We shall not give the explicit forms here but note only the following points: (i) the two-nucleon ground-state energy is for all three cases E D = - ~2 hZ/M; (ii) .the I = 0 phase shift is given by the expansion k cotg 6 = _ _1 + ½rok 2 _ P r ~ k 4 a

for the Y and B potentials, where a, r o and P are given in terms of ~t and/~ by Yamaguchi [ref. 2), eq. (20')]; (iii) the k cotg 6 expansion for the H potential does not break off after the k 4 term, but the parameters a, r o and P for this potential can be easily calculated using a procedure due to Ahlstr6m 4).

686

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et al.

KOK

We report here calculations for these three potentials for three different sets of parameters oc and ft. These sets are listed in table 1 with the corresponding values of a, r o, P and E D. We note that the values of a and r o for the Y and B potentials are practically the same as those for the H potential. This means that the low-energy phase shifts are almost the same. On the other hand, the shape parameters P differ considerably and even have an opposite sign. In fig. 1, we plot the H and B potentials for set 2, while in fig. 2, the phase shift for these same potentials is depicted as a function of the lab energy. The phase shift for the corresponding Y potential is of course the same as that for the B potential. We note that the H potentials are singular at the origin (r-~) and purely attractive, whereas the B potentials have a soft repulsive core, which has the finite value + 4 ( h 2 / M ) ( e + fl)2. Asymptotically, the B potentials behave as exp ( - 2 f i r ) , the H potentials decrease as exp(c~r-fir). 180

5(°)

T 120

H

O0

S

i

i

I

i

I00

ISO E=½Etab{MeV)

200

250

300

Fig. 2. S-wave phase shift (z0 for the generalized Bargmann (B) and Hulth~n (H) potentials from fig. 1 and for the Yamaguchi (Y) interaction as a function of E. The triton energy E r has been calculated exactly for the Y potentials using the method of Mitra 5). For the H and B potentials, upper bounds Eu and lower bounds EL were calculated. The values E u were obtained using the equivalent two-body method ETBM of Bodmer and Ali 6), cf. also Fiedeldey et aL 1). We are convinced that E u - E T is less than 0.1 MeV (probably even considerably less). This is based mainly on a comparison of ETBM results obtained by K o k v) and variational results of Humberston s) with exact results of Osborn 9) obtained with a Faddeev approach for various Yukawa and exponential potentials and also on the work of Homan et al. 1). The lower bounds EL are obtained by the method of Hall and Post 1o). For the H potential, it is easy to find an explicit expression for EL in terms of ct and fl, namely EL(H ) = h2 , 1 z ~ r (¥x/2~ +ax/2fl) •

NUCLEON-NUCLEON INTERACTION

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For the B potential, the appropriate eigenvalue differential equation must be solved numerically. TABLE 1 P a r a m e t e r sets for the v a r i o u s p o t e n t i a l s a n d t r i t o n e n e r g i e s (in M e V ) Potential number (fro -1)

1

2

3 0.23069894

0.0096140890

0.10036225

fl (fm -1)

1.45

1.45

a(H)(fm)

105.05007

ro (H) (fm) P(H) a(B, Y)(fm) ro (B, Y ) ( f m ) P(B, Y)

2.050683 +

0.039424 105,04964

1.45

11.004481

5.3502328

1.875775 +

0.054611 11.009923

1.601999 +

0.089338 5.3945465

2.050855

1.896167

1.716295

-- 0 . 0 1 8 7 6 4

-- 0.021043

-- 0.024146

ED

-- 0.00383297

- - 0.4176958

-- 2.207042

E L (H)

-- 11.6331

- - 19.7472

-- 35.1353

E U (H)

- - 7.265*

--14.59"

--28.91"

E T (Y)

- - 5.9659*

--12.4755"

--25.4083*

EL(B )

-- 6.612

--12.351

--23.644

Eu(B )

- - 5.146"

--10.90"

--22.24*

T h e m o s t i m p o r t a n t n u m b e r s , t o b e c o m p a r e d w i t h e a c h o t h e r , are i n d i c a t e d b y *.

The results of the calculations are given in table 1. All energies are in MeV, and 41.4686 MeV- fm 2. These results show clearly that E T depends very sensitively on various details of the nucleon-nucleon interaction. On the one hand, the results for the H and Y potentials indicate that the low-energy properties of the two-nucleon system including the detailed form of the bound-state wave function are insufficient to predict uniquely the triton binding energy, but that high-energy scattering data are needed. On the other hand, complete knowledge of scattering data for all energies equally fails to pin down the triton energy. Since the bound-state eigenfunctions and the phase shift as a function of energy completely determine the two-body T-matrix, and the phase shift alone defines its on-shell behaviour, E T sensitively depends on the of~-shell T-matrix elements. In view of the very large differences, one may hope to reduce the number of possible realistic nucleon-nucleon potentials by calculating the triton binding energy for each of them and comparing the results with the experimental values. However, a complication arises here because of the possibility of three-body forces. Louiseau and Nogami ll) have estimated an extra binding of 1.5 MeV due to three-body interactions. Nevertheless, even when allowing a generous uncertainty in this number, it is likely that certain of the current realistic potentials can be ruled out on the basis of their predicted ET values.

h2/M=

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Some preliminary results of this work (for another set ~ and fl) have been discussed by Noyes and Fiedeldey 12). We gratefully acknowledge stimulating discussions and suggestions of Professor Noyes and Dr. Fiedeldey on various occasions. The work forms part of the research program of the Foundation for Fundamental Research of Matter (F.O.M.), which is financially supported by the Netherlands Organization for Pure Research (Z.W.O.). One of us (G.E.) would like to thank the Shell Company (South Africa) and the South African Council for Scientific and Industrial research for financial support. References 1) T. Ohmura (Kikuta), M. Morita and M. Yamada, Progr. Theor. Phys. 15 (1956) 222, 17 (1957) 326, 22 (1959) 34; R. van Wageningen and L. P. Kok, Nucl. Phys. A98 (1967) 365; H. Fiedeldey, G. Erens, R. van Wageningen, D. H. Homan and L. P. Kok, Nucl. Phys. A l l 3 (1968) 543; D. H. Homan, L. P. K o k and R. van Wageningen, Nucl. Phys. A l l 7 (1968) 231 2) Y. Yamaguchi, Phys. Rev. 95 (1954) 1628 3) R. G. Newton, J. Math. Phys. 1 (1960) 319 4) P. E. Ahlstr6m, Royal Institute of Technology, Stockholm, Report, unpublished 5) A. N. Mitra, Nucl. Phys. 32 (1962) 529 6) A. R. Bodmer and S. All, Nucl. Phys. 56 (1964) 657 7) L. P. Kok, International Centre for Theoretical Physics, Trieste Internal Report IC/68/11 (1968) unpublished 8) J. W. Humberston, private communication 9) T. A. Osborn, SLAC-PUB-361 (1967); SLAC-79 UC-34 (1967) thesis 10) H. R. Post, Proc. Phys. Soc. A69 (1956) 936, A79 (1962) 819; R. L. Hall and H. R. Post, ibid. A90 (1967) 381 11 ) A. Louiseau and Y. Nogami, Nucl. Phys. B2 (1967) 470 12) H. P. Noyes and H. Fiedeldey, Proc. Conf. on three-particle scattering in quantum mechanics, College Station, Texas (April 1968) to be published