A microscopic optical potential for 3He

I-=-l

Nuclear Physics A241 (1975) 229-236; Not to be reproduced by photoprint

A MICROSCOPIC

@ North-Holland Publishing Co., Amsterdam

or microfilm without written permission from the publisher

OPTICAL

POTENTIAL

FOR ‘He

BIKASH SINHA, FEROZE DUGGAN and RICHARD J. GRIFFITHS Wheatstone Luboratory, King’s College, Strand, London WCZR 2LS Received 29 October 1974 (Revised 16 December 1974) Abstract: A three-parameter

optical model for JHe has been developed in terms of the optical potentials of its constituent nucleons. An effective interaction folding model including exchange is used to generate the real part, whereas the imaginary part is calculated using a forwardscattering amplitude approximation. The spin-orbit potential is calculated using the Blin-Stoyle prescription. This model reproduces the experimental data reasonably well.

1. Introduction

Several attempts ‘) have been made to calculate the optical potential of a composite projectile as a function of the optical potentials of its constituent nucleons. The first important calculation was due to Watanabe ‘), where the deuteron optical potential is considered as a sum of the neutron and proton optical potentials averaged over the internal motion of the deuteron. This method has been extended by Abul-Magd and El-Nadi “) to 3He and tritons. However, such a model does not include effects due to the break-up of the deuteron and consequently underestimates the imaginary potential. Johnson and Soper 4), using the adiabatic approximation in a coupledchannel formalism, estimated the effect due to deuteron break-up and found good agreement with experimental data. Mukherjee 6), and Samaddar et al. ‘), extending the Feshbach formalism 5), have calculated the deuteron and 3He optical potentials where they estimate the correction due to internal motion of the constituent nucleons. However the ambiguity problem of the phenomenological nucleon-nucleus optical potential seems to have caused difficulty in getting satisfactory agreement with experimental data. Austern and Richards “) have also considered the effects of internal motion in composite particle scattering. Other approaches include the eikonal approximation of Simbel “) who attempts to calculate the effects of 3He and triton break-up. In the present paper, following the formalism developed by Samaddar et al. ‘) we propose to calculate the 3He optical potential in terms of the nucleon-nucleus optical potential developed in refs. 9B13). The formalism is presented in sect. 2, the results of data analysis are given in sect. 3 and the conclusions are presented in the last section. 2. The model In this section the 3He optical potential is calculated in terms of its constituent nucleons following the formalism of Samaddar et al. ‘). The most important difference 229

B. SINHA ef al.

230

between the Watanabe type of model ‘) and a model such as developed by Samaddar et al. ‘) is that in the latter case the kinetic energy due to its internal motion of each nucleon of the 3He is taken into account explicitly. The total Hamiltonian of a system consisting of a 3He of c.m. energy E incident on a target nucleus is h2 V2 H=HT+ where

t -g

R

h2 - $r,-

2

2

R = +(r,-t-r2+r3),

h2 2 -VrfVpt,PZ+Up*,*+Yp2,n %3

r=ri

-r2

1

+v,l+‘t7p*+V,,(f)

and p=r,-)(r,+r,),

rl , r2

and r3 being the radial coordinates of the three nucleons constituting the ‘He. The reduced masses are pr = 3m, p2 = $rn and p3 = -fpn, where m is the nucleon mass; Hr is the target HamiItonian; z+,,,~~,9$X+and aPz+ are the respective protonproton and proton-neutron interactions inside the 3He;

are the sums of the interactions between each of the nucleons in the 3He and all the target nucleons. Using the Feshbach formalism “) the optical potential for the incident 3He becomes ‘*11) @JR,

E) = PVP+PVQ(E+

-QHQ)-“QVP,

(2)

where V = VP,+ VP,+ V,, and P is the ground state projection operator such that P!P = &(<)~~(r, p); Y, the eigenfunction of the total Hamiltonian H, is expressed as Y = C,,,, ~~(~)~~)(~,~~~~)(~) where #Jr) is the eigenfunction of the target Hamiltonian with e representing the internal coordinates of the target, whereas &‘(r, p) and B:)(R) d escribe the relative and cm. motion respectively of the projectile; (bc and x0 are the ground state wave functions of the target and the projectile respectively. The operator Q is defined such that P+ Q = 1. The first term of eq. (2), from the definition of P and V, is p’vp = (xo(r1 P)A#)lJ&

+ v,, + v,l~cl(M~~ P))t

(3)

which is simply the algebraic sum of three first-order nucleon optical potentials folded in with the 3He wave function x0. To facilitate the evaluation of the second term in eq. (2)let ~~defineGn(E) = Q(E’-QHQ)-lQandGi(E) = Qi(E’-QiHiQ,)-‘Q,, where Hi is the appropriate Hamiltonian of the ith nucleon (i = pl, p2 or n) of the 3He and target system with Qi the corresponding projection operator. Now expanding 12*’) G n( E) in terms of the Green function of any of the constituent nucleons of the 3He, we get G,(g) = Gi(E- T”-T,)+G,(E-T’-

Te)[v’+ Ffk+~PI,P*-t~pl,nf~gt,nlG~(E)t

(b)

3He OPTICAL POTENTIAL

for i # j # k, where 3He; rj = $V,” and nucleons in the 3He. jth and kth nucleons (4a):

+

231

i = pI , p2, n refer to the two protons and the neutron of the Tk = 2Vf are the kinetic energy operators of the jth and kth Similarly, Vi and V, are the sum of the interaction between the of the 3He with all the target nucleons. Thus from eqs. (2) and

& , ,n(xoh~~4o(~)lW,(&&vj+VK+~pt,pz+~p,,n+~PZ,n} (3

x Gu(E)Vl#ot~)~otr, P>>,

where Eefr = E- Tj-Tk and eq. (4b) contains the second- and higher-order real and imaginary terms. It should be noted that unlike the Watanabe model the Green function in this model is evaluated at E,,, rather than at E, thus taking into account the kinetic energy of the nucleons of the 3He. Samaddar et al. ‘), however, considered the second term of eq. (4b) to be small, and so we neglect it. Thus the 3He optical potential, in terms of the optical potential of the constituent nucleons of 3He is given by Cl&

E) =
+ v,,+ v,I#&)X&,

P)>

The model representation of the above formalism is obtained in the following manner using the one-body optical potentials previously developed ‘, r3). The real part of the one-body potential is generated by folding in an effective interaction t4) with the target nucleus density distributions r 6). An exchange term which arises from the antisymmetrization of the coordinates of the 3He nucleons and the target nucleons is included by using the energy-dependent equivalent local potential proposed by Thomas et al. ’ “). The effective interaction used is that due to Kuo I”). By neglecting the odd-state components the Kuo interaction is reduced to Serber type, given by

VK t.e. = v,f,t!--~V~‘~~

-4hJ*‘)

s

(6)

with K = 0.13; Y&. and VFe. are respectively the singlet-even and triplet-even parts of the Kuo interaction; Vsy_, Vt:. and V,, are respectively the long-range parts of the singlet-even central, triplet-even central and triplet-even tensor components of the bare two-body Hamada-Johnston ’ “) potential. The first-order terms in eq. (5) which have both a direct and an exchange component are calculated from the first-order components of the Kuo interaction, Vsyf. and VFz._ To estimate the second-order term in eq. (5) we make the following closure approximation as suggested by

232

B. SINHA ef al.

McKellar et al. 1“): P(‘V,G, 6) z -&lWI

- 4P/P0)‘19

(7)

where P stands for the principal value. Thus we consider that both the direct and exchange parts of the nucleon-nucleus optical potential arising from the second-order part of the effective interaction correspond to the second-order contribution to the optical potential. To facilitate the evaluation of the second-order term of eq. (5) we make the following further approximation:

= 6<~&, P)&(@I IrpG, I’,l#&)x(rt +3(x&,

~)#&)l V, G,

P)>

~l~~(~)~~(r,P)>,

(8)

wherep is either pi or p2. This has been possible by assuming (i) yP,(PZ~GPI(P2)VPt(P1) !SZ VPI(P2)GPI(P3)V*I(Pz); (ii) VPdP2)GPIdcl s VP)I(PI)GPI(PZ)VPI(P2) and (iii) VPdPdGnVll s V, G, V,,. The first assumption is trivial because of the functional form ‘) of ~e(r, p)

which has been chosen to be symmetric with respect to exchange of spatial coordinates of any two constituent nucleons of the ‘He. The remaining two assumptions are not drastic in second order since the difference V,- V,, is usually quite small. It is to be noted that the last assumption is not necessary for evaluating the first-order term. Eq. (8), similar to eq. (5), has contributions both for the direct and the exchange parts of the one-body optical potential. The energy dependence of the exchange component of the one-body second-order potential is now shared by the other two nucleons of 3He such that E + E- Tj - II,. The one-body exchange potential used in this work is linearly energy dependent “) so that by replacing E by E-TjTk, eq. (8) can now be written as

@> where, as indicated in eq. (8), &, = 6, gy-dependent part of the second-order the other hand is the one-body optical of the Kuo interaction. The first-order

j?, = 3; Uex&E) implies that only the enerexchange potential, Uejex,2j,is used; U,, , on potential derived from the second-order part term represented by eq. (3) is simply

Pb) are respectively the first-order direct and where ap = 2, a, = 1, and Udfl)i and UeXClji exchange parts of the one-body optical potential. As before, the spatial symmetry of x&, p) has been employed so that VP, = VP,, Finally therefore, the real part of the 3He optical potential becomes U@,

E) = U:(R, E)+ U;(R, E).

w

It is worth noting that if jIi = 0 then our potential reduces to that of ref. “). However

jHe OPTICAL POTENTIAL

233

the magnitude of pi indicates that the second-order corrections for composite particle scattering are of greater importance comparatively speaking, than those for nucleon scattering. The imaginary potential arises from the second-order part of eq. (5) when Gi goes through a singularity. Rather than estimating the imaginary potential in this fashion we have calculated the potential in first-order following the method proposed by Sinha and Duggan g), who used a modified version of the forward scattering amplitude approximation, often referred to as the “frivolous model”. Using such a model the imaginary part of the one-body potential is given by

where k is the local wave number given by the WKB approximately as k2 = @Pi’)@G), w h ere V, is the real potential and p&r) = VR(r)/Jd, J,, being the volume integral of the two-body interaction; (c)i is the average total nucleon-nucleon cross section calculated inside the nucleus. The WI, calculated in this way, it should be noted, is essentially a first-order term in the optical potential series using the impulse approximation ‘) and therefore the imaginary part of the 3He optical potential is obtained by calculating the matrix expressed in eq. (3), giving

W,H(R, E) = C

Yi
P)I%1lXo(~7 PI>,

i=p,n

(11)

with yp = 2andy, = 1. The spin-orbit part is the only other term required to give the composite 3He optical potential. The nucleon-nucleus spin-orbit potential is calculated following the BlinStoyle prescription 20) as in ref. 13) and on folding with the 3He wave functions yields

(12) where q is the product of the spin and angular parts of the 3He wave function and Vi:?. is the nucleon-nucleus spin-orbit potential. 3. Data analysis The final optical model for the incident 3He becomes UH(& E) = S,C@(R, g)] + S,C&“(R, E)l +SoCU:,,.(R, E)l,

(13)

as defined in sect. 2 and with the three parameters S,, S, and So being the three scaling parameters of the real, imaginary and the spin-orbit potentials. These three parameters are searched upon to get an optimum fit to the experimental data. The elastic scattering data used are from 40Ca (83.5 MeV and 51.4 MeV) 21) and “Zr (29.8 MeV) 22). Typical fits to the data are shown in fig. 1 which c!early shows

234

B. SINHA et al. 10

10.

20

I,

20

30

40

t

30

40

50

I

60

60

I

,60

70 1

I

70

60 I

I, -,80

90 I

0

100

I

10

110

I

20

em

Fig. I. Fits to the elastic scattering

120 I

L

!O

130 I

!

40 8cm

140 I

x

50

150 f

160 18

I

!

60

70

170

I

80

of 3He from p”Zr at 29.9 and 4oCa at 51.4 and 83.5 MeV.

the efficiency of the present model in reproducing the experimental results. In fig. 2 the theoretically derived real and imaginary parts of the potentials for 3He scattering from 4oCa at 51.4 MeV are shown. It appears that the corrections due to internal motion are small which in effect confirms the adiabatic approximation of Johnson and Soper 4), but conflicts with the findings of Samaddar et al. ‘). The latter authors used phenomenologi~l nu~Ieon-nucleus optical potentials so that the usual ambiTABLE1 The parameter? Isotope

Energy

so

(MN) 4oCa 4*Ca *OZr

51.4 83.5 29.9

0.57 0.67 0.84

1.26 1.23 1.38

0.87 0.89 1.68

JRIAR&

(MeV . fm3:

495.1 485.2 486.6

jHe OPTICAL

I

REAL

POTENTIAL

235

IMAGINARY I I

I

iO-

10 -

L 0

10

20

NUCLEAR

30 RADIUS

L.0

50

I 10

6.0

fm

Fig. 2. The terms in real and imaginary

I 20 NUCLEAR

potentials

I 5.0

I 40

I 30 RADIUS

for 3He scattering

8 6.0

I _ 70

fm

from 40Ca at 51.4 MeV.

guities of the phenomenological model might lead to the overestimation of the correction terms. In table 1 the parameters obtained by searching for the optimum fit are shown. The fact that S, is much less than one may be ascribed to the overestimation of the secondorder terms in the one-body optical potential “I 1s*23). As noted before ‘“) the second-order terms for scattering using the closure approximation of the Kuo interaction can be overestimated by as much as a factor of two. The fact that St is somewhat larger than one is probably because of the neglect of break-up and other channels, which would underestimate the total absorption. The volume integrals of the real and imaginary potentials respectively JR/AHAT and JT/AHAT indicate a variation with energy and target nucleus. The very weak energy dependence of JR/AH AT is in agreement with previous phenomenological work 24) although at present it is difficult to be definitive on this point. The last column shows (r’)! the mean square radius of obtained Su x JPIAHAT (MeV - fm3) 328.7 325.1 408.7

JdA&

(MeV - fm3)

S, x JIIA& (MeV * fm3)

525.1 457.0 365.4

168.7 187.3 148.0

x”/degr=

::

of freedom

(fm’)

14.9 25.0 14.0

18.50 18.13 25.10

236

B. SINHA et al.

the real part of the 3He potential which will satisfy the following equation: F = LS + L,+


(14)

where (r’>z, is the mean square radius of the interaction and (r2)~,,ss and Ls, are the mean square radii of the 3He and target nucleus respectively. The above equation is only valid if internal motion corrections are neglected. The strong Adependence of (r 2)eff reflects the A-dependence of the interaction. 4. conclusiolls The above model shows that the Feshbach formalism can be used to derive composite particle optical potentials from the nucleon-nucleus optical potential and that this approach reproduces the experimental data and the phenomenological results reasonably well. It appears that the internal motion of the incident projectile for a light ion like ‘He is negligible. The analysis shows that the scattering data are sensitive to the second-order terms. A more precise evaluation of the term will therefore be useful. References 1) P. E. Hodgson, Nuclear reactions and nuclear structure (Oxford University Press, 1971) ch. 10 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25)

S. Watanabe, Nucl. Phys. 8 (1958) 484 A. Y. Abul-Magd and M. El-Nadi, Prog. Theor. Phys. 35 (1965) 798 R. C. Johnson and P. J. R. Soper, Phys. Rev. Cl (1970) 976 H. Feshbach, Ann. of Phys. 5 (1958) 357; 19 (1962) 287 S. Mukherjee, Nucl. Phys. All8 (1968) 423 S. K. Samaddar, R. K. Satpathy and S. Mukherjee, Nucl. Phys. Al50 (1970) 653 M. H. Simbel, preprint IC/71/105, Trieste, 1971; N. Austem and K. C. Richards, Ann. of Phys. 49 (1968) 309 B. Sinha and F. Duggan, Nucl. Phys. A226 (1974) 31 T. Hamada and T. D. Johnston, Nucl. Phys. 34 (1963) 353 J. Testoni and L. C. Gomes, Nucl. Phys. 89 (1966) 288 J. R. Taylor, Scattering theory (Wiley, NY, 1972) p. 133 G. L. Thomas, B. C. Sinha and F. Duggan, Nucl. Phys. A203 (1973) 305 T. T. S. Kuo, Nucl. Phys. 90 (1967) 199 A. D. Mackeller, J. F. Reading and A. K. Kerman, Phys. Rev. C3 (1971) 460 J. W. Negele, Phys. Rev. Cl (1970) 1260 B. R. Barrett and M. W. Kirson, Nucl. Phys. Al48 (1970) 145; Al96 (1972) 638 B. R. Barrett, Nucl. Phys. A221 (1974) 299 A. K. Kerman, H. McManus and R. M. Thaler, Ann. of Phys. 8 (1959) 551 R. J. Blin-Stoyle, Phil. Mag. 46 (1955) 973 H. Chang, private communication J. W. Luetzelschwab and J. C. Hafele, Phys. Rev. 180 (1969) 1023 B. Sinha, Nucl. Phys. A203 (1973) 473 N. M. Clarke, C. J. Marchese and R. J. Griffiths, Phys. Rev. Lett. 29 (1972) 66 C. J. Marchese, N. M. Clarke, R. J. Griffiths, G. J. Pyle, G. T. A. Squier and M. Cage, Nucl. Phys. A191 (1972) 627; P. B. Woollam, R. J. Griffiths and N. M. Clarke, Nucl. Phys. Al89 (1972) 321