Nuclear surface diffuseness revealed in nucleon-nucleus diffraction

The nuclear surface provides useful information on nuclear radius, nuclear structure, as well as properties of nuclear matter. We discuss the relationship between the nuclear surface diffuseness and elastic scattering differential cross section at the first diffraction peak of high-energy nucleon-nucleus scattering as an efficient tool in order to extract the nuclear surface information from limited experimental data involving short-lived unstable nuclei. The high-energy reaction is described by a reliable microscopic reaction theory, the Glauber model. Extending the idea of the black sphere model, we find one-to-one correspondence between the nuclear bulk structure information and proton-nucleus elastic scattering diffraction peak. This implies that we can extract both the nuclear radius and diffuseness simultaneously, using the position of the first diffraction peak and its magnitude of the elastic scattering differential cross section. We confirm the reliability of this approach by using realistic density distributions obtained by a mean-field model.

Figure 1

Elastic scattering differential cross sections of (a) $N\text{−}{}^{120}\mathrm{Sn}$ and (b) $N\text{−}{}^{208}\mathrm{Pb}$ systems calculated with 2pF density distributions at 325, 550, and 800 MeV. The cross sections are multiplied by ${10}^5$ and ${10}^{10}$ for those at 550 and 800 MeV, respectively. The bottom panels plot the corresponding 2pF density distributions of (c) ${}^{120}\mathrm{Sn}$ and (d) ${}^{208}\mathrm{Pb}$, respectively, with various diffuseness parameter $a$. All density distributions give the same root-mean-square radius.

Figure 2

Scattering angles at the first peak positions of the elastic scattering differential cross sections incident at 325, 550, and 800 MeV as a function of mass number.

Figure 3

Elastic scattering differential cross sections at the first peak position incident at (a) 325, (b) 550, and (c) 800 MeV as a function of mass number.

Figure 4

(Top) Imaginary part of the spatial distribution of the scattering amplitude, and (bottom) its cumulative sum at the first peak of the elastic scattering differential cross section of (a), (c) ${}^{120}\mathrm{Sn}$ and (b), (d) ${}^{208}\mathrm{Pb}$, as a function of impact parameter $b$, incident at 325, 550, and 800 MeV. An arrow indicates the half-density radius of (c) ${}^{120}\mathrm{Sn}$ and (d) ${}^{208}\mathrm{Pb}$, respectively. See text for details.

Figure 5

Same as Fig. 4 but at the second peak of the elastic scattering differential cross section.

Figure 6

Relative deviations of the rms radius obtained by the HF+BCS method and the deduced rms radii incident at (a) 325, (b) 550, and (c) 800 MeV.

Figure 7

Deviations of the nuclear diffuseness deduced at (a) 325 and (b) 800 MeV from that at 550 MeV. See text for details.

Figure 8

Comparison of HF+BCS density and deduced 2pF density distributions of (a), (e) ${}^{120}\mathrm{Sn}$; (b), (f) ${}^{208}\mathrm{Pb}$; (c), (g) ${}^{120}\mathrm{Zr}$; and (d), (h) ${}^{140}\mathrm{Sn}$ at different incident energies, $E_1=325$, $E_2=550$, and $E_3=800\mathrm{MeV}$, drawn in (top) linear and (bottom) logarithmic scales. “Fit” denotes the 2pF density distribution deduced directly from the HF+BCS density. See text for details.

Figure 9

Nuclear diffuseness deduced from the HF+BCS density distributions incident at 550 MeV.

Figure 10

Comparison of the total reaction cross sections obtained by the Glauber (horizontal axis) and BS models (vertical axis). A solid-thin line indicates a $y=x$ line plotted to guide the eyes.

Figure 11

Difference of the scattering angles of the first peak position obtained by the Glauber calculation and the BS estimate incident at (a) 325, (b) 550, and (c) 800 MeV.

Figure 12

(a) Scattering angles of the first peak position and (b) its magnitude of the elastic scattering differential cross section at the peak position with the $pN$ and averaged $NN$ profiles functions. The HF+BCS density distributions with the SkM* interaction [35, 39] are used.