Reaction cross section described by a black sphere approximation of nuclei

Akihisa Kohama, Kei Iida, and Kazuhiro Oyamatsu

Phys. Rev. C 72, 024602 – Published 10 August 2005

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, $π a^{2}$ , 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 $\sqrt{3 / 5} a$ systematically deviates from the empirically deduced values of the root-mean-square matter radius for nuclei having masses 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.

Received 12 October 2004

DOI:https://doi.org/10.1103/PhysRevC.72.024602

Authors & Affiliations

Akihisa Kohama^1,*, Kei Iida^1,2, and Kazuhiro Oyamatsu^1,3,4

¹RIKEN (The Institute of Physical and Chemical Research), 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
²RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973-5000, USA
³Department of Media Theories and Production, Aichi Shukutoku University, Nagakute, Nagakute-cho, Aichi-gun, Aichi 480-1197, Japan
⁴Department of Physics, Nagoya University, Furo-cho, Chigusa-ku, Nagoya, Aichi 464-8602, Japan

^*Corresponding author: kohama@riken.jp (A. Kohama)

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Vol. 72, Iss. 2 — August 2005

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Figure 1
(Color) $r_{BS}$ as a function of mass number, A. For nuclei of $A > 50$ , we have plotted the values of $r_{m}$ adopted in Ref. 2. For the error bars in $r_{BS}$ and $r_{m}$ , see Ref. 2. For comparison we also plot the results for $r_{m}$ derived in the following references: For ${}^{12}C$ , in Refs. 5, 6, 7 ( $T_{p} = 800 MeV$ ); for ${}^{13}C$ , in Refs. 5, 7 ( $T_{p} = 800 MeV$ ); for ${}^{16}O$ , in Refs. 8, 9 ( $T_{p} = 1000 MeV$ ); for ${}^{28}{Si}$ , in Ref. 8 ( $T_{p} = 1000 MeV$ ); for ${}^{32}S$ , in Ref. 8 ( $T_{p} = 1000 MeV$ ); for ${}^{39}K$ , in Ref. 8 ( $T_{p} = 1000 MeV$ ); for ${}^{40, 48}{Ca}$ , in Ref. 10 ( $T_{p} = 796 MeV$ ), in Ref. 8 ( $T_{p} = 1000 MeV$ ); for ${}^{40, 42, 44, 48}{Ca}$ , in Ref. 11 ( $T_{p} = 796 MeV$ ), in Refs. 12, 9 ( $T_{p} = 1044 MeV$ ); for ${}^{46, 48}{Ti}$ , in Ref. 13 ( $T_{p} = 799.3 MeV$ ); for ${}^{48}{Ti}$ , in Ref. 12 ( $T_{p} = 1044 MeV$ ). For ${}^{4}{He}$ , the root-mean-square radius of the charge density distribution deduced from electron elastic scattering data [14, 15] is plotted instead of $r_{m}$ , because no $r_{m}$ is available for this case. The crosses (×) denote the root-mean-square matter radii of the point nucleon distributions, and the circles (○) denote those folded with the nucleon form factor. The dashed curve represents $0.94 A^{1 / 3}$ fm. The dotted line shows $A = 50$ .Reuse & Permissions
Figure 2
Difference, $r_{BS} - r_{m}$ , as a function of mass number, A. The crosses (×) and the circles (○) are calculated from the corresponding values of $r_{m}$ in Fig. 1. The dotted line shows $r_{BS} = r_{m}$ and $A = 50$ .Reuse & Permissions
Figure 3
(Color) Absorption cross section, $σ_{BS}$ , of a proton of $T_{p} ≳ 800 MeV$ by a target nucleus of mass number, A. For comparison, we plot the empirical data for the proton-nucleus total reaction cross section, $σ_{R}$ (○), which are listed in Ref. 46 for ${}^{9}{Be}$ , ${}^{27}{Al}$ , C, Cu, Sn, and Pb. For the last four elements the value of $σ_{R}$ is the average over the isotopic abundance in a target. For these data, we set A as the mass number of the most abundant isotope and assign the uncertainty in A due to the natural abundance. The dashed curve denotes $(5 / 3) π (0.94 A^{1 / 3})^{2}$ fm $^{2}$ . Inset: $σ_{BS}$ versus $σ_{R}$ for C, Sn, and Pb. The dotted line represents $σ_{BS} = σ_{R}$ .Reuse & Permissions
Figure 4
(Color online) Absorption cross section, $σ_{BS}$ , for a projectile stable nucleus of $A < 30$ and a ${}^{12}C$ target. We fix $r_{BS} = 2.086 \pm 0.05$ fm for ${}^{12}C$ . For comparison, we plot the interaction cross section, $σ_{I}$ (○), measured at $E / A ≳ 800 MeV$ for a projectile of ${}^{3, 4}{He}$ , ${}^{6, 7}{Li}$ , ${}^{9}{Be}$ , ${}^{10, 11}B$ , ${}^{12, 13}C$ , ${}^{14, 15}N$ , ${}^{16, 17, 18}O$ , ${}^{19}F$ , ${}^{20, 21}{Ne}$ , ${}^{23}{Na}$ , and ${}^{24, 25}{Mg}$ incident on a ${}^{12}C$ target [51]. The dashed curve represents $(5 / 3) π (2.086 + 0.94 A^{1 / 3})^{2}$ fm $^{2}$ . Inset: $σ_{BS}$ versus $σ_{I}$ for ${}^{4}{He}$ , ${}^{12, 13}C$ , ${}^{14}N$ , ${}^{16}O$ , ${}^{20}{Ne}$ , and ${}^{24}{Mg}$ . The dotted line represents $σ_{BS} = σ_{I}$ .Reuse & Permissions

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