Reaction cross section described by a black sphere approximation of nuclei

Abstract

We identify a length scale that simultaneously accounts for the observed proton-nucleus reaction cross section and diffraction peak in the proton elastic differential cross section. This scale is the nuclear radius, a, deduced from proton elastic scattering data of incident energies higher than ∼ 800 MeV, by assuming that the target nucleus is a "black" sphere. The values of a are determined so as to reproduce the angle of the first diffraction maximum in the scattering data for stable nuclei. We find that the absorption cross section, πa 2 , agrees with the empirical total reaction cross section for C, Sn, and Pb to within error bars. This agreement persists in the case of the interaction cross section measured for a carbon target. We also find that 3/5a systematically deviates from the empirically deduced values of the root-mean-square matter radius for nuclei having mass less than about 50, while it almost completely agrees with the deduced values for A > ∼ 50. This tendency suggests a significant change of the nuclear matter distribution from a rectangular one for A < ∼ 50, which is consistent with the behavior of the empirical charge distribution. The size of atomic nuclei is considered to be well deduced from empirical data for the proton-nucleus elastic differential cross section, dσ el /dΩ, and the total reaction cross section, σ R ≡ σ T − σ el , where σ T is the total cross section. So far, the analysis that respects both data in deducing the nuclear size has not been completed in particular for proton incident energies, T p , higher than 800 MeV. Various approximate theories based on optical potentials have been proposed to reproduce the elastic scattering data, while they usually tend to overestimate the reaction cross section for 800 MeV < ∼ T p < ∼ 1000 MeV (e.g., Ref.[1] and references therein).In Ref.[2], we constructed a method for deducing the nuclear size by focusing on the peak angle in the proton-nucleus elastic differential cross section measured at T p > ∼ 800 MeV, where the corresponding optical potential is strongly absorptive. In this method, we regard a nucleus as a "black" (i.e., purely absorptive) sphere of radius a, and determine a in such a way as to reproduce the angle of the observed first diffraction peak. If we multiply a by 3/5, a ratio between the root-mean-square and squared off radii for a rectangular distribution, the result for stable nuclei of A > ∼ 50 shows an excellent agreement with the root-mean-square radius, r m , of the matter density distribution as determined from conventional scattering theories so as to reproduce the overall diffraction pattern and analyzing power in the proton elastic scattering.In this paper, we extend such a previous analysis to the case of A < ∼ 50, and find out a systematic deviation between 3/5a and r m . We next show that the present method is effective at explaining the observed reaction cross sections for stable nuclei ranging from light to heavy ones. In the black sphere approximation of a nucleus, where the geometrical cross section ...

Formula for Proton–Nucleus Reaction Cross Section at Intermediate Energies and Its Application

“…2 we compare the formula (7) with the empirical data for the reaction cross section for targets such as C, Ca, Zr, and Pb. The formula (7) in which a 0 is set to be a value determined from the measured angle of the first diffraction maximum in proton elastic scattering [11,12] well reproduces the T p dependence of the measured reaction cross section [14,19] for T p down to about 100 MeV. Remarkably, it agrees almost completely with the 100-180 MeV data updated by Auce et al [19].…”

“…For this purpose we note the black sphere picture [11] in which the black sphere radius a is determined from 2pa sin(θ M /2) = 5.1356 · · · with the proton incident momentum in the center-of-mass frame, p, and the first peak angle for the measured diffraction in proton-nucleus elastic scattering, θ M . We then recall the known properties from the black sphere picture for T p = 800-1000 MeV [12]: (i) the black sphere radius a globally behaves as 1.214A…”

“…Recently we systematically analyzed the proton elastic scattering and reaction cross section data for stable nuclei at proton incident energy of ∼800-1000 MeV on the basis of a black sphere picture of nuclei [11,12]. This picture is originally expected to give a decent description of the reaction cross section for any kind of incident particle that tends to be attenuated in the nuclear interiors.…”

Erratum: Systematic analysis of reaction cross sections of carbon isotopes [Phys. Rev. C75, 044607 (2007)]

“…According to the NTG calculation, the total reaction cross section of 22 C is estimated to be 2200-2450 mb at 40 A MeV, and 1500-1600 mb around 900 A MeV. In order to see the implication of these results, we refer to the black-sphere picture [48,49] or the strong absorption model [16]: These pictures include only one scale, the nuclear radius, a. If one determine the values of a so as to reproduce the angle of the first diffraction maximum in the proton-nucleus elastic scattering data, the absorption cross section, πa 2 , agrees with the empirical total reaction cross section [49].…”

Extracting nuclear sizes of medium to heavy nuclei from total reaction cross sections

“…The reaction radius may somewhat be related to the one defined in the black sphere model [96,97]. The a R depends on E through the different energy-dependence of the np and pp total cross sections, σ tot np and σ tot pp [98], that enter the profile function (5).…”