# The n-th order Moment of the Nuclear Charge Density and Contribution from the Neutrons

## TL;DR

This paper derives relativistic expressions for the moments of nuclear charge density, highlighting the dependence on neutron density and implications for electron scattering in neutron-rich nuclei.

## Contribution

It provides a consistent derivation of relativistic and non-relativistic expressions for nuclear charge density moments up to second order using the Foldy-Wouthuysen transformation.

## Key findings

- Relativistic and non-relativistic msr expressions are equivalent up to 1/M^2.
- Higher moments depend on neutron density, especially the 4th moment.
- The 4th moment offers insights into neutron distribution in nuclei.

## Abstract

The relativistic expression for the $n$-th order moment of the nuclear charge density is presented. For the mean square radius(msr) of the nuclear charge density, the non-relativistic expression, which is equivalent to the relativistic one, is also derived consistently up to $1/M^2$ with use of the Foldy-Wouthuysen transformation. The difference between the relativistic and non-relativistic expressions for the msr of the point proton density is also discussed. The $n(\ge 4)$-th order moment of the nuclear charge density depends on the point neutron density. The 4-th order moment yields a useful information on the msr of the point neutron density, and is expected to play an important role in electron scattering off neutron-rich nuclei.