Moments of the point-neutron distribution observed through electron scattering in nuclei

The $n(le 4)$-th moment of the nuclear charge density depends on the $m(mle (n-2))$-th moments of the neutron charge-density, in addition to the $m(mle n)$-th moments of the proton onecite{ks}. The analysis of the moments makes it possible to explore the both point-proton and neutron distributions on the same basis through well-known electromagnetic interaction. There are experimental data of the 4th moments of charge densities for $^{48}$Cacite{emrich} and $^{208}$Pbcite{vries}, those of the $n(le 8)$-th moment for $^3$Hcite{mit} and of the $n(le 10)$-th one for $^3$Hecite{mai}, although the experimental errors of the higher moments are not small enough to be discussed in detail. We have analyzed the calculated values of the 4th moments with the $20$ mean-field(MF) models by the least squares method. The neutron skin thickness has been estimated to be $0.220(0.026)$ and $0.275(0.070)$ fm in $^{40}$Ca and $^{208}$Pb, respectively, in the relativistic MF models, while $0.121(0.036)$ and $0.162(0.068)$ fm in the non-relativistic onescite{kss}. Particularly interesting is the analysis of the moments of the few-body systems, whose wave functions are obtained almost exactly for less phenomenological nuclear interactions. These investigations are in progress, but rough estimation shows that a half of the 6th moment of the charge distribution in $^3$H stems from the neutron charge-density. Precise experimental data for the mirror nuclei may provide us with rich information on their neutron distributions and, moreover, on the neutron size itself, in addition to the proton ones. The Fourier-Bessel and the sum of Gaussian methods fail to estimate the 6th moments of $^{48}$Ca and $^{208}$Pbcite{toshio} from the experimental data available at presentcite{vries}, in contrast to those for the few-body systemscite{mit,mai}. More experimental values of the higher moments are expected to be obtained from low-energy electron scattering. In futurecite{suda}, we would be able to explore in terms of the moments how the neutron distributions play a peculiar role in the structure of neutron-rich unstable nuclei.