Physics of Nucleon-Nucleus Scattering - Chapter VIII

Chapter Eight

PREDICTING REACTION CROSS SECTIONS

Introduction

Reaction cross sections from the scattering of  nucleons by  nuclei (stable and radioactive) are required information in a number of fields of study; some being  of quite current interest [212]. An example of such a topical study concerns the transmutation of long lived radioactive waste into shorter lived products, which together with energy production, uses accelerator driven systems (ADS). These systems are being designed in the US, Europe, and Japan with the added objective of providing an intense neutron source to a subcritical reactor. The technology takes advantage of spallation reactions [213] within a thick high-Z target (such as Pb or Bi), where an intermediate energy proton beam induces nuclear reactions. The secondary nuclear products, particularly lower energy neutrons and protons [214], in turn induce further nuclear reactions in a cascade process. The total reaction cross sections of nucleon-nucleus scattering play a particularly important role since the secondary particle production cross sections are directly proportional to them. Also they are inputs to intranuclear cascade simulations that guide ADS design. As well, nucleon-nucleus (NA) cross section values at energies to 300 MeV or more are needed to specify important quantities of relevance to proton and neutron radiation therapy. Furthermore such cross sections are key information in assessing radiation protection for patients.

           

In more basic science, these total reaction cross sections are important ingredients to a number of problems in astrophysics, such as nucleosynthesis in the early universe and for aspects of stellar evolution especially as the density distribution of neutrons in nuclei are far less well known than that of protons. By considering the integral observables of both proton and neutron scattering from a given nucleus, one may seek direct information on the neutron rms  radius; a property sought in new parity-violating electron scattering experiments [194].

           

But most NA reaction cross sections cannot be, have not been, or are unlikely to be, measured.  Thus a reliable method for their prediction is required. The usual vehicle for specifying these NA total reaction cross sections has been the NA optical potential; a potential most commonly taken as a local parametrized function (of WS type). However, it has long been known that the optical potential must be nonlocal and markedly so, although it has been assumed also that such nonlocality can be accounted by the energy dependence of the customary (phenomenological) models. The results I have discussed in previous chapters have shown that such is problematic. Of more concern is that the phenomenological approach is not truly predictive. The parameter values chosen, while they may be set from a global survey of data analyses, are subject to considerable uncertainties and ambiguities.

           

But as I have shown, for energies at least between 40 and 300 MeV such phenomenology is no longer required to find quality reproduction of the angular observables; the differential cross sections and spin measureables. These calculations also give predictions of integrable observables which I consider now.

Results of calculations

All results I  show have been evaluated using the DWBA98 program [31], input to which  are density dependent and complex  effective NN interactions having central, two-nucleon tensor, and two-nucleon spin-orbit components. The form is that used in the NA scattering codes, DWBA98.

The effective interactions I use have been generated for energies from 10 MeV to over 300 MeV in 10 MeV steps by an accurate mapping to NN t- and g matrices found by solutions of the LS and BBG equations respectively and based usually upon the Bonn NN potentials. Details are given in the review [7].

Other inputs to DWBA98 are the ground state occupancies (or OBDME) and the associated SP state functions. The SP functions used in most  of the  calculations for   nuclei of mass 20 and above at best come from  a     shell model which has been adopted to describe their ground state occupancies. The HO SP functions used in those cases were obtained using oscillator lengths chosen by an A^-1/6 rule. For the  lighter mass nuclei considered, larger shell model spaces were used to define their ground states, and in some cases the interaction potentials defined as G matrix elements of a realistic interaction [171]. In shell model studies using those G-matrix elements, the value of  for the SP state functions also are specified. I have used values taken by the A^-1/3 (for  ) rule (for comparison of results from all masses) and also ones  that  give the appropriate rms  radii of the light mass nuclei considered. The case of ²⁰⁸Pb is special in that I have used structure information taken from recent Skyrme-Hartree-Fock (SHF) studies [194].

Energy variation of total reaction cross sections

In this subsection I present my predictions of the total reaction cross sections up to 300 MeV for diverse nuclei, ranging in mass from ⁶Li to ²³⁸U. In all cases, at least two calculations were made. The first of these used the effective interaction defined from the t matrices of the BonnB interaction while with the second, that built upon the associated g matrices was used. The ensuing   t- and  g-folding results are portrayed in the figures by the dashed and solid curves respectively. The data displayed in the next set of figures have been accumulated over many years and the relevant references have been cited  in reviews [7, 215].

           

As a base result with each nucleus, I have used ground state OBDME from simple (space) shell models. For the lightest, ⁶Li,  I have used as well a complete model of structure while   for ⁹Be and ¹²C I  have also used the OBDME given by a completeshell model calculations [87]. In addition I have calculated the reaction cross sections from ¹¹⁸Sn, ¹⁵⁹Tb and ²⁰⁸Pb allowing the outer (neutron) shell to have a smaller (15-20\%) harmonic oscillator energy. By that means, the neutron surface of each is   slightly more extended than with the base (packed shell) model forms. SHF calculations for ²⁰⁸Pb have been made giving SP states which vary somewhat from the conventional HO functions.

           

The results for scattering from ⁶Li are displayed in Fig. 8.1. Experimental data [202] [filled circle], [216][empty circle] are well reproduced by g-folding calculations (solid curve), but they are not with t-folding calculations (dashed curve). The results from  model (long dashed curve) underestimated the observation at all energies showing that for the light masses larger space calculations are suitable.

Figure 8.1: Energy dependence of s_R for p-⁶Li scattering. The solid curve and  the long dashed curve are  the predictions  from  g-folding model calculations using    and structures respectively. The dashed curve portrays the prediction obtained using the t-folding model potentials.

The results for scattering from ⁹Be are displayed in Fig 8.2. Experimental data were taken from Refs. [217] [empty square], [216] [empty diamond], [197] [empty up triangle], [218] [filled circle], [198] [filled square], [219] [filled diamond], [220] [filled up triangle], [221] [empty down triangle] and [199] [filled down triangle]. Clearly the data are well reproduced by the g-folding predictions (solid curve) resulting from folding with the ⁹Be ground state OBDME found with  spectroscopy. The results found with the simpler model with g folding (long-dashed curve) underestimates the data at all energies reflecting the too compressed density profile for the nucleus given by that model. Equally obvious is the fact that the results obtained with potentials formed by t folding (dashed curve) do not match observation.

Figure 8.2: Energy dependence of s_R for p-⁹Be scattering. The long-dashed curve is the prediction from  g folding model calculations and the dashed curve portrays the prediction obtained using the t folding model potentials. Both were obtained with OBDME from the  description of the nucleus. The solid curve is the result found when the OBDME from the    structure model was used.

Figure 8.3: As for Fig. 8.2 but for the energy dependence of s_R for p-¹²C scattering.

Next in Fig. 8.3, I compare my calculated p-¹²C reaction cross sections with the experimental data. Experimental data were taken from Refs. [222] [filled up triangle], [218][filled diamond], [221] [filled right triangle], [223] [cross], [224] [star], [225] [empty down triangle],  [226] [empty diamond], [200] [empty square], [227] [empty left triangle], [228] [filled square], [201] [filled down triangle], [197] [empty up triangle], [198] [empty circle], [220] [filled circle], [216] [filled left triangle], and  [217] [plus sign]. Results are displayed for proton energies from 20 MeV. Although experimental data exists to much lower energies, I do not consider the first order folding prescription for the optical potential appropriate in that lower energy regime of scattering from this nucleus. To 20 MeVexcitation, the spectrum of ¹²C is one of distinguishable states and such are not taken specifically into account in the optical potentials.

For ¹²C, the results found with the simpler model with g-folding (long-dashed curve) are very similar to the results from the other folding  made using the ground state wave function found from a complete  shell model (solid curve)  and with which successful analyses of the elastic differential cross sections and analyzing powers for protons of 40 to 800 MeV were made [76]. Likewise with that same (large space) spectroscopy, a number of inelastic (proton) scattering cross sections and analyzing powers as well as electron form factors were well explained [7]. Clearly the reaction cross sections obtained from those g-folding calculations are in very good agreement with the experimental data up to 300 MeV. Most evidently, the medium effects differentiating the g- from the t matrices used in the folding scheme defining the optical potentials are required for predictions to match observation. The t-folding model overestimates the data by 20-40 % within the energy regime below 200 MeV. Note that some data, at 61 MeV [228] and at 77 MeV [201] MeV, are in disagreement with the  calculated results. Comment on this mismatch is made later.

           

Predictions for p-¹⁶O and for p-¹⁹F scattering are compared with the data in Figs. 8.4 and 8.5. Experimental data were taken from Refs. [202] [open circles], [229] [filled circles], and [220] [filled square] are compared with predictions from 10 MeV. There are very many data points at the energies between 20 to 40 MeV and only the g-folding calculation replicates the data very well. But the data points at 13.1 MeV [229] and at 231 MeV [220] are in disagreement with that calculated results, the latter though  in agreement with the t-folding calculations. For ¹⁹F all data are well reproduced by the g-folding calculations.

        

Figure 8.4: As for Fig.8.1 but for the energy dependence of s_R for p-¹⁶O scattering.

  

Figure 8.5: As for Fig. 8.1 but for the energy dependence of s_R for p-¹⁹F scattering. Experimental data were taken from Refs. [218] [filled circles], and [197] [empty circle].

Figure 8.6: As for Fig. 8.1, but for the energy dependence of  s_R for p-²⁷Al scattering.

In Fig.8.6, the predictions for p-²⁷Al total reaction cross sections are compared with the experimental data. Experimental data were taken from Refs. [230] [empty circles], [224] [empty diamonds], [201] [empty left triangles], [227] [empty down triangles], [216] [empty right triangles], [197] [empty squares], [226] [star], [218] [filled circles], [200] [filled squares], [228] [filled diamonds], [231] [filled up triangles], [198] [filled left triangles], [232] [filled down triangles], [220] [filled left triangles] and [199] [empty up triangles]. Again only the g-folding calculations reproduce the data very well to 200 MeV. Three data points at 180 to 300 MeV though are in better agreement with the results of t-folding calculations. One data point at 61 MeV [231] is in disagreement with both calculations. This also happened for ¹²C and further comment on this mismatch is made later.

Figure 8.7: As for Fig.8.2 but for the energy dependence of s_R for p-⁴⁰Ca scattering.

My predictions for p-⁴⁰Ca and for p-⁶³Cu scattering are compared with the data (from 10 MeV) in Figs. 8.7 and 8.8. For ⁴⁰Ca, experimental data were taken from Refs. [202] [filled circles], [233] [empty circles], [217] [filled squares], [197] [empty squares], [234] [empty diamonds], and [216] [filled down triangles], while for ⁶³Cu they were taken from Refs. [235] [empty circles], [230] [filled circles], [224] [empty squares], [236] [empty diamonds], [201] [filled diamonds], [237] [empty up triangles], [197] [filled up triangles], [238] [empty left triangles], [198] [empty down triangles], [232] [empty right triangles], [220] [filled right triangles], and [199] [star].

Figure 8.8: As for Fig. 8.2 but for the energy dependence of s_R for p-⁶³Cu scattering.

With ⁴⁰Ca, the folding model approach is not expected to be reliable at the energies in the range 10 to 20 MeV since for excitation energies of that size, the nucleus has distinguishable modes of excitation. Indeed the reaction data from ⁴⁰Ca show rather sharp resonance-like features below 20 MeV. For ⁶³Cu however, no such sharp structures are evident in the reaction cross section data and my prediction with a g-folding potential at 10 MeV gives a value in quite reasonable agreement with observation. With both ⁴⁰Ca and ⁶³Cu, the g-folding results are in very good agreement with the data for energies above 20 MeV. That is in stark contrast to the t-folding results. The t-folding results underestimate the data below 20 MeV and overestimate considerably the data above 40 MeV. In the 20 to 40 MeV zone, the both forms I have used give results in reasonable agreement. Such trends are evident for most heavy nuclei.

Figure 8.9: As for Fig. 8.2 but for the energy dependence of s_R for p-⁹⁰Zr scattering.

In Fig. 8.9, the predicted total reaction cross sections from p-⁹⁰Zr scattering are compared with the experimental data taken from Refs. [236] [empty circles], [197] [filled squares], [200] [empty squares], and [199] [filled diamonds]. Results from g-folding calculations are in very good agreement with that data. The t-folding results however overestimate the data at and above 40 MeV and underestimate the data below 20 MeV.

Figure 8.10: Energy dependence of s_R for p-¹¹⁸Sn scattering. The solid and dashed curves designate predictions made using the g- and t folding optical potentials and with the basic model specification for the ground state. The long-dashed curve is the result of extending the h_11/2 neutron orbit by reducing the oscillator length for that shell by 20%.

The p-¹¹⁸Sn total reaction cross section results are given in Fig. 8.10 where three predictions are compared with the experimental data taken from Refs. [239] [filled  circles],  [236] [filled squares], [227] [empty squares], [217] [filled diamonds], [197] [empty diamonds], [228] [filled up triangles], [220] [empty up triangles], and [199] [filled down triangles]. Although not as markedly different as the results found for scattering from light mass nuclei, the g-folding potential still gives the better prediction (solid curve). The third result, portrayed in the figure by the long-dashed curves, was obtained from a g-folding optical potential formed by varying the surface neutron orbit (h_11/2) to be that for an oscillator energy reduced by 20 %. With the (slightly) extended neutron distribution that results, the g-folding potential total reaction cross sections then are in very good agreement with the data; save for the ubiquitous 61 MeV value about which comment is made later. Likewise there is a measurement at 32 MeV at odds with my results. But that point also is at odds with other data.

Figure 8.11: As for Fig.8.2 but for the energy dependence of s_R for p-¹⁴⁰Ce scattering.

Figure 8.12: As for Fig.8.10 but for the energy dependence of s_R for p-¹⁵⁹Tb scattering.

Predictions for p-¹⁴⁰Ce and for p-¹⁵⁹Tb scattering are compared with the data in Figs. 8.11 and 8.12. For ¹⁴⁰Ce the g-folding results are in good agreement with the data [240] at above 20 MeV. However, the data at 17.5 MeV is underestimated. For ¹⁵⁹Tb, the g-folding calculations (solid curve)  are still quite good replication of data [197, 241]  but the calculations obtained from g-folding optical potentials formed by varying the surface neutron orbit (h_9/2) to be that for an oscillator energy reduced by 20% (long-dashed curve), are much better.

           

In Figs. 8.13 and 8.14, I compare the calculated total reaction cross sections for p-¹⁸¹Ta and p-¹⁹⁷Au scattering with the data.

Figure 8.13: As for Fig.8.2, but for the energy dependence of s_R for p-¹⁸¹Ta scattering.

Figure 8.14: As for Fig.8.2, but for the energy dependence of s_R for p-¹⁹⁷Au scattering.

For ¹⁸¹Ta, data were taken from Refs. [241] [filled  circles], [197] [empty circle], and [199] [filled squares] and are well described by the g-folding calculations, except at 19.8 MeV  where the data point is underestimated by 20%.          For ¹⁹⁷Au, the g-folding calculations are in very good agreement with most data. In this case experimental data were taken from Refs. [241] [filled  circles], [230] [empty circle], [237] [filled squares], [216] [empty squares], [197]  [filled diamonds], [226] [empty diamonds] and [199] [filled up triangle]. The data point at 29 MeV is not matched by the calculations but again that data point is also at odds with others. 

Figure 8.15: Energy dependence of s_R for p-²⁰⁸Pb scattering. The solid and dotted curves designate predictions made using the g-folding optical potentials and with the basic model specification for the ground state. The long-dashed curve is the result of extending the i_13/2 neutron orbit by reducing the oscillator length for that shell by 15%.

           

The energy variation of the p-²⁰⁸Pb reaction cross sections is shown in Fig. 8.15 where I compare various g-folding optical potential results with the data. Experimental data were taken from Refs. [224] [star], [200] [empty square], [227] [empty left triangle], [228] [filled square], [201] [filled down triangle], [197] [empty up triangle], [198] [empty circle], [220] [filled circle], [217] [filled up triangle], [232] [empty diamond], [202] [filled diamond], and [234] [filled right triangle]. The t-folding results are not shown as they are as bad misfits as those displayed in previous figures for the scattering of other nuclei. The long-dashed curve in this case results when the oscillator energy for the outer neutron shell (i_13/2) of the simple packed shell model I have used to describe the nucleus is reduced by 15%. The associated increase in the matter profile brings the predicted reaction cross sections then in very good agreement with observation. Using the SHF wave functions [194], gives  the result displayed by the dot-dashed curve in this figure. Clearly using these new functions has made a slight change to the predictions found with the simple model (solid curve). That was a surprise given that those SHF wave functions do result in a better value for the neutron rms radius than does use of the simple packed shell model. As shown in the preceding chapter however, the differential cross sections and analyzing powers have been well reproduced by these SHF models. 

           

As with the analyses of ¹²C and ¹¹⁸Sn reaction cross sections, there are some data for p-²⁰⁸Pb scattering at odds with the calculated results (61 and 77 MeV specifically). But those data taken from Refs. [228] and [201], also do not agree with measurements made at 60.8 MeV [200] and at 65.5 MeV [217]; measurements which also gave reaction cross sections consistent with my  predictions. I note that Menet et al. [200] argue for a much larger systematic error in one of the earlier experiments.

     

      

Figure 8.16: As for Fig. 8.2 but for the energy dependence of s_R  for p-²³⁸U scattering.

In Fig. 8.16, I compare my predictions of p-²³⁸U scattering with the data [197, 198]. In this case, the difference between the g-folding (solid line) and t-folding results is slim. However, the available data are sparse and at high energies, nevertheless they are still well reproduced by the g-folding calculations.

Mass variation of reaction cross sections

The mass variations of reaction cross sections at 25, 30, 40, and 65 MeV are shown in Figs. 8.17 and 8.18 from which it is evident that the g-folding results are in quite good agreement with data while the t-folding results underestimate most of the 25 MeV data, are in reasonable agreement with the 30 and 40 MeV data but overestimate most of the 65 MeV data. Note that the extended matter SP states have been used with the ²⁰⁸Pb and the Sn isotopes calculations.

Figure 8.17: Mass variation of s_R for 25 and 30 MeV protons. The solid and dashed curves designate predictions made using the g- and t folding optical potentials and with the basic model specification for the ground state. The experimental data were taken from Ref. [215].

Figure 8.18: As for Fig.8.17, but for the mass variation of s_R for 40 and 65 MeV protons. The experimental data were taken from Refs.[217] and [215].

The disparities between the t- and g-folding potential results for the reaction cross sections are more evident at higher energies. In Fig. 8.19 I display the mass variation of  the total reaction cross sections measured [215] at 100 and 175 MeV. Again, the g-folding model predictions are in excellent agreement with the measured values, while the t-folding results overestimate observations typically by 150 mb. At 100 MeV proton scattering, total reaction cross sections from many nuclei in the mass range to ²³⁸U have been measured and it is very clear that the g-folding model predictions are in good agreement with them. Fewer measurements have been made at 175 MeV, but they too span the mass range to ²³⁸U and the results of those measurements also are in very good agreement with the g-folding optical model predictions.

Figure 8.19: As for Fig. 8.17 but for the mass variation of s_R for 100 and 175 MeV protons.

Conclusions

A microscopic model of the NA optical potential in coordinate space has been used to predict successfully the total reaction cross sections of nucleon scattering from nuclei. That optical potential has been formed by folding complex energy- and density-dependent effective NN interactions with OBDME of the target given by shell models of the nuclei. As the approach accounts for the exchange terms in the scattering process, the resulting complex and energy dependent optical potential also is nonlocal. It is crucial to use effective NN interactions which are based upon `realistic' free NN interactions and which allow for modification from that free NN scattering form due to nuclear medium effects of Pauli blocking and an average mean field. For optimum results and for the light masses in particular, it is essential also to use the best (nucleon based) model specification of nuclear structure available. Marked improvement in results were obtained when, for ⁹Be and ¹²C in this study, complete shell model calculations were used to define the OBDME required in the folding processes.

Chapter - IX