Neutron elastic scattering on T = 0 nuclei

Nrrckm Pl~ysfca A286 (1977) 232-242; © Nartb-Solland Prnblialring Co., Anrsterdone xa to be reproduoed by photoprlat or mia~o®]m wlehoat wrhean paemiaion tirom ths pablieber

NEUTRON ELASTIC SCATIEBING ON T a 0 NUCLEIC J. RAPAPORT, J. D. CARLSON tt, D. HAINUM, T. S. CHEEMA ttt and R. w. FINLAY Pbysks Departnreet, Olrlo Unioerslty, At/rens, Oblo 45701 Roceived 7 March 1977 Abr~act: Neutron elastic scattering on Si, S and Ca has been measured at 11, 20 and 26 MeV using tho Ohio University 11 MeV Tandem Van de Graafl: A time-0f-flight technique was used and the angular distributions covered an angular range from 15° through 155°. The measured cross sections were corrected for dead time, source anisotropy, detector efficiency, finite geometry, neutron flux attenuation and multiple scattering . Individual as well as global fits to the data using an optical-model search code are preeoated. The comparison of the opticalmodel analysis to the neutron and proton elastic ecattoring data in tho cax of 4°Ca allows an empirical determination of tha Coulomb correction term which may be parametrized as 0.46 Z/A~ . It is also shownthat the elastic scattering andinelastic scattering to the fltst 2+ states in ~°Si and '=S may be fitted using the name optical-model parameters obtained for '°Ca using tha coupled-channel formalism. E

NUCLEAR REACTIONS Si, 3, Ca(n, n), E m 11, 20, 26 MoV; measured a($ ~; deduced optical-model parameters

1. Introdactton The energy and isospin dependence of the nucleon optical-model potential has been studied at Ohio University by elastic scattering of 11, 20 and 26 MeV neutrons . It was soon realized that in order to separate uniquely both dependencies, a study of one dependence at a time was essential. In this paper a study of the energy dependence for T = 0 nuclei is reported. The elements chosen were Si, S and Ca which contain 92.2 ~, 95 ~ and 97 ~ T = 0 components in their natural abondances, respectively. A comparison of the optical-potential parameters obtained for neutrons to those obtained for protons for the case of Ca has resulted in an empirical determination of the Coulomb correction to the real well depth for protons. This correction has been parameterized as 0.46 Z/A} which is 15 ~ larger than the normally assumed correction . This work is intended to provide accurate neutron elastic scattering data in an energy region where such data are not available. Most neutron studies to date have been at incident energies below 9 MeV or at 14 MeV. The measured angular distributions have been analyzed in terms of an opticalpotential having up to 12 free parameters. Constraints imposed by physical considerations on the parameters, however, reduced this number. t Supported in part by a grant from the National ScIeaoe Foundation . tt present address; Lord Corporation, Erie, Pa . 16512. ttt present address: Physics Department, Paglab University, Chandigarh, India. 232

NEUTRON ELASTIC SCATTERING

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~ ~ iPP~ms and prooedare The measurements were performed at the Ohio University 11 MeV Tandem Van de Graaff Laboratory using standard pulsed-beam time-of-ßight techniques. The 11 MeV neutrons were produced by the ZH(d, n) reaction while the 20 and 26 MéV neutrons were produced by the 3H(d, n) reaction. In both cases a 3 cm long gas target cell 1) was used. The molybdenum entrance window of thé gas cell had a thickness of approximately S mg/cms. The pulsed deuteron beam had a repetition rate of 5 MHz, an average current of 2~ pA and a burst duration of approximately 0.7 ns (FWHM) ; the total width of the observed neutron groups was generally between 1.2 ns and 2.0 ns (FWHM) . The average energy of the neutrons and the energy spread introduced by the system were 11.0 f0.09 MeV, 20.0 t0.11 MeV and 26.0 t 0.06 MeV, respectively. A small NE 102 scintillation detector was used as a neutron ßux monitor . Scattered neutrons were observed by a "Colorado" style s) NE 224 liquid scintillator detector 5.1 cm thick and 18.4 cm diameter mounted on an RCA 4522 photomultiplier tube. Detector thresholds were set at alevel corresponding to one half the enorgy ofthe sourceneutrons . Thissettingproduced agoodcompromise between efficiency and background. Pulse shape discrimination was employed to eliminate y-ray events from the TOF spectra . The spectra were measured with a 6.62 m flight path every 5° in the angular interval 15°-15'5°. The overall energy resolution of the system was less than 300 keV at 11 MeV and approximately 750 keV for 20 and 26 MeV neutrons . The detector was locally shielded by placing it inside a large water tank. A collimator of iron and paraffin was located in front of this tank. Shielding from direct source neutrons, especially in the forward angle measurements, was achieved by using a properly tapered copper shadow bar positioned as close as possible to the neutron source') . A careful adjustment of the position of this shadow bar and the additional shielding provided a much more desirable reduction of the background as compared with a multidetector array as described in ref. 4). The scattering samples were right circular cylinders approximately 2.0 cm in diameter by 2.0 cm in height, containing between ~}-~ mole of the natural target material . The Ca was contained in a thin sealed Al casing to avoid oxidation . The samples were suspended by a fine thread 12.0 cm in front of the center of the gas cell. The detector was also placed at 0° with the scattering sample removed to measure the incïdent ßux per monitor count, This value was used to obtain the absoluté value of the measured cross sections. 3. Faperimental results

The data reduction was done in a manner similar. to that described previously a ) . . The extraction ofpeak areas was straightforward because in.all cases the elastic peaks were well. separ$ted from . the inelastic peaks . Monitor and detector yields were corrected. for dead time and the respective source anisotropy correction applied to the

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J. RAPAPORT ~t d.

measured angular distribution. This correction depends on the source reaction and was always loss than 4 ~. Tho orgy of the scattered neutrons depends both on the scattering angle and the atomic weight of the sample . A correction due to the change in detector efficiency at each incident neutron energy was iatroduood. Tho program DETEFF') was used to calculate relative ofiiciencies. This correction was always less than 1 ~. Tho t~dative errors in the extracted yields from counting statistics only, were kept at abort 3 %; except in some cases at the lnlck angles where a S-1 S ~ uncertainty was not uncommon. The resultant angular distribution was con+acted for finite angular geometry, neutron flux attenuation in the scattering sample and multiple scattering, using a Monte Carlo code 6~ Compound nuclear contributions to the cross sections were evaluated with the code HELENE ~) which performs a Hausar-Feshbach calculation together with a Porter-Thomas width fluc~nation correction. This correction was significant only at 11 MoV and it was as large as 1S ~ at points corresponding to back angle minima in the angular distributions . Tho optical-model parameters used for different (n, n), (n, p) and (n, a) channels were those given in rots. a -1°), respectively. The corrections wore done by subtracting the calculated cross sections from the experimental ones. Overall uncertainties in the relative angular distributions were generally in the vicinity of S ~ except in some back angle points where it was as high as 15 ~. The overall normalization uncertainty is estimated to be less that S ~. The uncertainty in the scattering angle is estimated to be approximately 0.3°. For optical-model parameter evaluation this angular uncertainty was transformed to a contribution to the uncertainty in cross section at each angle and was added to the estimated relative uncertainty at each angle. The data are presented in subsequent figares where they are compared with opticalmodel predictions . We do not present our data in tabular form. Such tables exist and they will be sent upon request to the authors . 4. Opäcal-model amlpris Within the spirit of the Lane model' 1) the isospin dependence in the depth of the nucleon optical potential is expressed as U(r,E)= U(E~(r)faUl(E)p(r)+dU~fi (r),

('1)

whore U is a complex quantity that in general may be written as the sum of a real term and an imaginary term, U ~ Y+iW. The nuclear asymmetry parameter is represented by a ~" (N-~Z)lA and dUc is the Coulomb correction corm which is usually parameterized 1=) only as a real term, dYrJ'1(r) a 0.4 ZlA}f(r~ The + sign applies for protons and the - sign for neutrons. The above equation (1) represents

NBUTRON B~I,AäITC SCAI'TSRINß

235

only a part of the optical potential. The complete potential incorporates a spin-orbit term sad a Coulomb potential term. In the present study for ?' e 0 nuclei the real part of eq . (1) may be written for protons and neutrons: Vn(r) ~ (yoo-~~(r)+dVcfi(r) . y.(r) ° (yoo-~).f(t).

(2)

where a linear energy dependence t has ban assumed in the potential depth. The form factorsf(r) andfi(r) are of the Woods-Saxon type and are usually assumed to be equal'=) ; they are characterized by a radius R and a surface-diffuseness parameter A. A similar set of equations may be written for t>u: imaginary part W, of the optical potential. It is clear from eqs. (2) that the optical-model analysis of neutron and proton elastic scattering on the same T = 0 nucleus at the same energy permits the extraction of information on the Coulomb correction term d UGfI(r). The optical model used in the present analysis is similar to that of Heochytti and Oreenlees is). The data were fitted using the global optical-potential search code GENOA i4) . Different starting values for the parameters were used, mainly those indicated in refs. is" i3, is, is) . It is clear that in a multi-dimensional space search with over ten different free variables it is possibly to obtain a reduced ~ with parameters which may be unphysical . The search was therefore guided to avoid ambiguities 4) and some constraints given by physical arguments were imposed. Since no polarization data were fitted, the strength and geometry of the spin-orbit part of the potential was kept constant to either the values of ref. i3) or ref. 16). Either choice as expected did not apprxiably change the results. As indicated previously all the samples used were of natural abundance; however, no correction was introduced by the fact that in all cases there is a small admixdire of T ~ 0 nuclei. An easy calculation assuming standard values for the real potential depth and its isospia contribution ~) shows that the correction is less than 0.1 ~ in the potential strengths. In many individual searches it was observed that the final imaginary radius (R,) was smaller than the real radius (Re). Since for physical consideration the opposite is expected I') a constraint was imposed such that a default value R, = Ra was used if the search required R, < R~ . The more rigid constraints Rl > RA > Rs and AR > d, > A s as mentioned is ref. 16) were not imposed. No search was done on either the spin-orbit geometry (Rs and As) or its strength. It is well known that the analysis of nucleon elastic scattering alone is rather insensitive to the values of the imaginary part of the potential. At low nucleon energies it is customary to use a surface-peaked absorption while at higher energies only a t here E = E, .

J. RAPAPORT et al.

236

volume absorption term is preferred. At intermediate nucleon energies both terms are frequently used . Following the work of Thomas and Burge ta) only a surface peaked absorption term was used in the 11 MeV analysis while both a volume and surface term were used in the 20 and 26 MeV analysis . The geometrical parameters were the same for both parts of the imaginary potential. It should be indicated that in the 20 and26 MeV analysis either ofthe two terms or a combination ofboth was attempted. A slightly reduced total Xz was obtained when both terms were present in the absorptive potential . The results obtained in the individual optical-model searches are indicated in table 1 and presented in fig. 1 . A review oftable 1 shows that the values obtained for the optical-model parameters are quite scattered and that no unique relationship between them, either in terms of potential depths or volume integral per nucleon may be easily obtained. As pointed out by Hodgson t9) the concept of the optical potential may be questioned whether it should give a good overall fit to many sets of data or whether it should be adjusted to fit each set of data as accurately as possible. In the latter cases, the deviations from average value sets may sometimes be due to physical processes. In particular the imaginary potential has been found to increase with increasing deformation t 9). In the present study the nucleus ~°Ca may be considered spherical while 3~S[~(2i ) = 0.37, ref. z °)] and se Si[~(2i ) = 0.40, ref. s °)] may be considered as deformed nuclei, indicating a possible reason why the obtained optical-model parameters did not show the expected relationship . The available elastic proton scattering data on 4°Ca has been extensively aaalyzed by van Oers si) in the energy range from 10 to 180 MeV. For proton energies less TAa~ 1 Results of the optical-anodel') parameter search with the code GBNOA ~) foreach nucleus Nuclei Neutron VR energy (MeV) (MeV)

Rs (fm)

Aa (fm)

Wu Wv (MeV) (MeV)

R, (fm)

A, (fm)

,~/N

(JlA),°°, (MeV ' fms) (fm)

3i S Ca

11 11 11

54.2 33 .4 53.3

1.13 °) 1.13 °) 1 .13 °)

0.743 0.69 O.t 47

0 °) 0 °) 0 °)

8.13 7.40 6.89

1.20 1 .296 1.269

0.62 °) 0.62 °) 0.62 °)

9.3 3 .6 7.4

480 441 411

3.822 3.773 3.837

Si S Ca 3i S Ca

20 20 20

48.17 49.4 47 .7

1 .17 °) 1.17 °) 1.136

0.68 0.767 0.81

0.89 0.49 1.21

9.13 10 .64 7.%

1.22 1.22 1 .20

0.49 °) 0.49 °) 0.573

3.8 1.3 1.6

4W 470 434

3.728 4.036 4.272 3.623 3.911 4.273

26 26 26

44.8 47.0 48 .2

1.13 1.18 1.17 °)

0.649 0.712 0.797

3.14 2.30 1 .96

4.34 7.64 8.93

1.20 1.22 1.29

0.818 0.541 0.49 °)

0.8 1 .7 1 .1

Y°,,, = 6.2 MeV, Rs ~ 1 .01 fm. and A, = 0.73 fm wero kept constant, with Rc = 1 .25 1m . ") Optical model used in the present aaelysis is similar to that of ref. ") . See ref. ' 4). ") Kept constant during search .

383 436 450

NEUTRON ELASTIC SCATTERINQ

23 7

b

15

55

95

135

15

55

95

135

ep , (degreee)

15

55

95

135

Fig. 1 . Results of optical-parameter search for individual nuclei . F7kd circles are data points whose errors are of the same size as the circles unless otherwise indicated. The solid lines are optical-model fits .

than 75 MeV he obtained the indicated average geometry for the optical-model parameters : Rs = 1.014, Rß = 1.152, Rt = 1.309, AR = 0.692,

A, = 0.549,

As = 0.526,

where all the values are in fm. A charge radius Rc = 1.32 A} was assumed zt) and where no polarization data were available, the values for the spin-orbit parameters Vs, Rs and As were kept fixed at the average values of a previously analyzed subgroup of the data . The above geometry was kept constant in the present study and the neutron elastic scattering data analyzed accordingly. The .final results are indicated in table 2. A graphical comparison between the "n+Ca" parameters and the "p+Ca." parameters obtained in ref. zt) is presented in figs. 2 and 3. Fig. 2 shows the real depth of the potential plotted versus incident nucleon energy . The circles represent the values from ref. 2t) obtained with the average geometry, while the crosses represent the present values obtained with the same geometry. The values of V for proton energies less than 22 MeV are quite scattered, probably due to the high (p, n) Q-value

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J. RAPAPORT et d.

(Q ~ -15.1 MeV) and resonance effects that make the analysis of the data in terms of an optical model unreliable . A linear relationship may be obtained between V and E for energies betwcen 26 and 76 MeV. (The point at Ep = 49.0 MeV was not included.) As shown in fig. 2, V docreases with E and a linear coefficient a = 0.34f0.02 may be obtained with a least square fit. A straight line with the same slope has also been drawn through the points obtained in the prenant neutron elastic scattering analysis. As indicated in eq. (2), the vertical separation between the two straight lines represents the strength of the Coulomb correction term, assuming it has the same radial dependence as the real part of the potential. A value d V~ = 2.7f 0.4 MeV is obtained for this term which parameterized in the usual way gives a value (0.46 t0.07) ~Z/A}. This empirical value, obtained for the first time, is 15 ~ larger than the commonly used 0:4 ~/A} as evaluated in ref. 12). The contribution of this term to the volume integral per nucleon is 22t 3 MeV " fm'. Jeukenne et al. sz) have recently performed microscopic calculations for the Coulomb correction term in the case of ~°aPb at 25 MeV. A value 25 ~ larger than the phenomenological value was obtained which agrees well with the present empirical value which was also 15 ~ larger than the phenomenological one. Tea~.s 2 Results of the optical-model parameter search for Ca using the same geometrical parameters as in ai)

Nucleonenergy Nnckon (MeV) 11 2t1 26

n n n

Y (MeV)

(MeV)

1Vv

Wu (MoV)

,t'~/1Y

J/A (MeV " fms)

52.23 49 .24 48.1ti

0 0 0.3

6.75 7.9 8.0

9.4 3.4 2.2

43ti 393 402

The analysis of the real part of the potential in terms of the volume integral per nucleon yields similar results because the potential depth values were all obtained with a fixod geometry . The values are:

The imaginary potential depths are compared in fig. 3. Aa indicated previously only a surface imaginary term (wn) is required for low energy nucleon elastic scattering while only a volume imaginary term (wv) is needed for high~nergy scattering. This trend is graphically displayed in fig. 3. There is no apriori reason why the Coulomb comction term should not also have an imaginary part. This term would be seeded if one assumed that the absorption also

NEUTRON ELASTIC 3CATTERINQ

4Co ;

`

Ca

f

P a ~

239

pmnt norfc

so

V

1

~

1

1

--1-

..

1

1

1

E (MeV1 Fit. 2. Variation with ever=y of Y, the depth of the real part of the optical potential. Open circles reprosent values for proton elastic scaüerlnt'i). Crosses represent the valves obtained in the present work for neutron elastic ecatterint.

depended on the local nucleon kinetic energy. This has bcen discussed is refs. ts, Zs), and recently Jeukenne et al. s2) computed the imaginary part of the Coulomb correction term in the case of 25 MeV protons on s °aPb. It was found in that particular case that this term is larger than the imaginary part of the symmetry component of the optical model for protons and that these two quantities have different signs. The W-valves reported by van Oera zt) using the average geometry and the present s results obtained with the same geometry are shown in fig. 3. In ref. t ) a Gaussian form factor was used for the surface absorptive term while the present results were obtained with a Woods-Saxon derivative form factor . There is no unique relationship between these two potential strengths but they are expected not to be too different. A reanalysis of the p+Ca data =4) at some of the energies corroborates this expectation . It is apparent from fig. 3 that the Wn values for protons and neutrons are about the same at the same energy. This is as indication that the Coulomb cort~ection term is mainly a term in the real part of the potential, or if it has an imaginary contribution,

240

J. RAPAPORT d al.

o.

3

E (MW) Fig. 3. Variation with energy of the imaginary term, W, of the optical-model potential. Open circles represent the surface tenor while closed circles represent the volume term in the proton elastic scattering al). Crosses represent the values obtained in the present work for matron elastic scattering.

this is rather small. It should be interesting to perform a microscopic calculation in the case of a°Ca and to compare it with the present results. From fig. 3 the following linear relationship may be obtained for the values of W: (a) E ~ 24 MeY:

WD ~ 2+0.27 E (MeV) Wv =0;

(b) E >~ 24 MeV:

WD = 8.5-0.27 E (MeV);

(c) E ~ 28 MeV :

Wv ~ -7.5+0.27 E (MeV).

A similar trend has been obtained in the analysis of neutron elastic scattering from the Mo isotopes and from s°epb [ref. s')] . It should be noted that for E < 24 MeV the obtained energy dependence of Wn is such that WD increases with energy. The global optical-model analysis of 8eochetti and Greenlces is) indicates that Wn decreases withincreasing energy for E > 10 MeV. The present results as well as other neutron elastic scattering data z') are in disagreement with these results of ref. is). The neutron elastic scattering on 32S and ZsSi was fitted with the same optical potential obtained for Ca using a coupled~hannel analysis t. No adjustable parameter except the value of the characteristic deformation parameter ßs was used. This type of analysis has been applied by Perey in the elastic scattering of 17 MeV protons from heavy spherical and deformed nuclei ")and by ßlendenning et al. ze) in the elastic t The calculation were done with the code CHUCK'6 ).

NEUTRON ELASTIC SCATTERING

241

scattering of 50 MeV a-particles by the even isotopes of Sm. More recently it has 29) and in the been used in the analysis of neutron scattering from even-A Se isotopes ao)] . analysis of neutron scattering near A = 208 [ref. The coupled-channel calculation of both the elastic and inelastic cross sections to the first 2 +~ state in Z sSi and 32S at 20 MeV neutron energy is shown in fig. 4. The

in l~l E _U

b 1~ 2

6~M (degrees) Fig. 4. DWBA calculation for neutron elastic and inelastic scattering on Ca at 20 MeV, using the parameters of table 2. CCBA calculations for neutron elastic and inelastic scattering on Si and S using the same Ca optical-potential parameters .

agreement may be considered rather satisfactory considering that there was no further adjustment of the optical-model parameters . The inelastic scattering to the 3 - state in a°Ca is also shown compared with a DWBA calculation t. 5. Conclusions In this study accurate measurements of 1 l, 20 and 26 MeV neutron elastic scattering differential cross sections for Si, S and Ca are presented. Prior to this work no reliable neutron data were available for these nuclei at these energies. The optical-model analysis and its comparison with proton elastic scattering has resulted in an empirical determination of the Coulomb correction term which may be r The calculations were done with the code DWUCK' 1 ).

242

J. RAPAPORT el al.

parannetri~ed as 0.46 Z/d}. Tho ooei8ciont 0.46, obtained for the first time, is 1S higher than the one usually usod for this term. It agrees well with a rooent calculation for this term 2s) where a value 25 ~ larger than the empirical one was computed. Itdoor not agree, however, with a priorcalculation') whero a value d Vc ~ 0.8 ~Z/A} was estimated. Microscopic calculations of the optical-model potential in symmetric nuclear matter using a Reid hard core nucleon-nucleon interaction roportod by Jeukenne etal. 33) glVe a Value Y = (56-0.3~ MeVfor the real part (10 MeV < E < 180 MoV) and W~ (0.19E+ 1.9) MeV for the imaginary term (10 MeV < E < SO MeV) . The above values agrce rather well with the present results. Coupled-channel calculations using the parameters obtained in the Ca analysis resulted in a good agreement to both the clastic and inelastic neutron scattering measurements on Si and S. Befa~aces 1) J . D . Carhwn, Nucl. Insu. 113 (1973) 341 2) D . A. Lind, R. F. 1lentiey, J. D . Carlson, S. D . Schary and C. D . Za&atas, Nnci. laftr. 130 (1973) 93 3) J . C. Hopkins, J. T. Martin and J. D . 3es~ve. Nacl. Insu. S6 (1%7) 175 4) J. C. Ferrer, J. D . Carhwn and J. Rapeport, Nncl . Phys. A27S (1977) 323 S) S . T. Thornton aad J. R . Smith, Nncl . Imtr. 96 (1971) 351 6) W. & Kiamy, Nuci . Insfrr. 83 (1970) 15 ; and private commnaication 7) S . K. Penny, ORNL-TM-2590, Oat RidOe National Laboratory, 1969, unpublished . 8) C. D . Ta&atos, T. A . Oliphant, J. S. Levin and L . CYanb~, Phys. Rev. Lett . 14 (1965) 913 9) C. M . Peray and F. ß. Perey, Nncl. Data Tables 13 (1974) 293 10) L. McFadden and ß. R. Safthier, Nuci . Phy: . 84 (1966) 177 11) A. M. Lace, Nuc1. PkYs" 35 (1962) 676 12) F. ß. Perey, Phys. Rev. 131 (1963) 743 13) F. D. Heochetti, Jr. sad ß . W. ßreenlees, Phys. Rev . 182 (1969) 1190 l4) F. ß. Peray, private wmmanication 15) F. ß . Piney. Phys. Lett . 26B (1968) 123 16) W. T. H. van Oers at al., Phys. Rev. C10 (1974) 307 17) P. E . Hodeson, Nuclear reacxions sad nuclear structura ((aaraadoa Pros, Oxford,1971) eh . 7, 8 18) ß. L. Thomas and & J . Buree, Nucl . Phys. AIZB (1969) 343 19) P. 13. HodBson, Natura Z49 (1974) 412 20) P. H . 3tehwn and L. ßrodzins, Null. Data 1 (1%S) 29 21) W. T. H. van Oars, Phys. Rev . C3 (1971) 1550 22) J.-P. Jeutemne, A. Leje~me sad C. Mahamy Phys. Left . âZ8 (1976) 256 : Pros. Iat. Cant on the interactions of neutrons with nuclei, Lowell, Mass ., 1976, ed., E. Stieldon, eonl: 760715-P-1 23) P. B. Hochecu, Null. Phys. A103 (1967) 127 24) N. S. Davison, private communication 25) T. Cheema at al., to be published Z6) P. D. Kunz, The code CHUCK, anpnblished 27) F. ß. Perey, is N~lear spectroscopy with direct rastdun:, ed. Throw, ANL-6848 (1968) 114, unpublished 28) N . K. ßlend~nin8, D. L. Hendrie and O. N. Jarvis, Pbys . Left . 26B (1968) 131 29) J. Lachkar, M. T. Mc>'iL~~ ß . Haouat, Y . Patin, J. SiBaud and P. Coca, Phys. Rav. Cl4 (1976) 933 30) P. ßuenther, D . Havel and A. Smith, ANL/NDM-22, Ar,onne National Lab. (1976), imP 31) P. D. Kunz, The code DWUCK, unpublished 32) C. B . Dover ead N. V. ßiai, Null . Phys. A190 (1972) 373 33) J.-P. Jeuloenne, A . Ijoune and C. Mahauz, Phys . Rev . C10 (1974) 1391