Thursday 22 August 2019
Physics of Nucleon-Nucleus Scattering - Chapter III
ANALYSES OF ELASTIC p-12C SCATTERING
Elastic scattering, the predominant event associated with the interactions of nucleons with nuclei, has been studied extensively over many decades. From experiment there now exists a vast data base. Extensive theoretical studies of such scattering also have been made and based upon inverse scattering theory, of global  and of numerical inversion form [8, 84], as well as on direct scattering theory. In the latter, optical potentials as specified in the preceding chapter have been determined in terms of underlying two-nucleon NN interactions. Note though such direct scattering theory optical potentials have been formulated in momentum space [85, 86] and also in coordinate space [7, 75, 87, 88].
Herein facets of the coordinate space formulation of optical potentials are investigated for the scattering of protons from 12C primarily to identify what NN interaction characteristics are required to specify appropriate optical potentials and what effect the attendant nonlocality of those potentials have. An appropriate optical potential is taken as one with which successful predictions of proton-nucleus (NA) elastic scattering data over a large range of energy is obtained. To this end, and as specified previously, complex, nonlocal optical potentials have been obtained by folding effective two-nucleon (NN) interactions with a ground state wave function from a large space shell model structure of 12C. The effective interactions have been derived from complex NN interactions which fit (complex) NN scattering phase shifts  to over 800 MeV with application to proton-12C scattering in mind.
With this approach and at energies of 65 and 200 MeV, successful predictions of observables from elastic proton-nucleus (pA) scattering from many nuclei have been made [75, 88]. But to do so, the inhomogeneous partial wave Schrodinger equations specified by the complex, nonlocal and energy dependent optical potentials had to be solved without any localization approximation. Medium effects of Pauli blocking and of the average fields in which the projectile and struck nucleon propagate had to be taken into account in specification of the effective NN interactions also.The process in which all of the features are included without further approximation is defined hereafter as g-folding.
To make predictions of NA scattering under the basic assumption that the scattering is due essentially to just pairwise interactions between the projectile and each and every nucleon in the target, three basic aspects of the system under investigation must be specified. Where possible those attributes should be defined independent of the pA scattering system being studied. First, the description of the nucleus, and in particular the OBDME, should be determined from a large scale structure calculation which describes well the ground state properties of the nucleus in question. This information in combination with the second ingredient, the single particle (bound state) wave functions, can be assessed further by the level of agreement their use gives in predicting elastic electron scattering form factors. Since these wave functions are energy independent, they should not be varied in further application such as seeking better reproduction of specific sets of pA scattering data. Of course the neutron distributions are not so well identified and this problem is considered later in an analysis of data from proton scattering from 208Pb. The final ingredient is a complex, energy and density dependent, effective NN interaction that describes the interaction between the incident and each and every struck nucleon.
For incident energies to 800 MeV, elastic scattering may be described by optical potentials though it has been suggested  that they should be formed by folding relativistic density dependent effective interactions (Lorentz invariant amplitudes) with relativistic nuclear structure wave functions. However, based upon the success of using g-folding (nonrelativistic scattering) optical potentials to analyze elastic scattering of 65 and 200 MeV protons from targets ranging 3He to 238U [75, 88], it has been considered herein just what may be achieved to 800 MeVwith that approach and allowing minimal relativity by using relativistic kinematics. The main interest is in what net effects arise when one uses an effective interaction that is linked to a good description of the NN scattering phase shifts. With this aim it is important to note that I make predictions to compare with the scattering data. All details entering the folding process were set a priori and single calculations made to specify the differential cross sections and analyzing powers.
The structure information for 12C
With most studies needing the nucleon based properties of 12C, 0p- or at best 0p 1s 0d-shell model calculations [90, 91] of the structure have been used, although they are known to be limited. Such models predict a spectrum with which large effective charges are needed to map measured electromagnetic transition rates. That is not the case now with current larger space calculations of structure. With a complete
model space, the positive
parity states of 12C have been specified while the negative parity
spectrum was found also but by using a
model space . All known states to 20 MeV
excitation were matched by candidates from this structure model and to within 2
MeV. Indeed use of this spectroscopy in
an analysis  of 200 and 398 MeV proton
inelastic scattering cross sections and analyzing powers permitted an
values for states in 12C that hitherto had
uncertain assignments. As a complete basis was used (for the
case at least) there is no spurious center of mass facet in
the state specifications. The ground state wave functions of that large space
shell model were used in the calculations whose results are presented herein.
In forming the optical potentials, besides the OBDME one needs single nucleon bound state wave functions. Frequently they have been chosen as harmonic oscillator (HO) wave functions but, and as seen previously  for 12C specifically, a more realistic representation is to use Woods-Saxon (WS) bound state wave functions. With the OBDME determined from the
shell model wave functions and the single nucleon bound
states appropriately specified, electron scattering form factors from both the
elastic and inelastic scattering of electrons from 12C were well fit
The effective NN interactions for 12C
As discussed previously, the effective interaction between two nucleons, one the projectile and the other a bound particle in the nucleus, is required in coordinate space and as a combination of central, tensor and two-body spin-orbit terms. Each term can have a linear combination of Yukawa functions as its form factor, and the (complex) strengths of those Yukawa functions may vary with the density of the nuclear medium.These effective interactions have been defined by optimally mapping [42, 93] to half-off-shell (momentum space) NN g-matrix elements determined from solutions of the Brueckner-Bethe-Goldstone (BBG) equations.
A realistic microscopic model of pA reactions is one that is based upon NN interactions whose on-shell values of t-matrices (solutions of the Lippmann-Schwinger equations) are consistent with measured NN scattering data to and above the incident energies of interest. Below pion threshold, the phenomenology of the NN interaction is relatively simple, and several one boson exchange potential (OBEP) models [21, 33, 35, 94, 95] exist with which very good fits have been found to NN phase shift data. Above pion threshold, that is no longer the case. Inelastic channels open and resonance scattering occurs. Simple potentials must be varied to account for the various meson production thresholds and also to account for effects of known [P33(1232) (∆) and P11(1440) (N*)] resonance structures in the NN system. There exist extensions to OBEP models which incorporate resonance  and particle production [36, 96], and with which some NN and NNπ data up to 1 GeV may be explained. The NN phase shifts above pion threshold found with these models are better than any from standard OBEP but as yet they are not adequate in a number of important channels. However the characteristics of the experimental NN scattering amplitudes to 2.5 GeV are consistent with an (NN) optical potential concept. The SM97 data  has been interpreted  very well by a basic OBEP supplemented by sensible complex optical potentials. With the OBEP component established by the fits its use gave with data below 300 MeV, the supplementing NN optical potentials reflected the effects of the P33(1232) and P11(1440) resonances in several partial waves. Otherwise those NN optical potentials are smooth, complex, energy dependent, and short ranged; consistent with the view of production processes being localised at and within the confinement surface of a nucleon. Those NN optical potentials also are consistent with the geometry of the profile function as it is known from analyses of high energy diffraction scattering. The optical potential approach  is used for the NN interaction for energies 300 to 800 MeV to define the effective interactions required in the g-folding process to give NA optical potentials. Specifically the coupled channel Bonn (BCC3) interaction [21, 33, 94] are used supplemented with complex, short ranged Gaussian potentials . Those Gaussians were parametrized so that the t-matrices of the modified BCC3 force on the energy shell match precisely the SM97 data sets. The one solitary boson exchange potential (OSBEP) [35, 95] are also used as the bare NN starting interaction.
This simple model prescription encompasses a plethora of terms that will be needed to adapt a more fundamental boson exchange model approach to adequately explain the NN scattering data to 1 GeV and higher. Not only do such boson model calculations increase in complexity with energy but also the number of adjustable parameters involved increase with every additional element incorporated in the theory, making the approach a less appealing way to treat NN scattering. Nevertheless it is a goal worth seeking and if, in the fullness of time, such a boson exchange model prescription can be found that meets the requirement that the empirical NN t matrices are reproduced then it would be appropriate for these studies. But such is not the case at present.
Nevertheless, use of an extending complex potential to adapt standard (below threshold) NN prescriptions so that the free NN scattering data to over 1 GeV are reproduced means that any specific effect of the ∆ and N* in specification of the NN g matrices will not be treated. Medium effects upon the ∆ propagation could differ from those set with the complex potential factors defining the effective g matrices. However, Ray  has studied the effects of including ∆ and N* Pauli blocking in a momentum space formulation of NN and NA scattering. He concludes that such variation results in only small changes in density dependent amplitudes at central nuclear densities (~ 1.4 fm-1). Ray states specifically ``The lack of sensitivity to ∆ blocking supports the Paris-Hamburg effective interaction model which does not treat virtual ∆ propagation explicitly". It is worthwhile noting that Ray observed that predicted NA elastic scattering observables were not very sensitive to the poorly known isobar-nucleus Interaction potentials.
Results of calculations
The results of proton-12C elastic scattering calculations obtained with the theoretically derived optical potentials are compared with data in this section. Such comparisons (of differential cross sections and analyzing powers) are made with data taken at 18 energies in the range 40 to 800 MeV. Specifically the data considered here taken at 40 MeV , 50 MeV , 65 MeV , 120 MeV , 135 MeV [101, 102], 144 MeV , 156 MeV , 160 MeV , 185 MeV , 200 MeV , 250 MeV , 300 MeV [109, 110], 318 MeV , 398, 597 and 698 MeV [112-114], 500 MeV , and at 800 MeV  for which medium modified, complex, effective interactions determined in the manner described earlier have been used to define the relevant proton-12C optical potentials. At each energy the OBDME specified by the
shell model wave
function and bound states are used with which good electron scattering form
factors were found. Thus there is no adjustable parameter considered with the
use of DWBA98 in finding cross sections and analyzing
powers. The solid curves in all figures shown in this section are the results
of making a single calculation in each case.The results of calculations made
with the full nonlocal g-folding optical potential defined for brevity hereafter as complete.
In Figs. 3.1 and 3.2 the results of complete calculations of the differential cross sections from proton-12C elastic scattering are compared with data.
In Fig. 3.1 the results for 40 to 250 MeV are displayed while in Fig. 3.2 the higher energies from pion threshold are given.
Figure 3.1: Differential cross sections from the elastic scattering of 40 to 250 MeV protons from 12C. The solid curves display our predictions obtained from single calculations with the complete nonlocal optical potentials formed by g-folding.
Figure 3.2: As for Fig. 3.1 but for the elastic scattering of 300 to 800 MeV protons from 12C.
The individual energies are listed with each set and clearly the energy variation of the cross sections is reproduced quite well. The associated analyzing powers are displayed in Figs. 3.3 and 3.4.
In the first of those, with energies below pion threshold, the data and its energy trend are quite well fit by the results of my complete calculations. Some improvement is needed at energies near 135 MeV, but the overall pattern of the results is very good. For energies 300 MeV and higher, the trend of the analyzing power data again is reflected in the results of my calculations but there is some mismatch in the magnitude of the analyzing powers at forward angles and most evidently with 400 MeV.
Figure 3.3: As for Fig. 3.1, but for the analyzing powers from the elastic scattering of 40 to 250 MeV protons from 12C.
The DWBA98 code allows one to make calculations also with the integral term of the nonlocal Schrodinger equation omitted. This corresponds to ignoring the exchange effects due to antisymmetrization of the A+1 scattering state. While that is a form of a gr approximation, it is far more severe than any localization procedure of coordinate space optical potentials  or of the approximations leading to the gr and tr models in momentum space [85, 86]. However, the purpose is not to find a best fit to data with the local term as that requires scaling and/or profile adjustment. Rather these approximate calculations are made to define just how large is the effect of the specific nonlocal component in the coordinate space optical potential.
Fig. 3.4: As for Fig. 3.1 but for the analyzing powers from the elastic scattering of 318 to 800 MeV protons from 12C.
It is substantial as will be seen at most energies and so, in most cases, localization or any approximation for nonlocality must be recognised as a representation of a large effect.
In Figs. 3.5 and 3.6 the results of calculations of the differential cross sections from proton-12C elastic scattering made using local direct potential (no exchange) are compared with data.
Fig. 3.5: Differential cross sections from the elastic scattering of 40 to 250 MeV protons from 12C. The solid curves display predictions obtained from single calculations with the local direct optical potentials.
The associated analyzing powers are presented in Figs. 3.7 and 3.8. Clearly predictions made ignoring exchange amplitudes in the scattering are quite disparate from those obtained using full folding nonlocal optical potentials. The essential difference between these results arises from the destructive interference between the direct and exchange amplitudes. Specifics will be considered shortly.
Fig. 3.6: As for Fig. 3.5, but for the differential cross sections from the elastic scattering of 318 to 800 MeV protons from 12C.
Fig. 3.7: As for Fig. 3.5, but for the analyzing powers from the elastic scattering of 40 to 250 MeV protons from 12C.
Fig. 3.8: As for Fig. 3.5, but for the analyzing powers from the elastic scattering of 318 to 800 MeV protons from 12C.
Other spin observables have been measured at 65 and 500 MeV. Those data and the predictions are compared in Fig. 3.9 where the 65 MeV spin rotation, R(θ), and the 500 MeV DSS(θ) and DSL(θ), are displayed in the top, middle and bottom segments respectively. Predictions agree as well with these data as they do with the analyzing powers at each energy.
Fig. 3.9: The spin rotation R(θ) from the elastic scattering of 65 MeV protons from 12C (top), and the observables, DSS(θ) (middle) and DSL(θ) (bottom) from the elastic scattering of 500 MeV protons from 12C.
These plots suffice to give credence to the optical potentials formed by g-folding at least as a good first order guess at the proton-12C interaction. Each of the energy results are discussed in more detail now.
The results of omitting exchange (knock-out) effects in the specification of the coordinate space optical potentials are displayed in the next set of figures by the dashed curves and are defined hereafter as no-exchange results.
In the ensuing set of figures, for energies above pion threshold (~ 300 MeV) data and results are presented to 40o scattering angle. There is little if any data at larger scattering angles. At lower energies studies are limited to 80o as by then the cross sections data are so small in magnitude that calculated results may be overly sensitive to small changes in phase shifts.
The results of calculations are compared with the 40 and 50 MeV data in Fig. 3.10.
Fig.3.10: Differential cross sections (top) and analyzing powers (bottom) from the elastic scattering of 40 and 50 MeV protons from 12C. The solid curves display predictions obtained from single calculations with the complete nonlocal optical potentials formed by g-folding. The dashed curves are the results obtained when exchange effects are ignored therein. The data were taken from Refs.  and .
The complete results are in good agreement with data but those from the no-exchange calculations are not. The complete calculations give especially good results for the forward angle cross sections (to ~ 40o). At the larger scattering angles the predictions have slightly more defined minima than evident with the data.
The analyzing powers at these energies are shown in the bottom segments of Fig. 3.10, and while complete calulations are quite reasonable in comparison with the data, there is room for improvement in those fits. That is the case for almost all of the energies studied and indicates that details of the scattering theory relating to the effective interactions currently used needs to be improved. Of course, as the analyzing powers are measures of differences between scattering probabilities and are normalized by the differential cross sections, predictions of analyzing powers are very sensitive to such details.
Fig. 3.11: As for Fig. 3.10 but for energies of 65 and 120 MeV. The data were taken from Refs.  and .
With both the cross sections and analyzing powers however, it is clear that exchange effects cannot be ignored in making any analysis. The exchange amplitudes destructively interfere with the direct scattering amplitudes to markedly change the shapes of the predictions as well as to reduce considerably the size of the calculated cross sections at larger scattering angles.
The results found for 65 and 120 MeV proton scattering from 12C are compared with the data in Fig.3.11. Again only the results found using the complete calculations match observation. As with the lower energies, while those predicted cross sections are in good agreement with the data there are slight differences in structure to what is observed at the larger scattering angles. The analyzing powers again are reasonable results in comparison to the data although the 120 MeV predictions do not give the relative sizes or exact angle locations of the peak values in the data. However, compared with the results obtained with the no-exchange potentials, the complete calculations are excellent fits to data. The cross sections found using the no-exchange local potentials are much too large and have the wrong fall off with scattering angle. The analyzing powers so calculated are completely at variance with observation.
In Fig.3.12, results equivalent to those discussed above are compared with the data taken with 135 and 144 MeV protons. Clearly the exchange amplitude contributions are extremely important. Without them the calculated cross sections become increasingly in disagreement with the measured data and by orders of magnitude with increasing scattering angle.
Likewise the analyzing powers from the no-exchange local potential calculations are totally wrong. The complete calculations are in stark contrast giving as good replications of the cross section data as found at the lower energies discussed above, but the analyzing powers need improvement. The 135 MeV analyzing power result, like the 120 MeV case, does not agree with the structure seen in the data, notably being too small compared to the forward scattering positive peak value and not giving the correct angle values for the maxima. The data at 144 MeV are too limited in momentum transfer but one can suspect that new data would reflect what has been found at 135 MeV.
Fig. 3.12: As for Fig. 3.10 but for energies of 135 and 144 MeV. The data were taken from Refs. [101, 102] and .
My predictions of proton-12C elastic scattering for 156 and 160 MeV, for 185 and 200 MeV, and for 250 and 300 MeV are compared with the data in Figs. 3.13, 3.14, and 3.15 respectively. In all cases it is very evident that omission of the exchange amplitudes is a serious problem in analyses.
Fig. 3.13: As for Fig. 3.10 but for energies of 156 and 160 MeV. The data were taken from Refs.  and .
Fig. 3.14: As for Fig. 3.10 but for energies of 185 and 200 MeV. The data were taken from Refs.  and .
Fig. 3.15: As for Fig. 3.10 but for energies of 250 and 300 MeV. The data were taken from Refs. , , and .
The complete calculations give cross sections in very good agreement with the data, albeit that for 300 MeV the resultant fit is not as good as those at all lower energies. But it is with the analyzing powers that results are better than have been found for the lower energies. With all six energies in these three figures, the locations of maxima in the observed analyzing powers, both positive and negative values, are correctly predicted. Also the magnitudes of the peaks in the analyzing powers are better reproduced, with the calculated value of the first positive peak increasing first to be almost complete asymmetry (1.0) and in agreement with the data at 200 MeV while slowly decreasing from that at the higher energies. In fact the data values of that peak decrease more rapidly than do the calculated values.
Fig. 3.16: As for Fig.3.10 but for energies of 318 and 400 MeV. The data were taken from Refs.  and [112-114].
Fig. 3.17: As for Fig. 3.10 but for energies of 500 and 600 MeV. The data were taken from Refs. [112-114] and .
Fig. 3.18: As for Fig. 3.10 but for energies of 700 and 800 MeV. The data were taken from Refs. [112-114] and .
The results of calculations for proton energies at and above pion threshold are shown in Figs. 3.16, 3.17, and 3.18 for the energy pairs 318 and 400 MeV, 500 and 600 MeV, and 700 and 800 MeV respectively. In these cases, the effective NN interactions have been formed by supplementing the bare BCC3 interaction with complex short ranged Gaussian NN optical potentials with strengths set to ensure a match with the NN scattering phase shifts at each relevant energy. Again the results found by using the complete (g-folding) optical potentials and the no-exchange local ones are displayed by the solid and dashed curves in each figure. At 318 MeV, as is evident in Fig. 3.16, the complete calculation gives a very good fit to the cross section data and to the analyzing power data, save that the forward peak valueis overpredicted.
The no-exchange potential calculation results are at great odds with the data as they were at the lower energies. But at 400 MeV and higher, the influence of the exchange amplitudes diminishes in size and quite rapidly with energy. The complete and no-exchange calculation results still differ however with the results found with complete calculations giving better agreement with observation.
In Fig. 3.16, the results of my 400 MeV calculations are compared with the data taken at 398 MeV. Of all results these give the poorest fit to data. The complete calculation nevertheless gives a cross section that agrees with the data to about 20o scattering angle, but the sharp minima observed is not reproduced. Likewise the gross features of the 398 MeV analyzing power data are reproduced, but the forward peak is overpredicted.
The results of 500 and 600 MeV calculations are compared with 500 and 597 MeV data in Fig. 3.17. While the results from the calculations made using the no-exchange local potentials now do not differ greatly from the complete calculation results, the latter remain the better predictions of the actual data. The differential cross sections are reproduced with the sharp minima at about 20o being matched quite well. Likewise the predictions of the analyzing powers are fits to the data that improve upon the results found at 400 MeV. The forward peak in the analyzing power is still overpredicted but the trend with energy of that peak decreasing in size is found. The 500 MeV analyzing power results at scattering angles larger than ~ 20o are not well reproduced, but this is where the cross section is small (order of a few tenths of a mb/sr) and so is sensitive to small details in the optical potential calculations.
Next, in Fig. 3.18, the complete and no-exchange local optical potential calculation results are compared with data for 700 and 800 MeV scattering. The cross sections are well fit by the results of the complete calculations with the no-exchange results being almost as good. The analyzing powers are reasonably well reproduced by both calculations with the calculated forward angle positive peak now in quite good agreement with the data, as are the analyzing power results for those scattering angles at which the cross sections are larger than a few tenths of a mb/sr.
Fig. 3.19: Differential cross sections (top) and analyzing powers (bottom) for 800 MeV proton scattering from 12C. In the left panel the data  are compared with predictions made using effective interactions based upon the bare OSBEP and BCC3 g--matrices (solid and dashed curves). In the right panel the data are compared with predictions obtained from folding the modified OSBEP effective interactions. The solid curves portray predictions with the complete optical potential. The dashed curves are the results when nonlocality is removed (the no-exchange approximation).
Other predictions of 800 MeV proton-12C elastic scattering are compared with data in Fig. 3.19. The differential cross sections are given in the top segments while the analyzing powers are shown in the bottom segment of this figure. In the left hand side panels the data are compared with the results of calculations made using the density dependent effective interactions obtained from the g matrices associated with the bare OSBEP [35, 95] and BCC3 [21, 33, 94] model potentials. Those results are portrayed by the solid and dashed curves respectively. In the right hand side panel the same data are compared with my predictions with and without the nonlocality. The solid and dashed curves in these cases depict the results have been found by using the modified OSBEP interaction. Neither the OSBEP nor the BCC3 interactions by themselves fit well the observed on--shell NN data at 800 MeV.
This is again evident from the comparisons given in left hand panel of Fig. 3.19 as neither lead to pA optical potentials, and thence scattering phase shifts from complete calculations, with which one can describe the observed proton-12C data. The (complex) nature of the BCC3 interaction improves the situation in comparison to use of the purely real OSBEP force, but not sufficiently to explain the proton-12C data.
In rather stark contrast, using the NN interactions modified by NN optical potentials, proton-12C optical potentials are obtained with which the cross section and analyzing power data are quite well reproduced. Most noticeable though is that the analyzing powers now are predicted well, and especially at forward scattering angles.
The differential cross section data of 800 MeV protons scattering from 12C have been very well reproduced to scattering angles of 25o by which the magnitudes have fallen to less than 0.1 mb/sr. The effects of the modulation of both the OSBEP and the BCC3 models are very noticeable. That is even more the case with the analyzing powers. Only with the modulations that tune OSBEP and BCC3 against the SM97 data set has satisfactory reproduction of that analyzing power structure been found. Specific medium effects in the effective NN interaction do not seem very important at 800 MeV, although such have been included in all of my analyses.
Fully microscopic model calculations of coordinate space optical potentials describing proton-12C elastic scattering at 18 energies in the range from 40 to 800 MeV have been made. Both differential cross section and analyzing power data have been analyzed at all energies considered. The complex optical potentials were formed by folding effective NN interactions with the density matrices of the ground state of 12C. A complete
calculation provided those density matrices. The effective interactions were
obtained by mapping half-off-shell NN g matrices (solutions of BBG equations) associated with Bonn potentials supplemented above pion threshold
by short ranged Gaussian NN optical potentials so that all NN
scattering phase shifts to over 1 GeV were reproduced. The results of the g-folding process are complex, nonlocal proton--12C interactions. Solution
of the integro-differential Schrodinger equations formed with those optical
potentials resulted in good to excellent fits to elastic scattering data at all
energies, both to the cross sections and to the analyzing powers.
The results confirm the large effect of the (knock-out) exchange amplitude in the elastic scattering process and which make the coordinate space optical potentials nonlocal. This is of import at all energies save for perhaps the highest. In almost all past coordinate space studies of pA elastic scattering, be they with a Schrodinger, Dirac or relativistic impulse approximation formulation, inherent exchange amplitudes either have been ignored or localized.
Although these predictions of the scattering are good at all energies they are better for energies below pion threshold than above. Notably at energies above pion threshold, my best results for the forward scattering angle analyzing powers overpredict the data. This may be due to the treatment of pion production and/or of the ∆ and N* resonance effects being too simplistic and not providing pertinent off-shell properties of the NN t- and g matrices at those energies to specify the appropriate details the effective interactions.
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