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A crystallographic phase is the angle of a nonzero complex structure factor. Writingseparates amplitude and phase for a reflection indexed by the integer Miller index vector $\mathbf{h}$. Measured diffraction intensities constrain squared amplitudes; reconstructing an electron density map also requires phases. This missing information is the phase problem.^[ref]

Using fractional crystallographic coordinates, one consistent Fourier convention isThe phase controls where a harmonic contributes maxima and minima. For example, the original positive profiles $3+2\cos (2\pi x)$ and $3-2\cos (2\pi x)$ have equal Fourier coefficient magnitudes, but opposite phases for their first harmonics. Their squared amplitudes cannot locate the density maxima.

Phase also depends on the coordinate convention. Translating every atomic position by a fractional vector $\mathbf{t}$ multiplies the structure factor by $e^{2\pi i\mathbf{h}\cdot \mathbf{t}}$. For $\mathbf{h}=(1,0,0)$ and $\mathbf{t}=(1/4,0,0)$, the phase increases by $\pi /2$ while the squared amplitude is unchanged. Shifting the coordinate origin by that vector has the opposite sign. A phase is defined modulo $2\pi$; when $F_{\mathbf{h}}=0$, its phase is undefined. Comparing crystallographic phases therefore requires matching origins and Fourier conventions.

A difference electron density map is an electron density map calculated to reveal disagreement between measured diffraction amplitudes and those predicted by a crystal structure model. The common $F_o-F_c$ map combines observed-minus-calculated amplitude information with phases, usually derived from the model. In practice map coefficients include weighting rather than simply subtracting raw measured intensities.^[ref]

For one simplified convention, let $F_c(\mathbf{h})=|F_c(\mathbf{h})|e^{i{\varphi }_c(\mathbf{h})}$. A schematic unweighted difference map isThe crystallographic phases remain necessary even though the coefficients represent differences. Positive features can indicate density missing from the model; negative features can indicate excessive modeled density. Errors in positions, crystallographic site occupancies or atomic displacement parameters can all contribute.

An original one-dimensional idealization takes ${\rho }_o(x)=3+2\cos (2\pi x)$ and ${\rho }_c(x)=3+\cos (2\pi x)$, with matching phases. Their difference isIt is positive near0 and negative near $1/2$, although its integral over one period is zero. Positive and negative residuals therefore need not mean that the total electron count is wrong.

Experimentally the difference electron density map guides corrections to a fitted macromolecular structural model. Finite diffraction data, uncertain phases and model-derived coefficients affect its interpretation; a peak alone does not uniquely identify an atom or chemical species.

An 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.^[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 formHere 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:Both profiles have the same Fourier coefficient magnitudes, but their maxima occur half a period apart. Squared magnitudes alone cannot distinguish them.

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.^[ref]

A macromolecular structural model is a representation of the three-dimensional organization of a biological macromolecule or complex. It can include atom identities, coordinates, crystallographic site occupancy, atomic displacement parameters and alternative conformations. A deposited coordinate set is an interpreted model supported by measurements, rather than the measurements themselves.^[ref]

For X-ray crystallography, an electron density map constrains where atoms can be placed, while chemical geometry and experimental uncertainty constrain interpretation. Poorly resolved regions can remain absent from the model or be represented less precisely. A numerical coordinate printed to several decimal places does not establish equally fine positional accuracy.

An original coordinate comparison illustrates the distinction. Suppose two models place the same atom at $(0,0,0)$ and $(0.2,0,0)$ in angstrom units. Their separation is $0.2\mathring{\mathrm{A}}$, but that number alone cannot say which model is better. One must examine agreement with the data and the relevant uncertainty, not merely the number of digits or the difference between the coordinates.

A concrete example is calmodulin structure1CLL in the Protein Data Bank. The record contains a refined model with four calcium ions, while its primary citation describes poorly defined terminal residues. Both the observed binding environment and the unresolved regions are part of interpreting the structure.^[ref]

Structural claims should therefore specify which components and conformations are modeled, what experiment supports them and what remains unresolved. A model useful for one question does not automatically establish the complete dynamics or every biochemical activity of the molecule.