A full finite-range CCBA analysis of the 118Sn(p, t) 116Sn reaction

Volume 87B, number 1,2

PHYSICS LETTERS

22 October 1979

A FULL FINITE-RANGE CCBA ANALYSIS OF THE 1 18Sn(p ' t) 116Sn REACTION T TAKEMASA 1, T TAMURA and T UDAGAWA Department of Physlcs 2, The Umverstty of Texas at Austin, Austin, TX 78712, USA Received 2 August 1979

Exact flmte-range coupled-channel Born approxlmatmn calculations, using a microscopic form factor, were performed for the 118 Sn (p, t) 116Sn reactmn Good agreement with experlrnent was obtained, both in the shape of the angular distributions and the relatwe magnitudes of the cross sectmns Corresponding results using a zero-range approxlmatmn were found to agree much more poorly with experiment

A number of coupled-channel Born approximation (CCBA) calculations, performed for hght-ion-mduced two-neutron transfer reactions, showed that the effects of lndtrect translttons via lnelastm processes are of importance for both spherical vtbrational [1,2] and deformed nuclei [3,4] However, all the CCBA calculations reported so far used the zero-range (ZR) approximation This is due mainly to the long computational time required when full fimte-range (FR) calculations are camed out m a straightforward manner In fact, there does not exist any known proof3ustffymg the use of the ZR approxlmataon Recently we developed a method to construct rather fast FR form factors for use in distorted-wave Born approximation (DWBA) calculations, for (p, t) and/or (t, p) reactmns [5] The method utlhzes an interpolatmn technique, which was origmally developed for heavy-ton-induced transfer reacttons [6] Thts technique allowed us to speed up the calculatmns by a factor of 15-20, without loss of needed accuracy in the resultant cross sectmn [5] Because of this mcreased speed, ~t is now practical to perform exact finite-range (EFR) CCBA calculations The purpose of the present paper is to report on the first result of such calculations, with the 118Sn (p, t) 116Sn reactton at Ep = 52 MeV [7] taken as an example We show the t On leave from Department of Physms, Saga Umverslty, Saga 840, Japan 2 Supported m part by the US Atomic Energy Commission

significance of using the EFR-CCBA method by comparing its results with those obtained with the ZRCCBA method As in ref [5], the interaction responsible for causmg the transfer was taken as the sum of two-body interactions between the incident proton and each of the transferred neutrons Q3 = V(rpnl)+ V(rpn2),

(1)

where

V(r) = PO Vo(r) + P1Vx(r)

(2)

and V0(1) = oo, 0 =-V~(1) exp[-~o(1)(r-rc)l,

if r ~
(3)

lfr>rc

Here P0 (P1) lS the spln-sInglet (triplet) prolectlon operator This potential and the parameters involved are those of Tang and Herndon [8] (TH) Thus r C = 0 45 fm, V0° = 277 07 MeV, V° = 549 26 MeV, ~0 = 2 211 fm -1 , and t~1 = 2 735 fm -1 The triton wave function used in our (p, t) calculation was obtained by a variational calculation, also discussed by TH (In the followmg we shall simply refer to the "TH triton", meaning the combined use of this triton wave function and the potential of eqs ( 1 ) - ( 3 ) ) Below we also present results obtained by assuming gausslan forms for both the triton wave function and the interaction potential (We shall simply refer 25

Volume 87B, number 1,2

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to the "gaussian triton" ) The parameters taken from refs [9] and [i0] a r e V O = 3 2 1 9 M e V , V 0 = 5 7 9 4 MeV,/52 (= square of the range of the interaction) = 0 3906 fm - 2 , and ,),2 (= square of the size of the triton) = 0 06 fm - 2 See ref [5] for further details Once the smgle-pamcle wave functions for the transferred neutrons are obtained, they are expanded in terms of oscillator wave functions The T a l m i Moshlnsky transformation then yields a series of terms containing products of two factors, one depending on the relatwe coordinate r of the transferred nucleons, and the other depending on their center o f mass (c m ) coordinates R 1 with respect to the residual nucleus Next a multxpole expansion IS performed of the product of the interaction potential of eq (1) and the triton wave function as a function of r and the c m coordinate R 2 with respect to the proton Integrating over r the product of the above two sets of expansions, the F R form factor emerges as a function of the coordinatesR 1 and R 2 See ref [5] again, for more details With this form of the form factor, the well-known approach o f Austern et al can be used to calculate the overlap mtegrals between the coupled-channel distorted waves and the form factor [11] In carrying out these overlap Integrals, use was made o f a modified version of the EFR-CCBA code MARS-2 [12], which was originally written for heavy-ion transfer reactions In obtaining the single-neutron wave functions, the W o o d s - S a x o n potential with r 0 = 1 25 A 1/3 fm and a = 0 65 fm was chosen A Thomas-type s p i n - o r b i t term was included having the same geometry and a strength VS = 8 MeV The binding energy for each neutron was taken to be half the two-neutron separation energy A simple model was assumed for describing the structure of each of the two tin isotopes involved con+ sldering only the ground 0g and the collective 2~ states The wave function of the former was given as that of the BCS vacuum, while that of the latter was generated

22 October 1979

in terms of the quaslpartlcle random-phase approximation (QRPA), assuming a q u a d r u p o l e - q u a d r u p o l e ( Q - Q ) Interaction The neutron single-particle levels [13] Included were l d 5 / 2 , 0 g 7 / 2 , 2 S l / 2 , 0h 11/2, and ld3/2 The coupling constant used for the pairing Inter. action was G = 21/,4 MeV, while the strength of the Q - Q interaction was determined by fitting, in terms + o f QRPA, the energy of the first excited 21 state In the F R form factor, only the total transferred splJ S = 0 was taken 1 ~to account, because the S = 1 contrlbu t:on could be completely ignored for a natural parity transfer [5,14] Also, for the 2~ ~ 2]- transfer, only the monopole transition was considered, because it dominated the others Its strength was weakened, however, by 15% compared with that for the 0g(A) -+ 0g(B] transfer, due to the blocking effect [15] The inelastic transitions were computed on the basis of a macroscopic vibrational model [16] The proton optical-model parameters used were those of Becchettl and Greenlees [17] We extrapolated their values to the present energy Ep = 52 MeV, and then increased slightly the depths of both the real and Imaginary potentials The triton parameters were taken from the work of Ball et al [18] However, the s p i n - o r b i t term was neglected, since Its effect on the (p, t) reaction is known to be small These optical parameters are summarized in table 1, together with the deformation parameter ~2 The latter was taken from the compilation by Stelson and Grodzins [19] The ZR-CCBA calculation was carried out by using the computer program MARS [20] There the form fac. tor was generated by the method o f Bayman and Kalho [21 ] For simphclty, the gausslan triton was used For comparison, ZR-DWBA calculations were also made In all these calculations, the parameters used were the same as those used in the EFR-CCBA calculations with the gausslan triton It is thus quite legitimate to compare the results of these calculations quantitatively to one another Table 2 summarizes the comparison of the magnl-

Table 1 OpUcal-model parameters used In the CCBA calculations The notation used m this table Is the same as that of ref [ 16]

26

Channels

V (MeV)

W (MeV)

WD (MeV)

rv (fro)

rw (fro)

rc (fm)

av (fro)

aw (fro)

WC(1)

WC(2)

32

proton

48 3

29

67

1 17

1 32

1 25

0 75

0 62

08

10

0 116

triton

170 0

19 0

-

1 15

1 52

1 40

0 74

0 76

08

10

0 113

Volume 87B, number 1,2

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22 October 1979

Table 2 Summary of normahzatlon constants for the 118Sn (p, t) ll6Sn reactmn with Ep = 52 MeV

jzr

EFR-calculatmns

ZR-calculatlons

Tang and Herndon gausslan

gausslan

CCBA

D W B A C C B A D W B A C C B A DWBA

02

0 20 (1 00)

0 29 (1 00)

1 02 (1 00)

1 52 (1 00)

13 70 (1 00)

17 60 (1 00)

2"~

0 14 (0 72)

0 80 (2 76)

0 75 (0 74)

4 17 (2 74)

7 54 (0 55)

38 46 (2 15)

tudes of the theoretical and experimental [7] cross sectmns One entry of this table is the normahzatmn factor N ( J rr) = o ( J ~r, O)exp/O(J 7r, O)th, which IS extracted after the predicted angular distribution for each level Is fitted to the experimental dlfferentml cross sectmns as shown in fig 1 Each N ( J ~r) is followed by another quantity in parentheses in the next row, which Is defined by R (j~r) = N(J~r)/N(Og-) By defmltmn, R (0"[) = 1 0 On the other hand, to have R (2"[) -- 1 means that the theory predicts the relative 2"[ to 0~ cross sections correctly It is seen m table 2 that the EFR-DWBA calculation, either w~th the TH or gausslan triton, gives an unsatisfactory R (2"[) = 2 7 However, if an EFR-CCBAcalculatlon is made, R (2"[) is reduced to 0 7 2 - 0 74, and ~s now rather close to unity The source of th~s Improvement is well known [ 1 - 4 ] In the CCBA calculation, the trans~tmn amphtudes vm indirect routes are added constructwely to that wa the direct route m th~s reac+ tmn, thus increasing the 21 cross section sufficiently The reductmn o f R(2"~) Is also achieved in the ZRCCBA calculation, but the obtained value, 0 55, is somewhat too small In fig 1, the sohd and d o t - d a s h hnes, respectwely, represent results o f EFR-CCBA and EFR-DWBA calculatmns, b o t h usmg the TH triton As seen, both calculations gwe angular distributions which are rather close to one another and agree fairly well with experiment, although some devaatlon takes place at larger angles (0 > / 4 0 °) It is not very surprising that the DWBA and CCBA calculations predict very similar an+ gular &stnbutlons for the 0g cross section That this is also the case for the 2"[ cross section, m spite of the large difference in the predicted R (2"[), is due to the similarity (in the CCBA calculatmn) of the one-step (0g(A) -~ 2"[ (B)) and the two-step (0g(A) ~ 0;(B)

(J

ZU

4U

60

Ocm (deg)

Fig 1 Comparison of angular dlstrxbutlons calculated by EFRCCBA, EFR-DWBA and ZR-CCBA with experimental cross sections [7] All curves are independently normahzed to fit the experiment best at forward angles 2 [ (B)) cross sections, when they are evaluated separately (The 0g(A) 2 +I ( A ) ~ 2 +I ( B ) cross section is small ) It is particularly worthwhile to emphasize m fig 1 that the dip at about 10 ° m the 0g angular distribution is very well reproduced Its slgmficance becomes still clearer b y noting that the ZR-CCBA result, presented m fig 1 by dashed hnes, completely falls to predict this dxp Note that the gausslan triton was used for the ZR calculation However, the use of the gausslan triton in the E F R calculatmn was confirmed to result m essentially the same angular &strlbutmns as those gwen in fig 1 by the sohd and d o t - d a s h hnes Therefore, the difference m the angular &strlbutlon m + the 0g cross sectmn in the 0 0 - 2 0 ° regmn is recogmzed as a genuine F R effect (The ZR fit ts poorer m the 300_40 ° region as well ) A tew years ago, we tried to reproduce the above dip within the framework of ZRDWBA and ZR-CCBA, and found it was impossible unless a very unusual choice was made for the bound state parameters [22] It lS thus very gratifying to find 27

Volume 87B, number 1,2

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in the present work that this long standing mystery is solved by considering FR effects Sumlnarlzlng these xesults, it is now clear that the EFR-CCBA calculation is needed in fitting the data of ref [7] The EFR part is needed in order to fit the + Og angular distribution The CCBA part, on the other + hand, is needed in making R(21)sufficiently close to unity We shall close this article by &scusslng the prediction of the absolute magnitude of the cross section As is seen in table 1, the N(J ~r)values obtained with ZR calculations are consistently too large, showing that the ZR theory systematically underestimates the cross sections This difficulty is, however, completely removed by switching to the E F R t h e o r y As is seen in table 2, we obtain N(O~-) = 1 02 for the FR-CCBA calculation with the gaussmn triton, although it becomes 0 20 when tire TH triton is used (This latter difficulty may, however, be removed, if an additional surface term that emerges at the hard core radius is included m the interaction This problem was discussed for a (d, p) reaction by Dobes [23] It has been believed for a long time that the DWBA theory will always underestimate the two-nucleon transfer cross sections Thus, much effort has been expended during recent years to increase the theoretical cross section, e g , by improving the calculation of the form factors [24] The present and earlier works [5] show, however, that the major source of the theoretical underestimate of the cross section hes In the consistent use of the ZR theory in the previous DWBA calculations It would thus be of interest to look at the problem of the absolute cross section once again, but now using EFR theory consistently The authors wish to thank Professor W R Coker for his careful reading of the manuscript

28

22 October 1979

References [1] R J Ascmtto and N K Glendennlng, Phys Rev 181 (1969) 1396,C2 (1970) 1260 [2] T Udagawa, Phys Rev C9 (1974) 270 [3] R J AsculttO, N J Glendennmg and B S~rensen, Nucl Phys A183 (1972) 60 [4] T Tamura, D R Bes, R A Brogha and S Landowne, Phys Rev Lett 25 (1970) 1507, 26 (1971) 156(E) [5] T Takemasa, T Tamura and T Udagawa, Nucl Phys A321 (1979) 269 [6] T Tamura and K S Low, Comput Phys Commun 8 (1974) 349, K S Low and T Tamura, Plays Rev Cll (1975) 789 [7] K Yagl et al, Phys Lett 44B (1973)447 [8] Y C Tang and R C Herndon, Plays Lett 18 (1965) 42 [9] GA Baker J r , J L Gammel, BJ Hill a n d J G Wills, Phys Rev 125 (1962) 1754 [10] R Hofstadter, Rev Mod Phys 28 (1956)214 [11 ] N Austern, R M Dnsko, E C Halbert and G R Satehler, Plays Rev 133 (1964) B3, T Tamura, Phys Rep 14C (1974) 59 [12] T Tamura, K S Low andT Udagawa, Phys Lett 51B (1974) 116 [13] R A Uher and R A Sorensen, Nucl Phys 86 (1966) 1 [14] B F Bayman, Nucl Phys A168 (1971) 1 [15] T Izumoto, Prog Theor Phys (Kyoto) 52 (1974) 1214 [16] T Tamura, Rev Mod Phys 37 (1965) 679,ORNL-4152 (1967), unpubhshed [17] F D Becchettl Jr and G W Greenlees, Phys Rev 182 (1969) 1190 [ 18] J B Ball, R L Auble, R M Drlsko and P G Roos, Phys Rev 177 (1969) 1699 [19] P H Stelson and L Grodzms, Nucl Data 1 (1965) 1 [20] T Tamura and T Udagawa, University of Texas, Center for Nuclear Stu&es Technical Report No 3 (1972), unpubhshed [21] B F Bayman and A Kalho, Phys Rev 156 (1967) 1121 [22] T Udagawa, T Tamura and T lzumoto, (1973), unpubhshed [23] J Dobes, Nucl Phys A235 (1974) 199 [24] R H Ibarra, M Valheres and D H Feng, Nucl Phys A241 (1975) 386, J Bang, C H Dasso, F A Gareev, M Igarashl and B S Nllsson, Nucl Phys A264 (1976) 157