Banerjee, S R

Chintalapudi, S N

Dasgupta, S S

De, A

Divatia, A S

Roychoudhury, T

Sen, D

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Indian J. Phys. 64A (1). 55-62 (1990)<-n final state interaction in the break-up of deuterons by «c-particles of 45 MeVA De, S S Dasgupta, D Sen, T Roychoudhury, S N Chintalapudi+,S R Banerjee*" and A S Divatia*Departmeiil of Physics, Burdwan University, Golapbag, Biirdwan-713 104, India Received 10 April 1989, accepted 3 October 1989A b stra ct: Alpha-neutron ( < - n )  final state interaction (PSI) has been in v es t i­gated in the bieak-iip of deutron by <-paitic les of 45 MeV. The line shapes  piedicted by single level i?-inatiix theo iy  is, in gencial,  acceptable. Kffective  range theory provides no better fit to the data.K eywords : Alplia-ncutron final state interaction, single level /^-matrix theory, effective range theoiy .P ACS N o :  35.104-s 1. IntroductionRecently considerable attention has been focussed on the study of alpha-dculeron interaction, both theoretically and experiinciilally, with an emphasis on fitting the data provided by kinematically complete experiments (Gaiser c t  a l 1988 and the references therein). Alpha-dcuteron system, being a six nucleon assembly, is of particular interest because at lc)w energies (below ‘t-break-up threshold) it could be treated as a simple three body system made up of a structureless oc-particle, a pioton and a neutron. Final state interactions which may be due to any one, two or all the three pairs {<p, <n and pn) of outgoing particles and which, often play an important role in understanding the break-up reaction mechanism, has been the subject of significant interest by several workers (Bruno c t  a l  1980, Sagara ct a l 1977, Warner and Bercaw 1968). Considerable success has been achieved by the authors in fitting the line shapes of FSI peaks in theTlight of single level /{-matrix calculations. Calculations based on effective range theory has also been used fruitfully in fitting the break-up data at low energies (Dasgupta ci al 1980a, h). The present paper is an endevour to carry out analysis of interaction at 45 MeV alpha-energy, using both the methods referred above.^Variable Bnergy Cyclotron Centre, BARC, Calcutta-700 C64, India.5556 A De et a!2. The experimentThe experiment was carried out at the Variable Energy Cyclotron Centre, Calcutta using «t-particles of 45 MeV. Experimental details and some of our experimental data have been reported elsewhere (Dasgupta et al 1989). Si(Li) detectors were used to detect "C-particles and protons in coincidence. The target used was deuterated polyethylene foil. The energy and correlated pairs of angles were chosen so as to favour highly the <-n interaction alone in the allowed phase space. Figure 1 shows32t• • • • • • • • • •T  ^ Ep(M»v/^ 16 20Figure 1. Bi'diniensioual lepiesentation of tlie coincident events at fl„=20'’, #/»=40“ with the nominal kinematic curve superimposed on it, (inc) =  45 MeV.one of the bidimensional spectra (20", 40 ) with the nominal kinematic curve superimposed on it.3. Method of analysisBefore applying any specific method to analyse the experimental data, we recall the particular kinematical situations we considered. The allowed phase space for each of the angular combinations favours the regions of “He-ground state formation (^an — 0-9 MeV) and at the same time, production of ''Li-ground state 1.9 MeV) and occurrence of «/> FSl (£ n p ~ 0  MeV) are almost discarded solely from the kinematical point of view. As shown by the relative energyK-n final state interactim  in the break-up etc 57curves in Figure 2, the latter two points are never reached in the angular combina­tions considered. In the figure, the relative energies Ena, Epa and Epn corres­ponding to the three final particles are shown by the dashed dotted curves marked accordingly. For the two values of E« where fTcn ~  0.9 MeV, theE„ ( MeV)F igure 2. Differential cross section d^a jdQ ^dO pdE ^  (in arbitrary nnit) against (MeV) for (a) «« =20°, e|) = 40“ and (b) e„ =  15°, 9|;=45° vritb $ „  =  180°, $ p = 0 °  and incident energy =  45 MeV (alpha). Da.sbed-dotted ciiives are the relative energies in MeV (right scale) of the outgoing particles as indicated. Dashed and solid curves are the PWIA calculations (arbitrary unit) corresponding to proton spectator and neutron spectator, respectively. The arrow marks indicate the kinem atically predicted ‘H e-grou n d  state positions.corresponding other two relative energies are much higher and far away from those needed for * Li-ground state or np FSI phenomena to manifest. From the curves, it is highly expected that the spectra of the cross sections might be influenced most by the <w FSI leading to ‘He-ground state formation.58 A De et alNow, the three body break-up amplitude in a rigorous theory like that based on Faddeev’s three body formalism, must include the Kp and impulse terms and all the three multiple scattering terms corresponding to the three pairs as mentioned. However, in this paper we have no intention of going into the ftill theoretical formulations. We are rather guided somewhat empirically to fit the shape of the measured spectra in terms of the standard approximations as used by other workers.Plane wave impulse approximation (PWlA) (Kuckes c / r t /1961, Warner and Bercaw 1968) is inadequate to reproduce the spectra in the present case as indicated by the dashed and solid curves in Figure 2. None of the PWIA contributions, Kn (dashed curve) or «t/» (solid curve) indicate any enhancement in the positions where sharp peaks are actually observed in the measured spectra. This is due to the very nature of the approximation (PWIA) where the distribution has been mainly governed by the spectator energies.As the experimentally observed peak positions (Figure 2) undoubtedly corres­pond to the kinematically predicted regions for ®He-ground slate, we were intended to fit the measured spectra assuming Kn FSI alone.We give an outline of the following two methods which we used for FSI analysis.3.L R-matrix calculation:In single level ii-matrix theory following the procedure of Werntz (1962) and as suggested by Sagara et al (1977), near strong Kn FSI which is assumed to be in State, the cross section may be written asd^o/dQadQf4Eci=Nsin^ ^i[Fl{ka)+G\{ka)]pl{ka)‘ (1)where JV is a factor dependent on the incident energy alone, Si is the Pg/a resonant phase shift, and Cj are the neutron wave functions and is the <n relative momentum. The phase shift 8  ^ is calculated using the relationtanSi= ir/(£o +  ^ i-£„n ) (2)where^ x = - i W i F ^ + G i < 7 i )  (3 )andi r = y » w ( £ j + < ; ; ) .  (4 )P is the phase space factor given by (Ohlsen 1965, Warner and Bercaw 1968)P = 8£p(£.)»'»/[(iE’p)"'*+(£:.)^'* cos 0.,-(£<)»'« cos M  (5)where Et, £« and Ep are the lab kinetic energies of the incident and scattered <-particles and the proton, respectively, 0p the proton scattering angle and 0„p,<-n fina l state interaction in the break-up etc 59the included angle between the two detected particles. Three parameters Eo, y* and a are as discussed in Section 4.Here, we mention that we have omitted the angular dependence factor, namely the cos* x term, in the right hand side of eq. (1) as was also done by Sagara et al (1977), represented by eq. (9) in their paper. Actually, it (cos® x) is taken as constant and the choice is somewhat empirical, x, the scattering angle of the neutron in the <n system, is a model dependent quantity characteristice ot single process model. In the presence of dominant multiple processes, at the lower energy like 8.9 MeV £a (r„ = 17.8 MeV). this choice (cos* x = constant) has been show'n to yield highly satisfactory result. Even at comparatively higher incident energy like 42 MeV (alpha), the same formalism has been seen to work better (Figure 3) than that including the x-dependence. In Figure 3,>2INI(A\JQ£aU140302010Ejnc (alpha) * a2 MeV60^ 40X)206<n•oFigure 3. Differential cross section (in mb/srV MeV) against(MeV) for (a) j„ = 19.8*, 9  ^=  40° and (b) 9 . - 1 4 .8 ” 9;, = 50° with incident energy of 42 MeV (alpha). The experim ental points ate from Warner and ercaw. As stated in the text, the dashed curves result from R-niatrix calcula­tions when cos % dependence is included w hile the solid curves are due to the same calculation but with cos* X"*Constant.60 A De et atthe dashed curves represent the calculations of Warner and Bercaw (1968) including the cos* \  factor while the solid curves display the present calculation using eq. (1). For the angular combination 0a = 14.8'’ and 0p=5O°, both the calculations just coincide at the higher peak, so far as the shape is concerned. P'or both the angular combinations (0o, 0») =  (19.8' , 40°) and (14.8", 50"), the situation is largely improved when eq. (I) is used and this motivate us to proceed in the same way for the present case at 45 MeV also.3.2. Effective range theory:Following Watson’s relation (Watson 1952) for the effective range approximation in final state interaction theory, we have for *c« FSI, the expression representing the squared matrix element assin* 8i—  («)where the phase shift 8j is given by the relationfc*‘+*cot8,=  - i + i r k » ,  (7)a ^a and r being the effective range parameters.4, Results and discussionsAs has been already shown in Figure 2, the Figure 4 (a and b) displays the three body differential cross section d^ojdQadQ^Ea (in arbitrary unit) against Ea (in MeV), for two correlated pairs of angles (0«, 0»)=(2O", 40") (Figure 4a) and (15", 45") (Figure 4b), The arrows indicate the position of FSI peaks corres­ponding to ®He-ground state (Figures a and b) and QFS peak (Figure b), solely determined by the kinematical considerations. The QFS peak is at £;»=0.19 MeV, the lowest allowed neutron kinetic energy. The solid curves are the results of i?-matrix calculations using the parameters from Dodder and Gammel (1952) which are£■0 = -4 .3  MeV, y® =  6.9 MeV and o=2.9 fm.So far as the FSI peaks at lower Ea are concerned, the theory predicts the line shapes fairly well, for both the angle pairs. However, the calculated peaks at higher Ea show a small displacement in energy as was also seen by Warner and Bercaw (1968). A slight change of E^ from —4.3 to —4.6 results in some improvement in these peak positions but the peak cross section degrades the situation (dashed curves), mainly for (20", 40") combination. Calculations with little change in other input parameters were performed but no significant improve­ment was achieved. The dotted curves are the prediction due to effective range<-« final state interaction in the break-up etc 61F igu re 4. Differential cross section d*oldS3^diipdE^  (in arbitrary unit) against (MeV) for (a) 9„=20". and (b) <»a =  15“, #  ^= 45*’ w ith ^ „  =  180°. $^,=0°and incident energy==45 MeV (alpha). Solid curves are the results of i?-niatrix calculation with 7*<=6.9MeV, a = 2 .9 fn i and B o= “ 4.3 MeV. Dashed curves depict the same calculations but with JSo=~4.6M eV . Dotted curves are the prediction due to effective range theory. The dashed-dotted curve indicates the im pulse approximation calculation.theory using the parameters from Arndt et al (1973). As is discerned from the figure at higher Ea, contrary to lower ones, results of effective range theoretical calculations become worse in comparison to that of iJ-matrix theory. Calculations including Sxi» and P i,, states were also peformed (not shown in the figure) using the effective range theory but no improvement was obtained. This, along with P-matrix calculation indicates that the reaction mechanism is mainly governed by the <n FSI in the P,,* state. Analysis using the more sophisticated theory based on Faddeev formalism would be welcome to understand the observed discrepancy in the present analyses.62 A De et alThough our data has rather poor statistics, we tried to fit the broad peak due to <p QFS in the light of impulse approximation formalism of Burress, as outlined in the paper of Warner and Bercaw. It is indicated by the dashed-dotted curve in Figure 4b. Measurements involving a large amount of data with better statistics is needed to make a systematic study of the problem.AcknowledgementFinancial support in the form of Research Fellowship from the University of Burdwan is gratefully acknowledged.ReferencesA rndt R A> Long D D and R oper L D 1973 N ucl. Phys. A 2 0 9  429Bruno M, C annata F. D’Agostino M, V annini G, Bongiovanni F, F risoni M and Lom bardi M 1980 N hovo CiiH. Lett. 29 1D asgupta S S, Roy R, Rioiix C, Laltlou F' and S lob idrian It J l9S0a Phys, Lett. 9 l B  32 D asgupta S S, Slobodriau R J ,  Roy R, Rim ix C and Lahlou F 1980b Phys. Rev. C22  1«15 D asgupta S S, De A, Sen D, Roychoiidhury T, C hintalapudi >S N, Banerjec S R and D ivatia A S 1989 Proc. o f Few Body X I I ,  Vancouver 1989 (to be published)Dodder D C and Gaininel J  L 1952 Phys. Rev. 88 520G aiscr N 0 ,  Darden S K, Luhn R C, Schieck H Faelz gen and Sen S 1988 Phys. Rev. C 3 8  1H9 Kuckes A F. W ilson R and Cooper P F, J r . 1961 A tm . of Phys. 15 193 Ohlsen G G 1965 Nuc l .  In s ir . Metb. 37 240SagataK , Ilaia M. Takahashi N. Molobayaslii T. Takeutclii F, Soga F and Noganii Y 1977 J , Phys. Soc. Jap a n  4 2  732W arner R F  and Bercaw R W 1968 N ucl. Phys. A 1 0 9  205 W atson K  M 1952 Phys. Rev. 88 1163 W ernlz C 1962 Phys. Rev. 128 1336

∝-N Final State Interaction in the Break-Up of Deuterons by ∝-Particles of 45 Mev

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∝-N Final State Interaction in the Break-Up of Deuterons by ∝-Particles of 45 Mev

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