# Electron-density distribution

The electron-density distribution is the three-dimensional realspace arrangement of the electrons in the material. Because electrons are quantum mechanically delocalized, each electron occupies a 'fuzzy' region of space (electron cloud). The total electron density--the summation of the electron distribution for every electron in every atom--is thus inherently spread spatially; e.g a map of electron distribution will have diffuse boundaries. Nevertheless, when the electron distribution is visualized, it is often shown as a surface, representing an isosurface of constant electron-density.

The exact electron-density distribution within a unit cell of a crystal can be reconstructed by carefully fitting the peak heights measured in an x-ray diffraction experiment (crystallographic 'structure solution').

The electron distribution at a larger scale can be similarly reconstructed by fitting small-angle scattering data (SAXS or GISAXS). In this case, one is probing the average electron-density distribution at the nanoscale, without resolving the exact arrangement of electron-distribution (atoms) within this nanostructure. The specific electron-distribution within a measurement volume (as opposed to the average structure defined by a unit-cell) can be reconstructed using coherent methods such as CDI or ptychography.

## Mathematical form

The electron-density (number of electrons per unit volume) can be computed from the atomic scattering factor, $f_1$; or from the refractive index:

\begin{alignat}{2} \rho_e & = \frac{\rho N_a f_1}{M_a} \\ & = \frac{2 \pi}{\lambda^2 r_e} \delta \end{alignat}