= Electron density map
{title2=$\rho(\mathbf r)$}
An \b[electron density map] is a spatial reconstruction of <electron density>, typically displayed as a grid or contours. In <X-ray crystallography>, a beam scattered by a <crystal> produces a diffraction pattern: measured intensities constrain <structure factor> magnitudes, while <crystallographic phases> are also required for reconstruction. Coordinates and experimental data are combined in common model-assessment maps.<Electron Density Maps and Coefficient Files at RCSB PDB>[Introduction, diffraction data, map coefficients and phases]{ref}
With <fractional crystallographic coordinates> $\mathbf r$, integer <Miller index> vector $\mathbf h$ and <unit cell> volume $V_c$, an ideal Fourier reconstruction has the form
$$
\rho(\mathbf r)=\frac1{V_c}\sum_{\mathbf h}F_{\mathbf h}
\exp(-2\pi i\mathbf h\cdot\mathbf r).
$$
Here the <structure factors> are complex <Fourier coefficients>. Finite data and uncertain phases limit the reconstructed detail; this is part of the <phase problem>, not simply an issue of drawing contours more finely.
An original one-dimensional example makes the phase information visible:
$$
\rho_0(x)=3+2\cos(2\pi x),\qquad
\rho_\pi(x)=3-2\cos(2\pi x).
$$
Both profiles have the same <Fourier coefficient> magnitudes, but their maxima occur half a period apart. Squared magnitudes alone cannot distinguish them.
\Image[/electron-density-phase-example.png]
{title=Equal Fourier magnitudes can give different density profiles}
{id=electron-density-map-phase-example}
{description=Original one-dimensional profiles, with phase0 and phase$\pi$ for the first harmonic. Both have mean3 and first-harmonic coefficient magnitude1. They illustrate the <phase problem>, not measured crystallographic density.}
An <electron density map> is also distinct from a <macromolecular structural model> fitted to it. A <difference electron density map> can highlight disagreement between observed and model-predicted scattering. Common map coefficients already incorporate information from the model, so the resulting map is not wholly independent confirmation of every modeling choice.<Electron Density Maps and Coefficient Files at RCSB PDB>[Introduction, observed/calculated maps and difference maps]{ref}
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