= Heavy-atom labeling
\b[Heavy-atom labeling] introduces atoms of relatively high <atomic number> at ordered crystal sites, giving a strong <X-ray> scattering contribution so that their signal can supply information for solving its <phase problem>. In <X-ray crystallography>, measured intensities constrain the magnitudes of <structure factors>, while their phases are also needed to reconstruct <electron density>. A <heavy-atom derivative> provides an additional measurement linked to a model of the label sites.
Under an ideal <single isomorphous replacement> model, the derivative keeps the parent structure unchanged apart from the added label contribution. For one reflection, write the complex amplitudes as
$$
F_{PH}=F_P+F_H.
$$
The heavy-atom model supplies $F_H$, while measurements supply $|F_P|$ and $|F_{PH}|$. These are vector constraints in the complex plane; subtracting intensity magnitudes does not directly reveal the protein phase.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[SIR, two-circle phase construction]{ref}
For an original example, take $|F_P|=5$, $|F_H|=3$ and $|F_{PH}|=7$ in consistently scaled amplitude units. If $\Delta\phi=\phi_P-\phi_H$, then
$$
49=25+9+2(5)(3)\cos\Delta\phi,
$$
so $\cos\Delta\phi=1/2$ and $\Delta\phi=\pm\pi/3$. The one-derivative measurement leaves two possible phases. In particular, $7-5=2$ is not the heavy-atom amplitude3: amplitudes must be combined with their phases.
<Multiple isomorphous replacement> adds measurements from further derivatives with consistently referenced label positions. These can distinguish the candidate phases in favorable circumstances.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[MIR, third circle and consistent origin/hand]{ref} Large changes in the crystal or protein would violate the stated parent-plus-label model, so derivatization must be assessed rather than assumed to be structurally harmless. Ordered occupancy and useful diffraction matter alongside the chemical act of adding a heavy atom.
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