The (p, t) reaction on 12C, 54Fe and 208Pb at 80 MeV

Nuclear Physics A322 (1979) 92-108; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

THE (p, t) REACTION ON lZC, S4Fe A N D 2°SPb AT 80 MeV J. R. SHEPARD, R. E. ANDERSON, J. J. KRAUSHAAR and R. A. RISTINEN

Nuclear Physics Laboratory, Department of Physics and Astrophysics, University of Colorado, Boulder, CO 80309, USA t J. R. COMFORT

Department of Physics, University of Pittsburgh, Pittsburgh, PA 15260, USA tt N. S. P. KING

Los Alamos Scientific Laboratory, Los Alamos, NM 87545, USA + and A. BACHER and W. W. JACOBS

Department of Physics, Indiana University, Bloomington, IN 47401, USA +t Received 10 January 1979 Abstract: Angular distributions have been measured for the low-lying levels of the residual nuclei for the

12C, S4Fe and 2°sPb(p, t) reactions at Ep = 80 MeV. The shapes of these angular distributions are generally well reproduced by the zero-range distorted-wave Born approximation (DWBA). Enhancement factors extracted from the data show that the DWBA predicts relative strengths consistent with those observed at lower bombarding energies. However, the overall empirical DWBA normalization at Ep = 80 MeV is observed to be ~ (¼) of that required at 40 MeV for 2°apb (54Fe).

E

NUCLEAR REACTION 12C, 54Fe, 2°aPb(p, t), E = 80 MeV. Measured a(Et, 0). 1°C, S2Fe, 2°6pb levels extracted enhancement factors. Self-supporting enriched (natural) 54Fe, 2°aPb(12C) target. Zero-range DWBA calculations, determined overall normalizations.

I. Introduction The (p, t) reaction occupies an intriguing position in nuclear physics at the present time. Given the simple picture of the reaction mechanism as the simultaneous pickup of a pair of neutrons with spins coupled to zero and relative angular momentum zero, the reaction can be expected to provide a rich variety of information on pairing correlations and other nuclear structure phenomena with a minimum of theoretical complexity 1). Such a picture, embodied in the zero-range distortedwave Born approximation, has been used extensively over the past decade to extract a great deal of nuclear structure information, especially (but not exclusively) the spins , Work supported in part by the US Department of Energy. +t Work supported in part by the National Science Foundation. 92

12C, 54Fe, 2°aPb(p, t)

93

and parities of levels in the residual nucleus. However, the Situation is greatly clouded by recent theoretical investigations 2-4) which suggest that the simple simultaneous pickup picture is naive and that several other processes may contribute significantly and must be explicitly taken into account before experiment and theory can be compared quantitatively. While some progress has been made in understanding the roles of such mechanisms as two-step processes - through both inelastic 2) and deuteron 3) intermediate channels - and direct pickup through exotic triton components 4), the overall theoretical picture of the (p, t) mechanism is complex and ill-resolved. It is therefore a reaction of proven utility that has been extensively exploited over the years, but which nevertheless is poorly understood theoretically (or, at best, difficult to treat). Given this state of affairs, (p, t) data can be analyzed in two ways: one can apply the old, over-simplified methods that have been proven empirically, at least for strong transitions, and thereby extract spins and parities of residual levels and perhaps also some of the gross features of the wave functions involved. Alternatively, when the nuclear structure aspects are understood, one can use the data to reveal features of the reaction mechanism either by noting discrepancies with the simple approach or by applying the latest and most comprehensive theoretical techniques directly. In this paper we adopt one of the latter approaches. Tile (p, t) reaction is studied on targets of 12C, 5~Fe and 2°apb and consideration is given to transitions to lowlying levels, whose nuclear structure aspects are reasonably well understood. The bombarding energy of Ep = 80 MeV is above that of all but a few limited measurements on light nuclei such as 12C [ref. 5)]. Hence, it is possible for the first time to investigate whether the simple DWBA approach, which works moderately well at lower energies for these targets, can be extended to higher energies. It is entirely possible that the reaction mechanism may be appreciably different here due either to a modification of the single-step simultaneous pickup process or to the relative enhancement of other reaction modes such as multistep processes 6). In addition, the 12C(p,t)~°C reaction was observed to provide a comparison with a study previously done at a high proton energy 5). Apart from this specific goal, there is the more general aim of documenting the behavior of (p, t) cross sections across a wide mass range as the proton energy goes beyond the limit of about 55 MeV used for previous measurements.

2. Experimental method The experiment was carried out at the Indiana University Cyclotron Facility. The quadrupole-dipole-dipole-multipole spectrograph of 2.55 msr solid angle was used in a dispersion matched mode for the analysis of the tritons from the reactions. The beam energy was chosen to be 80 MeV since the (p, t) reaction at that energy would yield tritons that were about at the limit of the magnetic rigidity of the

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J . R . SHEPARD et al.

spectrograph. A helical cathode position sensing proportional chamber was used in the focal plane of the spectrograph. Two plastic scintillators (0.63 and 1,27 cm thick) located in back of the proportional chamber provided two AE related signals for particle identification. The two scintillators were reversed for the ~2C(p, t) data so that the tritons would stop in the 1.27 cm detector and the final scintillator could be used as a veto to reject deuteron events. The targets of ~2C, S4Fe and 2°apb were of 11.3, 7.01 and 16.2 mg/cm 2 thickness, respectively. The carbon target was unenriched but the 54Fe was isotopically enriched to 97.2 ~o and the 2°Spb to 99.3 ~o. Representative triton spectra are shown in figs. 1-3. The energy resolution for the 54Fe and 2°SPb targets was about 130 to 140 keV (FWHM). The data were analyzed by fitting a skewed Gaussian line shape with a linear

II ill

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Fig. l. A 12C(p, t)l°C triton spectrum at a lab angle of 22°. This figure shows two momentum bites of the magnetic spectrometer spliced together. Energies shown are in keV and are taken from ref. 7).

i.~

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54Fe(p,t)52Fe

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Ep=280oMeV

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CHANNEL NUMBER

512

Fig. 2. A S4Fc(p, t) triton spectrum at a lab angle of 22°. Energies shown are in keV and are taken from ref. s).

12C, S4Fe, 2°SPb(p, t) 250

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Fig. 3. A 2°8pb(p, t) triton spectrum at a lab angle of 22 °. Energies shown are in keV and are taken from ref. 9).

background to the various peaks to determine their areas. After normalization to the integrated beam current with a dead-time correction of a few percent, the c.m. cross sections were computed. The error bars displayed on the data points in the angular distributions are based primarily on statistics while the uncertainty in the overall normalization of the data was estimated to be less than 15 ~.

3. The 12C(p, t)l°C reaction Triton energy spectra were measured for the 12C(p,t)l°C reaction at angles between 7° and 70° in the c.m. system. A region corresponding to about 6 MeV of excitation in ~°C was covered in two momentum bites of the spectrometer. A spectrum comprised of two bites joined together appears in fig. 1. Three states of °C are observed, the 0 ÷ ground state, the 2 ÷ level at 3353 keV and a level at 5280 keV which has a tentative spin-parity assignment of 2 + [ref. 7)]. All of these levels have previously been seen in a2C(p, t) reactions at proton energies ranging from 20 to 155 MeV [refs. 5,1o-12)]. Angular distributions for these levels are shown in fig. 4. The 0 ÷ angular distribution has the sharp diffraction structure which characterizes L = 0 transitions in the (p, t) reaction. The distinct stripping peak observed at about 16° for the 3353 keV angular distribution is also typical of L = 2 transitions. The shape of the 5280 keV angular distribution is appreciably different from that of the 3353 keV level at the extreme forward angle as well as for 0 > 30°. This suggests that either the broad level is a multiplet with some L 4= 2 strength or that the population of the level occurs via a mechanism which differs from that responsible for the excitation of the 3353 keV state. The former possibility has been advanced previously on the basis

96

J.R. SHEPARD et al. 104

I

I

I

I

tZC(p,t]mC Ep=80MeV

I

103

I0 z

I~...li,~ •

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60

7

Fig. 4. Experimental angular distributions for ~2C(p, t)~°C. The DWBA predictions shown were obtained by using the input parameters of table 1. The normalizations shown resulted in the enhancment factors

of table 2.

of comparison with levels excited by l°B(3He, t)~°C in the paper by Benenson et al. 11). Their measurements also show appreciable differences between the 3353

and 5280 keV angular distributions at back angles. The mirror nucleus X°Be has 7) four levels near 5 MeV excitation with J~ = 0 ÷, 2 ÷, 1- and 2-. The 2- analog in 1°C should not be populated by the (p, t) reaction, while a 1 - state could be excited weakly.

12C, 54Fe, 2°sPb(p, t)

97

Distorted-wave Born approximation (DWBA) calculations were performed by using the program DWUCK4 13). The calculations assume simultaneous transfer of a pair of neutrons coupled to S = 0 with zero relative angular momentum. The two neutron relative wave functions in the triton were assumed to be of a Gaussian form and the overlap with the nuclear wave functions was computed by using the technique of Bayman and Kallio x4). The interaction between the two-neutron c.m. and the proton is assumed to be of zero range. The method of calculation is presented in detail in the appendix of ref. ~5). Non-locality corrections for the distorted waves were not included. T^BLE 1 DWBA input parameters (the notation is as given in ref. t s)) V

r

a

W

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a'

V~.o.

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r~.o.

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re

x2C(p, t ) l ° C

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-8.51 1.45 0.45 1.96 0.571

1.33 1.30 d)

1.44 0.38

54Fe(p, t) S2Fe

p+S4Fe P6F e) -32.04 P8F e) -30.2 t+S2FeT6F s) -117.4 T7Fh) _138.4 n+*2Fe c)

1.17 0.75 -14.9 1.32 1.228 0.764 -7.29 1.456 1.165 0.802 --21.95 1.278 1.20 0.720 -14.6 1.43 1.29 0.65

0.536 --6.2 --1.01 0.75 0.505 --4.46 --0.40 1.052 0.611 0.755 0.840 d) 1.29 0.65

1.25 1.25 1.30 1.30

2°sPb(p, tj2°6pb

p+2°spb P5P i) -39.5 t+2°6PbT2PJ) -160.2 n + 2°6pb k) c) ') ') a) ")

1.211 0.769 1.20 0.672 1.27 0.90

-6.58

1.460 0.523 -23.39 1.095 0.931

4.61 d)

1.083 0.766 1.25 1.30 1.27 0.90

Ref. 16). b) Ref. 17). Well depth adjusted to give correct neutron separation energy. Spin-orbit potential of Thomas-Fermi form tied to central potential; proportionality factor g = 25. Ref. is). f) Ref. 19). J) Ref. 2o). h) Ref. 21). i) Ref. 19). J) Ref. 22).

k) Ref. 23). The parameters used in the calculations appear in table 1. The geometry of the well used to bind the neutrons in 12C was obtained via a best fit to e- + 12C electron scattering data by using a technique described elsewhere 23). The binding energies of the neutrons were chosen to be half of the two neutron separation energy appropriate to the excitation energy of the final state. For these calculations the pickup of two lp~ neutrons was assumed. The angular distributions calculated for the 0 + transition were extremely sensitive to the optical model parameters used, especially for the triton potential. While no calculation reproduced the data with precision, many choices of optical potentials

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J . R . S H E P A R D et al.

could be excluded. Furthermore, all triton potentials giving reasonable results were quite similar while variations among and sensitivities to proton parameter sets appearing in the literature were slight. A set of proton and triton optical model parameters providing the best agreement with the 0 ÷ data are shown in table 1. The DWBA calculation for the 0 ÷ transition generally reproduces the data adequately except in the vicinity of the local minimum at ,-~ 22° where the calculation shows only an inflection point. The depth and location of the primary minimum at -,~ 50° is also missed by the calculation. The quality of agreement is roughly equivalent to that obtained by Nelson et al. 12) at 49.5 MeV. Discrepancies between experiment and DWBA calculations for the 3353 and 5280 keV levels are much more pronounced than for the ground state transition. The stripping peak observed at about 16° for the lower level is displaced and flattened in the DWBA result. The experimental cross section is badly underpredicted for angles greater than 30° . Furthermore, Q-value and binding energy effects are predicted to be far too small to account for the shape differences between the upper two levels. It should be noted that Nelson et al. 12) found that finite-range effects treated in an approximate way increase back-angle cross sections for the 3353 keV level and substantially improve the agreement between theory and experiment. In the present work calculations were performed for all three levels in which finite-range effects were accounted for via the local energy approximation 13) using finite-range parameters of R = 0.35 and 0.7 fm. Inclusion of finite-range effects with either value of R destroyed the modest agreement for the L = 0 transition and resulted in much poorer overall agreement with the data although, in keeping with the findings of Nelson et al. 12), the discrepancies at back angles for the L = 2 transitions were somewhat reduced. Thus, on the basis of the limited calculations done here, it seems unlikely that the discrepancies between experiment and the DWBA results are wholly atrributable to improper treatment of finite-range effects. This is understandable, perhaps, since the large inelastic and single nucleon pickup cross sections in this mass region suggest that two-step mechanisms may be very important, especially relative to heavier targets. It is possible, too, that the details of the two-step mechanisms involved in the excitation of the 3353 and 5280 keV levels can account for the different shapes of their angular distributions and that it is not necessary to TABLE 2 Enhancement factors for ' 2C(p, t)~ °C

E+

J"

e ")

0 3353 5280

0+ 2+ (2 +)

0.863 0.255 0.255

") DWBA calculations used parameters oftable I without either finite-rangeornon-localitycorrections. Configurations of picked, up neutrons assumed to be (1p3/2) 2.

~2C, S4Fe, 2°SPb(p, t)

99

involve components of an angular momentum other than 2 for the excitation of the level at 5280 keV. Enhancement factors, ~, are given in table 2 for all three levels of 1°C. The quantity is essentially the renormalization factor required to bring the DWBA calculations into agreement with the data. Specifically we have

a~p = e( S½)Z(D~/lO *) O'DWUCK4 2L+ 1 ' where S~ is the two-neutron spectroscopic amplitude 1), 190 is the zero-range normalization factor and L is the angular momentum transfer. All calculations reported in this work use D2/lO 4 = 25 MeV 2" fm 3. The enhancement factors quoted in table 2 refer to DWBA calculations without finite-range (or non-locality) corrections.

4. The S4Fe(p, t)S2Fe reaction Cross sections were extracted for the ground state, 850 keV (2+), 2385 keV (4 +) and 3583 keV (4 +) states. The resulting angular distributions are shown in fig. 5. A number Of higher-lying states have been seen in 52Fe via the (p, t) reaction but the energy resolution of the present experiment did not permit their study in an unambiguous fashion. The weak state at 2762 keV (2 +) observed by others 24-26) is also in evidence in the spectrum shown in fig. 2 but no effort was made to obtain cross sections for it. The four states of interest are well resolved. The ground state angular distribution again shows the sharp diffractive structure associated with L = 0 (p, t) transitions. The other three angular distributions clearly show the L = 2 and 4 stripping peaks at about 8° and 20 °, respectively. Maximum cross sections are roughly 30 ~o lower at 80 MeV than at 40 MeV. The DWBA calculations for this reaction were carried out in the manner described in the previous section. The optical model parameters that are appropriate for the projectile and ejectile were not immediately obvious and a number of parameter sets were tried in order to describe the ground state transition adequately. Table 1 lists some of the various optical model parameter sets which were tried, as well as the neutron bound state geometry employed. Also, the configuration of the transferred neutrons was assumed to be (lf~) 2 for all four levels. Sensitivity to optical model parameters was found to be great as is illustrated in fig. 5 where calculations that use different proton and triton potentials are compared with the ground state angular distribution. A glance at table 1 shows that, numerically, proton potentials P8F and P6F are quite similar, yet the resulting DWBA cross sections are very different. Triton potential T6F has a real volume integral per nucleon pair of - 3 4 4 MeV.fm 3 and, hence, is a "shallow ''27) potential. The corresponding value for T7F is - 4 1 7 MeV. fm 3, classifying it as a "deep" 27) potential. It is obvious from the results shown in fig. 5 that the deep potential is

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~2C, 54Fe, 2°sPb(p, t)

101

preferred, in accord with the findings of several previous studies at lower energies 2a). The DWBA results also proved to be sensitive to the choice of the neutron binding well geometry. Agreement with the shape of the experimental angular distributions was optimized when the well radius parameter was set to ro = 1.29 fm. The value ro = 1.25 fm, as used in ref. 26), made the agreement in shape significantly worse, although the overall magnitude of the calculations was affected only slightly. Corrections due to the finite range of the interaction of the proton with the two neutrons were found to modify the shapes of the theoretical angular distributions somewhat while leaving the magnitudes substantially unchanged. In fact, a range parameter of R = 0.60 fm gave results very similar to those obtained without finite range but with a reduced bound state radius parameter of 1.25 fm. The similarity of results obtained by these two procedures may explain why Suehiro et al. 24), using ro = 1.25 fm, found finite range corrections to be important at Ep = 52 MeV while in the present work, where the radius parameter was freely adjusted, excellent agreement was achieved without them. TABLE 3 Enhancement factors for 54Fe(p, t)52Fe

g.s. 850 2385 3583

0+ 2+ 4+ 4+

(80 MeV) a)

(80 MeV) ")

(40 MeV) b)

(40 MeV) b)

0.966 0.173 0.019 0.021

0.179 0.020 0.022

3.72 0.488 0.092 0.087

0.131 0.025 0.023

") D W B A calculations used parameters oftable I without either finite-range or non-locality corrections ; optical model parameter sets P8F, T7F were used. Configuration of picked-up neutron assumed to be (11"7/2)2. b) Ref. 26).

Comparison of calculated and measured cross sections resulted in the enhancement factors quoted in table 3 where they are compared with those extracted in a similar analysis at Ep = 40 MeV. Two features are worth noting. The ratios of the enhancement factors for excited states to the enhancement factor for the ground state are in very good agreement for the two energies. However, the 80 MeV enhancement factors are roughly one-fourth as large as those for 40 MeV. The insensitivity of the magnitude of the zero-range DWBA cross sections to uncertainties in the neutron binding well or the approximate treatment of finite-range effects suggests that the discrepancy in enhancement factors is real and not an artifact of uncertainties in these DWBA parameters. Possible explanations of this effect will be discussed below.

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J.R. SHEPARD et al.

t)z°6pb reaction

5. The Z°Spb(p,

The nucleus 2°8pb has several features which recommend it as a target in the present study. There have been a number of comprehensive investigations of 2°Spb(p, t) at energies up to Ep = 40 MeV [refs. 29- 34)]. Further, there are many 10 2

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208pt)(p,t ) ZO6pb Ep= 80MeV

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12C, 54Fe, 2°Spb(p, t) 102 [E

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well-separated low-lying levels excited which are characterized by L-transfers ranging from 0 to 9. In addition, the shell model wave functions for these states have been studied extensively and are relatively well determined 35.36). In the present study, triton energy spectra were measured up to 4.5 MeV of excitation in 2°~Pb. However, due to the limited experimental resolution only eight states up to 3253 keV in excitation were analyzed. Angular distributions for these eight levels appear in figs. 6 and 7. The usual distinctive shape of an L = 0 transition is seen for the ground state, while the familiar behavior of the stripping peak moving out in angle with increasing L-transfer is observed for the other levels. The L = 2 stripping peak is forward of our lowest angle measurement while the L -- 4 peak is at about 9°; for L = 9, the peak has moved out to about 23 °. Many of these levels have been observed in previous studies and all eight were seen at Ep -- 40 MeV by

104

J . R . S H E P A R D et al. I.O

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Fig. 8. The ratios, R, of maximum experimental differential cross sections at Ep = 80 and 40 MeV are shown for 2°sPb(p, t)2°6pb transitions having various L-transfers. R = (dtr/dfl)m=z(80 MeV)/(da/dfl)m,x (40 MeV). Note that the most favorable angular transfer, classically, is A K R = Igp-g,I × 1.25 x 1081/3 9.4. Hence, as is observed in the data, an L = 9 transfer is strongly favored. At Ep = 40 MeV, A K R ,~, 6.3.

Smith et al. 34). Maximum cross sections are typically one order of magnitude smaller than those observed at 40 MeV, with the exception of the 9- level which, due to favorable angular momentum matching at 80 MeV, has roughly the same strength at the two energies. This effect is demonstrated in fig. 8 which shows the ratios of maximum cross sections at E v = 40 and 80 MeV for various L-transfers. DWBA calculations were performed as described in sect, 3. The optical model and neutron bound state parameters used are given in table I. The configurations of the transferred neutrons were determined by the leading components of the 2°6pb wave functions of True 36) and are quoted along with the extracted enhanceTABLE 4

Enhancement factors for 2°SPb(p, t)z°6Pb Ez

J~

Config.

e (80 MeV) =)

e/e~.s, (80 MeV) =)

g.s. 803 1684 1998 2200 2568 2924 3253

0+ 2+ 4+ 4+ 794+ 6+

(3p~-/2) - t 2f5/2) -1 (3pt/2, -1 2fs/2) -t (3p3/2, (2f~-/~) (3p~/~, I i~-~/2) (2f~/~, 1i~-3~/2) - 1 2f7/2) -t (3pt/2, (2f~/t2, 2f7/t,)

0.279 0.279 0.099 0.329 0.062 0.064 0.075-0.113 0.131

1.00 0.353 1.18 0.221 0.228 0.270-0.404 0.471

e e/e=.s. (40 MeV) b) (40 MeV) b) 3.36 2.81 1.! 7 6.56 1.00 0.795 0.506 1.37

0.835 0.348 1.951 0.297 0.236 0.150 0.407

=) DWBAcalculationsusedparametersoftable 1 without either finite-rangeornon-loca!itycorrections. b) Used data of ref. 34). DWBA calculations used optical model potentials o f ref. 34) ~and neutron bound state geometry of table 1. N o finite-range or non-locality corrections were used.

12C, 5'tFe, 2°spb(p, t)

105

ment factors in table 4. Agreement with the shapes of the experimental angular distributions is generally excellent except perhaps for the 2924 keV 4 + level where the shape and location of the forward angle maximum are poorly reproduced. Thus at the present energy the single-step DWBA is seen to be able to describe simultaneously (p, t) transitions which are very different dynamically in the sense that the L = 0 and 2 transitions are very much disfavored on the basis of semiclassical angular momentum arguments, while L = 9 is strongly favored. In table 4 the extracted enhancement factors for Ep = 80 MeV are compared with those obtained from the 40 MeV data of Smith et al. a4). The 40 MeV DWBA calculations employed the optical model parameters used by Smith et al. 34) and the neutron configurations and binding well geometry of the present work. As was the case for 54Fe(p, t), the ratios of enhancement factors to the ground state enhancement factor agree well for the two energies (except for the 2924 keV 4 + level). However, the 80 MeV values are smaller than those for 40 MeV, in this case, by a factor of 12. At this point, use can be made of the simplicity of the wave functions of the 2°6pb levels. According to the calculations of True 36), the 7- and 9- levels are essentially single configuration states, (3p~- 1, li~1)7_ and (2f~-t, li~1)9_, respectively. It is then reasonable to assume that if the (p, t) reaction is adequately described by the present DWBA calculations, enhancement factors for these two levels should be nearly unity. At Ep = 40 MeV such is the case. Furthermore, at 40 MeV, enhancement factors for other levels are consistent with what is gefierally expected of the (p, t) reaction; for example, the ground state enhancement factor is appreciably greater than unity as should be expected since all the configurations which have been omitted interfere constructively and would thereby increase the calculated cross section if included. At 80 MeV, however, the 7- and 9- enhancement factors are both about 0.064 while that for the ground state is 0.279, much less than unity. This result, taken together with the similar but less dramatic findings for the 54Fe(p, t) reaction, suggests that there is an overall energy dependence of the (p, t) cross section which is not reproduced by the single-step DWBA. Specifically, for 2°Spb, the observed (p, t) strength drops twelve times faster between 40 and 80 MeV than is predicted by the simple DWBA. Furthermore, this effect appears not to be a function of L-transfer or nuclear structure, but again suggests that there is an energy dependence to the overall (p, t) normalization. Variation with energy of the normalization of the zero-range DWBA has been examined in previous papers for the (p, d) [ref. 3~)] and the (3He, 0t) [ref. 22)] reactions and it is not unreasonable to expect a like phenomenon for the (p, t) reaction. Such an energy dependence, when observed, can be traced to the variation of momentum transfer to the ejectile, in this case the triton. Specifically, the Fourier transform of the quantity Vp.2n~p_2nwhich "drives" the (p, t) reaction is not a constant as implied by the zero-range assumption but is likely to decrease as the momentum transfer, q = [Kp-~Kt[, increases. (Such a f'mite-range effect is outside the region of validity of the simple first order corrections and must be treated in exact finite range.) At

106

J . R . SHEPARD et al.

40 MeV, q = 0.672 fm-1 for the 2°sPb(p, t)2°6pb(g.s.) reaction, while at 80 MeV q = 0.913 fm-1. These momentum transfers assume the asymptotic wave numbers and 0° scattering. Such an assumption is partially invalidated by distortion in the proton and triton channels since momentum components other than the asymptotic ones are thereby generated. Furthermore, these effects vary from nucleus to nucleus, e.g., distortion is more pronounced for a large nucleus like Pb than for a small one like C. However, if there is a minimum in the square of the Fourier transform in the vicinity of q = 1 fm- 1, a substantial reduction in the overall normalization of the DWBA could result. Such a speculation can be tested only via careful exact finiterange DWBA calculations utilizing a realistic triton wave function. Such calculations are beyond the scope of the present paper. It is also possible that the apparent change in overall DWBA normalization is due to other phenomena such as interference between multi-step and one-step reaction modes that the DWBA is unable to describe even phenomenologically. Exploration of this possibility awaits reliable techniques for calculating the two-step contributions.

6. Summary and conclusions Angular distributions for transitions to strongly excited, low-lying levels have been measured for the 12C, S4Fe and 2°SPb(p, t) reactions at a bombarding energy of 80 MeV. As at lower energies, L -- 0 angular distributions have a distinctive diffractive structure while for other transitions the relative location of the stripping peaks is dependent on the L-transfer in a straightforward way. Comparison with other measurements shows that maximum cross sections typically drop by about 30 ~o between bombarding energies of 40 and 80 MeV for S4Fe(p, t). For 2°Spb, cross sections drop by roughly an order of magnitude. Simple single-step DWBA calculations, without either finite-range or non-locality corrections, generally reproduce the shapes of the observed angular distributions very well. The exception is 12C(p, t) where agreement for the two excited levels is poor, particularly at back angles. The DWBA predictions of relative strengths for various transitions at 80 MeV are consistent with those extracted from 40 MeV data. For 2°sPb(p, t), the transitions cover a large range of angular momentum transfer ranging from kinematically favored (L = 9) to highly disfavored (L = 0). The DWBA consistently overpredicts the magnitude of the cross sections, that is, enhancement factors are typically less than unity even for ground state transitions where coherence effects usually require large enhancement factors when simple wave functions are employed. More significantly, enhancement factors for transitions in 54Fe(p, t) are about a fourth as large at 80 MeV as at 40 MeV. For 2°apb, the values at the higher energy are less than one-tenth as large as at the lower energy. Such a reduction is observed for all L-transfers. The ratios of enhancement factors for different levels are about the same for both bombarding energies for both 54Fe and 2°apb. Thus

12C, S4Fe, 2°sPb(p, t)

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the simple DWBA approach used here is seen to be able, over a wide range of dynamic conditions, to account for all major features of the data except for the overall normalization. The utility of the method as a phenomenological spectroscopic tool is thereby established for the bombarding energy used in the present work. We speculate that the normalization discrepancy may be a gross finite-range effect arising from an increased momentum transfer to the triton as a consequence of the high bombarding energy. Such a calculation should be tested in the future by exact finite-range calculations using a realistic triton wave function. Alternatively, a subtle energy dependence of the way in which one- and two-step reaction modes interfere might be responsible for the discrepancy. Future tests of this possibility depend on development of reliable methods of calculating the two-step contributions. We wish to thank Prof. R. Pollock and the staff of the Indiana University Cyclotron Facility for their help in the completion of the experiment. We are also indebted to Mr. M. Kirslaner and Mr. B. Campbell for their efforts in analyzing the data.

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