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Complement C3 is a central protein of the complement system. Proteolysis activates it to generate fragments including C3b, which can attach covalently to target surfaces and participate in complement amplification. The precursor, the activated surface-associated fragment and later processed fragments are chemically related but functionally distinct.^[ref]

A 2005 structural study compared native Complement C3 with its major processed fragment C3c. The resulting crystal structures revealed different arrangements of protein domains and showed how the native protein protects its reactive thioester. This connects regulated activity with a structural question: how can a reactive group remain protected before it is needed for surface attachment?^[ref]

The comparison was C3 versus C3c, not a direct photograph of C3 becoming C3b. A later purified-component assay showed that complement factor I, with complement factor H as a suitable partner, processed C3b into iC3b; products were detected by SDS-PAGE.^[ref]

Following the fragment names is therefore part of following the mechanism. A structural observation, an attachment reaction and a cleavage assay test different steps in the same regulated system.

Complement C6 is a plasma protein involved in assembly of the membrane attack complex, a multiprotein structure capable of creating a pore in a target cell membrane. It participates downstream of complement C5 activation, when the larger fragment C5b supplies an assembly platform. The intact circulating protein and the assembled pore are different molecular objects.^[ref]

A 2012 study determined a full-length complement C6 crystal structure and compared it with related pore-forming proteins. The authors used those structural relationships to propose a mechanism for sequential assembly. The measured structure and the proposed sequence of conformational changes are different kinds of result; a structure of isolated complement C6 is not a direct recording of a complete pore assembling.^[ref]

For an original stoichiometric bound, assume a model with one complement C6 per C5b-containing assembly. Nine available C5b molecules but only six available complement C6 molecules allow at most six such assemblies. This count does not guarantee six finished pores: later components, binding competence and assembly conditions also matter.^[ref]

A complementary functional readout is hemolysis. The 1965 isolation study of complement C5 found that its reported lytic function depended on complement C6. Combining component-dependence assays with structural evidence connects a molecular participant to a biological effect without equating protein amount with completed pore number.^[ref]

Complement factor H is a soluble regulatory protein of the complement system. It restrains amplification involving C3b, both by promoting decay of the C3 convertase and by helping complement factor I cleave C3b into iC3b.^[ref]

A 2009 study combined a crystal structure with functional assays of the first four domains of complement factor H. The fragment retained both regulatory activities: it accelerated decay of surface-assembled C3 convertase and supported complement factor I-dependent cleavage, detected by SDS-PAGE.^[ref]

The two assays answer different questions. Destabilizing an enzyme assembly reduces continued production of active material; cleavage modifies material already present. Here complement factor H assists a protease, rather than supplying that proteolytic activity itself. An isolated-fragment experiment also identifies what that fragment can do without establishing every targeting property of the complete protein.

Condensed matter physics studies the collective behavior of matter with many interacting constituents, especially solids and liquids. Crystal structure, electron interactions and thermal motion produce properties that are not apparent from one isolated atom.

A phonon describes a collective lattice vibration, while superconductivity involves a coherent many-electron state. The same microscopic ingredients can produce different phases as temperature, pressure or composition changes. Experiments connect structural, transport, magnetic and spectroscopic measurements with models of those phases.

A crystallographic unit cell chosen to display the symmetry or a standard setting of a crystal structure. It tiles the periodic arrangement but need not be primitive. A conventional FCC cube contains four lattice points and a BCC cube contains two; a primitive unit cell contains one. These are lattice-point counts, not a universal count of atoms in a chemical motif.

Crystal chemistry connects a crystalline material's composition, chemical bonds and atomic arrangement. It asks which coordination numbers, local geometries and substitutions are compatible with an observed crystal structure.

For example, the usual sodium chloride structure gives each ion six nearest oppositely charged neighbours, whereas the usual caesium chloride structure has eight. Charge balance constrains composition, while size, bonding and thermodynamic conditions influence which arrangement is stable. Ionic-size rules are useful guides rather than exact universal predictions. X-ray diffraction tests the arrangement experimentally, allowing composition-based expectations to be compared with measured periodic atomic structure.

The characteristic outward shape of a crystal; distinct from its internal crystal structure.

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.

A crystal lattice records the translation symmetry of a periodic crystal. With a primitive basis of three linearly independent lattice vectors $a_1,a_2,a_3$, its translation vectors formThe crystal structure additionally specifies the arrangement of atoms associated with each lattice point. A lattice point is therefore a symmetry marker, not necessarily one atom.^[ref]

For example, a one-dimensional chain with alternating species $A,B$ at spacing $a$ has translation period $2a$, not $a$: shifting by $a$ exchanges distinct species. It can be described by lattice $2a\mathbb{Z}$ with a two-atom basis at offsets zero and $a$. In a diffraction measurement, the translation lattice selects possible reciprocal-space peak positions, while the atomic basis affects their intensities and systematic absences. This distinction lets diffraction reveal internal structure rather than just the visible shape of the specimen.

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.

The crystallographic metric matrix is the Gram matrix of the three unit cell basis vectors. If $A$ has Cartesian columns $\mathbf{a},\mathbf{b},\mathbf{c}$, thenHere $\alpha ,\beta ,\gamma$ are the angles between $\mathbf{b},\mathbf{c}$; $\mathbf{a},\mathbf{c}$; and $\mathbf{a},\mathbf{b}$, respectively. The matrix converts fractional crystallographic coordinate differences into physical lengths: a fractional displacement $\mathbf{u}$ has squared length ${\mathbf{u}}^{T}G\mathbf{u}$.^[ref]

For an original oblique cell, choose $a=2$, $b=3$, $c=4$ angstroms, $\gamma =60^{\circ }$ and $\alpha =\beta =90^{\circ }$. ThenFor $\mathbf{u}=(1/2,-1/2,0{)}^T$,Ignoring the off-diagonal entries would instead give $13/4$, overestimating this distance. The cross terms encode the cell angles.

The reciprocal crystallographic metric matrix is $G^{-1}$ when the reciprocal basis has dot products ${\mathbf{a}}_i^{\ast }\cdot {\mathbf{a}}_j={\delta }_{ij}$, with no factor of $2\pi$. Thus the plane spacing for a nonzero Miller index column $\mathbf{h}$ satisfiesA basis change $A^{\prime }=AP$ gives $G^{\prime }=P^{T}GP$ and fractional coordinates ${\mathbf{u}}^{\prime }=P^{-1}\mathbf{u}$, preserving ${\mathbf{u}}^{\prime T}{G}^{\prime }{\mathbf{u}}^{\prime }={\mathbf{u}}^{T}G\mathbf{u}$. Distances do not change when the same crystal structure is described in another basis.^[ref]

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.

Crystallographic symmetry consists of the spatial transformations that leave an ideal crystal structure unchanged, including the identities of its chemical atoms. A symmetry of the crystal lattice alone need not preserve the atomic basis of a crystal. The full set of crystal-preserving transformations forms a space group; its translational part distinguishes it from a crystallographic point symmetry description.^[ref]

In fractional unit cell coordinates a space group operation has the form $x\mapsto Wx+w$. The matrix $W$ describes its linear action and $w$ its translation. A screw axis combines a rotation with a translation parallel to the axis; a glide plane combines reflection with translation within the plane. For example, a twofold screw can act asApplying it twice gives $(x,y,z+1)$, a full unit cell translation. Replacing it by a pure twofold rotation would lose this relation between orientation and position.

For ordinary three-dimensional periodic crystals, the possibilities are organized into 32 crystallographic point groups and 230 space groups. These counts concern spatial symmetry without additional magnetic labels. A square outline alone does not establish fourfold symmetry: decorating alternate corners with different atoms can break the rotation. In structure determination, the symmetry therefore constrains both the lattice geometry and which sites can carry equivalent chemical atoms.^[ref]

Crystallography studies crystal structures and their symmetries. A structure determination connects measurements such as crystal scattering patterns to a model of the unit cell and its atomic basis of a crystal.

X-ray crystallographic structure inference makes that connection explicit: compare proposed crystal structures through the positions and wave intensities of their predicted reflections. The main difficulty is that a diffraction pattern encodes wave interference, so it must be interpreted through a model rather than read as a direct picture of the chemical atoms.

The orientation of a crystal structure relative to a specimen or instrument coordinate frame. It determines which Bragg diffraction conditions are excited.

A solid thermodynamic phase with a particular crystal structure, among distinct crystalline forms of the same chemical substance. Crystal polymorphism names the existence of these alternatives; a crystal polymorph names one of the forms.

A crystal structure specifies the chemical species and spatial arrangement of the chemical atoms or ions in a crystal. For an ideal periodic structure, separate two ingredients: a Bravais lattice specifies the translations that repeat the arrangement, and an atomic basis of a crystal specifies the chemical motif repeated at every lattice point. The outward shape, or crystal habit, is a different description: the internal arrangement is what this article describes.

For a concrete two-dimensional model, choose primitive lattice translation vectors $a_1=(2,0)$ and $a_2=(0,1.5)$ in arbitrary length units. At every lattice point $R=m{a}_1+n{a}_2$, with integers $m,n$, place species A at $R$ and a different species B at $R+(0.65,0.55)$. Translating by $a_1$ or $a_2$ preserves both the positions and their chemical labels. Translating by $(0.65,0.55)$ sends one A position to a B position, so it is not a translation symmetry of the labeled structure. A picture of dots with the labels erased can therefore conceal essential chemical information.

A primitive unit cell contains one lattice point, but may contain several chemical atoms. A conventional unit cell can contain several lattice points to display the symmetry more clearly. Thus the conventional cube of a face-centered cubic lattice contains four lattice points, whereas a body-centered cubic lattice has two. Neither lattice name alone specifies which chemical species occupies the repeated motif. The NaCl conventional cell and coordination example makes the distinction explicit.

This separation is useful experimentally: interplanar spacings constrain the repeating geometry, while relative wave intensities also depend on the atomic basis of a crystal through the crystal basis structure factor. The Bragg crystal reflection experiment of 1913 describes the X-ray spectrometer, angular scans and detector checks in detail. Its conversion from angle to length is developed under Bragg's law; a geometrically possible reflection can still have zero wave intensity.

William Lawrence Bragg's 1913 structure determination supplied a complementary test. He directed X-rays through sodium chloride sections cut parallel to a cube face and recorded spots on photographic plates. Comparing section thicknesses tested how transmission altered the pattern: the relative spot strengths changed. Comparing related salts led him to two interpenetrating face-centered cubic lattices, one for each kind of constituent. He explicitly distinguished evidence for that arrangement from the still-needed identification of a single chemical atom with each scattering center. The photographs were evidence for a structural model, not direct pictures of labeled ions.^[ref]

The spatial transformations that preserve a crystal structure.

A cuprate superconductor is a copper-oxide material exhibiting superconductivity, with layered copper-oxygen units central to its crystal structure. Chemical composition and oxygen content must accompany a quoted superconducting transition temperature; the nominal recipe of a mixed-phase pellet is not the composition of its superconducting constituent. The formal hole count in a cuprate illustrates the distinction between composition and electronic interpretation.

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.