Superscaling in nuclei: A search for a scaling function beyond the relativistic Fermi gas model

We construct a scaling function $f\left({\psi }^{\prime }\right)$ for inclusive electron scattering from nuclei within the coherent density fluctuation model (CDFM). The latter is a natural extension to finite nuclei of the relativistic Fermi gas model within which the scaling variable ${\psi }^{\prime }$ was introduced by Donnelly and collaborators. The calculations show that the high-momentum components of the nucleon momentum distribution in the CDFM and their similarity for different nuclei lead to quantitative description of the superscaling in nuclei. The results are in good agreement with the experimental data for different transfer momenta showing superscaling for negative values of ${\psi }^{\prime }$, including those smaller than $-1$.

Figure 1

Superscaling behavior of inclusive electron scattering. The gray area represents experimental data [8] for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{27}\mathrm{Al}$, and ${}^{197}\mathrm{Au}$ at $q=1000\text{MeV}∕c$. The solid line is the RFG scaling function calculated using Eq. (24) with $k_F=1.191{\text{fm}}^{-1}$ from Ref. 8.

Figure 2

Results for the scaling function in the CDFM (solid line) calculated using Eqs. (30, 31) at $q=1000\text{MeV}∕c$ and for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{27}\mathrm{Al}$, and ${}^{197}\mathrm{Au}$ (with $b=1.0\text{fm}$ for the latter) compared with the data (gray area) from Ref. 8. The dotted line is the RFG result using Eq. (24). The dashed line in the case of ${}^{197}\mathrm{Au}$ corresponds to the CDFM result with $b=0.449\text{fm}$.

Figure 3

(a) Nucleon momentum distribution $n\left(k\right)$ calculated in the CDFM using Eqs. (21, 23) for ${}^{12}\mathrm{C}$, ${}^{40}\mathrm{Ca}$, and ${}^{197}\mathrm{Au}$ (for the latter with $b=0.449\text{fm}$ and $b=1.0\text{fm}$); (b) the weight function ${\mid F\left(x\right)\mid }^2$ of the CDFM calculated using Eq. (23) for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{40}\mathrm{Ca}$, and ${}^{197}\mathrm{Au}$ (for the latter with $b=0.449\text{fm}$ and $b=1.0\text{fm}$).

Figure 4

Results of the CDFM for the superscaling functions of ${}^4\mathrm{He}$ (solid line) and ${}^{197}\mathrm{Au}$ (dashed line) at $q=1650\text{MeV}∕c$ compared with the experimental data (gray area) from Ref. 8.

Figure 5

Results of the CDFM for the superscaling function of ${}^{12}\mathrm{C}$ at $q=500\text{MeV}∕c$ (solid line) compared with the experimental data (gray area) for $q$ in the interval from 500 to $600\text{MeV}∕c$ from Ref. 8.