= Crystal isomorphism
\b[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.<Isomorphous crystals in the IUCr Online Dictionary of Crystallography>[Definition]{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,
$$
F_{PH}=F_P+F_H.
$$
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 is
$$
F_{PH}=F_P+F_H+\Delta F_{\rm structure}.
$$
The 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 $\Delta F_{\rm structure}=1$, 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 gives
$$
F_{100}=f\left[1-e^{2\pi i\delta}\right],\qquad
|F_{100}|=2|f|\,|\sin(\pi\delta)|,
$$
where $f$ 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.<Phasing diagrams in CCP4 Basic Maths for Protein Crystallographers>[SIR and MIR constructions]{ref} Preserved cell metrics support that comparison, but cannot by themselves prove the unchanged-parent assumption.
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