Phase problem

The phase problem is the loss of the wave phases of the crystal basis structure factors when an experiment records their squared magnitudes. Write $F_{hkl}=|F_{hkl}|e^{i{\varphi }_{hkl}}$. A Fourier series reconstruction of a periodic electron density uses both $|F_{hkl}|$ and ${\varphi }_{hkl}$, whereas an ordinary wave intensity measurement supplies only $|F_{hkl}{|}^2$. Estimating the missing phases therefore needs additional constraints or measurements.^[ref]

There is also an unavoidable choice of origin. Translating all motif positions by the same fractional crystal coordinate vector $t$ multiplies $F_{hkl}$ by $e^{2\pi i(h,k,l)\cdot t}$ without changing $|F_{hkl}{|}^2$. The physical arrangement is unchanged. A reconstructed coordinate list must therefore distinguish a coordinate convention from information actually recovered from the measurements.

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.