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 E ∞k² 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) = 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

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

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 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 E^1/3. 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 d_i 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 ⁶Li– ²³⁸U 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.

Epilogue