Contribution of single-nucleon transfer reactions to

Volume 41B, number 3 CONTRIBUTION PHYSICS LETTERS OF SINGLE-NUCLEON TRANSFER 2 October 1972 REACTIONS TO 40Ar(r, t)40K * W.R. COKER, T. UDAGAWA and H.H. WOLTER ** Centerfor NuclearStudies, Universityof Texas, Austin, Texas 78712, USA Received 30 July 1972 An analysis of the 4°At(r, t)4°K reactions populating the 0+ analog state at 4.38 MeV and the 0+ antianalog state at 1.65 MeV with incident 3He energies of 35 and 18 MeV is made, including contributions from a pickup-stripping reaction mechanism as well as a conventional direct reaction mechanism. Agreement in magnitude and shape between calculated and experimental cross sections is best when both mechanisms are allowed to contribute. Experimental evidence has steadily accumulated [ 1 - 4 ] suggesting that the (~-, t) or (h, t) reaction to low-lying nuclear states with isobaric spin T = Tz does not occur via a simple direct mechanism. For example, the angular distributions for jTr = 0 + antianalog states (AAS) populated via (h, t) do not show characteristic l = 0 shapes [1]. In general, for states with j,r > 0, a similar persistent angular shift of the experimental angular distributions relative to any microscopic or macroscopic DWBA calculation is observed [2, 3]. The shift is to backward angles, so that for example the angular distribution for 4°Ar(h, t)4°K (0 + AAS, 1.65 MeV) can be fit in shape by an l = 1 DWBA calculation [ 1, 5]. An anomaly in magnitude is also noted; (h, t) cross sections for 4 + and 6 + states have magnitudes one or two orders of magnitude greater than one would expect on the basis of DWBA calculations using realistic microscopic form factors, normalized to agree with 0 + or 2 + angular distributions [4]. The simplest alternate mechanism for (h, t) is a sequence of single nucleon transfer reactions [6, 7]. In view of the small magnitudes of the cross sections for some nuclei, inelastic contributions may also be important. In the work reported here, we have studied the pickup-stripping contributions to the reaction 4°Ar(h, t)4°K populating the 1.65 MeV 0 + AAS or the 4.38 MeV 0 ÷ isobaric analog state (IAS). The pickup reactions considered are the 4°At(h, ¢t)39Ar * Research supported in part by the U.S. Atomic Energy Commission. ** Supported by Heinrich-Hertz Stiftung des Landes Nordrhein-Westfalen, Germany. reactions leading to one particle-two hole or two particle-three hole states in 39Ar, into which protons can be deposited by 39Ar(o~, t)4°K to give the expected two particle-two hole configurations which contribute to the AAS and IAS. These reactions are expected to make the largest contributions, with terms from other possible reaction pairs such as (h, d) and (d, t) being an order of magnitude smaller, or less. A detailed study of this point will appear shortly [8]. Our calculations have been done in terms of a coupled reaction channel (COREC) formalism [9, 10] embodied in the program JUPITER 4 [8]. Details of the formalism will appear elsewhere; we abstract here only a few pertinent details, sufficient to make our physical assumptions clear in terms of the COREC apporach. We have neglected the effects of non-orthogonality between the various reaction channels [9, 10]. The total system state function is taken to have the form x~(+)= ~ r - 1 xJ (ra)dpy, Ja (1) where a = {/a ]'a Sa ; aA}is the label for channel (la ]a Sa) and mass partition (a + A). Further, ~JM=[~i~Sa a ~a ® ~JA JIM ' (2) where cI,a is the (s-state) internal state function of projectile a, cI,j the internal state function of target nucleus A, andS'/{'a functions [11] 'Y.~aSa are . the . spin-angle . . Formal solution ot the complete Schrodmger 237 Volume 41B, number 3 PHYSICS LETTERS equation for the system then reduces to solution of a (generally large) set of coupled radial equations of the form D~(p~)x'~(P[3) ~ v~,r(P,y) J (3) where (4) = D#(r) - (h 2/2p#) {(d2/dr 2) -/b(/b + 1)/r2 ) - Ua(r) + E~, with Uo(r) the appropriate optical potential. The coupling potentials, v ~ , are taken as the effective (zero-range) interactions familiar from distored-wave Born approximation (DWBA) theory [12]. The zero-range assumption makes it convenient to define p,, - r, p~ = (Ar/B)p~ = (Ar/C), etc., where A stands for the mass of target nucleus A [ 12]. The coupling potential can be written in a more explicit form, using the notation of refs. [11, 12]. One has v/a(r)=~(B/A)2 " BA tsj X A(I b SbJbJb;laSa]aJA ; l s / J ) ~ ( r ) . (5) The quantity A is essentially the A-matrix defined by Tamura [11], which accounts for angular momentum conservation, namely A(lb Sb]b JB ;la Sa ]a JA ; Is]J) = .,_ Ja+J~ ~J l +l,-I . . . . (47r)ttz(_) ,~ (iao)lalb/a/bS](laOlbOllO) la Sa ] ! / x W(&/b& &;/j) lb sb l (6) s Finally, F BA and ~ s f are the form-factor and spectroscopic amplitude [ 12] familiar from DWBA Theory. In the calculations reported here, two types of couplings - and thus two types of form factors - are considered. The direct 4°Ar(h, t)4°K process is described by a mcroscopic form factor computed with a Gaussian two-body interaction, while the pickup and stripping reactions 4°Ar(h, ot)agAr and agAr(ot, t)4°K are described by the usual single238 2 October 1972 Table 1 Transition amplitudes3ff4siused in the (h, ~) and (~, t) calculations for 'mAr(h, t)'~°l(.'In colum 1, Th is the isospin of the hole-states, T is the total isospin. Intermediate agAr state fr(Th)T 7/2-( )3/2 3/2+ (1/2)3/2 3/2+ (3/2)3/2 3/2+ (3/2)5/2 ~sF(pickup)s~CB(stripping) Ex in 39Ar fo IAS toAAS (MeV) 1.4142 -0.70'71 0.7071 - 1.826 0.913 0.913 -0.5164 0.2582- 1.291 -0.6324 1.2648 0.0 0.0 1.5 3.9 9.0 nucleon-transfer form factors. For stripping, e.g., one has FlsBA / (r) = ~-1 CO O r-1 ulq(r) ' (7) where D O is the zero-range constant [12], C is the overlap of projectile and outgoing particle spin-isospin functions, and Uls/(r) is the radial wave function of the bound state occupied by the stripped particle. Two types of COREC calculations are conveniently distinguished. The first type includes only forwardgoing transitions, such as h - 0~and a ~ t, and might be called Second DWBA since it gives an amplitude equivalent to the second term in the distorted wave Born series [6]. That is, transitions back to the incident channel from intermediate channels, such as a -* h, or back to the intermediate channels from the outgoing channel, are neglected. In the second type of calculation, all couplings are included. Thus it might be called the fully-coupled reaction channel calculation. The differences between these two types of calculation are not as great as one might expect, and a comparison will be the subject of a forthcoming paper by the present authors. In the calculations reported here the Second DWBA is used exclusively because of the large number of intermediate pickup and stripping channels involved. Turning to the specific case of 4°Ar(h, t)4°K (0 +, IAS, 4.38 MeV and 0 + AAS, 1.65 MeV), we consider four intermediate states in agAr [13, 14]. These are the f7/2 ground state and three d3/2 twoparticle-three-hole states. The first of these is taken to correspond to the d3D state at 1.52 MeV, the second is taken to have an excitation of 3.9 MeV, roughly the centroid of higher d3/2 hole strength in agAr [14] and the third, with T = 5/2, is taken to lie Volume 41B, number 3 PHYSICS LETTERS 2 October 1972 I \~ I I J ' i ~, 35MeV, 0 ÷ IAS ~ (xl.O) -- I0-1 P-S , ~.~ i~(~ WBA (xO, 16) , , , , I I I i i I i i i i I 35 MeV, 0 " A A S _ :55 MeV, O+AAS - 10-2 P-S(,[.o) live,-- I0 -I 10-3 i÷l ~,\ E z o I i¢ I I ', I1 A e~ \ II [ "f'~ ii i ,".",.J ' V /', 10-2 o I,iJ o~ (I) o~ o t~ I 2O [I I hi II ',f! '? ,),_ I 4O ~1o-z ' 6O* i ~\ o'6o° i i i i i I Z 9 MeV, 0 + AAS -- P-S (xl.O) - - DWBA (xl.O) "J \J \ 17.9 MeV, O+IAS' ' / - " 10-2 -- P-S ' ' 4'o ' 6 '0 ° \ ;.9 'M.vo.' '. ,.'s 0 I 20 I I I I i i I I 40 dO* I 17.9 MeV, O* A A S ,-I ¢ • I0-2 10-2 I04 (xl.O) 10-4 ILl v re" UJ h LI. 1:3 10-2 i0-1 0 I¢j w 10-3 - - DWBA (xl.O) I ' ~o' Xo ' do* I I 20 40 60* Fig. 1. The 4°Ar(3He, t)4°K(0+, 4.38 MeV IAS and 0+, 1.65 MeV AAS) cross sections at 35 and 17.9 MeV. For each of the four cases, the absolute calculated cross section using the COREC formalism, assuming a pure (3He, a)-((x, t) or P-S mechanism, is shown as a solid line. The absolute cross sections calculated using standard DWBA are shown as dashed lines. Parameters are as given in the text. Data are from refs. [ 1, 5 ]. at around 9.0 MeV in excitation, in the continuum. Spectroscopic amplitudes for the (h, t~) transitions to, and (t~, t) transitions from these states are given in the table. The uppermost 3/2 + state cannot contribute by stripping to the ASS, and the small stripping contribution o f the 3.9 MeV 3/2 + to the IAS was neglected. The separation-energy procedure was used to compute the form factors appropriate to these transitions. However, since the intermediate states are unobserved and there are strong particle-particle, h o l e - h o l e , and p a r t i c l e - h o l e interactions which can shift the intermediate states drastically in energy, single-particle binding energies could be considered adjustable within wide limits. Results o f the calculations, using h and t optical parameters from the survey by Bechetti and Greenlees I 20 I I 40 I I 60* 20 40 60* Fig. 2. Comparison of 4°Ar(aHe, t)4°K(0 +, 4.38 MeV IAS and O+, 1.65 MeV AAS) cross sections at 35 and 17.9 MeV, with the results of calculations in the COREC formalism, assuming both direct and two-steppickup-sttipping processes occur. Parameters are as given in the text. Normalization is absolute except for the 0+ AAS calculation at 17.9 MeV, where the computed curve is multiplied by 6.25 to produce the agreement shown in the figure. Data are from refs. [1, 5]. [16] and the a-potentials used b y Toyama [6] are summarized in the figures. Similar results were obtained for a number of other potentials tried. The available data, which we compare to our calculations, are those o f Wesolowski et al. [5] at 17.9 MeV, and o f Hinrichs et al. [1] at 35 MeV. In fig. 1 are shown the conventional DWBA calculations (as dashed lines) and the pure pickup-stripping calculations, in Second DWBA (as solid lines). The microscopic form factor for the conventional DWBA calculations used a Gaussian two-body interaction with V = 3 MeV and /3 = 3.0 frn-1 , including configuration mixing. The D O valve for (h, ,v) and (ol, t) is based upon that suggested b y Stock et al. [15] namely D O = - 4.8 X X 102 MeV-fm 3/2 . It is seen that the pickup-stripping calculation is the only one agreeing with the shape o f the 0 + AAS angular distribution, at either incident energy. 239 Volumc 41 B, number 3 PHYSICS LETTERS On the otber hand, the relative magnitudes of conventional DWBA and Second DWBA are quite different and the AAS/IAS ratios for the pickup-stripping cross sections are much too small. One would expect, a priori, that both the simple direct and the pickupstripping mechanisms (among others) contribute to the observed angular distributions. Fig. 2 shows the result of a summation of the amplitudes for both processes. In these calculations, D O was reduced to - 3 . 9 X 10 2 MeV-fm 3/2 for (h, a) and (a, t) while V, for the microscopic (h, t) form factors, was reduced to 1.2 MeV. The resulting cross sections have magnitudes agreeing well with the experimental cross sections for the IAS and AAS at 35 MeV, and for the IAS at 17.9 MeV, as seen in fig. 2. The shape fits are also seen to be reasonably good in all cases, particularly considering that no effort was made to find optical potentials specifically appropriate to the nuclei and energies considered. One somewhat puzzling feature of this analysis is the magnitude of the experimental cross section for the AAS at 17.9 MeV. The calculated cross section shown in fig. 2 for this case is multiplied by 6.25, to produce the agreement in magnitude shown. Since the calculations agree in magnitude for the IAS cross section at both energies, qnd for the AAS cross section at 35 MeV, it seems possible that a sizable, near-isotropic background was inadvertently included under the very small (20 counts maximum) triton group for the AAS in the reduction of the data of Wesolowski et a l . see fig. 1 of ref. [5]. In summary, we have shown that a combination 240 2 October 1972 of pickup-stripping and charge exchange mechanisms is required to explain the observed shape and magnitude of the angular distributions for 4°Ar(h, @ ° K ( 0 + IAS and 0 + AAS). There is every reason [1, 2, 6, 8] to believe that this conclusion will be generally valid for all or most of the (h, t) cross sections, on a variety of nuclei, measured to date. References [1] R.A. Hinrichs et al., Phys.Rev.Lett. 25 (1970) 829. [2] J.R. Comfort et al., Phys.Rev.Lett. 26 (1971) 1338. [3] R.A. Hinrichs and G.F. Trentelman, Phys.Rev. C4 (1971) 2079. [4] P. Kossayni-Demay et al., Nucl.Phys. A148 (1970) 181; R. Schaeffer, Argonne Physics Division Informal Report RHY- 1970A (unpublished). [5] J.J. Wesolowski, L.F. Hansen and M.L. Stelts, Phys. Rev. 172 (1968) 1072. [6] M. Toyama, Phys.Lett. 38B (1972) 147. [7] R. Schaeffer and G. Bertsch, Phys.Lett.38B (1972) 159. [8] T. Udagawa, H.H. Wolter and W.R. Coker (unpublished). [9] T. Ohmura et al., Prog.Theor.Phys. 44 (1970) 1242. [10] L.J.B. Goldfarb and K. Takeuchi, Nucl. Phys. A181 (1972) 609. [11] T. Tamura, Reviews Mod.Phys. 37 (1965) 679. [12] G.R. Satchler, Nucl.Phys. 55 (1964) 1. [13] S. Sen et al., Phys.Rev. C5 (1972) 1278. [14] W. Fitz, R. Jahr and R. Santo, Nucl.Phys. A l l 4 (1968) 392. [15] R. Stock et al., Nucl.Phys. A104 (1967) 136. [16] F.D. Bechetti and G.W. Greenlees, in: Polarization Phenomena in Nuclear Reactions, ed. by H.H. Barscha[ and W. Haeberli (Univ. of Wisconsin Press, Madison, 1971) p. 682.