X-ray crystallography

X-ray crystallography infers a crystal structure from measured X-ray diffraction. Peak positions constrain the lattice geometry, while intensities constrain the atomic basis of a crystal; the inference also needs an experimental model and does not directly observe every atomic position.

An electron density map is a spatial reconstruction of electron density, typically displayed as a grid or contours. In X-ray crystallography, a beam scattered by a crystal produces a diffraction pattern: measured intensities constrain structure factor magnitudes, while crystallographic phases are also required for reconstruction. Coordinates and experimental data are combined in common model-assessment maps.^[ref]

With fractional crystallographic coordinates $\mathbf{r}$, integer Miller index vector $\mathbf{h}$ and unit cell volume $V_c$, an ideal Fourier reconstruction has the formHere the structure factors are complex Fourier coefficients. Finite data and uncertain phases limit the reconstructed detail; this is part of the phase problem, not simply an issue of drawing contours more finely.

An original one-dimensional example makes the phase information visible:Both profiles have the same Fourier coefficient magnitudes, but their maxima occur half a period apart. Squared magnitudes alone cannot distinguish them.

An electron density map is also distinct from a macromolecular structural model fitted to it. A difference electron density map can highlight disagreement between observed and model-predicted scattering. Common map coefficients already incorporate information from the model, so the resulting map is not wholly independent confirmation of every modeling choice.^[ref]

A difference electron density map is an electron density map calculated to reveal disagreement between measured diffraction amplitudes and those predicted by a crystal structure model. The common $F_o-F_c$ map combines observed-minus-calculated amplitude information with phases, usually derived from the model. In practice map coefficients include weighting rather than simply subtracting raw measured intensities.^[ref]

For one simplified convention, let $F_c(\mathbf{h})=|F_c(\mathbf{h})|e^{i{\varphi }_c(\mathbf{h})}$. A schematic unweighted difference map isThe crystallographic phases remain necessary even though the coefficients represent differences. Positive features can indicate density missing from the model; negative features can indicate excessive modeled density. Errors in positions, crystallographic site occupancies or atomic displacement parameters can all contribute.

An original one-dimensional idealization takes ${\rho }_o(x)=3+2\cos (2\pi x)$ and ${\rho }_c(x)=3+\cos (2\pi x)$, with matching phases. Their difference isIt is positive near0 and negative near $1/2$, although its integral over one period is zero. Positive and negative residuals therefore need not mean that the total electron count is wrong.

Experimentally the difference electron density map guides corrections to a fitted macromolecular structural model. Finite diffraction data, uncertain phases and model-derived coefficients affect its interpretation; a peak alone does not uniquely identify an atom or chemical species.

An atomic displacement parameter describes the spread of an atom's modeled position about its mean in a crystal structure. An isotropic atomic displacement parameter represents equal spread in all directions; an anisotropic atomic displacement parameter uses a directional matrix. These parameters can reflect thermal motion, unresolved static variation and limitations of the refinement model, rather than temperature alone.

For isotropic Cartesian displacement, let $U_{\mathrm{iso}}$ be the mean-square displacement of one component, in squared angstroms. The conventional alternative parameter isThe IUCr dictionary gives both conventions and prefers reporting $U$.^[ref]

For an original numerical example, $U_{\mathrm{iso}}=0.010{\mathring{\mathrm{A}}}^2$ givesThe second value is the root-mean-square spread along one Cartesian direction. Under isotropy the three-dimensional mean-square displacement is $3U_{\mathrm{iso}}$, so the corresponding radial root-mean-square spread is about $0.173\mathring{\mathrm{A}}$.

This spread is not the same as uncertainty in the refined mean coordinate. Nor does doubling $B$ by itself prove that an atom's temperature doubled. Crystallographic disorder, imperfect data and a different structural model can affect the fitted parameter. Interpretation requires the experimental and modeling context.

The equivalent isotropic atomic displacement parameter summarizes an anisotropic atomic displacement parameter by the average mean-square displacement along three perpendicular Cartesian directions:It has units of squared length. The corresponding equivalent $B$ parameter is $B_{\mathrm{eq}}=8{\pi }^2{U}_{\mathrm{eq}}$. The CIF field for isotropic displacement may contain either a genuinely isotropic atomic displacement parameter or this anisotropic summary.^[ref]

For an original example, take a Cartesian displacement covariance matrixThen $U_{\mathrm{eq}}=0.020{\mathring{\mathrm{A}}}^2$ and $B_{\mathrm{eq}}\simeq 1.579{\mathring{\mathrm{A}}}^2$. The principal root-mean-square displacements are0.1,0.1 and0.2 angstroms. A genuinely isotropic matrix $0.020I$ has the same equivalent parameter but a different directional distribution.

The trace is invariant under Cartesian rotations, so the average does not depend on the laboratory orientation. In an oblique crystallographic basis, however, the raw diagonal entries of the stored displacement matrix do not in general have this Cartesian interpretation. The cell geometry must enter the conversion before taking the physical trace.^[ref]

This scalar is convenient for comparisons and tabulation, but discards directional information displayed by an atomic displacement ellipsoid. Equality of equivalent parameters therefore neither establishes isotropy nor proves that two sites have the same displacement distribution.

An anisotropic atomic displacement parameter describes direction-dependent positional spread about an atom's mean position in a crystal structure. In Cartesian coordinates a physical displacement model has a symmetric covariance matrixThe six independent entries have units of squared length. The crystallographic components stored in CIF use a cell-dependent convention, including reciprocal-cell lengths in the structure factor attenuation; they must be transformed before being interpreted as Cartesian covariance entries.^[ref]

For an original Cartesian example, letThe principal root-mean-square displacements are0.1,0.2 and0.3 angstroms. An atomic displacement ellipsoid displays those preferred directions; its displayed boundary also depends on the chosen probability level. Rotating the coordinate axes generally introduces off-diagonal entries without changing the physical spread or eigenvalues.

Under a centered multivariate normal distribution for $\mathbf{u}$, its contribution to the scattering amplitude is multiplied bywhere $\mathbf{s}$ is the Cartesian reciprocal vector in cycles per unit length. Thus reflections probing different directions can distinguish directional spread. A physically valid covariance is a positive semidefinite matrix.

The equivalent isotropic atomic displacement parameter is $U_{\mathrm{eq}}=\mathrm{tr}(U)/3$ in Cartesian coordinates. In this example it is $0.04667{\mathring{\mathrm{A}}}^2$; that average does not make the original distribution isotropic. Positional spread can include unresolved static variation as well as motion, and differs from uncertainty in the refined mean coordinate.^[ref]

An atomic displacement ellipsoid visualizes the directional positional spread represented by an anisotropic atomic displacement parameter. For a centered three-dimensional normal distribution with positive-definite Cartesian covariance matrix $U$, its probability surfaces have the formThe eigenvectors of $U$ give the principal directions; if its eigenvalues are ${\lambda }_i$, the semiaxis lengths are $\sqrt{c{\lambda }_i}$. The atomic displacement probability level specifies how much modeled probability is enclosed.^[ref]

The squared Mahalanobis distance ${\mathbf{u}}^T{U}^{-1}\mathbf{u}$ has a chi-squared distribution with3 degrees of freedom in this Gaussian model. A50% ellipsoid uses $c\simeq 2.36597$. Taking $c=1$ instead encloses only about19.9%, so a three-dimensional probability ellipsoid cannot be read as a one-dimensional one-standard-deviation interval.

For an original example, $U=\mathrm{diag}(0.01,0.04,0.09){\mathring{\mathrm{A}}}^2$ gives50% semiaxes approximately0.154,0.308 and0.461 angstroms. An isotropic matrix with the same equivalent isotropic atomic displacement parameter, $U_{\mathrm{eq}}=0.14/3{\mathring{\mathrm{A}}}^2$, instead gives a sphere of radius about0.332 angstroms.

The ellipsoid represents modeled positional spread, including possible static disorder, rather than the atom's physical surface or a confidence region for its refined mean position. Its interpretation depends on the displacement model; the Gaussian probability construction requires a positive-definite covariance and does not turn a negative fitted eigenvalue into a physically meaningful radius.^[ref]

An atomic displacement probability level is the fraction of modeled positional probability enclosed by an atomic displacement ellipsoid. It specifies the size of the drawn surface for a given anisotropic atomic displacement parameter; changing the level does not change the fitted displacement matrix.^[ref]

For a centered three-dimensional normal distribution with positive-definite Cartesian covariance matrix $U$, defineThe variable $Q$ has a chi-squared distribution with3 degrees of freedom. To enclose probability $p$, choose $c$ satisfying $\mathrm{Pr}(Q\le c)=p$ and draw $Q=c$. If ${\lambda }_i$ are the eigenvalues of $U$, its semiaxis lengths are $\sqrt{c{\lambda }_i}$.

An explicit calculation uses the cumulative distribution functionwhere $\mathrm{erf}$ is the error function. Solving $F_3(c)=0.5$ gives $c\simeq 2.36597$; for $p=0.9$, $c\simeq 6.25139$.

For an original comparison, keep $U=\mathrm{diag}(0.01,0.04,0.09){\mathring{\mathrm{A}}}^2$. The50% semiaxes are approximately0.154,0.308 and0.461angstroms. The90% semiaxes are approximately0.250,0.500 and0.750angstroms. Every semiaxis grows by the same factor, about1.625, although the underlying directional spread is unchanged.

This also explains why a surface drawn at one root-mean-square displacement in each principal direction, $Q=1$, encloses only about19.9% in three dimensions. The probability inside a three-dimensional ellipsoid differs from the probability inside a one-dimensional standard-deviation interval. Comparisons of displayed atomic sizes therefore need the stated probability levels as well as the displacement parameters.

An isotropic atomic displacement parameter describes the same mean-square positional displacement along every Cartesian direction in a crystal structure. Its Cartesian displacement covariance matrix isThus $U_{\mathrm{iso}}$ is a one-component variance, in squared angstroms, rather than a displacement length. The alternate conventional parameter is $B=8{\pi }^2{U}_{\mathrm{iso}}$.^[ref]

For an original example, $U_{\mathrm{iso}}=0.020{\mathring{\mathrm{A}}}^2$ givesThe radial root-mean-square displacement is instead $\sqrt{3U_{\mathrm{iso}}}\simeq 0.245\mathring{\mathrm{A}}$ because it includes three components.

For a centered isotropic normal distribution, the structure factor amplitude attenuation at reciprocal-vector magnitude $s$, measured in cycles per angstrom, isAt $s=1{\mathring{\mathrm{A}}}^{-1}$, the stated example gives $T\simeq 0.674$ and squared-amplitude attenuation $T^2\simeq 0.454$. Amplitude and intensity attenuation are therefore distinct, even in this simple Gaussian model.

An equivalent isotropic atomic displacement parameter can also summarize an anisotropic atomic displacement parameter by its Cartesian trace divided by3. Such an average can appear in the same CIF field as a genuinely isotropic parameter, but does not assert equal directional spread.^[ref]

Crystallographic disorder is variation in atomic positions, orientations or chemical occupation that is represented by an average crystal structure rather than one perfectly repeated local arrangement. A split-site disorder model can describe alternative positions with fractional crystallographic site occupancies. The IUCr disorder-group convention distinguishes atoms present together in one configuration from mutually alternative configurations.^[ref]

For an original single-atom model, suppose an atom occupies position0 along an axis with probability0.6 and position $d$ with probability0.4. ThenAt $d=1\mathring{\mathrm{A}}$, the mean is $0.4\mathring{\mathrm{A}}$ and the variance is $0.24{\mathring{\mathrm{A}}}^2$. There need be no atom at the mean position in any individual configuration. Fitting only an average position with a large atomic displacement parameter can conceal this distinction.

The same average occupation could arise from different unit cells adopting different configurations, from motion among configurations during measurement, or from a mixture of these effects. A static average alone does not determine an interchange rate.

Experimental interpretation therefore combines the electron density map, the allowed chemical geometry, occupation constraints and other evidence. Weak or broadened density should not automatically be assigned one specific dynamic mechanism.

A split-site disorder model represents alternative atomic positions explicitly within an average crystal structure. Each alternative has its own coordinate and crystallographic site occupancy. For one atom distributed between exactly two alternatives, with no vacancy, the occupancies sum to1. Correlated alternative arrangements can be collected into disorder groups: atoms within one group occur together, while different groups in the same assembly represent alternatives.^[ref]

For an original scattering example, put the alternatives at fractional coordinates $(0,0,0)$ and $(1/2,0,0)$ with occupancies0.7 and0.3. Give both the same real atomic scattering amplitude $f$, and omit all other atoms and displacement attenuation. The coherent average contribution to a reflection $(h00)$ isFor an even integer $h$, this equals $f$; for an odd integer $h$, it equals $0.4f$. The corresponding squared amplitudes are $f^2$ and $0.16f^2$. Treating both alternatives as fully occupied would predict a different structure factor. This calculation concerns the periodic average contribution, not a complete description of diffuse scattering from local fluctuations.

A split-site disorder model is useful when an electron density map supports distinct alternative locations instead of one broad peak. An anisotropic atomic displacement parameter can describe directional broadening, but does not by itself encode two mutually exclusive chemical arrangements.

Occupancy constraints depend on the physical model: alternatives for a fully present atom sum to1, whereas a model admitting vacancies need not. An average crystal structure alone also cannot establish whether the alternatives exchange during the measurement, occur in different cells, or have particular local correlations.

Crystallographic site multiplicity is the number of distinct equivalent positions generated from one representative site by the space group within the conventional unit cell. It is a property of the Wyckoff position. If the site-symmetry group leaves a representative position fixed, fewer distinct positions are generated than at a general site.^[ref]

The relation can be writtenwhere $m_{\mathrm{general}}$ is the general-position multiplicity and $|G_{\mathrm{site}}|$ is the order of the site-symmetry group. For an original compatible example, general multiplicity8 and site-symmetry order2 give $m=4$.

Multiplicity is distinct from crystallographic site occupancy $q$. For one atom type represented at equivalent positions, its average contribution to the atom count per unit cell isA multiplicity8 site occupied to0.25 contributes2 atoms per cell on average. A fully occupied multiplicity4 site contributes4. If these are separate inequivalent sites for the same element, their total contribution is6.

Full occupancy at a symmetry-special site is still $q=1$, rather than automatically a fractional value. The symmetry reduction is already counted by $m$. Dividing occupancy again by the site-symmetry order would count that reduction twice. These distinctions connect a list of representative coordinates to the chemical contents of the unit cell.

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.

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.

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.

Crystallographic resolution describes the spatial detail represented by a diffraction data set and the resulting electron density map. In X-ray crystallography, its conventional numerical measure $d_{\min }$ is the smallest lattice-plane spacing represented by measured reflections. A smaller stated distance means finer nominal resolution.^[ref]

From Bragg's law, first-order diffraction at the largest measured glancing angle ${\theta }_{\max }$ giveswhere $\lambda$ is the X-ray wavelength. For an illustrative wavelength $1\mathring{\mathrm{A}}$ and ${\theta }_{\max }=30^{\circ }$, the nominal resolution is $1\mathring{\mathrm{A}}$. At the same wavelength and a maximum angle of15degrees, it is about $1.93\mathring{\mathrm{A}}$: collecting smaller-angle reflections alone omits finer spatial information.^[ref]

The2000 structure of a bacterial small ribosomal subunit provides a concrete use of the term: its primary report describes refinement to3Å resolution and interprets RNA–protein architecture. That number characterizes the data/model's spatial detail, not an assertion that every atomic coordinate has exactly3Å uncertainty. Noise, missing reflections, directional coverage and model errors still matter.^[ref]

Determination of a crystalline structure from the diffraction of X-rays by an oriented single crystal. It can distinguish cage connectivities and substitution patterns that share an elemental formula and the same number of NMR environments.

Use of an approximate structural model to obtain initial phase information for an experimental crystallographic dataset; the model must still be assessed against the diffraction data.