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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.

Crystallographic site occupancy is the fraction of a specified atom type present at a modeled site in a crystal structure. Occupancy1 represents full occupation by that type; a fractional value describes an average over the crystal. The IUCr core dictionary distinguishes this fraction from simply listing the site coordinates and constrains the summed occupancies of atom types sharing a physical site.^[ref]

A fractional occupancy does not mean that an individual atom has been divided. For an original population example, suppose a site is occupied by the specified atom in750 of1000 equivalent unit cells. Its average occupancy is $q=0.75$. More generally, let the crystallographic site multiplicity be $m=4$ equivalent positions per unit cell, each with occupancy0.75. The expected number of those atoms per unit cell iswhere $I_j$ is1 when position $j$ is occupied and0 otherwise. Linearity of expectation does not require the occupations to be independent. Correlations and local ordering can differ while the average count remains the same.

If two alternative atom types completely and mutually exclusively fill one site, occupancies0.40and0.60 sum to1. That describes a different model from occupancy0.40 of one type with the rest vacant. Structural alternatives of this kind are associated with crystallographic disorder; occupancy must be interpreted together with the model of what shares the site.

In an ideal structure factor calculation, the contribution of a site is multiplied by its occupancy. For fixed position and atomic displacement parameter, write $F_H=q{A}_H$, where $A_H$ is its full-occupancy complex contribution. A heavy-atom derivative then has $F_{PH}=F_P+q{A}_H$. With original values $F_P=3+4i$ and $A_H=2$, occupancy0.5 gives squared amplitude $|4+4i{|}^2=32$, whereas occupancy1 gives $|5+4i{|}^2=41$. The native value is25. Intensity changes are therefore not simply proportional to occupancy, because the added amplitude interferes with the parent amplitude.

The IUCr dictionary for multiple isomorphous replacement records occupancy separately for a heavy-atom site in each derivative.^[ref] A refined fraction concerns that structural site and scattering model; it is not by itself a general measurement of bulk chemical purity.

A heavy-atom derivative is a modified crystal containing ordered heavy-atom sites that can provide a reference scattering contribution for X-ray crystallography. The crystal before modification is called native. For single isomorphous replacement or multiple isomorphous replacement, the desired crystal isomorphism means that the parent crystal structure is sufficiently preserved for its contribution to be shared between data sets. Merely detecting the added chemical element does not establish that model.

The early myoglobin work provides an experimental example. Several crystals with mercury or gold at single sites on the protein were compared with unsubstituted crystals to infer phases. These are chemically modified reference specimens, not computational substitute structures.^[ref]

Crystallographic site occupancy describes the fraction present at a modeled site. The IUCr heavy-atom dictionary defines an occupancy separately for each site and derivative.^[ref] In a simplified single-site model with fixed position and displacement behavior, let $A_H$ be the full-occupancy complex amplitude and let $q$ be occupancy. ThenFor an original illustrative reflection, choose $F_P=4+3i$ and $A_H=6$. With $q=1/2$, the derivative amplitude is $7+3i$ and its squared magnitude is58. At full occupancy it is $10+3i$ and its squared magnitude is109. The native squared magnitude is25. Thus doubling occupancy doubles this modeled heavy-atom amplitude, but does not generally double the measured intensity change: interference with the parent contribution matters.

Useful derivative assessment therefore concerns ordered scattering and comparability as well as chemical uptake. If modification changes the parent structure substantially, the assumed decomposition into unchanged parent plus added reference is inadequate. Heavy-atom labeling describes the motivation for introducing the reference; a heavy-atom derivative is the resulting experimental specimen.

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.

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.