Codex @codex  0

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]

A diffracted beam or powder peak associated with a reciprocal-lattice vector satisfying the elastic scattering condition. Its position follows from lattice spacings and its amplitude from the unit-cell structure factor. A reciprocal point whose structure factor vanishes does not produce an allowed reflection in the kinematic model.

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.

For a finite set $S$ of crystal sites, the lattice structure factor at momentum transfer $Q$ isIt multiplies the single-atom scattering amplitude.

A crystallographic phase is the angle of a nonzero complex structure factor. Writingseparates amplitude and phase for a reflection indexed by the integer Miller index vector $\mathbf{h}$. Measured diffraction intensities constrain squared amplitudes; reconstructing an electron density map also requires phases. This missing information is the phase problem.^[ref]

Using fractional crystallographic coordinates, one consistent Fourier convention isThe phase controls where a harmonic contributes maxima and minima. For example, the original positive profiles $3+2\cos (2\pi x)$ and $3-2\cos (2\pi x)$ have equal Fourier coefficient magnitudes, but opposite phases for their first harmonics. Their squared amplitudes cannot locate the density maxima.

Phase also depends on the coordinate convention. Translating every atomic position by a fractional vector $\mathbf{t}$ multiplies the structure factor by $e^{2\pi i\mathbf{h}\cdot \mathbf{t}}$. For $\mathbf{h}=(1,0,0)$ and $\mathbf{t}=(1/4,0,0)$, the phase increases by $\pi /2$ while the squared amplitude is unchanged. Shifting the coordinate origin by that vector has the opposite sign. A phase is defined modulo $2\pi$; when $F_{\mathbf{h}}=0$, its phase is undefined. Comparing crystallographic phases therefore requires matching origins and Fourier conventions.

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.

The identical two-atom motif adds a factor $1+\exp [i\pi (h+k+l)/2]$ to the FCC lattice structure factor. Among even-index FCC diffraction reflections, only those with $h+k+l$ divisible by4 survive. The allowed FCC reflection200 therefore vanishes for equal motif scattering amplitudes.

For elastic scattering of a plane wave by a periodic crystal, a diffraction contribution can occur when the momentum transfer is a reciprocal lattice vector:Squaring ${\mathbf{k}}^{\prime }=\mathbf{k}-\mathbf{q}$ giveswhere $\theta$ is the angle between incident and outgoing wavevectors. This is the Ewald sphere construction. For planes with spacing $d$ and first reciprocal spacing $2\pi /d$, it becomes Bragg's law $2d\sin (\theta /2)=\lambda$.

The geometrical condition alone does not ensure an observed reflection. The structure factor must also be nonzero: different atoms in a motif can cancel the scattering amplitude, producing a systematic absence. The diamond-basis diffraction extinction is a concrete example.

For a powder or a crystal rotated to reach the required orientation, the smallest possible nonzero diffraction angle is therefore determined by the shortest allowed reciprocal lattice vector with nonzero structure factor and $|\mathbf{q}|\le 2k$. For a crystal at one fixed orientation, the incident direction additionally determines which reciprocal vectors satisfy the displayed elastic scattering condition.

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 researcher who coauthored a quantitative powder diffraction method using a surface diffractometer and an area detector. Extracting meaningful intensities requires corrections for polarization and measurement geometry as well as the crystal's structure factor. The work explains how an instrument designed for surfaces can also measure powders and textured samples, connecting detector images with reliable structural analysis.^[ref]

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.

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]

In the kinematic approximation, each incident wave scatters once, so the total scattering amplitude is the coherent sum of single-atom amplitudes. Multiple scattering and depletion of the incident beam are neglected. The structure factor then determines systematic reflection absences. Strong multiple scattering can violate an absence predicted in this approximation.

A magnetic Bragg peak is an elastic diffraction peak from periodic ordered magnetic moments. A single ordering wavevector $\mathbf{q}$ shifts chemical reciprocal vectors to $\mathbf{G}\pm \mathbf{q}$; the basis structure factor, magnetic form factor and moment component perpendicular to the neutron momentum transfer can impose additional extinctions. An allowed translation condition alone does not guarantee visible intensity.

The structure factor for magnetic neutron diffraction sums the magnetic form factor and the magnetic moment component perpendicular to the scattering vector over the magnetic atomic basis. For unpolarized neutrons, the magnetic contribution to the coherent diffraction wave intensity is proportional to $|{\mathbf{F}}_M{|}^2$. A periodic change of sign of magnetic moments can shift its allowed peaks away from the nuclear reciprocal lattice vectors without changing nuclear positions.

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.

A Gaussian field with free-energy kernel $A+\kappa q^2$ has structure factor proportional to $(A+\kappa q^2{)}^{-1}$ and correlation length $\sqrt{\kappa /A}$.

The useful feature is the selenium atom's wavelength-dependent anomalous X-ray scattering. Close to its absorption edge the atomic scattering factor is complex:Replacing a sulfur-containing side chain by its selenium analogue introduces relatively strong, spatially localized anomalous scatterers. This is the diffraction-physics reason for the label; no biological preparation procedure is needed for the calculation.

The Se substructure can be located from intensity differences or Patterson maps. Measurements at several wavelengths change its known complex contribution while leaving most other atoms' scattering nearly unchanged. Interference between that contribution and the unknown remainder constrains the otherwise missing structure factor phases. Both wavelength-dependent dispersive differences and differences between Friedel pairs contribute to multiwavelength anomalous diffraction. Several wavelength constraints resolve phase ambiguities that remain with one intensity difference. The resulting phased Fourier synthesis yields the electron-density map.

For reflection $222$, the positional phase factor is $e^{2\pi i(2x+2y+2z)}=1$ for each of the four sites just found. Treat the supplied anomalous increment as the imaginary term $i{f}^{\prime \prime }$, with $f^{\prime \prime }=4.8$ electrons per Se atom. ThenIts magnitude is $19.2$ electrons and its phase is $+90^{\circ }$ relative to the real axis in this structure-factor convention. Translating the origin rotates both this reference contribution and the ordinary structure factor consistently; the physically relevant phase difference is invariant.

Write the ordinary structure factor as $F_0=|F_0|e^{i\varphi }$ and use $F=F_0+19.2i$. The intensities implyConsequently $\sin \varphi =0.43893$, and the two possible ordinary phases areThe twofold ambiguity is expected from a single anomalous-intensity constraint. The calculation assumes that the specified imaginary increment is the only change requiring inclusion; an additional unspecified real dispersive change could not be reconstructed from these data alone.