Reaction mechanism of 32S(d, pα)29Si ground state reaction at Ed = 12.0 and 17.0 MeV

2.F

I

Nuclear Physics A123 (1969) 546--560; (~) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprmt or microfilm wxthout written permission from the publisher

REACTION

MECHANISM

O F S2S(d, p~)29Si G R O U N D

STATE REACTION

A T Ea = 12.0 A N D 17.0 M e V J. A. BURKE t, J. X. SALADIN and A. A. ROLLEFSON tt Nuclear Physws Laboratory, University of Ptttsburyh, Ptttsburyh, Pennsylvania ttt

Received 26 September 1968 Abstract: The 32S(d, pot)~Sl g.s. reaction was stu&ed at bombarding energies of 12.0 and 17 0 MeV.

Coincident protons and alpha particles were detected using sohd state detectors, and their correlated energy spectra were analysed by a 64 × 64 channel pulse-height analyser. Measurements were carried out at various angular combmatxons (0p, 0~). The angular dependence of the &fferentlal cross sechon and the shapes of the energy spectra were compared with the results of some s~mple model calculations. It was found that at a bombarding energy of 12.0 MeV, the reaction proceeds mainly by a double evaporation process, although for small proton angles a slgnLficant contribution of a direct (d, p) reaction followed by an alpha evaporation is visible. At a bombarding energy of 17.0 MeV', the reaction mechanism is predominantly the latter direct-evaporation process. i

E [

I

NUCLEAR REACTIONS azS(d, pot), E a : 12.0 and 17.0 MeV; measured g(Ed; Ev, 0p, 0~); deduced reaction mechanism. Natural target

1. I n t r o d u c t i o n

T h e d e t e r m i n a t i o n o f r e a c t i o n m e c h a n i s m s in t h r e e - b o d y b r e a k - u p reactions like (d, 2p), (d, 2~) a n d (d, p ~ ) etc. has for some time been a t o p i c o f interest. Originally, this t y p e o f r e a c t i o n was s t u d i e d b y m e a n s o f a c t i v a t i o n cross-section m e a s u r e m e n t s . These a c t i v a t i o n cross-section m e a s u r e m e n t s were c o m p a r e d with two types o f t h e o retical calculations. T h e first t y p e o f c a l c u l a t i o n is b a s e d on a r e a c t i o n m e c h a n i s m w h i c h involves three distinct steps. T h e first step consists in the f o r m a t i o n o f a highly excited c o m p o u n d nucleus b y the i n c o m i n g deuteron. In the second a n d t h i r d steps, this c o m p o u n d nucleus decays b y the sequential e v a p o r a t i o n o f two particles (in o u r case a p r o t o n a n d an a l p h a particle). W e shall refer to this m e c h a n i s m as the c o m p o u n d nucleus e v a p o r a t i o n ( C E ) model. T h e s e c o n d t y p e o f calculation assumes t h a t the r e a c t i o n p r o c e e d s in two stages, w h e r e the first stage is a direct reaction. W e m a y i m a g i n e t h a t at this first stage the d e u t e r o n interacts with the t a r g e t nucleus causing the n e u t r o n to be c a p t u r e d , while the p r o t o n is scattered b y t h e nucleus. The n e u t r o n c a p t u r e leads to a highly excited interm e d i a t e nucleus. T h e second step consists t h e n in the decay o f this nucleus v i a the t Present address: Knolls Atomic Power Laboratory, Box 1072, Schenectady, New York, 12301. tt Present address: Physics Department, UmversJty of Notre Dame, Notre Dame, Indmna. t** Work supported by the National Science Foundation. 546

a~S(d,p~t)SeS1 g.s REACTION

547

evaporation of a second particle. We shall refer to this mechanism as the directevaporation (DE) model. The amount of information that can be obtained from activation measurements is very limited. For a given bombarding energy and target nucleus, they provide only one single number, i.e. the activation cross section. This is the sum of all total cross sections which lead to those excited states of the final nucleus which are stable against particle emission, and any information that one obtains from such a measurement must be interpreted as an average over all the reactions which lead to different final states of the same nucleus. The relative contributions of various reaction mechanisms may however be different for different final states. This difficulty was overcome in another type of experiment 1,2) in which the energy of each of the reaction products was studied in coincidence using a two-parameter analysis. Therefore, only reactions leading to the ground state of the residual nucleus were considered, and data were taken as a function both of the angle and of the energy of each reaction product. In this manner the 35C1(d, 2p)35S g.s. reaction 1) at 14.5 MeV bombarding energy was found to proceed mainly by a DE mechanism at forward angles and by a CE riaechanism at backward angles. The 35Cl(d, 2~)29Si g.s. reaction 2) between E d = 9.0 and 12.0 MeV, on the other hand, was found to proceed by a CE process at all angles. These two experiments suffered from one basic difficulty, i.e. the detected particles were identical. Thus, if detectors 1 and 2 are at angles 01 and 02, respectively, the spectrum will be made up of the events in which the first emitted particle is detected in detector 1 and the second emitted particle is detected in detector 2 as well as the reverse situation, i.e. the second emitted particle detected in detector 1, while the first emitted particle is detected in detector 2. This introduces a considerable ambiguity in the interpretation of the spectra. In the present experiment, this difficulty is overcome, since by using detectors of suitable thickness, it is possible to distinguish between protons and alpha particles. Furthermore, since the 17.0 MeV deuteron beam recently became available, it was hoped that some information about the energy dependence of the reaction mechanism could be obtained by using bombarding energies of 12.0 and 1.70 MeV.

2. Experimental procedure The experiment was performed using the deuteron beam from the University of Pittsburgh tandem Van de Graaff accelerator at bombarding energies of 12.0 and 17.0 MeV. Protons and alpha particles were detected by surface-barrier solid-state detectors with depletion depths of 1000 pm and 70 pm, respectively. At 17.0 MeV, the 70 pm detector was replaced with one which had a depletion depth of 200 pro. Each had an active area of 200 mm 2. The angular spread of the detectors was _+7.5 ° in the reaction plane and in the direction perpendicular to it. The coincidence, i.e. Ep versus E~, spectra were recorded in a 64 × 64 channel aria-

~. A. BURKE et al.

548

lyser, which was gated by the output of a fast coincidence unit with a resolving time 2r = 40 nsec. The targets used consisted of 0.35 mg/cm 2 Sb2S 3 evaporated onto a 20/~g/cm 2 carbon backing. The product of the target thickness times the total charge acctrmulated was monitored by observing the 32S(d, p)33S ground state reaction using a solid-state detector. 3. Results for Ea = 12.0 M e V A typical two-dimensional spectrum as it appears at the output of the multi-channel analyser is shown in fig. 1. This corresponds to a beam energy of 12.0 MeV and detector angles o f 0p = 140 ° and 0= = 40'. The kinematic locus along which the events 2 I

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549

a2S(d,pct)Z°Slg.s. REACTION

c o r r e s p o n d i n g to the S2S(d, pa)295i g.s. lie appears as an almost straight line. Below 2.5 MeV, the a-detector can no longer differentiate between a-particles and protons, since it stops p r o t o n s up to 2.5 MeV. Therefore, the region below E~ = 2.5 MeV contains, in addition to the kinematic locus o f interest, the locus corresponding to protons stopped in the alpha detector and alphas detected in the p r o t o n detector. Furthermore, this region contains a large n u m b e r o f accidental coincidences caused b y protons and elastically scattered deuterons passing t h r o u g h the alpha detector. There are two types o f information which are o f interest, namely the differential cross section (d2a/df2pdQ~)(0p, 0~), which is proportional to the sum of all events on the kinematic locus, and the projection o f all the events on the kinematic locus o n t o the axis o f one detector. In the present case, the data were always projected onto the axis o f the p r o t o n detector. Differential cross sections were measured for various angular combinations (0p, 0~), where 0p and 0~ are the angles o f the p r o t o n and alpha detectors with respect to the incident beam. Table 1 lists the results o f these measurements. F o r all angular c o m binations studied, the p r o t o n and alpha detectors were on opposite sides o f the beam. The errors q u o t e d arise f r o m two sources, statistics and an extrapolation, which was necessary for some runs in which the kinematic locus was obscured in the region where E~ < 2.5 MeV. The listed errors refer to the relative cross sections, however the uncertainty in the absolute cross section is larger and is believed to be a b o u t 30 ~ . TABLE 1 Results ofthe differentml cross-section measurements 0p (deg)

30 40 50 130 140 150 40 140 50 130

0~ (deg)

30 40 50 130 140 150 140 40 130 50

dZtr d-Qpd-Q~ (lub/sr2) 18404-98 1045±67 952±35 5014-39 5334-56 6944-76 796±7l 723±22 7674-42 6334-16

Phase space factor

d2a d.Qpd.Q~/P

P(Op, 0~) (arbitrary umts)

(arbitrary units)

0.996 0.996 1.000 0.518 0.463 0.419 0.737 0.954 0.753 0.921

18475_ 98 1049± 67 9524- 35 9674- 75 1151±121 1656±181 10804- 96 7584- 23 10194- 56 6874- 17

C o l u m n 4 o f table 1 lists the available phase space in arbitrary units. The last c o l u m n gives the ratio o f the measured cross sections divided by the phase space. Fig. 2 shows the angular distribution o f (d2tr/df2pdf2,)(0) for symmetric counter configurations (i.e. 0p = 0, = 0), the corresponding phase space factors and the ratio d2tr/dI2pdf2~

550

J.A. BURRE et al.

divided by the phase space. Some of the projected energy spectra are shown in figs. 5, 7-9.

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Fng. 2. (a) Differential cross section (d=ty/dP-pd-Q=)(0p= O== O), (b) Phase space factor in arbitrary units, (c) (d2a/d-Qpd-Q~)divided by phase space.

4. Interpretation

In considering the data, one is immediately struck by the great difference in the appearance of the 12.0 and 17.0 MeV data. At 12.0 MeV, the projected spectra a r e rather smooth and cover most of the available kinematic locus. At 17.0 MeV, on the other hand, the projected spectra show a fairly sharp peak at a proton energy of about 10.5 MeV and only very few events below 7 MeV. This indicates that different reaction mechanisms dominate in these two energy regions. In this section, these data are compared with the predictions of simple CE and DE models for the reaction mechanism. Let us first discuss the Ed = 12.0 MeV data. Fig. 2(c) shows that the differential cross sections after being divided by phase space are roughly symmetric about 90 °. This is consistent with a pure compound nucleus-double evaporation picture. However, the error bars on the 0 = 30 ° and 0 = 150 ° points are quite large. Therefore within these errors, there may still be a small amount of forward peaking at 0 = 30 ° corresponding to a direct contribution. The differential cross sections do not, however, provide a very sensitive test of the details of the reaction mechanism. A much better test is provided by the projected energy spectra. In general, we can describe the projected spectra as being characterized by two

3 2 S ( d , pc~)29Si g . s . R E A C T I O N

551

broad peaks with a shallow minimum between them. In cases for which the proton is emitted at a backward angle, the peak corresponding to a low proton energy is generally about the same intensity or slightly more intense than the high-energy peak. However, when the proton detector is forward, the peak at high proton energy ( g 5-6 MeV) is considerably more intense. SECOND EMISSION COMPETITION

COMPETITION WITH FIRST EMISSION

SECOND EMISSION COMPETITION IN d, pet)

aN ( d , a p )

A

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34CI Fig.

3.

Energetics of the s~S(d, p0~)29Si reaction.

In the following, we compare these spectra with the predictions of a compound nucleus-double evaporation model and a direct reaction-evaporation model. The quantity which we wish to calculate is d3
__ 0"= (4n) 2

[PI(Ep)P2(E~) + PI(E~)P2(Ep)'],

(1)

where a c is the cross section for the formation of the compound nucleus, PI(Ep) the probability for its decay by the first emission of a proton of energy Ep and P2(E=) the probability for the decay of the intermediate nucleus by the emission of an alpha particle of energy E=. The evaporation probabilities P1 and P2 are calculated from standard compound nucleus decay theory 3). The energetics of the 32S(d, p~)29Si reaction are shown in fig. 3. In calculating P1 and P2, competing decays via proton, neutron, deuteron and alpha channels were taken .into account. Charged particle in-

552

s . A . BURKE

et al.

verse cross sections were taken from the tabulated values of Shapiro 4) for a radius parameter r o = 1.6, while the neutron inverse cross sections came from the expression of Blatt and Weisskopf 5) at(n) ~ rc(R + ;~)Z4kK/(k + K) z.

(2)

For a neutron channel energy En < 1 MeV, the value of at(n) at 1 MeV was used, since the expression becomes infimte for En = 0. The inverse cross sections used here are based on the assumpUon of a completely absorbing nuclear square well potential. This approximation should become increasingly better as the excitation energy of the residual nucleus increases 3), because the excited nucleus is less transparent to incident nucleons than is the ground state due to the decreased importance of the exclusion principle for excited states. The level density functions which enter into the calculation of the emission probabilities were assumed to be of the form given by the Fermi gas model 6)

p(U) = x~x/-~a- '/" (U') -s/4 exp

[2~/a U'],

(3)

where U is the excitation energy of the residual nucleus, and the effective excitation energy U' is expressed by U'= U-6. (4) The pairing energy correction 6 to the ground state is given by the following approximate expression 6- 8): 6 = 3 . 3 6 D ( 1 - 4 o1o A) MeV, (5) with 0 doubly odd nuclei D = ½ for odd-mass nuclei 1 double even nuclei. The Fermi gas model also predicts t x) that the level density parameter a is proportional to the atomic mass A of the residual nucleus, i.e.

a = A/y

(6)

In the present case, f = 3.33 agrees with the value of the level density parameter a found by Baron and Cohen 12) for the (d, ap) reaction on the nearby isotope 3SC1. The effect of the variation of the p a r a m e t e r f of eq. (6) on the shape of the spectra can be seen in fig. 4, in which the quantity d3a/dEpdf2pdf2, (Ep, 0p = 30 °, 0, = 30 °) is plotted versus the lab energy of the protons for a beam energy of Ed = 12.0 MeV and f o r f = 8.0 MeV, 3.33 MeV and 2.0 MeV, respectively. The curve at low proton energy corresponds to the 32S(d, ~p) reaction, while the high-energy curve corresponds to the aZS(d, pa) reaction. The sum of the two (solid curve) is proportional to the CE prediction of the shape of the projected spectrum. The dot-and-dash curve of fig. 4(c) shows the shape which would be expected if the reaction were governed solely by phase space. Figs. 4(a)-(c) are separately normalized.

azS(d,p~x)egSl g s REACTION

553

The obvious testing grounds for this type of calculation are the symmetric (0p = 0, = 0) measurements m the backward hemisphere. In fig. 5, the data at 0 = 130 °, 140 ° and 150 ° are compared with the results of a CE calculation. The data are shown as histograms plotted as the total number of counts versus the proton lab energy. The I0 8 6

-

MeV

.0

42

03

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2

5

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3

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3

4(b) 5

6

7

8

Ep(MeV)

f=20MeV

4 (C) 5

6

7

8 Ep(MeV)

Fig. 4. The CE prediction dao'/dEpdf2pd,,Q= (sohd curve) for 0p = 0= = 30 ° versus proton lab energy for various values of the parameter f ( e q u a t i o n (6)]. ( a ) f = 8.0 MeV, ( b ) f = 3.33 MeV, ( c ) f = 2.0 MeV. The dashed curves are the individual (d, pc) and (d, ep) predictions, The dot and dash curve in (c) is the shape predicted by phase space.

gross structure of these data shows in general a rather broad spectrum. This supports our choice o f f = 3.33 in eq. (6) for the level density parameter, since this choice gives the broadest prediction (fig. 4(b)). Actually, a reasonable shape would be obtained for any values f r o m f = 2.75 to 3.33. The dashed curves labeled CE in fig. 5 are the result of the above mentioned calculation. The difference in the shape of these curves

554

J.A. BURKE e t al.

arises from the difference in the relative kinematic shifting with angle of the calculated (d, p~) and (d, ~p) spectra. Fig. 2 shows a considerable angular dependence in the differential cross sections.

~

200 160 120 80 40

= S a = 130"

0 I

Z

5 (0)4

5

6

zoo- ~I! ~ ' ,

EptMeV}

240 -

Z 0u

ep = err = 1 4 0 '=

160 120 - 1 7 "

eo-

C E--'~' \

I/

',\

m

Z

DE

O=

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2

~

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s

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nrrL_ n I1 1]

,,o_

o°;o.=,oo.

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-

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0

I

2

3

4

5

6

Ep (MeV)

(c) Fig. 5. N u m b e r o f c o u n t s p e r c h a n n e l versus p r o t o n lab energy for angles in the b a c k w a r d h e m i s p h e r e for E a = 12.0 MeV. T h e d a s h e d curves are the C E a n d D E predictions o f the s h a p e , while the s o h d curve is their s u m . (a) 0p = 0= = 130 °, (b) 0p = 0~ = 140 °, (c) 0p = 0= = 150 °.

The calculation, on the other hand, takes no angle-dependent effects into account. It is therefore not meaningful to compare absolute values. At each angular combination, then, the calculated curves have been normalized separately. Fig. 5 shows that the CE curves are in good qualitative agreement with the data. The CE predictions of

asS(d, p~.)=~Si g s. R E A C T I O N

555

other angular combinations can be seen in figs• 7-9. One notes that the CE prediction becomes progressively worse as the proton detector is moved to forward angles• A pronounced peak appears at high proton energy, which cannot be reproduced by CE calculations. We now consider the predictions assuming a D E mechanism• F o r the D E process, we m a y write 1)

dao'(Ep, 0p, 0~)

d2o--

dEp dg'2pd ~ =

-

-

(d,

dEp d~'2p

p; 0p, Ep)P2(E,),

(7)

where the first factor is proportional to the cross section for stripping into the con-

tinuum, resulting in the emission of a proton of energy Ep. The second factor for the subsequent alpha evaporation is identical to P2(E=) or eq. (1). The plane wave expression

x/Ep-Ze2/r sin 0p , (8) Pd(Ep, 0p)dEpd0p O~ [I + 2Ep + e ~ - 3Ze2l r - 2x/2x/ Ep - Ze2/rx/ Ed -- ZeZlr cos 0p] z which was derived in ref. 1), was used to calculate the energy dependence of the stripping cross section into the continuum. The energies and angles appearing in eq. (8) refer to the center-of-mass system; ! is the deuteron binding energy. This expression was evaluated for various angles setting the reaction radius r equal to r = ro A'~ fm. (9) The functions Pd and P2 and the product fig. 6.

PaP2 calculated

for r o = 1.6 are shown in

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6O

Pd ---P.

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3

4

5

Ep (MeV)

6

7

8

Ep(MeV) (b)

Fig. 6. (a) Emission probabdities Pd and P~ for various lab angles as a function of Ep (channel energy). (b) The product PdP2 for various lab angles as a function of Ep (channel energy).

556

J.A.

BURKE

e t al.

A good testing g r o u n d for this calculation is the most forward a n g u l a r c o m b i n a t i o n studied, i.e. 0p = 0= = 30 °. The histogram in fig. 7(a) shows these data. The solid curve is the result of a CE calculation at this angle. The histogram in fig. 7(b) is the difference between the data and the CE calculation. The sohd curve in fig. 7(b) is the result of a D E c a l c u l a h o n for a radius p a r a m e t e r r o = 1.6, the d o t - a n d - d a s h curve for r o = 1.25 a n d the dashed curve for r 0 = 1.8. O n the basis of this type of calculation, the value of ro = 1.6 was chosen as being best, but a n y value from 1.5 to 1.8 would have been adequate.

200 160 120 u~

80

z

~c~

40 °

I

2

3 (0)4

5

6

7

5

6

7

EplMeVl

IM ,,n

160 Z 120

.... ....

to= i 25 to= 160 to= 1 8 0

8O 40 0

I I

I 2

I 3

I 4

(b)

Ep (MeV)

Fig. 7. (a) The number of counts per channel at 0p = 0= = 30° as a function of the proton lab energy. The solid curve is the CE prediction. (b) The histogram is the difference between the above energy spectrum and the CE calculation, while the curves show the DE predictions for various values of ro in eq. (9).

Thus, neither a pure CE nor a pure D E reaction m e c h a n i s m can explain the shapes of the measured spectra. By using a c o m b i n a t i o n of these processes, however, we can reproduce the gross features of the measured spectra rather well. The data a n d the results of CE a n d D E calculations for other a n g u l a r c o m b i n a t i o n s at Ed = 12.0 MeV are shown in figs. 5, 8 a n d 9. The solid curves in these figures show the sums of the C E a n d D E contributions. F r o m the above data, we note that the runs for wh]ch the p r o t o n detector is forward require a significant D E c o n t r i b u t i o n to fit the shapes, but when the p r o t o n detector is at a backward angle, little, if any, D E c o n t r i b u t i o n is necessary. As was m e n t i o n e d earlier, the appearance of the data at 17.0 MeV is qmte different

32S(d) pg.)29S1 g.s. REACTION

f r o m that at 12.0 MeV. The C E higher-energy case and were able A C E calculation at 17.0 MeV as large at Ed = 17.0 MeV as at

55'7

and D E reaction mechanisms were applied to the to explain the spectra very well. predicted that the C E probability is only a b o u t 1 12.0 MeV. A D E calculation, on the other hand,

280

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/L~

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/

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u3 I,Z 0 cJ IJ_ 0

120

)

80 40 0

IT_. IJ.I

I

2

3

4 (Q) 5

6

7

8

F'p(MeV)

I

2

3

4

6

7

8 F'p ( M e V )

200 Z 160 120 80 40

(b)

5

Fig. 8. Analogue o f fig. 5 for the symmetric forward detector positions. (b) 0 p = 0 = = 5 0 °.

(a) 0p = 0= = 40 °,

showed that the D E process is m u c h m o r e probable at 17.0 than and at 12.0 MeV. The D E probability for 0p(lab) = 20 ° at 17.0 MeV is more than 4.5 times as great as for 0p(lab) = 30 ° at 12.0 MeV. (At 12.0 MeV, 0p(lab) = 30 ° has the highest D E probability.) Even at 0p(lab) = 40 °, the D E probability at 17.0 MeV is more than twice the value for 30 ° at 12.0 MeV. The projected spectrum for 0p = 20 ° and 0~ = 40 ° is shown as the histogram in fig. 10(a), while that for the symmetric case o f 0p = 0~ = 40 ° is shown in fig. 10(b). The solid curves show the results o f the D E calculations normalized to the data in each case. The shape and position o f the calculations agree qmte well with the data. The low-energy p e a k of the CE contribution is expected to be between 2 and 5 MeV p r o t o n energy. In this area, we find a b o u t two counts per channel at 0p = 20 ° and

558

J . A . BURKE et al.

about six counts per channel at 0p = 40 °. This is in general agreement with the above mentioned expectations, which give about one count per channel at 0p = 20 ° and two counts per channel at 0p = 40 °. 320 280

0p = 40*

240

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200 r~ 160

#

120

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3

4

5

6

7

8 Ep(MeV}

3

4

5

6

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8

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(a)

i

0p = 130" 200 160 120

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0p = 140"

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Fig. 9. A n a l o g u e o f fig. 5 for s o m e a s y m m e t r i c detector po'sltions. (a) 0p = 40 ° , 0~ = 140°; (b) 0u = 130 ° , 0~ = 50°; (c) 0p = 140 °, 0~ = 40 ° .

a~S(cl, po029Sl g s. REACTION

559

The predominance of the D E mechanism is further supported by a comparison of the cross sections for the symmetric angular combinations 0 = 40 ° and 0 = 140 °. Due to a low counting rate and considerable background, few data were accumulated at 0 = 140 °. However, it was possible to estimate an upper limit on the number of counts at 0 = 140 °. Using this it was determined that the differential cross sections are definitely forward peaked even after dwision by the phase space.

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F i g . 10. N u m b e r o f c o u n t s p e r c h a n n e l v e r s u s p r o t o n l a b e n e r g y for Ed ~ 17.0 M e V . T h e s o l i d c u r v e s are t h e r e s u l t o f a D E c a l c u l a t i o n o f the e n e r g y s p e c t r a . (a) 0p = 20 °, 0= ~ 402; (b) 0p = 0= = 40 °.

Therefore, we conclude that the reaction mechanism at E d = 17.0 MeV is predominantly of the D E type. The authors would like to thank J. R. Kerns for the preparation of the CE computer program, P. A. Crowley and J. E. Glenn for their help in taking data and S. Gangadharan for weighing the targets. Special thanks are also due to W. McNamee for assistance in the design and maintenance of the electronics. Computations for this work were performed with the Univers,ty of P=ttsburgh computer which is supported by the National Science Foundation under Grant Number G-11309.

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s.A. BURKE et al.

References 1) P. F. Brown, J. X. Saladin, A. A. Rollefson, J. A. Burke, P. A. Crowley and J. R. Kerns, Nucl, Phys. A107 (1968) 529 2) A. A. Rollefson, J. X. Saladin, J. A. Burke, P. A. Crowley and J. R. Kerns, Nucl. Phys. A107 (1968) 545 3) T. Ericson, Advan. Phys. 9 (1960) 425 4) M. M. Shapiro, Phys. Rev. 90 (1953) 171 5) J. M. Blatt and V. F. Welsskopf, Theoretical nuclear physics (John Wiley and Sons, New York, 1952) p. 349 6) D. W. Lang, Nucl. Phys. 26 (1961) 434 7) A. Stolovy and J. A. Harvey, Phys. Rev. 108 (1957) 353 8) A. G. W. Cameron, Can. J. Phys. 35 (1957) 666 9) T. D. Newton, Can. J. Phys. 34 (1956) 804 10) A. G. W. Cameron, Can. J. Phys. 36 (1958) 1040 11) J. M. B. Lang and K. J. LeCouteur, Proc. Phys. Soc. A67 (1954) 586 12) N. Baron and B. L Cohen, Phys. Rev. 129 (1963) 2636 13) D. C. Peaslee, Phys. Rev. 74 (1948) 1001