Spin flip in the inelastic scattering of 12 MeV protons from 50Cr and 52Cr

2.B: 2.L / 3.A

Nuclear Physics Al77 (1971) 161-173; Not to be reproduced

by photoprint

SPIN FLIP IN THE INELASTIC

@ North-HoIIand Publishing Co., Amsterdam

or microfilm without written permission

SCATTERING

from the publisher

OF 12 MeV PROTONS

FROM “Cr AND ‘*Cr W. E. SWEENEY, Jr.t and J. L. ELLIS tt T. W. Banner Nuclear Laboratories ftt, Rice University Houston, Texas, USA Received 7 June 1971 Abstract: Angular distributions of the absolute spin-flip probability during inelastic scattering of 12 MeV protons have been measured for the reactions ““Cr(p, p’)50Cr*(0.78 MeV) and %r@, p’)%r*(1.43 MeV). The data are compared with collective-model DWBA calculations which include the effects of averaging over the y-ray detector acceptance angle. The effects of varying the spin-orbit distortion in the collective-model form factor and of changing the optical-model parameter set are discussed. The angular distribution for s°Cr exhibits a large peak at back angles characteristic of a direct reaction and agrees qualitatively with the DWBA calculations. The angular distribution for %r exhibits two peaks of nearly equal magnitude and does not agree with the theoretical calculations. E

NUCLEAR REACTIONS 5o*%r(p, p’y), E = 12.0 MeV; measured a(E,,, E,,, 0,); deduced spin-flip probability to first 2+ states. Enriched s°Cr target.

1. Introduction

The spin dependence of the inelastic interaction has been actively studied in the last few years. These studies have included the following types of measurements: (i) measurements of the analyzing power of the reaction using a polarized beam, (ii) measurements of the polarization of inelastically scattered particles using doublescattering techniques and (iii) measurements of the spin-flip probability using the (p, p’y) correlation method of Schmidt et al. ‘). The present paper reports on measurements of the latter type. Schmidt’s method can be applied straightforwardly to a y-ray transition between a 2+ first excited state and a O+ ground state. Let the quantization axis be along the normal to the scattering plane and let ki and k, be the incident and outgoing momenta of the scattered particle, respectively. For pure I = 2 multipoles, emission is forbidden along ki x k,-, except from the m = & 1 substates. Using the Bohr theorem ‘) one can show that for spin-) particles the m = + 1 substates can be populated only if the spins of the incident and outgoing particle are opposite. Hence, by detecting y-rays from the de-excitation of the 2+ state along ki x k. in coincidence with the inelastically t Present address: Harry Diamond Laboratories, Washington, DC, USA. tt Present address: Teledyne Isotopes, Westwood, NJ, USA. ttt Work supported in part by the US Atomic Energy Commission. 161

162

W. E. SWEENEY, Jr. AND J. L. ELLIS

scattered particle, one can determine what fraction of the inelastically scattered particles have undergone spin flip. The present paper reports on measurements of the spin-flip probability in the reactions 50Cr(p,p’)50Cr* (0.78 MeV) and 52Cr(p,p’)52Cr* (1.43 MeV) at an incident proton energy of 12 MeV. Although a considerable amount of spin-flip data is available for nuclei from 54Fe to 66Zn for incident energies below 20 MeV [refs.l, 3- ‘)I, the only measurement for an isotope of chromium has been the work of Ballini et al. ‘) for 52Cr at 11 MeV. They observed two peaks of nearly equal magnitude in their spin-flip angular distribution which is different from what is usually observed. Two large peaks in the angular distribution is also quite different from the usual DWBA predictions: a large peak at back angles (near 150”) and a much smaller peak near 80”. The measurements presented here are compared to DWBA calculations using the optical-model parameters of Becchetti and Greenlees “) and Perey et al. “). The success of Perey et al. lo) in fitting inelastic scattering with their average parameters indicates that a similar study including spin flip may be possible as more spin-flip data become available. 2. Experimental procedures 2.1. APPARATUS

The data were obtained with the Rice University Tandem Accelerator. After being accelerated and momentum analyzed, the beam was defined by two circular collimators, 2 mm in diam., located 40 cm and 70 cm from the center of the scattering chamber. Transmission through these defining apertures was typically 95-98 %. After passing through the scattering chamber, the beam was stopped in a paraffinand-lead shielded dump located about 10 m from the chamber. The y-rays passed through a 0.8 mm aluminum window in the scattering chamber and were detected using a 12.7 cm x 12.7 cm NaI(T1) crystal enclosed in a cylindrical lead shield 5.5 cm thick. An additional cylindrical shield provided a conical entrance aperture through 12 cm of lead. The angular opening subtended an 11’ half-angle of acceptance at the center of the chamber. The diameter of the conical aperture was 8.0 cm at the face of the crystal. With typical beam conditions, almost all of the detected y-rays originate from the target and not from the beam defining slits or other parts of the beam line. The shielded NaI crystal efficiency was measured in the same configuration as used in the experiment by replacing the target with calibrated y-ray sources. Since the crystal is larger than the entrance aperture, the full-energy-peak (FEP) efficiency is enhanced compared to a bare crystal. For y-rays in the energy range of interest, the FEP efficiency is about 50 “//,of the total efficiency. Therefore, at a relatively small cost in counting rate, it is possible to consider only FEP y-rays, which makes the efficiency independent of the y-ray discriminator setting and relatively insensitive to small

$0.52Cr@,p’)

163

gain changes. The FEP efficiencies were found to be (0.5O-t_O.O2)x lo-* for 0.78 MeV y-rays and (0.41 kO.03) x 10T2 for 1.43 MeV y-rays. The uncertainties quoted include an estimate of the possible error due to uncertainty in the placement of the sources and the detector. Scattered protons were detected by two 1000 pm surface-barrier solid-state detectors mounted on the rotating lid of the scattering chamber. Circular tantalum colGoring

Fig. I. Schematic diagram of the electronics.

limating slits were used to define a half-angle of acceptance of 2.3” for the solid-state detectors. The overall resolution in the free particle spectra was typically 80 keV FWHM. The targets were OS-l.0 mg/cm2 thick self-supporting foils mounted on thin aluminum frames. Enriched ‘*Cr and natural chromium were used for the 50Cr and “Cr measurements, respectively. 2.2. ELECTRONICS

Fig. 1 shows a schematic diagram of the electronics. The signals from the proton detectors and the NaI crystal were doubIe-delay-line amplified and fed to zerocrossing timing single-channel andyzers (TSCA). The discriminator of the y-ray detector TSCA was set to exclude low-energy y-rays. The fast negative signals from the TSCA’s were used as start and stop signals for a time-to-amplitude converter (TAC); the start signals originating from the particle detectors and the stop signal from the y-ray detector. Energy signals from the particle and y-ray detectors and an amplitude signal from the TAC were fed to four separate analog-to-digital converters (ADC). Another TAC output was used to gate the interface to an IBM 1800 computer which collected and stored the data on magnetic tape as a multi-parameter spectrum.

W. E. SWEENEY, Jr. AND J. L. ELLIS

164

FREE

PARTICLE

SPECTRUM

Q, (LAB) = 100”

d f

JOcrO.86) ‘2C

8000

I

3

5°Cr(a78)

'60

I

%.G

%r(O.O)

111

1

t

n

CHANNEL

NUMBER

Fig. 2. Partial singles spectrum at &.s = 100” for 12 MeV protons incident upon an enriched “Cr target.

TAC I

I

I

50Crq0.78

MeV)

SPECTRUM I I

160 %r(p,p’)

I

I

I

I

m

$8 (LAB) = 100’ m-

60-

CHAIWEL

NUMBER

Fig. 3. Pulse-height spectrum from the TAC obtained with a 50Cr target at a proton scattering angle of loo”. The start pulses were obtained from the proton-detector TSCA which was set to include only the inelastic proton peak plus any nearby contaminant peaks. The stop pulses were obtained from the y-ray detector TSCA which was set to exclude low-energy y-rays.

93.Wr(p,

I

800

165

p’)

COINCIDENCE PARTICLE SPECTRA I I I %(p,q) 50Cd0.78)

600

ii iz 9 "400 E E 2 200 0

0

I

I

520 NUMBER

CHANNEL

_-II 540

I

I

510

500

530

Fig. 4. Coincident particle spectra obtained with a 50Cr target at a scattering angle of 100”. The solid dots are the particle spectrum for a TAC window set around the time peak. The open circles show the same spectrum after accidental subtraction (see fig. 3). The horizontal bar around the inelastic proton group shows the energy window applied to the data to obtain the coincident y-ray spectra. GAMMA I

320

I

SPECTRA I

I

0.78 MeV “Cr (p,p’) 50Cr*(0.78 8p4LAf3)=500

4 .

MeV)

11

240-

BACKSCATTER 4

d 2 160-

0.51 MeV 4

5

i

\ \ \ ob t

I

0

I

I

20

40

I

CHANNEL

=NUMBER

I

a’

I

loo

3 I20

Fig. 5. Coincident y-ray spectra obtained with a “OCr target at a scattering angle of 50”. The solid dots are the coincident y-ray spectrum obtained without accidental subtraction. The open circles show the same spectrum with accidental subtraction. The horizontal bar over the peak indicates the region of the spectrum taken as the full-energy peak.

166

W. E. SWEENEY,

Jr. AND J. L. ELLIS

An event consisted of a three-parameter data record comprised of the energy signals of the coincident charged particle and y-ray as well as the TAC signal corresponding to the coincidence event. In order to obtain the free spectra of inelastically scattered protons, the slow positive logic signals from the particle TSCA’s were used to gate two separate 256channel analyzers. The windows on the particle TSCA’s were set to accept only the inelastic peak of interest plus any nearby contaminant peaks. The total counting rate in the solid-state detectors was of the order of lo3 counts per set and the number of start signals was generally less than 10’ per sec. Thus, deadtime and pile-up effects were insignificant in the particle branch. The rate at which data could be collected was determined by the intensity of y-rays striking the NaI crystal. The total number of y-rays detected by the NaI crystal was kept less than 3 x lo4 per set and the number of pulses exceeding the discriminator of the TSCA was kept below 2 x lo4 per sec. The only pile-up effect observed in the y-ray spectra was a slight broadening of the FEP. A typical free particle spectrum obtained at 100” from 5oCr is shown in fig. 2. The presence of the large “C and 160 contaminant peaks introduces large uncertainties in extracting the number of inelastically scattered protons at some angles. 2.3. DATA REDUCTION

Fig. 3 shows a TAC spectrum obtained with a 50Cr target at a proton scattering angle of 100”. The FWHM time resolution was 9 nsec which is typical of the results achieved. The horizontal bars on fig. 3 show the windows which were set about the time peak and about an equal-time slice of the accidental region in order to obtain the particle spectra shown in fig. 4. The solid dots in fig. 4 are the particle spectrum for the TAC window set around the time peak. The open circles show the same particle spectrum after the accidentals were subtracted. As expected, the elastic protons from 160 do not yield any real coincidences. Finally an energy window was set about the inelastic proton group, as shown in fig. 4, in addition to the time windows, to obtain the y-ray spectra. The y-ray spectra shown in fig. 5 were obtained with a ‘OCr target at a proton scattering angle of 50”. This angle has one of the worst real-to-accidental ratios and is shown for that reason. The solid dots are the coincident y-ray spectrum without accidental subtraction. Some accidentals due to annihilation radiation and the backscatter peak are visible. The open circles show the same spectrum with accidentals subtracted. The bar over the peak in fig. 5 indicates the region of the spectrum taken as the full-energy peak. The spin-flip probability is calculated from the experimentally measured quantity

(1) where Npy(O,,)is the number of proton y-ray coincidences for which the y-ray energy lies in the full-energy peak and Np(ep) is the number of inelastic proton counts corrected for dead-time losses, which are small in the present experiment. E7 is the FEP

m 52Cr@,p’)

167

y-ray detection efficiency measured as described above and er, is the direction of the inelastically scattered proton. 3. Analysis 3.1. SOLID-ANGLE

CORRECTIONS

In the limit of infinitesimal solid angles the spin-flip probability (S) is equal to N(B,), however, for finite detector apertures one detects true coincidence events due to de-excitation y-rays from the m = 0, +2 magnetic substates. Since the half-angle of acceptance of the proton detectors was small compared to that of the y-ray detector and since measurements were not made at extreme forward or backward angles it is necessary to correct only for the finite solid angle of the y-ray detector. It has been shown “) that, to first order in E2, the spin-flip probability lies within the limits smin(ep>

=

N(ep)

-

m

~max(~p)

=

w,)

- ET4

- e+~~(~,~i~ - wvm

where E is the half-angle of acceptance of the y-ray detector. The percentage correction due to the finite solid angle is small when N(0,) is large but can be significant at forward angles when N(8,) is small. We have chosen to include the effects of a finite-sized y-ray detector in the calculated fits rather than adjusting the data. Rybicki et al. 11) have given the following expression for the angular correlation function for a finite-sized cylindrical y-ray detector: We, p 4,) = K~hd+4KQ ~~(4 The attenuation function,f,(s), and is given by

y 43.

(2)

depends only on the characteristics of the detector,

fz(.s) = /~r(cc)P,(cos CX) sin a da//Ir-(a)

sin ads

(3)

where r(c1) is the response function across the face of the detector. The correlation coefficients A&k,, kb) are given by

where the subscript B refers to the excited state of the target nucleus. The C,,(B,, 4,) are renormalized spherical harmonics and for the spin-flip geometry, 8, = 0, we have CKQ = &0. The average correlation function becomes

(5)

168

W. E. SWEENEY,

The function P,,(J,) &e(Jn)

Jr. AND J. L. ELLIS

can be expressed in terms of the statistical tensors: =

C

P~~~~(_ l)‘B-“‘=(Je

Jx &fn _ x;iKO

IF\

j,

(061 MBM’B

Poo(JB)

=

Tr p/(2J, + 1)“.

(7)

In the case of a single-multipole transition, the Rx(y) reduce simply to

where L is the multipolarity of the y-ray transition, JB is the initial-state spin and J, the final-state spin. The coefficients &(LLJ,J,) are tabulated by Rose and Brink ’ “). The Clebsch-Gordan coefficient in eq. (6) requires that MB = &. Combining eqs. (5)-(7), the expression for the averaged spin-flip correlation function becomes

is

W(OY4%) =,F

fx(s) ‘!%!@!s Tr D (2Js + l)*( - l)‘“-+“(Js

JBn/i, - M,IRO)R,(LLJ,

Jc).

I e

(8)

We note that the quantity pMBMB /Trp is the relative probability of forming the state B with magnetic substate Mu (i.e., the substrate population parameters). We have calculated theoretical average correlation functions, W(0, &), assuming that the detector response function, r(c), is a constant across the face of the detector. This assumption is valid because the conical shielding exposes only the center portion of the NaI crystal. The averaged spin hip, s (t?,), is given by S(6,) = $W(O, 4$), and is independent of 4, but, of course, depends on the angle of the inelastically scattered proton, 0,. It is this averaged spin-flip function, s (e,), which should be directly compared with the measured quantity N(6,) given in eq. (1). 3.2. CALCULATIONS

Collective-model calculations where performed using the DWBA codes DWUCK and HELMY. The optical-model potentials for the entrance and exit channels were of the form

+a-Z

A ‘+ ( m,c )

~{l+exp[(r-r,.Ai)/a,,]~-‘, ..

..

(9)

plus the Coulomb potential of a uniformly-charged sphere of radius rcA*. The values of the parameters were obtained from the work of Becchetti and Greenlees “) (BG) and Perey et al. ‘) and are listed in table 1. These parameters were used

s”~szCr(p, p’)

169

because of the lack of available cross-section and polarization data for proton elastic scattering from 5o*‘*Cr at 12 MeV. Becchetti and Greenlees present a general formula to calculate optical-model parameters for nuclei with A > 40 and for bombarding energies less than 50 MeV. Because of the averaging involved in obtaining their formula, it can be used with reasonable confidence to generate standard optical potentials in the range of its validity. In order to obtain information about how the calculated spin flip depends on the potential parameters, the calculations for soa 52Cr were also made using Perey’s F-parameters for 52Cr. It should be noted that 50Cr was not included in Perey’s study. These parameters were chosen for the comparison because they were obtained from 11 MeV data, which is close to the energy of the present experiment, and because of the success of Perey et al. lo) in fitting inelastic cross sections using the F-parameters. The collective-model form factors were obtained by deforming the optical-model potentials for the scattering states. In addition to the deformation of the real and imaginary parts of the potential, the deformation of the spin-orbit term was treated three different ways. Calculations using the code DWUCK did not include any spin-orbit term in the collective form factor. Calculations using the code HELMY included either a deformed spin-orbit potential of the full Thomas (FT) type 13) or a simplified version of the spin-orbit distortion used by Fricke et al. 14) (OR). TABLE 1

Optical-model Nucleus 50Cr 52Cr %r

Source BG BG Perey F

VR 53.13 54.58 47.11

rR 1.17 1.17 1.285

parameters used in DWBA calculations uR

W,

rD

0.75 0.75 0.65

9.28 9.72 8.96

1.32 1.32 1.285

Vs.0. rs.0. 0.538 0.564 0.53

6.2 6.1 9.95

1.01 1.01 1.285

as...

rc

0.75 0.75 0.53

1.285 1.283 1.25

The DWBA codes were used to generate transition amplitudes in the coordinate frame where the z-axis is along the direction of motion of the incident particle. Using the prescription of Satchler 15), these amplitudes were rotated to the coordinate frame where the z-axis is along the normal to the reaction plane. The rotated transition amplitudes were used to calculate substate population parameters. The solid angle corrections discussed in subsect. 3.1 were then applied to obtain the averaged spin flip, S (0). The effects of the various deformations of the spin-orbit potential in the collectivemodel form factors are shown in fig. 6. The calculations were made using the 50Cr parameters of Becchetti and Greenlees. The solid curve shows the calculated values of s (0) for no spin-orbit distortion in the collective form factor. The dashed curve shows the results for the simplified spin-orbit distortion (OR) while the dot-dashed and dotted curves show the results using the full Thomas distortion with the relative spin-dependent deformation parameter, /3_/j?, having the values one and two, respectively. For

I

I

40”.

80”

I

I

120”

160”

e cm Fig. 6. Collective-model predictions for the averaged spin flip, S(0). using the S°Cr parameters of Becchetti and Greenlees (table 1). The solid curve is the result with no spin-orbit deformation in the collective form factor. The dashed curve is the result using a simplified spin-orbit deformation while the dot-dashed and dotted curves are the results using the full Thomas deformation with the relative spin-dependent deformation parameter, /I..../& having the values one and two, respectively. I

0.40 -

50Cr(p,p9

I

50Cr*(0.78

I

I

I 120”

I 160”

MeV)

E, = 12.0 MeV

0.30 -

s^ IE 0.20 -

0.10-

I 40°

I 80”

8cm Fig. 7. Averaged spin-fiip angular distribution for the reaction 5oCr(p, p’)50Cr* (0.78 MeV) at E* = 12.0 MeV. The solid and dotted curves are DWBA predictions using the 50Cr parameters of BG (table 1) with no spin-orbit deformation in the collective-model form factor and with a full Thomas spin-orbit deformation with &,./B = 2, respectively. The dashed curve was obtained using the %!r Perey F-parameters (table 1) and no spin-orbit deformation.

171

50*52Cr(p, p’)

0.401

=‘cr(p,p’J 52Cr”(l.43MeV)

-I

Fig. 8. Averaged spin-flip angular distribution for the reaction 52Cr(p,p’)52Cr(1.43 MeV) atE, = 12.0 MeV. The solid and dotted curves are DWBA predictions using the 52Cr parameters of BG (table 1) with no spin-orbit deformation in the collective-model form factor and with a full Thomas spin-orbit distortion with /?,.J@ = 2, respectively. The dashed curve was obtained using the %r Perey Fparameters (table 1) and no spin-orbit deformation.

the present calculations, it is clear that the main effect of increasing distortion of the spin-orbit potential is to increase the predicted spin flip at the forward maximum, near 90”, and to shift the peak at back angles toward larger angles.

4. Results and discussion Figs. 7 and 8 show the experimental angular distributions for the averaged spinflip correlation function, s(0) for the reactions “Cr(p, p’)“Cr* (0.78 MeV) and 52Cr(p, p’)52Cr* (1.43 MeV). The error bars include only the statistical uncertainty in the number of coincidences added in quadrature to a realistic estimate of the uncertainty in subtracting the background under the inelastic peak of interest. Uncertainties in the absolute efficiency measurement and possible systematic errors such as an effective half-angle of acceptance different from the measured value of 11” have not been taken into account and result in absolute uncertainties in the ordinate scale of 5 % for 5oCr and 8 ‘Afor 52Cr. The theoretical predictions of S(0) for 50Cr are plotted in fig. 7. The solid curve was obtained using the 50Cr BG parame t er s with no spin-orbit deformation in the collective form factor while the dotted curve represents the “Cr BG parameters including the full Thomas spin-orbit deformation with fiS.,./p = 2. Finally the dashed curve was obtained using the 52Cr Perey F-parameters with no spin-orbit deformation.

172

W. E. SWEENEY, Jr. AND J. L. ELLIS

three predictions give reasonable fits to the shape of the angular distribution, but the predicted values are all too low. This could be attributed either to the uncertainty in the ordinate scale mentioned above or to the possible addition of a compoundnuclear contribution to the measured spin flip, The theoretical predictions of S (8) for 52Cr are plotted in fig, 8. The solid curve was obtained using the 52Cr BG parameters with no spin-orbit deformation while the dotted curve represents the “Cr BG parameters with the full Thomas spin-orbit deformation with &_/B = 2. Again the dashed curve was obtained using the “Cr All

I

50Cr(p,p’l

5oCi1078

,

MeV)

I

I

I

I

I_

’

%r(p p’l 52Cr’(l.43MeV) l $& Q

0.30-

i

-L

s IE

i

1

0.20-

e,=ks

-1-f f O.lO-

I It.0

I 11.5

I 12.0

I I25 PROTON

ENERGY

I

t

I

If.0

II.5

12.0

I i25

(MeV)

p’) Fig. 9. Spin-tIip excitation function for the reactions s°Cr@, p’)50Cr* (0.78 MeV) and %r(p, 5zCr* (1.43 MeV). The curves are calculated using the BG parameters (table 1) with no spin-orbit deformation in the collective-model form factor. They have been arbitrarily normalized to show only the predicted trend for the spin-flip probability.

Perey F-parameters with no spin-orbit deformation. It is apparent that although the theoretical curves predict the maxima and minimum in the appropriate positions, they fail to predict the magnitude of the maximaor their relative values. One notes that for both “Cr and 52Cr the choice of a different optical-model parameter set has a larger effect on the predicted spin flip than going from no spin-orbit deformation to a rather large spin-orbit deformation. The results of the present experiment for the reaction 52Cr(p, p’)52Cr* (1.43 MeV) are in qualitative agreement with the results of Ballini et al. ‘) for the same reaction at 11 MeV. Ballini observed two peaks of nearly equal magnitude while the present data indicate a stronger peak at back angles.

5o*52Cr(p, p’)

173

This apparent decrease in magnitude of the forward peak in the “Cr spin-flip angular distributions with increasing energy suggests the possibility of compoundnuclear contributions. To look for such effects we measured spin-flip excitation functions from 11 to 12.5 MeV at angles near the maxima in the spin-flip angular distributions for 50Cr and 52Cr. The results are shown in fig. 9. The curves are calculated using the BG parameters with no spin-orbit deformation in the collectivemodel form factor. The predictions have been normalized arbitrarily and are only included to show the predicted trend for the averaged spin-flip probability at the respective angles. One observes that the predicted trends for “Cr agree fairly well with the data, as one might expect since the angular distribution for 5oCr at 12 MeV had the typical direct reaction shape. The 80” data for 52Cr indicates a decreasing trend in the spin-flip probability with increasing energy. However, one cannot definitely say that the 80” data is inconsistent with the DWBA trend line. The excitation functions for so, 52Cr do not exhibit any strong resonance effects at 12 MeV, therefore, the angular distributions measured here are probably typical for this energy range. One should note that the Q-values for the reactions 5‘Cr(p, n)“Mn and 52Cr(p, n) 52Mn are -8.41 MeV and -5.49 MeV, respectively. Although the value of the (p, n) threshold only indirectly determines the compound-nuclear contribution, it is interesting that in the present experiment the nucleus with the higher (p, n) threshold has a spin-flip angular distribution that is more like what one would expect from a direct reaction. This suggests the desirability of studying other nuclei in this mass region. It should also be noted that the availability of more data may make it possible to invoke some procedure to handle the compound-nuclear contribution to the spin flip. The authors wish to acknowledge the assistance of Dr. E. V. Hungerford and the support and encouragement of Professor G. C. Phillips. We are also indebted to Dr. W. Braithwaite for providing us his modification of DWUCK to include the calculation of spin-flip probabilities. The authors are indebted to R. Howell of Michigan State University for performing the calculations with H. Sherif’s code HELMY. References 1) F. H. Schmidt, R. E. Brown, J. B. Gerhart and Wojciech A. Kolasinski, Nucl. Phys. 52 (1964) 353 2) A. Bohr, Nucl. Phys. 10 (1959) 486 3) J. Delaunay, J. P. Passerieux and F. G. Perey, as quoted in F. G. Perey, Proc. of the 2nd Int. Symp. on polarization phenomena of nucleons, Karlsruhe, September 6-10, 1965, eds. P. Huber and S. Schopper (Birkhaiiser Verlag, Basel, 1966) 4) W. A. Kolasinski, J. Eenma, F. H. Schmidt, H. Sherif and J. R.Tesmer, Phys. Rev. 180 (1969) 1006 5) D. L. Hendrie, C. Glashausser, J. M. Moss and J. Thirion, Phys. Rev. 186 (1969) 1188 6) M. Ahmed, J. Lowe, P. M. Rolph and 0. Karban, Nucl. Phys. Al47 (1970) 273 7) R. Ballini, N. Cindro, J. Delaunay, J. Fouan, M. Loret and J. P. Passerieux, Nucl. Phys. A97 (1967) 561 8) F. D. Becchetti, Jr. and G. W. Greenlees, Phys. Rev. 182 (1969) 1190 9) C. M. Perey, F. G. Perey, J. K. Dickens and R. J. Silva, Phys. Rev. 175 (1968) 1460 10) C. M. Perey, R. J. Silva, J. K. Dickens and F. G. Perey, Phys. Rev. C2 (1970) 468 11) F. Rybicki, T. Tamura and G. R. Satchler, Nucl. Phys. Al46 (1970) 659 12) H. J. Rose and D. M. Brink, Rev. Mod. Phys. 39 (1967) 306 13) H. Sherif, Nucl. Phys. A131 (1969) 532 14) M. P. Fricke, E. E. Gross and A. Zucker, Phys. Rev. 163 (1967) 1153 15) G. R. Satchler, Nucl. Phys. 55 (1964) 1