A crystal is a solid with long-range spatial order in its constituents. In a periodic crystal, translation by a lattice vector reproduces the structure; a unit cell and its constituent arrangement describe the repeating pattern. The atomic structure is distinct from the visible external shape: an irregularly cut specimen can remain a single ordered crystal.
Diffraction probes that order because radiation scattered from many regularly arranged sites can reinforce at selected directions. The resulting pattern reveals structural spacings and orientation, rather than merely showing the specimen's outer faces. A single crystal maintains one coherent orientation across the specimen, while a polycrystalline material contains many crystal grains. Crystal orientation can make properties anisotropic, so a test along one direction need not predict response along another. Surface shape additionally depends on equilibrium surface free energy and growth kinetics.
A crystal lattice records the translation symmetry of a periodic crystal. With a primitive basis of three linearly independent lattice vectors , 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]
Matrices, Mappings and Crystallographic Symmetry by Hans Wondratschek. Section 1.5, Definition D 1.5.2 and primitive bases
For example, a one-dimensional chain with alternating species at spacing has translation period , not : shifting by exchanges distinct species. It can be described by lattice with a two-atom basis at offsets zero and . 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.
A magnetic unit cell repeats the ordered magnetic moments as well as the atomic positions. It can exceed the chemical primitive unit cell when translations interchange oppositely ordered magnetic sublattices. For a simple-cubic Néel pattern , translations preserving the sum's parity form an FCC lattice with conventional parameter twice the chemical spacing and a two-spin primitive basis.
A macroscopically planar crystal boundary characterized by a crystallographic orientation. Its Miller index specifies that orientation; visible facet area depends on surface thermodynamics and growth kinetics. A low surface energy and a slow growth rate are different properties.
A solid containing many differently oriented crystalline crystal grains. grain boundaries impose compatibility and act as transport paths or sinks for mobile defects.
A grain in a polycrystal is a region whose crystal structure has approximately one common orientation, separated from differently oriented regions by grain boundaries. The atoms remain ordered within each grain even when a macroscopic specimen contains a complicated mosaic of orientations. A grain is therefore different from an amorphous patch or a separate chemical phase.
The boundaries enforce geometric compatibility between neighboring crystals and can obstruct dislocation glide, provide rapid transport paths or act as sinks for defects. A mechanical measurement over many grains averages both the intragrain response and those boundary effects. Observing a prepared cross-section can reveal grain shape, but a single two-dimensional section need not directly give every grain's three-dimensional size. This distinction matters when comparing an elongated grain structure with one described by a single mean grain size.
A measure of crystal grain elongation. It changes directional boundary spacings and can influence anisotropic defect transport independently of average grain size.
A measure of the spatial scale of crystal grains. Its definition must be stated for elongated crystal grains; size, aspect ratio and orientation distribution have distinct physical effects.
Crystal packing describes how constituents are arranged and make contact within a crystal. It includes both the repeated geometry and, for molecular crystals, the orientations and intermolecular contacts of the molecules. Composition alone does not determine this arrangement: the same substance can have different crystal polymorphs with different packing.
A useful geometric model replaces constituents by equal hard spheres. Hexagonal close packing and face-centered cubic packing put each sphere next to twelve nearest neighbours but arrange successive close-packed layers differently. Their local contact geometry can therefore agree while their longer-range order differs. Real molecular shapes and directional intermolecular forces supply further constraints. Packing affects mass density, available slip geometry and directional response, making diffraction a way to connect structural measurements with macroscopic properties rather than merely assign a repeating pattern.
A cocrystal is a single crystalline phase containing two or more distinct molecular components in a shared ordered structure. Its components occupy defined roles in the repeating arrangement, rather than simply forming separate crystals mixed together. A stylized arrangement of molecules A and B illustrates the idea without assuming that every cocrystal has that ratio.
Intermolecular forces and molecular geometry influence which combined structure is favorable. A diffraction measurement tests the shared ordering, while a phase diagram distinguishes a stable combined phase from coexistence of separate A-rich and B-rich phases. This is useful because a shared lattice can change properties such as melting behavior and mass density even though the identities of the molecular components remain recognizable. The word describes a structural phase assignment and does not by itself predict which component combinations will form one.
Close packing of triangular layers with ABAB stacking. Its nearest-layer geometry agrees with face-centered cubic stacking, but the further-layer registry differs.
A cubic crystal structure with lattice sites at cube corners and face centres. Its close-packed layers are (111) planes, stacked ABCABC.
The fixed-volume equilibrium crystal shape obtained by minimizing orientation-dependent surface free energy. Facet support-plane distances are proportional to their interfacial free energies under the construction assumptions.
An equilibrium construction for a supported particle that modifies the Wulff construction using particle-support and support-medium interfacial energies. The support truncates the corresponding free-particle shape.
A single crystal is a crystalline specimen whose ordered structure has one continuous crystallographic orientation, rather than being divided into differently oriented crystal grains. It can still contain dislocations, impurities and other defects; the term does not mean an absolutely perfect lattice.
A diffraction pattern from one well-oriented specimen resolves its crystallographic directions, while rotating it changes which reflections are observed. Mechanical or thermal measurements along different directions can reveal intrinsic anisotropy that would be averaged over many orientations in a polycrystalline material. The absence of internal grain boundaries also changes transport and deformation mechanisms that depend on those boundaries. A specimen's outline is not the criterion: cutting a piece from a single crystal changes its external shape without necessarily changing its internal crystallographic identity.
