Sensitivity analysis of neutron total and absorption cross sections within the optical model

Distinct maxima and minima in neutron total and absorption cross-section uncertainties when optical-model parameters are varied have been observed in large-scale covariance calculations. These features were seen over a wide mass range (20–210) and for energies up to 20 MeV. Here we investigate the physical origin of the observed patterns over an extended energy range (1 keV to 200 MeV). We have calculated the sensitivity of the cross sections for a specific nucleus (${}^{56}\mathrm{Fe}$) to variations of the 15 parameters of a standard global optical potential parametrization, and have also carried out calculations for alternative global optical potentials over the original wide mass and energy ranges. We find that simple physical descriptions can be found in two energy ranges. Below approximately 100 keV, the patterns arise from the interplay of the $s$- and $p$-wave single-particle resonances. Above approximately 4 MeV, a single-phase-shift approximation (the Ramsauer model) describes the observed behavior. We discuss the potential importance of such sensitivity studies for further development of optical potentials.

Figure 1

Relative uncertainties $\Delta \sigma /\sigma$ for neutron total neutron cross sections showing the patterns obtained by propagating uncertainties in the parameters of the Koning-Delaroche spherical optical potential [4].

Figure 2

The total neutron cross section of ${}^{56}\mathrm{Fe}$. An optical-model calculation using the global potential of Ref. [4] is compared to the ENDF/B-VII.0 evaluation [7]. For clarity the cross section is divided by the black nucleus cross section, where $\lambda$ is the reduced wavelength and the radius $R$ is 1.415$A^{1/3}$ fm.

Figure 3

Sensitivity of the ${}^{56}\mathrm{Fe}+n$ total and absorption cross sections to perturbations of the potential strengths, radii, and diffuseness parameters. $D$ is the ratio of the fractional change in the cross section to the fractional change in the perturbed parameter; see Eq. (12). For energies at which a curve passes through $D=0$, the cross section is completely insensitive to the corresponding parameter.

Figure 4

Fractional uncertainties for neutron total cross sections on 75 target nuclei obtained with three different global spherical optical potentials.

Figure 5

Left column: Total cross section using the optical potential of Ref. [4] for 1-keV neutrons (solid lines); the dotted and dashed lines are calculated with $+2$% and $-2$% variations of $V_v$, respectively. Right column: Sensitivity of total cross section to $V_v$, as defined by Eqs. (13) and (14). The $s$- and $p$-wave parts of these quantities are shown, along with their sum.

Figure 6

Same as Fig. 5, but for 100 keV instead of 1 keV.

Figure 7

The solid lines are the $s$- and $p$-wave parts of the total cross section using the optical potential of Ref. [4] for 1-keV neutrons. The dotted and dashed lines are calculated with $+20$% and $-20$% variations of the surface imaginary potential strength $W_s$, respectively.

Figure 8

Neutron total cross sections ${\sigma }_{\mathrm{tot}}$ of ${}^{56}\mathrm{Fe}$ defined in Eq.(16), divided by the black-nucleus value. Bottom: Total cross-section sensitivities obtained by perturbing the parameters $\alpha$(solid line) and $a$ (dashed line) in Eqs. (16) and (17). The radius $R$ is 1.415$A^{1/3}$fm.

Figure 9

Sensitivity of the ${}^{56}\mathrm{Fe}+n$ total cross section to a perturbation of the real central potential strength, $V_v$. The upper panel shows the partial-wave sensitivities for two groups of $l$ values, and the bottom panel shows the total sensitivity. For clarity, the curves are multiplied by $\mathit{kR}$, where $k$ is the wave number and $R$ is 1.415$A^{1/3}$ fm.

Figure 10

Fractional total cross section uncertainties of ${}^{56}\mathrm{Fe}+\mathcal{n}$ obtained by perturbing the optical model parameters for different global potentials [4, 9, 10, 12]