Physics of Nucleon-Nucleus Scattering - Chapter IV
25 to 40 MeV
PROTON ELASTIC SCATTERING
Using a fully microscopic, coordinate
spaceoptical model, successful analyses have been made of
proton-nucleus (pA) elastic scattering data taken at 65 and200 MeV [75, 88] and from targets of diverse mass
and elastic and inelastic p-12Cscattering data at 200 MeV have been
understood . 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 (6Lito 238U) are presented and made
using coordinate spaceoptical potentials formed by g
folding.A select few of the results
presented in this chapter have been used in a brief report , 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-foldingoptical potentials are required to define the
distorted waves in `no parameter' DWAanalyses 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
 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  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 potentialsufficed to give the bulk of the (elastic)
scattering results in that past study . 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 interactionone should use in the g-foldingprocess. 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
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 differentialcross 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,
Results of Calculations
The results of calculations of the
elastic scattering of 25, 30, and 40 MeV protons frommany 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
In most of the cases the HO functions are
used for the bound state single particle functions. But for light nuclei, 6Heand 6Liin 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
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 sectionsare 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 modelcalculations 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, 27Aland 28Siare compared with the experimental data. Data
were measured at 25.9 MeV for 6Li, at 24.4 MeV for 7Li , at 24 MeV for 12C
, at 26 MeV for 14N , at 24.5 MeV
for 18O , at 27 MeV for 24Mg , at 28 MeV for 27Al
, and at 25 MeV for 28Si
Figure 4.2: The differential cross
sectionsfrom the elastic scattering of 25 MeV protons
from 6,7Li, 12C, 14N, 18O,
24Mg, 27Aland 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  and40,42,44,48Ca , at 28 MeV for 58Cu , and at 24.6 MeV for 86,88Sr
The 32S, 40,42,44,48Ca
and 58Cu results agree reasonably with observation although mypredictions 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 for208Pbnucleus are also presented in this figure, but
no data at this energy is available.
Data were measured at 25 MeV for89Y ,92Mo  and 152Sm
, and at 26 MeV for 232Th
and 238U .
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 powerdata compared with the optical modelcalculations for targets mass from 12 to 152.
Target mass identifies each result.
The 25 MeV elastic proton scattering analyzing
powerdata are compared with the results obtained
from my optical modelcalculations 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, at 24 MeV for 18O
, at 25 MeV for 24Mg, 28Si, and 32S , at 24.6 MeV
for 86,88Sr , and at 24.5 MeV for 118Sn
 and 152Sm
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
Results of the optical modelcalculations 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 modelcalculations for target mass 9 to 209. The
target mass identifies each result.
In Fig. 4.7, differential cross sectionsof 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
sectionsfrom the elastic scattering of 30 MeV protons
from 9Be, 10,11B, 13C, 16O, 19F, and 20,21Ne.
In Fig. 4.8, the differential cross
sectionsfrom 9Be, 10,11B, 13C, 16O, 19Fand 20,21Neare compared in flat perspective with the
Differential cross section data were measured
[141, 142] at 30.3
MeVfor 9Be, 10,11B, 16O, 19Fand 20,21Ne. For13C, data were taken at 30.5 MeV . 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, 63Cuand 64Zn.
In Fig. 4.9, the differential cross
sectionshave been calculated for scattering from 40Ar, 54,56Fe, 58,60Ni, 59Co, 63Cuand 64Znare displayed and compared with the relevant
Data were measured at 30 MeV for 40Ar and 63Cu, at 30.3 MeV for
56Fe,58,60Niand59Co, at 30.4 MeV
for 54Fe  and at 30.5 MeV
for 64Zn. 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
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, 112,114,116Snare compared with the respective differential
cross section data. Data were measured at 30 MeV for 65Cu  and 90Zr
, at 29 MeV for 104Ru , at 30.5 MeV for 66,68Zn
, and at 30.4
MeV for 112,114,116Sn .
For the 65Cuand 66,68Zncases, the first order minima are
underestimated and the higher order minima are overestimated by my
calculations. The shapes are still well reproduced.
The 90Zrdata 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, 208Pband 209Bi are compared with the
respective differential cross section data, which have been measured at 30.3
MeV for 120Sn and 208Pb , at 30.4 MeV
for118,122,124Sn , at 29.32 MeV for 139La
and 141Pr , at 30 MeV for 144Sm
 and 176Yb
 and at 31 MeV
for 209Bi .
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, 176Yband 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
Figure 4.12: The 30 MeV elastic proton
scattering analyzing powerdata compared with the optical modelcalculations 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
Figure 4.13: The analyzing powers from
the elastic scattering of 30 MeV protons from 9Be,10,11B, 13C, 16O, 40Ar, 40Ca, and54Fe.
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 9Bethrough 54Fe are shown in Fig.
4.13. Data were measured at 30.3 MeV for 9Be ,10,11B, and 16O, at 30.4 MeV
for 13C and 54Fe , at 30 MeV for 40Ar and at 29 MeV
for 40Ca. 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, 63,65Cuand 64,68Zn.
That improved agreement is retained with
target mass 56 through 68 as shown in Fig. 4.14, in whichpredicted analyzing powers for 56Fe,
58,60Ni, 59Co, 63,65Cuand 64,68Zn are compared with the
Data were measured at 30.3 MeV for 56Fe
, at 29 MeV for 58,60Niand 59Co, at 30 MeV for 63,65Cu and at 30.5 MeV
for 64,68Zn . 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,
In Fig. 4.15, the calculated analyzing
powers from 90Zr, 92,100Mo, 120Sn,
176Yband 208Pbare compared with the experimental data. Data
were measured at 30 MeV for 90Zr  and 176Yb , at 30.3 MeV
for92,100Mo , and at 29 MeV
for 120Sn and 208Pb . The quality of
the fits to data with 90Zr through 120Sn is good while
the predictions for 208Pb in particular underestimate the observed
In summary,mycalculated analyzing powers show the trend of the data for the lightest
mass targetsandquite good agreement is found for targets
ranging from16Oto 58Ni. That agreement remains
with the data from 65Cuto 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 176Yband 208Pbanalyzing powers are matched in location
but the peak magnitudes are at best half
what is observed.
Results of the Scattering of 40
The results obtained from the optical modelcalculations 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
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, 27Aland 28Si, are compared withexperimental data that were measured at 40
MeV for 6Li , 12C
, 24Mg , 27Al
and 28Si , at 39.8 MeV
for 15N  and at 39.7 MeV for 16O
. For 6He, data
wereobtained  by inverse
kinematics for an experiment in which a radioactive beam of 40.9A MeV 6He
scattered froma 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 modelcalculations for target mass 6 to 208. The
target mass identifies each result.
Figure 4.17: The differential cross
sectionsfrom the elastic scattering of 40 MeVprotons
from 6He, 6Li, 12C, 15N, 16O, 24Mg, 27Al, and28Si.
The results found for 40 MeV proton
scattering from 40Ca, 58,60,62,64Niand 64,66,68Znare compared with the data in Fig. 4.18. Data
were measured at 40 MeV for 40Ca and 58Ni , and at 39.6 MeV for60,62,64Ni and 64,66,68Zn
. 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,
116,118,120,122,124Snand 208Pbare compared with the data.Thatdata were measured at 40 MeV for 90Zr, 92Zr  and 208Pb , and at 39.6 MeV for116,118,120,122,124Sn .
Such quality of results of predictions
were found previously for higher energy studies, and with 65 MeV in particular . The case of 208Pbis 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 powerdata compared with the optical modelcalculations 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, 58,60,62,64Niand 64Znin Fig. 4.21, andfrom 66,68Zn, 90,92Zr, 116,118,120,122,124Snand 208Pbin Fig. 4.22.
Data were measured at 40 MeV for 208Pb, 90Zr, and92Zr , and at 39.6 MeV for66,68Zn and116,118,120,122,124Sn.
Figure 4.22: As for Fig. 4.21 but for 66,68Zn, 90,92Zr, 116,118,120,122,124Snand 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 at65 and 200 MeV. However the 208Pbresults are slightly at odds with observation;
a feature considered again being due to inadequacy of the assumed target
However, the degree of compression of
the 208Pbdata (from analyzing powerpeak 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
The energy variation of the
The energy variations of cross section
and analyzing powerdata and of the results obtained using
g-foldingoptical potentials, for proton elastic
scattering from 40Caat 25 and 40 MeV and from 58Ni at
30 and 40 MeV, are shown in Fig. 4.23. Those for 90Zr, 120Sn, and 208Pbfor 30 and 40 MeV protons are given in Fig.
In Fig. 4.23, the 25 and 40 MeV results
are shown for 40Caby the solid dots and curves while the 40 MeV
data is presented by the open dots with calculated results displayed by the
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
powerdata from these nuclei is reproduced with the
calculations; the more so with 58Ni.
In the 40Cacase, 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
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 90Zrand 120Sn quite well
predicted.Those in the cross section
data from 208Pbare slightly shifted at both energies.
Figure 4.23: Energy variation of the
differential cross sectionsand analyzing powers for proton scattering for
25 and 40 MeV from 40Ca(top) and for 30.3 and 40 MeV from 58Ni
powerdata 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 208Pbanalyzing 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 40Caresults 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 . 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 208Pbappear 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
Variations with isotope of the target
nucleus of data and calculated results are presented in Figs. 4.25, 4.26, and
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.
The cross section and analyzing powerresults obtained from the coordinate spacenonlocaloptical 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 .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 DWAanalyses of inelastic scattering from stable
nuclei , or of radioactive beam ions , as well as of other reaction calculations