the phase problem

To rebuild where the atoms are, you have to add up a great many waves, and every wave needs two things: how big it is (its amplitude) and its timing relative to the others (its phase). A diffraction experiment faithfully records the brightness of each reflection, which gives the amplitude — but it throws the timing away entirely. It is like being told exactly how loudly each instrument in an orchestra plays, but never when each note sounds: you cannot reconstruct the music. That missing timing is the phase problem, and it is the central obstacle in crystal structure determination.

Precisely: each reflection is described by a structure factor F(hkl), a complex number with an amplitude and a phase, F = |F| e^(i phi). The electron density is recovered by a Fourier sum, rho(r) = (1/V) times the sum over hkl of F(hkl) e^(-2 pi i (hx + ky + lz)), which needs both |F| and phi for every reflection. But the measured intensity gives only I proportional to |F|^2, so you can extract |F| = sqrt(I) and nothing about phi. The phases — which, uncomfortably, carry most of the structural information — are simply not measured.

This is why solving a structure is a genuine puzzle rather than an automatic readout. The whole art of structure solution is a set of tricks to recover the lost phases: direct methods use statistical relations among strong reflections, the Patterson function sidesteps phases by working from |F|^2 alone to find heavy atoms, and isomorphous replacement and anomalous dispersion (MAD, at a synchrotron) add extra measurements that pin the phases down. Be blunt about the stakes: without the phases you cannot compute the electron-density map at all, so there is no shortcut around this problem — only ways through it.