Optical model for 30 MeV proton scattering

[ I

2.E

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Nuclear Physics A92 (1967) 273--305; (~) North-Holland Publishiny Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

O P T I C A L M O D E L F O R 30 MeV P R O T O N SCATTERING G. R. S A T C H L E R

Oak Ridge National Laboratory, Oak Ridge, Tennessee t Received 4 July 1966 Abstract: T h e differential cross sections, polarizations a n d a b s o r p t i o n cross sections for the elastic scattering o f 30 M e V p r o t o n s f r o m the nuclei Si, Ca, Fe, Ni, Co, Cu, Sn a n d Pb have been analysed in t e r m s o f the optical model. W e find a smaller radius a n d a larger surface diffuseness t h a n h a v e previously been inferred for the real potential. A v o l u m e absorption strength o f a b o u t 3 M e V is required, while the surface a b s o r p t i o n t e r m increases in strength as A increases. T h e diffuseness o f the i m a g i n a r y potential is also larger for the heavier nuclei. Fits to the polarization reveal evidence for a spin-orbit coupling with radius a n d diffuseness different f r o m those o f t h e real central well, a n d the t r e n d is for the spin-orbit diffuseness to be smaller rather t h a n the radius. T h e s y m m e t r y or ( N - - Z ) / A dependence o f the potential m a y be ascribed either to the real well depth or to its radius; an average potential o f b o t h types is f o u n d which gives a good overall a c c o u n t o f the data.

1. Introduction

Extensive studies have been made of the elastic scattering of protons of 9 to 22 MeV (ref. 1)) and of 40 M e r (ref. 2)) using the optical model, with particular emphasis on finding smooth trends in the parameters as a function of the bombarding energy and the mass of the target nucleus. There have since been published data 3- 5) for a series of nuclei taken at 30.3 MeV and more recently new data 6) have become available for an energy of 40 MeV. These new data include cross section and polarization measurements. The 40 MeV results have been analysed by Bassel 6, 7) and the present paper extends the analysis to the 30.3 MeV measurements. Particular attention is paid to the nuclei Fe, 58Ni, 6°Ni, Co Cu, Sn and Pb because they yield a coherent set of data from which one may hope to extract information about any systematic trends in the potential parameters. The data 8.9) for Ca and Si are discussed separately. For completeness, some other sets of data which are available for energies close to 30 MeV are considered as well. These include measurements on 6°Ni at 30.8 MeV (ref. to)), Cu, Ag, Ta, Au and Pb at 31.5 MeV (ref. 11)) and Pb at 31 MeV (ref. 10)). The scattering from elements lighter than Si will be discussed in a later communication. An earlier analysis of the 30.3 MeV data has been reported 12), but considerable restrictions were imposed upon the optical potentials used and only qualitative fits to the data were obtained. Another analysis, in many ways similar to that given here, has been reported by Greenlees and Pyle 13). t R e s e a r c h s p o n s o r e d by the U.S. A t o m i c Energy C o m m i s s i o n u n d e r contract with U n i o n Carbide Corporation. 273

274

G.R. SATCHLER

2. The optical model The optical-model potential used in this analysis is local and has the usual form U(r) = - V(e x + 1) - ~ - i ( W - 4 W D d/dx')(e ~' + 1) --1

+(h/m~c)2Vscr'Lr-l(d/dr)(eX~+l)

-I,

(1)

where

x = (r-roA+)/a, x' = ( r - r ' o A ~ ) / a ', x~ = (r-r~A~)/a~, to which is added the C o u l o m b potential from a uniformly charged sphere of radius 1.2 A ~ fm. The spin-orbit coupling term is taken to be real because no convincing evidence has been found that an imaginary part is needed. Then the potential (1) is specified by ten parameters. Previously the restrictions r s = ro, as = a have usually been applied, leaving eight parameters, and we shall give some consideration to this restricted potential also. The parameters which best fit the elastic scattering data were determined by use of the automatic search computer routine H u n t e r which minimizes the quantity

z~ = Y~ {[~,h(O,)-~oxp(Oi)l/A~ex~(Oi)} ~, i=1

where C%xp(0i) is the measured, and ath(0 i) the calculated, differential cross section at angle 0 I, while AO'exp is the " e r r o r " associated with CrCxp. When polarization data are being considered also, a similar quantity 2 2 is constructed. The search routine is able to minimise either X~ " 2 or Zp . 2 or their sum. The errors Aa~xp and APe~p may be chosen to weight certain features o f the data and hence prejudice the o p t i m u m fits to be found. Except where otherwise noted, in the present work we chose the quoted experimental errors for APex p and uniform errors for AO'exp; that is, AO'~xp/O'ex p was chosen constant for all 01, usually as 3 O//o. W h e n cross-section and polarization fits are being made, varying this constant varies the relative weight given to the two types o f data. The experimental Ac%xp/a~ p are not constant, but usually are larger at large angles. In m o s t cases this had little effect on the results, and uniform errors have the slight advantage o f making the ) i 2 values for different nuclei more directly comparable. The polarization measurements are made with a relatively large angular acceptance ( + 3 ° for the data considered here) while the polarization values often vary rapidly with angle. The X2 values as defined here m a y then be misleading since they do not take account o f this angular smearing. More sophisticated treatments ~3, ~4) include the angular resolution effects, but it was found that for most o f the cases considered quite g o o d fits to the polarization were obtained even when only the cross-section fits were being optimised. Including the polarization data in the search generally produced on small changes (although we shall see that there are exceptions), so we feel 2 A similar effect that no gross error is being committed by using the simple form for Zv.

OPT[CAL MODEL

275

arises from the energy spread due to the thickness of the targets used in the polarization measurements (about 1 MeV). Since the diffraction structure of the polarization angular distribution varies smoothly with energy (except possibly with extreme back angles), this energy averaging has results similar to the finite angular resolution. However, another more important consequence is that the mean energy of the polarization measurements (quoted as 29 MeV) differs from that for the cross-section measurements (quoted as 30.3 MeV) and the change in polarization between these two energies is not insignificant. This means that simultaneous fits to cross-section and polarization data require computations at two different energies, and also some assumptions as to the energy variation of the optical-model parameters. The latter point is unlikely to be serious in the present case, but the first is because the computer search codes currently available will only handle one energy at a time. Thus we need, for example, to obtain a fit to the cross section at one energy, adjust the parameters to fit the polarization at the other energy and then recycle until some convergence is obtained. We discuss these matters in more detail below.

3. Nuclei Fe, Ni, Co, Cu, Sn and Pb The main emphasis was placed on the analysis of the data 3-s) for the nuclei 56Fe, 58Ni, 6°Ni, 59C0, Cu, 12°Sn and 2°spb. These were all taken at an energy of 30.3 MeV, with the same equipment, with a high accuracy and covering a wide range of angles (15 ~ to 165°). This is a very favourable situation is) for any analysis which hopes to find a single "universal" potential, or at least to establish any trends in the parameter values with changes in N , Z and A. Data for 4°Ca were taken in the same series of measurements, but present some special problems which will be discussed below. Nuclei lighter than this are more liable to show individual characteristics, and will be considered separately also (Si in sect. 5 below and others in a later communication). Table 1 lists the number of angles at which cross sections N, and polarizations Np were measured and also gives the values of the reaction or absorption cross sections a a which were measured 4) at an energy of 28.5 + 1.5 MeV. Calculations indicate that a a d e c r e a s e s with increasing energy at the rate of about x1 o~ or less per MeV for A ~ 60, whereas it increases at about ½ % per MeV for Sn and 17I o% per MeV for Pb. A "geometry" which has been used with considerable success at lower energies 1) is defined by r o = % = r s = 1.25 fro, a = as = 0.65 fm and a' = 0.47 fm. However, it was never claimed that this was a unique prescription, and ambiguities in the parameter determinations would allow small changes without affecting the quality of the fits to experiment. It has been shown 2) that this particular choice is inadequate at 40 MeV; at that energy the optimum real geometry appeared to be more like r 0 ;5 1.18 fm, a > 0.7 fm. Similar results emerged from the earlier analysis 12) of the 30 MeV scattering and have been confirmed by studies 7) of the new 40 MeV data 6). During the analysis of the earlier 40 MeV cross-section data (which did not extend to as large angles as the 30 MeV results considered here) it was found that equally

0

No

~L) Measurcd at mean energy of 30.3 MeV.

6n(mb)

21

30

N a

2t

2sSi ~)

28Si ~)

Target

TABLE 1

915,38

21

75

4°Ca

1140--43

0

75

~6Fe

2l

21 1169!39

22

72

59Co

0

29

63.5Cu

1638!68

24

75

l~oSn

') Measured at mean energy of 29 MeV.

1053 151

75

75

1038_t-32

OONi

~SNi

/865 ± 9 8

21

72

~ospb

Targets and numbers of data considered in present analyses, also the measured reaction cross sections ~A at a mean energy of 28.5 MeV

1.108 0.977 1.172 1.125 1.086 1.072 1.101 1.101 1.099 1.098 1.116 1.168 1.162 1.158 1.157

r (fm)

0.705 0.755 0.703 0.764 0.782 0.801 0.776 0.775 0,764 0.762 0.760 0.725 0.703 0.713 0.696

a (fin) 5.49 1.07 1.78 0.08 3.91 4.11 3.47 3.15 3,34 3.23 1.993 2.59 3.00 2.01 2.62

W (MeV) WD

2.03 6.50 4.83 6.55 4.95 4.20 5.00 5.37 5.21 5.35 6.03 7.49 7.23 8.40 7.74

(MeV) 1.407 1.274 1.288 1.289 1.339 1.382 1.328 1.323 1.311 1.306 1.277 1.314 1.328 1.314 1.340

r o' (fro) 0.521 0.599 0.653 0.593 0.556 0.497 0.589 0.589 0.602 0.606 0.676 0.647 0.643 0.761 0.756

a" (fm) 5.92 7.92 5.59 5.81 6.22 6.36 6.19 6.26 6.35 6.41 5.94 6.12 6.19 5.13 4.82

Vs (MeV) 720 758 904 881 1107 1082 1147 1149 1141 1141 1210 1636 1636 2040 2070 (1622) (1623) (1997) (2028)

(1086) (1151) (1152) (1144) (1145)

(729) (766) (912) (885)

era (mb) 19 ( 4 8 ) 87 (131) 28 120 411 404 208 248 213 216 111 442 573 169 252

Za 2

450 375 130 140

307 630 597 414 411

660 378 640 416

Zp2

a,b) b,e) a) e,a) e) a, ¢) a) ¢) a) e) e) a) c) a) e)

Notes

The 22 given were evaluated using 29 MeV and experimental errors for the polarizations, 30.3 MeV and 3 ~ errors for the cross sections, except that 10 ~ errors were assumed for the Ca and Si cross sections. The reaction cross sections crA are given for 30.3 MeV, and (in parentheses) for 29 MeV. a) Minimizes 2o 2 alone. b) Value o f z a z in parentheses is for 29 MeV cross-section data. e) Minimizes 2a2÷Zv 2, with Zp2 evaluated using the theoretical predictions at 30.3 MeV. a) 50 ~0 errors were assigned to the 20 cross sections at angles greater than 122 °. e) Polarization data not available.

2°8Pb

c3"~Cu 12°Sn

~Co

~Fe 5SNi ~°Ni

54.8

50.4 64.9 47.2 51.1 56.2 56.6 54.8 55.0 55.5 55.7 53.8 51.4 52.0 55.4

28Si

~°Ca

V (MeV)

Target

TABLE 2

O p t i m u m optical potential parameters with the constraints rs -- r o, as = a.

278

G.R.

SATCHLER

good fits could be obtained with only surface (W = 0) or only volume (WD = 0) absorptive potentials, and it did not seem possible to determine unambiguously a mixture of surface and volume absorption. The volume term required a radius of r; ~ 1.4 fm, and the surface term needed r; ~ 1.04 fm. The diffuseness parameters varied considerably, but averaged to about a' ~ 0.7 fm. However, on physical grounds one expects both volume and surface terms to be present, and it was hoped the more complete 30 MeV data would be sufficient to determine the relative amounts. For this reason, both W and WD were allowed to differ from zero in the studies reported here. There have been a number of indications 16) that optimum fits to polarization data require a spin-orbit coupling with parameters different from those for the real, central, 08

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Fig. 1. Polarizations for Pb predicted by the second potential given in table 3 at energies of 29 and

30.3 MeV compared to the data measured at a mean energy of 29 MeV. well; in particular the spin-orbit strength appears to be concentrated somewhat inside the surface of the central potential. These previous indications have mostly come from lower-energy experiments; the data at 30 and 40 MeV should provide a more favourable situation for studying this effect. Indeed preliminary analyses 6,7, ~3) of these data do confirm the earlier indications, and have been interpreted as showing r s < r 0. Here we shall first discuss the results obtained by keeping rs = to, as = a, and then turn to the improvements, if any, obtained by allowing the spin-orbit parameters to be different. 3.1. OPTIMUM FITS WITH CONSTRAINED SPIN-ORBIT COUPLING Searches for optimum fits were made both by minimizing Z~ " 2 alone and by mini2 + 2Zp. The results are given in table 2. An energy of 30.3 MeV was used for mizing Z~ the theoretical calculations, so that, in view of the discussion above, only the former search mode is strictly consistent. Fig. 1 shows how the predicted polarization for Pb varies as the energy changes; the experimental measurements are included for illustration. The change is large and suggests that values of Zo2 obtained by using the energy of 30.3 MeV may be in serious error. Indeed, the Zp2 values obtained for Sn

279

OPTICAL MODEL

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OPTICAL MODEL

281

and Pb from searches using 30.3 MeV were found to be a factor of 2 larger than the "true" values obtained with the same potentials but using the energy 29 MeV. Fortunately, the overall features of the polarization are determined by the diffraction 2 in this way structure of the cross section, so the incorrect bias introduced by using Xp may not be large. This conclusion is supported by the results in table 2, which are very similar whether or not the polarization data are included in the search. However, this also implies that minimizing Z2 alone is sufficient, at least when the constrained spin-orbit coupling is being used. Further the effect shown in fig. 1 for Pb is an extreme example; the variation is much less for Ni and Co and is almost negligible for Si and Ca. In view of this, a more sophisticated treatment of the energy dependence of the polarization was not attempted until the independent spin-orbit coupling, discussed below, was introduced. However, although the searches described here used the energy of 30.3 MeV, the g z values quoted in table 2 were evaluated at 29 MeV. The cross sections and polarizations predicted by the potentials of table 2 are shown as solid curves and compared to the measured values in figs. 2 and 3. Only one curve for each nucleus is shown because minimizing XZ+Zp2 and 7~2 alone gave indistinguishable results. The fits to the cross-section data are very good over the entire angular range. The predicted polarizations are generally satisfactory, although they are consistently more positive than the measurements in the forward hemisphere. The optimum parameters given in table 2 show remarkably little fluctuation from element to element, and several trends are already clear. We have r o ~ 1.10 fm for the lighter elements increasing to r o ~ 1.16 for Sn and Pb. Subsidiary studies showed that this increase significantly improves the fit to the cross sections at small angles for these two elements. The values of a are between 0.7 and 0.8 fm. The absorptive potential has a volume component of strength W between 2 and 4 MeV and a surface component of strength Wo which is larger for the heavier nuclei. The imaginary radius varies little from its average value of l"o = 1.325 fm, but the imaginary diffuseness a' also increases for the heavier nuclei. The latter trend had been noted before at lower energies 2, 27). The spin-orbit strength Vs is very constant at approximately 6 MeV except for Pb which requires Vs ~ 5 MeV. This property of Pb will be seen to persist throughout the present analysis. It was thought this might be due to a small error in the absolute values of the cross sections for Pb; however, although reducing the quoted values by 5 ~ did result in an appreciable reduction in g z values, particularly for angles in the forward hemisphere, there was no significant change in the optimum parameters. The measured reaction cross sections aA (table 1) are quite well reproduced except that the prediction for 60Ni is about 50 mb (or 5 ~ ) above the upper limit placed on the measurement. The predictions for SSNi and Pb are also a little high, but both are only 2 ~ outside the quoted experimental errors. Since both 63Cu and 65Cu have a spin o f ~ , and 59Co has a spin of~, the possible effects of some quadrupole contribution to the elastic scattering ~8) were explored.

48.1 50.9 50.2 51.1 53.5 54.5 53.9 52.5 53.9 55.9 53.5 55.8 52.8 56.0 52.8 51.4 51.5 51.0 56.1 56.1 55.8

1.160 1.106 1.121 1.098 1.083 1.076 1.075 1.112 1.099 1.083 1.109 1.099 1.122 1.097 1.110 1.172 1.168 1.177 1.150 1.144 1.152

ro

(fro)

V

(MeV)

a

0.640 0.687 0.674 0.711 0.807 0.831 0.814 0.732 0.764 0.783 0.753 0.772 0.724 0.763 0.692 0.713 0.720 0.718 0.742 0.724 0.705

(fro)

W

3.00 3.92 4.28 5.44 2.02 3.11 2.69 3.71 3.86 3.82 3.36 2.86 3.39 3.25 2.62 2.07 2.57 2.21 1.03 2.65 2.25

(MeV)

WD

5.02 3.63 3.42 1.79 4.73 3.98 4.15 4.90 4.21 4.55 5.01 5.58 4.73 5.38 6.23 7.47 7.53 7.30 9.63 7.82 8.03

(McV)

r o'

1.201 1.339 1.326 1.442 1.369 1.404 1.414 1.341 1.364 1.357 1.330 1.286 1.311 1.298 1.280 1.301 1.314 1.312 1.292 1.343 1.320

(fro)

a'

0.616 0.543 0.546 0.500 0.548 0.498 0.509 0.555 0.529 0.524 0.586 0.633 0.627 0.611 0.602 0.684 0.650 0.675 0.763 0.751 0.761

(fro)

Vs

6.29 6.01 6.56 5.80 3.83 3.52 3.83 7.42 7.08 6.58 6.77 5.74 7.60 6.13 7.46 6.50 6.17 6.35 5.66 5.07 5.15

(MeV)

rs

0.892 0.930 0.899 0.953 1.011 1.015 0.989 1.024 0.952 1.005 1.094 1.022 0.956 1.077 1.269 1.067 1.159 1.056 1.079 1.182 1.183

(fro)

a~

0.690 0.546 0.665 0.495 0.362 0.273 0.391 1.048 0.950 0.844 0.921 0.688 1.028 0.690 0.971 0.828 0.744 0.774 1.085 0.580 0.592

(fro)

aA

715 716 711 716 883 883 884 1091 1078 1082 1138 1152 1136 1140 1140 1644 1637 1652 2026 2078 2055 (1632) (1624) (1640) (1985) (2035) (2014)

(1081) (1086) (1141) (1156) (1139) (1143)

(724) (724) (718) (724) (890) (891) (892)

(rob) 17 24 20 44 115 105 50 234 349 389 164 311 189 300 28 326 413 368 163 230 188

(46) (46) (53) (43)

)~a 2

478 452 454 545 87 68

348 264 1307 309 920 256

153 45 96 20 169 174 156

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e,h) a,~) a) e) a) e) a) e) a, ~) a) e) ~) a) e) ~)

e,g)

a,b) b,c) b,d) b,e) e,g)

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Minimizes Za" alone. Value of Za 2 in parentheses is for 29 McV cross section data. Minimizes Za2+Zv 2 using the theoretical polarizations for 30.3 MeV. As e), but using 3 o / e r r o r s for the cross sections, Minimizes Z,r~+Zv 2, using the 29 MeV cross-section data.

~1 Polarization data n o t available. g) 50 °/o errors were assigned to the 20 cross sections at angles greater t h a n 122 °. h) A s g), but 100 ~, errors used. i) As e), but polarization angles adjusted for 30.3 MeV.

The Z 2 given were evaluatcd using 29 MeV and experimental errors for the polarizations, 30.3 MeV and 3 ~,, errors for the cross sections, except that 10 % errors were a s s u m e d for the Ca a n d Si cross sections. The reaction cross sections c~a are given for 30.3 MeV, and (in parentheses) for 29 MeV.

2°8Pb

~3.~Cu a2°Sn

5~Co

6°Ni

66Fe 5SNi

4°Ca

'-'ssi

Target

TABLE 3 O p t i m u m optical potential parameters with independent spin-orbit coupling

OPTICAL MODEL

283

To lowest order this contribution has the same angular distribution as the inelastic scattering to the lowest 2 + state in neighbouring even nuclei, so ~ of the cross section for this inelastic transition in 6°Ni was subtracted from the Cu elastic cross section. This fraction corresponds, for example, to an admixture of about 15 ~ of one-phonon core-excitation into the ground state of Cu and results in an almost uniform reduction of about 5 ~o in cross section for scattering angles greater than 120 °. The resulting cross sections were again fitted with an optical potential, but there was no qualitative change in the optimum parameters obtained. We conclude that these effects are not important for the present analysis. 3.2. OPTIMUM FITS WITH INDEPENDENT SPIN-ORBIT COUPLING Searches were now made for optimum parameter sets without the constraints r~ = r 0, as = a. At lower energies, or for lighter nuclei, small deviations of the spinorbit parameters could be treated as perturbation which only affected the polarization, leaving the cross section unchanged to a good approximation 19). Unfortunately this cannot be done here; changes in rs, and especially as, also produce significant changes in the cross sections at angles in the backward hemisphere. Small re-adjustments in the other parameters are required to restore the fits. Thus we require simultaneous fits to cross section and polarization and immediately encounter the difficulty discussed above that the two types of data were taken at different energies. First, the difficulty was ignored and fits were obtained both by minimizing (Z~2 + 2Zp) and Z2 alone, using the energy 30.3 MeV. Starting values were chosen close to the best fits given in table 2, and the results are given in table 3. Of course, only the parameters obtained by minimizing Z~ ~2 are strictly valid, but the changes in polarization with energy for Ni and Co are sufficiently small that the results of minimizing (Z~2 + 2Zp) are also significant. (The actual values of Z~ quoted in table 3 were evaluated at 29 MeV). The changes in polarization for Sn and Pb are large as shown in fig. 1 for Pb; Sn is similar. However this figure also suggests that the difficulty can be circumvented for these two nuclei by scaling the angles at which the polarization measurements were made by the ratio of the wavelengths for the two energies, namely (29/30.3) ~. This was done and optimum fits obtained with the third parameter set listed for each target in table 3. This procedure was not followed for the other targets, partly because for them an energy change gave rise to some change in shape of the polarization distribution as well as a simple scaling in angle. When only the cross sections are being fitted, table 3 shows a trend for rs ~ as ~ 1 fm. However, the corresponding polarizations, although in qualitative agreement with the data, actually yield larger Zzp than those obtained with the constrained spinorbit coupling. This is illustrated in fig. 3, where these polarization predictions are shown as dashed curves and are compared to the measurements. In particular, the fits at forward angles are generally worse. When the ()G2_t_2 Zp) are minimized (both being evaluated at 30.3 MeV), the polarization fits are somewhat improved over those given by the constrained spin-orbit coupling. The values of as remain smaller than the

284

G. R. S A T C H L E R

values of G, and fluctuate about tile values of a. The other parameters given in table 3 do not show any marked change from those given in table 2 with the constrained spin-orbit coupling. Further, the predicted reaction cross sections ~rA are closely the same for both types of potentials. 4

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Using polarization angles for Sn and Pb scaled according to the classical diffraction law yields somewhat similar results. Surprisingly the fit to the Sn data is no better than that obtained using the actual angles and the incorrect energy, and the improvement for Pb is not large. The theoretical predictions are compared to the measurements in fig. 4. ,2 The origin of these various results seems to be that while Zz~ ~ Zp when both are minimized, Zo ~ 2 is more sensitive than Zp .2 to variations in the parameters and hence the

OPTICAL MODEL

285

cross-section fit tends to dominate the search. The main exception to this appears to be the spin-orbit diffuseness as; the Z2 m i n i m u m favours a large value as ~ 1 fm, whereas the polarization favours a smaller value. Auxiliary studies showed that it is at the forward angles, as expected 19), where variations in a s have most effect on the

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.

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~,,~ , . ...2/ 1t

it

I I

, ',

M/7

tl

~\

,

,f

~1 ",

'~,

"

-04

,/!\ Ill

i

t,

/ ~ ,

/

ff

n l\

:

~,,,,, r.,

B, -

o ~

L f

1\

",,,/ &

t~

,t

',

o

.

1

'

----

I

'\ I ~ ',\

~\

I

" X.~ l . t

.

.

.

.

.

g

.

"1 -0.6 10

20

30

40

. . . . . . . . 50 60 70

80

90

100

HO

t20

t50

~c M (deg)

Fig, 5. Comparison with experiment of polarization predictions for the second (full curves) and third (dashed curves) potentials for Ni and Co given in table 4.

47.2 56.6 56.6 61.5 54.8 54.8 59.2 55.5 55.5 65.3 51.4 51.4 48.8 55.4 55.4 53.0 53.9

4°Ca 58Ni

1.172 1.072 1.072 1.036 1,101 1.101 1.062 1.099 1.099 0.999 1.168 1.168 1.202 1.158 1.158 1.199 1.173

ro (fro)

0.703 0.801 0.801 0.855 0.776 0.776 0.842 0,764 0.764 0.886 0.725 0.725 0.704 0.713 0.713 0.694 0.682

a (fill) 1.78 4.11 4.[ [ 4.43 3.47 3.47 3.92 3.34 3.34 4.44 2.59 2.59 2.58 2.01 2.01 2,38 2.28

W (MeV) 4.83 4.20 4.20 4.42 5.00 5.00 5.20 5.21 5.2l 4.10 7.49 7.49 7.22 8.40 8.40 7.85 7,66

IVD (MeV) 1.288 1.382 1.382 1.355 1.328 1.328 1.322 1.311 1.31l 1.395 1,314 1.314 1,322 1.314 1.314 1.287 1.325

r o" (fro) 0.653 0.497 0.497 0.508 0.589 0.589 0.574 0.602 0.602 0,565 0.647 0.647 0.644 0,761 0,761 0.751 0.764

a' (fro) 4.03 5.73 5.18 5.18 5.73 5.68 5.68 5.88 6.86 6,86 5.15 4.96 4.96 5.17 5.85 5.85 5.11

V~ (MeV) 1.009 0.959 1.066 1.066 0.987 1.103 1.103 1.046 1.129 1.129 1.058 1.142 1.142 1,067 1.152 1.152 1.211

rs (fm) 0.446 0.801 0.451 0.451 0.776 0.468 0.468 0.764 0.517 0.517 0.725 0.508 0.508 0.713 0.514 0.514 0.519

as (fro) 904 1082 1080 1093 1148 1144 1161 1140 1136 1186 1637 1634 1644 2040 2038 1971 2036

(912) (1087) 11084) 11096) 11150) 11147) (1164) (1144) 11140) (1190) (1624) (1621) (1631) (1998) (1996) (1932) (1994)

~rA (rob)

80 870 1596 963 391 1797 940 295 2754 1293 2394 1528 772 361 816 381 211

Za e

199 181 39 43 400 155 166 354 100 368 317 248 292 85 22 110 56

Zo~

Except for Ca the first potential for each nucleus was obtained by \ a r y i n g I,~, and rs to optimize the fit to the polarization data at 29 MeV with the other parameters fixed at the l;.rst set given l\w each nucleus in table 2. The second potential \~as obtab,,cd by ~ar?ing I% rs and a~. The third was then obtained from the second by optimizing ti~c lit to the cross sections at 30.3 McV. l h e last Pb potential results from three such iterations. The Ca potential derives Iron: the first o~c gi., ct~ in table 2 with V, r~: a~d as ~.aricd to lit the polarization. The rcacti(m cross sections %t are gixen for 30.3 MeV and (in parentheses) for 29 MeV.

'°sPb

12°Sn

~9Co

~Ni

V (MeV)

Target

Iterated optical potentials

TABLE 4

OPTICAL MODEL

287

polarization and where the improvements in fit are mostly found. Variations in rs have rather similar effects on the polarization and there is some interplay between the two parameters r s and as. If anything, the potentials in table 3 which minimize X] +Xp2 favour r s ~ ro, as < a, but it would be dangerous to draw any firm conclusions at this stage. In a further attempt to elucidate the properties of the spin-orbit coupling, searches were made varying only Vs, r s and a s, minimizing Zp 2 alone and using the correct energy of 29 MeV. The potentials with constrained spin-orbit coupling given in table 2 which optimize the fits to the cross sections were taken as starting points. Doing this then spoils the fits to the cross-section data, so a second step was taken of re-adjusting the other parameters to optimize the cross-section fits again. The results of these two steps are given in table 4, and the optical-model predictions from the first step (fit to polarization alone) for Sn and Pb are shown as the dashed curves in fig. 4. The polarization predictions for Ni and Co for both steps are shown in fig. 5. The contents of table 4 emphasize the weight given the parameters as by a polarization fit; while the value of r s changes little from that of r o, in every case a s takes a value considerably less than that of a. In each case, the greatest improvement this brings about is at forward angles, where the polarization is generally made more negative and brought into better agreement with the measurements. When the other parameters are re-adjusted to minimize )(], leaving Vs, rs and as fixed, the cross section fits so obtained are not even as good as those shown in fig. 2 for the constrained spinorbit coupling (r s = r0, as = a). This is not entirely unexpected in view of the evidence from table 3 that optimum cross s e c t i o n fits favour as > a rather than a s < a. This iterative process was continued for a few more cycles. Unfortunately, only for Pb was there any sign of convergence; the potential obtained after three cycles is included in table 4. The results for the other nuclei oscillate about those obtained in the first cycle. In particular there is a tendency for Ni and Co for r o to become small and a to become large. The reduction in as induced bythe polarization search cannot be compensated for by small changes in the other parameters during the cross-section search. Next a study was made to see if the improvements in the polarization at forward angles could be obtained by variation of r~ alone, keeping as = a. Again the searches were started with the first set of parameters for each nucleus given in table 2, using an z energy of 29 MeV, and V~ and rs were varied to minimize Zp alone. The results are given as the first potential for each nucleus in table 4. In each case, rs is reduced below the value of r 0 and the polarization fit is improved. However for Co and the two Ni nuclei the peak at 20-25 ° is not fully reproduced, and this contributes about 25 % of the 7.2 values. A similar effect is seen for the peak at 35-40 ° for Sn. It appears necessary to reduce as < a in order to fit the polarization data in this region also (see fig. 5). 3.3. A V E R A G E

POTENTIALS

If we wish to see the presence of, for example, a symmetry dependence 2 o) in the potential depth V it is important that the fluctuations in the other parameters be

288

G.R.

SATCHLER

TABLE 5 Optical potentials with fixed geometry, r 0 ~ rs = 1.12 fro, a -- as = 0.75 fro, r 0, -- 1.33 fro, a' = 0.58 fm (except for Sn, a' = 0.65 a n d for Pb, a" ~ 0.75) with W = 3 MeV Target

V (MeV)

a~Fe

a b a b c d a b c d a b c d a b a b c d a b c d

5SNi

~UNi

59Co

Cu ~2°Sn

'-'usPb

52.4 52.4 51.4 51.9 52.4 52.7 52.4 52.6 53.1 53.2 52.8 53.1 53.3 53.7 53.0 53.5 56.6 56.0 55.9 56.7 59.4 60.4 59.1 59.5

WD (MeV)

Us (MeV)

6A (mb)

Za ~

5.64 5.40 5.05 4.85 4.93 5.00 5.57 5.42 5.48 5.60 5.86 5.43 5.45 5.59 5.91 5.91 7.21 7.35 7.25 7.46 7.94 7.10 8.12 7.98

6.40 6.40 6.40 6.40 6.01 4.80 6.40 6.40 6.15 5.16 6.40 6.40 6.35 5.21 6.40 6.40 6.40 6.40 6.36 5.23 6.40 6.40 5.20 4.89

1120 1109 1097 1088 1093 1096 1147 1141 1144 1148 1157 I 138 1140 ! 146 1193 1194 1651 1655 1652 1658 2058 2026 2066 2061

1034 704 1508 871 758 1145 576 336 329 1123 1755 366 359 913 330 297 1322 1142 1158 208l 1093 702 395 464

Potential a is potential 1, with V - = 47.5+0.4Z/A~-+30(N--Z)/A a n d W--4.5+I6(N--Z)/A. F o r potential b, V a n d W D are varied to minimize Za 2. F o r potential c, V, WD a n d V~ were ~aried to minimize Z ~ -i-Zv2, with Z92 evaluated using the theoretical polarizations for 30.3 MeV. Potential d has a s = 0.5 @ a with V, WD a n d Vs varied to minimize Za z alone. T h e reaction cross sections aA are given for 30.3 MeV. averaged out. In view of the results of the previous sections we shall concern ourselves here mainly

with a constrained

sidiary calculations and

were made

V s. T h e p a r a m e t e r

rather than attempt

spin-orbit

coupling

with various

t e r m w i t h r S = ro, a s = a. S u b -

values for the parameters

r o, a,

W, r o

a' definitely needs to be larger for the heavier elements,

but

to correlate this with A, etc., we chose to take a' = 0.58 fin for

Fe, Ni, Co and Cu, a' = 0.65 fm for Sn and

a'

suming to find an optimum

in the absence of a computer

set of parameters

= 0 . 7 5 f m f o r P b . I t is v e r y t i m e - c o n -

w i l l fit t h e d a t a f o r s e v e r a l d i f f e r e n t n u c l e i s i m u l t a n e o u s l y .

However,

code which

limited studies

indicated that a good choice was ro = with

a'

1.12 f i n ,

a = 0.75 fm,

ro =

1.33 f m ,

W = 3 MeV,

fixed at the values just discussed. With this choice, V and

minimize

Z] for each nucleus, with

(2)

Wo were wtried to

Vs fixed at 6.4 MeV. In addition,

V, WD a n d

Vs

~p

D

D

D

D

~) D

0

0

S

0

O3

D

D

D

-0

5

t~

0

p o

o

o

p D

0

do-/do- R

0

D

0

0

>

0

"'

=-=

,,.~

=.+= ml

tb 0

~-~ ~

iI~

•

£?

if:z, o~t

-~

o

m

o

bo

I

.

~

4~

o

.

.

.

o

-0

.

.

k~

.

.

".~

o

.

.

b0

~,

o

.

~

o

....----y-~s ~ "

-~----L.

.

i-o

o

.

ro

o

.

.

~

og

.

.

~

o

.

.

.~

o

L

..sJY~-'-f

o

o

.

.

.

.

''--

o

o ~

.

o°

4~

o

POLARIZATION

~ ' _

.

ro

~

P (8) ~

;

o

~

o

2

"

.

~

o

m

o

.

.

~

u .....

~ - ~ m -

, ~

r~

,

o

o ~

--L..-~'"-

~

cl

2

"

•

~

o

r

•

o c0

~.....i

"

.

m

o

o

OPTICAL MODEL

29l

were varied to minimize (X2~+ Z2) for those nuclei where polarizations had been measured, using the energy 30.3 MeV. The results are given in table 5. These optimum V and WD are closely given by V (MeV) = 47.5 + 0.4(Z/A ~) + 3 0 ( N - Z)/A, } WD (MeV) 4.5+ 16(N-Z)/A. _ potential 1

(3)

The cross sections and polarizations predicted by potential 1, eqs. (2) and (3), are shown as the solid curves in figs. 6 and 7, and are seen to fit the measurements satisfactorily. It should be remembered that one expects small variations in V and WD at least, from nucleus to nucleus, reflecting variations in the strength of the coupling to the collective quadrupole and octupole oscillations for example ~' 17, 21). Table 5 confirms that quite small deviations from the values of eq. (3) may lead to dramatic reductions in )G2, which mainly arise from improvements in the fits at large angles. For example, a change of less than 1 To in V and less than 10 ~ in WD reduces Z~z for Co by a factor of 5 and results in a fit as good as that shown in fig. 2. The conventional choice 1) has been made of including the term proportional to Z/A + in the expression (3) for V. Such a term arises for charged particles from the energy dependence of the potential z2). If, aside from the symmetry dependence on ( N - Z ) , a proton experiences the same energy-dependent interaction with all nuclei then the numerical value of the potential at a given energy would vary with the average Coulomb interaction (and hence Z/A ~) in the manner shown in eq. (3). The value 0.4 for the coefficient may be obtained by assuming that the Coulomb potential inside the nucleus is that from a uniformly charged sphere of radius 1.3 A + fm, and that the energy dependence of V is given by OV/aE = --0.3. The latter is close to the value observed in this energy region. Because the quantities Z/A ~ and ( N - Z ) / A vary in a similar fashion for stable nuclei (it is the balance between Coulomb and symmetry potentials which determines the line of stability) the presence of the Coulomb correction term in eq. (3) has a strong influence on the strength of the symmetry term inferred from data analyses. If the Coulomb term were omitted, the coefficient of the symmetry term would be roughly doubled in value. It would then be incompatible with the values found from studies (ref. 23)) of a chain of isotopes or isotones for which Z/A ~ is essentially constant. Further it would then be much larger than the value inferred 24) from (p, n) reactions between analogue states at lower bombarding energies. The symmetry coefficient of 30 MeV adopted in eq. (3) is subject to error; the data are consistent with 30_+ 5 MeV. Also the choice in eq. (2) of expressing the variation of WD in terms of ( N - Z ) / A must not be taken too seriously. While the empirical values of WD are well correlated in this way, as has also been found at lower energies 1), it would be premature to assume this is more than coincidental. Many of the nuclear properties which we expect to influence the value of the absorptive potential, such as the strength of the coupling to collective motions, increase in importance as we to go heavier nuclei and could be responsible for the observed trend 17).

292

G.

R. S A T C H L E R

The f o r m o f potential we have discussed so far assumes that the symmetry dependence can be simply expressed as a v a r i a t i o n in the well d ep t h V as in eq. (3). H o w e v e r , if this same s y m m e t r y dependence (expressed as a potential p r o p o r t i o n a l to t • T where t, T are the isospins o f the p r o t o n and target nucleus, respectively) is assumed to be responsible for the (p, n) reactions observed between analogue states, then there is evidence that this interaction is c o n c e n t r a t e d near the nuclear surface z4). In its simplest f o r m this could be interpreted as a dependence o f the radius r o u p o n ( N - Z ) / A with a constant well d e p t h V. F o r this reason, an average potential o f this type was sought also. Starting f r o m the values given in eq. (2) and table 5, V, Vs, ro and WD TABLE 6 Optimum optical potentials when V -- 51 +0.4 Z/A~ MeV, a -- as -- 0.75 fm, W 3 MeV, ro.... 1.33 fro, a' = 0.58 fm (0.65 for Sn, 0.75 for Pb) and Vs = 6.l MeV. Mode 1

Mode 2

Target

ro (fm)

crA (rob)

Z~°"

ro (fm)

WD (MeV)

crA (mb)

Z~2

5~Fe 5SNi ~°Ni 59Co Cu ~2°Sn z°spb

1.105 1.096 1.106 1.118 1.114 1.132 1.145

ll03 1077 1131 1148 1181 1648 2063

1122 897 760 449 517 1014 788

1.108 1.101 1.112 1.114 1.120 1.132 1.147

5.50 5.01 5.57 5.48 5.98 7.23 7.17

[ll0 1090 1145 1138 1198 1657 2044

583 675 439 332 308 833 703

Only r 0 -- r s was varied for the first mode, with WD = 4.25+16 (N--Z)/A; both r o and WD were optimized for the second mode. Only Za2 was minimized. The reaction cross sections ~A are given for 30.3 MeV. were varied to o b t a i n o p t i m u m fits. T h e resulting values o f V - 0 . 4 Z/A s clustered a r o u n d 51 MeV, those o f Vs a r o u n d 6.1 M e V (except for Pb, which again preferred V~ ~ 5 M e V ) , while the radius p a r a m e t e r s scattered a b o u t the line r o = 1.1 + 0 . 2 5

( N - Z ) / A . Th e surface absorptive strengths fall close to the line WD = 4 . 2 5 + 16 ( N - Z ) / A (MeV). Because the scatter in r o values was partially correlated with the scatter in values o f V, the searches were repeated with V fixed at 51 + 0 . 4 Z/A ~ MeV. Also fixed was V~ = 6.l M e V and, initially, WD = 4 . 2 5 + 1 6 ( N - Z ) / A (MeV). O p t i m u m values o f r o were then found, and finally o p t i m u m values o f t"o and WD together. The results are given in table 6; it is seen the Z 2 values are c o m p a r a b l e to those o b t a i n e d by varying V and WD with ro fixed. T h e values o f ro and WD are n o w given closely by r o ( f m ) = 1.09+0.25 ( N - Z ) / A , t WD(MeV) 4 . 2 5 + 16 ( N - Z ) / A , J potential 2,

(4)

which we call average potential 2. (Once again b o t h ;~ and Z2 for Pb are reduced by half if V~ is reduced to 5.1 M e V ; the c o r r e s p o n d i n g o p t i m u m values are then r o =

OPTICAL MODEL

293

1.141 fm, WD = 7.9 MeV.) The cross sections and polarizations predicted by this potential 2 are shown as the dashed curves in figs. 6 and 7. Again, as table 5 indicates, quite small adjustments in r o and WD away from eq. (4) can reduce )~2 significantly. It is not surprising that the scattering can be described well by varying either the well depth V, eq. (3), or the radius ro, eq. (4). It is known that there exist ambiguities such that small changes can be made in V and r o provided the product Vr~ is kept constant. Eqs. (3) and (4) imply n ~ 2.6, and the differences between the two prescriptions represent at most about 2½ % change in r o or 6 % in V. It may be significant that when both V and r 0 were optimized, the results led to r o varying with (N-Z)/A rather than V, indicating a preference for a surface-type of symmetry potential. However, this conclusion rests heavily on the results for Sn and Pb because (N-Z)/A has rather similar values for the other targets. There is clear evidence that one or the other parameter has to vary in such a way. As already discussed in the previous section, better fits to the polarization data in the forward hemisphere can be obtained by allowing r~ and as to take values differing from r o and a. Included in fig. 7 are dot-and-dash curves for the polarizations predicted by potential 1 when we put rs = 1.00 fm; it was found that doing this had rather little effect on the cross-section predictions, merely deepening the minima slightly at back angles. There are significant improvements for the polarizations, however, although not enough to bring the theory into full agreement with the measurements. If instead we put as = 0.5 fm and leave rs = ro (as suggested by table 4), the polarization fits are greatly improved; in particular the negative maximum occurring around 25 ° for Ni and Co and around 40 ° for Sn is reproduced. Unfortunately while the cross sections are essentially unchanged in the forward hemisphere, they are affected appreciably at angles greater than about 100 ° for Ni and Co, or 125 ° for Sn and Pb and the agreement with the measured values is poorer in these regions. An attempt was made to remedy this by varying V, WD and Vs once more to minimize X2, keeping as = 0.5 fm. The results are included in table 5; WD changes very little, Vs is reduced to approximately 5 MeV, and V is now given quite well by V = 48.8+0.4

Z/A~+26 (N-Z)/A.

(The difference in value of the coefficient of (N-Z)/A from that given in eq. (3) is within the uncertainties associated with this quantity.) The polarization fits are not changed appreciably by doing this; the cross-section fits achieved are improved but are still inferior to those obtained with as = a. Again we are encountering the preference of the cross sections for as > a, in opposition to the polarization requirement a s < a. Nonetheless, the quality of fit is of the same order as for the other curves shown in fig. 6.

4. The nucleus 4°Ca The angular distribution of proton scattering from 4°Ca displays a marked minim u m at approximately 140 ° which presents difficulties for an optical-model analysis.

294

G.

R.

SATCHLER

(Similar difficulties are encountered 6, 7, a3) with protons of other energies.) It was not found possible to reproduce this minimum using the potential (1) and at the same time preserve the fit at other angles. A fit to this minimum results in a theoretical curve whose oscillations are slightly out of phase with the oscillations in the measured ant0 5

.

2

/

b~

.

/

,, _

0.5

+

,..

•

,

V

°

,~ .

•

,

.

~

.

-.,

"~,'"

| + •

\

.

.

.

_

"\"., . ' / / ~ 1 •

.

0.2

. . . .

0.6

, t

.

# i

'\

I

."~\

0.4

/

k ' "

i

0.2 ._J 0

a_

0

"7',

-0.2

40Ca + p

-0.4

-0.6

0

20

40

60

80

t00

~20

t40

160

#80

8C M (deg) Fig. 8. Comparison with experiment of the predictions of the first potential in table 2 for ~°Ca (full curve). The dashed curve is obtained from the same potential with the spin-orbit parameters varied (table 4).

gular distributions at other angles. The origin of this anomaly is not understood. An imaginary term in the spin-orbit coupling, for example, brings no improvement, it may be due to some other inadequacy of this particular potential model. Since scattering data are often only available for angles less than this, it is amusing to speculate

OPTICAL

295

MODEL

how many other apparently excellent optical-model fits would be in trouble if more extensive data had been taken. It was confirmed that for Ca the polarization fits were influenced to a negligible extent by the difference in energy for the cross section and polarization. For example,

i %

7--7t : ; f j

I~

~

],- %::~- :

/ / / !

0.2

k~2_i.-----'~_-- --'~

0.6

t

i"

~

:

~

1

I

i

i

i

i i

0.4

iI

0.2 d 0 n

\ l

I

-0.2 40Ca + p -0.4

t -

•

~

i

;

;

!

i

.

I .

.

.

.

I -0.6

I

0

20

40

60

i

__

8O

t00

t20

140

460

480

OC.M.(deg) Fig. 9. Comparison with experiment of the predictions for ~oCa using independent spin-orbit coupling. The solid curve is for the third potential of table 3 and the dashed curve for the second.

for a given potential gp2 changes by a few per cent only when the energy is changed. (The 7,2 are much more sensitive, however. A potential which optimizes the fit at 30.3 MeV yields a Z2 which is five times larger when evaluated using the theoretical cross sections for 29 MeV.) The polarization only changes significantly at angles greater than 100 °, where there are few data and the errors are large. Hence the polari-

296

O.R.

SATCHLER

z a t i o n d a t a were usually treated as t h o u g h they h a d been m e a s u r e d at 30.3 MeV. F u r t h e r , the difficulty with the cross section m i n i m u m at 140 ° gives a large contrib u t i o n to Z 2 which biases the search. Since it was felt t h a t a g o o d fit at smaller angles was m o r e significant, m o s t studies were m a d e with 10 ~ errors for the 55 d a t a for fO

i .~,,,., !

5

~ .,,

I

~

~,.~

!

i ,,

i

i

°co +p

•

2

#

%

'e %

/I\

i

/i

~

~ "

x'~'-----(~

0.5 •

!

/

..... :: I

'.

[

-

0.2

r-

"

0.6

I.

0.4

I

,,71

x,

~

Ik,

,

40

60

\

lilt

,~,

!

,\

]

\

; / t l l 1.

~",,_~,'!Y

1-,

\

; ',

\ /

',,

0.2

0

-0.2

-0.4

-0.6

-0.8

0

20

80

400

420

#40

460

48O

OC.M.(deg)

Fig. 10. Comparison with experiment of the predictions for 4°Ca of potential 2 of eqs. (2) and (4). Also shown are the predictions when r s and a8 are allowed to differ from r o and a, respectively. angles less t h a n 124 ° a n d 50 ~ (or s o m e t i m e s 100 ~ ) errors for the 20 d a t a for larger angles. Potentials are given in table 2 with c o n s t r a i n e d s p i n - o r b i t coupling which mini2 a n d Z~2 alone; the theoretical curves for the latter are c o m p a r e d to the m i z e d (Z~2 + Zp)

OPTICAL MODEL

297

measurements in fig. 8. The former potential gives similar results. The cross sections are fitted very well, but the predicted polarization is too positive for 0 < 65 °. Independent spin-orbit coupling was f o u n d to give no improvement in the crosssection fits but greatly improved the agreement between the experimental and theoretical polarizations. Three such potentials are given in table 3; they are rather similar, and as fig. 9 shows they give very similar predictions. The cross-section fits are slightly inferior to those obtained with a constrained spin-orbit term, but the polarization is made more negative at forward angles as required. I n an attempt to discover whether variation o f r s or as was most effective in improving the polarization fits, the constrained spin-orbit potential which minimizes Z2 (table 2) was taken and either rs or a s varied with Vs to minimize Z2 at 29 MeV. Varying r s yielded the largest reduction in Zp2 (V s = 4.87 MeV, rs = 0.983 fro, Z2 = 253), although judged qualitatively varying as gave as m u c h improvement (Vs : 4.70 MeV, as = 0.559 fro, X~ = 546). The latter yielded a better fit for 0 > 80 °, but was somewhat poorer at the peaks near 30 ° and 4 0 o, where the Zp2 values m a y be influenced by the neglect o f angular resolution effects. Varying both parameters led to the potential given in table 4 and the dashed curves shown in fig. 8. The crosssection fit remains good, but there are still discrepancies with the polarization between 70 ° and 110°; it is necessary to re-adjust the parameters o f the central potential to remove these (as in table 3 and fig. 9). The average potentials 1 and 2 discussed in subsect. 3.3 reproduce the measured cross sections quite well in the forward hemisphere. The solid curves in fig. 10 show this for potential 2; even the polarization is given qualitatively. Also shown in fig. 10 are curves for which r s and as are each reduced f r o m the values r o = 1.09 fm, a = 0.75 fm, respectively. The same features just discussed are seen; varying r s improves the agreement on the peaks at 30 ° and 40 ° and worsens it beyond 80 ° . This figure also shows h o w the cross section is more sensitive to a variation in as than one in rs; varying as significantly affects the cross section as far forward as 70 °. The conclusion from these and similar studies seems to be that is it advantageous to make both rs < ro and a s < a, although the o p t i m u m potentials given in table 3 indicate that the reduction required in r s is only 7 or 8 ~ , while that in a s is by a factor o f 2 or more.

5. The nucleus 2Ssi There are two sets o f cross-section data available for 2aSi, those taken at 29 MeV in conjunction with the polarization measurements 9) and at 30.3 MeV (ref. 8)). The analysis followed the same pattern as described for the other nuclei. M o s t work was with the 30.3 MeV cross sections because they covered a wider angular range (15 ° to 150 °) but the 29 MeV data gave very similar results. Two potentials 2 with constrained spin-orbit terms were found which minimized g 2 alone and (g,2.~_ Zp); the parameters are included in table 2. The former gives a very g o o d fit to the cross

298

G.R. SATCHLER

sections and its polarization predictions differ from experiment in much the same way as those for the analogous potential for Ca (fig. 8, solid curve). The second potential improves the polarization fit for 0 < 100 ° but at the expense of a poor fit to the crosssection data. t0

.

t

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. .

,.

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.

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.

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28Si + p

-02

-04

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80

40

60

80

!00

t20

440

t60

~80

C~C M (deg)

Fig. 11. Comparison with experiment of the predictions of the second (full curve) and third (dashed curve) potentials of table 3 for 2sSi. Table 3 presents potentials with independent spin-orbit coupling. The first minimizes ZZ~ (30.3 M e V ) alone, but still gives a quite g o o d fit to the polarization. The next two minimize Z] (30.3 M e V ) + x 2 ; the first was obtained assuming 10 ~ crosssection errors, the second assumed 3 ~ (hence giving the cross sections more weight). The corresponding curves are compared to the data in fig. 11 ; the agreement is good.

OPTICAL MODEL

299

Finally, the fourth entry in table 3 minimizes )~2 (29 M e V ) + x 2, and yields fits very similar to those shown in fig. 11. In all respects, these potential parameters found for Si are very similar to those obtained for heavier nuclei. Their spin-orbit components exhibit the same trends as were discussed above for other nuclei; rs is about 20 % smaller than r o. When the polarization fit is being optimized, as is about 30 % smaller than a, but when only the cross sections are considered, as ~ a. To test the relative sensitivity to variations in r s and as, the first constrained spin-orbit potential of table 2 was taken, and 22(30.3 M e V ) + Z2 was minimized by varying Vs and rs, Vs and as, separately, keeping the other parameters fixed. A large reduction in Z2 is induced by reducing rs (Vs = 7.18 MeV, r~ = 0.874 fm, Z~ = 84), and a smaller reduction by decreasing as (Vs = 5.53 MeV, as = 0.466 MeV, Z2 = 300). The latter also worsens the cross-section fit. The smaller as improves the fit to the polarization minimum near 70 °, while the smaller rs gives better agreement at the negative maxima near 30 ° and 110 °. Varying both rs and as ~2 yields both improvements with a further considerable reduction in Xp (Vs = 5.98 MeV, r s = 0.929 fro, as = 0.542 fin, Zz = 28) without changing the cross-section fit. Once again it appears that a reduction in both r s and a s from the values of ro and a is required to obtain an optimum fit.

6. Other nuclei

There are available some other sets of data for proton scattering at energies close to 30 MeV; these include 6°Ni at 30.8 MeV (ref. so)), Cu, Ag, Ta, Au and Pb at 31.5 MeV (ref. it)) and Pb at 31 MeV (ref. lo)). (Other nuclei lighter than Si will be discussed in a later communication.) Preliminary studies with these data indicated that they gave results consistent with those obtained from the nuclei discussed above. Not much further analysis was performed; rather, we compare these data with the predictions of the two average, constrained spin-orbit, potentials discussed in subsect. 3.3. The comparison is shown in fig. 12, and the agreement is seen to be good. The measured cross sections for 60Ni are consistently low for scattering angles of less than 50 ° and especially for less than 20 ° . There is probably some inelastic contamination in the data for 31.5 MeV. Further, the nucleus Ta is known to be permanently deformed, while low-lying excited states are quite strongly coupled to ground in both Ag and Au. We would thus expect the optimum values of at least WD and V to depart to some extent from those of the average potentials.

7. Discussion

In addition to the desire to determine the overall potential for 30 MeV protons as well as possible, there was particular interest in this analysis to see if more could be learnt about the nature of the spin-orbit coupling term. The conclusion is that, while the cross sections can be fitted very well with the constrained form with rs = ro and

10 5 •

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480

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Fig. 12. Comparison with experiment of the predictions of the average potential 1 (solid curves) and 2 (dashed curves) for proton scattering from other nuclei with energies close to 30 MeV.

OPTICAL MODEL

301

as = a, the p o l a r i z a t i o n d a t a d e m a n d t h a t one or b o t h o f these p a r a m e t e r s take on different values, p a r t i c u l a r l y in o r d e r to r e p r o d u c e the m e a s u r e m e n t s at f o r w a r d angles. T h e p o l a r i z a t i o n d a t a b y themselves require r s g r 0, as < a (a s ~ 0.5 fm is f a v o u r e d ) except for Si, a n d to a lesser extent for Ca, where some r e d u c t i o n in r s is also indicated. H o w e v e r , fits to the cross sections alone show the o p p o s i t e t r e n d for as, leading to as ~ 1.0 fm. A possible way to a v o i d these conflicting tendencies is to keep as = a a n d have r s < ro, since the cross sections are less sensitive to v a r i a t i o n s in rs t h a n to v a r i a t i o n s in as. Fig. 7 and table 4 show t h a t i m p r o v e m e n t s in the fits to the p o l a r i z a t i o n measurements can be o b t a i n e d this way, b u t the d a t a for Ni, Co a n d Sn at the m o s t f o r w a r d angles are stilll n o t r e p r o d u c e d correctly. The feeling o f the present a u t h o r f r o m the m a s s o f results he has studied is t h a t b o t h rs < ro a n d as < a are required, a l t h o u g h it is h a r d to present i n c o n t r o v e r t i b l e evidence for this *. P a r t o f the difficulty lies in the different energies at which cross section a n d p o l a r i z a t i o n m e a s u r e m e n t s are m a d e ; in the absence o f a m o r e general o p t i c a l - m o d e l search code one is forced either to a p p r o x i m a t i o n s or to a relatively clumsy iterative procedure. (The v a r i a t i o n in p o l a r i z a t i o n between the two energies was shown in fig. 1. It is also d a n g e r o u s to take some m e a n energy because o f the sensitivity o f ;(]. The first p o t e n t i a l for Pb in table 2 yields X] = 169 at 30.3 M e V ; if we use the theoretical cross sections for 29 MeV, we o b t a i n X] = 7104!) A n o t h e r w o r d o f c a u t i o n is advisable. It is n o t k n o w n to w h a t extent the p o l a r i z a t i o n studies are being biased by our neglect o f the finite angular resolution effects. T h e features o f the p o l a r i z a t i o n d i s t r i b u t i o n which seem to require the changes in r s a n d as d o n o t a p p e a r in regions where there are very r a p i d changes with angle; nonetheless, further investigation o f this p o i n t w o u l d seem to be necessary. A n o t h e r difficulty is that, by the nature o f things, the p o l a r i z a t i o n d a t a are m u c h less c o m p l e t e a n d subject to larger errors t h a n the cross-section data. One is t e m p t e d to ask whether it is n o t w o r t h w o r k i n g h a r d to o b t a i n p o l a r i z a t i o n d a t a at least for a few representative nuclei, a n d particularly at f o r w a r d angles, which is o f a quality c o m p a r a b l e to t h a t o f the cross-section measurements. Only in this way does it seem likely t h a t the nature o f the s p i n - o r b i t interaction can be d e t e r m i n e d u n a m b i g u o u s l y . Some studies were also m a d e in which an i m a g i n a r y p a r t was included for the spino r b i t coupling; c o n t r a r y to previous reports 25), however, no evidence was f o u n d o f any i m p r o v e m e n t s that this introduced, even for the a n o m a l o u s b e h a v i o u r o f the nucleus 4°Ca. Other features o f the p o t e n t i a l revealed by the present analysis include the expected increase o f V or r o, with increasing n e u t r o n excess (N-Z)/A. There are indications that an increase in r o for Sn a n d P b improves the cross-section fits at small angles. It would be useful to have m o r e d a t a for nuclei in this mass region to see if these indicat It should be noted that the value r~ ~ 1.0 to 1.1 fm found here is not in conflict with the values obtained from lower-energy data. The result r S m r o found here is then a consequence of our requiring a smaller value ofr 0 than at lower energies. We are indebted to R. H. Bassel for drawing our attention to this. Also, F. G. Perey has presented in) some evidence for both r s < r o and a, < a at lower energies.

302

G.

R.

SATCHLER

tions o f a surface symmetry potential z4) are confirmed, especially since these two nuclei m a y be a typical, Sn being singly closed shell and Pb being doubly closed shell. The real parts o f the potentials deduced here are also characterized by smaller radii (r o between 1.1 and 1.2 fro) and larger surface diffusenesses (a between 0.7 and 0.8 fro) than have previously been used. The real and imaginary parts of the average potential 2 for SSNi and 2°8pb are shown in fig. 13. At lower energies, below 20 MeV, the tendency has been 1) to find values of r o between 1.2 and 1.3 fm and of a between 0.65

///

0

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........... ~ " -

~

6O 0

E

5

-~0 0

2

4

6

8

t0

a2

• (fro)

Fig. 13. Plots of the real and imaginary parts of the average potential 2 of eqs. (2) and (4) for

58Ni and 2°sPb. and 0.75 fm. In a few cases 26), a radius parameter closer to ro ~ 1.15 fm has been needed. A n earlier study 2) suggested r o between 1.15 and 1.20 fm and a between 0.7 and 0.8 fm at an energy of 40 MeV. At energies between 100 and 200 MeV, a radius o f r o ~ 1.0 fm seems to be preferred, but the diffuseness tends to be smaller 2v). Presumably the main characteristics of the real part o f the optical potential are the same as those o f a Hartree-Fock potential evaluated with some effective nucleonnucleon interaction. It is then of interest to see the kind o f Hartree potential that results from folding a Y u k a w a nucleon-nucleon interaction into the nuclear density distribution p(r). A Fermi shape was taken for the latter, with parameters taken from analyses o f electron scattering 28), namely

p(r)

= po(eX°+ 1) -1,

xe =

(r-rcA~)/ae,

OPTICAL MODEL

303

with ro = 1.08 fm, ae = 0.5 fm. A Yukawa of range 1 fm results in a Hartree potential shape which is closely similar to the Woods-Saxon form of eq. (1) with r o ~ 1.05 fm and with a between 0.8 and 0.9 fm. Increasing the range scarcely changes the equivalent r o, but it increases the diffuseness a. The empirical potentials, if derived in this way, would require a range shorter than 1 fm. It is interesting to note that the inelastic scattering 26, 29) of protons from 9°Zr and other nuclei has been interpreted in terms of an effective interaction of the Yukawa type with a range of about 1 fm and a strength of approximately 200 MeV. This strength, if used here, would imply an optical potential of depth V ~ 400 MeV! However, this strength was deduced for the quadrupole and higher multipole parts of the interaction; in the case of the monopole 0 + to 0 + transition in 9°Zr, it was found that a much weaker interaction was implied (ref. 29)). We require this monopole part for the optical potential. The interpretation of strengths obtained by calculations of this type is open to question, but the results may be indicative of the shape to be expected for the potentials. They imply that the radius and diffuseness parameters found here are not unreasonable. Further, it has been suggested by Volkin 16) that the shorter range of the nucleon-nucleon spin-orbit coupling may be responsible for the spin-orbit term being inside the main real part of the optical potential. If this is so, calculations of the type just described imply that this would be manifested as as < a rather than rs < r0. It should also be remembered that we are using a model potential which is local. We have every reason to believe the "true" potential is non-local, and that at least part of the energy dependence observed for the equivalent local potential is due to this non-locality. It is entirely possible that a local potential which is equivalent to this non-local one will show some energy dependence of the shape parameters r o and a as well as of the depth V. Further, even if the spin-orbit term in the non-local potential were strictly of the Thomas form 30), the local equivalent spin-orbit coupling might show departures from this form. Theoretical calculations of the imaginary part of the optical potential always involve severe approximations, but they agree that the absorptive potential should exhibit peaking at or near the nuclear surface. At lower energies 1) the scattering is insensitive to the volume to surface ratio of the absorption, although there is a general preference for a predominantly surface form. There are indications ~) that a small volume component improves the fits to 22 MeV proton scattering data, and the present analysis very definitely demands a volume absorption W of between 2 and 4 MeV. One might expect this volume term to be characteristic of nuclear matter and hence largely independent of mass number, and this is in rough agreement with our results. The specifically finite size effects should mainly show in the surface absorption. An increase in the width a' of the surface absorption term, first noticed at lower energy, is also found to be necessary here. The strength Wo of the surface absorption also increases as one moves to heavier nuclei (see fig. 13). This increase has been presented here [eqs. (3) and (4)] as a correlation with the symmetry parameter (N-Z)/A, but it should be stressed again that this is only a choice of convenience. Although it is

304

G . R . SATCHLER

possible, especially if the symmetry potential is concentrated on the surface, that the average behaviour of WD should be correlated with the neutron excess, other features such as the strength of coupling to collective oscillations are also important in determining the surface absorption. The closed-shell nature of Sn and Pb could be important here also, and again it would be useful to have similar data for other nuclei in this mass region or heavier. Finally it is of interest to comment on the scattering matrix elements qLJ obtained with these potentials. The magnitude and phase of the qs predicted by the average

~.o [---

i

i~__ ,.

.

.

.

.

soo,,

/

5oo

....

,~

300

06

:

,"

•

o

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i' . . . . . . .

•

=

a

i

:

~

........

"';,~

' ', 200 "@ , ~ :

v

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400 :

I !

!

7

i

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! .... J r / - r V2 ..,fx~.

'

:

:

. ~

i

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o:

',,,

/

~,~.

- .....

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,

L

L

Fig. 14. M a g n i t u d e a n d p h a s e o f scattering matrix elements predicted by average potential 2 o f eqs. (2) a n d (4) for 30.3 M e V p r o t o n s on ~SNi a n d 2°sPb.

potential 2 for Ni and Pb are plotted in fig. 14. The phases were determined by assuming that Re tl and Im q are smooth functions of L. We shall not discuss them extensively because a more detailed examination of the very similar results for 40 MeV protons will be given elsewhere 7). However, we do note that the matrix elements are far from exhibiting the simple characteristics of strong absorption. Even for Pb, whose Irt~l do show the trend for strong absorption, the magnitudes approach the relatively large value of 0.2 for small L and have considerable fluctuation with changing L. This behaviour is even more marked for Ni and neighbouring nuclei. In conclusion we may say that the analysis of 30.3 MeV proton scattering leads to very consistent results for different nuclei. The general properties of the optical potentials are summarized by the average potentials 1 and 2 given by eqs. (2)-(4). In addition, there is evidence that the spin-orbit coupling is characterized by shape parameters differing from those of the real central potential, either r~ < ro or a~ < a, or

OPTICAL MODEL

305

probably both. The measured reaction cross sections are well reproduced by these potentials. The author is indebted to R. M. Drisko for making available the optical-model search code "Hunter" and for helpful discussions. The analyses reported here were guided by the results of R. H. Bassel for 40 MeV proton scattering and have been greatly assisted by his comments. Lynne McFadden assisted with the calculations. References 1) F. G. Perey, Phys. Rev. 131 (1963) 745; L. Rosen et aL, Ann. of Phys. 34 (1965) 96 2) M. P. Fricke and G. R. Satchler, Phys. Rev. 139 (1965) B567 3) B. W. Ridley and J. F. Turner, Nuclear Physics 58 (1964) 497 4) J. F. Turner et al., Nuclear Physics 58 (1964) 509 5) R. M. Craig et al., Nuclear Physics 58 (1964) 515 6) E. E. Gross et al., Int. Conf. on polarisation phenomena of nucleons, Karlsruhe (1965), paper 11/12-5 and to be published 7) R. H. Bassel, to be published 8) R. K. Cole et al., Nuclear Physics 75 (1966) 241 9) R. M. Craig et al., to be published 10) D. W. Devins et al., Nuclear Physics 35 (1962) 617 11) J. Leahy, University of California Radiation Laboratory Report UCRL-3273 (1956) unpublished 12) R. C. Barrett et al., Nuclear Physics 62 (1965) 133 13) G. W. Greenlees and G. J. Pyle, to be published 14) C. H. Poppe and G. J. Pyle, Ann. Progress Report, University of Minnesota Linear Accelerator Laboratory (1965) unpublished; M. A. Melkanoff et al., Methods of computational physics, Vol. V (Academic Press, New York) to be published 15) J. K. Dickens, Phys. Rev. 143 (1966) 758 16) H. C. Volkin, Bull. Am. Phys. Soc. 9 (1964) 439; J. A. R. Griffith and S. Roman, Phys. Lett. 19 (1965) 410; D. J. Baugh et al., to be published; L. J. B. Goldfarb et al., Int. Conf. on polarisation phenomena of nucleons, Karlsruhe (1965) paper 12-1 ; F. G. Perey, ibid., invited paper 17) F. G. Perey, in Nuclear spectroscopy with direct reactions I, Argonne National Laboratory Report ANL-6848 (1964) unpublished 18) G. R. Satchler, Nuclear Physics 45 (1963) 197 19) L. J. B. Goldfarb, G. W. Greenlees and M. B. Hooper, Phys. Rev. 144 (1966) 829 20) G. R. Satchler, Phys. Rev. 109 (1958) 429; A. E. S. Green and P. C. Sood, Phys. Rev. 111 (1958) 1147 2l) F. G. Perey and G. R. Satchler, Phys. Lett. 5 (1963) 212 22) A. M. Lane, Revs. Mod. Phys. 29 (1957) 191 23) J. Beneveniste et al., Phys. Rev. 133 (1964) B323; J. B. Ball et al., Phys. Rev. 135 (1964) B706 24) G. R. Satchler et al., Phys. Rev. 136 (1964) B637 25) D. J. Baugh et al., Nuclear Physics 65 (1965) 33; Phys. Lett. 13 (1964) 63 26) W. S. Gray et al., Phys. Rev. 142 (1966) 735; R. L. Robinson et al., Phys. Rev., to be published 27) G. R. Satchler and R. M. Haybron, Phys. Lett. 11 (1964) 313; P. G. Roos and N. S. Wall, Phys. Rev. 140 (1965) B1237 28) L. R. B. Elton and A. Swift, Proc. Phys. Soc. 84 (1964) 125 29) M. B. Johnson et al., Phys. Rev. 142 (1965) 748 30) L. H. Thomas, Nature 117 (1926) 514