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The crystallographic metric matrix is the Gram matrix of the three unit cell basis vectors. If $A$ has Cartesian columns $\mathbf{a},\mathbf{b},\mathbf{c}$, thenHere $\alpha ,\beta ,\gamma$ are the angles between $\mathbf{b},\mathbf{c}$; $\mathbf{a},\mathbf{c}$; and $\mathbf{a},\mathbf{b}$, respectively. The matrix converts fractional crystallographic coordinate differences into physical lengths: a fractional displacement $\mathbf{u}$ has squared length ${\mathbf{u}}^{T}G\mathbf{u}$.^[ref]

For an original oblique cell, choose $a=2$, $b=3$, $c=4$ angstroms, $\gamma =60^{\circ }$ and $\alpha =\beta =90^{\circ }$. ThenFor $\mathbf{u}=(1/2,-1/2,0{)}^T$,Ignoring the off-diagonal entries would instead give $13/4$, overestimating this distance. The cross terms encode the cell angles.

The reciprocal crystallographic metric matrix is $G^{-1}$ when the reciprocal basis has dot products ${\mathbf{a}}_i^{\ast }\cdot {\mathbf{a}}_j={\delta }_{ij}$, with no factor of $2\pi$. Thus the plane spacing for a nonzero Miller index column $\mathbf{h}$ satisfiesA basis change $A^{\prime }=AP$ gives $G^{\prime }=P^{T}GP$ and fractional coordinates ${\mathbf{u}}^{\prime }=P^{-1}\mathbf{u}$, preserving ${\mathbf{u}}^{\prime T}{G}^{\prime }{\mathbf{u}}^{\prime }={\mathbf{u}}^{T}G\mathbf{u}$. Distances do not change when the same crystal structure is described in another basis.^[ref]

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.

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]