Chapter Nine
SIMPLE
FUNCTIONAL FORM
FOR
NUCLEON-NUCLEUS TOTAL REACTION CROSS SECTIONS
Introduction
The values of total reaction cross
sections of nucleon scattering from nuclei (stable as
well as radioactive) which are the sum of all the reaction processes induced
during nuclear collisions are required in a number of fields of study ranging
from fundamental nuclear physics, through radiation effects in biology, to the
radiation shielding design for future space exploration vehicles [242]. Some of quite
current interest concern the transmutation of long lived radioactive waste into shorter lived products using accelerator
driven systems (ADS), and in predicting dosimetries for
patients in radiation therapy. To be able to specify those total reaction cross
sections in a simple functional form then has great utility for any associated evaluation
such as of nucleon production in spallation calculations. As shown in the
preceding chapter, nucleon-nucleus total reaction cross sections now can be
predicted in good agreement with data. But each such prediction of nucleon
scattering is an involved calculation that culminates with use of large scale
computer programs, and those of DWBA98 [7, 31] as discussed in
the earlier chapters. It would be very utilitarian if aspects of such
scattering were well approximated by a simple convenient functional form. I
demonstrate herein that for the total reaction cross sections such a form may
exist. Herein I have considered the elastic scattering of rotons for energies
ranging from 20 MeV to 300 MeV, and from 15 nuclei ranging in mass from 9 to
238, to demonstrate the possibilities.
Simple functional form
The total
reaction cross sections for proton scattering from nuclei can be
expressed in terms of partial wave scattering (S) matrices specified at
energies E ∞ k2, by
where 
are the (complex) scattering phase shifts and 
. Total reaction cross sections then follow from
are the (complex) scattering phase shifts and . Total reaction cross sections then follow from
Despite the
successes with use of the g-folding approach to define NA optical
potentials [7] we note that there are discrepancies
between the predictions of total reaction cross sections found thereby and actual data. Those usually
are due to limitations with the structure model used to describe the ground
state densities of some nuclei. However, there have been so many successes with
the approach [7, 212, 243, 244] that the functional form developed here
on the basis of matching the values obtained with g-folding potentials
is a pertinent initial guess to start a refinement (of the parameter values) to reproduce actual measured total reaction cross sections. Of
course, any optical potential model scheme could be used for this purpose
and the paper [245] gives a detailed specification of global
phenomenological potentials that would serve very well indeed. Their results
for proton reaction cross sections to 200 MeV are very good; better
representations of the data than some of our g-folding results. However,
we reiterate, our focus is upon finding a simple functional and utilitarian
form by which such data may be represented, so that any reasonable set of
partial cross sections serve as the initiating set.
Note that the
Coulomb potential we have used in the optical potentials for all nuclei is
assumed to be that for a uniform charged sphere of radii 1.05A1/3.
Otherwise, any Coulomb screening effect is taken as subsumed in the values of
the S matrices and/or partial total reaction cross sections.
As evident
from the figures presented in the paper [246], the partial total reaction cross
sections,
, can be described by the simple functional form,
with l0(E,A),
a(E,A), and e(E,A) varying smoothly with energy and mass.
The summation
giving the total reaction cross section can be limited to a value lmax
and the associated form tends appropriately to the known high energy limit.
With increasing energy, lmax becomes so large that the
exponential fall of the functional form, Eq. (9.3), can be approximated as a
straight vertical line (l0 = lmax). In that
case, the total reaction cross section equates to the area of a triangle, and
taking lmax
~ kR at high energies to get the geometric cross section when R is
the ‘‘black disk’’ radius.
Although using
Eqs. (9.2) and (9.3) to match values of (theoretically) calculated total
reaction cross sections led [246] to the three parameters l0(E,A),
a(E,A), and e(E,A) having smooth variations with energy,
there are discrepancies between those predictions and the actual measured data.
Herein we modify the method of selection of those parameter values to produce
more accurate reaction cross section values, while keeping as smooth a
variation with energy of those parameters as possible. Specifically, in Eq.
(9.3), we have set the e as a constant
(-1.5) and so independent of energy and of mass. Further we assume that a(E,A)
varies linearly as
Then l0(E,A)
were adjusted to ensure that all measured total reaction cross section values
are matched by using the function form, Eq. (9.3).
Results
The resultant
optimized values for the parameter l0 are presented in
Fig.9.1. In this figure, the points represent the calculated values of l0
at particular energies where the experimental values of total reaction cross
sections are available for the considered nuclei. We
have used the lower and upper uncertainty limits to get the error bars for l0
values.We call these l0 values with uncertainties as ‘‘data-l0’’
hereafter. The curves result on using a spline interpolation on those data
points constrained to give an optimal smoothness variation of l0(E,A)
with E. The figure displays the results for the entire energy range (up
to 300 MeV) in the left hand panels, while the right hand panels emphasize the
low energy variations up to 50 MeV since most experimental values lie in that
range of energy.
Figure 9.1: The
parameter values for l0 in the simple functional form for the p-A reaction cross
sections. Points shown with error bars are the
values required to match actual measured data values.
In panel (a)
of Fig. 9.1, the solid, dotted, dashed, longdashed, and dot-dashed lines depict
the values for 9Be, 12C, 16O, 19F, and 27Al, respectively, as do circles, squares,
diamonds, up triangles, and down triangles for the data-l0
results.The same legends apply to the 10–50 MeV plots shown in panel (b). In
panels (c) and (d), the energy variations of l0 values for 40Ca, 63Cu, 90Zr, 118Sn, and 140Ce
are presented by solid, dotted, dashed, long-dashed, and dotdashed lines,
respectively. The corresponding data-l0 points are also presented by circles, squares,
diamonds, up triangles, and down triangles. Finally , in panels (e) and (f), we
display the variations of l0 with energy for 159Tb,
181Ta, 197Au, 208Pb, and 238U by solid, dotted,
dashed, long-dashed, and dot-dashed lines, respectively, with the corresponding
data-l0 points presented by circles, squares, diamonds, up
triangles, and down triangles.
Figure 9.2:
The parameter values for l0 in the simple functional form for the p-A reaction cross
sections as
a function of target mass.The numbers near select curves are the relevant proton
incident energies (in MeV).
Numerical
values for l0 that result from the simple functional form calculations, and as displayed by the curves
in Fig. 9.1, are also available in tabular form. For energies 50 MeV and higher,
these l0 values monotonically increase with mass and energy
as might be anticipated. At lower energies, however, and while the l0
values do still vary monotonically with energy, there is additional structure
in their mass variation.This is emphasized in the plots given in Fig. 9.2. But
that may simply reflect the scattered (in energy) nature of the actual data
from which starting values of l0 have been taken to generate
the functional forms.
In Figs. 9.3,
9.4, and 9.5, we show the total reaction cross sections generated using the simple functional form and tabled values of l0 ,
and displayed by the solid curves, with those obtained from calculations made
using g-folding optical potentials [244]. Dashed lines represent the predictions
obtained from those microscopic optical model calculations.The experimental data [215] are depicted by circles.
Figure 9.3: Energy dependence of sR for proton scattering from the nuclei identified in each
segment. Dashed lines represent the results obtained from g-folding optical potential calculations, while the solid curves portray
the values obtained by using simple functional form.
The results
for scattering from 9Be, 12C, 16O, 19F, 27Al, and 40Ca are displayed in Fig. 9.3.
Note that to
get the 9Be data reproduced by the g-folding predictions it was essential to fold with the
ground state OBDME found with
spectroscopy. The reaction cross sections obtained from g-folding calculations of
p-12C reaction cross sections are in good agreement
with the experimental data, but only in the energy range above 20 MeV. On the
other hand, the results obtained from the simple functional method are
excellent for all energies, replicating the average trend of data well even in
the lower energy range from 10 MeV. There are two data points, at 61 MeV and at
77 MeV, in disagreement with the calculated results, however. But, as noted
previously [212], these data points might be discounted.
For p-16O case, there are many data points at the
energies between 20 and 40 MeV. Predictions from g-folding calculations, while replicating the data well
at and above 25 MeV, overestimate at lower energies. That g-folding
result also underestimates the datum at 250 MeV; the sole datum above 50 MeV.
In contrast, and by design, the results obtained from the simple functional
method are in excellent agreement with the experimental data at all energies.
For p-19F, although g-folding calculations
reproduce the data well, the simple functional form method gives slightly more accurate
predictions.
However, while
the total reaction cross section predictions for p-27Al and p-40Ca found from g-folding calculations reproduce the data quite well to
200 MeV, three data points between 180 and 300 MeV are not matched. The g-folding
results underestimate them noticeably. But predictions from a simple functional
form approach replicate the data well at all
energies. One data point at 61 MeV is exceptional in the set. With 40Ca,
the folding model approach is not expected to be reliable at the energies in
the range 10 to 20 MeV, as is the case with 12C, since for excitation energies of that
range, both nuclei have clearly discrete spectra. That is true for most light
mass nuclei but little or no total reaction cross sections have been reported for them. Indeed, the
reaction data from both 12C and from 40Ca show rather
sharp resonance features below 20 MeV. Both the g-folding calculations
and functional form calculations reproduce the rest of the 40Ca data
well.
The results
for scattering from 63Cu, 90Zr, 118Sn, 140Ce, 159Tb,
and 181Ta are displayed in Fig. 9.4.
Figure 9.4: Same as Fig. 9.3, but from the nuclei identified.
For 63Cu, predictions at low energies may be
slightly too small and the parameter sets driven too severely by the sole datum
at 30 MeV in the range 20 to 70 MeV. Also, the data in the range 100 to 300 MeV
are quite scattered but the simple functional form gives a good average result. The predicted
total reaction cross sections from p-90Zr scattering compare well with experimental
data, although the data value at 30 MeV is overestimated. The results obtained
from the simple functional form, and by dint of construction, are in excellent
agreement with the experimental data at the few energies measured, but the
shape is not optimally smooth. This we believe is the prime cause for the kink
shown at A = 90 in the mass variation plots in Fig.9.2 and most
noticeably at 200 MeV. Lack of data meant that we had to use the g-folding values to specify the functional form. That is
also the case with masses 140, 159, and 181.
The p-118Sn
total reaction cross section results in Fig. 9.4 do not compare with data as
well as do the results found for scattering from light mass nuclei. The g-folding potential still gives reasonable shape
prediction; the data are underestimated by 5–10%. Again, by construction, the
simple functional form model form gives an excellent reproduction of
the data at all energies except 61 MeV. This 61-MeV data point is again
exceptional, being much smaller than other data and the predictions as in the
cases of 12C and 27Al.
Predictions
for p-140Ce, p-159Tb, and for p-181Ta
scattering are compared with the (limited amount of) data in Fig. 9.4. The g-folding calculations give good agreement with the data
for p-140Ce, slightly underestimate the data for p-159Tb,
and for p-181Ta, underestimate data at the energies up to 20
MeV and overestimate data in the energy range 40 to 60 MeV. In all cases,
results predicted by the simple functional form are excellent reproductions of the
experimental data.
For 197Au,
the g-folding optical potential calculations are in good agreement with most
data, though the 29-MeV datum is grossly underestimated. But that datum is also
at odds with the energy trend of the other data. Save for that 29-MeV value,
the simple functional form gives even better predictions of the data. The
g-folding optical potential calculation results for scattering from 208Pb and the variation set by the simple functional
form are compared with a fairly extensive set of experimental data in the
middle panel. In making the g-folding potentials, we have used
Skyrme-Hartree-Fock wave functions [194] that have been shown to be more
realistic [244] than simple oscillator model ones. Still
such g-folding calculations underestimate the data up to 50 MeV.
Naturally, the simple functional form calculations are in excellent agreement
with the experimental data, save that data values at 30, 61, and 77 MeV again
are exceptional. Finally, the predictions of total reaction cross sections for p-238U scattering, from g-folding
optical potential calculations and from using the simple functional form, are
compared with the few data available. Given the lack of data, the two results
are virtually identical.
Figure 9.5:
Same as Fig. 9.3, but from the nuclei identified.
In Fig. 9.5 we
compare with available data the calculated total reaction cross sections for protons from 197Au, 208Pb, and 238U.
As a final
note regarding many of the exceptional data values so defined in the foregoing,
Menet et al. [200] argue that there may be a systematic
error in the studies reported in those experiments.
Conclusions
The measured
reaction cross sections for 10–300 MeV proton scattering from nuclei
ranging in mass from 9Be to
238U are well reproduced by calculations made using a three parameter
function. The values of the parameters (l0 , a, e ) are set in a simple manner and when set by enough actual
data, can be used to predict the total reaction cross sections at any energy
for a given nucleus. The mass variations of those parameters at fixed energies
also are smooth, and very much so when data exist to control the results, and
we believe that the simple functional form also can be used to find reasonable estimates
of the total reaction cross sections of protons from any stable nucleus in the
mass range.
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