Superscaling, scaling functions, and nucleon momentum distributions in nuclei

The scaling functions $f({\psi }^{\text{'}})$ and $F(y)$ from the ${\psi }^{\text{'}}$- and $y$-scaling analyses of inclusive electron scattering from nuclei are explored within the coherent density fluctuation model (CDFM). In addition to the CDFM formulation in which the local density distribution is used, we introduce a new equivalent formulation of the CDFM based on the one-body nucleon momentum distribution (NMD). Special attention is paid to the different ways in which the excitation energy of the residual system is taken into account in $y$ and ${\psi }^{\text{'}}$ scaling. Both functions, $f({\psi }^{\text{'}})$ and $F(y)$, are calculated using different NMDs and compared with the experimental data for a wide range of nuclei. The good description of the data for $y<0$ and ${\psi }^{\text{'}}<0$ (including ${\psi }^{\text{'}}<-1$) makes it possible to show the sensitivity of the calculated scaling functions to the peculiarities of the NMDs in different regions of momenta. It is concluded that the existing data on ${\psi }^{\text{'}}$ and $y$ scaling are informative for NMDs at momenta not larger than $2.0–2.5{\text{fm}}^{-1}$. The CDFM allows us to study simultaneously and on the same footing the role of both basic quantities—the momentum and density distributions—for the description of scaling and superscaling phenomena in nuclei.

Figure 1

Scaling function $f({\psi }^{\text{'}})$ in the CDFM (solid line) at $q=1560$ $c$ for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{27}\mathrm{Al}$, and ${}^{197}\mathrm{Au}$. The results are obtained using Eqs. (53) and (49) and equivalently by Eqs. (54) and (51) with $n(p)$ from the CDFM [Eq. (31)]. The experimental data from [22] are given by the shaded area. The RFG result [Eq. (20)] is shown by the dotted line.

Figure 2

Nucleon momentum distribution $n(k)$ from (i) CDFM: the calculated results using Eqs. (31) and (36) for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, ${}^{27}\mathrm{Al}$, ${}^{56}\mathrm{Fe}$, and ${}^{197}\mathrm{Au}$ are combined by the shaded area ($n_{\mathrm{CDFM}}$); (ii) $y$-scaling data [12] given by open squares, circles, and triangles for ${}^4\mathrm{He}$, ${}^{12}\mathrm{C}$, and ${}^{56}\mathrm{Fe}$, respectively; (iii) $y_{\mathrm{CW}}$ analyses [13, 14] for ${}^{56}\mathrm{Fe}$ [Eq. (9)] ($n_{\mathrm{CW}}$), $n_{\mathrm{MFA}}$ [Eq. (10)], $n_{\mathrm{corr}}$ [Eq. (11)]; (iv) LFD approach [25] [Eqs. (76)–(79), (81), and (82)] for ${}^{56}\mathrm{Fe}$ ($n_{\mathrm{LFD}}$); and (v) MFA calculations using Woods-Saxon single-particle wave functions for ${}^{56}\mathrm{Fe}$ ($n_{\mathrm{WS}}$).

Figure 3

The $y$-scaling function $F(y)$ for ${}^4\mathrm{He}$ and ${}^{56}\mathrm{Fe}$ calculated in the CDFM [Eq. (65)] (solid and thick dashed lines), from the $y_{\mathrm{CW}}$-scaling approach [13, 14] [Eqs. (9)–(11)] for ${}^{56}\mathrm{Fe}$ (dash-dotted line), and from the approach [25] within the LFD method [Eqs. (76)–(79), (81), and (82)] (thin solid and dashed lines). The results are compared with the $y_{\mathrm{CW}}$-scaling data taken from [13, 14].

Figure 4

The ${\psi }^{\text{'}}$-scaling function $f({\psi }^{\text{'}})$ at ${}^{56}\mathrm{Fe}$ and $q=1000$MeV/$c$ calculated from Eq. (75) using $n(k)$ from (i) the $y_{\mathrm{CW}}$-scaling analysis [13, 14] [Eqs. (9)–(11)] (solid line) and (ii) the approach [25] within the LFD method [Eqs. (76)–(79), (81), and (82)] (dashed line). The CDFM result obtained using Eqs. (45) and (46) is given by the dotted line. The experimental data given by the shaded area are from [21].