Cross sections and analyzing powers for fast-neutron scattering to the ground and first excited states of 58Ni and 60Ni

Nuclear Physics @ North-Holland

A438 (1985) Publishing

187-211 Company

CROSS SECTIONS AND ANALYZING POWERS FOR FAST-NEUTRON SCATTERING TO THE GROUND AND FIRST EXCITED STATES OF %Ni AND @Ni P.P. GUSS*, R.C. BYRD**, C.E. FLOYD***, K. MURPHY, G. TUNGATE **** , R.S. PEDRONI

C.R. and

HOWELL, R.L. WALTER

Department

of Physics, Duke University, Durham, NC 27706, USA and Triangle Universities Nuclear Laboratory?, Duke Station, NC 27706, USA J.P. DELAROCHE Centre d’Etudes

de Bray&es-le-Cha^tel, Service de Physique Neutronique 91680 Bruyeres-le-Cha^tei, France

et Nucleaire,

BP No. 12,

T.B. CLEGG Department

of Physics, University of North Carolina, Chapel Hill, NC 27514, USA and Triangle Universities Nuclear Laboratory, Duke Station, NC 27706, USA Received 6 April 1984 (Revised 10 October 1984)

Abstract: Differential cross sections for neutron scattering from 58Ni and 60Ni to the ground state and first excited state have been measured at 8, 10, 12 and 14 MeV. In addition, analyzing powers were measured for scattering to the same states for 58Ni at 10 and 14 MeV, and for 60Ni at IO MeV. The data were analyzed in the framework of a coupled-channel formalism in which the vibrational model was assumed with deformed central and spin-orbit potentials. A spherical-optical-model analysis of the elastic scattering data was also performed following the coupled-channel analysis. Predictions for (p,p) and (p,p’) scattering observables have been made and compared with measurements previously published. This approach permits neutron and proton deformation parameters to be deduced similarly from (n, n’) and (p, p’) scattering measurements for 58,60Ni. These deformation parameters are compared in the framework of the core-polarization model of Madsen, Brown and Anderson.

* Present address: Department of Physics, The College of William and Mary, Williamsburg, VA 23 185, USA. ** Present address: Indiana University Cyclotron Facility, Bloomington, IN 47405, USA. *** Present address: Department of Radiology, Duke University Medical Center, Durham, NC 27706, USA. **** Present address: Max-Planck-Institut fur Kernphysik, Heidelberg, West Germany. ’ Work supported by the US Department of Energy, Director of Energy Research, Office of High Energy and Nuclear Physics, under contract no. DE-AC0576ER01067. 187

188

E

P.P. Guss et al. / 58s60Ni(n, n), (n, n’)

NUCLEAR REACTION ‘sNi, 60Ni(n, n), (n, II’), E = 8, 10, 12, 14 MeV, measured w( 0, E), A,(& 15); deduced deformation parameters for central and spin-orbit potentials; coupledchannel and spherical-optical-model calculations.

1. Introduction In the present paper extensive experimental and optical-model investigations of neutron scattering from 58Ni and 60Ni are reported. Previously, relatively few data existed for Ni(n, n) for incident neutron energies above 8 MeV. The present measurements of the differential cross section (+( f3) at 8, 10, 12 and 14 MeV are part of a series of neutron scattering studies underway at the Triangle Universities Nuclear Laboratory (TUNL). Of primary interest, lately, have been cross sections for elements which would be essential components of controlled fusion reactors of the US energy program. In addition, the present ~(0) measurements are of basic interest in that elastic and inelastic scattering data permit .a detailed study of the interaction of fast neutrons with nickel isotopes - nuclei which may be considered as soft vibrational nuclei ‘). In order to make our study more complete, the a@) measurements have been complemented with measurements of analyzing powers A,(B) for elastic and inelastic scattering from 58Ni and 60Ni in the same energy range. These A,(B) measurements for neutrons are the first ever reported above 4 MeV for separated isotopes of nickel and the only ones for inelastic scattering from these nuclei. The experimental setup used for the a(0) and A,(8) measurements, as well as the data reduction, are described in sect. 2. A coupled-channel (CC) analysis of the data is presented in sect. 3, where the spin-orbit deformation effects 2*3) are also discussed. The CC analysis has been carried out in a manner which also requires a good representation of other neutron scattering and reaction observables measured over a broad energy range (10 keV to 80 MeV), a range much wider than that covered in the present

scattering

ted into a proton

experiments.

potential

The neutron

in an isospin-consistent

optical potential manner

has been conver-

in order

to compare

with elastic and inelastic proton scattering data 4-7) and to determine proton quadlengths 6,,,. The rupole deformation parameters &,+ and thereby, deformation comparison between the values of the deformation lengths S,,, and 6,“. deduced from the present proton and neutron analyses is shown in sect. 3 and is discussed in the context of the core-polarization model ‘), according to which Pnn, should be different from &,,’ for nuclei with one closed shell. In sect. 4 we transform the CC model into a spherical optical model (SOM) for neutron elastic scattering in order to provide a parametrization with a simple model. Such SOM representations can be useful for applied purposes, DWBA calculations, and for comparison with nuclear matter predictions. One special feature of the SOM analysis was that it was constrained to use almost the same parameters as those derived in the CC study. The only significant change was to increase the depth of the absorptive potential in order

P.P. Guss et al. /

to account integrals

for channel-coupling

effects which are ignored

for the CC and SOM optical

2. Experimental 2. I. CROSS-SECTION

5*.60Ni(n,n), (n, n’)

potentials

189

in the SOM. The volume

are presented

in sect. 5.

method and data-handling technique

DATA

Considerable detail about the experimental technique for the neutron cross section measurements has been presented in other TUNL reports 9-l’). Briefly, a deuteron beam is chopped and bunched at a 2 MHz rate and is injected into the tandem Van de Graaff. The deuteron pulse width is about 2 ns. The beam is directed onto a gas cell which is 2.9 cm long and is pressurized to 2 bar of deuterium. The zero-degree flux from the ‘H(d, n)3He reaction is incident on cylindrical samples, 1.6 cm in diameter and 2.4 cm in height, with enrichments of 99.93% and 99.79% for the ‘*Ni and 60Ni samples, respectively. The cylinders are hung vertically 7.9 cm from the center of the gas cell. The scattered neutrons are observed with two detectors located on opposite sides of the incident beam axis. The left detector, 13 cm in diameter and 5.1 cm thick, is located 5.7 m from target; the right detector, 8.9 cm in diameter and 5.1 cm thick, is 3.7 m from target. The detectors, liquid organic scintillators, are encased in massive shields which rotate in a horizontal plane about the scatterer. Pulse-shape discrimination for excluding -y-ray interactions is employed. The threshold of the detector electronics is set at a proton recoil energy of about 1.9 MeV. A third detector, which looks directly at the 2H(d, n) source reaction, is located at 50” above the horizontal plane of the detectors and serves as a monitor for normalization purposes. At each angle two time-of-flight spectra were obtained. The SAMPLE-IN spectra were accumulated

for each detector

with the sample in place, and the SAMPLE-OUT

spectra were taken with the sample removed. The SAMPLE-OUT spectra were subtracted from the SAMPLE-IN spectra to produce DIFFERENCE spectra, several of which are shown in fig. 1. A complete description of the data reduction is given in ref. ‘I), so only an outline of the procedure is given here. From fig. 1 it is seen that after having subtracted the SAMPLE-OUT spectra, there still remains a residual background. The influence of this background on the determination of (r(0) was carefully studied with two specialized fitting programs, and eventually a statistically weighted mean was found for all of the estimates of the yields of a single peak. To produce the reported uncertainty, the standard deviation of these estimates of the yield was added in quadrature to the conventional statistical uncertainty of the measurement. An error which depended on the shape of ~(6) was also added to cover an 0.3” angle uncertainty d13 in the location of the detectors. In order to determine the absolute cross section scale, scattering from hydrogen contained in a well-characterized polyethylene sample was also measured. For this

P.P. Guss et al. /

190

58*60ivi(n, n), (?I, ?I’)

4;;

Fig. I. Time-of-Bight

“DIFFERENCE”

determination

the SAMPLE-OUT

that contained

the same number

spectra

700

8

for a(O) measurement

at a flight path of 5.7 m.

measurement of carbon

nuclei

1

was made with a carbon as are present

scatterer

in the polyethylene

scatterer. The scattering angles, which were near 30” in the lab system, were selected for minimizing the overlap between the peak for the n-p scattering and the n-“C peaks for elastic scattering and inelastic scattering to the first excited state. The yields for the nickel samples were thereby normalized to the absolute cross section for n-p scattering. Multiple scattering effects in the nickel and the polyethylene scatterer were treated by Monte Carlo simulation utilizing the TUNL code EFFIGY, which is outlined in ref. ‘I) and references therein. The final values obtained for a(0) are shown in fig. 2 as the solid circles. The curves shown in fig. 2 are Legendre polynomial fits to the data. These curves were calculated according to ~(0, E) =I, a,(E)P~(cos 0); the coefficients a,(E) are

PP. Gusset al. /

58,601?i( n, n),

(n, n’)

109 104 104 T c 13 104 E 6

103

c I02

IO’ n

0 IO2

60

120

180

0 102

60

120

180

60Ni(n,n,) 2+ STATE

10’ y IO’ T 13

-510’ 6 -5

IO’

loo

Fig. 2. Presentation of measured values for u( 0). Solid curves are polynomial fits. The uncertainties the data are typically the size of the symbols or less, except when shown.

on

available from the authors. For elastic scattering, the points plotted near 8 = 0” are the values of Wick’s limit calculated from the total cross section values reported I*) in ENDF/B-V. These points were included for the Legendre polynomial fitting process in order to guarantee that the prediction for the region between 0” and 20” would fall into a physical range. For this analysis the Wick’s limit values were assigned uncertainties of about f 10%. The error bars shown on the data points in fig. 2 only account for the relative uncertainties in the cross-section values. These uncertainties include: counting statistics (1 to 14% for (n, n) and 3 to 32% for (n, n’)), relative efficiency (less than 2.5%) and Monte Carlo calculations (less than 1%). The same relative uncertainties

192

P.P. Guss et al. / 58*60Ni(n, n), (n, n’)

were used in the fitting procedure coefficients.

In addition

to relative

and for determining uncertainties,

the errors on the polynomial

the overall

normalization

uncer-

tainty is estimated to be about 5% ; this includes an 0.8 to 1.4% uncertainty in the hydrogen scattering yield and a 2 to 3% uncertainty in the Monte Carlo correction factor for the polyethylene and carbon scatterers. Tables of our data and the estimated uncertainties are available from the National Nuclear Data Center at Brookhaven National Laboratory. The average neutron energies E for the measurements, as calculated by EFFIGY, are 7.90, 9.95, 11.95 and 13.94 MeV with respective energy spreads AE of 0.19,O. 14, 0.12 and 0.10 MeV. The principal contribution to AE comes from the energy loss in the deuterium gas.

2.2. ANALYZING-POWER

DATA

For the analyzing-power A,,( 0) measurements the polarization transfer reaction ‘H(d, ?Q3He at 0” was employed to produce the polarized neutron beam. The TUNL Lamb-shift polarized ion source 13) provided the deuteron beam, and the quenchratio method 14) of determining the deuteron polarization was used. About 90% of the magnitude of the vector polarization of the deuteron beam is transferred to the neutron beam at 0”. To permit the use of neutron time-of-flight spectroscopy, the polarized deuteron beam is pulsed into 2 ns wide bursts 15). Approximately 0.1 p,A of polarized beam is obtained at the target, compared to the 1.5 CIA of unpolarized beam used in the a(0) measurements mentioned above. Two changes are employed to compensate for the lower beam intensity. First, the amount of deuterium is increased fourfold by doubling the cell length to 5.08 cm and doubling the pressure to 4 bar. (For these measurements, the center of the cell is 9.3 cm from the nickel target.) Second, for the left and right detectors, the flight paths were shortened to 3.7 and 2.7 m, respectively. In order to reduce instrumental asymmetries in the A,(O) measurements, the detectors are positioned at equal scattering angles (on opposite sides of the beam axis), and data are obtained in several pairs of runs with the polarization axis alternately directed along the spin-up and spin-down orientations. Spectra obtained in the A,,(B) measurements are presented in fig. 3. As in the a(0) experiments, several background selections were chosen in the analysis procedure. All consistent values for the yields were combined to obtain the final values. Corrections to the yields closely follow the procedure of ref. I’). The effects due to the finite size of the sample and the multiple scattering within the sample have been treated by Monte Carlo simulation with the code JANE, which originated at the University of Tiibingen 16). The code accounts for multiple scattering up to and including triple-scattering events. The A,(8) results for 58Ni are presented in fig. 4. The curves are derived from the associated Legendre polynomial fits of the product ~(0, E)A,( 6, E) =

193

Pi? Guss et al. / “*“Ni( n, n), (n, n’)

1000

900

CHANNEL Fig. 3. Time-of-flight

“DIFFERENCE”

spectra for A,(O) measurement

at a flight path of 3.7 m.

C, b,(E)Pj(cos 0). The values of the energy spreads AE were obtained from JANE and are 0.53 and 0.50 MeV for the 9.92 and 13.91 MeV measurements, respectively. One A,,(B) measurement was made for 60Ni in order to see whether a sizeable difference exists between A,,( 0) values for the two isotopes. The measurement was made for this nucleus at 0 = 100” and E = 9.92 MeV, and the results are -0.68 f 0.03 and -0.19 f 0.05 for elastic

and inelastic

scattering,

respectively.

These values

are

shown in fig. 4 as the open circles and are to be compared to the corresponding values of -0.58 f 0.02 and -0.20 f 0.05 for 58Ni. The more negative value for (n, n,,) for 60Ni suggests that the deep valley near 120” shifts slightly to a more forward angle. One expects such an effect from a simple diffraction picture in which the nuclear radii are proportional to A”3. The uncertainties shown as error bars in fig. 4 and used to generate the uncertainties in the polynomial coefficients are based only on the relative uncertainties in the data: counting statistics (0.5 to 17% for elastic scattering and 3 to 3 1% for inelastic scattering), Monte Carlo corrections (less than 1%) and deuteron polarization instability (less than 1%). Also included was the uncertainty (up to 4%, but usually less than 1%) in the correction for that fraction of the unpolarized background component of the deuteron beam which quenched during the calibration of the

194

P.P. Cuss et al. / 58~60Ni(n, n), (n, n’)

I 5*Ni (n,n,) 05 > 14MeV

Fig. 4. Measured A,(O) values for ‘“Ni (indicated by the solid circles) and “Ni (open circles near 100” at 10 MeV). The curves were obtained from fitting a polynomial expansion of the product c+(B)A,(B) to the ‘sNi data. (See subsect. 2.2.)

polarization. The normalization uncertainties are 2% for the uncertainty 14) in our method of measuring the deuteron beam polarization and 1 to 2% for the uncertainty “) in the ‘H(& ti)3He polarization transfer coefficient. Furthermore, there may exist up to an additional 1% normalization uncertainty in the “effective polarization”

due to the anisotropy

(across the scatterer)

of the polarization

of the source reaction I*). In preparing for the CC model calculations, an adjustment measured A,(8) data for elastic scattering. This was necessary interaction interaction

transfer

was made to the as there exists one

that the CC code was not equipped to consider - the Mott-Schwinger 19) of the magnetic moment of the neutron moving in the Coulomb field

of the nucleus. As far as a( 0) is concerned, we determined with spherical-opticalmodel (SOM) calculations ‘“) that inclusion of this term makes an insignificant change in a( 0) at the angles for which we have data available; the change in a( 6) exceeds 1% only at the deepest minimum and at scattering angles much smaller than the most forward angle (20”) for which data were measured here. The effect on A,,( 0) is noticeable *‘) primarily at angles forward of 40”, although it does cause a slight shift (- 1”) with angle in the A,,( 0) function in the regions of the steepest slopes. Therefore, the A,(B) data were adjusted by a slight amount, as estimated from SOM calculations. These modified data are shown alongside the CC calculations illustrated in sect. 3, whereas the actual experimental values are shown in sect. 4.

P.P. Guss et al. / 58.mNi(n, n), (n, n’) 4.47

3-

3-

2f

'+

195

1.45

58

0.0

2+

1.33

0'

0.0

60Ni

Ni

Fig. 5. Low-lying

4.05

excited

states of ‘*Ni and 60Ni from ref. ‘I).

3. Coupled-channel

description

3.1. INTRODUCTION

As suggested by the excitation energy (see however, the structure differences in nuclear

ordering of excited states “) and the systematics of the low fig. 5), the 58Ni and 60Ni nuclei are presumed to be vibrators; properties of the nuclei do not seem to be identical. These structure are apparent, for instance, when considering the

quadrupole moment Q(2+) of the first 2+ state. The values of Q(2+) deduced from Coulomb excitation measurements 22) are Q(2+) = 0.03 f 0.07 and -0.15 f 0.08 e - b for 60Ni and 58Ni, respectively. We have accounted for these differences by inserting into the CC calculations the reorientation matrix elements of the first 2+ state in 58,60Ni inferred from these Q(2+) values. With the phase conventions of the CC code ECIS79 written by Raynal 23), the relative values ME’ used for these reduced matrix elements are 0.00 and +1.16 e * b for 60Ni and “Ni, respectively. Using the CC formalism 24) which includes complex and deformed potentials, a unique parameter set which can describe the a( 0) and A,,( 0) data has been sought. Moreover, the optical potential has been varied from one isotope to the other through the quadrupole deformation parameter p and the asymmetry parameter E= (N -2)/A. The isospin-dependent term U, of the potential U has been assumed to be complex: U, = V, + i W,. In order to remove part of the ambiguities underlying the potential parametrization, the present analyses have been conducted while constraining the CC calculations to reproduce also the s- and p-wave strength functions So and S,, the potential scattering radii R’ of ref. 25) and the energy variation of the total cross section a,(E) of refs.26s27) between 0.1 and 80 MeV. This method of analysis is an extension of the SPRT method described in ref. *‘), since analyzing powers for elastic and inelastic scattering are presently considered along with a(0) and the other observables. Since it is required that the deformed-optical-potential parametrization provides a fair representation of neutron scattering and reaction observables over a wide energy range, a volume absorption term is included along with the usual surface

196

P.P. Guss et al. / s8*mNi(n, n), (n, ?I’)

absorption term. The neutron optical potential is expressed as follows: U=--(V+iW,)f(

r,

Rv)+4iaDW,~f(r,aD,

a~,

-2iX~K.,.Vf(r,%,., R,.,.~xv 1s. In this expression, the form factor is a Woods-Saxon

RD)

(1) type:

f(r, ai,Ri)={1fexp[(r-Ri)/ail}-‘, where

L

Ir

1

_I

is a “nuclear” radius and J2 refers to the center-of-mass system. The operator (Y+ is related to the vibration amplitude & as described in ref. 24). In addition, it has been assumed that the deformation lengths Si = PiRi (fm) have the same value 6, for all the central terms in eq. (1) and that the spin-orbit potential is a full Thomas type ‘). The deformation length of this term is defined as S,,,. = &,,R,,. (fm) and is permitted to be different from 6, for the central potential. The CC calculations were performed assuming a coupling basis (O+, 2+). In eq. (1) only a real spin-orbit potential is included, since the insertion of a small imaginary spin-orbit component in the optical potential does not significantly improve the fits presented below.

3.2. ANALYSIS

METHOD

AND

RESULTS

The potential parameter search was initiated by assuming the p-values compiled by Stelson and Grodzins “) and the geometries determined previously 30) at TUNL for iron and copper. In the early stage of the search for the best geometrical parameters it was found that the values quoted in ref. 30) were also suitable for the nickel potential. Therefore, our analysis has been specialized to optimizing the values of the real and imaginary potential depths. Moreover, as it was possible to fit the various measurements at incident energies E s 14 MeV assuming no volume absorption, WV was set to zero in the first stage of our analysis. This specific parametrization of the absorptive potential is labelled step I in table 1, which also contains the defo~ation parameters Pnnr. As can be seen in table 1, for E 9 14 MeV the real potential V displays an energy dependence which is linear and identical to that found earlier 30) for 63*65Cu.On the other ha&, the absorptive potential W, has an energy dependence which behaves as JE; this behavior was seen earlier 30) for 54,56Fe.A fair agreement with the measured ‘“) a,(E) is obtained with this potential from 0.1 to 14 MeV. The S, and R’ values calculated at 10 keV incident energy are 2.76 X 10e4 and 6.57 fm for 58Ni and 1.93 x 10e4 and 6.82 fm for 60Ni. These values are in satisfactory agreement

P.P. Cuss eta/. / 58*60Ni(n,n), (n, n’)

197

TABLE 1 Coupled-channel

step I:

parametrization of potential depths for neutron nickel for three stages of analysis “)

(OSEGll) (ll~E~24) (OSESll) (ll~E~24)

step.2 "): W,=4.04-

15.54s+ l.lJZ W,=7.69-15.54&-0.15(E-11) W” = 0.0 Wv=0.23(E-11)

and deformation

(OsEs20)

-0.46.E

E

V=76.12-21.75&-9.2l~ Wo=4.04-15.54e+l.lJE wo=7.73-15.548-0.1 w,=o W,=O.l (E-11) V,,, = 6.50 - 0.035E geometries

(E-12)

(20~E~80) (OS E =s12) (12~E~80) (OsE=~ll) (llSES80)

(O=sES80)

parameters

rv= 1.165, ov = 0.656, &(‘*Ni)=0.19+0.01,

from

(OS ES 14) (0 s E c 14) (OSE=zl4) (OSESl4)

V=57.75-21.75&-O&E wo=4.04-15.54e+l.l& W” = 0.0 Vs.,, = 6.50

step3'): v= 57.75 -21.758

scattering

for steps 1, 2 and 3:

r,= 1.261, r_ = 1.017 a, = 0.593, a,.,, = 0.600 pJ6’Ni) = 0.21 * 0.01

“) Energies and potential well depths are expressed diffusenesses are in fm; E is incident energy. b, Here V and V,,,are the same as in step 1. ‘) Relativistic kinematics used throughout.

in MeV;

radii

and

with the experimental values 25) of (3.1 kO.8) x 10e4 and 7.5 kO.5 fm for 58Ni and (2.4rt0.6) x 10e4 and 6.7kO.3 fm for 60Ni. The real and imaginary components Vi and W, of the symmetry potential have the values of 21.75 and 15.54 MeV, respectively. Within

the estimated

uncertainties

- 15% and -30%

attached

to VI and W,,

respectively, these values are very close to those found 30) for iron. The second part of the present discussion is devoted to the competition between surface and volume absorption. For this study, the (n, n) and (n, n’) scattering measurements 3’) obtained at 24 MeV for 58*60 Ni by the Ohio University group as well as total cross sections 26*27 ) up to 30 MeV have provided useful information. In preliminary CC calculations we found that these measurements cannot be properly fit assuming just a surface absorptive potential. Therefore, in a more extensive study, these measurements were combined with the present TUNL a(0) and A,,( f3) measurements in a simultaneous search for best WD and W, values, using the values from step I in table 1 for V(E), VS.,., p and all the geometrical parameters. Overall, good fits are achieved using the absorption parametrization labelled step 2 in table 1. This combination of W,(E) and W,(E) preserves the goodness of fits obtained

198

P.P. Guss et al. / 5**60Ni(n, n), (n, n’)

58Ni 104

(n,n)

l

TUNL

0

Yamonouti

Dota et 01.

I

B ,.,.fdeg)

I

I

120

60 Q,.,.

I

180

(deg)

Fig. 6. Differential crass sections for neutron elastic scattering from 58Ni and 60Ni. The full lines represent coupled-channel calculations. The full circles are the present measurements: the open circles are measurements from ref. 3’).

earlier below - 14 MeV in step I, and leads to good fits to c~(6) at 24 MeV and to fair agreement with the measured 26*27) values of CT={ E) over the whole energy range of 0.1 to 30 MeV. Although the parameter set derived in step 2 for the absorptive potential is still not unique, we list it here to illustrate the effect on the parameters as the data base included in the determination is increased. A precise determination of W,(E) and W,(E) requires more (n, n) and (n, n’) scattering measurements above 14 MeV. An alternative method, which consists of fine-tuning the parameters labelled step 2, can be performed by constraining the CC calculations to reproduce also +(.E) up to 80 MeV. This was done using the unpublished measurements from ref. 27). The potential parameters which were obtained in this manner are quoted as step 3 in table 1. Note that V now has a logarithmic energy dependence at energies above 20 MeV and that V’.,. is slightly energy-dependent. Figs. 6 and 7 show the data obtained for the neutron elastic and inelastic cross sections for 58*60Ni.The full lines in the figures represent the direct interaction (DI) cross sections given by the CC calculations that used the parameters of step 3. For elastic scattering the overall structure and the magnitude of the cross sections are

P.P. Gus et al. / 5s*mNi(n,

.

TUNL

Dota

X Yamanoufi

-cc

199

n), (n, n’)

l

X

et 01.

-

Colc”lotlon

-1,~ 5 2

TUNL Data Yomonouti et ol.CC Colculofion

10.0

-

5.0

“1

E

5

T’

lO.O-

10.0

5.0

ttt’

5:

12.0

5.0

1.0

h

0.5

“ttt

1.0 500


?l

‘“‘“W

1.0

1 lf3”

I -.-

10.0

1

0.5

I

4

-+-?I 74 V.-l

‘ftt’

***

ro.o5.0 -

t+ ‘*t

-

‘t

t

l.O-

8.0 ttttt

+I

10.0 5.0

10.0 5.0-

1.0

1.0

o.50’lo Fig. 7. Differential

t

b

0.5 6,.,,(W)

cross sections for neutron inelastic and 60Ni. For explanation,

‘t

ttlt”fttt~p

t*t++

1.0 5 5.0

1.0 I

60 h.,,

I

I

120

180

(d4

scattering from the first 2+ excited see caption of fig. 6.

states of 5sNi

well reproduced by the calculations. For inelastic scattering the calculations shown in fig. 7 for 10, 12 and 14 MeV reflect most of the features seen in the TUNL measurement and for 24 MeV agree excellently with the data from Ohio University. It is perhaps noteworthy that at 10, 12 and 14 MeV the calculations for inelastic scattering fail to explain the data near 120”.

P.P. Giisf et al. / ~*~~*Ni(R, n), (n, n’)

200

Furthermore, at 8 MeV the systematic deviations between the curves and the measured 2+ differential cross sections in fig. 7 suggest that compound nucleus (CN) processes are also involved in the reaction mechanism. That is, assuming that the CN cross sections for the 2+ states of 58Ni have magnitudes comparable to those determined 32) for 54Fe at the same incident energy, much better agreement between the (DI + CN) calculations and the TUNL data at 8 MeV is achieved. We would expect a comparable CN component for elastic scattering at 8 MeV, but such a

kJ

I

nat .

-

I

Ni

I

-cc

l

I

200 E (keV) t

1

notNi

10.0,

t

cc

-

:.

,

0.6 E (MeV)

8 ,

I

I

1

I

I

I

1

notNi

1 “1’1 -

---

6-

-‘-

._,_._.--20

I

I

2.5

3

E

I

I 4

I

I

f+

20

30

cc

SOM BNL

I”IIL1 5

7

IO

(MeV)

Fig. 8. Total neutron cross section of nickel between 0.1 and 30 MeV. Comparison among available data [ref. r6)]; coupled-channel calculations (CC) performed using the parameters step 3 of table 1; and sphe~cal-optical-model calculations from sect. 4. Note that the scale along the ordinate, which is loga~thmic for the top two panels, makes a visual comparison of the quality of agreement difficult.

PP.

cusset al. / 5s*aNi(n,

201

n), (n, ?I’)

contribution would spoil the quality of the agreement shown in fig. 6 around 130”. The reason for this apparent inconsistency is not understood, but might be tied to the limited coupling scheme adopted in the present CC calculations. A comparison of the published total cross section a, and reaction cross section uR of ref. 26) to the predictions based on parameters of step 3 is shown in fig. 8. Here the cross section labelled a,,, corresponds to the evaluation ‘*) for the reaction cross section minus the cross section for CN elastic scattering. There is good agreement across the whole energy range for a=, and vR and a,,,, come into agreement as the energy increases, which is expected as the compound-nucleus elastic scattering becomes relatively small. Furthermore, the predictions are within the uncertainty (3%) of the unpublished a,(E) measurements of ref. *‘). The fits obtained for the analyzing power measurements are shown in fig. 9. The curves represent our CC calculations with S,.,. = 6,. It can be seen that the calculations are in close agreement with the data. In summary, our final potential parameters found in step 3 provide a good representation of the present TUNL measurements as well as the other neutron scattering and total cross section measurements available below 80 MeV.

0

60

120

180 9 c.rn.

0

60

120

180

(deg)

Fig. 9. Analyzing powers for (n, n) and (n, n’) scattering at incident neutron energies of 10 and 14 MeV for ‘sNi (solid circles). The curves represent CC calculations described in the text. The experimental values for (n, n) scattering have been corrected for Mott-Schwinger effects as explained in sect. 2. The data obtained at 10 MeV for 60Ni are represented by the open circles near 100”.

P.P. Guss et al. / 58*60Ni(n, n), (n, ?I’)

8,,,,,

(ded

Fig. 10. Differential cross sections and analyzing powers for neutron scattering from ‘*Ni at 10 MeV. The curves are different CC predictions obtained when the deformation length a,.,, is varied (see text). For elastic scattering the three calculations are nearly identical.

3.3. DISCUSSION

The sensitivity of the CC predictions to the size of 6,,, is illustrated for ‘*Ni in fig. 10, where the curves represented as dashed, full and dotted lines represent CC calculations in which a,,, has the values zero, S, and 2S,, respectively. The first observation which can be made is that the sensitivity of the CC predictions to changes in the values of 6,.,, is higher for A,,(8) than for the cross sections. For 58Ni the best fits of A,,(O) at 10 and

14 MeV for the 2+ state are achieved

when

S,.,. = 6,. As indicated in our earlier report 33), it is difficult to determine a precise value of a,,. for 60Ni from the single data point for A,,( 0) at 10 MeV for the first 2+ state; our calculations suggest, however, that S,.,.- 8,. In the following discussion, the sensitivity of CC predictions to a variation of the values assumed for the reorientation matrix element A4$ is presented for 58Ni. All the previous calculations were performed under the assumption that Q(2+) # 0 for “Ni and Q(2+) = 0 for 60Ni. New calculations for 58Ni were conducted assuming Mg’= 0. In fig. 11 these calculations are compared to the earlier ones in which M:2! = 1.160 e - b. Both predictions assumed that 6,,.= 6,. First, as can be seen in fig. 11, the values of (+( 0) predicted for the 2+ state are not affected much by

P.P. Gusset 50

I

I

I

I

“Ni

1.0 ----.----.

0

203

al, / 58*60Ni(n, n), (n, n’) I

(n,n,)

Mzz#O M22

= 0

120

60 9 c.m.

160

(dd

Fig. 11. Cross section and analyzing power for 58Ni(n n’) at incident energies of 10 and 14 MeV. The curves represent CC calculations in which S,,,, = 6, and’ the reorientation matrix element of the 2+ state is set either to zero (dashed curves) or to the value inferred from the measured value ‘*) of the quadrupole moment Q(2+) (solid curves). (See subsect. 3.3.)

variations in M$:‘. In contrast, fig. 11 illustrates that A,,( 13)calculations are sensitive to the value of ME’. It is seen that the fit achieved for A,( 0) at 10 MeV using the value i@’ = 0 is worse than that obtained with M$’ = 1.160 e - b in the angular region 0 = 70”-loo”, the region where the four higher accuracy measurements were performed. Although the A,,(6) predictions are also quite sensitive to 6,.,. in this angular range, the variations of 6,.,. produce an overall raising or lowering of the curve (see fig. lo), whereas the introduction of i@ # 0 amplifies the structure

204

P.P. Guss el al. / 58*“Ni(n, ?I), (n, n’)

everywhere, in particular by the right amount to explain the high-statistics A,,(B) data in the 70” to 100” region at 10 MeV (see fig. 11). Based on this agreement at 10 MeV, we conclude that the inclusion of the quadrupole moment Q(2+) is significant for describing our high-statistics data. Finally, we have also tested the influence of M$) on the predictions for S,, R’, and a,. When compared with our results obtained with M$;’ # 0 (see subsect. 3.2), the new prediction for S, is lower by about 20% and for R’ is higher by about 8%, and the predicted total cross section is changed by less than a few percent in the whole energy range. 3.4. COULOMB

CORRECTION

DEFORMATION

TERMS AND COMPARISONS

OF (n, n’) AND

(p,p’)

LENGTHS

An advantage of neutron scattering data over proton data lies in the fact that extraction of the symmetry potential U, from data for different isotopes is not obscured by Coulomb correction terms AU, existing in proton potentials. One can then use the neutron potential along with (p, p) and (p, p’) scattering data to convert the neutron potential into a proton potential in the framework of the Lane model 34). This has been done with the neutron potential specified in table 1 using published proton data 4-7) near 20 MeV for a(e), A,,( 0) and the spin-flip amplitude S( 0). This exercise was indeed useful for determining quadrupole deformations /I,,, as well as Coulomb correction terms which are denoted here as AV, and A WC for the real and imaginary parts, respectively. For the calculations a deformed Coulomb potential is inserted and an electromagnetic quadrupole deformation parameter&m. which reproduces the Coulomb excitation measurements quoted in ref. 35) is assumed along with a sharp-edge form factor with the Coulomb radius of Rc = 1.2A”3 fm. The deformation length &,.Rc is denoted as a,.,,. for the Coulomb potential. Calculations were performed using the potentials of step 1 and step 3. Both sets were used in order to study the sensitivity of A WC to the interplay between W, and W,. We emphasize that the same geometry parameters are used for the proton potential as for the neutron potential. Some of the calculations for the proton scattering observables that are based on the potentials of step 1 are shown in figs. 12 and 13. The CC predictions at 20.4 MeV represented by dashed, full, and dotted lines, correspond to the assumptions that the spin-orbit deformation length a,,. is equal to zero, 6, and 2S,, respectively. Here 6, is the deformation length of the central potential for (p, p’) scattering. From the comparison with the measurements, it can be seen that the best overall fits are obtained for a,,,. = 6,. The value S,,,, = S, is also presently found from the (p, p’) scattering measurements 4*5)at 18.6 MeV. This relation, S,.,. = 6,, which is close to that found in subsect. 3.2 from the (n, n’) scattering data analyses, contradicts recent suggestions 3, concerning the properties of the spin-orbit deformations, as discussed in ref. 33). The Coulomb correction terms are estimated to be AV, = 0.5 Z/ALf3 MeV and -3.5 MeV< A WC < -2.5 MeV.

P.P.

Guss et al. / “@Ni(q

n), (n, n’)

205

8 c.m. ideq) Fig. 12. Differential cross sections, analyzing powers, and spin-flip amplitude for 20.4 MeV protons inelasticaIIy scattered from the 2’ state of “Ni. The data are from refs. 6*7).The curves are CC calcuiations performed using three different values for the deformation length of the spin-orbit potential and with Ir*,*.= 4.4 MeV. (See subsect. 3.4.)

The calculations based on step 3, which are not shown here, represent equally well the data and give similar results for &. and A V,, but indicate a value for A WC = -1.5 MeV. Therefore, there is an intrinsic ambiguity in determining d WC from the proton data available at energies near those of our work. However, in both cases the term A WC has a negative sign, which is gratifying 36). The value of - 1S MeV is comparable to that found 37*38)a t similar incident energies for the nearby

206

P.P. Guss et al. / 58.60Ni(n, ?I), (n, n’)

1 O+STATE

.fi

I _.*.

,,...”

.

.

%o=O - sso. IS,

---

q

(&

2

(3,

1

Q4.33MeV

-ID!0

8,.,,Wg)

Fig. 13. Differential cross sections and analyzing powers for 20.4 MeV protons scattered from 60Ni. The data are taken from ref. 5). The curves were calculated with V, o, --4.4 MeV. (For other comments, see caption for fig. 12 and subsect. 3.4.)

nucleus 40Ca. Finally, in order to obtain the quality of the fits illustrated, we had to reduce the value of Vs.,. to 4.4 MeV. However, if we ignore the S( f3) measurements, of the other measurea K.,. value of 6 MeV gives almost as good representations ments. The comparison between the deformation lengths 6,,. and 6,,,, for (n, n’) and interaction 35) is shown in table (p, p’) scattering and 6,.,. for the electromagnetic 2. It can be seen that &,p > &I,*a &In. .

P.P. Guss et al. / 58,60ivi(?I, n), (n, n’)

207

TABLE 2 The deformation

lengths “) for (II, II’), (p, p’) and (e, e’) scattering and 60Ni 6 PP’

Nucleus 58Ni 60Ni “) Expressed

These

results

comparison shell. For

from

58Ni

S”“,

6 e.m.

0.95 f 0.05 1.09* 0.05

0.86 * 0.06 0.96 * 0.06

0.82kO.02 0.92 f 0.01

agreement

with the conclusions

in fm.

are in good

given

in ref. 3’). For

to nuclear models, we note that isotopes of nickel have a closed proton such nuclei, in the extreme shell model picture, inelastic scattering

produces a vibration of the valence neutrons. Since the n-n interaction is much weaker than the n-p interaction, in this model the deformation parameter for neutron scattering should be smaller than that for proton scattering. At the opposite extreme, the simplest vibrational model predicts that the deformation parameter is independent of the probe producing the excitation. Madsen, Brown and Anderson “) have introduced a more realistic model in which contributions due .to mixing of high-lying collective particle-hole excitations are taken into account through a microscopic treatment containing isoscalar and isovector components of the deformation parameter. Their “core-polarization model”, when applied to proton-closed-shell nuclei, predicts the same ordering of the quadrupole deformation lengths as we observed in the present analysis for 58Ni and 60Ni. Furthermore, these authors predict that S,,, is much closer to the value for a,.,.,. than to S,,,. This too is in agreement with our findings.

4. Spherical optical model The neutron data have also been used to develop a spherical optical model (SOM) for elastic scattering for neutron energies above 4 MeV. After extensive preliminary searching it was decided to tie the SOM analyses to the CC solution of step 3. We were able to find a solution

in which

only the depth

W, needed

to be altered.

Therefore, the geometries and strengths of the SOM are identical to the CC results, except for the values of W,, and minor differences in the radius and diffuseness of the WV potential caused by the limitations of our code. The results for W,, are E -=c10 MeV) ,

W,=4.04-15.5~+1.63&

(4 MeV<

W,=9.2-15.5&-O.l(E-10)

(E 2 10 MeV) .

Sample results of this energy-dependent SOM analysis are presented for A,,(B) in fig. 14. Here the large dip at forward angles is the effect of the Mott-Schwinger interaction, mentioned above. The calculations for ~(0) (for elastic scattering),

208

P.P. Gum et al. / 58-60Ni( ?I, n),

(n, ?I’)

0.5

-0.5

0.0

-0.5

-1.0

0

60

120

I9c.m. (deg) Fig. 14. The A,,( 0) data for s8Ni compared to the SOM calculations. The Mott-Schwinger is responsible for the large dip in the calculation near 1”.

interaction

which are not shown, agree with the data as well as those for the CC model. The predicted (TV(for the region above 4 MeV) is shown in fig. 8. The SOM predictions of a,, which are not shown, agree with the CC values quite well in the energy range of our model, 4 MeV< E < 80 MeV.

5. Volume integrals The volume integrals per nucleon of the central real and imaginary potentials, (Jv/A and Jw/A, respectively) for the SOM neutron parametrization are shown as solid curves in fig. 15 for 58Ni, taken for illustrative purposes. The volume integrals calculated 3’) from the CC parameters of step 3 in table 1 give identical results for J,/A. Understandably, the CC results for Jw/A fall below the SOM values, for instance, by -20% for E a 10 MeV. In order to investigate the uncertainty in the depths of the SOM potentials, further SOM calculations were made. Starting with the parameter set of sect. 4, at each energy where a(0) data were available the V and W,, values were allowed to vary until a local minimum in x2 was reached. The solid circles shown in fig. 15 were deduced from the final values. A comparison phenomenological

between our results for J,/A and Jw/A and predictions model 39) of Rapaport, Kulkarni and Finlay (RKF)

from the and the

P.P. Gusseial./58~60Ni(n,

I

I

500-

I

n),(n,

I SOM

IOE

350-

-

Present ____. R K F

> zQ, 300-

JLM

I

209

n’)

I

PREDICTIONS

work

+ L&aje”“e

I Q

4s

150-

>

_' 100 -

50-

0 0

,,,_,.__,..__..............~~~~-~~ y ----

W

1 4

I 8

1 12

E Fig. 15. Comparisons

.-_______

e;....:....

I 16

I 20

I 24

28

(MeVl

of volume integrals evaluated for the present SOM and models The solid circles are from sensitivity tests discussed in sect. 5.

of refs. 36,39*40).

microscopic model 36) of Jeukenne, Lejeune and Mahaux (JLM) is shown for ‘*Ni in fig. 15. It can be seen that the energy dependence for J,/A of the JLM model agrees well with our result; however, this model slightly overestimates our empirical J,/A results by 5%. For Jw/A, the dotted curve for E > 14 MeV is also based on ref. 36), while the dotted curve for E < 14 MeV is based on an extension of the JLM model by Lejeune 40). However, for both of these latter calculations, recommendations from ref. 4’) were employed; by the renormalization

that is, the predictions

factor to account

have been scaled downward

for the k-mass of nucleons

in nuclear

matter.

6. Summary An extensive set of a( 13) and A,,( 0) data in the energy range from 8 to 14 MeV has been obtained for neutron elastic and inelastic scattering from 58Ni and 60Ni using the TUNL neutron time-of-flight facility. Coupled-channel calculations have been performed for the elastic and inelastic scattering data within the framework of the vibrational model. A good overall representation of the data was achieved. The quadrupole deformation lengths S,,,, which were derived in this formalism, agree with the results of the analysis of Yamanouti et al. for neutron scattering by nickel at 24 MeV. When compared to the S,, I values deduced from the (p, p’) work of van Hall et al. 6, and Eccles et al. “) on 58Ni and 60Ni, our results are in complete accord with the core-polarization model for proton-closed-shell nuclei proposed by Madsen, Brown and Anderson ‘).

210

PP. Guss et al. / 58@‘Ni(n, n), (n, n’)

The full Thomas form of the spin-orbit potential was used in these CC analyses. The CC calculations for A,(B) for neutron inelastic scattering require a spin-orbit deformation

length

CC analyses

illustrate

are valuable Furthermore,

S,,,, = 6,,. to describe

the present

how A,,( 0) data for inelastic

A,(6)

scattering

data.

In addition,

of neutrons

the

and protons

tools in determining the deformation of the spin-orbit the four A,( 0) points obtained with high statistical accuracy

potential. at 10 MeV

for ‘*Ni(n, n’) indicate a non-spherical shape for the 2+ excited state, a result consistent with Coulomb excitation measurements. The transformation from the CC framework to an SOM has been investigated. The data for neutron elastic scattering are well predicted with the SOM. A comparison of the results for the SOM and the CC models shows explicitly that increasing the SOM surface absorption by 20% simulates quite well the coupling to the 2+ state. In the SOM case also, the A,,(B) measurements reported here have led to a new determination of the spin-orbit parametrization for neutron scattering. Our values, Vs.,, = (6.5-0.035E) MeV, r,.,, = 1.017 fm, and as._. = 0.6 fm, are in the vicinity of those of van Hall ef al. obtained for proton scattering at 20.4 MeV, the worst disagreement being that their value for a_. is 0.54 fm. The volume integrals per nucleon J,/A and J,/A for our SOM study have been compared to values from a previous phenomenological global SOM analysis of RKF and predictions of JLM based on nuclear-matter calculations. Our volume integral values are seen to be qualitatively consistent with these earlier results. We gratefully acknowledge the cooperation of other members of the neutron group, in particular, C.R. Gould, L.W. Seagondollar and F.O. Purser, for providing assistance with the computer system and the time-of-flight apparatus. We are indebted to the Ohio University group for providing their cross-section data for 24 MeV neutron

scattering

and

to D.C.

Larson

and

J.A. Harvey

at ORNL

for

providing their a=(E) data prior to publication. Discussions about multiple scattering with W. Tomow, E. Woye and S.M. El-Kadi are also appreciated. We also are indebted to G.M. Honore and H.G. Pfutzner for assistance in the experiments and to J. Dave for calculating the volume integrals for the JLM model. We are grateful to F. Perey of ORNL

for providing

us with the SOM code GENOA. References

1) M. Girod and P.G. Reinhard, Nucl. Phys. A384 (1982) 179 2) H. Sherif and J.S. Blair, Phys. Lett. 26B (1968) 489; H. Sherif, Nucl. Phys. 131 (1969) 532; J. Raynal, Structure of nuclei, Trieste Lectures (IAEA, Vienna, 1971) p. 75 3) J. Raynal, in Proc. Int. Conf. on neutron physics and nuclear data for reactors and other applied purposes, Harwell, United Kingdom, 1978 (OECD/Nuclear Energy Agency, Paris, 1978) p. 372 4) S.F. Eccles, H.F. Lutz and V.A. Madsen, Phys. Rev. 141 (1966) B1067 5) C. Glashausser, R. de Swiniarski, J. Thirion and A.D. Hill, Phys. Rev. 164 (1967) 1437; P. Kossanyi-Demay, R. de Swiniarski and C. Glashausser, Nucl. Phys. A94 (1967) 513

P.P. Gum et al. / ss,60iVi(n, n), (n, n’)

211

6) J.P. van Hall, J.P.M.G. Melssen, S.D. Wassenaar, O.J. Poppema, S.S. Klein and G.J. Nijgh, Nucl. Phys. A291 (1977) 63; SD. Wassenaar, J.P. van Hal1,‘S.S. Klein, G.J. Nijgh, O.J. Poppema, J.H. Polane and J.F.J. Dautzenberg, in Proc. 5th Int. Symp. on polarization phenomena in nuclear physics, Santa Fe, NM, 1980, AIP Conf. Proc. no. 69 (AIP, New York, 1981) p. 523 7) W.A. Kolasinski, J. Eenmaa, F.H. Schmidt, H. Sherif and J.R. Tesmer, Phys. Rev. 180 (1969) 1006 8) V.A. Madsen, V.R. Brown and J.D. Anderson, Phys. Rev. Cl2 (1975) 1205 9) D.W. Glasgow et al., Nucl. Sci. Eng. 61 (1971) 521: H.H. Hogue et al., Nucl. Sci. Eng. 68 (1978) 38 10) SM. El-Kadi, C.E. Nelson, F.O. Purser, R.L. Walter, A. Beyerle, CR. Gould and L.W. Seagondollar, Nucl. Phys. A390 (1982) 509 11) P.P. Cuss, Ph.D. dissertation (Duke University, 1982), available from University Microfilms, Ann Arbor, Michigan; P.P. Cuss et al., to be published 12) R. Schenter, F. Schmittroth, F. Mann and C.Y. Fu, ENDF/B-V Nickel Data File (1980), available from National Nuclear Data Center, Brookhaven National Laboratory 13) C.E. Busch, T.B. Clegg, S.K. Datta and E.J. Ludwig, Nucl. Phys. A223 (1974) 183 14) G.G. Ohlsen, J.L. McKibben, G.P. Lawrence, P.W. Keaton and D.D. Armstrong, Phys. Lett. 27 (1971) 599 15) S.A. Wender, C.E. Floyd, T.B. Clegg and W.R. Wylie, Nucl. Instr. Meth. 174 (1980) 341 16) E. Woye and W. Tornow, University of Tiibingen, unpublished 17) P.W. Lisowski, R.L. Walter, C.E. Busch and T.B. Clegg, Nucl. Phys. A242 (1975) 298 18) W. Tornow, E. Woye, G. Mack, R.L. Walter, C.E. Floyd, P.P. Guss and R.C. Byrd, Nucl. Instr. 190 (1981) 523 19) J. Schwinger, Phys. Rev. 73 (1948) 407 20) C.E. Floyd, R.L. Walter and R.G. Seyler, to be published 21) C.M. Lederer and VS. Shirley, ed., Table of isotopes, 7th ed. (Wiley, New York, 1978) 22) A. Christy and 0. Hausser, Nucl. Data Tables 11 (1972) 281, and references therein 23) J. Raynal, Computing as a language of physics (IAEA, Vienna, 1972) 24) T. Tamura, Rev. Mod. Phys. 37 (1965) 679 25) S.F. Mughabghab and D.I. Garber, BNL report no. BNL-325 (1973), 3rd ed., vol. 1; S.F. Mughabghab, CERN report RIT/FIS-LDN(80) 1, NEANDC(E) 209 “L” (1980) 179 26) D.I. Garber and R.R. Kinsey, BNL report no. BNL-325 (1976), 3rd ed., vol. II 27) D.C. Larson, D.M. Hetrick and J.A. Harvey, Bull. Am. Phys. Sot. 25 (1980) 543 28) J.P. Delaroche, Ch. Lagrange and J. Salvy, in Nuclear theory in neutron nuclear data evaluation, vol. II (IAEA, Vienna, 1976) p. 251 29) P.H. Stelson and L. Grodzins, Nucl. Data Al (1965) 21 30) J.P. Delaroche, S.M. El-Kadi, P.P. Guss, C.E. Floyd and R.L. Walter, Nucl. Phys. A390 (1982) 541 31) Y. Yamanouti, J. Rapaport, S.M. Grimes, V. Kulkami, R.W. Finlay, D. Bainum, P. Grabmayr and G. Randers-Pehrson, in Proc. Int. Conf. on nuclear cross sections for technology, Knoxville, 1979 (NBS publication NBS SP 594, 1980) p. 146 32) E. Sheldon, Univ. of Lowell, private communication; see also ref. 30) 33) P.P. Cuss, C.E. Floyd, K. Murphy, C.R. Howell, R.S. Pedroni, G.M. Honore, H.G. Pfutzner, G. Tungate, R.C. Byrd, R.L. Walter and J.P. Delaroche, Phys. Rev. C25 (1982) 2854 34) A.M. Lane, Phys. Rev. Lett. 8 (1962) 171; Nucl. Phys. 35 (1962) 676 35) M.A. Duguay, C.K. Bockelman, T.H. Curtis and R.A. Eisenstein, Phys. Rev. 163 (1967) 1259 36) J.P. Jeukenne, A. Lejeune and C. Mahaux, Phys. Rev. Cl6 (1977) 80 37) J. Rapaport, Phys. Lett. 92B (1980) 233 38) W. Tornow, E. Woye, G. Mack, C.E. Floyd, K. Murphy. P. P. Guss, S.A. Wender, R.C. Byrd, R.L. Walter, T.B. Clegg and H. Leeb, Nucl. Phys. A385 (1982) 373 39) J. Rapaport, V. Kulkarni and R.W. Finlay, Nucl. Phys. A330 (1979) 15 40) A. Lejeune, Phys. Rev. C21 (1980) 1107 41) C. Mahaux, Phys. Rev. C28 (1983) 1848