PHYSICAL REVIEWER C UOLUME 49, NUMBER 2

Rainbowlike efFects in ('He, a) reactions

FEBRUARY 1994

V. Burjan, J. Cejpek, J.Fojtu, V. Kroha, and I. PecinaNuclear Physics Institute, Academy ofSciences of Czech Republic, 250 68 Rez, Czech Republic

A. M. Mukhamedzhanov and N. K. TimofejukNuclear Physics Institute, Uzbek Academy ofSciences, 702132 Tashkent, Uzbekistan

(Received 24 July 1992; revised manuscript received 6 April 1993)

The angular distributions of the elastic scattering and ('He, a) cross sections for 37.9 MeV 'He parti-cles incident on target nuclei ' C and ' C were analyzed from the standpoint of possible presence of therefractive effects in one-nucleon transfer reactions. The analysis of experimental data is given in theframework of the distorted wave Born approximation using external microscopically calculated formfactors. The presence of the rainbowlike mechanism in both elastic and reaction channels was observed.The effect is apparent mainly for transitions where strong orbital mismatching occurs.

PACS number(s): 24.50.+g, 25.55.Hp

I. INTRODUCTION

It is well known that in the elastic scattering of Heand a particles on lp-shell nuclei (for A ) 10 and incidentenergies of tens MeV/nucleon) rainbow effects are ob-served [1,2]. These effects clearly manifest optical prop-erties of the quantum scattering of light ions and allowone to get information about the nuclear optical potentialin the nuclear interior. The same effect was found incharge-exchange reactions ( He, t} [2] which can also beconsidered (in the isospin formalism) as elastic scattering.

Let us briefly outline characteristic features of the"rainbowlike" mechanism: (1) In the angular distribu-tions of elastic scattering the domain of the Frauenhoferinterference is followed by a broad bump with an ex-ponential descent towards large angles. This behavior isdescribed by the Airy function [3]. (2) The form of theangular distribution about the main maximum does notdepend on the imaginary part of the optical potential—only its absolute value changes. (3) This whole domain isdetermined essentially by the far-side component of thereaction amplitude. For the rainbowlike mechanism, apartial transparency of the nucleus is necessary also forsmaller orbital momenta than values corresponding tothe surface domain 1=kR, [where k is given by the ener-

gy of the particle in the corresponding reaction channeland R, is the strong absorption radius defined as R, =1.5(A ,'"+A,'"}].

It has been shown that for reactions with good match-ing of the initial and final angular orbital momentum themain contribution to the reaction amplitude comes fromthe external surface region of the nucleus-target interac-tion [4]. The contribution from the nuclear interior canbe usually neglected because of the strong volume absorp-tion for such incident energies.

The extent of refractive effects in the case of severelymismatched reactions such as ( He, a), when the contribu-tion of the nuclear interior cannot be neglected [4], is notfully understood. In this case one can hope that if rain-bow scattering is seen in elastic scattering then it could

be observed in the nuclear reaction, too. This expecta-tion is based on the following qualitative consideration:If there exists a rainbowlike elastic scattering mechanismin the entrance channel He+C, it gives evidence, in theclassical interpretation, for an amplification of the contri-bution of some trajectories (caustic) corresponding to acertain interval of reaction angles. It also indicates thatthere is some transparency for lower partial waves asthere occurs a typical interference characterized by anAiry function. If it is so in the H+C channel, and if weknow that there exists a nonnegligible influence of aninner domain of the nucleus onto the ( He, a) reactionamplitude (because of strong momentum mismatching)[4], then it is interesting to investigate if these cir-cumstances manifest themselves in the mechanism of thisreaction, too.

The aim of this work was to test this idea by measuringthe differential cross sections of the ( He, a) reaction onthe '3C and ' C targets at energies E=38 MeV (in thelaboratory system}. At the same time we measured theangular distributions of the elastic scattering of He onboth nuclei. The rainbow effect was observed both inelastic scattering and transfer reactions.

The rainbow efFects in the ( He, a} reactions in such anenergy region could be seen for the first time in our pre-liminary communication [5] for ' C( He, a) ' C(g.s) only.A study [6] performed recently at the energies 50 and 60MeV seems to confirm our conclusions. In Ref. [2] an at-tempt to interpret the results of the paper [7] in terms ofrainbow scattering is mentioned which, however, was notmade in the original paper. The first observation of therainbow effects in a one-neutron transfer was in a heavyion reaction ' C( C, ' C)' C [8], but it seems that furtheranalysis for this case is needed. Some features similar tothose observed by us might be identiied in the reaction' 0( He, a)' 0 [9],namely some traces of rainbow for theground state transition which disappear as one goes tothe highly excited states with better momentum match-ing. This previous work was a part of a series of studiesmade for several light targets from Be up to Fe, see e.g.

0556-2813/94/49{2)/977(8)/$06. 00 49 977 1994 The American Physical Society

978 V. BURJAN et al.

[10], [11], and [12], but in all these cases nobody haslooked for rainbow effects.

For analysis of reactions with a large Q value it is veryimportant to have reliable form factors (overlap integrals)to determine the reaction amplitude. Therefore a de-tailed discussion of the calculation of these quantities ispresented in this work.

In Sec. II the experimental method is briefly described.An overview of the analysis, techniques, and theoreticalmethods to calculate the overlap integrals are given inSec. III. In Sec. IV the experimental results and theiranalysis are presented.

II. EXPERIMENTAL METHOD

The experiments were performed on the He beam withthe energy 37.9 MeV using the isochronous cyclotron U-120M of the Nuclear Physics Institute of Academy ofSciences of Czech Republic at Rei. The angular distribu-tion measurements were carried out by means of fourE-hE telescopes placed in a target chamber. Both hEand E silicon detectors were the surface-barrier type ofthicknesses 150 and 2000 JMm, respectively. These detec-tors were cooled to —35 'C. The on-line massidentification of reaction products was accomplished us-

ing a special fast digital peripheral processor. Errors inthe angular measurements were reduced by stabilizingthe beam position on the target. This stabilization wasachieved with a correcting feedback system which con-sisted of microchannel detectors inside a diagnosticchamber. These detectors allowed the visualization ofthe beam trace on a screen. The picture of this tracefrom a TV camera was digitized and mathematically pro-

cessed by a computer which then provided the error sig-nal for the correcting magnet. Details of this method arebriefly described in Ref. [13]. The average error of the re-action angle was kept to about 0.1'.

The thickness of the self-supporting targets was care-fully measured by the shift method using the change inenergy of collimated a particles of the Th nuclide aftertransition through targets. The central area of each tar-get was systematically scanned and the thicknesses werefound to be 0.165 mg/cm for ' C and 0.180 mg/cm for' C. The average errors of measurements did not exceed5%. The ' C and ' C targets were enriched to 76.4% and80%, respectively.

The evaluated average error of absolute differentialcross sections which included uncertainties due to thetarget thickness, charge measurement, solid angle deter-mination, dead time counting, statistical errors, etc. wasof the order of 5% for laboratory angles up to -50' andless than 10% for other angles.

III. OVERLAP INTEGRALSAND VERTEX FORM FACTORS

The distorted wave Born approximation (DWBA} am-plitude of the nuclear transfer reaction A (a, b}B, whereA=B+n, b=a+n, and n is a transferred neutron, isgiven by the expression

The 4',-+' and 4f ' are the distorted waves in the en-

trance and exit channels, respectively; I'f; is the form fac-tor

Ff, =(Bb~V,„~An)=(+J M 'pj I ~V ~% J ~ 'pj M )

(jm, JaM~ ~ J„M„)(1m&—,'e~jm )(j 'm', J,M, ~ Jl M&)ll' jj'

m, m,

X (1'mr' ,'0 ~j'm ')I„a—„(1'ra„)@b,„(1',j ', r,„)Y&~ (ra„)Y*, , (r,„),

where V,„ is the interaction potential of particles a and n,

r; =r, —r. , r, being the radius vector of the center ofmass of a particle i, r=r/r; I„~„and 4b,„stand for theradial parts of the overlap integral (%z M ~%J ~ ) and

the "potential overlap" integral (%J ~ ~ V,„~%J~ ) re-b b a a

spectively, %JI is the internal wave function of the nu-1

cleus i with the spin J; and its projection M,-. The in-

tegration is performed over all coordinates of the nu-cleons of nuclei B and a, respectively.

The calculation of these quantities is the crucial pointin the theory of transfer reactions. The well-known ap-proximation in the standard DWBA is used (for brevitywe omit quantum numbers 1,j in overlap integrals}:

(3)

where y(r) is the radial single-particle wave function ofthe nucleon bound state in a nucleus X calculated in theWoods-Saxon potential and S is the spectroscopic factorof that state. To find the parameters of such potential thewe11-depth procedure is usually used. However, in suchan approach the significant dependence of the overlap in-

tegrals on the geometrica1 parameters of the Woods-Saxon potential is hidden. The many particle characterof the problem is ignored and the Pauli principle is nottaken properly into account. Additionally, there is aprincipal shortcoming of such a procedure: the overlapintegral is calculated up to the normalizing constant only,which then calibrates the absolute value of the differentialcross section. Usually, a search for this constant is themain task of the DWBA analysis of experimental data.The reliability of such spectroscopic factors depends on

49 RAINBOWLIKE EFFECTS IN ( Hc,a) REACTIONS

Ixr (r)-Cxr~exp( ar~r)/r, r~ (4)

where ~z„=2p z„cz„.c&„ is the binding energy of particlesFand n in the bound state (Fn), pr„=mrm„l(mr+m„)and Cx~„ is the asymptotic normalization coeScient(ANC) which is proportional to the vertex constant (VC)Gx„„ for the neutron removal from the nucleus [14]

IGxr„l =m{filpr„c) Cxr (5)

Gxr„defines the normalization of the external regioncontribution to the reaction amphtude. This quantity,which is the fundamental nuclear characteristics, can beextracted, in principle, in a model independent way. Forexample, it could be deduced from sub-Coulomb nucleontransfer reactions when the contribution from the nuclearinterior is negligible and the cross section is proportionalto IGzr„ I only, from heavy-ion reactions at low energies

I

the correctness of the approximation used in the deriva-tion of Eq. {3)which may be, however, quite dubious inthe nuclear interior.

In the external region which gives the dominant contri-bution to the cross section at small angles the overlap in-tegral can be approximated by its asymptotic behavior:

Ixrn {q) = 4e—rnx

Gxr. {q)~{q'+&'r.»

Ixr„(r )=—2Prn

X I dqq j,(qr)Gxr„(q)l(q +sr„),0

(7)

N„resulting from the identity of nucleons is equal to theatomic number of nucleus X according to isospin formal-ism. To determine Ixr„(r) or Ixr„(q) it is suScient tocalculate the vertex form factor Gxr„(q) which is definedby the expression [14]

when the absorption is strong enough or by extrapolationof the differential cross sections to the nearest singularitypoint in the complex cost plane {8is the reaction anglein the c.m. system) [15]. The problem of calculation ofthe overlap integrals in the nuclear interior has not yetbeen solved.

In our DWBA analysis we used the microscopicmethod. The idea of the method is based on the connec-tion between Ixr„(q) and the vertex form factors Gxr„(q)appearing in the dispersion theory of nuclear reactions[14,16]:

GXrn{q)={4~N )'" 2 (jmj'I&~&I J.~.){imIT'o ljmj)x~1/2) X(1/2H'f «jI{q~)1~m ( pJ M I1 r I pJ M ~ ' (8)

m, m. M

Here g&+, /2& and y&+, /2&, are the spin and isospin nucleonwave functions. It is worth noting that the on-shell valueof Gzr„(q =i~r„) is the vertex constant mentionedabove, Gxr„(iaN ) =Gxr„. To calculate Gxr„(q) weshould select the wave functions of nuclei X and Y. Inthis work we used the wave functions of the translational-ly invariant oscillator shell (TISM) model, which is themost suitable from the practical standpoint. The mainshortcoming of these functions is an incorrect asymptoticbehavior. For this reason they are usually not used forthe calculation of cross sections of transfer reactions,which are thought to occur at the nuclear surface. Nev-ertheless, using Eq. (8) we can obtain the overlap in-tegrals Ixr„(r) with a correct radial dependence forr-+ oo even in such a case [16]. This assumption followsfrom the presence of the nuclear potential Vr„ in Eq. (8)which damps the contribution to the integral over r' inthe outer region where the oscillator wave functionswould not be suitable. The correct asymptotic depen-dence of Ixr„(r) is provided by the existence of a pole atq = —a. in the integrand of Eq. (7) [14].

Let us consider the potential Vz„. It is the sum of NNpotentials and in the self-consistent problem it is neces-sary to take into account such potentials to calculateinternal wave functions of nuclei X and Y. In this workwe used TISM wave functions of Boyarkina [17] whichare not the sohitions of the Schrodinger equation with acertain NN potential. However, the proper choice ofsuch a potential is of great importance because, as follows

from calculations [16], difFerent NN potentials givedifferent values of Gx&„. In our work we used the poten-tial M3YE [18] containing central, spin-orbit and tensorparts optimized according to matrix elements of E11iot[19]. Therefore, such a potential is closely related toself-consistency. This potential was reported to be thebest of all the tested ones in describing all reliably estab-lished VCs for the lp-shell nuclei [16]. Similar values ofVC, IG, 3 „ I

=0.39+0.02 fm were obtained for exam-Il

pie, for the virtual decay ' C~' C(g.s.)+n from variousexperiments. We obtained from Eq. (8) with the potentialM3YE, for q=iar„, the value IG&3» I

=0.41 fm. Areliable value of this quantity for the vertex a~ He+nis not known. The range of various estimates is between5.7 to 17.6 fm. Present calculations of VC by the solutionof Yakubovsky equation with the XN potential from thecomposed quark bag model gave the value IG 3 I

=5.7a Henfm [20]. Our calculation with the M3YE potential givesI G 3 I

=6.9 fm, which is in reasonable agreement withthe preceding value. Therefore in our DWBA analysisdescribed below, the same potential M3YE was used forcalculation of both overlap integrals.

In summary, in the DWBA analysis of experimentaldata the overlap integrals containing both light andheavy vertex in the form factor of the reaction amplitudewere calculated microscopically according to Eqs. (5) and(7). The wave function of TISM and the NN potentialM3YE were used. Such an approach is free of any fitting

980 V. BURJAN et al. 49

parameters, like the geometrical ones of a Woods-Saxonpotential and spectroscopic factors which calibrate theabsolute value of differential cross sections. Use of mi-croscopic form factors allows us, to some extent, to takeinto account the many particle effects as well as theeffects of antisymmetrization. The correct asymptoticnormalization of overlap integrals ensures an accuratedetermination of the contribution of the external regionto the reaction amplitude. In our approach, the only"free" parameters in DWBA are those associated withuncertainties of the optical parameters in the entranceand exit channels.

IV. ANALYSIS OF THK EXPKRIMKNTAI. DATA

Angular distributions from 8' to 160' (lab) for thedifferential cross section of the elastic scattering and( He, u) reaction on ' C and ' C were measured using a37.9 MeV He beam. In the reaction channel, data wereextracted for the 0.00, 4.44, 7.66, and 15.11 MeV states in' C and 0.00, 3.68, and 15.11 MeV states in ' C. Analysisof the elastic scattering data was performed using thecode ECIS 79 [22] while DwUCK 5 [21], modified to allowthe use of the external form factors, was employed in the( He, a) studies.

overlap integrals with the correct asymptotic behaviorwere used in the analysis of the experimental data in aDWBA framework. Both integrals I(a Hen) andI(' C' Cn) were calculated with the same M3YE poten-tial. In Fig. 3 the radial dependence of "heavy" overlapintegrals is presented. As noted in the previous part,there are no free parameters connected with the structureof nuclei in this approach. The quality of the agreementbetween experimental data and the theory depends on theadequacy of the DWBA and on a proper choice of theoptical potentials. Unfortunately, we did not have exper-imental data for the elastic a scattering in the relevantenergy region. Therefore, we used several different po-tentials which covered both deep and shallow real centralpotentials from the literature, some of which are given inTable I as B1-B4. The potentials listed describe the ex-perimental a scattering by the ' C nucleus for alpha ener-gies of 56 MeV (Bl—B3) and 65 MeV (B4) [26]. The a-particle elastic scattering angular distribution does notexhibit an unambiguous rainbowlike feature at these en-ergies, but its presence cannot be excluded. In Fig. 2(b)

A. The reaction ' C( He, a)' C(g.s.)Q

2 Eiob=&7. 9MeV. .---- "

In Fig. 1(a) the angular distribution of the scattering isshown. The distribution is characterized by the rainbow-like behavior: oscillations at small angles and deepminimum followed by a wide peak which decreases grad-ually towards large angles. Different optical potentialsets were tested to describe elastic cross section. It wasfound that there exist several potentials which give al-most equally good descriptions of the cross section in theentire angular region. However, all of these potentialshad to incorporate both volume and surface absorptionterms, indicating a more complicated radial dependence.Potentials without such a combination are not able to de-scribe the angular distribution; in particular the "rain-bow" region of angles is poorly reproduced. Similar re-sults were obtained in Ref. [2]. A sample of two such po-tentials are given in Table I as set Al from Ref. [2] andA2 from the present work. The best results were ob-tained with the potential A2 which corresponds to theoptimal fit of the experimental data. The spin-orbit partwas neglected as unimportant in the effect on thedifferential cross section. This A2 potential gives the bestdescription of the reaction cross section, as seen in Fig.2(a). The corresponding near side —far side decomposi-tion of the scattering cross section as well as the angulardistribution (multiplied by factor —„) for no absorption(W=O) is also shown in this figure. This calculationconfirms the presence of the refractive mechanism in theelastic scattering at E -38 MeV. From Fig. 2(a) it is evi-dent that the angular distribution of the analyzed reac-tion has the typical characteristics of such a mechanismand the rainbow peak approximately coincides with thewide maximum in the scattering channe1.

As mentioned previously, microscopically calculated

10

) p—1

10b

to '=-

/

I/

~ N ~'~]II /

C( He, He) C

E~,b=37.9 Mev ...------. ——"---."""r~

10-t-I

]O—- (b)

O 2O

I/

/ /y/

I I I I I I I I I I i I i I I I I I I I i I I I i I I I I

40 60 80 100 120 140 160 180

0,„(deg)

FIG. 1. (a) Differential cross-section angular distribution forthe elastic scattering of 37.9 MeV He particles on ' C. Full cir-cles: experimental points; solid line: full cross section; dashedline: far-side component; dash-dotted line: near-side component;dotted line: calculated far-side component without absorption.All the curves were obtained with the potential A2. (b) Same as(a) for ' C. Theoretical curves were calculated with the poten-tial C1 (see Table I).

49 RAINBOWLIKE EFFECTS IN ( He,a}REACTIONS 981

10 'C( He, a)" C(g.s.)

9MeV

1 0 —1

- (a)

0

~ W

/

b

13C(3 "'C(g.s.).9MeV

the DWBA calculations with different combinations ofoptical parameters are given. It is seen that the calculat-ed angular distributions are strongly dependent on a-particle optical model parameters especially when com-paring the "rainbow" angular region (8, m -40—100 ).The region of this sensitivity corresponds to nuclear dis-tances about 3 fm. The radius of the strong absorption isin this case R, -5 fm.

The best description of experimental angular distribu-tions was achieved by the A2 —B2 potential combination.The shallow potentials in the output channel (Bl and B4as the example) do not describe experimental data in the"rainbow" region. In Fig. 2(b) the theoretical cross sec-tion with the standard form factor (by the "well-depth"procedure) is also given. The corresponding curve has tobe normalized to the experimental points (by the usualspectroscopic factor). Nevertheless, its shape does notseriously differ from a microscopic one. This similarity 0

~ IIII

VegV

V

0~ &

V5

CO

4P

~ &

OJ

V8CO

CO

C4

V'008COV

~ &

0

Ooo

O

oo e4oo

QOQ

ChQ

Oooo ln Q

oom ch% O

Oo

Q OOO~ '4) ~ Oom ~ W rt.OOOO

O O 'P '+@~44cV ~ cV

W rt Oonl W

QOo ChCV

oo Ooo

Oo

Q

~~nOtOooom 4)m~ooooOOOOOOO

10

- (b)-3 « I I

0 20 40I I I I I I I I I I I I I I I I

~ ~~~I I I I I I

60 80 100 120 140

5,„(deg)

FIG. 2. (a) Differential cross section angular distribution forthe ' C( He, a)' C(g.s.) reaction. Full, dashed, and dash-dottedlines are the full, far-side, and near-side components of the reac-tion cross section, respectively. Calculated curves correspondto the potentials A2-B2; see Table E. The dotted line representscalculation with W =0 in the entrance channel. (b) Calculationsshowing the effect of using the different optical model parame-ters for both entrance and exit channels (see Table I): full line:potentials A2-B2; dashed line: potentials A1-B2; dash-dottedline: potentials A2-B4; dotted line: potentials A2-B3; shortdashed line: potentials A2-B2 with the standard "well-depth"form factor. This curve is normalized to experimental points.

M

ClooQ

e Q&OOOVl

M O m ah oo tlgj

lpj O ~ I/j

OQ

cV

CP

0y$ Ce

y$

~ 8O

8~~ II

C4

ch

0bPgc 80

a 0 Q

08 3

4J

g ~ g g oCO rII CO

eg t CCN

pc &pc

49V. gURJA~ et gl.982

0.10

0.08

0.06

0.04

0.02

0.000 6 10 'l2

ithout the tyP'ca—60 but wthe effect of

sta~t~ng from c.~.. In other wordst smaller ang es.

tron as for t eminimum at

is not so sO+

~

m if Presen,states-

is mechanis,' F other excite

+round state tran

d the analog statensjtion or o

~+, 15.g

V (Q 7 97 MeV} anf the rainbovvlike

7 66 MeV)—the eff'ect oV (Q ()52 M

isa ears.-

11 calculatedmechanism Pp .

bl microscopica y'1-

Unfortunately,d states of C are

reliab e mi2 not avai-'

te rais for excitein Fig. 4 are cal-ble. Therefore, t e

11-depth procedure aculate d b the standar. C rresponding spto the experimenta p'

this way armalized to

obai d i iop c seto

f h o i 1combination o t e6+ The bestials corresponds to the set 2A-

arison of the expe'

erimentalhat detailed compari1

gerly ca cu a

ic factors is quethe stan ard d spectroscopicverlap integrals. hhe fullof the nuclear over ap

'

d hd li' O' Cpg) while the as eline is I(I("c"cn).

'ution to the amp

'litude ofhe main contribu

'

where thet is r'

es from t eimilar.th om

ction wi op o. I '

een as in theo iven. t is sll behavior o. 1(a}],that the overal

1 lid

rib

, oTh f

litative y

b th i bo anism'

1'

robably driven y1

c 's ec-1 1 t

g

r. In terms o mictroscopic factor. n

(9

10

10

10

10

10

10

E

10O

~ ~E„=7.66 MeV

rla integral in quest'stion. Thef n"'1"n"""'1"d'd

(9) obt dgXYB '

h h

a 8 —. The spectroscopicBoyarkina . rtson, t e

re seelid h o db the usual wel- ep5 The spectroscop

e a article given byi . ' =0.45+0.0 . Ic a

He+ n state in the a par'

while TISM gives S=

„0 -I

10 '-.

10

10

E„='1 5.106MeV

I I i I I I I I I I I II i g g I i & i I

18020 40 60 80 10

(deg)

' C Hea)' CB. The reaction C

tions for di6'erentental cross sections tFi . 4(a) the experimenIn Fig. astates of

eV) state the angu-M V (Q =11.2hast ec alar distribution ha

n ular distri-cross section angtal differential crpu xcited states, (a) in the

'ns using a well-depond to standard D

'1 sets A2-82 for the rs

sponhe o tical potentia secedure and the op ic

nd reaction.82 for the second

49 RAINBOWLIKE EFFECTS IN ('He, a) REACTIONS 983

cially for analog states. The generalization of existingmethods of the form factor calculations will be a subjectof further investigations.

Nevertheless, we can make the qualitative conclusionthat the rainbow mechanism is observed in ( He, a)transfer reactions, where a strong angular momentummismatching is present. For higher excitations (withbetter matching) this mechanism is not so important andfor the best matching of the orbits (Q -0) we can observethe diffraction typical for the strong absorption mecha-msm.

C. The reaction '4C('He, a) "C(g.s.)

The differential cross section angular distribution forthe elastic scattering of He on the ' C nucleus is present-ed in Fig. 1(b) and the ( He, a) reaction difFerential crosssection in Fig. 5(a). The He optical parameters used aregiven in Table I as C1 and C2. The best fit of the elasticscattering corresponds to parameters C1. As in the pre-

10

vious case of the scattering on the ' C nucleus we can seethe typical rainbowlike behavior with a characteristicAiry maximum. Since in the literature there are no avail-able data for a scattering on the C in our energy region,we used the same optical model parameters as for the' C( He, a)' C reaction, see Table I. The microscopicallycalculated form factors were also used both for the"light" and for the "heavy" vertices. Note that ourM3YE potential gives a value of the vertex constant~G(' C' Cn)~ =3.18 and S=1.69. The spectroscopicfactor deduced by means of the standard well-depth pro-cedure [see Fig. 5(b)] is 1.1+0.1. Theoretical values of Sare 1.76 from TISM [17] and 1.73 from Cohen andKurath [24].

The best description of the angular distribution was ob-tained with potential parameters Cl-B2. The calculateddifFerential cross section is in reasonable agreement withthe experiment both qualitatively and in the absolutevalue. As in the case of the ' C target, the only free pa-rameters in calculations were the optical potential ones.Differences between exit channel optical-model parame-ter sets are manifested mainly in the "rainbow" region ofangles while in the vicinity of the main maximum theygive practically the same results. These differences aredemonstrated in Fig. 5(b). In this figure the effect of theform factor is presented with the same normalization ofthe "well-depth" curve as is shown in Fig. 2(b).

10

10

10J3E

10

bO

10

~ /

~ M

//

//

14( (3

D. The reaction ' C(3He,a)' C

The measured angular distributions for excited statesof '3C——', , 3.68 MeV (Q =8.72 MeV) —and the analogstate —', , 15.11 MeV (Q = —2.70 MeV) are given in Fig.4(b). Again we see that rainbowlike effects for excitedstates in these reactions gradually disappear and for the15.11 MeV state transition the angular distribution exhib-its the diffraction pattern. For the same reasons as in theSec. IVB the theoretical differential cross section hasbeen calculated in a standard DWBA framework. Theoptical potentials used correspond to sets C1-B2. Corre-sponding spectroscopic factors are S(—,')=1.3+0. 1 forthe 3.68 MeV state and S(—', )=4.0+0.4 for the 15.106MeV analog state.

10V. CONCLUSIONS

- (b)~o '

0 20l I i I I I i I I I i I I 1 i I I I i I I I i I I 1 i 1 I I

40 60 80 100 120 140 160 180

(deg)

FIG. 5. DifFerential cross section angular distribution for the' C( He, a)' C(g.s.) reaction. (a) Same as Fig. 2(a), calculatedcurves correspond to the optimum set of optical potentials C1-B2 (see Table I). The dotted line is the calculation with 8'=0for the entrance channel. (b) Same as Fig. 2(b), full line: poten-tials C1-B2; dashed line: potentials C1-B3; dash-dotted line: po-tentials C2-B4; dotted line: potentials C2-B2; short dashed line:potentials C1-B2 with the standard "well-depth" form factor.The normalization is as in Fig. 2(b).

The differential cross section of the ( He, u) reactionson the ' C and ' C nuclei as well as the elastic scatteringin entrance channels were measured for the incident ' Heof energy E=37.9 MeV.

For the case of the ground-state transition with themaximum momentum mismatching the possible presenceof the rainbowlike mechanism was established. A similarmechanism manifests itself in the entrance elastic chan-nels.

A theoretical analysis of reaction channel angular dis-tributions for ground-state transitions was carried out inthe framework of DWBA with a microscopically calcu-lated form factor. An adequate quantitative descriptionwas obtained without any free structural parameters.

984 V. BURJAN et al. 49

It was shown that the optical potentials which optimal-ly describe the rainbow mechanism of the elastic scatter-ing could be further selected according to the shape ofthe reaction differential cross section. Therefore, al-though the we11-known ambiguity of the optical-modelpotentials is not eliminated, it may be essentially reduced.

The mechanism of both transfer reactions was ana-lyzed using a coupled-channel basis (using the programCHUCK [25]). It was found that the coupling with inelas-tic channels is negligible.

It was also found that the rainbowlike mechanism oftransfer reactions seems to occur for the case of strong

orbital momentum mismatching only. Decreasing the Qvalue leads to a gradual reduction of this effect. We cantherefore assume that for smaller Q (better matching) thetransfer reaction is more quasielastic, i.e., more peri-pheral in an output channel and the mechanism is morediffractive. This effect will, however, be dependent on en-

ergy of the incident particles. It is, of course, only a qual-itative conclusion. More extensive experimental andtheoretical study of the refractive processes in transfer re-actions would be very desirable because they can givevaluable information about the interactions at small dis-tances in the region of the nuclear form factors.

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