Use of polarized deuterons to determine the total angular momentum transfer in stripping reactions

Nuclear Physics A117 (1968) 1--26 ;

Q North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

T

USE ®F POLARIZE DEUTERONS DETERMINE THE TOTAL ANGULAR MOMENTUM TRANSFER' IN STRIPPING REACTIONS T. J. YULE t and W. HAEBERLI

University of Wisconsin, Madison, Wisconsin tt

Received 13 May 1968 Abstract : Polarized deuterons have been used to initiate stripping reactions to determine vector analysing powers for transitions for which the orbital angular momentum t and total angular momentum j of the captured neutron are known. Measurements have been performed at a deuteron energy of about 8 MeV for the target nuclei ®Be, i2C, s4Mg, "Ca and G$Cr . For a given 1, there is pronounced difference between the vector analysing powers forj = l+I andj = 1-I reactions. The measured vector analysing powers are compared with predictions based on the distorted-wave Born approximation. Effects of variations of optical parameters are discussed. If optical potentials which fit the elastic scattering data are used, the measured vector analysing powers are reproduced for the intermediate-weight nuclei . E

NUCLEAR REACTIONS ®Be, 1,C, s4 Mg, 4 °Ca, s2Cr (polarized d, p), E = 8 MeV; measured vector analysing power. Natural targets.

1. Introduction During the past decade, the (d, p) stripping reaction has been the most extensively studied direct reaction process and has become an important tool of nuclear spectroscopy. From the shape of the differential cross section of the outgoing protons at forward angles, it is possible to determine the orbital angular momentum 1 of the captured neutron 1 ). Thus, the parity change in the reaction is known, and the total angular momentum transfer j is determined to within one unit, i.e. j = 1± . In the case of spin-zero target nuclei, the spins of the final states are thus determined to within one unit. A number of extensive analyses 2 - 5) has been performed of the differential cross sections for incident deuteron energies of about 10 MeV in terms of distorted-wave Born approximation. The measured cross sections are in general well represented, except at very backward angles. Measurements of the polarization of the outgoing protons in stripping reactions have been suggested as a possible method to determine whether the spin of the captured neutron is parallel or anti-parallel to its orbital angular momentum 6 ). In t Present address: Argonne National Laboratory, Argonne, Illinois. tt Work supported in part by the U.S. Atomic Energy Commission . September 1968

T. J. YULE AND W. HAEHERLI addition, comparison of sucl wasurements with calculations would serve as an additional test of the theory and v, Ad yield information about the reaction mechanism . ecause polarization measurements are difficult to perform, there are not many experimental data'). Most of the measurements have been performed for light-weight target nuclei, which, from the experimental point of view, are favorable targets because of the widely spaced levels in the residual nucleus and the relatively large cross sections. However, measurements on some light target nuclei have shown a sudden change in the shape of the polarization angular distribution in a fairly narrow energy range, which indicates that interference effects with compound-nucleus formation may be present ' -1). Therefore, these light targets are not favorable for comparison with stripping theory . Some measurements are available for heavier target nuclei'- '°). At present there are not enough data to establish general rules to determine j. The 11) data are not well represented by distorted-wave Born approximation calculations 12) . except in the case of I = 0 transitions on intermediate-weight target nuclei Instead of studying the outgoing proton polarization, one may investigate the effects when polarized incident particles are used to initiate stripping or pick-up reactions 13). A measurement of the deuteron left-right asymmetry for a (p, d) pickup experiment with polarized protons is equivalent to a measurement of the proton polarization in the corresponding (d, p) reaction at the same center-of-mass energy and angle 14) . At this time only one such experiment has been performed 11). The reactions "C(p, d)"C and 16 ®(p, d)" were studied using a 30.3 MeV polarized proton beam. Poor agreement between distorted-wave Born approximation calculations and the measured asymmetries was found. In the experiment to be described here, (d, p) stripping reactions were initiated with a polarized deuteron beam from a polarized-ion source . Some of the results were reported in a recent short publication "). The vector analysing power was measured for a number of (d, p) reactions on light- awl intermediate-weight target nuclei for which j and I had been determined. The vector analysing power is identical to the vector polarization of deuterons in the inverse reaction initiated with unpolarized protons ''). The only previous measurements which used polarized deuterons to initiate stripping reactions were those by a group at Saclay l') . The vector analysing powers were measured at 22 MeV for a number of transitions for the target nuclei 12 C and 2 "Si . No analysis of these data has been reported . Whether a measurement of the vector analysing power can determine j cannot be learned from the Saclay data, since transitions with different j were not measured for the same l. The usual method to determine the spin of a final state for intermediate-weight nuclei is to study the electromagnetic radiation emitted in the decay of the state. The measurement of the directional correlation in (n, yy) reactions initiated with thermal neutrons or in (d, py) reactions have been used to determine spins ' a). However, such measurements are difficult and time consuming . ecently, a method to determine j from stripping reactions has been proposed by

Lee and Schiffer et al.

19),

who observed that for a given I, the shape of the angular

"PPMG MMM

distribution in ( , p) reactions shows a systematic dependence on j. Detailed analyses have not been successful in reproducing the observed dependence. 2

t®f M mtor wal

Since the deuteron is a spin-one particle, the description of a reaction initiated with polarized deuterons is more complex than that for a reaction initiated with polarized protons. For a given scattering angle, four parameters are required to describe the dependence of the differential cross section on the polarization of the incident beam, e.g. a vector analysing power and three second-rank tensor analysing powers ' 0). In this experiment, only the vector analysing power Pa (8) was measured 1. If the reaction is time-reversal invariant, the vector analysing power has a simple physical interprctation . It is identical to the vector polarization of the deuterons in the inverse reaction initiated with unpolarized protons at the same center-of-mass energy (ref. ")). As a consequence of parity conservation, the vector analysing power has only a component perpendicular to the scattering plane. We choose to have the values ofPd (0) positive when the vector polarization of the deuterons in the inverse reaction is parallel to k P x k. This choice of sign is consistent with the Basel convention for the polarization of the deuterons for the inverse reaction. The polarized deuteron beam was obtained from an atomic beam type polarizedion source "). For this type of source, a neutral beam polarized in electron spin is obtained by passing a neutral atomic beam through a strong inhomogeneous magnetic field . The neutral bem is then ionized by electron bombardment in a uniform weak magnetic field. No r.f. transitions are used . The spin system is symmetric about the z-axis, which is taken along the magnetic field direction at the point of ionization. In this case, the polarization state of the beam can be described by the fractional populations N, No and N.- of deuterons with spin projection mr of 1, 0 and -®1 along the z-axis. Since only relative populations are relevant, two numbers describe the polarization. One of these is the vector polarization

p., = N+-N- ;

(1)

and the other, which devends on the deuterons in the state m, = 0, is the tensor polarization (2) p.,.. = (I - 3NO). For our source, the polarization is can be set in any direction, but the magnitudes ofp. and p,,.. are fixed 20,22) p, = 0.274±0.010,

A.- = -0.293±0.010.

In the present experiment, measurements were performed at a fixed reaction angle 0 with a spin-up deuteron beam (polarization axis parallel to kd X kp) and spin-down t The vector analysing power Pd (O) is related to 1T11(0) of ref.,20) by Pd(O) = (2/-%/3) ff',(O) . With this definition, the maximum value of Pd(O) is 1 .

4

T. 1. YULE AND W. HAEBERLT

borain. The spin-up and spin-down differential cross sections are related to p = p, and to P,j (0) by the equations 2 °)

where aunp (0) is the cross section when the incident beam is unpolarized, and C(9) depends on the tensor polarization of the beam and tensor analysing powers of the reaction. From eq. (3), Pd (0) can be expressed in terms of polarized and unpolarized cross sections with C(0) eliminated 1 Pd(O) ~ 3p

dup(0) ,~, trdown(®) aunp(0)

aunp(0)

Eq. (4) indicates that if Pd (0) is to be determined, it is necessary to measure the polarized cross section relative to the unpolarized cross section, i.e. it is also necessary to perform measurements with an unpolarized beam. The vector analysing power can also be expressed in terms of the polarized cross sections and C(0) with aunp eliminated Pd(O) ® 2 «u 1)( 0) ~° edown(0) ~~ ~, 5 C(0)1 . 3p dup(o) ~- ~down(~) q. (5) shows that the vector analysing power is proportional to the left-right a>>ymmetry if C(0) is equal to zero, i.e. when a purely vector polarized deuteron beam is incident . In the present experiment usually only aup (0) and 6rdown (0) were measured, and thus Pd (0)J[1 +C(0)] was determined. However, the tensor polarization of our beam is small enough that C(®) can be almost neglected. Some measurements were performed with an unpolarized beam for normalization, and they indicated that C(0) is small, usually < 0.03 . The distorted-wave calculations mentioned in sect . 7 also indicate that the uncertainty in Pd(O) which results from neglecting C(©), is within the statistical uncertainty of the present measurements. The values of Pd(0) given in this paper were calculated from eq. (5) assuming ®~(0) _ 0. 3. Preliminary results The first measurements of the vector analysing power were for the 40Ca(d, p)41Ca reaction . This reaction is rather attractive because the final states in 41Ca which are strongly excited are well separated, and the values of d and., for these levels are known. Moreover, calculations have indicated that the, compound-nucleus contribution to the reaction cross sections is relatively small 1 ). Measurements had been performed of the elastic deuteron scattering and strivping cross sections over a wide energy and angular range 2,11, 2 3, 24) . The cross sectio;,às are fitted over a wide energy range by DWBA calculations 2, ` 11, 11,24,25) except at backward angles.

IP IN

REACTIONS

The measured vector analysing power for two 1 = 1 transitions with different values ofj e sow in . . pronounce difference in d(O) is seen for the two transitions . e the stripping peak, which is located at approximately 2®°, the slopes of d() are o opposite sign for the two values ofj. The solid line is drawn through the data points for the j = transition . The dashed line results from assuming that _ 2 d(O) i = I , i . . Such a relation holds if there are no spin JO), dependent interactions d if the transitions have the same 1-value and excitation

energy. The dashed line is seen to provide an excellent fit to the data points for the j = transition . Whether the fit is simply fortuitous is investigated in sect. 7.

eV ®Ex` i .95 MeV 1=1 ® E x - 3-95 eV :t =1

m3®2 ® 112

®;

Fig. l. Angular distributions of the vector analysing powers for two transitions with the same 1 but different j for the '°Ca(d, p)"Ca reaction . The solid line is drawn through the data points for the j = transition . The dashed line results from assuming that Pd(O)j =-. = -2Pd (8)j = 2.

The strikingbehavior ofPd(O) seen for the two 1=1 transitions for the 4° Ca(d, p)41 Ca reaction led us to consider other reactions. Reactions were chosen for which j and 1 had been previously determined and for which the strongly excitdd levels in the residual nucleus are rather well separated . For the transitions studied, stripping cross sections were available at or near the energy at which the present measurements were made. The transitions for which we have measured Pd(O) are listed in table 1 . These reactions permitted an investigation of Pd(O) for 1 = 1, 2 or 3 transitions with at least one transition with j = 1+ and one with j = 1- I for each 1.

T. J. YULE AND W. HA

6

E LY

TABLE 1

Fia states for which 1'Q(®) was measured Final nucleus

Incident deuteron energy ( eV)

Excitation (MeV)

l,

i2 ( eV)

1°Be lac

7.0

7.0 and 10.0

1,

4.57

5 6M g

8.0

7.0

2, 0, } 0, 2, 1,

5.11

61Ca

0.00 0.00 3 .09 0.00 0.58 2.56 2.80 3.41

63 ( ;r

8.0

0.00 0.57 1.00 2.32 3.61

1, 1, 3, 1, 1,

i, 0,

0.00 1 .95 2.47 3.95

Ref.

2.72

3, 1, 1, 1,

6.14

5.72

b)

$) The total angular momentum transfer j is equal to the spin of the correspondin final state except in the case of 1°B . b) The j-assignment is from the present work.

er me

meth

The experimental arrangement is similar to that, used at this laboratory for measurements of the analysing powers for elastic scattering of deuterons on intermediateweight target nuclei a o) . The accelerated, momentum-analysed deuteron beam from the University of Wisconsin tandem Van de Graaf accelerator was steered and focussed through energy control slits and beam defining slits onto the target. The unscattered beam was collected in an electrically suppressed Faraday cup. The polarized beam current was about 10 pA. ecause of the low intensity of the incident polarized deuteron beam, relatively thick targets were used. The targets were of natural isotopic composition and approximately 5 mg/crn 2 thick. The e target was obtained commercially 1. The C target

was made by painting a colloidal suspension of graphite in water tt onto a sheet of tantalum. After the solution dried, the tantalum sheet was heated which allowed the foil to be easily removed from the tantalum . The evaporation. The Cr target was rolled.

g and. Ca targets were produced by

t Brush Beryllium Co., Cleveland, Ohio. It "Aquadag" 11rom Acheson Collu'ds, Port Huron, Michigan.

STRIPPING

ACTIONS

he targets were oriented such that the spread in proton energy caused by the finite target thickness was a minimum . Therefore, measurements cof ild be made on only one side of the beam . o measure relative cross sections for spin-up and spin-down deuteron beams, it was necessary to integrate the beam. Integration was performed y detecting the elastically scattered deuterons in two silicon surface-barrier detectors placed at a scattering angle o 1Y. The two monitor detectors were set either in the scattering plane to the left and the right of the deuteron beam or above and below the scattering plane. The sum of the counts for both monitor detectors is insensitive 4.6

3 .68 3.i6

®LAB 240®

Soo

12 C(d,p) 13 C

600r

e2Cr (d,p)s 'Cr E d = B.0 MeV

E d =10.0 MeV =77.50

e

500

.61 3 .70

40

400 4,12 4 .21

-120 keV

300

232

i

200 r. 100

~- 300 Z

qs

.a

,

000 57

j

v 220 240 260

I

200 x-180 keV

:1 d.

.

1

1

. . i ' . 280 300 320 340

360

100

120

; 140

'60 (d,p)° 0 gs 160

180

v

200

B

220

i

240

CHANNEL NUMBER CHANNEL NUMBER Fig. 2. Typical proton spectra for the two detector systems employed in the measurements . The spectrum in the right-hand frame was obtained with a silicon surface-barrier detector and a 50,um TaW foil and the one in the left-hand frame with a lithium-drifted silicon detector. The excitation energies in MeV c -the the levels in the residual nucleus are indicated.

to small changes in beam direction and is independent of the vector analysing power of the elastic deuteron scattering . Since at 13° the elastic scattering is mostly Coulomb scattering, the effect of the tensor analysing power is negligible . The beam energy was reproducible to better than 10 keV, and thus the change of Coulomb cross section with energy had a negligible effect on tt. accuracy of the beam integration . The protons from the (d, p) reactions were detected either with silicon surfacebarrier detectors t with 1 mm maximum Depletion thickness or with a lithiumt Nuclear Diodes, Inc., Prairie View, Ill., Model .,H7-70-8X50 .

8

T . J . YULE AND

w.

HAEHERI,I

drifted silicon detector t of 2 mm depletion depth. With the surface-barrier detectors, it was necessary to use foils in front of the detectors since otherwise the protons would not have been stopped in the detector. A typical pulse-height spectrum for each detector system is shown in fig. 2. For both detector systems the angular accept13° at 90ö. At far ance was about ±2.5° and the azimuthal angular acceptance forward reaction angles, the vertical height was reduced to avoid too large a spread in scattering angle . For each target angle setting, the beam energy was adjusted to give the desired mean scattering energy in the target. Measurements were then performed with a spin-up and with a spin-down deuteron beam. For the intermediate-weight target nuclei, about 2 000 counts were collected in each peak for each spin direction. For the measurements on C, about 5 000 counts were collected. The time for data acquisition for each angle varied from 1 to 30 h. For the measurement on 'Be, a Lamb-shift polarized-ion source was used 21 .29). This source became operational near the end of the present measurements . For this source, the current on target was about 5 x 10 -1 A. This source is several hundred times more efficient for vector analysing power measurements than our atomic beam type source .

6.

esults

The measured vector analysing powers as a function of c.m. reaction angle are shown in figs . 3-5, 9 and 10. The curves are the result of WBA calculations which are discussed in sect . 'I. Fur reactions on the intermediate-weight target nuclei 24Mg, 4°Q) and "Cr, there were unresolved levels, which come from the isotope under consideration and from the presence of other isotopes in the target. The effect on the measurements caused by the unresolved levels c. a be estimated because unpolarized cross sections for these reactions are available at or near the energy at which the present measurements were performed 2, 3, 3 0 _ 3 3) . Using the unpolarized cross sections and the known isotopic abundances, the contributions to a proton peak from unwanted levels were calculated and subtracted out. An uncertainty of 25 % was assigned to these contributions to take into account the uncertainty in the unpolarized cross sections and the uncertainty resulting from neglect of the polarization dependence of the cross sections . For the light-weight target nuclei, no such corrections were necessary. n addi`ion to the uncertainties caused by unresolved levels, the error bars shown in the figures contain counting statistics and the uncertainty in the beam polarization . Moreover, an uncertainty of 3.5 % of Pd(O) was assigned because C(9) in eq. (5) was neglected. t

Sirntec Ltd., Montreal, Canada, Model LC-500-2.0-110.

STRIPPING REACTIONS

The vector analysing powers for reactions on intermediate weight target nuclei are shown in figs. 3--5. Eight I = 1 transitions were investigated . There is a consistent and pronounced difference between d(®) for j = and j = transitions. The behaviour clear o d(O) is very similar for all I = 1 transitions. In each case, Pd (O) has a maximum near ° for the j = transitions and a minimum for j = transitions. 24Mg(dtp)25Mg

Ed= 8.0 MeV

24 Fig. 3. angular distributions of the vector analysing powers for the Mg(d, p) a1 Mg reaction with 8.0 MeV incident deuterons . The curves are the theoretical predictions in zero-range approximation for 10.01AeV incident deuterons on the target nucleus $®Si using the potentials listed in table 2. The full curve in the left-hand frame is for the ground state, l = 0, j = j transition. The dashed curve in the right-hand frame is for the F, = 1 .28 MeV, I = 2, j = transition, while the full curve is for the E. = 2.03 MeV, I = 2, j = 1 transition .

40CO 0.8

(d, p)41

Ed= 7.0 MeV

Ca

0.8

.

0.6

, i lEx, 1 .95 MeV Ir =I " Ex2I2 .4TMeV .1 =1 0.6 l1

0.4

0.4

0.4

0.2

0.2

0.2

0.8

Ex-3.95 MeV R =I p m 1/2

v

pd 0

Pd

-0.2

-0.2

-0.4

' L./_ -0.4

-0 .J0

!oil, 11111 -0 .6 t 80 0 20 40 60

%m(deg)

v F

=3/2 =3/2

I ' 1 Ex--0.00 MeV t. =3 j = 7/2

0.6

f i j

\ ô

I

r-j

r 0

L

T 0

Pd0

~,

-0.2 -0 .41.

1 20

.

1 40

'

Bcm(deg)

1 60

'

I 80

-0 .6 1 0

'

I 20

'

1 40

'

®cm(deg)

I 60

i

1 80

Fig . 4. Angular distributions of the vector analysing powers for the 4°Ca(d, p)41 Ca reaction with 7.0 MeV incident deuterons . The full curves are the theoretical predictions in zero-range approximation using the potentials listed in table 2.

10

T. ' . YVLE Arte w. HAEAERU

The I = 1 transition to the 3.61 MeV state in "Cr had been tentatively identified as a 34) . e j = reaction by thermal neutron capture y-ray directional correlations measurement of Pd(O) for this transition strongly suggests that j =- , and it is placed for comparison with that for a knownj = reaction. I4 (d, p) The third frame in fig. 3 shows Pd(0) for two 1 = 2 transitions for the 2 SMg reaction . ere the difference between the two different j-values is not as striking but still clearly visible. Near the stripping peak, which occurs at appj : oximately 30°, e third frame in fi4~. and the third frame the sign of Pd is different for dif erentj. in fig. 5 show Pd (0) for 1 = 3 transitions for the 4° Ca(d, p)4 'Ca reaction and for the 52 Cr(d, p) $3 Cr reaction, respectively . Again, near the stripping peak, which occurs at approximately 40°, the sign ofPd is different for different j. "C r (d,p)'3 C r

Ed- 8.0 MeV

Fig. 5. Angular distributions of the vector analysing powers for the 8'Cr(d, p)88Cr reaction with 8.0 MeV incident deuterons. See text concerning the spin of the Ex - 3.61 MeV state in "Cr. The curves are theoretical predictions in zero-range approximation using the potentials listed in table 2. No spin-orbit coupling was used in the deuteron potential, and volume absorption was used for both the deuteron potential and proton potential. The full curves and dashed curves differ from each other only because of the different Q-values of the transitions.

The vector analysing powers for reactions on the lightweight target nuclei are shown in figs . 9 and 10. The behaviour of Pd(O) for the 1 = 1, j --- transition on e is quite similar to that for the intermediate-weight target nuclei. The behaviour of Pd(0) for the ground state 1 = 1, j = I transition on ' IC at 7.0 and at 10.0 eV ins. -dent deuteron energy is similar at backward angles, but at forward angles the shape j- quite different at these two energies. Measurements taken in 0.5 eV steps (fig. 10) indicate that the shape at forward angles is a rather sensitive function of the incident deuteron energy. Only at 10.0 eV does the shape of the distribution begin to approach that seen for intermediate-weight target nuclei . The behaviour of Pd(O)

STRIPPING REACTIONS

11

for the ,, = 3. e ,1= 0, j = transition on 12C is also strongly dependent on the incident deuteron energy (fig. 1)® There is little similarity between Pd (O) at 7.0 n at .

Si

8

)28

Si

28

Si

Si

24

(p .P)28

Mg (d "

r.TY~ .. . ..T~.Y.._T_

SC (68), E-9.O MeV -- CALC , E -10 .0 MeV

" 1

-

,

_

-

i1\- ""

ao .1I

Q SC (68), E®9 .0MeV --® CALC, E - 10 .0 MeV '

,"

,

M I (62) . Ed=10.1 MeV CALC, Ed=10.0 MeV

1ook

cr

p)25Mq

~;E x

=2 .80MeV 1.=

ee

_

t=3/2 ~;

11

01~

0 ® .m (deg)

30

®Cm (deg)

60

90

120

150

18n

8c m (deg)

Fig. 6. Comparison between the data for elastic scattering of deuterons s°) and of protons", 45) on "Si and for the differential cross sections $') of the "Mg(d, p)25 Mg reaction and predictions of the potentials used in the vector analysing power calculations . The calculated curves in the right-hand frame are for 28Si(d, p)29Si. The full curve at the top is for the ground state, t = 0, j = I transition . The full curve in the middle is for the F., = 1 .28 McV,1= 2, j = I transition, while the full curve at the bottom is for the Ez = 2 .03 MeV, 1= 2, j = transition in $ 8Si(d, p)s9Si .

tort -wave

ysis

As was seen in sect . 6, the measured vector analysing powers are sizeable .

For a

given 1, there is a clear difference between Pd(O) forj = 1+ I andj - 1- I transitions. This section describes an analysis in terms of DW13A theory . Two aspects of the comparison between experiment and theory are of importance . In the first place, one wants to

investigate the reliability of using measurements of Pd(O)

to

determine

j-values . For this purpose, one is interested in the general features of the calculated Pd(®), such as the positions of maxi

and minima.

e other aspect concerns the

12

T . J. YULE AND W. HAEBERLI

question whether measurements of Pd(O) might lead to a better understanding of the mechanism ofstripping reactions. For both aspects of the comparison, one is.interested in the sensitivity of Pd (O) to such features as uncertainties in the central parameters of the optical potentials, the presence of spin-orbit coupling in the optical potentials and corrections for finite range and non-locality of the optical potentials. Similar analyses have been performed for the differential cross sections for the 4 °Ca(d, p) 41 Ca reaction a) and the "Cr(d, p)-"Cr reaction ;1- 4). 4oA 4oCa 4oCa 4° A (p . p) (d .d)4oCa . (d .p,a~Cat " BA (64) . E = 7.0 MeV - CALC . E = 7 .0 MeV

" LE (64) . E $12 .OMeV - CALC, E -12 .0 MeV

" YU (67) . E=7 .0 MeV - CALC, E= 7 .0 MeV

" RO (65), E-14 .5 MeV -- CALC,E -14 .5 MeV

I --"

..

LE (64), Ed-7-OMeV CALC, E d - 7.0 MeV

Fig. 7. Comparison between the data for elastic scattering of deuterons $$- 41 ) on 4°Ca, of protons',`a) on 40Ar and for the differential cross sections 2) of the 40Ca(d, p)41Ca reaction and predictions of the potentials used in the vector analysing power calculations .

In this section, a comparison of predictions from DWBA with the experimental data are presented . The effects of variations and uncertainties in the parameters are considered in the next section. 7.1 . FORM OF OPTICAL POTENTIALS

The calculations were carried out at Oak Ridge National Laboratory using the computer program JULIE 3s) . Spin-orbit terms were included in the optical-model potentials . Unless indicated, no corrections were applied for non-locality of the optical

s°rnaergxa 52Cr

(

~~~Cr

.

acr>
13

53 52 Cr (d,p) Cr ~T...f~..,._.r. .T_~. .-..._,.._,

~3Cr ( , p)S ~Cr

" AN (64), E " 8.0 MeV --- CALC . E " .0 MeV

-- CALC , E =10 .0 MeV

~ - CALC, Ed=8 .0 MeV

0; - "' ; r"

0.1 ~. . I

~

~ 10 ~ r r ® :i, __ _

,

I

" E =0 .5?MeV R=I ~=I/2~ ~~ "E x =3 .61 MeV R=I $=1/2

" .E =2 .32 P~1eV R=I ~=3/2 x ~""~ J " "â ; v" ,

®CALC, E=10.0 MeV

b

1= L

F

" 1

". " .~,Ex= I .OOMeV R=3 $=512-i .:i ",. ti "" " ~"

0.1~

30

c ~,ddeq)

i

6iJ

90

_

_

i

I50 . I80 ~

_

120

0

1 iJ_1jy. t

30

60

90

120

i 50 I80

®Cm (deg) ®eR.,fdeg) a:) dig. 8. Comparison between the data for elastic scattering of deuterons on bgCr, of protons ~s, a6) on °~Cr and for the dilfferential cross sections g) of the 8'Cr(d, p)BaCr reaction and predictions of the potentials used in the vector analysing power calculations. A1o spin-orbit coupling was used in the deuteron potential, and volume absorption was used for the deuteron potential and the proton potential.

E~ o.oo

9 Be cd,~)'°ee

..®

ev

E d = 7 .0 MeV 0.5

Pd -0 .51-

t

-LO

0

t 20

,

t 40

,

t 60

,

t

~

80

t

100

(deg)

.

I 120

.

t 140

~

t 160

I i80

®Cm

F'ig. 9. Angular distribution of the v~tor analysing power for the'Be(d, p)1°l3e ground state reaction with 7.0 MeV incident deuterons. The full curare is the theoretical prediction in zero-range approximation for 8.0 MeV incident deuterons using the potentials listed in table 2.

14

T. J . YULE ANI! w. HAEBERLI

potentials or for the finite range of the interaction. The deuteron and proton optical potentials had the form U(r) = Uc(r) - YAX) - i dx ( z xr d " , J(x )L Q, m,c r dr

where

i)_ f (x) = (e+ l' x' -= (r - rô A*)Ia',

(6)

x = (r _° ro A*)1a

x" = (r-r0 A*)Ja 1' 9

and where Uc (r) is the Coulomb pot--ntial of a uniformly charged sphere of radius 12 C(d,p) 13C E x =3.09 MeV E=0 ~=1/2 --T---r --.,--r-.*

I

E d = 7.0 MeV 0.5

Pd

e o

o

e

0

0.5

I 40

1 Ed = &OMeV

o

0

-0 .5 0

20

Pd

1 .o r--1

4

'

1 1 60

I 80 100 120 eecm (deg) ~I I I I Ed=8.5MeV Ed=9.OMeV 0 ®

¢

0

160 T`

e

0

~ e

® *

20

40 0 20 ecm(deg)

400

20

0

180

. Ed=9.5MeV

®

e

40 0

140

40

Rd

0

-0 .5

A

A

0 .5

20

40

-0 .510

60

so

too

140

120

160

180

9cm(deg) Ea~B .OMeV

a o d

Ed- 8.5MeV FE d~9.0MeV® 0 P Ô4~1$

I 1 1 I 20 400

I 20 1

Ed": 10-0 MeV 0.5

E d - 7 .0 MeV

-0 .5 1 20

=I/2

-.,..,

Pd o

e

0

E x -0.00 MeV 1= I

0 .5

-0.5 - 1 .O

12C(d,p) 13C

1 .or--r .--,--r,

1 1 40 0

1 20

%m(deg) , 1 1 1 1 E d " 10 .0 MeV

1 I 400

,

1 20

1 40

1

0 .5

Rd

o

-0 .5

18 Fig. 10 . Angular distributions of the vector analysing powers for the l'C(d, p)13 C reaction with 7 .0 and 10.0 MeV incident deuterons . In the center frames, the behavior of the vector analysing powers at forward angles for intermediate energies are shown . The full curves in the bottom frames are the theoretical predictions in zero-range approximation using the potentials listed in table 2 .

STRIPPING

15

ACTIONS

rC ill . For protons, a is the Pauli spin- operator, and for deuterons it is the spin-one

operator . In the calculations, either a surface absorption or a volume absorption was used. For the calculations on carbon, a Gaussian form was used 36) for the imaginary part of the proton potential .)a G(r) = _ i WG e-(x . The neutron was assumed to be captured in a potential well defined in analogy with the potential for elastic proton scattering U.(r) = - V f (x) + a(l, J) where a(l'1) = t _(l+1)

d

"S.C. h f(x) r 211ßp c dr for

1+1 i = t_1. j

The depth of the well was adjusted to give a binding energy 1f the neutron equal to the empirical separation energy . Parameters of the optical potentials for the calculations were those which best reproduce the measured elastic scattering cross sections and polarizations . ®f course, the potentials are not unique. There are the well-known ambiguities between correlated parameters in the potentials . The values of the parameters used in the calculations are listed in table 2. 7.2. INTERMEDIATE-WEIGHT TARGET NUCLEI

In figs . 3-5, the predicted vector analysing powers for intermediate-weight target nuclei are compared with the measured one. In figs . 6-g, the fits to the elastic scattering cross sections and polarizations and the predicted stripping differential cross sections are shown. Recent analyses of elastic proton scattering cross sections and polarizations on intermediate-weight nuclei have shown that average potentials exist which reproduce the data over a wide variation in target mass and energy 40,45,46) Therefore, even though potentials are not available at the relevant energies for elastic scatteri_ - d of protons for some of the present calculations, potentials for neighboring nuclei a* -imilar energies should be satisfactory . The parameters of the deuteron potential are known with less certainty than those of the proton potential 2 °, ") . During the present experiment, some measurements were performed of the vector polarization for the elastic scattering of deuterons on Mg. It was not possible to find a potential which represents both the cross section and polarization . Potentials similar to those of Mayer-Wricke and Siemssen 47) in their analysis of elastic scattering of deuterons on Mg in the energy range of 6-13 Met' were tried. It was felt that it would be more in the spirit of the present analysis to perform calculations for a target nucleus with similar mass for which potentials that reproduce both the cross section and vector polarization are available. The nucleus Z 8Si was chosen 2°). Before the present (d, p) measurements were begun, the vector

T.

16

a. YuL~ nrrta w. x~H~aRt,t .M

a

w r. l~

M N

M N

N f° v I M ra r+

~i r+

h O O

~-;°II .-~ .-+ "F.,

~MIIII r;r!II11II O "-+ ~ "^ O ~-+ ".,, " ^ " ,~,

~m â

â

~o ~w

>

N a O

`.r

b

w C,

~+

h~Nd'

~ ~ °°»> a~ a~ Ci

°°»»> a~ ai C? u ai

~-+ 00 p h o N ~ ~D,d:O~OM~O . . . r+~O~-+N ~~-+OO^+NM . .. .~0 II II II II II II II II II II II II~ AA~~~ AQ II A~ IaA~~~ ~~~~~~~ O 00 ~ et N ~ ,~ C~ O \, `.r `.r C~ ~ r.r ~.r ~.r ~ ~ v ~.r e.~ ~..r v O ~ Oi 00 00 .~+ r1 .~, aô ~ O .-~ .-a N o ~e0+~0~ .~ .-iäM

p

hNONÔ

p

w?NO~~ . . . .-~ .-~O~M

O l'~

C!

o~0 ~0 ~0 o00

.-~ ~n h cA ~O ~G ä

op .~+ W h

~o

tph0 ~NN

V'1hh ONN

$OON N

N~-+O NNM

M

..~ .-~ ^

v a wCi a> A b

~r

~ M .-~ ."+ ,,u II II II ti w ti 1.

~t`w

C~ C~ 0~0 ~0 ~D ôôôô ®®

I

~+

.~ N ~ II ~+,.

~ .~r

äo » ääs» u u u u a~

_

NIIiIlIIIII

M .-+ ."~ 00 il II il ä ~ ä ti ~.. ~..

~N

" Îa

~NIIIiII O !rl "~,

O N N Oh ~: ,~ II I) il ti ~. ti

oä > a~

O

~

O 00 NO ,; ,~ II ,~ ,~ II II wr ti ~ ~ ..r

" ~i

O N~

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y

r"+ r+

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h ~D O O

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s~w r~e

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.-+ .-i

ri

ä

ä

.-, ."; ~-r

OOooerO ~G?d'O~l` ~ 00 O~ ~"-i N ~ M ~! M O~ ^+ d' d' h er ~ h h ~+`~ h

äo

0

~i II O

O

a o~ 3 y II N~ N V w

~ O

.

a .â H Ô w _

a 40

'o ,~ û

wy .ov r+

~~NONM~n . . . 00 h C~ 00 h

bas~vaaabaaaabaaaabaaaaaa

0 a

II

~i N

a

w

O

. .,

O N '^"H ~i e~ ci il

a

v

Ep w 

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ä m ®i ~

~O U ri ~

~+O oô

U b

a

9Be (p, p)

30

60

90

9

Be

120 150

Acm(deg)

9 Be

180

0

30

(d,p )1 °Be

60

90

120

. 150

180

ecm(deg) Fig. 11 . Comparison between the data for elastic scattering of deuterons 411 ) and of protons 49,50) on OBe and for the differential cross section 51) of the 9 Be(d, p)1O Be round state reaction and predictions of the potentials used in the vector analysing power calculation. 12C "

(d .

12C

d ) 12C

12C

(p .p)12C

" NA (61) . E =12 .1 MeV ® CALC, /" " -~ E=12 .214leVj

OH (63) . E-10.0 MeV CALC . E-10.0 MeV

(d,p)13C

SC (64) . Ed=11 .6MeV

"

-- CALC, Ed=10 .0 MeV

I

1

E X =3.09meV .2 =0 e=Il2 fz

ZA (65) . Ed29 .6MeV CALC, Ed =10.0 MeV E x =0.00MeV k=1 9=112

t

60

.

,

t

90

.

.

t

~a

120

6cm(deg)

t

.

150

.

t

180

1 .0 1 0

i . . . . . . i . . , . . , . . 1 30 60 90 120 150 180

ecnm(deg)

01 ' 0

30

60

90

.~i_ .__ ..-~

120

®cm(deg)

150

1

180

53,54) Fig. 12. Comparison between the data for elastic scattering of deuterons 52) and of protons ss, 12C(d, predictions of the on '2C and for the differential cross sections ") of the p)13C reaction and potentials used in the vector analysing power calculations .

18

T. 1. YULE AND

w.

HAERERLI

polarization and two of the three tensor polarizations were measured for the elastic scattering of deuterons on Ca at 7.0, 9.0 and 11 .0 MeV. Potentials were found which 41). For the Cr calculations, a potential of fit the cross sections and polarizations 42) was used, which does not include spin-orbit coupling and has Andrews et al. volume absorption . The calculations represent the measured analysing powers rather well. The observed j-dependences are reproduced . Even the small shift in the measured j = angular distributions with change in excitation energy is in agreement with the calculations (see fig. 5). For a given j, the calculated Pd(O) for the l = 1 transitions for the target nuclei 4°Ca and 52 Cr are quite similar. The calculations also represent rather well the measured cross sections for the target nuclei 4°Ca and "Cr . The significant difference between the present analysis and that of Lee et al. 2) for the 40Ca(d, p)41Ca reaction is that a deuteron potential which reproduced both the cross section and polarization data has been used . For the present calculations, thej-dependence observed by Lee and Schiffer 2 - 1 g) is reproduced . There is a pronounced dip at backward angles for the 1 = 1, j = transition, while no such dip appears for the 1 == l, j = transition . There is very little difference between the present predictions for the 52 Cr target nucleus and those of Andrews et al. 3). The main difference is that no cut-off has been used in the present calculations . The measured cross sections for the 24 Mg target nucleus are not reproduced as well as those for the heavier target nuclei. There is some indication that the j-dependence which Lee and Schiffer 19) observed near 60® for l = 2 transitions for the 1d shell are reproduced. For the j = transition, there is a dip about 60®, whereas for the j = I transition no such dip occurs . 7.3 . LIGHT-NMIGHT TARGET NUCLEI

In figs . 9 and 10, the predicted vector analysing powers for the light target nuclei are compared with the measured ones . In figs . 11 and 12, the fits to the elastic scattering cross sections are shown. For the Be calculations, the proton potential was taken from an analysis of proton scattering from 'Be in the energy range from 5-15 MeV by Siemssen 3s) . The excitation function for elastic scattering and polarization of protons on 12C show a considerable amount of structure below about 12 MeV [ref. ")] . Although measurements were performed of Pd(O) at 7.0 MeV and 10.0 MeV, an analysis has been made only at 10.0 MeV. For the calculations, the potential was taken from an analysis of proton scattering from "C in the energy range from 7-20 MeV by Nodvik et al. s6), which fits the cross section at 12.2 MeV. The deuteron potential for the Be calculations is from a group of potentials for Be which Satchler 3") had found in an analysis of elastic-scattering data on light nuclei. There is no assurance that this is a physically meaningful potential, since data in this energy range exist at only one energy. For the calculations on cat-bon, a deuteron potential from an analysis by Satchler of the elastic scattering on 12C from 3-34 MeV was used 39) . For both the Be and C potentials, the amount of spin-orbit coupling is

19

STRIPPING REACTIONS

not known very well, since no angular distributions of the vector polarization are available. e fits to the measured Pd(O) for the light-target nuclei are considerably poorer than those for the intermediate-weight target nuclei . For "C, the agreement between the calculated and measured Pd(O) for the ground state transition is poor. Tphe most disturbing feature is the presence of a maximum of Pd(O) at 100° in the measurements but none in the calculations . The same maximum also appears in the 7.0 MeV measurements. The fits to the measured cross sections are also poorer for the light-target nuclei. For the ground state transition for carbon, the magnitude of the (d, p) cross section at angles beyond 0° is seriously over-estimated, and the positions of maxima and minima are incorrect . A similar difficulty was encountered by Schiffer et al. 5 ) in attempting to fit the cross section for this transition at 12 .0 MeV. The difficulties experienced in fitting the carbon data are not unexpected . The measured Pd(O) for the two transitions as well as the measured angular distributions of the outgoing proton polarization 9) show a rapid change with energy . s2,5") and polarAlso, the energy dependence of the elastic scattering cross sections ization 57,58) show a considerable amount of structure. One cannot expect that the present calculations would be able to reproduce such rapid variations . Forcing the optical model to fit the effects of isolated resonances leads to unphysical parameters . Furthermore, calculations with different values of deuteron spin-orbit strengths showed that for these transitions Pd (O) is rather sensitive to this parameter . This is especially the case for Pd (O) for the ground state transition at angles greater than 60°. ia o

ce ta' fies 1

ara e ers

In sect. 7, it was seen that for intermediate-weight target nuclei rather satisfactory fits are obtained for the cross sections and vector analysing powers. It was also seen that a change of a few MeV in Q-values produces only a small change in Pd(®), and that its dependence on the target mass is rather weak. In this section, the effects on the calculated vector analysing powers of variations and uncertainties in the various parameters are systematically investigated . 8.1 . SPIN-ORBIT COUPLING It is not possible to determine the strength of the spin-orbit coupling in the optical potentials from cross-section data alone. For proton elastic scattering, the spin-orbit coupling has been determined from fits to a large number of polarization measurements 45,46 ) . However, for deuteron elastic scattering, there are only a few measure20,37 ). Therefore, if the analysing power of the ments of the polarization parameters (d, p) reaction is a sensitive function of the amount of deuteron spin-orbit coupling, it would be necessary to measure elastic cross sections and polarizations before reliable j-assignments could be made from the measured Pd(®). However, this is not the case .

20

T. J. YULE AND

w. HAE®ERLT

In fig. 13, the effects of the presence of spin-orbit coupling in the deuteron and proton channels are shown for the 2 $Si(d, p)"Si and "Ca(d, p)"Ca reactions. For both reactions the effects of spin-orbit coupling are small at forward angles. The effects are more significant at back-ward angles for 2 'Si(d, p)29 Si than for " OCa(d, p)"Ca. 28S

i

(d,p)24 S i

"'Ca (d,pl4' Ca

E d=100 MeV

Ed= 7.0 MeV

E x -3 .95MeV 06

0 .6

0 .4

0.4

0 .2

0.2

Pd°

Pd 0

-0 .2

-0 .2

-0 .4

-0 .4

0 .4

0 .4

0.2

0 .2

Pdo

Pd o

-0 .2

-0 .2

-0 .4

-0 .4

E x -1 .28MeV

1-2 j-5/2

0 .4

0 .4

0 .2

0 .2

Pd 0

c

v

-0 .2-

(MeV) (MeV) 8.0 7.5 --- 0 7.5 ------- 0 0

-0 .4-0-6L 0

Pd °

VP

"

20

40

.

60

80

100 "

% ,(deg)

-0 .4

"

~ 140 .

v 160

.

I 180

-0,6

.Q-1 j" 3/2

E x -O.OOMeV

A-3 j-7/2

VP Vd (MPV) (MeV) 5.0 8.0 --- 0 8.0 ........ 0 0

-0 .2

120

Ex -1 .95MeV

0

20

40

60

80

100

eWdeg )

120

140

160

Fig. 13. Effect of spin-orbit coupling on the vector analysing powers for the IBSi(d, p)29 Si reaction with 10.0 MeV incident deuterons and the 4°Ca(d, p)41Ca reaction with 7.0 MeV incident deuterons. The central optical potential parameters are similar to those listed in table 2. As one goes to lighter target nuclei or higher energies, the sensitivity of Pd(8) to spin. orbit coupling increases. If the effects of spin-orbit coupling are negligible, there is a simple relationship

180

STRIPPING REACTIONS

21

between Pd(8) for two transitions of different total angular momentum transfer as long as the excitation energy and the orbital angular momentum transfer are the same 19) CPd(®)l,i=,+, _ _ o l l+ 1 CPd(O)lj=I -*

The analysing powers have the same angular dependence, but the sign is different for different j. 8.2. VARIATIONS IN THE CENTRAL POTENTIALS

Although the elastic scattering cross sections are rather effective in limiting the

Fig. 14. Effect of varying the strengths of the deuteron and proton central potentials on the vector analysing power for an 1= 1, j = j transition for the 4°Ca(d, p)°"Ca reaction with 7 .0 MeV incident deuterons. The unvaried optical potential parameters are listed in table 2.

values of the central parameters, sometimes there are sizeable differences between the parameters for neighboring nuclei or for different energies 37 ) . Also, there are arguments which suggest that a different absorptive strength should be used for elastic scattering calculations than for stripping calculations 11 ) . Fig. 14 shows how the

22

T. J. YULE AND

w. HABHBRLT

vector analysing power for the E$ = 3.951VJCeV, l = l, j = transition for the target nucleus "Ca changes when the depth of the real potential is varied by 10 % and the depth of the imaginary potential by f 25 %. Such variations produce significant changes in the elastic scattering cross sections and polarizations. It is seen that changes in V have a larger effect than changes in . In all cases, the J-dependence of Pd(O) remains unambiguous. 40Ca 41 Ed-12.0 MeV (d .p) Co A-1 j-1/2

\,,__ \'

-------°

LOCAL / ZERO RANGE NONLOCAL / FINITE RANGE LOWER CUT -OFF . 4 f m

120

140

160

ig. 15. Effect of non-locality and finite-range corrections on the vector analysing powers for the "Ca(d, p)InCa reaction with 12.0 eV incident deuterons. The effect of using a lower cut-off on the radial integrations is also shown. The optical potentials are similar to those in ref. ').

RIPPING REACTIONS

23

nother uncertainty associated with the absorptive potential concerns its shape. It is known that equally good fits to deuteron elastic scattering data and stripping cross sections n obtain with either volume or surface absorption") . It appears that this uncertainty makes little difference for the vector analysing powers. The calculations for the -"Cr target nucleus were performed with a volume absorption, while those for the ® target nucleus were with a surface absorption . 3.3. INTERIOR DAMPING AND

DIAL CUT-OFF

All the calculations shown so far have been with a zero-range interaction and without corrections for non-locality. The effects offinite range and non-locality corrections have n discussed in detail, and approximations have been developed for them "61). The effects of both ofthese corrections are to reduce the contributions from thv nuclear interior compared to those from zero-range calculations . For example, a reasonable estimate of the effects of non-locality lead to the wave function of the proton being reduced by about 15 % and that of the deuteron by about 25 % in the nuclear interior (ref. 2)). One way to investigate the contributions from the nuclear interior is by use of a radial cut-off. This is a rather extreme step and not physically reasonable, but it does place an upper limit on the effects of the contributions from the nuclear interior. In fig. 15 a comparison is made for the 4® Ca(d, p) 4 'Ca reaction t at 12 .0 MeV tween local zero-range calculations and non-local finite-range calculations as well as calculations with a cut-off. The corrections for finite range and non-locality are not very significant at this energy and fall between the zero-range calculation and those with a cut-off. It should also noted that the calculated Pd(O) at 12.0 MeV are quite similar to those at 7.0 eV. Just as in the case of spin-orbit effects, the effects become more significant for lighter target nuclei or higher energies. 9. Conclusion The present investigation of stripping reactions with polarized deuterons demonstrated the existence of appreciable magnitudes for the vector analysing powers . For a given 1, there is a pronounced and systematic diference at forward angles between Pd(O) for j = 1+ and j = 1-1, transitions . The most striking differences were observed for I = 1 transitions for the intermediate-weight target nuclei. The analysis in terms of DWBA has been successful in reproducing the measured vector analysing powers at forward angles for the intermediate-weight target nuclei . Furthermore, there is some indication that the j-dependences in the cross section observed by Lee and Schiffer can be accounted for if potentials are used which represent the elastic scattering cross sections and polarizations . For the light-target nuclei, there are too many uncertainties in the parameters of the optical potentials and complications from the presence of non-optical model effects to make a meaningful comparison between theory and experiment. t These calculations are part of a previous unpublished analysis by Satchler 2).

?4

T . ,ï . YULE AND W . HAEBERLI

The general features of the predicted vector analysing power at forward angles are quite insensitive to the parameters of the model. Reproduction of these features requires only the correct selections of orbital angular momentum and total angular momentum transfer. This comes about because of the strong /-space localization at energies around 10 MeV for intermediate-weight target nuclei 61). Recently, different approaches besides the DW A have been suggested for strippin reactions 6 °, 6 3), and some calculations have been carried out for the cross section . The various approaches attempt and polarization of the outgoing proton 60,63,64) to treat the motion of the deuteron in the nucleus in a more realistic way than is done in the distorted-wave theory . It would be worthwhile to extend the present measurements to higher energies and to backward angles . t may then be possible to make meaningful comparisons between the various approaches for stripping reactions and their refinements . Measurements on heavier target nuclei also may serve this purpose as well as determine if spectroscopic information can be gathered on these nuclei by this method. The authors are indebted to Dr. G. R. Satchler for making available preliminary calculations and in particular for providing the opportunity to perform calculations at Oak Ridge National Laboraory and to Dr. . Schwandt for helpful discussions . We should like to thank Dr. J . Schiffer of the Argonne National Laboratory for the loan of the chromium target. e erences 1) S. T. Butler and O. H. Hittmair, Nuclear stripping reactions (John Wiley and Sons, New York, 1957) ; W. Tobocman, Theory of direct nuclear reactions (Oxford University Press, London, 1961) ; N . Austern, Fast neutron physics, Vol . 2 (Interscience Publishers, New York, 1963) p. 1113 2) L. L . Lee, Jr., J . P. Schiffer, B. Zeidman, G. R. Satchler, R. M. Drisko and R. H. Bassel, Phys. Rev. 136 (1964) B971 ; erratum Plays. Rev . 138 (1965) AB6 3) P. T. Andrews, R. W. Clifft, L. L. Green and J. F. Sharpey-Schafer, Nucl. Phys. 56 (1964) 465 4) J. L. Alty, L. L. Green, G. D. Jones and J. F. Sharpey-Schafer, Nucl . Phys . 86 (1966) 65 ; J. R. Mines, Nucl. Phys . 86 (1966) 89 ; J. C. Legg, H. D . Scott and M . K. Mehta, Nucl. Phys. 84 (1966) 398 5) J. L. Alty, L. L. Green, R. Huby, G. D. Jones, J. R. Mines and J. F. Sharpey-Schafer, Nucl. Phys. A97 (1967) 541 ; J. S. Forster, L. L . Green, N. W. Henderson, J. L. Hutton, G. D. Jones, J. F. Sharpey-Schafer, A. G. Craig and G. A. Stephens, Nucl . Phys . A101 (1967) 113 ; C. A. Wiedner, A. Heusler, J. Solf and J. P. Wurm, Nucl . Phys. A103 (1967) 433 ; J. P. Schiffer, G. C. Morrison, R. H. Siemssen and 13. Zeidman, Phys. Rev . 164 (1967) 1274 6) H. C. Newns, Proc. Phys . Soc. A66 (1953) 477 7) D. W. Miller, Proc. Int. Symp. on polarization phenomena of nucleons (Birkhduser Verlag, Basel, 1966) p. 410 8) R. A. Blue, J. H. Stout and G. Marr, Nucl . Phys . 90 (1967) 601 9) J. E. Evans, J. A. Kuehner and E. Almqvist, Phys. Rev. 131 (1963) 1632 10) K. A. Kuenhold, P. L. Beach, G. A. Bokowske and T. R. Donahue, Bull. Am. Phys. Soc. 13 (1968) 117 ; C. T. Kelley, Jr., W. E. Maddox and D. W. Miller, Ibid. ; A. A. Rollefson, P. F. Brown, J . A . Burke, P. A. Crowley and J. X. Saladin, Phys. Rev. 154 (1 1-167) 1088

STRIPPING REACTIONS

25

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26 50) 51) 52) 53) 54) 55) 56) 57) 58) 59) 60) 61) 62) 63) 64)

T.

r.

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