Saturday 24 August 2019

Physics of Nucleon-Nucleus Scattering - Chapter X


Chapter Ten

SIMPLE FUNCTIONAL FORM
FOR NUCLEON-NUCLEUS 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 6Li and 238U. 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 g-folding 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 partial-wave scattering (S) matrices specified at energies Ek2 by

, .........................(10.1)

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

,.......(10.2)

,.......(10.3)

and

................(10.4)

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 l0(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, energy-dependent, optical potentials generated from a g-folding formalism [7]. While those g-folding 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 6Li, 12C, 19F, 40Ca, 89Y, 184W, 197Au, 208Pb, and 238U, 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 cross-section values themselves can be used to fine-tune the details and of the parameter l0 in particular. We chose to use results from g-folding 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 g-folding prescription for the nucleon-nucleus 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 g-folding 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 = l0(E) = lmax, giving a triangle in angular momentum space. In that case, the total reaction cross section equates to the area of that triangle and

.  ................................(10.6)

Then with lmax~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 l0 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 g-folding 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 nucleon-nucleus scattering above 300 MeV. Also relativistic effects in scattering, other than simply the use of relativistic kinematics in the distorted-wave 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 l0, a, and e from which to find ones that reproduce the measured total cross-section data. One must also note that the g-folding 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 g-folding 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, so-called nucleon halo, nuclei were studied [7, 121].
           
The results from analyses of 40A MeV scattering of 6He 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 6He-hydrogen scattering was 16%–17% larger than found using the standard shell model prescription. That and the momentum transfer properties of the 6He-P differential cross section were convincing evidence of the neutron-halo nature of 6He. 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 208Pb 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 g-folding optical model calculations. Each curve shown therein is the result of a search for the best fit values of the three parameters l0, 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

..........................(10.7)

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 l0 in each case so that actual measured neutron total cross-section data were fit using Eq. (10.5). Numerical values for l0 from that process are presented in Table I.


Figure 10.1: The partial total cross sections for scattering of neutrons from 208Pb 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 l0 increase monotonically with both mass and energy and that is most evident in Fig. 10.2 where the optimal values l0(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 197Au (opaque diamonds) and for 208Pb (solid circles). The curves are the shapes deduced by a function of energy for the l0(E), which will be discussed subsequently.


TABLE I: l0 values with which the function form, Eq. (10.5), fits neutron total cross sections.

E(MeV)
6Li
12C
19F
40Ca
89Y
184W
197Au
208Pb
238U
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 l0 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 l0(E).


Plotting the values of l0 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 l0(A) at each energy, and that too will be discussed later.




Figure 10.3: The values of l0 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 l0(A).


The total neutron scattering cross sections generated using the function form for partial total cross sections with the tabled values of l0 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 g-folding 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 6Li, 12C, 19F, and 40Ca. The results have been scaled as described in the text to provide clarity.



Therein from bottom to top are shown the results for 6Li, 12C, 19F, and 40Ca 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 89Y (unscaled), 184W (unscaled), 197Au (shifted by 2b), and 238U (shifted by 3b) are compared with the base g-folding optical potential results and with the function forms with the optimal parameters. Again the g-folding 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 89Y, 184W, 197Au, and 238U. 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 208Pb. In this case we used Skyrme-Hartree-Fock model (SKM*) densities [194] to form the g-folding 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 208Pb.


Using the SKM* model structure, the g-folding 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 g-folding 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 g-folding 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 l0(E,A) vary smoothly and monotonically with both E and A in giving the partial cross-section sums that perfectly match measured values of the total cross sections.
           
For energies above 250 MeV, the l0 values approximate well as straight lines and a likely representation of all of the sets of l0 values is found with the energy-dependent function

. .................................(10.8)

The values of the parameters that lead to the curves depicted in Fig. 10.2 are listed in Table II. The result for 208Pb nonetheless is as good a fit as found in the other eight cases.




TABLE II. Values of parameters defining l0(E).

A
c1
c2
c3
E0
b
χ2
χ2
(<100)
6Li
4.665 x 10-3
3.582
1.537
13.87
3.670 x 10-2
0.025

12C
9.103 x 10-3
4.865
3.449
21.35
3.285 x 10-2
0.30

19F
1.374 x 10-2
5.808
4.794
24.42
2.880 x 10-2
0.89

40Ca
2.272 x 10-2
6.820
4.896
25.97
1.937 x 10-2
0.73

89Y
3.31 x 10-2
8.357
5.256
29.47
1.470 x 10-2
2.59
2.0
184W
4.27 x 10-2
11.50
7.574
43.73
1.310 x 10-2
4.2
3.0
197Au
4.41 x 10-2
11.65
7.635
43.96
1.277 x 10-2
4.5
3.2
208Pb
4.067 x 10-2
13.43
9.402
62.51
1.400 x 10-2
5.0
3.3
238U
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 χ2 which in this case are defined by

, ...............................................(10.9)

with the sum extending over the 24 energies used. For the heavier masses the values of χ2 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 c1, c2, c3, E0, and b themselves.
           
Of the parameter values for the , those for 208Pb 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 c1 (solid circles) and of b (open down triangles) have been multiplied by 10 for convenience of plotting. The other parameter values are identified as c2 (open squares), c3 (solid diamonds), and E0 (solid triangles). Clearly there is a smooth mass trend of these values with the exception of the entries for 208Pb. But the 208Pb values are based only on achieving the smallest χ2 value as defined by Eq. (10.9). Using parameter values consistent with the smooth mass trend, the χ2 for the fit to the 208Pb 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 l0(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 mass-dependent 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 three-parameter prescription with parameter values set by the energy function forms for l0(E), a, and e. Those differences for 40Ca and heavier nuclei are depicted in the top segment, while those for 6Li, 12C, and 19F are given in the middle segment. In the bottom segment are the differences between the optimal data fit values of l0(E) and those specified by using Eq. (10.8) for 40Ca and heavier nuclei.
           


In the top segment, those differences for 40Ca and heavier nuclei are shown. Curiously these variations look sinusoidal with argument proportional to E1/3. In the middle segment the differences for the three light mass nuclei are given with the solid, dashed, and long-dashed curves depicting the values for 6Li, 12C, and 19F, 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 l0(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 40Ca 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 40Ca, 89Y, 184W, 197Au, and 238U are shown in the bottom segment, respectively, by the solid curve connecting solid circles, the long-dashed curve connecting solid diamonds, the dashed curve, the solid curve connecting opaque diamonds, and the dot-dashed curve. Of particular note is that the differences between these fit and function values of the l0(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 first-order 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 l0 as a function of mass

As noted above, the l0 parameter values vary smoothly with mass. In fact we find that a good representation of those values is given by

. ............................................(10.10)

With this mass variation form, the coefficients di are as set
out in Table III.

TABLE III. Values of parameters defining l0(A).
Energy
d1
d2
d3
d4
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 l0 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 l0(A) that give best fits to total cross-section data. Details are specified in the text.



Specifically the coefficients are shown by the solid circles (d1), by the solid squares connected by the long-dashed lines (d2), by the opaque diamonds (d3), and by the opaque up-triangles connected by dashed lines (d4). Again for clarity the actual values found for d1 and d4 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 6Li238U and for energies ranging from 10 to 600 MeV. That function was predicated upon the values of partial total cross sections evaluated using a g-folding 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 l0 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 l0 varied smoothly with both energy and target mass. The energy variations l0(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 l0 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 cross-section data show large-scale oscillatory structure.
           
Nonetheless, on the basis of the limited set of nuclei and energies considered, there is a three-parameter 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 l0(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.

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