the diffracted intensity
A diffraction pattern is not just a set of spot positions; each spot has a brightness, and that brightness carries the information about what sits where inside the unit cell. The diffracted intensity is simply how much energy comes out in a given reflection — how dark that spot is on the film, or how tall that peak is on the chart. Positions tell you the shape and size of the unit cell; intensities tell you the arrangement of atoms within it. You need both to know a structure.
At its heart, the intensity of reflection (hkl) is proportional to the squared magnitude of the structure factor: I is proportional to |F_hkl|^2. But several other factors multiply in before you reach the number a detector actually reads. In a powder pattern these include the multiplicity (how many symmetry-equivalent planes overlap at that angle), the Lorentz-polarization factor (geometric and polarization effects that vary with angle), the Debye-Waller factor (thermal vibration weakening high-angle peaks), and absorption in the sample. Stripping these known factors away to recover clean |F|^2 values is a standard, if fiddly, part of the analysis.
The single hard truth about intensity is what it loses. Because I is proportional to |F|^2, the measurement keeps the amplitude of F but discards its phase — squaring a complex number throws its direction away. This is the phase problem, and it is why a structure cannot simply be read off the pattern: you measure amplitudes and must reconstruct the missing phases (by direct methods, Patterson maps, or other tricks) before a Fourier synthesis can turn the data into a map of electron density.
In an X-ray powder pattern of aluminium the (111) peak is the tallest, not because those planes are special mirrors but because |F_111|^2 is large and eight equivalent {111} planes (multiplicity 8) overlap. The nearby (200) peak is weaker: similar |F|^2 but multiplicity only 6 and a slightly higher angle.
Peak height is |F|^2 times multiplicity and geometric factors — decoding it, not just reading positions, reveals the atomic arrangement.
Intensity is proportional to |F|^2, so it preserves the amplitude of F but not its phase. That single squaring is the origin of the phase problem — the reason structure solution is an inference, not a direct readout.