**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^{208}Pb 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

^{6}Li to^{238}U. 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,

^{6}Li, I have used as well a complete model of structure while for^{9}Be and^{12}C I have also used the OBDME given by a completeshell model calculations [87]. In addition I have calculated the reaction cross sections from^{118}Sn,^{159}Tb and^{208}Pb 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^{208}Pb have been made giving SP states which vary somewhat from the conventional HO functions.
The results for scattering from model (long dashed curve) underestimated the observation at
all energies showing that for the light masses larger space calculations are
suitable.

^{6}Li 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
Figure 8.1: Energy dependence of
and structures respectively. The dashed curve portrays the
prediction obtained using the

*s**for p-*_{R}^{6}Li scattering. The solid curve and the long dashed curve are the predictions from*g*-folding model calculations using*t*-folding model potentials.
The results for scattering from
spectroscopy. The
results found with the simpler
model with

^{9}Be 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^{9}Be ground state OBDME found 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
description of the
nucleus. The solid curve is the result found when the OBDME from the
structure model was
used.

*s**for p-*_{R}^{9}Be 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
Figure 8.3: As for Fig. 8.2 but for the
energy dependence of

*s**for p-*_{R}^{12}C scattering.
Next in Fig. 8.3, I compare my calculated
p-

^{12}C 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^{12}C is one of distinguishable states and such are not taken specifically into account in the optical potentials.
For
model with
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

^{12}C, the results found with the simpler*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*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-

^{16}O and for p-^{19}F 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^{19}F 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**for p-*_{R}^{16}O scattering.
Figure 8.5: As for Fig. 8.1 but for the
energy dependence of

*s**for p-*_{R}^{19}F 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**for p-*_{R}^{27}Al scattering.
In Fig.8.6, the predictions for p-

^{27}Al 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^{12}C and further comment on this mismatch is made later.
Figure 8.7: As for Fig.8.2 but for the
energy dependence of

*s**for p-*_{R}^{40}Ca scattering.
My predictions for p-

^{40}Ca and for p-^{63}Cu scattering are compared with the data (from 10 MeV) in Figs. 8.7 and 8.8. For^{40}Ca, 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^{63}Cu 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**for p-*_{R}^{63}Cu scattering.
With

^{40}Ca, 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^{40}Ca show rather sharp resonance-like features below 20 MeV. For^{63}Cu 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^{40}Ca and^{63}Cu, 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**for p-*_{R}^{90}Zr scattering.
In Fig. 8.9, the predicted total
reaction cross sections from p-

^{90}Zr 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**for p-*_{R}^{118}Sn 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*neutron orbit by reducing the oscillator length for that shell by 20%._{11/2}
The p-

^{118}Sn 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*) to be that for an oscillator energy reduced by 20 %. With the (slightly) extended neutron distribution that results, the_{11/2}*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**for p-*_{R}^{140}Ce scattering.
Figure 8.12: As for Fig.8.10 but for the
energy dependence of

*s**for*_{R}*p-*scattering.^{159}Tb
Predictions for p-

^{140}Ce and for p-^{159}Tb scattering are compared with the data in Figs. 8.11 and 8.12. For^{140}Ce 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^{159}Tb, 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-

^{181}Ta and p-^{197}Au scattering with the data.
Figure 8.13: As for Fig.8.2, but for the
energy dependence of

*s**for*_{R}*p-*^{181}Ta*scattering.*
Figure 8.14: As for Fig.8.2, but for the
energy dependence of

*s**for*_{R}*p-*^{197}Au*scattering.*
For

^{181}Ta, 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^{197}Au, 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-^{208}Pb 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-
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.

^{208}Pb 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*) 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_{13/2}
As with the analyses of

^{12}C and^{118}Sn reaction cross sections, there are some data for p-^{208}Pb 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**for p-*_{R}^{238}U scattering.
In Fig. 8.16, I compare my predictions of
p-

^{238}U 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^{208}Pb and the Sn isotopes calculations.
Figure 8.17: Mass variation of

*s**for 25 and 30 MeV protons. The solid and dashed curves designate predictions made using the*_{R}*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**for 40 and 65 MeV protons. The experimental data were taken from Refs.[217] and [215].*_{R}
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^{238}U 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^{238}U 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**for 100 and 175 MeV protons.*_{R}

**Conclusions**

A microscopic model of the
shell
model calculations were used to define the OBDME required in the folding
processes.

*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^{9}Be and^{12}C in this study, complete
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