= Crystallographic phase
{title2=$\phi_{\mathbf h}$}
A \b[crystallographic phase] is the angle of a nonzero complex <structure factor>. Writing
$$
F_{\mathbf h}=|F_{\mathbf h}|e^{i\phi_{\mathbf h}}
$$
separates 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>.<Electron Density Maps and Coefficient Files at RCSB PDB>[Introduction, diffraction data and map coefficients containing amplitudes and phases]{ref}
Using <fractional crystallographic coordinates>, one consistent Fourier convention is
$$
F_{\mathbf h}=\int_{\mathrm{cell}}\rho(\mathbf r)
e^{2\pi i\mathbf h\cdot\mathbf r}\,dV,
\qquad
\rho(\mathbf r)=\frac1{V_c}\sum_{\mathbf h}F_{\mathbf h}
e^{-2\pi i\mathbf h\cdot\mathbf r}.
$$
The 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.
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