Central nucleon-nucleon potentials and the behaviour of the ground state wave function of 6Li

Nuclear Physics Al57

(1970) 363-368;

@

North-Holland

Publishing

Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

CENTRAL

NUCLEON-NUCLEON

POTENTIALS

AND THE BEHAVIOUR

OF THE GROUND STATE WAVE FUNCTION S. A. AFZAL Atomic

Energy

Centre,

Received

and

S. AL1

Dacca-2,

29 May

OF ‘Li

East Pakistan 1970

Abstract: We have analysed the central nucleon-nucleon potentials which have usually been employed in recent years in the resonating group studies of light nuclei. In the course of this analysis, it has been observed that although some of these potentials produce the low-energy nucleon-nucleon scattering data not too unsatisfactorily, there is still scope for a somewhat better fit to the data with central potentials. The sensitivity of the radial wave function of an assumed a-d structure of the ground state of 6Li to different central potentials which fit the low-energy two-body scattering data with varying degrees of accuracy has been explored and the significance of the explicit distortion effect in terms of the behaviour of the potentials used in the 0c-d model of 6Li has also been discussed.

1. Introduction In recent years, the method of resonating group structure has assumed an important role in the structural studies of light nuclei. This is mostly because in systems which consist of two-cluster configurations (tl-ti model of 8Be, a-d model of 6Li, etc.) one can obtain an effective integral equation of motion of the two clusters, without violating the Pauli principle and which is also free from a smooth projectile-target interaction ansatz. The basic ingredient which characterises the resonating group formalism is the nucleon-nucleon interaction from which the interaction between the clusters is developed. Although the nature of the nucleon-nucleon interaction is not completely known, phenomenological potentials constructed from the two-body scattering data do however exist [for example, Hamada-Johnston potential ‘), Tabakin potential ‘) etc.]. These phenomenological potentials which are fairly realistic are difficult to use in the resonating group studies due to the reason that for the calculations to be analytical, the two-nucleon potential has to be necessarily assumed to be of a central Gaussian shape. Thus all resonating group studies of light nuclei have been performed t with this restricted form of the N-N interaction. This restriction is, however, not too severe so long as the potential with all its limitation is constructed as well as possible to represent the two-body scattering data at least in the energy region of O-20 MeV (lab) which concerns most of the resonating group systems. + For an excellent review of these studies, see the invited papers of Wildermuth and Tang presented at the international conference on Clustering phenomena in nuclei, held at Bochum, Fed. Rep. Germany during 21-24 July 1969. 363

364

S. A.

AFZAL

AND

S. ALI

In the cr-d model study of 6Li, the central Gaussian N-N potential has already been used by Hasegawa and Nagata “). But this potential grossly overbinds the deuteron (& = 5.39 MeV) and the singlet N-N phase shifts given by this potential are far from being satisfactory. A potential which has been in use for quite a long time is that employed, amongst others by Schmid and Wildermuth “) and Tang and his collaborators “). Although this potential, which has been the best available so far, reproduces the deuteron binding, the singlet effective range parameters are not reproduced correctly +. This means that the ratio of the s-wave singlet to triplet interaction is not quite correct and consequently the exchange mixture parameter (v) which determines the mixture of Serber and Rosenfeld forces is also not very accurate, although it is seen that the u-d binding energy is very sensitive to this parameter. In the present investigation we have studied the behaviour of the ground state wave function of the assumed cl-d system of 6Li for different central N-N potentials which fit the scattering data in varying degrees of accuracy. By doing so we have been able to pick out a potential which reproduces fairly well the triplet and singlet effective range parameters. The implication of the distortion effect in the u-d model of 6Li has also been studied in terms of the behaviour of the two-body potentials used. 2. The N-N interaction The N-N interaction to be used in the resonating group method calculations is assumed to be of the form ~j = (~+mP~j+bP;-hP~j)V~

exp (-fir;)+

e2Eli r,

(1)

where w, m, b and h satisfy the relation w + m + b + h = 1 and the ratio of the singlet to triplet force is determined by w +m -b - h( = x). In eq. (l), E,~ = 1 for protons and 0 otherwise. The values of V,, and /I used by previous authors and the triplet and singlet n-p effective range parameters calculated for these values are shown in table 1. It can be seen from table 1 that of the three potentials A, B, C used by previous authors ‘94, ‘), potential B obviously merits more consideration. A slight adjustment in the x-values for these potentials brings 4 in agreement with the experimental value but the value of the singlet effective range is somewhat below the experimental one (see potential G). It was thus felt necessary to make a more careful determination of the force parameters. We therefore searched through a wide range of values of the range and depth parameters /I and I’, respectively. The initial guesses of these parameters were obtained from the early work of Feenberg ‘). A variation of the potential of ref. “) was also made and the potentials which finally merited considert This potential has often been quoted [see e.g. refs. 4-6)] as reproducing the deuteron binding, triplet effective range, n-p singlet scattering length and n-p singlet effective range. However we find in the course of the present investigation that not all these claims are true ( as shown later).

-72.98 -80.00 -48.50

--51.39 62.20 - 72.98

B [refs. 4-6)] C [ref. 3)] D

E F G

0.300 0.379 0.460

0.460 0.412 0.279

0.266

( fm’ “)

5.60 5.45 5.36

5.36 3.85 5.63

6.09

Triplet scattering length at (fm) (5.43 fO.004)

The figures in traces are the experimental values of ref. s).

-45.00

(M7V)

A [ref. ‘)I

Potential

2.01 1.84 1.70

1.70 1.57 2.07

2.16

Triplet effective range r1 (fm) (1.76&0.005)

2.224 2.230 2.223

2.223 5.390 2.234

1.850

Deuteron binding energy (MeV) (2.226&0.002)

n-p parameters

TABLE 1

0.598 0.630 0.656

0.630 0.640 0.586

0.600

X = w+m--b-h

-23.92 -23.73 -23.70

- 14.66 16.56 -23.50

-22.83

Singlet scattering length a, (fm) (-23.715&0.013)

2.81 2.50 2.23

2.29 2.10 2.93

3.01

Singlet effective range r. (fm) (2.66 sO.09)

5 t:

2 2! ;;I

P

2

5 P 2

L?

366

5.

A. APZAL

AND

S. ALI

ation are D to G in table 1. While potential D gives more weight to the triplet parameters than to the singlet ones, the reverse is true for potentials E and G. An excellent overall fit is however obtained for potential F which in the given class of central Gaussian potentials considered here seems to provide as good a representation as possible of the low-energy scattering data. The p-p singlet phase shifts calculated for potentials D-G in the energy region of O-10 MeV (c.m.) were found to agree with the experimental values ’ “) to within 10 %. Thus we have a set of potentials which represent the two-body data with varying degrees of adequacy and we are now in a position to explore the sensitivity of the wave function of relative motion of a-d system to these potentials. On assuming the u-d cluster structure of 6Li, we have taken the 6Li ground state wave function as where

4, = exp[-fai~I(ri-Ra~21~ 4d= ev I-tYj~~~j-4d219 R, and Rd being the position vectors of the c.m. of the clusters and r their relative separation; xe and xd are the spin functions of the a- and d-clusters, F(r) is the a-d relative wave function and A is the antisymmetrization operator. Following the procedure of the resonating group formalism and resolving F(r) into partial waves F(r)

=

5 f~r

P,(cos O),

z=o

an integro-differential --_

equation Z(Z+1) + E - V,(r)] fi(r) = JI&(r, r2

(2)

r’)Sl(;)dr’,

was obtained. In this equation E is the relative a-d energy and V,, is the direct interaction between the clusters (corresponding to the unity part of the antisymmetrization operator) and is obtained as l&(r) = 2Vo

1

&Y *(40-m+2b-2h)exp 2c$ + 4ay + 3/?y

-

hP:Ifi+3PY

r2] . (3)

The expression for the kernel k,(r, r’) which includes the effects of all possible nucleon exchanges between the clusters is a rather complicated one and hence is left out. It is often customary to write eq. (1) in the form %j =

YSerbcr

+

(4

(1- Y) VRosentc~ 9

are obtained by putting w = m, b = h and whereVSerberand VRooenfeld

m =

26,

6Li NUCLEON-NUCLEON

POTENTIALS

367

h = 2w respectively in eq. (1). The parameter y, as explained earlier, may be treated as an adjustable parameter which compensates for any specific distortion effect present in the system besides that which is given by the antisymmetrization procedure. This seems to be reasonable especially when one is dealing with a one-channel approximation. 3. Results and discussions The integro-differential eq. (2) was solved numerically as an eigenvalue problem for 1 = 0 in which there was only one variational parameter y. We have taken a = 0.543 corresponding to the rms radius of 1.44 fm for the a-particle and y = 0.2 corresponding to an rms radius of 1.93 fm for the deuteron. The value of y has been adjusted to give the relative a-d binding energy of 1.47 MeV for the ground state which is obtained as a 2s state wave function. It is worth mentioning here that in previous a-d model 11*12) studies of 6Li in which the a-d interaction is developed from phenomenological a-nucleon potentials, the wave function of relative a-d motion was chosen rather on an ad hoc basis as a 2s state. It is gratifying that this structure of the ad grwnd d&e is k&xxi cutdid here by fundamental cons&rations. The probability density off( r ) was obtained for each of the potentials D to G and it was found that the peaking of the a-d wave functions for potentials D and E takes place at about the same separation distance (of the order of 1 fm). This is reasonable in view of the not too large difference in the range of the two potentials. The wave functions for the potentials F and G, however, were found to peak a little earlier as expected and were found to be relatively more concentrated in the inner region than those for the potentials D and E. The rms radii of the relative a-d wave functions for the potentials D to G are 3.47 fm, 3.46 fm, 3.38 fm and 3.28 fm respectively. These radii are comparable to the combined radii of 1.44 fm for the a-particle and 1.94 fm for the deuteron, the a-d clusters overlapping for the potentials F and G. This suggests that as we pass from the potentials D to G, the need for the inclusion of a specific distortion effect in the a-d system makes itself progressively felt. The above points are illustrated in an interesting way by the values of y which are given below: potential D: y = 1.196, potential E:

y = 1.228,

potential F:

y = 1.357,

potential G:

y = 1.520.

The value of y is found to increase monotonically with the range of the two-body interaction. However, a large positive deviation of the value of y from unity points to the presence of appreciable specific distortion effect. This is because, the nuclear force, although predominantly of Serber type has a small admixture of Rosenfeld force and hence the value of y should be only slightly less than unity. It is seen that

S. A. APZAL

368

AND S. ALI

of the four potentials considered, potential D produces the value of y closest to unity. Thus one might use either potential D, neglecting the distortion effect (although it would not be quite correct to do so) or use potential F which gives the best fit to the two-body data and include the distortion effect rather implicitly by using more flexible trial wave functions 5*13), thereby reducing the value of y. We strongly feel that the latter approach is more sensible than the former if one attaches, as one should, due importance to the basic two-nucleon data. Thus an analysis of the type presented here not only discerns between the features of different central potentials of a given family but also gives an indication of the amount of flexibility one must include in the trial wave functions, for each of these potentials, in the resonating group formalism of light nuclei. We are grateful to the computer staff of the Atomic Energy Centre, Dacca for their help with numerical calculations which were performed by the Centre’s IBM 1620 machine. References 1) T. Hamada and I. D. Johnston, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13)

Nucl. Phys. 34 (1962) 382 F. Tabakin, Ann. of Phys. 30 (1964) 51 A. Hasegawa and S. Nagata, Prog. Theor. Phys. 38 (1967) 1188 E. W. Schmid and K. Wildermuth, Nucl. Phys. 26 (1961) 463 D. R. Thompson and Y. C. Tang, Phys. Rev. 179 (1969) 971 S. Okai and S. C. Park, Phys. Rev. 145 (1966) 787 S. Hochberg, H. S. Massey and L. H. Underhill, Proc. Phys. Sot. A67 (1954) 957 J. C. Davis and H. H. Barschall, Phys. Lett. 27B (1968) 636 E. Feenberg, Phys. Rev. 47 (1935) 850 L. Hulthen and M. Sugawara, Handbuch der Physik, vol. 39 (Springer Verlag, Berlin) S. Ali and S. A. Afzal, Nuovo Cim. 49 (1967) 103 J. L. Gammel, B. J. Hill and R. M. Thaler, Phys. Rev. 119 (1960) 267 H. Jacobs, K. Wildermuth and E. J. Wurster, Phys. Lett. 29B (1969) 455