E

Jup~ 1978

Nuclear Physics A305 (1978) 1-14 ; © North-Holland PublWtinp Co., Mtat~ndarxNot to be reproduced by photoprint or microfilm without written permission from the publisher

BREAKUP OF 15 MeV POLARIZED DEUTERONS ON 4He t

HIROSHI NAKAMURACollege ofScience and Engineering, Aoyama GakurYt University, Tokyo

HIROSHI NOYADepartment ofPhysics, Hosei University, Tokyo

and

S. E. DARDEN and S. SEN tr

Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556

Received 21 November 1977(Revised 17 April 1978)

A6etract : Differential aoss sections and vector analyzing powers have been measured for the`He(d, p)`He n reaction at Fb = 15 MeV for twelve angles between 15° and 85°. The mostprominent feature ofthe cross sections as a function ofproton energy is the maximum correspondingto the n `He interaction in the °He ground state . The vector analyzing power is for the most partnegative, with a broad minimum in the vicinity of the peak in the cross section . An analysis of thedata has been made using the modified impulse approximation. The theory succeeds fairly well inreproducing the cross-section data for angles up to 40°. For larger angles, the magnitude of thecalculated cross section is too large for small EP. The negative analyzing power is also predicted,but the agreement y~ith the measured energy variation is in need of improvement at some angles .

NUCLEAR REACTION `He(d, p), E = 15 MeV; measured a(Ep, B~ and iT, l(Ep, B~ .Nâtural target .

1. Introduction

.Breakup of the deuteron on °He has been considered one of the more tractablethree-body problems in nuclear physics. For deuteron bombarding energies below30 MeV, the a-particle may, in the first approximation, be treated as an inert body,so that the breakup process involves only the relatively well-known low energynucleon-alpha interaction and the neutron-proton interaction, mainly in the isospin-zero state. However, two of us (H.N ., H.N.) suggested in a previous work t ) that athree-nucleon transfer reaction might occur even in the low-energy region andchange the reaction mechanism to a certain extent . Measurements of the various

T Work supported in part by the National Science Foundation .rr Present address ; Indian Institute of Technology, Kanpur, India .

2

H. NAKAMURA et al.

observables in a-d breakup permit a test for the validity of the above a-n-p three-body model and also for the existence of such specific reaction mechanisms .

Experimentally, there has been a good deal of work 2- ") on the breakup of thea-d system, most of it concentrated on the measurements of proton or a-spectra .The structure which has been observed in these spectra has been interpreted in termsof quasi-free scattering and final-state interactions in mass-5 andn-p systems. Muchof the data obtained to date has been reproduced using a simple modified impulseapproximation (MIA) 's - i s) in which the suppression ofthe low partial waves in theinitial state interaction is taken into account phenomenologically . This method hasbeen used successfully to reproduce the ZH(a, n)4He p measurements of Knoxet al. t') at E, = 39.4 MeV and the vector analyzing power in the 4He(~, p)4Hen reaction at 14-11 MeV [refs. s ' ls )] .

In the analyses of recent data of neutron polarization at 33 MeV c.m . energy `e)and the differential cross section at 5 MeV c.m . energy ') some modifications to theMIA have been introduced . The difference (~) between sHe-p (sLi-n) and a-p (a-n)phase shifts has been taken into account in the former. In the latter, a specific term(the Cr term), which may be related to a three-nucleon transfer reaction, has beenintroduced into the reducing factor `s - t s) as well as some other minor corrections(the R, term, R, term, Coulomb correction) .

In order to provide a somewhat larger base ofdata on which to test the validity ofthese modifications, the differential cross section and vector analyzing power forthe 4He(c~, p)4Hen reaction are presented here for Ed = 15 MeV over a considerableangular range.We have already obtained good agreement '4) between calculated and measured

differential cross sections for the ZH(a, pa)n reaction at 29.2 MeV [ref. s)] and42 MeV [ref. ')] without using the C~ term, and there was in these data no indicationof the existence of a three-nucleon transfer reaction . Therefore, the C~ tenor is nottaken into account in the present analysis .The analysis of neutron polarization and differential cross section at 33 MeV c.m .

energy 'e) indicated that the polarization depends strongly on the phase difference ~while the differential cross section is rather insensitive to ~. Since vector analyzingpower has properties similar to polarization, it is expected that some valuableinformation on ~ can be extracted from the present data. For this reason, we fo~usour attention mainly on the phase difference ¢ in the present analysis . We alsodiscuss somewhat in detail another correction term R� which is related to 6Li res-onating states and the n-p final state interaction .

Experimental details and results are given in sects. 2 and 3 and the theoreticalanalysis is presented in sect . 4.

DEUTERON BREAKUP

3

2. Experiment

All of the measurements were carried out at Notre Dame using a °He gas targetin the 43 cm scattering chamber. A target cell 2.5 cm in diameter having 4 ~m thickstainless steel foils and filled to a pressure of 1 .2 atm was used . The center-of-targetdeuteron energy was 15.0 MeV . Outgoing charged reaction products were detectedwith an array offour dE-Ecounter telescopes employing a collimation system similarto that used previously ") . A two-parameter data-handling program permittedsimultaneous acquisition of deuteron and proton spectra. Measurements werecarried out over the angular range 15-85° in 5° steps. Background spectra weremeasured for several representative angles with the gas pressure in the cell reducedto a few percent of 1 atm. After correction for the gas remaining in the cell, back-ground contributions to the proton spectra were generally less than a few percent,except for the smallest angles investigated, where, for example, the amount ofbackground at the low energy portion ofthe spectrum amounted to 20 %for 9P = 15°.Corrections for this background were made in the 15° and 20° spectra . Analyzing-power measurements were carved out using the vector-polarized deuteron beampolarized alternately in directions parallel and anti-parallel to the scattering plane.Polarization of the beam was monitored using the °He(~, d)4He spectra acquiredsimultaneously with the 4He(~,p)°He n data. For this calibration, the iTl , data ofHardekopf et al. ' $) were used .The proton spectra were converted into lab differential cross sections by correcting

for the energy loss of the protons in the gas cell and for the slight distortion of thespectra produced by the dependence of this energy loss on proton energy . Absolutenormalization of the cross sections was effected at each angle by comparison to thenumber ofcounts in the elastic deuteron peak . For this purpose, a separate measure-ment of the 4He(d, do) cross section was made using two telescopes and a monitordetector . The relative cross sections obtained in this way were nonmalized to the160(p, p) cross-section data of Hiddleston et al . 19 ) for B~,b = 75° at EP = 15 MeV.For some of the measurements, at lab angles between 40° and 65°, there was a

discrepancy between the energy calibrations of the deuteron and proton spectraobtained in the same measurement. The origin of this discrepancy is not known,but for these angles the energy calibration used was that obtained from the protonspectra . The position of the peak in the cross section was assumed to correspond tothe calculated proton energy for the reaction 4He(d, po)sHe. In view of the dis-crepancy in the energy calibration, an uncertainty estimated to be about 200 keVmust be attached to the energy scale of figs . 1~.

3. ResultsThe results of the measurements are presented in figs . 1~. On the left in each

figure the lab cross sections dZQ/dEpdl2p are given as a function oflab proton energy .The corresponding values of the vector analyzing power iTl1 are given on the right

4

H. NAKAMURA et al.

4He(d,p)n4He

0

-0.2

-0.2

10 12

Fig . 1 . Differential cross sections for the `He(d, p)`Hen reaction (in mb/sr ~ MeV) and the vectoranalyzing powers for polarized deuterons plotted against proton energy EP . The theoretical curves arethe results of the MIA analysis with the reducing factor (1). Solid curves correspond to parameter set (6),where four parameters a(1), a(2), ¢(0, }) and ¢(1, }) are chosen so as to fit the present data and thedifferential cross sections for the =H(a, pa)n reaction at 29 .2 MeV [ref. °)] . Short dash curves are theresults for (~ and (8), where two parameters a(1) and a(2) are chosen to fit the vector-analyzing-powerunder condition (~ and, therefore, the agreement is worse for the data of the ZH(a, pa)n reaction at

29 .2 MeV .

of each figure . Each point corresponds to a proton energy interval of approximately230 keV. Statistical uncertainties in the cross-section measurements are generallyequal to or smaller than the size of the. points . However, the data are subject touncertainties arising from both the relative and absolute normalization, and thisuncertainty is estimated to be of the order of 15 ~. In the region of the high energypeak in the spectrum, the cross-section values may depend sensitively on the energyinterval over which the data are averaged in obtaining each point, so that somewhatlarger uncertainties may be present in these regions. The uncertainties shown foriT, t are statistical. Data .are shown in figs . 1-3 down to a proton energy of about3 MeV. For proton energies lower than this value, the background was generallyobserved to increase rapidly with decreasing proton energy, so that data were notconsidered usable.

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DEUTERON BREAKUP

5

'He(d .p)n'He

0

'He(d,p)n'He

-0.2

-0.40

-0.2

Fig. 2. Sce caption to fig, I .

-0.2

10 2 4 6 9 10

Ep ( ~I~V )

Fig. 3 . See caption to fig. 1 .

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ff ______

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fff` "f `

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6

H. NAKAMURA et al.

200 r

160

WZ<120U

W0-OZ 80

OU

40

Fig. 4. The differential cross section of the `He(d, p)`Hen reaction at 9P = 85°. The solid curve has thesame meaning as in figs . 1-3. In calculating the dashed curve, the R, term was used, i .e ., the reducingfactor (11) with (12) and (13) and the same parameter set (6) as used in calculating the solid curve. The

proton energy at which the maximum occurs is 5.2 MeV.

Both the cross-section and analyzing-power data are very similar in appearance tothe results of Keller and Haerberli e) for Ea = 10 and 11 MeV.

4. Analysis

Before discussing the analysis of the present data, we summarize here briefly thefundamental ideas of the MIA. In the MIA, a description similar to the distortedwave Born approximation (DWBA) is adopted. Namely, the final state interactionbetween spectator proton (neutron) and quasi-bound SHe (°Li) as well as the initialstate interaction in the entrance (a-d) channel is taken into account. However, theMIA is different from the DWBA in some features .

Since many partial waves take part in reactions with a heavy target nucleus, thereaction strength is distributed among many partial waves. This indicates that thereaction strength in aqy one partial wave is relatively weak, and then the empirical

DEUTERON BREAKUP

7

optical potential can be used to describe the final and initial state interactions . Thisis one of the important reasons why the DWBA gives good results in tfie strippingand pick-up reactions for heavy target nuclei . However, in the breakup reactionsbetween light nuclei such as the a-d breakup reaction, the reaction strength in theplane wave impulse approximation (PWIA) is mostly concentrated on a few partialwaves with low angular momentum andthe reduction ofthePWIA partial amplitudesin these partial waves is very strong . It is very difficult to describe this extremereduction using the wave distortion produced by the empirical optical potentialalone. In fact, even a fully antisymmetrized DWBA theory z°) still overpredictsthe magnitude of the quasi-free-scattering peaks in absolute coincidence crosssections for 3He-d breakup reaction . Fortunately, in the breakup reactions betweenlight nuclei, a good deal ofthe measurements of differential cross section, polarizationand analyzing power are available and the number of the partial waves, in whichstrong modification for the PWIA occurs, is few. Therefore, by an appropriateparameterization of the partial amplitudes in these partial waves, one can carry outa good phenomenological analysis. The MIA is based on this idea .

In the MIA, the final and initial state interactions in respective SHe-p (SLi-n) anda-d channels are introduced through a reducing factor r(Y) [= c(Y)lC(Y)], wherec(Y) represents the partial amplitude for state Yinvolving the final and initial stateinteractions, while C(Y) is that ofthe PWIA. A part of the n-p final state interaction(n-p FSI) is involved in the SHe-p (sLi-n) final state interaction .

In the analysis i s) of vector analyzing power of the °He(~, p)4He n reaction atEd = 10-11 MeV [ref. e)], we obtained reasonably good results by the introductionof an additional term (the Ra term) into the reducing factor. Since the present dataare very similar to the previous data in appearance, it is expected that, by the use ofthe same method, good results will be obtained for the present data, also. However,the agreement with the present data obtained by this method is not as good, indi-cating that a part of the reaction mechanism maychange between6MeV and 10 MeVc.m . energy . For this reason, in the present analysis, we try the approach which wasused successfully earlier in analyzing neutron polarization at high energy 16) .

Here we summarize the method of ref. 16 ) . In the MIA, reducing factors are param-eterized~on the basis of the a-n-p three-body model . The values of parameters aredetermined so as to reproduce experimental data . The reducing factor for the'He-pexit channel is given by

~a = na(LaJa}expC2l(aâ(LaJ~+Q(L~) ].where the quantities with a bar and subscript p represent the quantities in the SHe-psystem, whereas those with subscript d are for the a-d system . The superscript Rmeans the real part, and L and J are respectively the orbital and total angular mo-

8

H. NAKAMURA et al.

menta. Notations q(LJ), S(LJ) and o~(L) represent the absorption coefficient, phaseshift and the Coulomb phase shift, respectively . The reducing factor for the SLi-nexit channel is given in the same way.The reduction parameter a(Ld) represents the reduction of the partial amplitudes

of the PWIA with orbital angular momentum La due to the initial state interaction.The special cases a(Ld) = 0 and 1 correspond respectively to a complete reductionand zero reduction . For simplicity, we assumed that a(Ld) does not depend on totalangular momentum Jd. As was discussed in ref. i<), formula (1) is obtained from thegeneral expression for the partial amplitude when an appropriate cut-off of reactionstrength in the central region is introduced. In other words, the a-d breakup reactionis assumed to occur in a rather external region . With this assumption, we can use theasymptotic forms for SHe-p and a-d wave functions distorted by respective final andinitial state interactions . Thedynamical factor gyp+~a in (1) comes from these simplifiedwave functions reflecting the effects ofthe final state interaction for the SHe-p systemand the initial state interaction for the a-d system .

Since SHe-p phase shifts are not available, a-p phase shifts Sp(LpJp) were used inrefs . ' 1-'a) instead Of ~p(~~r), where the quantity with subscript p but without thebar refers to the a-p system . However, in the analysis of neutron polarization at33 MeV c.m . energy ie), we showed that the neutron polarization depends stronglyon b~(~p.Tp) with ~p = 0, 1 ; it is not appropriate to neglect the difference betweenSP(Lp,,T~ and b P(LPJp). Therefore, in the present analysis, we assume an expression

~pl~l = apl~J+~(~

~(~ = Snl~/ +~(~)~

l2)and try to determine the phase difference ~(L,n phenomenologically, where we putLp = Lp = L, Jp = Jp = J, neglecting the spin of sHe (5Li).The phase difference ¢(LJ) is a function of the kinetic energy Eof the'He-p.(a-p)

system, and satisfies

~(LJ) -~ 0° or 180°,

forE -" 0.

(3)

However, in the MIA, the contributions from the region E ~ 0 are not importantin general. Therefore, neglecting the energy dependence, we treat the ¢(LJ) as realparameters .For simplicity, we assume that ~(LJ) = 0 for L Z 2. Thevector analyzing power

as well as the differential cross section is not sensitive to ~(1, }) in general. This isbecause ~(1, ~) relates mainly to D1.Z waves in the entrance channel, and the contri-butions from these partial waves to the matrix element are rather less important in theMIA. Therefore, it is difficult to extract information on ¢(1, }) from the present data.Hereafter, for simplicity, we adopt the following relation

~b(l, ~) _ ~(l, ~)~From the analysis of elastic a-d scattering z'), it is inferred that the breakup

reaction occurs only weakly in the S1 state. This indicates the extreme suppression

DEUTERON BREAKUP

9

of the partial amplitude for the S 1 state by the initial state interaction. As for thestates with La z 3, the reduction of the partial amplitudes may not be importantaround 10 MeV c.m . energy, as discussed in ref. '4) . Therefore, we may assume thefollowing values :

Using four free parameters a(1), a(2), ~(0, ~) and ~(1, ~), we search for the bestfit to the present data together with the data for the zH(a, pa)n reaction at 29.2 MeV.The best fit is obtained with

a(1) = 0.2, a(2) = 0.4,

The results are illustrated by the continuous curves in figs . 1-3. For comparison, wealso make the MIA analysis without including any phase difference, i.e .,

In this case, the best fit to the vector-analyzing-power data is obtained witha(1) = 0, a(2) = 0.4 .

(8)These results are illustrated by the short-dash curves in figs . 1-3 . We remark thatparameter set (7), (8) is not favorable concerning the fit to the ZH(a, pa)n reactionat 29.2 MeV [ref. s)].

Evidently, the agreement between theory and data is much improved by theinclusion of the phase difference ~. However, the discrepancy in the differentialcross section for small EP evident at larger angles indicates that further correctionsmust be introduced . We illustrate in fig . 4 one typical example, 6P = 85°. Note thatthe theoretical curve hac a large peak around EP ~ 1 .5 MeVwhile the data show notendency for peaking. It is very difficult to remove this discrepancy because this largepeak comesfrom neutron quasi-free scattering (QFS) as well as the (~) -SLi level and,in fact, the data for the ZH(a, pa)n reaction at 29 .2 MeV [ref. S)] show prominentpeaking at the corresponding geometry . Though this may be a very interestingproblem, it may be very difficult to extract definite conclusions from the presentdata alone. Therefore, in the present article, we will only show that the introductionof an additional term (the R~ term) in the reducing factor (1) is one way to improvethe agreement in the differential cross section .

In formula (1), we considered that the reduction parameters a(La) do not dependon h, where h represents the relative momentum between the neutron and a-particlein the final state. However, for partial waves with Ld <__ 2, a(L~ may depend on hrather strongly, because the suppressions of these partial amplitudes are very strongand a prominent reduction may occur around the (~)- °He level. Besides, as wediscussed in ref. zz), the contributions from 6Li resonating states may induce furtherstrong h-dependence for the reduction parameters. Thus, hereafter, we consider

1 0

H. NAKAMURA et al.

reduction parameters forLa <_ 2 as functions of h with the notation a(La : h) but, asin (5), a(0 : h) is taken to be 0.

Let us introduce the following function

R~(La : h) _

where hl is the value ofh at the (~)- SHe level. Evidently, this function satisfies

In the analysis based on (1}{6), reduction parameters a(L~ as well as ~(L,n havebeen determined by the useofthe experimental values at the (~)- SHe level. Therefore,onecan consider that a(La : hl ) is identical with a(L~ given in (1}-(6). Then, by usingnotation a(L~ instead of a(La : hl ), the reducing fäctor can be written as

Note that the function R,(La : h) in (11) is a kind of R, term which was discussedextensively in the appendix of ref. ') . In fact, in the latter half of ref. s2), we usedalmost the same R, term as in (11) .

In general, enhancement of the differential cross section related to QFS becomesprominent when the reduction parameter is large . In the analysis based on (1}{6),we have obtained pronounced peaking in the differential cross section for tke geo-met~y forQFS [and the (~)- SLi level] . The actual data, however, do not show such atendency . This indicates that the value of a(La : h) for QFS is considerably smallerthan a(La : h,). Since the value of h for QFS is larger than h l , it is inferred stronglythat a(La : h) is a decreasing function of h and possibly vanishes at a certain value ofh . Let us consider this possible zero point h(La) (> hl ) as an additional free param-eter. In this article, we take the following simple approximation for R,(La : h) :

0,

for h < h 1- [h-h,]/[h(La)-h, ],

for h, -~ h < h(La)

(12)

-1,

for h(L~ < h.

This function induces a cut-off for the reducing factor for h ? h(La) .The following values improve the agreement in the differential cross section,

reducing the peak at Bp = 85°, Ep = 1 .5 MeV, while keeping a good fit to the dataofthe ZH(a, pa)n reaction at 29.2 MeV [ref. S)]

h(1) = 0.50 fm-1 ,

h(2) = 0.40 fm-1 .

(13)

The results are illustrated by the dashed curve in fig . 4 . We have tried the sameanalysis adopting the following simple linear approximation instead of (12) :

~(La~ h) _ - Ch - hi]/Ch(La)-hi]~

(14)

But there is not any essential change of the results.

DEUTERON BREAKUP

1 1

In the above analysis, the ~(L.n forL >_ 2 have been taken to be zero, for simplicity.The agreement with the data is improved considerably, when the ~(L.n for L = 2are included . If we put ~(2, ~) _ ~(2, ~), the optimum value is ¢(2, ~) _ -20°.In this case, the optimum value of ~(0, ~) is x 2Q-30° .

5. Concluding remarks

We have obtained reasonably good results by taking into account the difference~ between SHe-p and a-p phase shifts in the MIA analysis. It is interesting that theoptimum values of ~ obtained in the present analysis are quite consistent with theresults of the analysis of neutron polarization at high energy ' 6) . It is also inferredthat an R, term plays an important role in the present enérgy region .We have carried out a semi-phenomenological analysis using some parameters

which have clear physical meaning. However, it is also possible and interesting tomake the analysis parameterizing the reducing factor directly without consideringthe physical meaning. We discuss this problem here in some detail .

In the energy region ofthe present analysis, the impulse approximation gives fairlygood results for the partial amplitudes with Ld Z 4. Since the imaginary part of thea-d phase shift is experimentally very small for the S-state, one can consider thereducing factor for the S-state to be very small. Then the partial waves in questionare P-, D- and F-waves. Let us take the following approximation for the reducingfactors of these partial waves

where zo, 1 (Y) are complex constants, and ho is the value ofh around which the mostimportant contribution to the matrix element comes from . One can interpret theR, term in the analysis of the preceding section as corresponding to the second term,although the R, term given by (12) is not a simple linear function. In this case, param-eter ho corresponds to h, of (12) .

Regarding zo,1(Y) as free parameters, one can determine them by searching forthe best fit to the experimental data as in the usual phase-shift analysis . Note thatthe matrix element is a linear function of zo. ,(Y) and then the analysis can be easilyperformed. The problem is that the number of adjustable parameters is not small.Therefore, one must try to eliminate minor or unnecessary parameters by somereasonable or plausible speculation. In the following, we show one reasonable ex-ample of such a procedure.

Firstly, the contribution from Lp ~ Ld-1 states to the matrix element may besmall. In fact, the amplitudes with Lp ~ Ld -1 are quite small in the impulse ap-proximation. Therefore, it is reasonable to treat zo, 1(Y) for Lp ~ Ld-1 as minorparameters, neglecting them for P- and D-waves in the first approximation . As forthe F-wave, the impulse approximation is reasonably usable, considering that thecorrection for the F-wave is not as strong as for P- and D-waves in the present energyregion . Hereafter, we restrict the state Y to LP = Ld-1 cases.

12

H. NAKAMURA et 4l.

Secondly, the parameters z,(Y) for F-waves may not be important . As we havediscussed in the preceding section, the reduction parameter for F-wave depends onh rather weakly . So we take zt(Y) for F-waves to be zero in the first approximation .

Thirdly, parameters zo.1 (Y) for P- and F-waves may be almost independent of thespin states of the entrance and exit channels, because the initial and final state inter-actions for these stat�°s do not depend strongly on the spin states . Therefore, one caneliminate the spin dependence of zo.l(Y) for P- and F-waves.Fourthly, for zo.1(Y) of D2. 3 waves, the exit channel is uniquely determined in the

first approximation . To show this, we take the impulse approximation with a 3S 1deuteron wave function, neglecting the 3D, component. Then, when the neutron(proton) in the exit channel is in a P~ orbit corresponding to the sHe (SLi) resonatingstate, the proton (neutron) must be in P,~ and P~ orbits for DZ and D3 waves, re-spectively, i.e ., the open exit channels for DZ and D3 waves are Pt and P~ states,respectively . The D, wave is the exceptional case and one must specify the exitchannels . Hereafter, we use the simple notation zo,1 (LdJd) .for zo, l(Y), except for theD1 wave . For theD1 wave, we use the notation z<o~l(2, 1) and zô X1(2, 1) correspondingto the exit channels for P,~ and P~ states, respectively .Thus, the number of free parameters is reduced to the following eleven

zo(l, 2)

[= zo(1, 1) = zo(1, 0)],

z~o'(2, 1),

~ô ~(2, 1)

zo(2, 2),zo(2,3),

zo(3, 4)

[= zo(3, 3) = zo(3, 2)],

z1(1, 2)

[= z1(1, 1) = z1(l, ~)],

zi' (2, 1),zi3' (2, 1),z1 (2, 2),

z1 (2, 3).

Since the contributions from D1. a states to the matrix element are rather small inthe impulse approximation, it is inferred that parameters zbl'1( 2, 1), zb3'1(2, 1) andzo,1(2, 2) are less important. Among the rest, zo(3, 4) is possibly close to unity, andz1(1, 2) and z1(2, 3) may be less important than zo(1, 2) and zo(2, 3) . Thus, we canextract two major parameters, zo(1, 2) and zo(2, 3) . We remark that the analysis inthe preceding section corresponds to the analysis using zo(1, 2) and zo(2, 3) only.In fact, one can easily see that the two pairs of real parameters [a(1), ~(0, ~)] and[a(2), ¢(1, ~)] correspond to two complex parameters zo(1, 2) and zo(2, 3), re-spectively, neglecting the contributions from minor parameters as well as the implicith-dependence of the reducing factor (1) via the a-p phase shift . The example givenin the latter half of the preceding section corresponds to the analysis by four param-eters, zo,1(1, 2) and zo,1(2, 3), taking the ratios z1(1, 2)/zo(1, 2) and z1(2, 3)/zo(2, 3)to be real numbers.

Since fairly good agreement with the present data as well as that of the ZH(a, pa)nreaction at 29 .2 MeV [ref. s)] is obtained by the major parameters zo(1, 2) andzo(2, 3) only, it is very difficult to determine minor parameters from these data

DEUTERON BREAKUP

13

alone. Besides, in detailed analysis, one must introduce several other minor param-eters, which represent the n-p FSI zs). Therefore, much more new information isrequired to determine these minor parameters. We emphasize that if the vectoranalyzing power and polarization were to be measured taking a-p (a-n, n-p) coinci-dence, the analysis in this approach could be developed much further .

Finally, we would like to propose an interesting problem. In the above analysis,it may be very reasonable to start from the solution of the Faddeev equation, i.e .,

where [i{ Y)]Faddeev represents the reducing factor calculated by the use ofthe Faddeevequation based on an a-n-p three-body model. The same procedure is used for then-p FSI. Let us perform the above analysis for Sr(Y) instead of r(Y) and searchfor the optimum values of the free parameters . If the a-n-p three-body model is valid,the optimum Sr(Y) must be very small. Then, one can test well the validity of thea-n-p three-body modél in the a-d breakup reaction . Our conjecture that a part ofthe reaction mechanism of the breakup reaction changes between 6 and 10 MeVc.m . energy may be examined by this method. We remark that this calculation ismore reliable than the direct analysis by the Faddeev equation, even if 8r(Y) is takento be zero . In fact, for Sr(Y) = 0, the partial amplitude of this analysis is given by

Since [r( Y)]Faaae~ may be rather insensitive to the details of the two-body forceemployed and C(Y) is calculated exactly by the use of realistic potentials includingthe Coulomb force, the partial amplitude thus calculated maybe much more reliablethan the one calculated directly by the Faddeev equation using separable potentialor the variants .

Two of us (H.N ., H.N.) would like to acknowledge the cooperation of Mr. M.Kadoi and the computer staff of Hosei University where the numerical calculationshave been performed.

References1) I. Koersner, L. Glantz, A . Johansson, B. Sundgvist, H. Nal~amura and H . Noya, Nucl . Phys . A2B6

(1977) 4312) G. G . Ohlsen and P. G . Young, Phys . Rev . 136 (1964) B16323) K. Nagelani, T . A . Tombrello and D . A . Bromley, Phys. Rev . 1+0 (1965) B8244) H . G . Pugh, D . I. Bonbright, D. A . Goldberg, P . G . Roos, R. A . J . Riddle and J . W . Watson,

Bull . Am. Phys. Soc. 1 3 (1968) 13675) T . Tauabe, J. Phys. Soc . Jap. 25 (1968) 216) K . Fukunaga, H. Nakamura, T . Tauabe, K . Hosono and S. Malsolo, J. Phys. Soc . Jap. 2Z (1967) 287) R . E . Warner and R. W. Bercaw, Nucl . Phys. A109 (1968) 2058) L . G. Keller and W. Haeberli, Nucl. Phys . A172 (1971) 6259) T. Rausch, H . Zell, D. Wellenwein and W. von Witsch, Nucl. Phys . A21.2 (1974) 429

10) E. Hourany, H. Nahemus, F. Takeutchi and T . Yuasa, Nucl . Phys AZZ'2 (1974) 537

14

H. NAKAMURA et al .

11) H. D. Knox,R. G. Graves, F. N. Rad, M. L. Evans, L. C. Northcliffe, H. Nakamura andH. Noya,Phys. Lett . 56B (1975) 33

12) H. Nakamura, Prog. Theor. Phys . 48 (1972) 695l3) H. Nakamura, Nucl . Phys . A208 (1973) 20714) H. Nakamura, Nucl . Phys . A223 (1974) 59915) H. Nakamura and H. Noya, Nucl . Phys. A264 (1976) 5416) P. Leleux, M. Bosman, P. Lipnik, P. Macq, J. P. Meulders, R. Petit, C. Pirart, G. Valenduc, H.

Nakamura and H. Noya, Phys. Lett. 70B (1977) 16317) K. W. Corrigan, R. M. Prior, S. E. Darden and B. A. Robson, Nucl . Phys . A188 (1972) 16418) R. A. Hardekopf, D. D. Armstrong, L. L. Catlin, P. W. Keaton, Jr. and G. P. Lawrence, Report

LA-5051, Oet. 1972, Los Alamos Scientific Laboratory, Loa Alamos, New Mexico, 8755419) H. R. Hiddleston, J. A. Aymar and S. E. Darden, Nucl . Phys. A1r12 (1975) 32320) R. E. Warner, R. L. Ruyle, W. G. Davies, G. C. Ball, A. J. Ferguson and J. S. Forster, Nucl . Phys.

A2SS (1965) 95 .21) W. Grüebler, P. A. Schmelzbach, V. König, R. Risler and D. Boerma, Nucl . Phys. A242 (1975) 26522) K. Sagara, T. Motobayashi, N. Takahashi, Y. Hashimoto, M. Hara, Y. Nogami, H. Nakamura and

H. Noya, Nucl. Phys . A299 (1978) 7723) K. Sagara, T. Motobayashi, N. Takahashi, Y. Hashimoto, M. Hara, Y. Nogami, H. Noya and

H. Nakamura, Nucl . Phys. A273 (1976) 493