Chapter Ten
SIMPLE
FUNCTIONAL FORM
FOR
NUCLEONNUCLEUS TOTAL CROSS SECTIONS
Introduction
Total cross
sections from the scattering of neutrons by nuclei are required in a number of
fields of study which range over problems in basic science as well as many of
an applied nature. It would be utilitarian if such scattering data were well
approximated by a simple convenient function form with which predictions could
be made for the cases of energies and/or masses as yet to be measured. Recently
it has been shown [246, 247] that such forms may exist for proton
total reaction cross sections. Herein we consider that concept further
to reproduce the measured total cross sections from neutron scattering for
energies to 600 MeV and from nine nuclei ranging in mass between ^{6}Li and ^{238}U. These suffice to show
that such forms will also be applicable in dealing with other stable nuclei
since their neutron total cross sections vary so similarly with energy [245].
Total
scattering cross sections for neutrons from nuclei have been well reproduced by
using optical potentials. In particular, the data (to 300 MeV) from the same
nine nuclei we consider compare quite well with predictions made using a gfolding method to form nonlocal optical potentials [244], though there are some notable
discrepancies. Alternatively, in a study Koning and Delaroche [245] gave a detailed specification of
phenomenological global optical model potentials determined by fits to quite a vast
amount of data and, in particular, to the neutron total scattering cross
sections we consider herein. However, as we show in the case of the total cross
sections from 10 to 600 MeV, there is a simple function form one can use to
allow estimates to be made quickly without recourse to optical potential calculations. Furthermore, we shall show that
the required values of the three parameters of that function form themselves a
trend sufficiently smoothly with energy and mass suggesting that they too may
be represented by functional forms.
Formalism
The total
cross sections for neutron scattering from nuclei can be expressed in terms of
partialwave scattering (S) matrices specified at energies E ∞k^{2}
by
where
are the (complex) scattering
phase shifts and
are the moduli
of the S matrices. The superscript designates j = l±1/2.
In terms of these quantities, the elastic, reaction (absorption), and total
cross sections, respectively, are given by
and
Therein the
are defined as partial cross sections of the total
elastic, total reaction, and total scattering itself. For proton scattering,
because Coulomb amplitudes diverge at zero degree scattering, only total
reaction cross sections are measured. Nonetheless, study of
such data [246, 247]
established that partial total reaction cross sections
may be described by
the simple function form
, .................(10.5)
with the
tabulated values of l_{0}(E,A), a(E,A),
and e(E,A) all varying smoothly
with energy and mass. Those studies were initiated with the partial reaction
cross sections determined by using complex, nonlocal, energydependent, optical potentials
generated from a gfolding formalism [7]. While those gfolding
calculations did not always give excellent reproduction of the measured data
(from ~20 to 300 MeV for which one may assume that the method of analysis is
credible), they did show a pattern for the partial reaction cross sections that
suggests the simple function form given in Eq. (10.5). With that form excellent
reproduction of the proton total reaction cross sections for many targets and
over a wide range of energies was found with parameter values that varied
smoothly with energy and mass.
Herein we
establish that the partial total cross sections for scattering of neutrons from
nuclei can also be so expressed and we suggest forms, at least first average
result forms, for the characteristic energy and mass variations of the three
parameters involved. Nine nuclei ^{6}Li, ^{12}C, ^{19}F, ^{40}Ca, ^{89}Y, ^{184}W, ^{197}Au,
^{208}Pb, and ^{238}U, for which a large
set of experimental data exist, are considered. Also those nuclei span
essentially the whole range of target mass. However, to set up an appropriate
simple function form, initial partial total cross sections must be defined by
some method that is physically reasonable. Thereafter the measured total
crosssection values themselves can be used to finetune the details and of the
parameter l_{0} in particular. We chose to use results from gfolding optical potential calculations to give those starting values.
Results
and Discussions
That a
function form for total cross sections is feasible has been suggested
previously in dealing with energies to 300 MeV from a few nuclei [244] and by using a gfolding prescription
for the nucleonnucleus optical potentials. At the same time, studies of the
partial reaction total cross sections for proton scattering [246, 247] found that a form as given in Eq. (10.5)
was most suitable. A similar form can be used to map the partial total cross
sections given by the gfolding potential calculations and thence by
suitable adjustments for their sums to give the measured total cross sections.
Of note is that, with increasing energy, the form of the simple function [Eq. (10.5)]
can be approximated by a sharp fall at l = l_{0}(E)
= l_{max}, giving a triangle in angular momentum space. In that case,
the total reaction cross section equates to the area of that triangle and
Then with l_{max}~kR
at high energies,
the geometric cross section as required. Furthermore, for
high enough energies then, the total cross section is double that value. This
is an asymptotic behavior one can assume for the l_{0 }values to
be used with the total cross sections.
The function
form results we display in the following set of figures were obtained by
starting with gfolding model results at energies of 10–100 MeV in
steps of 10 MeV, then to 350 MeV in steps of 25 MeV, and thereafter in steps of
50 MeV to 600 MeV. The g matrices used above pion threshold were those obtained
from an optical potential correction to the BonnB force [248] which, while approximating the effects
of resonance terms such as virtual excitation of the D, may still be somewhat inadequate for use in nucleonnucleus
scattering above 300 MeV. Also relativistic effects in scattering, other than
simply the use of relativistic kinematics in the distortedwave approximation
(DWA) approach, are to be expected.
Nonetheless, the DWA results are used only to find a sensible starting set of
the function form parameters l_{0}, a, and e from which to find ones that reproduce the measured total
crosssection data. One must also note that the gfolding potentials for
most of the nuclei considered were formed using extremely simple model
prescriptions of their ground states. A previous study [244] revealed that with good spectroscopy the
gfolding approach gives much better results in comparison with data
than that approach did when simple packed shell prescriptions for the structure
of targets were used. That was also the case when scattering from exotic,
socalled nucleon halo, nuclei were studied [7, 121].
The results
from analyses of 40A MeV scattering of ^{6}He ions from hydrogen targets [121] lead to a note of caution for the use of
the trends we set out here. Our results are for a range of energies and for a
diverse set of stable nuclear targets. Total cross sections with
unstable halo nuclei may be considerably larger than one expects if they were
assumed adequately described by standard shell model wave functions. Indeed at
40A MeV the total reaction cross section for ^{6}Hehydrogen
scattering was 16%–17% larger than found using the standard shell model
prescription. That and the momentum transfer properties of the ^{6}HeP
differential cross section were convincing evidence of the neutronhalo
nature of ^{6}He. We proceed then with the caveat that specific
structure properties may be needed as variation to the functional forms we
deduce. But given the results found with the diverse (nine) nuclei considered,
we believe that such would need be very significant structure aspects, such as
a halo, to be of import.
While we have
used the partial total cross sections from DWA results for neutron scattering from all the
nine nuclei chosen and at all of the energies indicated, only those obtained
for ^{208}Pb are shown in Fig. 10.1. The results from
calculations of scattering from the other eight nuclei have similar form. The
“data” shown as diverse open and solid symbols in Fig. 10.1 are the specific
values found from the gfolding optical model calculations. Each curve shown therein is the
result of a search for the best fit values of the three parameters l_{0},
a, and e that map Eq.
(10.5) (now for total neutron cross sections) to these “data.”
From the sets
of values that result from that fitting process, the two parameters a and
e can themselves be expressed by the
parabolic functions
where the
target energy E is in MeV. There was no conclusive evidence for a mass
variation of them. With a and e so fixed, we then adjusted the values of l_{0}
in each case so that actual measured neutron total crosssection data were fit
using Eq. (10.5). Numerical values for l_{0} from that process
are presented in Table I.
Figure 10.1:
The partial total cross sections for scattering of neutrons from ^{208}Pb with the set of energies between 10 and 600
MeV specified in the text. The largest energy has the broadest spread of
values.
The values of l_{0}
increase monotonically with both mass and energy and that is most evident in
Fig. 10.2 where the optimal values l_{0}(E) are presented
as diverse solid or open symbols. The sets for each of the masses (from 6 to
238) are given by those that increase in value, respectively, at 600 MeV. While
that is obvious for most cases, note that there is some degree of overlap in
the values for ^{197}Au (opaque diamonds) and for ^{208}Pb (solid circles). The curves are the shapes
deduced by a function of energy for the l_{0}(E), which
will be discussed subsequently.
TABLE I: l_{0} values with which
the function form, Eq. (10.5), fits neutron total cross sections.
E(MeV)

^{6}Li

^{12}C

^{19}F

^{40}Ca

^{89}Y

^{184}W

^{197}Au

^{208}Pb

^{238}U

10

3.330

3.650

3.838

4.925

6.297

6.916

6.883

6.892

7.397

20

4.016

4.974

5.573

6.088

7.657

9.838

10.059

10.169

10.508

30

4.292

5.589

6.677

7.675

8.671

11.013

11.337

11.578

12.241

40

4.432

6.039

7.329

8.898

10.141

11.993

12.212

12.393

13.184

50

4.447

6.200

7.672

9.822

11.602

13.418

13.526

13.635

14.392

60

4.435

6.296

7.873

10.331

12.791

15.001

15.143

15.181

15.950

70

4.404

6.348

7.979

10.718

13.629

16.439

16.632

16.634

17.506

80

4.353

6.305

8.000

10.922

14.221

17.591

17.857

17.996

18.884

90

4.324

6.255

8.003

11.036

14.631

18.438

18.808

18.982

19.940

100

4.292

6.259

8.040

11.071

14.891

19.058

19.459

19.541

20.726

125

4.261

6.284

8.067

11.241

15.190

19.924

20.427

20.596

21.900

150

4.303

6.315

8.189

11.404

15.461

20.432

20.960

21.167

22.584

175

4.387

6.436

8.362

11.597

15.771

20.871

21.441

21.843

23.129

200

4.515

6.686

8.610

11.981

16.256

21.567

22.125

22.304

23.870

225

4.648

6.847

8.921

12.307

16.850

22.313

22.910

23.112

24.735

250

4.767

7.113

9.226

12.756

17.543

23.255

23.866

23.981

25.745

275

4.883

7.369

9.593

13.196

18.250

24.226

24.866

25.076

26.814

300

4.974

7.621

9.967

14.008

19.071

25.249

25.894

26.297

27.961

325

5.143

7.850

10.312

14.501

19.794

26.262

26.962

27.221

29.069

350

5.265

8.131

10.658

15.069

20.569

27.277

27.966

28.236

30.180

400

5.456

8.677

11.399

15.915

22.015

29.255

30.007

30.319

32.327

450

5.656

9.159

12.102

17.091

23.482

31.173

31.946

32.202

34.398

500

5.966

9.674

12.751

17.953

25.011

32.971

33.887

33.978

36.510

550

6.069

9.559

13.146

19.341

26.362

34.624

35.574

35.749

38.425

Figure 10.2: The
values of l_{0} that fit neutron total scattering cross section
data from the nine nuclei considered and for energies between 10 and 600 MeV.
The curves portray the best fits found by taking a function form for l_{0}(E).
Plotting the
values of l_{0} against mass also reveals smooth trends as is
evident in Fig. 10.3. Some actual energies are indicated by the numbers shown
in this diagram. Again the curves shown in the figure are the results found on
taking a functional form for l_{0}(A) at each energy, and
that too will be discussed later.
Figure 10.3:
The values of l_{0} depicted in Fig. 10.2 as they vary with mass
for all of the energies considered. Some of those energies are indicated in the
diagram and the curves are splines linking best fit values for each mass
assuming a function form for l_{0}(A).
The total
neutron scattering cross sections generated using the function form for partial
total cross sections with the tabled values of l_{0} and the
energy function forms of Eq. (10.7) for a and e are shown in Figs. 10.4 –10.6. They are displayed by the
solid lines, which closely match the data which are portrayed by opaque
circles. The data that were taken from a survey by Abfalterer et al. [249] which includes data measured at
LANSCE that are supplementary and additional to those
published earlier by Finlay et al. [250]. For comparison we show results obtained
from calculations made using gfolding optical potentials [244]. Dashed lines represent the predictions obtained
from those microscopic optical potential calculations. Clearly for energies 300 MeV and
higher, those predictions fail.
The total
cross sections for neutrons scattered from the four lightest nuclei considered
are compared with data in Fig. 10.4.
Figure 10.4:
Total cross sections for neutrons scattered from ^{6}Li, ^{12}C, ^{19}F, and ^{40}Ca. The results have been scaled as
described in the text to provide clarity.
Therein from
bottom to top are shown the results for ^{6}Li, ^{12}C, ^{19}F, and ^{40}Ca with shifts of 1b, 2b and 3b made
for the latter three cases, respectively, to facilitate inspection of the four
sets. A slightly different scaling is used in Fig. 10.5 in which the total
neutron scattering cross sections from the nuclei ^{89}Y (unscaled), ^{184}W
(unscaled), ^{197}Au (shifted by 2b), and ^{238}U
(shifted by 3b) are compared with the base gfolding optical potential results and with the function forms with the
optimal parameters. Again the gfolding potential results are displayed
by the dashed curves while those of the function form are shown by the solid
curves.
Figure 10.5:
Total cross sections for neutrons scattered from ^{89}Y, ^{184}W,
^{197}Au, and ^{238}U. The results have been scaled as
described in the text to provide clarity.
Finally, we
show in Fig. 10.6 the results for neutron scattering from ^{208}Pb. In this case we used
SkyrmeHartreeFock model (SKM*) densities [194] to form the gfolding optical potentials. That structure when used
to analyze proton and neutron scattering differential cross sections at
65 and 200 MeV gave quite excellent results [243]. Indeed those analyses were able to show
selectivity for that SKM* model of structure and for the neutron skin thickness
of 0.17 fm that it proposed.
Figure 10.6:
Total cross section for neutrons scattered from ^{208}Pb.
Using the SKM*
model structure, the gfolding optical potentials gave the total cross
sections shown by the dashed curve in Fig. 10.6. Of all the results, we believe
these for Pb point most strongly to a need to improve on the gfolding
prescription as is used currently when energies are at and above pion
threshold. Nonetheless, it does do quite well for lower energies, most notably
giving a reasonable account of the Ramsauer resonances [245] below 100 MeV. However, as with the
other results, these gfolding values serve only to define a set of
partial cross sections from which an initial guess at the parameter values of
the function form is specified. With adjustment that form produces the solid
curve shown in Fig. 10.6 which is an excellent reproduction of the data, as it
was designed to do. But the key feature is that the optimal fit parameter
values still vary smoothly with mass and energy.
Without
seeking further functional properties of the parameters, one could proceed as
we have done this far but by using many more cases of target mass and
scattering energies so that a parameter tabulation as a database may be formed
with which any required value of total scattering cross section might be
reasonably predicted (i.e., to within a few percent) by suitable interpolation
on the database and the result used in Eq. (10.5).
Parameters
as functions of energy
As noted
previously, the two parameters a and e can be chosen to have the parabolic forms in energy as given
by Eq. (10.7). Once they are set, the required values of l_{0}(E,A)
vary smoothly and monotonically with both E and A in giving the
partial crosssection sums that perfectly match measured values of the total
cross sections.
For energies
above 250 MeV, the l_{0} values approximate well as straight
lines and a likely representation of all of the sets of l_{0}
values is found with the energydependent function
The values of
the parameters that lead to the curves depicted in Fig. 10.2 are listed in
Table II. The result for ^{208}Pb nonetheless is as good a fit as found in the
other eight cases.
TABLE II. Values
of parameters defining l_{0}(E).
A

c_{1}

c_{2}

c_{3}

E_{0}

b

χ^{2}

χ^{2}
(<100)

^{6}Li

4.665 x 10^{3}

3.582

1.537

13.87

3.670 x 10^{2}

0.025


^{12}C

9.103 x 10^{3}

4.865

3.449

21.35

3.285 x 10^{2}

0.30


^{19}F

1.374 x 10^{2}

5.808

4.794

24.42

2.880 x 10^{2}

0.89


^{40}Ca

2.272 x 10^{2}

6.820

4.896

25.97

1.937 x 10^{2}

0.73


^{89}Y

3.31 x 10^{2}

8.357

5.256

29.47

1.470 x 10^{2}

2.59

2.0

^{184}W

4.27 x 10^{2}

11.50

7.574

43.73

1.310 x 10^{2}

4.2

3.0

^{197}Au

4.41 x 10^{2}

11.65

7.635

43.96

1.277 x 10^{2}

4.5

3.2

^{208}Pb

4.067 x 10^{2}

13.43

9.402

62.51

1.400 x 10^{2}

5.0

3.3

^{238}U

4.75 x 10^{2}

12.56

8.081

46.51

1.235 x 10^{2}

4.1

2.9

In Table II,
the last two columns give values of χ^{2} which in this case are
defined by
with the sum
extending over the 24 energies used. For the heavier masses the values of χ^{2}
that result when the sums are restricted to energies below 100 MeV (ten points)
are given in the last column. They reveal that the mismatch occurs at those low
energies particularly. Note, however, that the function for the parameter
variation was chosen solely by inspection. No particular physical constraint
was sought and so alternate function forms are not excluded. This is one reason
why we have not proceeded further and sought a mass dependence in the
coefficients c_{1}, c_{2}, c_{3}, E_{0},
and b themselves.
Of the
parameter values for the
, those for ^{208}Pb differ
most from smooth progressions in mass as is evident in Fig. 10.7. Therein the
values of the parameters defining
are plotted with the connecting lines simply to guide the
eye. The values for c_{1} (solid circles) and of b (open down triangles) have been multiplied by 10 for
convenience of plotting. The other parameter values are identified as c_{2}
(open squares), c_{3} (solid diamonds), and E_{0}
(solid triangles). Clearly there is a smooth mass trend of these values with
the exception of the entries for ^{208}Pb. But the ^{208}Pb
values are based only on achieving the smallest χ^{2} value as defined
by Eq. (10.9). Using parameter values consistent with the smooth mass trend,
the χ^{2} for the fit to the ^{208}Pb values doubles at most.
Figure 10.7:
The coefficients of
for each nucleus. The separate results are identified in the
text.
But use of the
function form of Eq. (10.8) for l_{0}(E), along with
those of Eq. (10.7) for a and e, with Eq. (10.5), as yet do not replicate the measured total
cross sections well enough at all energies, another reason why we do not as yet
seek massdependent forms for the coefficients in Eq. (10.8). We consider that
an appropriate criterion is that the measured cross sections should be
replicated to within ±5%. The percentage differences in cross sections for each
nucleus considered are displayed in the top two segments of Fig. 10.8.
Figure 10.8:
The percentage differences between actual total cross section values and those
generates using the threeparameter prescription with parameter values set by
the energy function forms for l_{0}(E), a, and e. Those differences for ^{40}Ca and heavier nuclei are depicted in the top
segment, while those for ^{6}Li, ^{12}C, and ^{19}F are given in the middle segment. In the bottom
segment are the differences between the optimal data fit values of l_{0}(E)
and those specified by using Eq. (10.8) for ^{40}Ca and heavier nuclei.
In the top
segment, those differences for ^{40}Ca and heavier nuclei are shown. Curiously these
variations look sinusoidal with argument proportional to E^{1/3}.
In the middle segment the differences for the three light mass nuclei are given
with the solid, dashed, and longdashed curves depicting the values for ^{6}Li, ^{12}C, and ^{19}F, respectively. Clearly the reasonable
fit criterion has been met for the light masses for all energies. That is so also
for the heavier nuclei but only for energies above 100 MeV. There is too large
a mismatch for the heavy nuclei at lower energies, however. This mismatch
reflects the differences between the actual best fit values of l_{0}(E)
and those defined by the function form, Eq. (10.8), and which differences for
just the heavy nuclei are shown in the bottom segment of Fig. 10.8.
Only the
values for ^{40}Ca and heavier nuclei are shown as the
differences for the light mass nuclei are very small for all energies, being
less than ±0.1 and usually less than ±0.01. The results for each nucleus ^{40}Ca,
^{89}Y, ^{184}W, ^{197}Au, and ^{238}U are
shown in the bottom segment, respectively, by the solid curve connecting solid
circles, the longdashed curve connecting solid diamonds, the dashed curve, the
solid curve connecting opaque diamonds, and the dotdashed curve. Of particular
note is that the differences between these fit and function values of the l_{0}(E)
mirror those of the total crosssection differences shown in the top segment,
both in energy and with different mass. It is most likely then that the
function form, Eq. (10.8), is a firstorder guess and may be improved to meet
the reasonable fit criterion we have set. That is the subject of ongoing study
in which many more targets and more numerous values of energy in the region to
100 MeV are to be used.
Parameter
l_{0} as a function of mass
As noted
above, the l_{0} parameter values vary smoothly with mass. In
fact we find that a good representation of those values is given by
With this mass
variation form, the coefficients d_{i} are as set
out in Table III.
TABLE III. Values of parameters defining l_{0}(A).
Energy

d_{1}

d_{2}

d_{3}

d_{4}

10

0.0034

6.4

3.62

0.020

20

0.0224

5.51

3.06

0.108

30

0.023

6.74

5.18

0.114

40

0.0198

8.37

6.57

0.080

50

0.018

9.99

7.91

0.057

60

0.0205

11.08

8.81

0.046

70

0.0247

11.68

9.38

0.041

80

0.0297

11.92

9.64

0.039

90

0.0337

12.05

9.77

0.037

100

0.037

12.02

9.75

0.037

125

0.043

11.71

9.56

0.038

150

0.0464

11.68

9.58

0.039

175

0.0484

11.70

9.64

0.040

200

0.0497

12.17

9.97

0.04

225

0.0514

12.62

10.32

0.039

250

0.0532

13.19

10.82

0.039

275

0.056

13.65

11.25

0.039

300

0.0585

14.23

12.11

0.041

325

0.0607

14.81

12.59

0.040

350

0.0628

15.41

13.22

0.040

400

0.068

16.40

14.14

0.040

450

0.072

17.53

15.47

0.041

500

0.0748

18.85

16.50

0.039

550

0.077

20.13

18.39

0.040

That mass
equation with those tabled values of the coefficients gave the nine values of l_{0}
for each energy that are connected by a spline curve in Fig. 10.3. The optimal
values for these parameters (listed) are shown by the diverse set of open and
solid symbols. The coefficients defining
are portrayed by the various symbols in Fig.10.9.
Figure 10.9:
Parameter values of l_{0}(A) that give best fits to total
crosssection data. Details are specified in the text.
Specifically
the coefficients are shown by the solid circles (d_{1}), by the
solid squares connected by the longdashed lines (d_{2}), by the
opaque diamonds (d_{3}), and by the opaque uptriangles
connected by dashed lines (d_{4}). Again for clarity the actual
values found for d_{1} and d_{4} have been
multiplied by a scaling factor. This time that factor is 100. These mass
formula coefficients vary smoothly with energy and one might look for a
convenient function of energy to describe them as well. However, as we noted
earlier with the energy function representation, the choice of this mass
equation resulted solely from inspection of the diagram and so alternate
formulas are not excluded. Therefore it was not sensible to seek a function form
for the coefficients themselves. In any event, one needs results from a much
larger range of nuclei to study further such mass variations.
Conclusions
We have found
that a simple function of three parameters suffices to fit observed neutron
total scattering cross sections from diverse nuclei ^{6}Li– ^{238}U and for energies ranging
from 10 to 600 MeV. That function was predicated upon the values of partial
total cross sections evaluated using a gfolding optical potential for scattering. The patterns of the calculated
partial cross sections suggested that two of the parameters, a and e, could be set by parabolic functions of energy for all
masses. Then allowing the third parameter l_{0} to vary, values
could be found with which the appropriate sum over partial cross section given
by the function form exactly match measured data. The optimal values of l_{0}
varied smoothly with both energy and target mass. The energy variations l_{0}(E)
could be characterized by yet another simple function form as could the mass
variations
. However, the reasonable fit criterion that final results
remain within ±5% of observation showed that refinement of the functional
dependences of the parameter l_{0} in particular awaits the
results of a far more complete study involving as many target masses as
possible and for many more energies, particularly below 100 MeV where the total
crosssection data show largescale oscillatory structure.
Nonetheless,
on the basis of the limited set of nuclei and energies considered, there is a
threeparameter function form for partial total cross sections that will give
neutron total cross sections as required in any application without recourse to
phenomenological optical potential parameter searches. One may use tabulations of
l_{0}(E) and interpolations on that table or indeed a
better database formed by considering many more energies and many more nuclear
targets, to get cross sections satisfying the reasonable fit criterion. A
caveat being that any special gross nuclear structure effect, such as a halo
matter distribution for example, must be separately considered.
No comments:
Post a Comment