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A periodic crystal cannot have a fivefold crystallographic rotation axis. The crystallographic restriction theorem permits rotation orders $1,2,3,4,6$ for a periodic Bravais lattice. A $72^{\circ }$ rotation cannot preserve such a lattice. This restriction concerns exact symmetry of the infinite crystal structure; an isolated finite cluster with fivefold symmetry is a different object.

An energetic neutron can collide with a lattice atom and create a primary knock-on atom. That recoil transfers energy to other atoms in a displacement cascade. Some atoms leave their normal lattice sites, creating Frenkel pairs: a vacancy and an extra atom at an interstitial position.

Many nearby defects recombine during the cascade and its thermal relaxation. The surviving point defects are therefore fewer than a simple count of all displacement events. Some defects are already clustered within the cascade; others migrate and aggregate later.

A collection of vacancies can collapse into a platelet bounded by a vacancy-type dislocation loop. An aggregation of self-interstitial atoms can create an extra atomic platelet bounded by an interstitial-type dislocation loop. The boundary is a line defect because a circuit around it has a nonzero Burgers vector. The allowed Burgers vectors and habit planes depend on the material's crystal structure and fault energies.

Loops can grow by absorbing mobile point defects, shrink by emitting or absorbing the opposite defect type, and interact with existing dislocations. Some remain faulted, while suitable loops can unfault or become part of a larger dislocation network. Repeated cascades and subsequent defect transport therefore create a population of extended defects even in material with a low initial dislocation density.

This describes the main displacement-damage mechanism. Ionizing energy deposited in a material does not automatically create the same defect population as a neutron recoil cascade; recoil spectrum, displacement thresholds and material response matter.

Synroc immobilises waste ions in selected durable crystalline hosts, rather than trying to put the whole inventory into one universal site. The underlying philosophy is that minerals retaining analogous elements over geological times suggest useful crystal structures for engineered containment. These natural analogues motivate the choice; their survival does not prove that every synthetic composition, damaged microstructure or groundwater environment will behave identically. This was the rationale of the original mineral-analogue proposal.^[ref]

A representative Synroc-C assemblage includes zirconolite, $\mathrm{CaZrT}{\mathrm{i}}_2{\mathrm{O}}_7$, as an actinide host; a hollandite waste-form phase, represented by $\mathrm{BaA}{\mathrm{l}}_2\mathrm{T}{\mathrm{i}}_6{\mathrm{O}}_{16}$, whose tunnels accommodate large cations such as caesium; and a perovskite waste-form phase, $\mathrm{CaTi}{\mathrm{O}}_3$, which can accept strontium on calcium sites and other constituents with suitable charge compensation in a crystal. Titanium-oxide phases provide additional accommodation and composition flexibility. The actual phases are solid solutions, not simply pure end-member minerals mixed with an inert waste powder.^[ref]

Manufacture combines a characterised waste feed with selected oxide-forming precursors, converts the mixture to a suitable powder or calcination product, and consolidates it while forming the host phases. hot pressing and hot isostatic pressing are consolidation approaches; the latter uses pressure acting approximately equally in all directions. The objective is a dense ceramic with the intended phase assemblage and low connected porosity, contained in an appropriate radioactive waste package. Verify density and phase composition, use microscopy to find segregated constituents or pores, and test leaching of the actual product. Successful densification alone does not establish radionuclide retention.^[ref]

To determine a crystal structure, scatter probes whose wavelength is comparable to atomic spacings, typically $0.1\mathrm{nm}$. X-ray diffraction, neutron diffraction and electron diffraction probe periodic order through coherent elastic scattering. With incident and outgoing wavevectors ${\mathbf{k}}_i,{\mathbf{k}}_f$, define $\mathbf{Q}={\mathbf{k}}_i-{\mathbf{k}}_f$. Elastic scattering has $|{\mathbf{k}}_i|=|{\mathbf{k}}_f|$ and constructive interference occurs when $\mathbf{Q}=\mathbf{G}$, a reciprocal lattice vector. The equivalent Bragg's law iswhere the detector scattering angle is $2\theta$, not $\theta$. Peak positions reveal plane spacings and the unit cell; systematic absent reflections constrain centring and the atomic basis.

For X-ray diffraction, a laboratory tube or a synchrotron supplies a collimated beam. In a monochromatic single-crystal experiment a goniometer rotates the specimen and an area detector records reflection positions and integrated wave intensities. A powder experiment uses many randomly oriented crystallites; the reflections form cones recorded as rings or peaks versus $2\theta$, permitting indexing and determination of lattice constants without selecting one crystallite. For a cubic cell, for example, $d_{hkl}=a/\sqrt{h^2+k^2+l^2}$. The white-beam Laue method is especially useful for crystal orientation.

The crystal basis structure factor isafter geometric and thermal corrections, including the Debye–Waller factor. For X-rays, $f_j$ describes scattering from the atomic electron density. Thus positions provide lattice geometry while relative wave intensities constrain atomic positions and species. Wave intensities do not directly give the phases of $F_{\mathbf{G}}$: structure refinement or other constraints must address the phase problem.

Neutron diffraction uses a reactor or spallation source, wavelength selection and a detector bank or time-of-flight detection. Neutrons interact with nuclei through nuclear scattering lengths that vary between elements and isotopes rather than simply with atomic number. This makes them useful for light atoms, isotope contrast and distinguishing some neighbours in the periodic table; their magnetic interaction also probes magnetic order. Electron diffraction in a transmission electron microscope gives strong signals from very small thin crystallites, although strong multiple scattering often complicates quantitative wave intensity analysis. These techniques complement one another.

A phonon dispersion relation requires both momentum and energy transfer, so elastic diffraction alone is insufficient. In coherent inelastic neutron scattering, the measured transfers for phonon creation areHere $\mathbf{q}$ is the phonon wavevector reduced into a Brillouin zone, and $s$ labels an acoustic phonon branch or optical phonon branch. Negative energy transfer corresponds to annihilating an already occupied phonon. For neutrons $E={\hslash }^2{k}^2/(2m_n)$, so wavelength and energy selection are linked. Typical thermal-neutron energies of a few to tens of millielectronvolts ($1\mathrm{meV}=10^{-3}$ electronvolts) are well matched to many lattice vibrations.

A triple-axis spectrometer uses a monochromator crystal to select ${\mathbf{k}}_i$, the sample as the second scattering axis, and an analyser crystal to select ${\mathbf{k}}_f$ before detection. Rotate these elements and the specimen to hold $\mathbf{Q}$ fixed while scanning $E_i-E_f$; inelastic peaks locate ${\omega }_s(\mathbf{q})$. Repeat at different $\mathbf{q}$ along symmetry directions to map the phonon dispersion relation. The dynamic structure factor governs peak weights, including the selection of phonon polarization by the scattering geometry. A time-of-flight spectrometer instead infers outgoing neutron energy from travel time over a known flight path and records many directions simultaneously.

Inelastic X-ray scattering can also resolve phonons when very high energy resolution is available. Optical Raman scattering and Brillouin scattering measure vibrations near the zone centre because photon momentum transfers are small compared with a typical reciprocal lattice spacing; they complement, rather than generally map the entire zone like, momentum-resolved neutron measurements. Ultrasound measurements give the long-wavelength sound speed, the initial slope $d\omega /dq$ of an acoustic phonon branch.

Oscillatory response. A material with viscoelasticity stores some deformation energy and dissipates some over a loading cycle. In small-amplitude oscillatory shear, represent the strain by $\gamma (t)=\mathrm{Re}({\gamma }_0{e}^{i\omega t})$. The shear-stress amplitude is related to the strain amplitude by the complex viscoelastic modulusHere $G_1=G^{\prime }$ is the storage modulus, the component in phase with strain, and $G_2=G^{\prime \prime }$ is the loss modulus, the dissipative quadrature component. Both have units of pressure. The stress leads the strain by an angle $\delta$, withFor this $e^{i\omega t}$ convention, a passive material has $G_2\ge 0$ at positive frequency. Integrating $\tau \dot{\gamma }$ over one period gives dissipated energy per unit volume $\pi {\gamma }_0^2{G}_2$, and the average input power per unit volume is $\omega {\gamma }_0^2{G}_2/2$. Thus the loss tangent compares the dissipative and elastic parts of the response. The frequency dependence reflects which internal relaxation processes can follow the imposed deformation.

Torsional measurement. Use $\rho$ for radial position inside the rod, reserving $r$ for its outer radius. A small angular displacement $\theta$ of one end relative to the fixed other end gives shear strainIn harmonic amplitude notation, the shear stress is $G^{\ast }\rho \Theta /\ell$. Its restoring torque is obtained by multiplying by the lever arm and integrating over the circular section:The factor $J={\int }_A{\rho }^{2}dA=\pi r^4/2$ is the polar second moment of area; it is not the disc's mass moment of inertia $I$. The torsional spring constant becomes the complex stiffness $K^{\ast }=J{G}^{\ast }/\ell$. Angular Newton's second law therefore gives the paper's compact representationIts precise interpretation is a torsional measurement of a complex shear modulus. At a specified frequency, with complex angular amplitude $\Theta$, the driven response obeysSetting the external torque to zero supplies the free-mode condition, whose damped frequencies are generally complex. A dissipative free oscillator does not execute a sustained nonzero oscillation at a real frequency without driving. In real time, a general frequency-dependent complex viscoelastic modulus represents a causal memory response, rather than a complex instantaneous force on a real displacement. The derivation assumes a uniform slender circular rod, small twist, negligible rod inertia relative to the disc, and no additional bending, slip or external damping.

Frequency regimes and sketches. A permanently crosslinked elastomer has a low-frequency rubbery plateau $G_R>0$. Slow chain rearrangements can follow the strain, so the network is relatively compliant but still elastic. At high frequency the relevant segmental motions are frozen, giving a much larger unrelaxed or glassy stiffness $G_U$. Between these regimes the motions lag behind the loading and dissipate strongly.

A standard linear solid model supplies a consistent illustrative sketch of this process. Put $x=\omega {\tau }_r$, where ${\tau }_r$ is the stress viscoelastic relaxation time, and $\Delta G=G_U-G_R>0$. ThensoAt very low frequency, $G_1\to G_R$, $G_2\sim \Delta Gx$ and $\tan \delta \to 0$. Through the relaxation region $G_1$ rises, $G_2$ has a maximum and $\tan \delta$ has a loss peak. At high frequency, $G_1\to G_U$, $G_2\sim \Delta G/x$ and $\tan \delta \to 0$ again. Real polymers usually have a spectrum of relaxation times, so these features are broader and can have further structure.

The two kinds of loss peak need not coincide. The peak frequencies in a standard linear solid areThe first follows by differentiating $x/(1+x^2)$; the second by differentiating $x/(G_R+G_U{x}^2)$. This is why the sketch marks the peaks separately. If the polymer is an uncrosslinked melt rather than a permanent network, there is an additional ultimate low-frequency terminal-flow regime, with $G_1\propto {\omega }^2$ and $G_2\propto \omega$; a permanent elastomer does not have that zero-frequency fluid limit.

Interstitial atoms in a metal. A suitable mechanism is the Snoek effect, a form of anelasticity. An interstitial defect distorts its local crystal structure anisotropically. Different site orientations become unequally favorable under an applied stress, so hopping atoms redistribute towards the preferred orientations and supply additional strain. The hopping time is thermally activated, approximately ${\tau }_a={\tau }_0{e}^{E_a/(k_{B}T)}$, with activation energy $E_a$.

At low frequency, $\omega {\tau }_a\ll 1$, the populations nearly equilibrate during each cycle. The defect contribution to strain is nearly in phase with stress, giving a relaxed compliance and little dissipation. At intermediate frequency, $\omega {\tau }_a\sim 1$, redistribution lags behind the stress and energy dissipation is largest. At high frequency, $\omega {\tau }_a\gg 1$, the populations are effectively frozen; the extra relaxation strain cannot develop, and the response approaches the stiffer unrelaxed elastic limit. For weak single-mode relaxation, the loss tangent is approximately proportional to $\omega {\tau }_a/[1+(\omega {\tau }_a{)}^2]$. The metal remains a solid in the low-frequency regime: the extra delayed strain is recoverable, not unrestricted viscous flow.

A continuous family of normal modes labelled by the wavevector and a branch index $s$. For $r$ chemical atoms per three-dimensional primitive unit cell, the dynamical matrix has $3r$ eigenvalues, giving $3r$ branches: three acoustic phonon branches and $3r-3$ optical phonon branches for a mechanically stable ordinary crystal structure.

Physical metallurgy connects the internal structure of metals and alloys to their properties and the processes used to change them. It studies crystal structures, crystal defects, phases and their transformations, including changes caused by deformation and heat treatment.

A controlled comparison heats samples of the same steel and cools them differently, then measures hardness and examines their microstructures. Differences in properties can then be related to phase formation and transformation history rather than to a change in overall chemical composition. This is why selecting an alloy and selecting its thermal/mechanical processing are coupled design decisions.^[ref]

Crystal polymorphism is the existence of different crystal structures for the same chemical substance. Each crystal polymorph can have its own mass density, melting behavior and range of stability on a phase diagram. A different external shape of the same crystal structure alone does not establish a different crystal polymorph.^[ref]

At fixed temperature and pressure, compare the chemical potentials of the possible crystal polymorphs. The lower value determines the stable bulk thermodynamic phase; equality permits phase coexistence when neither form is beaten by another thermodynamic phase. A higher-chemical potential form can persist through metastability, so finding it in a sample does not by itself locate an equilibrium boundary.

Consider a chosen primitive cubic lattice $a{\mathbb{Z}}^3$ with $a=0.400\mathrm{nm}$ and one identical point scatterer per primitive unit cell. Retain the incident wavelength $\lambda =0.154\mathrm{nm}$. This is an ideal point model. Its reciprocal lattice vectors are $q=(2\pi /a)(h,k,l)$ for integer triples other than $(0,0,0)$. Write $S=h^2+k^2+l^2$. The associated interplanar spacing and first-order Bragg angle satisfyThe reflection labels $(hkl)$ use the Miller index convention, keeping unreduced triples to encode diffraction order. With that convention, set $n=1$ in Bragg's law for each label. The $(200)$ reflection is equivalently the second diffraction order of $(100)$: $d_{200}=a/2$ does not imply extra scatterers halfway between the occupied $(100)$ planes. Counting both descriptions as distinct reflections would count the same direction twice.

For $(100)$, $(110)$, $(111)$ and $(200)$, respectively, $S=1,2,3,4$ and the calculated interplanar spacings are $0.400000$, $0.282843$, $0.230940$ and $0.200000\mathrm{nm}$. Their calculated deflections $2\theta$ are $22.197435^{\circ }$, $31.594744^{\circ }$, $38.952948^{\circ }$ and $45.287481^{\circ }$. Equal steps in $S$ give equal steps in ${\sin }^2\theta$, not equal steps in $2\theta$. This regularity makes an angular pattern a test of a proposed crystal structure.

A real crystal structure has a Bravais lattice together with an atomic basis of a crystal. The crystal lattice structure factor selects the periodic directions; the crystal basis structure factor combines the scattering amplitudes of atoms within a unit cell and can cancel a reflection. Consequently Bragg's law alone predicts neither allowed reflections nor their wave intensities. ^[ref] Systematic absences further restrict observed families; for example, face-centered cubic lattice centering permits only indices all even or all odd in conventional cubic indexing. The four strips above apply to the stated primitive cubic lattice, not to sodium chloride or a face-centered cubic lattice. ^[ref]

Pyrolytic carbon is a form of deposited carbon produced by thermal decomposition of a carbon-containing precursor. The name specifies a preparation route, not one uniquely fixed crystal structure or degree of ordering.

The deposited material can contain differently oriented graphitic domains. Texture therefore matters: properties measured parallel and perpendicular to preferred layers can differ strongly. Highly ordered pyrolytic graphite is a particular well-oriented material, whereas the broader term pyrolytic carbon also covers less ordered deposits. Preparation conditions, porosity and microstructure are needed to interpret a measured mechanical or thermal property.

The primary experimental-methods paper's section 4, equation 5, separates powder diffraction multiplicity, squared structure factor, polarization and geometric corrections. Sections 4.1–4.2 explain why powder diffraction intensity corrections depend on the measurement geometry rather than only the crystal structure.

Richard B. Frankel coauthored research on crystal structure.^[ref]

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.

A tetragonal $A{B}_2$ crystal structure with sixfold cation and threefold anion coordination numbers. A common fractional-coordinate description has A at $(0,0,0)$ and $(1/2,1/2,1/2)$, and B at $\pm (u,u,0)$ and $\pm (1/2+u,1/2-u,1/2)$, interpreted modulo integer lattice translations. The cation coordination is a distorted octahedron and the anion coordination is a planar triangle. Titanium dioxide in its rutile polymorph is an example.

R. W. G. Wyckoff compiled crystallographic structures and lattice data in a major reference work. A documented magnesium entry gives the two independent lattice parameters of its hexagonal structure.

Such tables connect X-ray diffraction measurements with an atomic model: lattice dimensions specify periodic geometry, while atomic positions specify how a unit cell is occupied. Both are needed to describe a crystal structure.^[ref]

A secondary phase is a distinguishable material phase present alongside a principal matrix phase. It can differ in chemical composition, crystal structure or both. Particles of a secondary phase within an alloy are therefore different from atoms simply dissolved in a solid solution.

Their effects depend on size, distribution and interface properties. Fine particles can obstruct dislocation motion and increase mechanical strength, while coarse or brittle regions can assist crack initiation. Heat treatment can change a secondary phase's amount and morphology, explaining why overall composition alone does not determine an alloy's behaviour.

A serpin is a member of a protein family best known for inhibiting serine proteases through a large change in protein structure. In an inhibitory serpin, a protease attacks the exposed reactive-center loop; subsequent movement of that loop can trap the attached protease and disrupt its active site.^[ref]

A 2000 crystal structure of a serpin–protease complex gave direct structural evidence for this trap. The attacked region moved about $71\mathring{\mathrm{A}}$ across the serpin, carrying and deforming the tethered protease.^[ref] A static plug at an active site would give a very different picture: the inhibitory reaction here uses a moving protein scaffold.

Antithrombin gives a regulated example. Binding a specific heparin fragment changes its reactive-center loop exposure and increases inhibition of suitable coagulation proteases.^[ref] Thus, membership in a protein family, availability of a reactive loop, and the measured inhibitory activity under particular conditions are related but distinct observations.

An aluminium-bearing silicate mineral, one of several minerals sharing this composition with different crystal structures.

The site-symmetry group of a point $\mathbf{r}$ in a crystal structure is the subgroup of its space group $G$ that fixes that point:It is the stabilizer subgroup for the action on points. Although the space group contains infinitely many lattice translations, the site-symmetry group is finite and is isomorphic to a subgroup of the corresponding crystallographic point group.^[ref]

For an original example, take a lattice with inversion symmetry and no other point operations, the space-group type conventionally denoted $P\bar{1}$. At the origin, both the identity and inversion $\mathbf{r}\mapsto -\mathbf{r}$ fix the site, giving a group of order2. A generic site has only the identity in its site-symmetry group; inversion instead produces a distinct site at $-\mathbf{r}$.

The equality in the definition fixes the actual point. At $\mathbf{r}=\mathbf{a}/2$, inversion alone moves it to $-\mathbf{a}/2$. Inversion followed by translation by $\mathbf{a}$ fixes it. Reducing fractional coordinates modulo integers can conceal that translation, but the full space group operation retains it.

The crystallographic site multiplicity consequently satisfiesIn this primitive inversion example the general multiplicity is2, so the origin has multiplicity1 while a generic site has multiplicity2. This is why high site symmetry reduces the number of distinct sites generated in a cell.^[ref]

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.

A state of matter is a broad category of material behavior. A solid resists a sustained change of shape, a liquid flows while remaining difficult to compress, and a gas expands to fill its container. These distinctions are useful first descriptions, but they do not enumerate all thermodynamic phases. Plasma adds collective behavior of charged particles, while liquid crystals show that flow and structural order need not disappear together.

Distinguish three questions: what broad state of matter does the material resemble, which thermodynamic phase is present, and what are its temperature, pressure and other state variables? Two samples at different temperatures can be in the same liquid thermodynamic phase. Conversely, crystal polymorphism gives different solid thermodynamic phases of the same substance, distinguished by their crystal structures. A phase transition between them need not change the broad state-of-matter category.

For an imagined substance with crystalline forms A and B, melting A produces a solid–liquid phase transition, while converting A to B produces a solid–solid phase transition. A sample containing A and B has two thermodynamic phases even though all its material is solid. Cutting one uniform crystal into two pieces does not create a second thermodynamic phase: count distinct equilibrium forms, rather than pieces or grains. This distinction matters when applying the phase rule.

A phase diagram locates stable thermodynamic phases using temperature and pressure. Its liquid–gas phase coexistence curve ends at a liquid-gas critical point, around which a path connects liquid and gas without a first-order phase transition. Broad categories can therefore remain useful without a sharp boundary.^[ref]