Spin-dependent interactions and polarization observables in elastic scattering of deuterons at intermediate energies

NUCLEAR PHYSICS A

Nuclear Physics A533 (1991) 574-600 North-Holland

EN E INTERACTIONS AN POLARIZATION LES IN ELASTIC SCATTERING O DEUTERONS AT INTERMEDIATE ENERGIES Y. ISERI Chiba-Keizai College, Chiba 260, Japan M. TANIMI Department of Fhrsics and Research Center of Ion Beam Technology, Hosei University, Tokyo 102, Japan H. KAMEYAMA' and M. KAMIMURA

Depariment of Ph1"sics, Kvusyu Universi~y, Fukuoka 812, Japan M. YAHIRO Shimonoseki University of Fisheries, Shimonoseki 759-65, Japan Received 12 November 1990 (Revised 18 March 1991) Abstract: Deuteron-nucleus spin-dependent interactions are investigated in elastic scattering at intermediate energies. Spin-space tensor amplitudes which characterize spin-dependent interactions are introduced by the invariant-amplitude method so that effects of TR - and TL -type tensor interactions are described separately. Through these amplitudes, numerical calculations based on the folding model clarify that the spin-orbit interactions produce TL -tensor like effects on the scattering amplitude as the higher-order effect and the virtual breakup contribution is only a minor part of this effect in most angles . An additional T,-type tensor interaction, phenomenologically introduced, is found to improve the fit to the experimental data of A,.,. considerably and A,. in less magnitude. The origin of this interaction is discussed . The theoretical prediction is made of which observables enhance the effect of a particular spin-dependent interaction .

1. Introduction Deuterons are the simplest composite projectile of nucleons and their scattering and reactions by nuclei are an important source of information on nuclear interactions as well as on reaction mechanisms, where the projectile internal structure is reflected. The feature of deuteron interactions with spinless nuclei which makes a marked difference from that of the nucleon interactions is characterized by the presence of the second-rank tensor interactionss. l-'arlier, three types of the tensor interaction, i.e. the coordinate-dependent TR , the angular-momentum-dependent TL and the momentum-dependent Tp, have been proposed') in the viewpoint of phenomenology but the real effects of these tensor interactions have been known only partially. ' Present address : Chiba-Keizai College, Chiba 260, Japan. 0375-9474/91/$03 .50 c: 1991 - Elsevier

Science Publishers B .V. All

rights reserved

Y. Iseri et al. / Spin-dependent interactions

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One simple and reasonable way of obtaining the deuteron-nucleus interaction is the so-called folding model, where proton-nucleus and neutron-nucleus interactions are folded into the relevant state of the deuteron 2). For the ground state, the interaction obtained consists of central, spin-orbit and second-rank tensor interactions; the former two originate in the central and spin-orbit parts of the nucleonnucleus interactions respectively, while the last one, the tensor interaction, arises from the D-state admixture in the deuteron wave function through the nucleonnucleus central interactions and thus is of TR -type. On the other hand, it is known that the deuteron-nucleus spin-orbit interaction explains a part of measured tensor analyzing powers. This suggests that the spin-orbit interaction produces effectively a kind of tensor interaction . Based on the folding model, ref. 3) has derived a TL-type tensor interaction as the effect of virtual breakup of the deuteron due to nucleonnucleus spin-orbit interactions in a two-step approximation but has claimed that the properties of the TL-type interaction thus obtained contradict those required for the explanation of experimental data 4). Later, we will discuss these problems in more detail . The Tp-tensor interaction has been investigated in the standpoint of a microscopic nuclear model, where the antisymmetrization between the nucleons of the deuteron and those of the nucleus is essential for the derivation of this tensor interaction . The quantitative studies have found that the strength C fthe Tp interaction is fairly weak 5) . In principle, effects of such spin-dependent interactions can be investigated through experiments on polarization phenomena, for example those in elastic scattering of deuterons . In practice, however, the effects of these interactions are usually mixed up with each other even in one polarization observable and this complication disturbs the finding of the real effect of each spin-dependent interaction. Such situations have brought some confusion into the conventional analyses on understanding ofthe effect of spin-dependent interactions. To solve such difficulties, the invariant-amplitude method 6) has been proposed previously, which is similar to the later work in red:') but is more general and thus has wider applicability. In this method, the scattering amplitude is decomposed according to the tensorial character in spin space so that the components obtained, the spin-space tensor amplitudes, represent the effect of the corresponding spin-dependent interactions, respectively . For example, the scalar amplitude describes the scattering by the central interaction, and the vector amplitude that by the spin-orbit interaction, in the sense of effective ones . Since the observables are composed from these spin-space tensor amplitudes, the method is quite useful for identifying the effect of each spindependent interaction in the observables . lit fact, scattering of polarized 6Li and 6 .1 °) and d + p -> 3 He + ,;r° reactions ") have 7 Li projectiles 8,9), some (d, p) reactions been analyzed by this method with remarkable success. In the present paper, we will study the deuteron-nucleus spin-dependent interactions through the elastic scattering, by applying the invariant-amplitude method . The scattering amplitude is decomposed into the spin-space tensor amplitudes

57 6

Y. Iseri et al. / Spin-dept dent interactions

discussed above. As will be seen later, we can choose two independent second-rank tensor amplitudes, for which we will assign one to the TR-type tensor interaction and the other to the TL -type one according to the prescription provided in ref. ' 2) which proposed an essential idea on the assignment . Thus it becomes possible to identify the effects of these interactions separately in the observables, when the Tp interaction is neglected. The validity of such decomposition of the scattering amplitude is quantitatively examined by numerical calculations . The success in the separation of the TR- and TL-tensor amplitudes allows us to reexamine the previous work ;) on the TL interaction in the new light of the present method, where the virtual breakup of the deuteron is treated by the method of coupled discretized continuum channels 1-1) (CDCC `,. Further, we will investigate the effect of each spin-dependent interaction on the observables by describing them in terms of the spin-space tensor amplitudes. Considering that the experimental data of the polarization transfers will become fully available in the near future, we will treat the analyzing powers and polarization transfers for the polarized deuteron beam, for example, and discuss in particular what observables are suitable for identifying the effect ofa particular spin-dependent interaction. These theoretical predictions are confirmed by the numerical calculations. Such information will be useful in planning the experimental setup. Since the invariant-amplitude method is essentially a non-relativistic treatment and on the other hand the validity of the TR -T, separation is justified at some high incident energies ''), the theory can be reasonably applied to the intermediate-energy region. Here, the deuteron incident energy for the numerical calculations is chosen to be 700 MeV as the standard, and the results of 400 MeV calculations are presented as the complement . The deuteron-nucleus interaction is basically that which is obtained by folding the proton-nucleus and neutron-nucleus potentials, which are the Schrödinger equivalent of the so-called Dirac phenomenological potentials. The validity of the folding interaction thus obtained has already been examined ' 3) by comparing the calculated observables with the experimentally measured. Since this interaction includes the spin-orbit and TR-tensor parts but not the TL -tensor one, surface-type TL -tensor potentials are phenomenologically introduced to examine the TR-TL separation and to see their effects on physical observables. In the next section, a short review of the invariant-amplitude method and the decomposition of the scattering amplitude, which includes the separation of the effects of the TR- and TL -tensor interactions, are presented. Sect. 3 describes the result of the numerical calculations which show the decomposition to be successful . Also, it is shown that the nucleon-nucleus spin-orbit interactions produce the second-rank tensor amplitude of T1.-type as their higher-order, dominantly secondorder, effect and the virtual-breakup contribution to this TL-type tensor effect proposed in ref. 3 ) is small. In sect. 4, contributions of the genuine TL -tensor interaction p henomenologically introduced are examined and found to have the tendency of improving the theoretical fit to the experimental data of the analyzing

Y. Iseri et al. / Spin-dependent interactions

577

powers. The origin of this TL interaction is discussed. Sect. 5 discusses which observables are the good measure of a particular spin-dependent interaction. Sect. 6 is devoted to the summary. Finally, it will be noted that in ref. "4), some low-energy-scattering data have been analyzed by the use of the spin-space tensor amplitudes. The analyses have taken account of a part of the present results in advance and have shown that the present theory is qualitatively successful even at low energies . 2. Invariant-amplitude method in deuteron elastle -a-r¢e*inb

The transition matrix of deuteron elastic scattering from spinless nuclei is given by designating the row and column by the z-component v ofthe deuteron spin s as ") /A

M=~ D C

B

E -B

C\

(2.1)

A

where the row denotes the initial state, vi = 1, 0, -1 from left to right and the column the final states, vf = l, 0, -1 from top to bottom. The elastic scattering restricts the matrix elements as C=A-E-vr2-(B+D)

cot 0 .

(2.2)

The matrix elements in the plane-wave states, A - E describe the exact scattering amplitude in the non-relativistic form. To decompose the amplitudes according to the tensorial property in the spin space, we will expand into the spin-space tensor operators, the K component of the rank-K tensor being denoted by SK,, KK

K - ) SK-,cRKoc s

(2.3)

where RKK is the counter part, the coordinate-space tensor. By taking the matrix element of (2.3), we get') (Pf ;

kfI M I vi ; ki) _ Y_ (-)'-'f(11 v i - vf) KK ) K

X

[C,(ki) E r=K-K

x

cK-r(kf)JKaFK,(E,

cos 0),

2.4

where P's are the z-components of the deuteron spin, ki and kf are the initial and final momenta and ki and kf are their solid angles. The quantity K is K for K = even and K + 1 for K = odd. In (2.4), the geometrical factor of the matrix element of SK -,, appears as the CG coefficient and that of RKK which is described by the K component of the rank-K tensor in the ordinary space is constructed by C,(ki) and C,.(kf) . The remaining factor FK,(E, cos 8) is invariant under rotation of the coordinate axes and thus called the invariant amplitude. The amplitude is designated

Y. Iseri et aJ. / Spin-dependent interactions

57 8

by , which is the rank of the associated spin-space tensor operator, and is a function of the c.m. energy E and the scattering angle 0 but, in the following, we will skip these arguments for simplicity . Using (2.4), one gets, in the reference frame with the z-axis //k; and the y-axis// ; X kr, A=fFoo +A(3 cos2 ® -1)F2o +3 cos ®F21+fF22, B = f sin ®Ff 1 +f cos 0 sin 0F2o + ,/ù sin

OF2,

C = f sine 0F2o ,

D = - ' R siii OF, 1 +vr cos 0 sin 0F2o+f sin OF,, , E =~ Uoo -- f(3

0 -1)F2o - ~ cos OF,, -fF22

(2.5)

F,2

(2.6)

COS 2

Eq. (2.2) gives 1,0

Thus we can choose four independent amplitudes, i.e. one scalar, one vector and two tensor ones. Referring to the consideration on the second-rank tensor amplitudes given later, we will define the four amplitudes as U=2A+E= vfN Foo,

(2.7)

S

(2.8)

-D= f sin OF,,,

T« --- E + D = f sin 0(vI6_ cos 0F20+ F2,) ,

(2.9)

Tß = C+ .%(B+ D) cot ;0 = 2(v~F2o+ F,,) cos 2 20,

(2.10)

where U is proportional to Foo and is the scalar amplitude, S is proportional to F and is the vector one and T,, and Tp consist of F2o and F2, and are the tensor ones . The scalar amplitude is associated with the scalar interaction in spin space. Thus U describes the scattering amplitude due to the central interaction in the sense of an effective interaction ; that is, it includes, in addition to the original central interaction, higher orders of spin-dependent interactions in the perturbation-theoretical sense, when they form the scalar in the spin space. Throughout this paper, the effective interaction is defined so as to include such higher-order effects. Other amplitudes, S, Ta and Tß , are similarly related to the respective spin-dependent interactions . As will be discussed below, Ta and Tß describe the effect of the TR -type tensor interaction and that of the TL -type one separately in the high-energy limit. Hereafter, we will call these amplitudes U - T,, the spin-space tensor amplitudes. The TR - and TL-type tensor interactions are defined as 1 ) TR = (S24, s) - R.,( R,

)) UR(

),

Ti- =( S2(59 .,) - ®2(s.,, ))Ut_( )'

(2.11)

(2 .12)

Y. Iseri et al. / Spin-dependent interactions

57 9

where S2 and R2 are the second-rank tensor operators constructed by their arguments, s, and being the deuteron spin, its space coordinate from the target and the corresponding orbital angular momentum . To define the two tensor amplitudes so as to describe the scattering by the TR -tensor interaction and that by the TL-tensor interaction separately, we need some guidelines for defining these amplitudes . The guideline has been proposed in ref. ' 2) and we will follow their prescription . In the following, we will introduce the plane-wave Born approximation (I'WBA) and a classical concept, which will be used only as a guide. These app, rnximate treatments have validity at the high energies . However, the final justification is provided by the quantitative numerical calculations at given energies. The F WBA provides the following relation s ) for the TR -type tensor interactions ,/6-F20 = - F2,

(213)

for a projectile of any spin. This relation has been numerically examined for scattering of 7Li by SsNi and has been found to be valid even at Eaab = 20 MeV to a good approximation'). Due to (2.10) and (2.13), Tp = 0 for TR tensor interactions at high energies . To consider the property of the TL -type tensor interaction in a simple way, we will treat the operator L in the framework of the classical concept. The quantummechanical correction will be discussed later. The q component of the space tensor R2(L, L) in (2.12) is written explicitly as R2q = Y_ (11 mm' 12q)L,L,- . m

(2 .14)

Since, classically, L is perpendicular to the momentum, Lo = 0 we get tl,

(2 .15)

v; 0 ~ 1

(2.l6)

q5-1-

which leads to Pf -

for

( vfI S2,-q I v,) '* 0 .

The restriction on v in (2.16) means that, for the TL -type tensor interaction,

B=n=o .

(2.17)

Combining this to (2.9), one gets for a TL -type tensor interaction .

Therefore, when only the two types, TR and TL , are considered as tensor interactions, one will distinguish approximately -the effect of one type from that of the other type by using é,, and Tij ; i .e. T,, describes predominantly the effect of the

SSO

. lseri et aL / Spin-de, Y

ndent interactions

describes that of the TL-type ones. The quantumr u s a correction ") to (2.17); that is, the Ti-type ve on- er contribution to the matrix elements, B and nie e corrections are estimated by the I' ly small factor 1/L for each partial wave of ition, it v at the correction vanishes n si to between an into ent distortions a t the numerical calculation is quantitatively small. ly small an allows sA

k=alc l w"ies

s for "Ni to ets a

carried out to examine whether the define by (2.7)-(2.10) desdzribe scattering by the central interaction, the spin-orbit one, the TR-type e e I -type tensor one, respectively. -Me incident deuteron energy sen to be 400 and 7 eV, as typical intermediate energies. The relativistic tics is taken into account but the corrections due to the Thomas precession oren contraction ") are discarded for simplicity. Tie basic deuterontarget interaction is the folding one, where the inpui nucieon-taide` optical potentials the so-called wine- ottie type obtained by Dirac phenomenology 13); i.e. they are derived f the Schr®inger equivalent of the Dirac equation . For the proton it is given by Up( rp) = Uo+

Ep 1

Us -

1

2Ep

(U0

dB

2)_

- Us

1

2Ep

Vc( Vc + 2 Uo) + Upa,in

,

(3.1)

+Vc 2EpBrp d rp ( p lp) with B=(Ep +m+U,-Uo -Vc)l(Ep +m),

(3.2)

and UO(rp) =

with

VO.Î(XR)+ iWof(x °) ,

Vs(rp)= Vsf(XR)+iWsf(x;), f(x;)=(1+expx) -',

. xj =(rp_r;A'/3)/a'

(3.3) (3.4) (3 .5)

)°. iseri et al. / Spin-dependent interartïons

S

the Here, lrc is the Coulomb potential. The neutron potential is assumed t is as the proton one except for VC r ® for the neutron. More details, referred to ref. "). e the numerical values of the parameters, will of the folding potential of the deuteron, the spin-orbit potentials of the factors of which are abbreviated , the neutron, the o transformed l) . s

,P.0. & A C rd)P , t

i i

+0-S'

where is the proton-neutron relative coordinate a angular momentum. p~(r~

~~Crn) ,

s=51("

is the n) s

s"® :!!(

®

)

,, - part f the and the In the following, we will keep the first to term in the r. .s. ofeq. (3.6). e - s part of the first term corresponds to the us, discussed later. spin-orbit interaction of the deuteron . Effects of other terms will e wave function of the deuteron is given by the Reid soft-core potential '9) for potentials for the breakup states "). e the ground state and by the gaussianuse ofthe Reid potential for the breakup states will give corrections to the calculated observables . These corrections will be discussed elsewhere 2®). In the folding procedure, the variables of Up(rp) and U.(rn) are changed by the use of the multi and the deuteron-nucleus interaction thus obtained expansion') into p and consists of the central, spin-orbit and TR-type tensor parts. This interaction reproduces fairly well ') the experimental data of the cross section and the vector and tensor analyzing powers even without the virtual-breakup corrections, showing that the interaction is almost realistic. 3.1. SEPARATION OF EFFECTS OF TR- AND TL-TENSOR INTERACTIONS

Since the purpose of the numerical investigation in this subsection and the next is limited to examining if the spin-space tensor amplitudes describe the scattering due to each spin-dependent interaction separately, the virtual breakup of the deuteron is tentatively neglected. The present folding interaction has no TL-type tensor interaction in the hamiltonian and, thus, we will introduce phenomenologically a TL -type tensor potential, for which two kinds of form factor and many sets of potential parameters are examined. On the other hand, as will be seen later, the spin-orbit interaction produces strong tensor erects of the 1'i type on the scattering amplitude. Therefore, for the examination of the TL -tensor-potential effect, we will

582

Y. Iseri et al. / Spin-dependent interactions

discard the spin-orbit interaction to avoid the disturbance due to such TL effects of the interaction. Fig. 1 shows the magnitudes of U, S, Ta and Tp at Ed = 700 MeV as functions of scattering angle 0 up to 8 = 25°, to which the observables have been measured. The interaction used contains the TL interaction but not the spin-orbit one. The form factor of the TL potential is assumed to be a surface-gaussian type, )2 ]

(3.7) UL (R) =(VL+iWL)exp a [ z with VL = - WL =1 .0 x 10_ MeV, Ro = 4.0 fm and a =1 .4 fm. These parameters are determined so as to simulate I Tp I produced by the folding spin-orbit interaction in order of magnitude. To see the dependence of I Ta I and I Tp I on the TR and TL interactions, the calculation is compared in fig. 1a to the modified case I where the strength of the TR interaction is reduced by 50®îo through the D-state amplitude, while in fig. lb it is compared to the modified case 11 where the strength of TL is reduced by 50%. In case 1, the reduction of the TR strength decreases I Ta l by about 50% except for very small angles, while produces very little effect on I Tß l in most angles . On the other hand, in case 11, the reduction of the TL strength decreases I Tß I by about 50% except for small angles, while affects IT,, I very little. The latter indicates that the TL interaction hardly contributes to Ta and eq. (2.18) is valid in a good approximation. In the sense that the TR interaction predominantly affects the T,,, amplitude and the TL interaction similarly does the Tp one, T,, and Tp succeed in describing the

Fig. 1 . Magnitudes of spin-space tensor amplitudes U, S, T,, and Tß versus scattering angle 0. The curves are calculated for the elastic scattering by the "Ni target at Ed = 700 MeV . (a) compares the standard calculation by the folding potential minus the spin-orbit (LS) one plus the phenomenological TL one (solid lines) with the calculation by reduction of A by 50% (dotted lines) . (b) compares the standard calculation (solid lines) with the calculation by reduction of the strength of the T,, potential by 50% (dashed lines) . The T,, potential parameters are V,. = - W,, = 1 .0 x 10 -; MeV, R = 4.0 fm and a = 1 .4 fm.

Y. Iseri et al. / Spin-dependent interactions

58 3

scattering by the TR interaction and the one by the TL interaction, separately. Further, the approximate linear dependence of T,, and Tß on the strengths of their respective interactions indicates that T,, and Tp are a good representation of the TR and TL interactions, respectively. Moreover, ISI is quite small in both of figs . la and 1b. This of course happens because of the absence of the spin-orbit interaction. However, it will be easily understood that this small ISI itself is produced by the second order of the TR-type tensor interaction because, as fig. la shows, the reduction of the TR strength causes the decrease of ISI to about â of the original. This effect is quite similar to that observed in polarized 7Li scattering by nuclei at low energies 9), where the second order of the TR-type tensor interaction prods;ces are effective spin-orbit interaction and plays an important role in explaining the vector-analyzingpower data. Also, in fig. lb, one will find ISI to be independent on the variation of the TL strength . This may be explained by considering that the second order of the TL interaction has the matrix element only for the cases of 4v --- vf- v; = 0, :±2 due to eq. (2.16), when the spin-dependent interaction of the propagator is neglected. The scattering amplitudes for these dv have no vector component as is seen in eqs. (2 .1) and (2.5). Thus the second order of the TL interaction does not contribute to S in this approximation. The quantum-mechanical corrections to B and D do not change the conclusion because their effects are cancelled with each other. The amplitude U is expected to arise from the central interaction and remains unchanged for the reduction of the TR strength or that of the TL strength. As the other choice of the functional form of UL, the square of the form factor of the deuteron-nucleus spin-orbit interaction is assumed by taking account of the result of the nex: sulssection. Through the examinations for the functional forms adopted and the associated many parameter sets, the characteristic feature of the calculation observed in fig. 1 is commonly seen, although the linearity of the amplitudes to the strength of the respective interactions is a little bit poor for some parameter sets. At Ed = 400 MeV, the same natures are found for U, S, T,, and Tp as those at Ed = 700 MeV. 3.2. PROPERTIES OF U AND S, AND TL -LIKE EFFECTS OF THE SPIN-ORBIT INTERACT ION

In order to see the contribution of the spin-orbit interaction to the spin-space tensor amplitudes, the calculation of I UI, ISI, I T,, I and I Tp I are carried out by including the spin-orbit interaction. The TL-tensor interaction is neglected to avoid the competition between the TL interaction and the spin-orbit one in Tß . The results are shown for Ed = 700 MeV in fig. 2. The calculated I UI, I SI , I Ta I and I Tß I are compared in fig. 2a to the modified case III where the strength of the TR interaction is reduced by 50%, while in fig. 2b compared to the modified case IV where the strength of the spin-orbit interaction is reduced by 50%. In case III, the reduction of the TR strength decreases IT,, I by about 50% except for very small angles while other amplitudes are affected very little in most angles. On the other hand, in case

384

Y Is-ri et aL 1 Spin-dependent interactions

R& 2. Magnitudes of spin-s ce tensor amplitudes , %, and Tp versus scattering angle é) e curves are calculated for the elas scattering by the *5"Ni target at d ® 7 MeV. (a) compares the standard (dotted calculation by t folding tential (solid lines) with the calculation by reduction of ® by 5 lines) . Ib) compares the standard calculation (solid lines) with the calculation by reduction of the strength of the spin-orbit (LS) interaction by 501% (dashed lines).

V, a is scarcely affected by the reduction of the spin-orbit interaction . These features of T,, are very similar to those in cases I and II and indicate that T. is the

representative ofthe TR interaction even in the presence ofthe spin-orbit interaction. oreover, in e IV, the reduction of the spin-orbit interaction induces the reduction of SI y about 501% showing that S describes the scattering by the sin-orbit interaction. This reduction of the spin-orbit interaction also decreases I p to about 41 of the original except for small angles. Thus one speculates that the higher orders of the spin-orbit interaction produce the tensor amplitude To and presumably the second order makes the dominant contribution . Since Tß describes the scattering by the TL interaction as was discussed in the previous subsection, it will be concluded that the higher orders of the spin-orbit interaction, mainly the second order, produce the effective TL -type tensor interaction. Similar TL-tensor effects of the spin-orbit interactions are observed at Ed = 400 MeV, as is shown in fig. 3. These TL effects of the spin-orbit interaction will be explained in a way similar to that in ref. 3). In the folding model, the spin dependence of the second order terms ofthe nucleon-nucleus spin-orbit interactions is characterized by the operator ( - S)2, in both cases, with and without the virtual excitation . This operator is transformed to the TL-type tensor interaction according to the following identity, 3L2 (3 .8) (L - S)2 = (S2(S, S) - R2(L, L)) -1(L - s) + , which shows that the spin-orbit interactions possibly produce an effective TL-type tensor interaction as the second-order effect. Finally, referring to analyses in sect. 5, one will understand why the spin-orbit interactions can contribute to tensor

Y. Iseri et al. / Spin-dependent interactions

585

Fig. 3. Magnitudes ofspin-space tensor amplitudes U, S, T. and Ta versus scattering angle 0 The comes

are calculated for the elastic scattering by the S"Ni target at Ed = 400 MeV. Others are identical to t

in the caption of fig. 2.

analyzing powers; the key to understanding is this effective TL-type tensor interaction . e The magnitude of U is governed predominantly by the central interaction . reduction of the strength of the central interaction reduces the magnitude of U in e most angles and shifts the angular distribution of J UI toward larger angles. I UI. The results increase of the central strength causes just the opposite effects on for Ed = 700 MeV are shown in fig. 4, for example in the case ofthe 20% modification. The variation of the central interaction can also affect the effective spin-dependent interactions keeping their tensorial character. This effect appears in the figure and will be understood by considering the DWBA amplitudes of the spin-dependent interactions with the spin-independent distortion . Further, both figs . 2 and 3 show that I UI is influenced slightly by the reduction of the spin-orbit interaction . This will be interpreted as the effect of the additional central interaction induced by the higher orders of the spin-orbit interaction, as is seen in eq. (3.8). 3.3. EFFECTS OF VIRTUAL BREAKUP TO Q, S, T,,, AND Tp

To see how each effective interaction is modified by including the virtual breakup process, we will derive I U1, ISI, I T,, I and I Tß I from the CDCC calculation performed in ref. 13), where 3S,, and 3 D.r (J=1, 2, 3) states in the continuum are taken into account, up to 1 .0 fm- ' for the proton-neutron relative momentum . The calculated spin-space tensor amplitudes are shown in fig. 5 for Ed = 700 and 400 MeV. In both cases, all of the amplitudes are affected by non-negligible amount and, in particular, I UI, ISI and I TQI at large angles, ®> 15°, receive contributions of considerable

Y lseri et nL / Spin-de ndent interactions

586

Fig. 4. Magnitudes of spin-space tensor amplitudes U, S, T. and Tq versus scattering angle 0 The curves are calculated for the elastic scattering by the 5$ Ni target at Ed = 7 MeV. The standard calculations by the folding interaction (solid lines) are compared to those by the modified interaction, where the strength of the central interaction is increased by 20~l0 for the dotted lines and is decreased by 20®/® for the dashed lines. 58NMA5

Ed =400 taeV

Ed= 700 MeV

Ux10

Ux13

v

v

ITPIx1 (~ v

vY' v;

vV.

4

16

100,

IT, ' IX 102

" with breakup --- without breakup

20

- with breakup

0

W --- without breakup 10r

20°E)CM30'

Fig. 5. Contributions of virtual breakup to magnitudes of spin-space tensor amplitudes U, S, T,,, and T,, in elastic scattering by "Ni target . The interaction is the folding one and the virtual breakup is taken into account by the CDCC mcthod. (a) is for Ed = 700 MeV and (b) for Ed = 400 MeV. The solid lines include the virtual-breakup effect and the dashed lines do not.

Y. Iseri et al. / Spin-dependent interactions

58 7

magnitude. Since the dominant part of Tp is produced by the spin-orbit interaction when the TL interaction is absent as was shown in the previous subsection, t modification of I Tp I by the virtual breakup is attributed to the TL-lie e of e spin-orbit interaction through the breakup process. Because the -modification is rather minor compared to the original at most angles, the effective L inte induced by the breakup 3) is a small fraction of the entire TL-lie e of the spin-orbit interaction at these intermediate energies .

e realistic prescription for the numerical investigation of the physical ables in the present model may be the last case in the preceding section, that is,, to calculate the observables by taking account of the central, spin-orbit an T tensor interactions derived by the folding procedure and, at the e time, nsi e ing the effect of the virtual breakup of the deuteron to the continuum states . In this realistic case, however, it will also be interesting to investigate effects ofan additional TL -type tensor interaction on the observables, cause the contribution of this interaction to the observables has scarcely been clarified so far, particularly in the realistic case, although the interaction had been proposed many years ago'). Since the TL erect of the spin-orbit interaction is already taken into account, the ori n of the additional TL interaction is not clearly known at the present stage. lthin the framework of the present model, the origin is attributed to the supplement to the insufficient treatment of the spin-dependent interactions. We will go back to this problem at the end of this section. In the first stage of the calculation, we will tentatively neglect the virtual-breakup effect to save computer time. This will not omit the essentials of the investigations. The form factor of the interaction is assumed to be surface-gaussian type as in eq. (3.7). Several values of the depth parameters are examined within the restriction shows VLI = I WL I, other parameters being fixed to R o = 3.0 fm and a = 0.5 fm. Fig. 6 the calculated tensor analyzing power A,,y at Ed = 700 MeV for ! VLi=IWL I=5 .0x10-3 MeV where the four combinations of the signs of VL and WL are examined. The calculated Ayy for ®r 15° is quite sensitive to the signs of VL and WL . Among them, the negative values for both of VL and WL (fig. 6d) show preferable tendency ; that is, the valleys of Ayy at 15°- ® -- 25° are considerably filled by the TL interaction . As will be seen later, this improves the theoretical fit to experimental data. The cross section and the vector analyzing power receive very small contributions for all combinations of the sign of the depth parameters. These features are common for other values of the depth parameters. The sophistication of the calculation is enhanced by the use of the CDCC method 3 to include the virtual breakup of the deuteron to 3S, and D,J(J =1, 2, 3) continuum

Y 06 a

588

a / Spin-dependent interactions

cc of calculated tenser analyzing power A,, an signs of depth parameters of TL 'es are calculated for the elastic scattering by the `"Ni target at Ed = 700 Mev. The standaM cakulation by the folding interaction without virtual-breakup effects (solid lines) is compared to the calculations which include the additional T,,-type tensor interaction phenomenologically introduced. The shert dashed line ( 0, the dash-dotted one (b), the long-dashed one (c) and the dotted one (d) are the case for %7,_ > 0 and WI >0, the one for VL > 0 and IVL <0, the onz for VL < 0 and WL > 0 be one W 11 < 0 and Wu < 0, respectively, V, and It', being the real depth and the imaginary one. The magnitudes of parameters are I VIJ = il W1_1 = 5.0 x 10` Mev, RO = 3.0 fin and a = 0.5 fin.

states . Choosing the signs of I and 1VL to be negative and keeping the magnitudes

f the potential parameters the same as above, the calculation is performed for eV . The cross section and the vector and tensor analyzing powers at

700 MeV are displayed in fig. 7 . The calculated quantities, including the virtual

breakup but not the TL interaction, reproduce most of the experimental data .

owever, the TL contribution improves the fit to the data of A,, around 0 = 20'

without changing the success in the cross section and the vector analyzing power.

is result may encourage one to get better agreement by adjusting the potential

parameters in more detail . The large effect of the TL interaction only on Ayy can be explained as follows. Using the spin-space tensor amplitudes,

9 Ay = A

=

11 U12

2-%1290-

3u +

3

lm

+3

1(

IS12

012 + +81T

U-2Tj3+

4 I T0,12_ 4 vr2 Re (T* a T13 ) 20 sin sin 0 11 1

(12)

f sin 0 T- ) S* 1 '

e3 Re (UT*)+ Re (UT*) 0 3 sin 0 2 T0,12+IS12+ S020 1

3 sin 0

3

(4 .1)

13 12

Re (T,,T*) 0 ), J-

(13)

Y. Iseri et al. / Spin-dependent interactions

589

1.

0

Fig. 7 . Effects of TL -type tensor interaction on cross section or (a) and vector and tensor analyzing powers A, and A (b) in elastic scattering by 5'Ni target at Ed = 7 MeV . e solid lines are for the CDCC calculations by the folding interaction, which include the virtual-breakup effect. e dashed lines are calculated by taking into account the additional TL -type tensor interaction, the parameters of which are VL = WL _ _ 5.0 x 10-3 MeV, Ro = 3.0 fin and a = 0.5 fin. The experimental data are taken from ref. `-' ).

The spin-space tensor amplitudes U, S, T~ and Tp for the present case are displayed in fig. 8, where the largest amplitude in most angles is U and the next one is S and, on the other hand, the effect of the TL interaction is appreciable only in I Ta I at larger angles . Combining these features of the amplitudes with eqs. (4.1)-(4 .3), the following will be understood reasonably. One of the leading terms of A.,,, is proportional to Re (UT,) which is easily influenced by the TL interaction through Tß. On the contrary, A}. does not contain Tß il. the dominant term though Tß appears as the form of Im (TsS*), which is the second in magnitude, and thus Ay is less affected by the TL interaction compared to Ay,,. Finally the dominant term of tT is I UI2 , compared to which, I Tp(` is very small. These circumstances appear more clearly in similar calculations at Ed = 400 MeV. Because the energy dependences of the parameters are unknown, the same parameter values for Ed = 700 MeV are applied for the TL potential at Ed = 400 MeV. The results are shown in fig. 9. Again one can see the large contribution of the TL interaction in Ayy and the lesser one in Ay , while the effect on ~ is almost negligibly small . The effect on Ay is larger in this case than at 700 MeV. In particular, the TL contribution improves the fit to the data of Ayy at the valleys around 0 =19° and L6°. "7he feature of the TL contribution is to make these valleys in A,.,, deeper and a similar but less remarkable tendency is seen in the calculated A,, . The figure

Y. iseri et al. / Spiti-dependeni interactions

Fgig. & Effects of T, -type tenser interaction on magnitudes of spin-space tensor amplitudes U, S, T and T,, in elastic scattering by 5 'Ni target at E, = 700 MeV. The solid and dashed lines are explained in the caption of fig . 7.

displays the additional calculation which uses larger magnitudes of I Q and I ELI ; that is, the previous values are multiplied by 1 .5. The tendency of making the valleys deeper is enhanced by increasing the TL strength . The effects on the spin-space tensor amplitudes at this energy are shown in fig. 10 for the same parameters as those at E d = 700 MeV. The effect is remarkable only in I T"ß I at 0 19*. Since the additional TL interaction can give such important contributions to the observables, we will proceed to the discussion ofthe possible origin ofthe interaction. In the numerical calculation, we have discarded several terms in eq. (3.6) which involve the transformation of the nucleon-nucleus spin-orbit interactions to the deuteron-nucleus ones. Among these terms, however, the U'-*O , L - s' one will produce a TL-type tensor effect on the scattering amplitude. In the multipole expansion, the O. dominant contributions of QpA "jQ and UnA '- (r,,) are additive in U'--O - for transitions between different-parity states of the pn relative motion and s' causes a transition between the singlet and triplet states . Therefore, the U'--° .L - s' term gives rise to the breakup of the deuteron to P, states in the continuum as being dominant, which will produce an effective second-rank tensor interaction in the elastic channel as the result of the second-order virtual excitation . This will be understood by the simple analogue of the effective "L-tensor interaction due to the second order of the spin-orbit interaction, which was discussed in the preceding section. That is, the spin dependence of the effective interaction due to the two-step virtual excitation I

Y. Iseri et a0. / Spin-dependent interactions

59 1

anal-Zing Fig. 9. Effects of TL-type tensor interaction on cross section rr (a) and vector and ten powers A, and A,_,. (b) in elastic scattering by S "Ni target at Ed = 400 MeV. e solid and dashed lines are explained in the caption of fig. 7. The dotted lines are calculated by the following EnetieF-s of the TL -type tensor interaction; VL = WL = -7.5 x 10-3 MeV, = 3.0 fm and a = 0.5 fm. e experimental data are taken from ref. ").

by Us-°- s' will be characterized by the operator (L - s')` and (L - s°) 2 = -(S2(s, s) - R,(L, L)) - ;(L - s)+1L2 .

(4 .4)

Thus the second-rank tensor interaction induced by the virtual breakup to 'P, states by the L - s' interaction can be treated as if it is the usual TL-type tensor interaction . During the preparation of this work interesting reports have been published 21 ), which evaluate the erect of such singlet breakup contribution on the observables by the two-step model with an adiabatic closure approximation in the deuteron elastic scattering at intermediate energies . At Ed = 400 MeV, one of the effects is characterized by the appearance of the deeper valleys around at 0 =19° and 26° for both of A.. snd A, compared to the case without this effect. As is seen in fig. 9. a similar tendency is produced by the TL interaction in the present calculation. Therefore, the virtual breakup due to the L - s' interaction will be an important candidate of the origin of the phenomenological TL-type tensor interaction. To examine the validity of such speculations, straightforward coupled-channel calculations which include the L - s' term are now in progress. However, other possibilities of the origin of the TL interaction, for example the relativistic effect "), cannot be excluded at the present. These will be discussed elsewhere. In the present analysis, we have not tried to look for the best-fit parameters of the TL interaction by their full variation. To do this without ambiguities, we need

Y veri ei at. / Spin-dependent interactions

Rja. and

.

cts of T;,,-type tenor interaction on magnitudes of spin-space tznsor amplitudes U, S, T«,, e solid and dashed lines are explained in the caption o g. 9.

in elastic scattering by 5""Ni target at Ed = 4ev.

a icula ly for different kinds of observables. From this viewpoint, it will be worthwhile to investigate the effect of the TL -tensor interaction on other observables t eoredcally in order to stimulate the experimental investigations . Tak ing this into consideration, we will discuss in the next section which observable is appropriate for examining of a particular spin-dependent interaction. 0

tributions of spin-dependent interactions on polarization observables Since the spin-space tensor amplitudes, U, S, Ta and Tß describe the scattering by the central, the spin-orbit, the TR-type and TL -type tensor interactions, respectively, one will be able to identify the effect of each spin-dependent interaction in the observables by describing them in terms of these spin-space tensor amplitudes. ere we will study for the analyzing powers and polarization transfer coefficients of the polarized deuteron beam what kinds of observables are convenient for examining of the effect of a particular spin-dependent interaction. Since the central interaction is usually stronger than the spin-dependent ones, the magnitude of U is larger than those of S, T,, and Tp in most angles. Hence the terms of US*, UTI and UTß are predominantly larger than the quadratic terms of û, T,, and Tp . This matter will provide an approximate scale of the effect of

Y. Iseri et aL / Spin-dependent interactions

593

spin-dependent interactions. In the following, the numerical calcul t ns on the folding interaction which neglects the virtual breakup effects. i the TL -type tensor interaction, the phenomenological TL one in the revio is additionally introduced when necessary. e parameters are = -5. x 5.1 . THE SPIN-ORBIT I NTE

-3

eV ,

O=

3. f ,

= .5

r v ss

a ti ate

.

TI

e effect of the spin-orbit interaction is exhibited through the s e tude n thus the vector analyzing wer , is measure f this i te because the dominant te o . is the term, is see i eq. (4.2). This t is demonstrated in .11, here , is quite sensitive to the variatio oft e interaction while insensitive to the variation o the T -ty tensor i te it is affected very little when the phenomenological L interaction is additionally introduced . Some linear combinations of the polarization transfer coefficients are ex cted to exhibit the effect of the spin-orbit interaction selectively, although, eac c ponent o the combination has complicated dependence on multiple kinds of

Fig. 11 . Effects of spin-dependent interactions on vector analyzing power A, in elastic scattering by S"Ni target at Ed = 700 MeV. The calculation by the folding interaction (solid line) is compared to the one by reducing the strength of the spin-orbit (LS) interaction by 50% (dash-dotted line), to the one by reducing the D-state amplitude AD by 50% (dotted line) and to the one by including the TL-type tensor interaction (dashed line) . The potential parameters of the TL interaction are V,. = WL = -5.0 x 10-3 MeV, Ro = 3.0 fm and a = 0.5 fm.

~;

interactions .

e e t e

Iscr~ et a~ / Spàn-depeaadc®at dnteractioras

e ning

as

antities w ic ~ e phasize t e spi -o bit e ect are, or exa

le,

(5 .

)

(5 .3)

(5 .

titi

a

~

~a ~ ,

~~

vm ® 8 ~ ®.,. m~

~ ,~ ~,

- e~ m_-

~IT~~' c®t ere, ~~ ~ -

)

.

(s.~)

;._ are the co plementary quantities because the for er rove es the infor anon of the real part of aS~ and the latter that of the aginary part of the sa e uantity . eir co bination gi~~es the magnitude and base of ~ ~ separately. Also, it is interesting that the linear combination of the olarization transfer coe cie ts and the vector analyzing power, d{ r,. + ~._ + 3A,. in e . (~. ), re acts rather pure spin-orbit effect, that is, independently frown the e act of other spin-dependent interactions . These will be useful in the determination of the details of the spin-orbit interaction . j.? .

THE

' ~ and

T~-TYPE

TENSviè

_,. ~

INTERACTION

°The tensor analyzing power t1C is well known as a good measure of the D-state admixture of the deuteron and thus of the TR-type tensor interaction . This feature can e understood by the fact that the dominant term of AXZ is proportional to e ( UT~) as ~

t~ x ~=~I~e

N

U+3v SCO$

® - 2T~-~-

1 Ta Tâ . ~ sin 8

5 .6

The other two tensor analyzing powers Akx and Avv contain both Re ( UTq) and e ( UTß ) as the dominant terms and thus are easily influenced by both the TR -type tensor interaction and the ?'~-type tensor one. ~-iowever, by linear combination of these analyzing powers, one can eliminate one of the dominant terms. The following

Y. Iseri et al. / Spin-dependent interactions

595

Fig. 12. Effects of spin-dependent interactions on tensor analyzing power A,- in elastic scattering by s8Ni target at Ed = 7 eV. e solid, dotted and dashed lines are explained in the ion of fig. 11.

X2, which is the well-known quantity, is o e of the o the TR-tensor interaction.

is e

asizes the effect

X2 =\/3' (2 ,, -12- U cot ® - T,,,

cos 2%f2- Te cot + 3%I T*` sin' ® +

(5.7)

The sensitivity of these quantities to the I)-state amplitude, A® , is shown in figs. 12 and 13 . They are strongly affected by a decrease of ® by 54% but affected very little by adding the pl;eno=nenological TL-type tensor interaction, except for ®> 22°.

Fig. 13. Effects of spin-dependent interactions on a linear combination of tensor analyzing powers X2 = f (2A .;,, + A_,.,.) in elastic scattering by S8Ni target at Ed = 700 MeV. The solid, dotted and dashed lines are explained in the caption of fig. l l.

Y. Iseri et at / Spin-dependent interactions

e following polarization transfer coefficients are expected to be good measures of the -state amplitude and the TR-type tensor interaction . K' (5.g) ~

1

J(

cot

- 2 s cot

-3

S)

},

(5.9)

(5.10) ere

is given by eq. (5.2), and 2~

Im ((U-2 B + 3N/!S cot )T*} .

(5.11)

s, the quantities (K- -) and z( ô~. + `) are the real and the imaginary parts of the e quantity, ( -2 / 3 N) T~* cot . Similarly, Yr and !( Y - K'~) are the real an imaginary parts o (\/-2/N) (U + 3\As cot ®- 2 Tp +(1/,12- sin O) T,,) T* , ., and --!,N an _ are the real and imaginary parts of (2/ N) x - 3\/7' S tan - 2 o + (1/%f2- sin ) T,) T* cot r. The measurement of these complementary parts will provide useful information not only for T,, but also for other related spin-space tensor amplitudes . e quantity ;.'(or "`) is not suitable for the investigation of the D-state admixture, because it contains two dominant terms, US* and UT*, and the effect of the -state admixture is mixed up with that ofthe spin-orbit interaction . However, the following linear combination with the vector analyzing power is expected to be a good measure of the D-state amplitude, I

N

Im{(VGU-3S cot ®-2V'2_Tß )Tq} .

(5.12)

Fig. 14 compares K :,' and ,::,2 +2A,. on their AD-dependence and justifies the above expectation . The magnitude of the latter quantity is approximately proportional to A® at most angles, while the effect of variation of AD is rather minor in K ;,Z . Experiments at low energies on this linear combination are now in progress 23). The polarization transfer coefficients K ~ti and K xZ will emphasize the erect of the D-state admixture, because they contain neither US* nor UT* . However, their linear combination given in the following is expected to be more sensitive to the -state amplitude because it does not contain S even as minor terms.

1(

Kyy+Kxi= 2~ Re U-2Tp + 1 Ta T* . (5 N -,/2- sin

.l3)

Y. Iseri et al. / Spin-dependent interactions

597

Fig. 14. Comparison between -motor to tensor polarization transfer K , and its linear combination with vector analyzing power K ;' + ;A ® in effect of D-state admixture, in elastic scattering by 5'Ni target at Ed = 7 MeV. The solid lines are calculated by the folding interaction, while for the dotted lutes the D-state amplitude A® is decreased by 50°/®. 5.3. THE TL -TYPE TENSOR INTERACTION

As was discussed in the previous section, the TL-type tensor interaction, phenomenologically introduced, gives important contributions to the tensor analyzing power A,.,. and thus further investigations will be required on the effect of this interaction . The information of the observables which enhance the contribution of this TL interaction, which also includes the TL effect of the spin-orbit interaction, will be useful in planning of experimental investigations on such interactions . Noticing that the TL interaction affects the observables through the amplitude T.., the following will be candidates for a good measure of the TL interaction . X3 = -2

=

cos ®Axx + (3 + cos O )A_,_, - 2 sin OA.,,

8 Re -UT*0 - ~Tß12+g1 SI2+ sin N (

e

T«,, T*

(5 .14)

and K2 = (1-cos

®)K - ,+(1 +cos ®)K z

T,, Tß cos ® . = -4 lm U N sin «

(5.15)

Y. Iseri ei al. / Spin-dependent interactions

15. Effects of spin-dependent interactions on a linear combination of tensor analyzing powers 2 m OA, + 13 +cos &)A,, - 2 In ", in elastic scattering by 5'Ni target at E. = 700 MeV. The ssolid, dotted and dashed lines are explained in the caption of fig. 11 .

V and the reduction of the -state amplitude but they are st stable against retiebly affected by the presence of the TL interaction for 0 :,--- 15". e linear combination which emphasizes the effect of the TL interaction can also be constructed by the use of the tensor-to-tensor polarization transfer ve predictions are justiRed numerically in figs. 15 and 16, where

Fig. 16. Effects of spin-dependent interactions on a linear combination of tensor to vector polarization transfers K2 - cos O)K .,,,. + (I + cos O)K -,'- in elastic scattering by "Ni target at Ed = 700 MeV. The solid, dotted and dashed lines are explained in the caption of fig. I I -

599

Y. dseri et al. / Spin-dependent interactions

coe cients . An example is y_`, ® 2

1 + 3 cos 0)A,,y + 1

® 4c®s

® IZe

T~+ I TJSI2 -31S1 2 -

sin

3 T~To + cos

T~

.

5.16)

e measurements of these quantities will be helpful in the examination if the TL interaction has a real existence and, when it is accepted, ill useful in the determination of the parameters of this interaction.

e present work investigates the elastic scattering of the deuteron y a s inl s nucleus at intermediate energies . e scattering amplitude is decomposed y the invariant-amplitude method into spin-space tensor amplitudes, such the la`, the vector and the two second-rank tensor amplitudes i sin space. e present theory predicts these amplitudes to represent respectively the contributions of the central interaction, the spin-orbit one, and the TR- and TL-type tensor on to the scattering amplitude, in the sense of the effective interaction. e separation of the effects of the two tensor interactions is derived in the high-energy limit. Such a correspondence between the spin-space tensor amplitudes and the related intereV by introducing the actions is numerically justified at Ed = 400 and 7 in addition to the interactions from the folding phenomenological TL-interaction model. Referring to these results, one finds that the second order term for the spin-orbit interaction produces effects on the scattering amplitude similar to those of the TL-type tensor interaction. This effect is usually taken into account in the calculation by the use of distorted waves which contain the spin-orbit interaction. The virtual 3), breakup of the deuteron by the spin-orbit interaction, as proposed in ref. contributes to the above TL-like effect . However, at most angles, it is only the minor part of the total TL erect of the spin-orbit interaction, the major part being produced by this interaction in the deuteron ground state. The effect of the phenomenological TL-type tensor interaction on the observables is studied at 400 and 7001VIeV by taking into account simultaneously the virtual breakup effect by the CDCC method. The calculated observables, in particular the tensor analyzing power Ayy , are improved appreciably in comparison with the experimental data at both incident energies by the same potential parameters . At Ed = 400 MeV, the feature of the contribution of the TL interaction to A,, and A,,,, is quite similar to that obtained in ref. 2' ), where the virtual breakup to the singlet states is taken into account by the adiabatic two-step approximation . Thus, this mechanism is speculated to be an important candidate for the origin of the phenomenological TL interaction.

Y. Iscri et al / Spin-dependent interactions

he theory also predicts that many linear combinations of the analyzing powers an for polarization transfer coefficients exhibit exclusively one of the effects of the in-orbit interaction, the TR-type tenser interaction (or the D-state admixture) and the ~ -type tenser interaction. he validity of this prediction is confirmed by the erical calculation at 7 eV for some typical cases. To investigate the TL-type ion, the comparison of X3, and 2-(l + 3 cos O)A,, - K ` between and the measured will be articularly useful. î .;9

e aut ors thank Professors M. Kawai and R.C. Johnson and Dr. J . A. Tostevin for iscussions . One of the authors (M.T.) thanks Professor H. Karwowski for valua le communications on his exneriment. eferences 1) 2) 3) 4) 5)

G,R, Satchler, Nucl, Phys, 21(1960) 116 PW. Keaton Jr, and D.D. Armstrong, Phys . Rev. C& (1973) 1692 A.P. Stamp . Nucl, Phys. A159 (1970) 399 P. Schwandt and W. Haeberli, Nud. Phys. AIIO (1968) 585 ; A123 (1969) 401 AW loannides and R.C. Johnson, Phys. Rev . C17 (1978) 1331,

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10) M. Tanifuji and Y. Iseri, contribution to the 12th Int . Conf. on few body problems in physics. Vancouver (1989) W 11) Nt Tanifuji and K. Yazaki, Czechoslovak J. Phys. 839 (1989) Suppl. p. 67, Prog. Theor. Phys. 84 (1990) 1160 12) M . Tanifuji, H. Kameyama, M. Kamimura, Y. Iseri and M. Yahiro, Phys. Lett. B217 (1989) 375 13) M. Yahiro, Y. Isen, H. Kameyama, M. Kamimura and M. Kawai, Prog. Theor. Phys. Suppl. 89 (1%6)32 14) Y. Ised, H. Kam,sma, M. Kamimura, M. Yahiro and M. Tanifuji, Nucl. Phys. A490 (1988) 383 15) B.A. Robson, The theory of polarization phenomena (Clarendon, Oxford, 1974) p.62 16) R.C. Johnson, private communication 17) H. Kameyama and M . Yahiro, Phys. Lett . 8199 (1987) 21 18) M. Tanifuji, Proc. of the Tsukuba Symposium on polarization phenomena in nuclear reactions, NSSRP-31 (1980) 154 19) R.V. Peid, Ann, of Phys. 50 (1968) 411 20) Y. Iseri, Y. Aoki, M. Tanifuji and M. Kawai, Contribution to the 7th Int . Conf. on polarization phenomena in nuclear physics, Paris (1990) ; Y. Iseri, M. Tanifuji, Y. Aoki and M. Kawai, Phys. Lett. 8265 (1991) 207 21) J.S. Al-Khalili, LA. Tostevin and R.C. Johnson, Phys. Rev. C41 (1990) R806; Nucl . Phys. A514 (1990) 60

22) N. van Sen, J. Arvieux, Ye. Yanlin, G. Gaillard, B. Bonin, A. Boudard, G. Bruge, J.C. Lugol, R. Babinet, T. Hasegawa, F. Soga, J .M. Cameron, G.C. Nailson and D.M. Sheppard, Phys. I.ett. 8156 (1985) 185 and private communications 23) H . Karwowski, privaia communication