Friday 23 August 2019

Physics of Nucleon-Nucleus Scattering - Chapter IV


Chapter Four

25 to 40 MeV PROTON ELASTIC SCATTERING



Introduction

Using a fully microscopic, coordinate space optical model,  successful analyses have been made of proton-nucleus (pA) elastic scattering data taken at 65 and  200 MeV [75, 88] and from targets of diverse mass and elastic and inelastic p-12C scattering data at 200 MeV have been understood [117]. Now two nucleon (NN) effective interactions have been specified, which when folded with wave functions from a complete  shell model calculation of 12C, give (g folding) optical potentials for proton energies from 40 to 800 MeV. With those potentials, as shown in the preceding chapter elastic p-12C scattering with energies 40 to 800 MeV were reproduced quite well.
           
Herein the results of analyses of the elastic scattering of 25, 30, and 40 MeV protons from many nuclei (6Li to 238U) are presented and made using coordinate space optical potentials formed by g folding.  A select few of the results presented in this chapter have been used in a brief report [77], the purpose of which was to establish that one could define appropriate NN effective interactions in this energy regime. The interest to find a credible prescription of the optical potentials at these energies lies with current and future analyses of data from the scattering of 25A, 30A, and 40A MeV radioactive ions from hydrogen targets.  Such experiments are being made at many facilities throughout the world [118-120]. Also g-folding optical potentials are required to define the distorted waves in `no parameter' DWA analyses of the cross sections from the inelastic excitation of the radioactive ions. Measurements and subsequent analyses of such inelastic excitations are feasible and have been made recently [121] for the excitation of the 2+ (1.8 MeV) state in 6He.
           
At the energies considered in this chapter (25, 30 and 40 MeV), collective structures in the response function of a nucleus may contribute above any specific microscopic description based on an effective NN multiple scattering theory. For example, if the energy is consistent with excitation of a giant resonance, virtual excitation of that resonance could contribute to the scattering.  Indeed past studies [122] indicated that such virtual excitation of the giant resonances gives energy-dependent signatures in cross sections. Those effects however are of the order of 1 mb/sr at most and so are evident, basically, only at large momentum transfers for elastic scattering. The usual (phenomenological) optical potential sufficed to give the bulk of the (elastic) scattering results in that past study [122]. Hence, notwithstanding interference effects, a first-order microscopic description of the optical potential based on single-site NN scattering in medium could still produce good agreement with elastic scattering data of magnitude greater than a few tenths mb/sr taken for energies in the range 25 to 40 MeV.
           
Still, at these energies the specific character of the target response may be needed to specify appropriately the effective NN interaction one should use in the g-folding process. If so, the standard prescription have been used to date to define the effective interactions may need some modification. Calculations at these energies using that standard prescription and comparison with data would calibrate any such required modifications. Of course, if the specific response function effects in the definition of the effective NN interaction are of sufficient import, their omission should be evident in the comparisons of current model results with data from light mass targets first, and at 25 MeV in particular, given the excitation energies of the giant resonances and the variation of those excitation energies with target mass. Therefore proton elastic scattering data have been analyzed taken in the range of energies 25 to 40 MeV and from a number of nuclei in the mass range A = 6 to 238.  The method used was that with which successful analyses of cross section and spin-dependent data from 65 and 200 MeV proton scattering  have been made from many nuclei ranging in mass from 3He to 238U [75, 87, 88, 123, 124]. As with those studies, all details of the effective interactions and structure required to define the (complex, nonlocal) optical potentials are preset and no a posteriori adjustment or simplifying approximation is made to the complex nonlocal optical potentials that result from the g-folding process.
           
Herein only the elastic scattering channel are considered. At and about 25 MeV proton energy  22 targets have been considered for elastic scattering cross sections, namely 6,7Li, 12C, 14N, 18O, 24Mg, 27Al, 28Si, 32S,  40,42,44,48Ca, 58Cu, 86,88Sr, 89Y, 92Mo,  152Sm,  208Pb, 232Th, and 238U. At that energy the data from 10 targets have been analyzed for which analyzing powers have been taken, namely 12C, 18O, 24Mg, 28Si, 32S, 58Cu, 86,88Sr, 118Sn, and 152Sm. At and about 30 MeV proton energy, 34 targets are considered for elastic differential cross sections, namely 9Be, 10,11B, 13C, 16O, 19F,  20,21Ne, 40Ar, 54,56Fe, 58,60Ni, 59Co,  63,65Cu, 64,66,68Zn, 90Zr, 104Ru, 112,114,116,118,120,122,124Sn, 139La, 141Pr, 144Sm, 176Yb, 208Pb, and 209Bi. Also 22 targets are used for analyzing powers, namely 9Be, 10,11B, 13C, 16O, 40Ar, 40Ca, 54,56Fe, 58,60Ni, 59Co, 63,65Cu, 64,68Zn,   90Zr, 92,100Mo, 120Sn, 176Yb, and 208Pb. At 40 MeV, 24 targets are considered for differential  cross sections, namely, 6He,  6Li, 12C, 15N, 16O, 24Mg, 27Al, 28Si, 40Ca, 58,60,62,64Ni,  64,66,68Zn, 90,92Zr, 116,118,120,122,124Sn, and 208Pb, and 18 targets for analyzing powers,  namely 12C, 24Mg,  40Ca, 58,60,62,64Ni, 64,66,68Zn,  90,92Zr, 116,118,120,122,124Sn, and 208Pb.



Results of Calculations

The results of calculations of the elastic scattering of 25, 30, and 40 MeV protons from  many target nuclei are displayed in five subsections; the first three dealing with data for each particular energy separately. In the fourth and fifth energy and isotope variations are discussed respectively.
           
In most of the cases the HO functions are used for the bound state single particle functions. But for light nuclei, 6He and 6Li in particular, WS potential functions have been used. The oscillator length for the HO functions was set by an A1/6 rule as indicated as reasonable by electron scattering studies.



Results of the Scattering of 25 MeV Protons

The results of my calculations of the elastic scattering of 25 MeV (and adjacent energies) protons from different nuclei are shown in Figs. 4.1 to 4.6.
           
In Fig. 4.1, the mass variations of the 25 MeV elastic proton scattering differential cross sections are shown in 3D form. The mass numbers identify each result. Calculated results are compared with the experimental data. Clearly the mass trends of the data, if not precise detail, are reproduced by the predictions made here. Each of the results are shown in flat perspective in the following figures.

Figure 4.1: The 25 MeV elastic proton scattering differential cross section data compared with the optical model calculations for targets from mass 6 to 238. The target mass identifies each result.


  
In Fig. 4.2, calculations of proton scattering from the nuclei, 6,7Li, 12C, 14N, 18O, 24Mg, 27Al and 28Si are compared with the experimental data. Data were measured at 25.9 MeV for 6Li [125], at 24.4 MeV for 7Li [126], at 24 MeV for 12C [127], at 26 MeV for 14N [128], at 24.5 MeV for 18O [129], at 27 MeV for 24Mg [130], at 28 MeV for 27Al [131], and at 25 MeV for 28Si [132].
           

Figure 4.2: The differential cross sections from the elastic scattering of 25 MeV protons from 6,7Li, 12C, 14N, 18O, 24Mg, 27Al and 28Si. Data (dots) are compared with the results of microscopic model calculations (solid curves).



For 6Li, the calculated results are in very good agreement with the experimental data up to 120o scattering. For the other cases however, while the shapes of the calculated results are quite similar to those of experimental data, minima are overaccentuated. And this overaccentuation increases with the target mass.


Figure 4.3: As for Fig. 4.2 but for 32S, 40,42,44,48Ca, 58Cu, 86,88Sr.


In Fig. 4.3, my predictions for 25 MeV proton scattering from the nuclei 32S, 40,42,44,48Ca, 58Cu, and 86,88Sr, are compared with the experimental data. Data were measured at 25 MeV for 32S [130] and  40,42,44,48Ca [133], at 28 MeV for 58Cu [131], and at 24.6 MeV for 86,88Sr [134].
           
The 32S, 40,42,44,48Ca and 58Cu results agree reasonably with observation although my  predictions again give too sharp a structure and have the maxima and minima at slightly too large scattering angles.
           
For 86,88Sr the data are well reproduced to quite large scattering angles.


 
Figure 4.4: As for Fig. 4.2 but for 89Y, 92Mo, 152Sm, 208Pb, 232Th and 238U.



In Fig. 4.4, predictions for 25 MeV proton scattering from the nuclei 89Y, 92Mo, 152Sm, 232Th and 238U, are compared with the data.  The calculations for   208Pb nucleus are also presented in this figure, but no data at this energy is available.

Data were measured at 25 MeV for  89Y [135],  92Mo [136] and 152Sm [137], and at 26 MeV for 232Th and 238U [138].
           
For 89Y and 92Mo, the data are well reproduced to quite large scattering angles. That is the case also with 152Sm up to 90o. For the heavy nuclei, 232Th and 238U, the predictions are in good agreement with the experimental data up to 60o. For most cases at the larger scattering angles, these results depart from observation, though the shapes of the cross section predictions remain quite similar to the data.




Figure 4.5: The 25 MeV elastic proton scattering analyzing power data compared with the optical model calculations for targets mass from 12 to 152. Target mass identifies each result.


 The 25 MeV elastic proton scattering analyzing power data are compared with the results obtained from my optical model calculations in Fig. 4.5 and Fig. 4.6.
           
In Fig. 4.5, the calculated analyzing powers are poresented compared with experimental data as a variation of mass, from which the mass trend of data and results is quite evident.
           
These results are now shown in flat perspective in Fig. 4.6. The target nuclei are indicated in each segment.




Figure 4.6: The analyzing powers from the elastic scattering of 25 MeV protons from 12C, 18O, 24Mg, 28Si, 32S, 58Cu, 86,88Sr, 118Sn and 152Sm.

Data were measured at 24.1 MeV for 12C  [139], at 24 MeV for 18O [129], at 25 MeV for 24Mg [130], 28Si [132], and 32S [130], at 24.6 MeV for 86,88Sr [134], and at 24.5 MeV for 118Sn [140] and 152Sm [137].
           
For the light mass nuclei (A ≤ 40), the shape and size of the data are very similar to my predictions. For heavier nuclei, those predictions tend to underestimate the magnitude variation in the data, particularly so for 152Sm.

Results of the Scattering of 30 MeV Protons

Results of the optical model calculations of 30 MeV proton scattering from different nuclei are presented in Figs. 4.7 through 4.15.



Figure 4.7: the 30 MeV elastic protons scattering differential cross section data compared with the optical model calculations for target mass 9 to 209. The target mass identifies each result.


In Fig. 4.7, differential cross sections of 30 MeV proton scattering from all considered nuclei are presented to show the mass trend of differential cross sections. The predictions, shown by the solid curves, are compared with the respective experimental data. Again the mass trend of data is reproduced by the predictions with some details differing.



 

Figure 4.8: The differential cross sections from the elastic scattering of 30 MeV protons from 9Be, 10,11B, 13C, 16O, 19F, and 20,21Ne.


In Fig. 4.8, the differential cross sections from 9Be, 10,11B, 13C, 16O, 19F and 20,21Ne are compared in flat perspective with the experimental data.

Differential cross section data were measured [141, 142] at 30.3 MeV  for 9Be, 10,11B, 16O, 19F and 20,21Ne. For  13C, data were taken at 30.5 MeV [143]. The shapes of the data agree with the calculated results quite reasonably but there are slight differences in the magnitudes. In particular, the minima are overestimated. For 19F and 21Ne though, calculated results are in good agreement with the data up to 75o.


 
Figure 4.9: As for Fig. 4.8 but for 40Ar, 54,56Fe, 58,60Ni, 59Co, 63Cu and 64Zn.


In Fig. 4.9, the differential cross sections have been calculated for scattering from 40Ar, 54,56Fe, 58,60Ni, 59Co, 63Cu and 64Zn are displayed and compared with the relevant experimental data.
           
Data were measured at 30 MeV for 40Ar [144] and 63Cu [145], at 30.3 MeV for 56Fe,  58,60Ni and  59Co [146], at 30.4 MeV for 54Fe [147] and at 30.5 MeV for 64Zn [148]. Again the shapes of the experimental data are well reproduced by the results of calculations, but the minima are more sharply predicted. These effects concur with what was found from the 25 MeV data analyses. For 56Fe, 58,60Ni, 59Co, 63Cu and 64Zn, the first order minima at 30o are underestimated in my calculations while the higher order minima are overestimated.


 
Figure 4.10: As for Fig. 4.8 but for 65Cu, 66,68Zn, 90Zr, 104Ru, and 112,114,116Sn.


In Fig. 4.10, the results of calculations made from 65Cu, 66,68Zn, 90Zr, 104Ru, 112,114,116Sn are compared with the respective differential cross section data. Data were measured at 30 MeV for 65Cu [145] and 90Zr [149], at 29 MeV for 104Ru [150], at 30.5 MeV for 66,68Zn [148], and at 30.4 MeV for 112,114,116Sn [151].
           
For the 65Cu and 66,68Zn cases, the first order minima are underestimated and the higher order minima are overestimated by my calculations. The shapes are still well reproduced.
           
The 90Zr data are well replicated in the range of scattering angles from 30o to 100o. 
           
For 104Ru, however, calculated results are in very good agreement with the experimental data but that has been measured only to 45o. Data to larger scattering angles have been measured with the light Tin isotopes and my predictions give good results at least to 40o scattering. At larger scattering angles, the shapes of the calculated results are similar to the data but the calculated minima are more sharply defined.
           
In Fig. 4.11, the results of calculations of scattering from 118,120,122,124Sn, 139La, 141Pr, 144Sm, 176Yb, 208Pb and 209Bi are compared with the respective differential cross section data, which have been measured at 30.3 MeV for 120Sn and 208Pb [146], at 30.4 MeV for  118,122,124Sn [151], at 29.32 MeV for 139La and 141Pr [152], at 30 MeV for 144Sm [152] and 176Yb [153] and at 31 MeV for 209Bi [154].
           
Again, with the heavier tin isotopes, data are well reproduced up to 40o scattering, with the shape of the calculated results at larger scattering angles being similar to the data but with the successive minima more sharply defined than observed. But the calculated results are in excellent agreement with the experimental data from   139La, 144Sm, 176Yb and 209Bi.
           
For 208Pb, the predictions overestimate the cross section data at large scattering angles which may reflect an inadequacy of the chosen model of structure.  Such will be considered later in more detail.




 

Figure 4.11: As for Fig. 4.8 but for 118,120,122,124Sn, 139La, 141Pr, 144Sm, 176Yb, 208Pb, and   209Bi.



In Fig. 4.12, the calculations for the analyzing powers of the 30 MeV proton scattering from all the nuclei considered are presented. 




Figure 4.12: The 30 MeV elastic proton scattering analyzing power data compared with the optical model calculations for target mass 9 to 208. The target mass identifies each result.



This figure shows that the analyzing powers change with mass in a smooth way and one which the predictions emulate well.




Figure 4.13: The analyzing powers from the elastic scattering of 30 MeV protons from 9Be,  10,11B, 13C, 16O, 40Ar, 40Ca, and  54Fe.

          
The results of calculations for the analyzing powers from the elastic scattering of 30 MeV protons from all the nuclei are presented in figs. 4.13, 4.14, and 4.15 in flat scale.
           
The results from targets 9Be through 54Fe are shown in Fig. 4.13. Data were measured at 30.3 MeV for 9Be [141],  10,11B, and 16O [142], at 30.4 MeV for 13C [143] and 54Fe [147], at 30 MeV for 40Ar [144] and at 29 MeV for 40Ca [139]. The comparisons between that data and my predictions improve with increasing mass.




Figure 4.14: As for Fig. 4.13 but for 56Fe, 58,60Ni, 59Co, 63,65Cu and 64,68Zn.



That improved agreement is retained with target mass 56 through 68 as shown in Fig. 4.14, in which  predicted analyzing powers for 56Fe, 58,60Ni, 59Co, 63,65Cu and 64,68Zn are compared with the data.
           
Data were measured at 30.3 MeV for 56Fe [155], at 29 MeV for 58,60Ni and 59Co [139], at 30 MeV for 63,65Cu [145] and at 30.5 MeV for 64,68Zn [148]. The sharp changes from negative to positive values of the analyzing powers are well predicted in particular.




Figure 4.15: As for Fig. 4.14 but for 90Zr, 92,100Mo, 120Sn, 176Yb and 208Pb.


In Fig. 4.15, the calculated analyzing powers from 90Zr, 92,100Mo, 120Sn, 176Yb and 208Pb are compared with the experimental data. Data were measured at 30 MeV for 90Zr [149] and 176Yb [153], at 30.3 MeV for  92,100Mo [156], and at 29 MeV for 120Sn and 208Pb [139]. The quality of the fits to data with 90Zr through 120Sn is good while the predictions for 208Pb in particular underestimate the observed values.


In summary,  my  calculated analyzing powers show the trend of the data for the lightest mass targets  and  quite good agreement is found for targets ranging from  16O to 58Ni. That agreement remains with the data from 65Cu to 120Sn, although for these targets, predictions slightly underestimate the data in the angle range 20o to 60o and also the characteristic forward (negative value) peak. While the data from heavier nuclear targets are not as sharply structured as those from lighter nuclei, these calculations gave more compressed values. The general structure of the 176Yb and 208Pb analyzing powers are matched in location
but the peak magnitudes are at best half what is observed.


Results of the Scattering of 40 MeV Protons

The results obtained from the optical model calculations of the elastic scattering of 40 MeV protons from targets of different nuclei are compared with data in Figs. 4.16 to through 4.22, with the cumulative set of results and data shown in the first.
           
In Fig. 4.16, all of the elastic 40 MeV proton scattering differential cross section data are compared with the calculations for target mass 6 to 208 to show the trend with mass at this projectile energy. Clearly the trend of the data is reflected in the calculated results. With the lower energy results, there are differences in detail.
           
In Fig. 4.17, the calculated cross sections from 6He, 6Li, 12C, 15N, 16O, 24Mg, 27Al and 28Si, are compared with  experimental data that were measured at 40 MeV for 6Li [157], 12C [97], 24Mg [158], 27Al and 28Si [159], at 39.8 MeV for 15N [160] and at 39.7 MeV for 16O [161]. For 6He, data were  obtained [118] by inverse kinematics for an experiment in which a radioactive beam of 40.9A MeV 6He scattered from  a hydrogen target. Calculated results are in good agreement with the data; much better in fact for most than for the 25 and 30 MeV studies. The case of 28Si is exceptional but would need data analyzed at more energies to comment further. At large scattering angles however, the predictions still have slightly more defined minima than are observed.


 
Figure 4.16: The 40 MeV elastic proton scattering differential cross section data compared with the optical model calculations for target mass 6 to 208. The target mass identifies each result.



 
Figure 4.17: The differential cross sections from the elastic scattering of 40 MeVprotons from 6He, 6Li, 12C, 15N, 16O, 24Mg, 27Al, and  28Si.



The results found for 40 MeV proton scattering from 40Ca, 58,60,62,64Ni and 64,66,68Zn are compared with the data in Fig. 4.18. Data were measured at 40 MeV for 40Ca and 58Ni [97], and at 39.6 MeV for  60,62,64Ni and 64,66,68Zn [162].             Although most calculated results have too sharply defined minima, the agreement between predictions and data shown is very good. That is also the case with heavier mass targets as shown in Fig. 4.19, in which results found for 40 MeV proton scattering from 90,92Zr, 116,118,120,122,124Sn and 208Pb are compared with the data.  That  data were measured at 40 MeV for 90Zr [97], 92Zr [163] and 208Pb [97], and at 39.6 MeV for  116,118,120,122,124Sn [164].
           
Such quality of results of predictions were found previously for higher energy studies, and with 65 MeV in particular [88]. The case of 208Pb is again special and will be considered later.



Figure 4.18: As for Fig. 4.17 but for 40Ca, 58,60,62,64Ni, and 64,66,68Zn.






Figure 4.19: As for Fig. 4.17 but for 90,92Zr, 116,118,120,122,124Sn, and 208Pb.


  
In Fig. 4.20 the analyzing powers have been calculated for 40 MeV proton elastic scattering from 16 of these nuclei are compared with data. The mass identifies each result. As with the cross sections there is a definite mass trend to this data and again it is one that is reproduced by the predictions.



Figure 4.20: The 40 MeV elastic proton scattering analyzing power data compared with the optical model calculations for targets ranging in mass from 12 to 208. The target mass identifies each result.


Figure 4.21: The analyzing powers from the elastic scattering of 40 MeV protons from 12C, 24Mg,  40Ca, 58,60,62,64Ni, and 64Zn.



In flat scale, the analyzing powers associated with 40 MeV proton scattering are compared with the experimental data from 12C, 24Mg, 40Ca, 58,60,62,64Ni and 64Zn in Fig. 4.21, and  from 66,68Zn, 90,92Zr, 116,118,120,122,124Sn and 208Pb in Fig. 4.22.
           
Data were measured at 40 MeV for 208Pb, 90Zr [97], and  92Zr [163], and at 39.6 MeV for  66,68Zn [162] and  116,118,120,122,124Sn [164].



Figure 4.22: As for Fig. 4.21 but for 66,68Zn, 90,92Zr, 116,118,120,122,124Sn and 208Pb.



From these two figures, it is clear that the predictions and data from all targets at 40 MeV are in almost as good agreement as found [75, 88] with studies at  65 and 200 MeV. However the 208Pb results are slightly at odds with observation; a feature considered again being due to inadequacy of the assumed target structure. 

However, the degree of compression of the 208Pb data (from analyzing power peak sizes of ± 1) now compares quite well that predicted.  It is the mismatch of the angle values at which the zeroes occur that noted as possible evidence of the inadequacy of the simple packed orbit model of structure that has been used.


The energy variation of the scattering data

The energy variations of cross section and analyzing power data and of the results obtained using g-folding optical potentials, for proton elastic scattering from 40Ca at 25 and 40 MeV and from 58Ni at 30 and 40 MeV, are shown in Fig. 4.23. Those for 90Zr, 120Sn, and 208Pb for 30 and 40 MeV protons are given in Fig. 4.24.
           
In Fig. 4.23, the 25 and 40 MeV results are shown for 40Ca by the solid dots and curves while the 40 MeV data is presented by the open dots with calculated results displayed by the dashed curves.
           
With the 58Ni results, the notation differs only in that the energies are 30.3 and 40 MeV. It is evident that the general pattern of change in both the cross section and analyzing power data from these nuclei is reproduced with the calculations; the more so with 58Ni.
           
In the 40Ca case, the 25 MeV results are most at odds with observation. Of particular note in the cross sections is that the positions (and in case of 58Ni particularly the peak sizes) are correctly found.
           
That is also the case with the analyzing powers. The calculated results for 58Ni reproduce the positions and size variations of the maxima in that data very well. Such is evident with the results given in Fig. 4.24, with magnitudes and positions of the peaks in the cross sections from 90Zr and 120Sn quite well predicted.  Those in the cross section data from 208Pb are slightly shifted at both energies.


Figure 4.23: Energy variation of the differential cross sections and analyzing powers for proton scattering for 25 and 40 MeV from 40Ca (top) and for 30.3 and 40 MeV from 58Ni (bottom).



The analyzing power data trends are well followed also, particularly the relative size changes of the data with energy to 60o in the case of 120Sn. As with the cross section comparisons, the calculated 208Pb analyzing power mismatch observation; the calculated maxima at both energies occurring at slightly larger scattering angles than is seen in the data. But the general variation of sizes of the analyzing power data at the two energies is evident with the calculations.


Figure 4.24: As for Fig. 4.11 but for the scattering of 30 and 40 MeV protons from 90Zr (top), 120Sn (middle), and 208Pb (bottom).



The problem with the 25 MeV 40Ca results could be attributed to effects such as virtual excitation of giant resonances, given that the giant dipole excitation is near 25 MeV in mass 40 nuclei [165]. At 30 MeV for the other nuclei, and 40 MeV for all five nuclei considered, such competing processes in elastic scattering are not favored. Since the discrepancies between data and results found for 208Pb appear constant with energy, it seems that my choice of (simple shell) model for the structure of the nucleus is poor. This has been considered specifically later.



The isotope variation of the scattering data

Variations with isotope of the target nucleus of data and calculated results are presented in Figs. 4.25, 4.26, and 4.27.



 
Figure 4.25: Differential cross sections from the elastic scattering of 25 MeV protons from 40,42,44,48Ca (top), and of 30.3 MeV protons from 112,116,120,122,124Sn (bottom).


In Fig. 4.25, the cross sections for 25 MeV proton scattering from 40,42,44,48Ca are shown in the top segments while those from 30.3 MeV protons scattering from 112,116,120,122,124Sn are given in the bottom panels. In both cases the data are shown in the left hand sectors with lines drawn through them to guide the eye, while the calculated cross sections are presented on the right.




Figure 4.26: Differential cross sections (top) and analyzing powers (bottom) from the elastic scattering of 40 MeV protons from 58,60,62,64Ni.


With 40Ca, the calculated result is not in as good agreement with the data as it has been found in almost all other cases. But these results demonstrate that the trend with mass is viable. It is expected that any competing process, e.g. virtual excitation of giant resonances, would be similar for all of these calcium isotopes at 25 MeV.



Figure 4:27: As for Fig. 4.26 but for the elastic scattering of 39.6 MeV protons from 116,118,120,122,124Sn.


With the Tin isotopes, the trend with increasing neutron number seen in data is reflected in the calculated results with only the 112Sn result being slightly out in angular form.
           
In Fig. 4.26, the 40 MeV cross sections and analyzing powers for the Nickel isotopes, 58,60,62,64Ni, are shown. Again the data with lines drawn to guide the eye are given in the left panels while the results of calculations are shown on the right.
           
Albeit that thecalculated results have more sharply defined structure than the data, they do show the mass variation trend of the data and now with very reasonable peak values in both cross sections and analyzing powers.
           
That is also the case with 39.6 MeV proton scattering from the tin isotopes, 116,118,120,122,124Sn, as is evident in Fig. 4.27. Again the data are given in the left panels and the calculated results in the right side ones.


Conclusions

The cross section and analyzing power results obtained from the coordinate space nonlocal optical potentials formed by g folding at 25, 30, and 40 MeV are in quite reasonable agreement with the data obtained with targets of mass 6 to 238; the 40 MeV results the more so. In general the cross section predictions give the magnitudes and trends of the peaks in the data but the minima often are too sharply defined.  The comparisons between the calculated results and the data for 25 and 30 MeV proton elastic scattering remain reasonable but the disparities are more pronounced than at higher energies [7].  Nevertheless, the g folding optical potentials remain a credible first approximation, sufficiently so that the results may still select between different structure inputs. Also the associated distorted wave functions and effective interactions still should be appropriate for use in DWA analyses of inelastic scattering from stable nuclei [123], or of radioactive beam ions [121], as well as of other reaction calculations [166].

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