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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.

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 asThe 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.^[ref]

For an original example, take $|F_P|=5$, $|F_H|=3$ and $|F_{PH}|=7$ in consistently scaled amplitude units. If $\Delta \varphi ={\varphi }_P-{\varphi }_H$, thenso $\cos \Delta \varphi =1/2$ and $\Delta \varphi =\pm \pi /3$. 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.^[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.

Multiple isomorphous replacement combines a native X-ray diffraction data set with data from two or more heavy-atom derivatives to estimate the native structure factor phases. In the ideal crystal isomorphism model, each derivative adds a constraint $|F_P+F_{H,j}|=|F_{PH,j}|$. Consistently referenced heavy-atom models can distinguish the two alternatives left by single isomorphous replacement.^[ref]

Continue the original example in single isomorphous replacement, where $F_P=4\pm 3i$ both fit the native amplitude5 and first derivative with $F_{H,1}=3$. Choose a second reference $F_{H,2}=2i$. Its predicted squared amplitudes differ:A measured second-derivative amplitude $\sqrt{41}$ therefore selects $4+3i$ in this exact model. If the second reference had instead been real, both candidates would still have equal magnitudes after adding it. An additional derivative must contribute useful geometry, rather than merely repeat a constraint.

The heavy-atom models must share a consistent coordinate origin and handedness. Resolving a circle-crossing ambiguity does not by itself determine absolute handedness: conjugating all the amplitudes preserves their magnitudes. CCP4's diagrams explicitly retain this distinction.^[ref]

The method has a concrete experimental history. The opening of the1960 myoglobin structure report describes several mercury- or gold-containing derivatives whose X-ray diffraction patterns were compared with the unsubstituted protein. These measurements had supplied phases for the earlier map at spacings greater than6Å; the new report advanced the analysis to2Å.^[ref] Different labeled crystals supplied information unavailable from native intensities alone.

Single isomorphous replacement estimates phases in X-ray crystallography by comparing a native crystal with one heavy-atom derivative. Under the ideal crystal isomorphism assumption, a reflection's derivative structure factor is $F_{PH}=F_P+F_H$, where the heavy-atom model supplies the complex reference $F_H$. The measured magnitudes constrain $F_P$ to two circles in the complex plane: one centered at zero with radius $|F_P|$, the other centered at $-F_H$ with radius $|F_{PH}|$. A general intersection gives two phase choices.^[ref]

An original exact example uses $|F_P|=5$, $F_H=3$ real and positive, and $|F_{PH}|=\sqrt{58}$. Writing $F_P=x+iy$ givesSubtracting the equations gives $6x+9=33$, hence $x=4$ and $y=\pm 3$. The alternatives areBoth alternatives predict exactly the same native and derivative intensities. Greater precision in these two magnitudes alone cannot select between them. The phase is the missing direction of a complex amplitude, not an unmeasured additive intensity.

Multiple isomorphous replacement can supply an additional circle from another derivative. The numerical example there resolves these particular candidates. In actual measurements, uncertainty, imperfect crystal isomorphism and an imperfect heavy-atom model broaden the constraints, so exact circle crossings are an explanatory idealization.