Tensor interactions and polarization phenomena in heavy-ion scattering

Nuclear Physics A415 (1984) 27 I-303 0 North-Holland Publishing Company

TENSOR INTERACTIONS AND POLARIZATION PHENOMENA IN HEAVY-ION SCATTERING H. OHNISHI Department of Physics, Hosei University, Tokyo, Japan

M. TANIFUJI Department of Physics and Research Center of Ion Beam Technology, Hosei University, Tokyo, Japan

M. KAMIMURA

and Y. SAKURAGI

Department of Physics, Kyushu University, Fukuoka, Japan

M. YAHIRO Department of Physics, Shimonoseki University of Fisheries, Shimonoseki, Japan

Received 1 August 1983 Abstract: Elastic and inelastic scattering of ‘Li by ‘*Ni at Eub = 14.2 and 20.3 MeV is investigated theoretically, special emphasis being laid on polarization phenomena. A parameter-independent study shows second-rank tensor interactions to be the main origin of tensor analyzing powers for both elastic and inelastic scattering. Coupled-channel (CC) calculations using cluster-folding interactions which include the tensor terms are found to be successful in reproducing the data for cross sections and vector and tensor analyzing powers, when projectile excitation effects are sufficiently taken into account. Scattering of ‘Li by ‘sNi at E,,, = 20.0 MeV is also investigated by the CC calculation, where successes similar to the ‘Li case are obtained in understanding experimental data.

1. Introduction In recent years, interest has been concentrated on scattering of polarized 6Li and ‘Li by nuclei’ ’ - 18), among which scattering of ‘Li by 58Ni targets has attracted particular attention because of interesting features observed in the experimental data for cross sections and vector and tensor analyzing powers 5*6*8*q,14).A global sketch of the above features is that in elastic scattering the tensor analyzing powers have a fairly large magnitude while the vector analyzing power has a small one, exhibiting a remarkable contrast to the case of deuterou scattering, where the latter analyzing power is much larger than the former. Furthermore, precise investigation of the vector analyzing power observed at ELab=: 20 MeV has 271

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H. Ohnishi et al. / Tensor interaeti~ns

revealed a discrepancy: the spin-orbit potential derived by double-folding models has the same sign for the 6Li- “Ni and ‘Li-‘*Ni systems ‘I), the strength of the potential being larger for the former system, while the experimental vector analyzing powers have different signs and similar magnitudes at most angles for these projectiles g,11*13). The first theoretical approach is based on a rather phenomenological viewpoint, where the data are analyzed in terms of quadrupole deformation effects of the projectile in a semi-classical way. The analyses 5*6,‘4) have been successful in understanding the tensor analyzing powers observed in elastic scattering. Quantitative calculations by this model have been carried out ‘*14), where both the ex~rimental cross sections and tensor analyzing powers are well reproduced simultaneously by the use of the deformation parameter p = -0.14 at Elab = 20.3 MeV. Investigations of particular linear combinations of the tensor analyzing powers i6) seem to support such calculations of the tensor analyzing powers in elastic scattering. However, theoretical parameters which give fits to the crosssection and tensor analyzing power data at Em, = 14.2 MeV differ considerably from those at I&,, = 20.3 MeV ; for example, at Elab = 14.2 MeV /I is - 0.25 and the depth of the real central potential between ‘Li and ‘sNi is about twice that at Elab = 20.3 MeV. Furthermore, the model has not been successful so far in explaining the vector analyzing power data. These results suggest that the problems are still unsolved in such deformation models. Another approach is to derive interactions between the projectile and the target from some internucleon forces, investigating whether they can explain the above experimental data. Interactions obtained by single or double folding are examples. Before carrying out actual calculations, in ref. I’), the tensor analyzing powers observed in ‘Li elastic scattering at Em, = 14.2 MeV have been analyzed to get information on the spin dependence of the interaction by the invariant-amplitude method 19). which shows second-rank tensor interactions to be the main origin of the observed tensor analyzing powers. The conclusion is quite general, because the method is free from interaction parameters. Referring to this result, a clusterfolding potential for the ‘Li- 5ENi interaction has been derived 12) by assuming that ‘Li consists of an a-particle and a triton. For reasonable a-‘*Ni and t-58Ni interactions “t21), a complex tensor potential of considerable strength and a very weak spin-orbit potential have been obtained in addition to a complex central potential. These results are consistent with the global sketch of the features of the experimental data mentioned above. However, the spin-orbit potential obtained can hardly explain the vector analyzing power data similarly to the case of the doublefolding model ” ). To overcome this difficulty, ref. 12) has proposed to take account of projectile excitation effects in the calculation by the coupled-channel (CC) method. Quantitative calculations r3.tf) of the potential scattering which have not included the CC effect have been performed for the elastic scattering by the use of

H. Ohnishi et al. / Tensor interactions

213

the cluster-folding-model potential similar to that obtained above ‘*) at ELab= 20.3 MeV, where the tensor analyzing power data are well reproduced as was expected. The calculated vector analyzing power has the same sign as that measured, although the theoretical magnitude explains only a part of the measured one. Such a partial success in the vector analyzing power is mainly due to the tensor potential ; that is, the dominant contribution to this analyzing power arises from higher-order terms of the second-rank tensor interaction as has been shown in later analyses “,i’). The magnitude of the calculated cross section is considerably small i3) compared to that of the measured one except for very forward angles using the shallow a- 58Ni potential **). Improvements in the calculation have been attempted 17,18) by including projectile excitation effects in the CC method, where the excited states taken into account are the 3- bound state at E,= 0.478 MeV in ref. l’) and the $- and $- resonance states at E, = 4.63 and 6.68 MeV, in addition to the $- bound state, in ref. is). The channels which include these nuclear states are coupled by the tensor part of the folding interaction, dominant coupling effects arising from the second-rank tensor ones. In these calculations, the contribution of the 3- state increases the magnitude of the vector analyzing power but not the cross section and thus the renormalization of the interaction by the factor 0.5 is required to get a reasonable lit to the cross-section data “). The inclusion of the $- state together with the *- state improves the calculated vector analyzing power, giving nice agreement with the measured value and, at the same time, increases the cross section considerably, which explains almost all the observed cross section without any renormalization factor. The contribution of the $- state is small but not negligible, which improves both the cross section and the vector analyzing power by a small amount. In this CC calculation 18), most of the observed tensor analyzing powers are again explained as effects of the second-rank tensor potential in the elastic-scattering channel. Summarizing these results, one may conclude that the folding-model interaction gives a reasonable explanation for all observables, i.e. the cross section and the vector and tensor analyzing powers, at Elab = 20.3 MeV, when the projectile excitation effects are sufficiently taken into account. In this approach, the second-rank tensor interaction plays an important role in reproducing the data even for the vector analyzing power through the projectile excitations. The main purpose of this paper is to describe the details of the CC calculation 18) at Elab = 20.3 MeV, in addition to giving its theoretical foundation by the invariant-amplitude method, so as to provide a deeper insight into the reaction mechanism and to extend the calculation to the elastic scattering of ‘Li at Elab = 14.2 MeV and the inelastic scattering of ‘Li leading to the first f- excited state of ‘Li at Elab = 20.3 MeV. Also, 6Li-58Ni elastic scattering is similarly analyzed, where, as excited states of 6Li, the 3+ state at E, = 2.19 MeV, the 2’ state at E, = 4.31 MeV, the 1+ state at E, = 5.65 MeV, and some non-resonant continuum states are taken into account. The intention of such applications is to

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H. ~hn~shi et al. / Tensor interaction

examine the validity of the folding interaction accompanied by the projectile excitations and, particularly, to clarify the role of the second-rank tensor interactions. In the next section, simple formulae for the tensor analyzing powers in elastic and inelastic scattering are derived under the assumption that the spin-dependent force is only the second-rank tensor one. The result is quite general because the derivation does not need detailed knowledge of the interaction. The formulae are shown to reproduce the observed tensor analyzing powers in ‘Li-“Ni scattering. In sect. 3, the expression for the cluster-folding interactions is derived explicitly, where qualitative estimations are made for the projectile excitation effect by the interactions. The details of the CC calculation by these interactions, which includes the projectile excitations, are described in sect. 4. The cross section and the vector and tensor analyzing powers in the elastic scattering are compared between the calculated and the measured at the incident energies, Elab = 14.2 and 20.3 MeV. The calculations are successful in reproducing the data at both incident energies by the same folding interactions. The third-rank tensor analyzing powers which have not been measured are theoretically predicted and found to be small at this energy. Also, it is shown that the effects of the excited state of ‘Li strongly depend on the excitation energy of that state. The cross section and the vector and tensor analyzing powers in the inelastic scattering leading to the $- state of 7Li at Elab = 20.3 MeV are calculated by the same folding interactions, where the calculated values are in adequate agreement with the data. In the last part of this section, theoretical predictions are made for the cross sections and the analyzing powers in scattering at higher energies, where the third-rank tensor analyzing powers at EIab = 30 MeV are found to be much larger than those at Elab = 20 MeV. In sect. 5, the cross section and the vector analyzing power in the 6Li-58Ni elastic scattering are investigated, where the calculated cross section is increased by including the projectile excitations, improving the agreement between the calculation and the experiment. This effect is quite similar to what we have seen in the ‘Li case. However, the magnitude of the calculated cross section is still smaller than the measured one. The discrepancy is removed by taking account of 6Li break-up into non-resonant continuum states in the CC calculation. The calculated vector analyzing power is in reasonable agreement with the data. Throughout the present studies, the CC calculation containing the projectile excitations is quite successful in explaining the experimental data, and the folding interaction including the nondiagonal elements is found to be a good description of the interaction between the 6Li or ‘Li projectile and the 58Ni target. The last section is devoted to concluding remarks.

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H. Ohnishi et al. / Tensor interactions 2.

Interpretations of tensor analyzing powers by invariant amplitudes

The analyses in ref. I’) suggest that, in the elastic scattering of ‘Li by 5*Ni, the dominant contribution to the tensor analyzing powers arises from tensor interactions. In this section, simple formulae for the analyzing powers are derived in the framework of the invariant-amplitude method 19) for both elastic and inelastic scattering, where only second-rank tensor interactions are assumed for the spinde~ndent interactions. The derivation of the formulae does not require details of the interaction and the results are free from such details, for example potential parameters. The analyzing powers for the polarized beam are defined by the use of the transition matrix M as TKQ = WMrKQM+

)P,

(2.1)

with N = Tr (MM+).

(2.2)

where K(Q) denotes the rank (z~om~nent) of the beam polarization and rrg is the corresponding tensor operator composed of the spin of the incident particle, the matrix element of which is given by

Si(Vi)being the spin (z-component) of the particle. The transition matrix M can be expanded in terms of the tensor in the spin space pfjpkK, M = z(-Y%-,$%,

(2.3)

kr

where B,, is the ordinary space tensor and the tensor of the same rank constructed momenta. In the case where the spin of the one are zero and no parity change happens M is given by 19)
its matrix element can be described by from ki and kr, the initial and final target nucleus and that of the residual in the transition, the matrix element of

Vi ; ki) k

=

(-pq

(SiSf k

vi

-

VflW

c

t=li-k

[wa

x

k CE-,(ln,W&

(2.41

where &?rand 62, are the solid angles of ki and k, respectively, sf(vr) is the spin (zcomponent) of the emitted particle and 1Gis k for k = even and 8% + 1 for k = odd.

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H. O~~is~~et al. / Tensor in~eracrions

The quantity C,,(B) is related to the spherical harmonics I&,(a) as

The Clebsch-Gordan coefficient (sisfvi-v&) arises from the matrix element of Y,_, and the factor constructed from C, and CE,_, describes that of &&. Other trivial factors like the physical part of the matrix element of Y,_, are included in the coefficient F,,. The amplitude F,, is invariant under rotations of the coordinate axes and is a function of 0, the angle between ki and k,, the cm. energy and the Q-value. The invariant amplitude F,, is designated by the rank of the spin tensor, with which F,, is associated; for exampIe, F,,, F,, and F&r = 0, 1, 2) are the scalar, vector and second-rank tensor terms, respectively. In the first order of to the scattering interactions~ Foe, Fll and F,, represent the contributions amphtudes from the central, spin-orbit and tensor interactions, respectively. In principle, they contain any higher-order contribution of these interactions within the restriction due to their tensorial properties. Inserting (2.4) into (2,1), one obtains (2.5) with

Setting K = Q = 0,

Using eqa (2.1}, (2.5) and (2.71,

Gw In the scattering of ‘Li by “Ni, the folding model predicts a very weak spin-orbit interaction and a fairly large magnitude for the second-rank tensor interaction, as was discussed in the preceding section, the former interaction arising from the t58Ni spin-orbit potential while the latter one from the t-58Ni and c+58Ni central potentials. The third-rank tensor interaction is also weak because it is supplied by the t-58Ni spin-orbit potential. The higher-rank tensor interactions may not be effective at such low energies. These circumstances allow us to assume only the

H. Ok&ski et al. / Tensor interactions

277

second-rank tensor interaction as the spin-dependent one. Accordingly, we will keep only the scalar term, Foe, and the second-rank tensor one, F&r = 0, 1 and 2), in the invariant-amplitude expansion (2.4). For the present force assumption, the three tensor amplitudes satisfy the following relations in the PWBA, as is shown in appendix, for both elastic and inelastic scattering: Fzr = -,/%%I,

(2.9)

F 22 = p'F20,

(2.10)

where

The PWBA is a crude approximation and is of doubtful practical use. However, the tensor interaction is not so strong and may be treated by the Born approximation. In fact, eq. (2.9) does hold, within errors of about 10 %, for most angles in the full calculations in sect. 4. Also, for elastic scattering, eq. (2.10) is derived exactly by the use of the time-reversal theorem, that is, inde~ndently from the use of the PWBA. Under the assumptions discussed above, one derives a rather simple expression for the second-rank tensor analyzing power (K F 2). For elastic scattering, with si = Sf = s, TZQ

2Re(Bf$ + 5$W(s2s2; s2)Bzi =

B;;+,/%;;

(2.11)

’

where P

IL

1,

For inelastic scattering leading to an excited state of the projectile (Sf # T

Eli

W(Si2S,2; Si2)B:2,

2Q =

B22 00

,

Si),

(2.12)

where P#

The quantities B$$

and BP;

1.

can be written explicitly as follows in the reference

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El. CJhnishiet al. / Tensor interactions

frame where z/l& and yllki x k,:

.B:; = (&3 cosz 8 - 1) -2p cos 8+pZ)F,p&, B:y = ,/‘&.I -cos 0) sin 0 F,,F& Bf; = fi

sin’ 8 F$,F,,,

B;; = ~~l-4p~ose+2~z(2cos2e+l~-4~3cose+~4~~F~~~z, B:; = &

cosze)+2pcose(1+3cos2e)-p2(1+11cos2e)

((l-3

+ 8p3 cos e -2p4)IF2& B$I = ~sine~~ose-~(l+2cos2e)~3p2cose-~3~~F2~J2, - 1+2pco~e-p~)IF,,~~,

(2.13)

B;;+J;BB;;

(2.14)

Bij: = &&sin’e(

which satisfy = -flB;;.

Using (2.13), the 7&&j = 0, 1 and 2) for elastic scattering are obtained as

T2, = & (cos B - 1 f(3 cos 0 - 1If@, cos O), T21

=

-

$(cose-l)sint3f(s

7

c0se)

9

(2.15)

Tz.2= - ~fi(c0se-i)(~0se+~)~(~,~0~8),

and another tensor analyzing power which is defined in ref. 14), TTzo 3

-&T,,+$$,),

(2.16)

is given by vZo = $ (COSe - i )f(s, cos

(2.17)

81,

where N = lF,,f2 +4 (cos 8- 1)2~F,,~2, j+, cos ej = Re(&,F&,)+

,/~W(s2s2;s2)(mO-

(2.18) i)IF,,12.

(2.19)

ii. Okniski et al. / Tensor interaction

279

Eliminating N and f by the use of (2.17), eqs. (2.15) become T,, = (1 - 3 sin2 $)TT,,, T2i = - 3

sin OTTzo,

Tz2 = -&os2+@‘TZ0.

(2.20)

These equations predict Tzo, T,, and TZZ, when TT20 is measured, in a way independent from theoretical parameters and thus can be applied to cases of any spin and any incident energy when the approximations used have validity. Also, eqs. (2.20) are identically the same as eqs. (7) in ref. i4), which have been derived by the semi-cfassical theory under the deformation model. In figs. 3 and 4 of ref. i4), it is emphasized that eqs. (2.20) are very good descriptions of the tensor analyzing powers observed in elastic scattering at Elab = 14.2 and 20.3 MeV. The case of the latter incident energy is shown in fig. 1 as an example, which is taken from ref. 14). From such successes it will he reasonably concluded that most of the tensor analyzing powers in elastic scattering are effects of the second-rank tensor interactions, when considered from the viewpoint of the spin~e~ndent interactions. This conclusion is consistent with the one derived in ref. 12). In inelastic scattering, the relation (2.14) gives TT20 = 5 ~~

$i W(Si2S,2 ; Sj2).

(2.21)

Thus *TZo is a constant number, which depends only on the spins, si and st, and is independent of other quantities, for example ki or k,. In the case of ‘Li scattering where 7Li is excited to the i- state, Si = 3 and s, = i and one gets from (2.21) rT20 =

4,

(2.22)

Using (2.12) and (2.21), we obtain (2.23) where both B$$ and B$$ include the invariant amplitudes only as the factor JFzo12 and thus the right-hand side of (2.23) is free from F,, and is a function of 0 and p. Particularly, when the excitation energy is small compared with the incident one, one can approximately set p = 1, for which eqs. (2.23) are automatically reduced to eqs. (2.20). That is, in such cases, eqs. (2.20) can be applied to both elastic and inelastic scattering, except for the difference in TTzo- In inelastic scattering of ‘Li by Wi, T,, and TT20 have been measured at Elab = 20.3 MeV. As will be seen in sect. 4, eqs. (2.22) and (2.23) reproduce the essential &atures of these measured quantities and also agree with the results of more quantitative calculations. Such successes may justify the unrefined treatments in the present theory. It will be

280

H. Ohnishi et al. I Tensor interactions

03 _

I_ bo

a2 0.1 0 ,II

Fig. 1. Second-rank tensor analyzing powers in ‘*Ni(%, 7Li)58Ni at E,,, = 20.3 MeV. The curves for TzQ(Q = 0, 1 and 2) are calculated by eqs. (2.20) [eqs. (7) in ref. ‘*)I while the dashed curve for ‘Tzo connects the data. This figure is taken from ref. I“).

remarked that the angular distributions of these tensor analyzing powers calculated above seem to be similar to the lines denoted by AM, = 0 in fig. 6 of ref. 14). The ratio T,,/TT,,,, which is given by (2.20) for the elastic scattering and by (2.23) for the inelastic one is only a function of 8 or a function of 0 and ki/k,. Also, Y20 in the inelastic scattering is a constant number. Such simplicities of these expressions extend their applications to a wide range of scattering, although the underlying assumptions are not justified in some cases. At high energies, for example, the validity of the PWBA is increased while the higher-rank tensor

H. Ohnishi et al. / Tensor interactions

281

interactions will become effective. In such cases, however, it is possible to extend the present method so that the higher-rank tensor interactions are taken into account. Through the analyses described above for both elastic and inelastic scattering, it is shown that the tensor analyzing power data can be explained as effects of the tensor interactions in a way free from the details of the interactions. In the next section, the 7Li-58Ni interactions are derived explicitly by a folding model and they include tensor parts as was expected. In sect. 4, CC calculations of these interactions are performed for ‘Li+%li scattering, where the available crosssection and vector and tensor analyzing power data at Elab = 14.2 and 20.3 MeV are analyzed quantitatively. Such analyses will examine the conclusions of the present section more closely.

3. F’olding interactions between ‘Li and ‘*Ni and their second-order effects One of the simplest ways of obtaining the interaction between the projectile, ‘Li, and the target, ‘*Ni is to fold a-‘*Ni and t-58Ni optical potentials into the ‘Li states concerned, considering that the ground state and the low-lying T = $ excited states of ‘Li are described by well~eveloped a-t cluster ~on~g~ations. This method suffers inevitably from the following disadvantages: the Pauli principle is taken into account only partially and the ima~nary part of the interaction obtained is not reliable because of the differences in open channels between the real 7Li-58Ni system and the c+‘*Ni plus t-58Ni system. Fu~hermore, the a-58Ni and t“Ni interactions at the proper energies are not always available for use because of the lack of scattering data. However, the numerical calculations in the next section show that the results obtained by the folding interaction depend on the choice of the a-“Ni and/or t -‘*Ni interactions only weakly, which will guarantee the validity for the practical use of the above method. In the present section, the folding interaction is derived for the 7Li-58Ni system. The diagonal element for the internal state of ‘Li will give the optical potential for this state and the non-diagonal one the interaction for excitations or deexcitations of ‘Li. The successive excitation and deexcitation of ‘Li yield effective potentials in the elastic-scattering channel. The effective spin-orbit potentials thus obtained are qualitatively discussed in this section. Denote the internal wave function of ‘Li having spin (z-component) S(M,) by I# 1; S M,), which is composed from the spin function of the triton x+ and the wave function of the a-t relative motion u,,(p)I;(Q,) as usual:

where p is the relative coordinate between the a-particle and the triton and 1 is the

282

H. ~hnish~ et al. / Tensor inreruc~io~

corresponding orbital angular momentum. The folding interaction matrix element of V,

is given by the

(3 I’ ; S’ M;Iw,) f V,(r,)+I, * s,v,,,.(r,)I~ 1;s M,),

(3.2)

=

where the first term of the interaction hamiltonia~ V is the a-58Ni interaction and the second and third terms are the t-58Ni central and spin-orbit interactions, respectively. The potential in each ‘Li state and the interaction for excitation or deex~tation of ‘Li between different states are specified by the proper choice of I’, 1, S and S. The expression (3.2) can be calculated explicitly by using multipole expansions of V,, V, and V,,,,. Changing variables r, and r, to p and R, the 7Li-5*Ni relative coordinates, rl = R+ap,

r, = R-BP,

with a = $,

we have (3.3) where j stands for 01,t and so on, and s2, and GP are the solid angles of R and P, respectively. Furthermore,

with

for j = a and t. For the triton spin-orbit potential I,*s, =

~(R+aP)x~~va+v~).s~,

where V, and V, describe the differentiation

(3.6)

by R and p, respectively. Using (3.3)

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H. Ohnishi et al. / Tensor interactions

and (3.6) and denoting the ‘Li- ‘*Ni relative angular momentum by L,

=

I-i (-

)k+P+l/%.o.,k

tR, P)([ck(Q,) ’ L]p*[ck(Q,) ’ %I”)

kp

5

+

(- JkfPa k.,k(Ry

+

(- )“+”$

d(ck(%)'

[CCk@p)

x I]” x %lk)

c {RI/,.,,,,(R,P)G(k! 0 ‘WW’(l~lk

; d)(c,(&t) *[[ck@,)

X

VP]’x $1’)

4

1

-@PI/,.o.,,(h’)h(k 1OOlpO)W(lplk; ql)([c,(B,)xv,]q’ [c,(a,)x s,]‘)} .

(3.7)

The general form of the matrix element of this interaction is quite complicated and thus the following will be given for examples; that is, the expressions for the resultant spin-orbit potential L. IF, for the $- ground state and the +- excited state and for the resultant third-rank tensor potential T3F3 for the 3- ground state. They are for the $- state, with I = $,

z..IF,

= L* 1

s

{~V,,,,(R,P)+~~V,.,.,,(R,P)+~~ ‘V,.,.,o(R,p))lu,3(P)12p2dp, (3.8)

i5.F~ = -&([IxIxZ13.

CBxRxL13)

s

~.o.,2(R,p)lu,t(p)12p2dp,

(3.9)

and for the 3- state, with I = f, .

L.ZF,

= -Z_..Z

J-{‘I’ 7

s.o.,oW,

PI+&

L,2@9

PI+&;

K.o.,dR,

P)IIuI~(P)I~P~ dp. (3.10)

In the present model, the $- ground state and the &- excited state are bound Pstates and the $- and $- excited states are resonance F-states. Each pair is split by a proper spin-orbit potential. By such specifications, one can see in eq. (3.4) the effective value of k for the potential in each state or the interaction for the transition between a particular pair of states of ‘Li. For example, in eq. (3.4), the terms of k = 0 and 2 are effective for the potential in the ground state. They are the central and second-rank tensor potentials. In some states, k = 4 and 6 can contribute to the above interactions but, as will be discussed in the next section, the real effect of such terms on physical observables investigated turns out to be small through numerical calculations. The most remarkable feature of the folding potential thus obtained is that the spin-orbit interaction is very weak, while the

H. Ohnish~et al. / Tensor int~~~~ia~s

284

second-rank tensor interaction is fairly strong in the ground state of ‘Li. These interactions have been compared to the d- 60Ni interactions 24) in ref. I’), where the relative strength of these two kinds of interactions changes places with each other between the ‘Li case and the deuteron case at most projectile-target distances. More details of the numerical calculation of the ‘Li interactions will be given in the next section. As was discussed in the preceding section, the data for the tensor analyzing powers are considered to be explained by the above tensor interactions. However, the spin-orbit potential obtained gives small and positive vector analyzing powers for the elastic scattering, which cannot explain the experimental data 9). In the following, the effective spin-orbit potential arising from the second order of tensor interactions will be investigated to look for the possibility of explaining the vector analyzing power data, where projectile excitations by the tensor interaction are taken into account. Denoting the total angular momentum (z-component) of the ‘Li + 58Ni system by J,(M), the effective potential in the partial wave ILS ; JM) due to the second order of the second-rank tensor interaction is given by

= ‘&!,S;

L

JN[V’“‘[B’;

JM)G,&R,

R’)(CS’;

JMIT/‘2’ILS; JM),

(3.11)

where S and L: are for the intermediate state and G,,(R, R’) is the Creen function for this state. The interaction V2’ is the k = 2 part of the folding interaction given by (3.2) and (KS’; JIvIIV(~)ILS; JM) is calculated by the use of eqs. (3.4) and (3.5). When the ‘Li ground state is chosen as the intermediate state, the above effective interaction is just the ordinary second-order term of the tensor potential and its contribution to the scattering amplitude is taken into account automatically in the usual potential-scattering calculations. When the intermediate state is one of excited states of 7Li, eq. (3.11) describes the effective potential due to excitations or break up of the projectile. Explicitly, such effective interactions are given by

x (I’2 00~~O)2(E200JLO)2W2(~S1S;~2)W2(~SLS;

where ‘VLil.s,(R) is essentially the sum of VJ,2,s(R) and l@,(R). vector term from the above, F,(R,R’)

= ,,hc(LS;

JMI&(LS;

(3.12)

J2),

JM)(-)L+S-J~2W(LLSS;

Extracting

1J).

the

(3.13)

J

Setting S = $ and I = 1, one obtains

the effective spin-orbit

potential

for the

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H. Ohnishi et al. / Tensor interactions

ground state of 7Li, F,(R, R’) = c l’-&(R)G,.,.(R,

R’)V&+4J5

9Zf2~2(1’200~10)2

L’

x (E200)L0)2W(C2Ll; L2)W*(lrsl~; 32)W(s’2$1; 32).

(3.14)

In eq. (3.14), the second-order contribution of the ground-state tensor potential is specified by S’ = 4 and I’ = 1 and other choices of S’ and I’ represent the contributions of the excitation or break up of 7Li. Using eq. (3.14), one can evaluate qualitatively the contribution of the excitation of 7Li by the ratio to that of the ground-state tensor potential ; that is, neglecting the dependences of l’&,G,.,. V’fs).Lt on the spin and the internal energy and angular momentum of 7Li, one obtains by eq. (3.14), F,(R, R’),. = +,’ = JFL(R, R’),, = +r’= I = 3,

(3.15)

F,(R, R’),, = %r’= JFL(R, R’),. = +1>= 1 = - y,

(3.16)

F,(R, R’),. = *I’

=

JFL(R, R’),,=St,

=

1

=

&.

(3.17)

Since the observed vector analyzing power is small, the contribution of the effective interaction to the analyzing power will be proportional to the above form factors. Thus, eqs. (3.15k(3.17) will become a measure of the contribution to the vector analyzing power for each excited state of 7Li. For example, the contribution from the $- excited state will have the same sign and a magnitude of 1.5 times that from the ground-state tensor potential. Since the ground-state tensor potential gives a vector analyzing power of considerable magnitude with the correct sign, as discussed in sect. 1, the virtual excitation of the $- state is a hopeful candidate for reproducing the experimental data. In this case, it should be emphasized that the approximation made in the derivation of eq. (3.15) seems to be valid because the 1state has very small excitation energy and the same internal orbital angular 2 momentum as that in the ground state. This is not the case, however, for the 3state. The fairly large excitation energy yields a different effect on the analyzing power expected from eq. (3.16). In the next section we will discuss the effects of the $- and $- states referring to the numerical calculations.

4. Numerical results of coupled-channel calculations for ‘Li-58Ni scattering The validity of the cluster-folding interactions for ‘Li derived in the preceding section will be quantitatively examined by comparison of numerical calculations with the experimental data for the cross sections and the vector and tensor

H. Ohnishi et al. / Tensor interactions

286

analyzing powers for scattering from 58Ni targets. Such investigations are performed for Elab = 20.3 MeV and 14.2 MeV. Additional predictions are made for features of the calculation at Elab = 30 MeV which includes the third-rank tensor analyzing powers. In all cases, the effects of the projectile virtual excitations are taken into account by the coupled-channel method, where the first 4-, $- and $- states are considered as the excited states of 7Li. In the calculation of the internal wave function of 7Li, the wave function of the a-t relative motion I.+~,a real Woods-Saxon type potential is assumed for the central interaction and the Thomas type potential for the spin-orbit one, as usual. The parameters of these potentials are given in table 1, which are determined by referring to the a-3He potentials 25). These potential lit the data 26) for the binding energy and the quadrupole moment for the ground state and those of the binding energy and the B(E2) strength for the &- excited state. Form factors are calculated for the ground-state charge distribution and for the transition charge density to the *- state by the use of uls obtained above. They are compared with the data from electron scattering 27) in fig. 2, where the calculation agrees with the data satisfactorily. Also, the potential parameters produce the phase shifts of the $-, $-, $- and $- partial waves in t-a scattering, which are compared with the empirical values’s) in fig. 3, where the calculation agrees with the data reasonably well except for the rather poor splitting between the $- and $- phase shifts; in particular, the resonance energies and the widths of the $- and $- states are very well reproduced. The folding interactions are calculated using eqs. (3.lt(3.10) with uIs obtained above. In the calculation, phenomenological optical potentials are assumed for the t-58Ni and a-58Ni interactions. It would be reasonable to choose the potential parameters which lit the t-58Ni and a-5*Ni scattering data at some energies

TABLE1 Potential parameters for d-a and t-a relative motion

RO Rc a0

d-a parameters

t-a parameters

1.90 1.90 0.65

2.05 2.05 0.70

.I” 0+

o-, 1-, 2If, 2+, 3+

“IS

- 77.5 - 75.0 - 78.5

-4.1 -4.1

*+ +-, j$-, )-

- 76.0 - 93.0 -90.5

-2.0 -3.5

The unit of length is fm and that of energy MeV. The quantities R, and R, are the radius of the nuclear potential and of the Coulomb potential, respectively. Other notations are defined as usual.

H. Ohnishi et al. / Tensor interactions

q2[fme21

287

q2 [fmo21

Fig, 2. Charge form factor of ‘Li. The charge form factors calculated by the use of u,s and the intrinsic charge distribution of e and t obtained from the experimental data are compared with those derived from ei~~on-~tte~ng data “). Here (a) is for the ground-state charge dist~bution where M, Q and T refer to monopole, quadrupok and total contributions, respectively, and (b) for the ratio of the form factor for the transition charge density associated with the excitation of the $- state to that of the ground-state charge distribution.

Fig 3. Phase shifts of a-t scattering. The phase shifts calculated by the potential parameters in table 1 are compared with measured values **) for the partiai waves of .fn = *‘, f-, j-, $- and $-.

N. Ohnishi et al. i Tensor interactions

288

TABLE2 Parameters for d-SsNi, t-58Ni and a-58Ni optical potentials

d-58Ni t-‘*Ni x_S*Ni (:

- 107.1 - 151.0 1 ‘69:i

- 14.8 - 18.2 1’2

-9.6 -4.0

1.05 1.20 1.43 1.65

1.42 1.60 1.43 1.65

0.84 1.10

1.30 1.30 140 1.65

0.85 0.66 0.50 0.52

0.716 0.830 0.500 0.520

0.46 0.83

The unit of length is fm and that of energy MeV. The notations are detined as usual. For the deuteron, the imaginary potential is assumed to be of the surface type with depth Wo.

determined by sharing properly the ‘Li energy between the triton and the alpha. Because of the lack of the scattering data at these energies, several sets of which are determined at rather higher parameters of the potential energies 20-22T2g-31) are employed for the calculation, Since the results are found to depend only weakly on the choice of the above parameters, the results are presented in the following for two typical sets of a-58Ni potentials “g31) with the fixed t-‘*Ni one 2g), the numerical values of the parameters being listed in table 2. In fig. 4, the form factors of the ‘Li- 5sNi interactions obtained from set A in the table are shown as examples for the central, spin-orbit and second- and third-rank tensor interactions for the ground state of 7Li and for the second-rank tensor one

Fig. 4. Form factors of the cluster-folding interaction between 7Li and “Ni. The real part V”) and the imaginary one W co)of the central interaction are shown in (a). The second-rank tensor interactions in and those for the excitation of ‘Li to the s- state the ground state V3(f&z and W$& P(Z) are shown in (b), where the P’s and Ws refer to the real and the 7,23:2 and W#& imaginary parts, respectively. The interactions for the excitation to the p- state and to the j- state cannot be discriminated from those in the ground state and those for the excitation to the $- state, respectively. The spin-orbit interaction V(rr and the third-rank tensor one Y@rwhich are real are shown in (c). The parameters used are set A in table 2.

H. O~~is~j et al. / Tensor ~~er~fions

289

for the transition of ‘Li to the 3-, $- and $- states from the ground state. In this figure, it will be emphasized that the form factors of the second-rank tensor interactions V(‘) are quite similar to each other, i.e. one cannot distinguish the ground-state interaction, say V(2)t+ from the interaction for the transition to the $state, k’$, and the interaction for the transition to the i- state, Vi”:, from the one to the $- state, Vi?; furthermore, the difference between the former two and the latter two is very small. For the interactions concerned with the resonance states, the form factors are averaged over energy points of suficiently large number around the resonance energy, using the weight determined by the one-level formula. The detailed results of the numerical calculations for the elastic scattering are given for Elab = 20.3 MeV, and appear in figs. 5, 6 and 7. The theoretical curves in fig. 5 are for the potential scattering calculation which neglects projectile excitations, the CC calculation which takes into account the ground state and the i- excited state of ‘Li, and the CC one which takes into account the ground state and the f-, $- and $- excited states of ‘Li, where the potential parameters employed are set A in table 2. The first of the above calculations is similar to the calculation in ref. r5) and, as was pointed out in ref. ‘*), the negative value of the vector analyzing power iT,, in the calculation arises mainly from the second, or higher, order of the second-rank tensor potential. From the comparison between the three curves, it is seen that the coupling with the channels of the projectile excitations increases the magnitude of iT,, at most angles. Particularly, the inclusion of the 3- state approximately doubles the magnitude compared with the potential scattering calculation. The detailed explanation of such effects will be given later. The contribution of the j- excited state increases considerably the magnitude of iT’, at forward angles and decreases slightly that at backward angles, and the contribution of the s- excited state increases the magnitude at backward angles, com~nsating the decrease by the )- state. The resultant of all of these effects is described in the last curve in fig. 5, which agrees well with the empirical data r4). The cross section is also affected by the virtual excitations of the projectile in an interesting manner; the inclusion of the $- state decreases the cross sections at large angles by small amount while further inclusion of the $- state increases them considerably, giving a good agreement with the experimental data. The $state contributes to the cross sections by a small amount. The data for the tensor analyzing powers T&, T,, and T,, are well reproduced by the calculation, where the dominant contributions arise from the potential scattering and the effect of the projectile excitations is relatively small. The stability of the result is examined in fig. 6, where the calculation for set B in table 2 is compared with set A. There are very small differences between them, In this figure, simpl~ed c~culations with set A, which use only the second-rank tensor interactions for the spin-dependent interaction, are also compared with the full calculations by the same set. The simp~~tion seems to be a good approximation and the theoretical successes in

I

I

I

I

//

__.’

-

”

60”

T21

90” 120’ 150” 180” 8 cm

d:Y$-_____I..

...’

/#dg+qO.$ ;f + ++

0” 30°

0.1 -

,) 0.2-

0

0 -O.I-

30” 60” 90” 120’ 150° 180” 8cm

I

ELAB = 20.3 MeV

0.2 0.1 i

0.3

&

.___ :,.,__,,,_, :c..,.! (,,,_,,, ;?

0” 30” 60” 90” 120” 150“ 180” 0cm

I

Fig. 5. Cross sections and vector and second- and third-rank tensor analyzing powers in 58Ni(‘&. ‘Lil’sNi at E,,, = 20.3 MeV. The effects of the projectile virtual excitations are compared for various excited states of ‘Li. The lines are obtained by the use of the cluster-folding interactions derived from the parameters of set A in table 2. The dotted lines are the potential-scattering calculation. The dashed and solid lines are the CC calculation taking account of the ground state and the i- excited state, and the CC one taking account of the ground state and the f-, SW and $- excited states, respectively. I he calculations are also compared with the experimental data 14).

0”

-0. I c

-0.05

0

0.01

0.1

I

291

0.0 I

El&e =20.3M

i

Fig. 6. Cross sections and vector and second-rank tensor analyzing powers in s*Ni(‘~i,‘L~)ssNi at E tab= 20.3 MeV. The lines are the CC calculations which take account of the ground state and the t-, j- and j- excited states for 7Li. The solid and dashed lines employ the folding interactions derived from the parameters of set A in table 2 and those derived from the parameters of set B in the table, respectively. The dotted iines use the former interactions allowing only the second-rank tensor interactions. The calculations are compared with the ex~rimental data 14).

reproducing the data are not spoiled by this procedure, Thus the most important spin-de~ndent interaction is the second-rank tensor interaction in the present system. From these results it will be emph~ized that the cluster-fol~ng interactions are successful in explaining the data when the virtual excitation effects are taken into account s~ciently. Figs. 7a and 7b show how the effects of the excited states depend on the excitation energies E, of the states concerned. In fig. 7a, E, of the i- state is artificially varied from 0.0 to 8.5 MeV, where the channel coupling is considered only between the ground state and the $- state and the spin~e~nd~nt interaction which is limited to the second-rank tensor one is assumed only for the transition between these states. In this case, the cross sections at large angles increase with increase of E, up to E, = 4.5 MeV and decrease with the further increase of -E, and ~rres~~din~y the vector a~aly~ng powers at most angles decrease from large positive values to negative vaIues and finally approach zero. The realistic

90” 180”

2.5 4.5 6.5 8.5

---*-se-=-

0”

!.,

...., E&=20.3MeV

58Ni(7n ,7Li)sNi

1 eR

-

b) _..... .“..._

0.05

I

t iT I

( C)

I

I

I

5eNi( 7fi,7Li)58Ni

1

Fig. 7. Ekts of projectile excitations on cross sections and vector analyzing powers in ‘*Ni(‘Li,- 7Li)58Niat E,,, = 20.3 MeV. Parts (a) and (b) show variations the effects in the adiabatic Part (c) shows of the effects by the excitation energy E, in the case of the f- state and of the )- state, respectively. lines, dotted ones and solid ones, respectively. The dashed lines are the potential approximation (see text) for the f-, t- and )- states by the dash-dotted scattering. In the upper part of (c), the dotted line and the dash-dotted one cannot be discriminated from each other. The spin-dependent interaction IS

0"

,

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H. Ohnishi

et al. / Tensor

interactions

293

E, of the $- state is 4.6 MeV, at which the channel coupling fortunately

increases the backward cross section and the.magnitude of the vector analyzing powers at most angles. Similar effects are seen for the +- state which are shown in fig. 7b. The effects of the various states considered are compared to each other in an adiabatic approximation in fig. 7c, where all states are assumed to be degenerate with the ground state and the channel coupling is considered only between the ground state and the excited state concerned, the assumption for the spindependent interaction being similar to the one in fig. 7a. As is easily seen, the calculated vector analyzing powers qualitatively satisfy the relations predicted by eqs. (3.15)-(3.17) both in sign and magnitude. From these analyses, the characteristic features, of the real contribution of each excited state in fig. 5 will be understood as follows. The approximations made in the derivation of eqs. (3.15)(3.17) are appropriate for eq. (3.15) because the form factor of the *- state is quite similar to that of the $- as is seen in fig. 4 and the excitation energy of the $- state is very small. Thus, fig. 7c explains the contribution of the t- state in fig. 5 both in sign and magnitude. Eq. (3.16) predicts that the contributions of the I- state to the vector analyzing pokier have opposite sign to those of the $- state and thus have positive sign, which is confirmed by the adiabatic numerical calculation in fig. 7c. In this case, however,' the excitation energy is not small and the actual contribution changes its sign and magnitude according to the real value of E, as seen in fig. 7a. The smallness of the contribution of the $- state is due to the small factor (& in eq. (3.14) and the large excitation energy. It will be emphasized from the above that the excitation energy is the important factor in considering effects of the virtual excitation and thus the adiabatic approximation looses its validity except for very low excited states. The importance will be demonstrated in another way: the sum of the contributions to the vector analyzing power from the $-, 9-, $- and $- states becomes zero by the cancellation in eqs. (3.15k(3.17) but the real contributions do not cancel with each other. The calculations are also carried out for Elab = 14.2 MeV using the same potential parameters, the results of which are compared with the experimental data in fig. 8. In the measured data, the contributions of the inelastic scattering to the i- state of ‘Li are mixed up with the elastic scattering. Fig. 8 contains two kinds of calculations: one is the ordinary calculation and another takes account of the inelastic-scattering effects by the following formula 32) : (4.1) (4.2) The effects of the admixture of inelastic-scattering quantities are not negligible and the resultant of the calculation agrees with the data fairly well but the agreement is

294

H. O~n~h~ et al. j

Tensor interactions

O_ -O.i0” 30’

60’

90” @cm

120°1500

180”

0”

I

I

I

30”

60°

90”

1

I

I

120°15001800

0 cm

Fig. 8. Cross sections and vector and second-rank tensor analyzing powers in ‘sNi(‘6, 7Li)58Ni at E,,, = 14.2 MeV. The effects of the projectile virtual excitation are compared for various excited states of ‘Li. The lines are obtained by the use of the cluster-folding interaction derived from the parameters of set A in table 2. The dotted lines are the ~tential-pattering calculation. The dashed and solid lines are the CC calculation taking account of the ground state and the f- excited state, and the CC one taking account of the ground state and the f-, f- and $- excited states, respectively. The dashdotted lines take into account the effect of the inelastic scattering by eqs. (4.1) and (4.2) for the last case of the above. The calculations are also compared with the experimental data r4).

worse compared with that at Elab = 20.3 MeV. The kinetic energy of ‘Li in the present case is too low to apply the present potential parameters straightforwardly. Suitable choices of the parameters will improve the agreement. The figure also shows that the effects of the virtual excitations of the projectile in this energy are similar to those at Em, = 20.3 MeV. In fig. 9, the calculated cross section and the vector and tensor analyzing powers for the inelastic scattering leading to the &- excited state of ‘Li are shown for Elab = 20.3 MeV, where the interactions and the 7Li states considered are the same as those in fig. 5. The effects of the virtual excitations of ‘Li to the z- and SWstates are not remarkable in the analyzing powers although they increase considerably the cross section at large angles. The calculation reproduces the experimental data 14) satisfactorily well except for the cross section at small angles. In the figure, the predictions by the invariant-amplitude method for the tensor analyzing powers, eqs. (2.22) and (2.23), are compared with the CC calculations, where the CC calculations differ from the calculations by the invariant-amplitude method at small angles, while the former approach the latter asymptoti~lly at large angles.

H. Ohnishi et al. J Tensor interactions

295

[mbl

IO2

58Ni(7ti, l!h~s8Ni Eu8=20.3MeV

O0

90” 8cm

I800

90”

e cm

Fig. 9. Cross sections and vector and second-rank tensor analyzing powers in 58Ni(7~i,‘Li*(~-))58Ni at E,,, = 20.3 MeV. In g(0) and iTI,, the effects of the virtual excitatiort of the projectile in the CC calculation are compared between various excited states of ‘Li, where the cluster-folding interactions derived from the parameters of set A in table 2 are employed. The dotted and solid lines take into account the ground state and the $- excited state of ‘Li, and the ground state and the t-, I- and $excited states of ‘Li, respectively. In TT2,,, T2,, and Tzl, the above two cases are hardly discriminated from each other and both are described by the solid lines. The dash-dotted lines are for the parameterfree calculation by the invariant-amplitude method (see text). The calculations are also compared with the experimental data 14).

The cross section and the vector and tensor analyzing powers are calculated for both elastic and inelastic scattering at Elab = 30 and 40 MeV, using the same parameters of the a-58Ni and t-‘*Ni optical potentials as in fig. 5. Although the parameters do not take account of the energy dependence of the potentials and accordinglv the detail of the results of the calculation are nor always realistic, the following features are observed in the calculation, i.e. the effects of the virtual excitations of the projectile are remarkable in the vector and tensor analyzing

30’

iTll

0cm

90”

120°

150°

Fig. 10. Cross sections and vector and secondthe virtual excitations of ‘Li are taken into account, lines are for the ootential-scattering

O”

-0.:

60°

ELAE = 30 MrV

2L

-0.

0

Id

IO

lo’

lo’

lo‘

I

0”

‘-

30’

60’

@cm

90”

120”

150”

-0.02

0

0.02

-0.02

0

0.02

-0.02

0

0.02

0”

30’

60’

@cm

90”

120’

150°

and third-rank tensor analyzing powers in 58Ni(7ci, ‘Li)‘sNi at E,,, = 30 MeV. The effects of the f- excited state in the dashed lines and the f- and $- excited states in the solid ones. The dotted calculation. The interaction is derived bv the parameters of set A in table 2.

-0.4

0.8

H. Ohnishi et al. / Tensor interactions

291

powers in both elastic and inelastic scattering, while the cross sections are less affected. Typical examples of these investigations are shown in fig. 10. Elastic thirdrank tensor analyzing powers iTaQ(Q = 1, 2 and 3) which have not been measured so far are calculated at Elab = 20.3 MeV and Elab = 30 MeV, and are shown in fig. 5 and in fig. 10, respectively. The predicted magnitudes of the analyzing powers are very small at the lower energy due to serious virtual excitation effects but seem to increase with incident energy. At Elab = 30 MeV, they look measurable.

5. Elastic scattering of 6Li by ‘*Ni Considering that the projectile excitations are particularly important for the cross section and the vector analyzing power in the ‘Li-‘sNi scattering, calculations similar to the ‘Li case are performed for the elastic scattering of 6Li by ‘*Ni at Elab = 20.0 MeV, attention being focussed on the cross section and the vector analyzing power. Analogously to the ‘Li case, the 6Li is assumed to consist of an a-particle and a deuteron which interact with each other by real WoodsSaxon central potentials and Thomas-type spin-orbit ones, as usual, the parameters being given in table 1. These potentials reproduce the empirical data for the binding energy and the charge form factor in the ground state of 6Li and of the phase shifts in the d-u elastic scattering. The interactions between 6Li and 58Ni are calculated by folding a- ‘*Ni and d-58Ni optical potentials 31,33) into the internal state of 6Li concerned. The parameters of these potentials are given in table 2. The states of 6Li coupled to the l+ ground state are the first 3+, 2+ and l+ excited states at E, = 2.19, 4.31 and 5.65 MeV, respectively, which are observed as resonance states in the d-cr elastic scattering 26). The configuration is assumed to be a pure S-state for the ground state and pure triplet D-states for the excited states, the latter being split by the spin-orbit interaction. In this case, the main spindependent interactions are the second-rank tensor parts of the folding interaction similar to the ‘Li case, which allow us to expect some important projectile excitation effects. The results of the CC calculations are shown in fig. 11, where two remarkable contributions of the projectile-excitation process are observed: most of the calculated vector analyzing power arises from the virtual excitations of the projectile which yields a good agreement with the experimental data 34) while the cross section is increased by a considerable amount by these processes but is still smaller than the measured. One of the candidates to improve the theoretical cross section would be to take account of projectile excitations of other modes: namely, the break-up of 6Li into non-resonant continuum states of the d-a system, for example in S- and P-waves. Although some problems must be solved in order to take into account such continuum states exactly in the CC calculation [for example, discretizations of the continuum spectra and limitations of the excitation

298

H. Ohnishi et al. / Tensor intermtians

90”

.

180”

8 cm Fig. 11. Cross sections and vector analyzing powers in ‘*Ni(%, 6Li)S8Ni at EIab = 20.0 MeV. The lines are obtained by the calculation using the cluster-folding interactions for the parameters in table 2. Here, A designates the ~tential-s~ttering calculation and 8, C and D, the CC ones. They take account of the ground state and the first 3’ excited state, the ground state and the first 3+ and 2+ excited states, and the ground state and the first 3+, 2+ and 1+ excited states, respectively. In line E, these three excited states are artificially assumed to be degenerate at the center of their energies. In the cross section, E cannot be discriminated from D and both are represented by a single line. Line F includes the effect of the break-up of 6Li into the non-resonant continuum states (see text) in addition to those of the resonant 3 +, 2+ and 1+ states. The calculations are compared with the experimental data 34).

energy 35)], a simple calculation is attempted tentatively to see the trends of the contributions of such non-resonant break-up of the projectile. The continuum states between E, = 0.0and 5.65 MeV are represented by one level for each spinparity state, O’, O-, I- and 2-, t h e energy of which is assumed to be centered in the momentum space of the above energy region. The d-a relative wave functions for these partial waves are calculated by the use of the potential parameters in table 1. The results of the calculation where the ground state, the three resonance states and these four non-resonant break-up states are coupled with each other are shown in fig. 11, where an appreciable increase of the calculated cross section accompanied by a small improvement of the calculated vector analyzing power is seen, which is quite promising for more sophisticated calculations taking account of the non-resonant break-up states. Fig. 12 shows how the coupling effects depend

H. Ohnishi et al. / Tensor interactions I

I

I

299

I

I

@ii&5eNi(6fi~Li)5eN

i

I

10-l : I o-’ - -

-*-

2.26 3.75 5.00

\\

L.

IO-! 0.1

0

5;s -J

-0.1 0"

90'

l8d

8cm Fig. 12. Excitation-energy dependences of effects of projectile excitations in 5SNi(6ci, 6Li)5*Ni at El,, = 20.0 MeV. Variations of the cross section and the vector analyzing power are shown for those of the excitation energy of the first 3+ state, I&(3+), from 0.0 to 5.0 MeV. The spin-dependent interaction is limited to the second-rank tensor one and the parameters employed are given in table 2.

on the excitation energy of the states. In this figure, the 3+ state is coupled to the ground state, the excitation energy of which is varied artificially from 0.0 to 5.0 MeV. The spin-dependent interaction is only the second-rank tensor part of the interaction acting between the ground state and the 3+ excited state. It is easily seen that the coupling effects depend seriously on E, which is similar to the case of 7Li. This property is demonstrated in fig. 11, which presents the calculation where the three resonance states are assumed to be degenerate at the center of their energies and compares wieh the realistic calculation. The degenerate case gives a different sign for the vector analyzing power in most angles.

6. Concluding remarks In the ‘Li-‘sNi

scattering, the invariant-amplitude

method explains that the

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observed tensor analyzing powers are mainly the effect of the second-rank tensor interactions: namely, in the elastic scattering, eqs. (7) in ref. 14), which are very good descriptions of the experimental data for the tensor analyzing powers, are derived by this method in a way free from theoretical parameters under the assumption that the spin-dependent interaction is only the second-rank tensor one. In the inelastic scattering, similar formulae for the tensor analyzing powers are derived by the method, which are shown to be in good agreement with the experimental data. The quantitative calculation by the CC method demonstrates that the clusterfolding model is successful in understanding the 7Li-58Ni scattering data. The second-rank tensor interactions, which are derived from the central parts of the CIs8Ni and t-58Ni optical potentials in this model, reproduce the main part of the observed tensor analyzing powers. The interactions also give rise to the virtual excitations of ‘Li which yield the important contributions to both the cross section and the vector analyzing power. Similar virtual excitations have remarkable contributions in the 6Li-58Ni system, where the additional non-resonant break-up processes have indispensable effects on the theoretical cross section in order to explain the data. Such results in 6Li scattering suggest examining effects of the non-resonant break-up of the projectile in 7Li- 58Ni scattering. The investigation is carried out where the S and P break-up states are tentatively taken into account in the CC calculation, the x-t potential parameters for these states being given in table 1. In this case, however, the non-resonant break-up processes have no appreciable contribution to the cross section both in the elastic and inelastic scattering, although the magnitude of the vector analyzing powers are decreased in some way. Earlier it has been pointed out that in the scattering of soft heavy ions, doublefolding potentials are required to be renormalized by some factors, for example 0.5, in order to fit the cross-section data 36). The present analyses suggest that such difficulties will be solved by taking account of the virtual excitations of the projectile which include break-up to non-resonant continuum states. In fact, it has been shown recently 35) for high-energy scattering that the renormaIization procedure is not necessary when the break-up mechanism is taken into account. Finally, the tensor analyzing powers in 6Li- ‘8Ni scattering have not been studied here, because the D-state admixtures in the 6Li internal motion which are important in the tensor analyzing powers 37) have been neglected. We are indebted to Prof. D. Fick for sending us the data and useful suggestions. We also wish to thank Prof. M. Kawai for valuable discussions. A part of this work was performed by the financial support of Research Center for Nuclear Physics, Osaka University.

H. Ohnishi et al. / Tensor interactions

301

Appendix

In the framework of the invariant-amplitude method, the scattering amplitudes due to second-rank tensor interactions satisfy simple relations in the plane-wave Born approximation (PWBA), which will be derived for both elastic and inelastic scattering. An interaction which includes a spin tensor of rank k can be written as (5k. %J = c (- )KSk?JkG K

(A.1)

where tlkK

=

Y,,(S2)T/,(R)*

64.2)

Here, tkK and qkK are the tensor operators of rank k in the spin space and the coordinate space, respectively, and R and Sz are the variables of relative motion between ‘Li and ‘*Ni, T/,(R) being the form factor of the interaction. In the PWBA, elements of the transition matrix Mck’ are given by (SrVr

;

k,lM’k’lsiVi;ki) = C (- )“(s,v,lrk_.Isivi) ei’.R Yk,(Q)T/,(R)dR K s

= (-

)s’-“‘(SiSfVi-

V,~klC)Yk,(~,)x‘ik(~),

(A.3)

with Ak(q)

=

47Cik~-'($I15kIlsi)

jk(4R)V,(R)R2

(A.4)

dR,

s

where si(vi) and sr(vr) are the spins (z-component) of the projectile, before and after scattering, respectively. The quantity q is the momentum transfer in this scattering, q = k,-k Iv

(A.5)

and 52, is the solid angle of q. On the other hand, the above matrix elements can be described by the invariant amplitude lg) F,, as
=$_k [Cr(Qi) X C,_,(Q,)]EFk,-

64.6)

The notations are similar to those in (2.8) of the text. Choosing the coordinate axis so that the z-axis is parallel to ki and the y-axis is perpendicular to the reaction plane and comparing (A.6) with (A.3), (- )Si-SrY,,(S2,)A,(q)=

c (d-r I

0 KlkK) /z

pi%=S

@F,,.

(A-7)

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H. Ohnishi et al. / Tensor interactions

Fixing k = 2, (A.7) gives

(-)“l-S’A,(c?)

J

&(3

co? e,- 1) = 3(3 cos2 e4- l)F,, +&os

OF,, +F22,

(A.8)

for ic = 0.

(- )Sl-S'A2Gl)

J

g co.58, sin 8, = - J$ cos e sin OF,, -+ sin OF,,,

for K = 1, (A.9)

for K = 2.

(- )“ieSfA2(q)

(A.lO)

Combining eqs. (A.9) and (A.lO), F 21=_ F2o

sin e

cos 8 cos 8, sin e4+ e4 sin e

fi-- sin’

(A.ll)

Eq. (A.5) gives

k:

sin’ e4 = 1 sin2 8,

4

(A.12)

cos 0, sin e4 = 5 (ki - k, cos 0) sin 8. q2 Inserting (A. 12) into (A. 1l), F LL&

(A.13)

F 20

f

Further, the use of eqs. (A.8), (A.12) and (A.13) gives F 22 F 20

;

0 f

2*

(A.14)

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