Crystal isomorphism

Crystal isomorphism is a relation between crystal structures that preserves the relevant lattice and structural arrangement while permitting specified atomic replacement or addition. The IUCr definition requires the same space group and unit cell dimensions, with corresponding atom positions and types apart from those substitutions or additions.^[ref] Similar external crystal shapes or equal chemical formulas alone do not establish those conditions.

The relation matters experimentally in single isomorphous replacement and multiple isomorphous replacement. A heavy-atom derivative is useful when the unlabeled parent contribution is sufficiently unchanged to write, in a common coordinate convention,Here $F_P$ is the native structure factor and $F_H$ is the reference contribution from the added heavy atoms. If derivatization also changes the parent structure, a more honest decomposition isThe additional term cannot generally be inferred from the heavy-atom label alone. For original complex amplitudes $F_P=4+3i$ and $F_H=3$, the ideal derivative has squared magnitude58. If a structural change contributes $\Delta F_{\mathrm{structure}}=1$, the squared magnitude becomes73. Interpreting that increase solely as extra label scattering would misattribute the change.

Matching unit-cell dimensions is only one test. An original periodic two-site model illustrates why the atomic basis of a crystal matters too. Put identical point scatterers at fractional coordinates $(0,0,0)$ and $(1/2,0,0)$. For the $(100)$ reflection their amplitudes cancel. Moving the second site to $(1/2+\delta ,0,0)$ without changing the cell dimensions giveswhere $f$ is the common atomic scattering factor. At $\delta =0.01$, the magnitude is about $0.0628|f|$ rather than zero. The stipulated model is not an experimental derivative; it shows how unchanged lattice spacing can coexist with changed diffraction intensities from shifted atoms.

In actual phasing, crystal isomorphism is an approximation to assess across the measured X-ray diffraction data. The ideal vector-circle constructions assume comparable native and derivative amplitudes and consistently referenced heavy-atom positions.^[ref] Preserved cell metrics support that comparison, but cannot by themselves prove the unchanged-parent assumption.