Crystal structure
A crystal structure is a highly ordered, periodic arrangement of atoms, ions, or molecules in three-dimensional space that defines the microscopic organization of a crystalline solid.[1] This repeating pattern arises from the minimization of energy during solidification, where particles pack as densely as possible while respecting electrostatic and bonding interactions.[2] The fundamental components include a Bravais lattice, which provides the geometric framework of translationally repeating points, and a basis or motif—a group of atoms attached to each lattice point—that specifies the atomic content.[1] Together, these elements form the unit cell, the smallest volume that, when translated via lattice vectors, reproduces the entire structure.[3] Crystal structures are classified into seven crystal systems based on the symmetry of their unit cell parameters (edge lengths a, b, c and angles α, β, γ): triclinic (no symmetry constraints), monoclinic (one 2-fold axis), orthorhombic (three perpendicular axes), tetragonal (two equal axes perpendicular to the third), trigonal (a = b = c, α = β = γ ≠ 90°), hexagonal (two equal axes at 120° with a third perpendicular), and cubic (all edges equal and angles 90°).[4] These systems encompass 14 distinct Bravais lattices, accounting for variations like primitive, body-centered, face-centered, and base-centered arrangements that maintain translational symmetry without altering the overall point group.[5] Common examples include the face-centered cubic lattice in metals like aluminum and the hexagonal close-packed structure in magnesium, both achieving high packing efficiency near 74%.[6] The arrangement in a crystal structure profoundly influences macroscopic properties, such as mechanical strength, electrical conductivity, thermal expansion, and optical anisotropy, making it central to fields like materials science, mineralogy, and solid-state physics.[7] Techniques like X-ray diffraction exploit the periodic nature of crystals to determine structures at atomic resolution, revealing bonding connectivity and intermolecular interactions essential for understanding phase transitions and defects.[1] While perfect crystals are idealized, real materials often feature imperfections like vacancies or dislocations that modify behavior without disrupting the underlying lattice.[6]Basic Elements
Unit cell
In crystallography, the unit cell is defined as the smallest volume element of a crystal lattice that contains all the structural information necessary to describe the entire crystal, such that repeating this volume by pure translations fills the space without gaps or overlaps.[8] This parallelepiped-shaped building block serves as the fundamental repeating unit, encapsulating the positions of atoms, ions, or molecules relative to lattice points.[9] Unit cells are classified as primitive or non-primitive based on the number of lattice points they contain. A primitive unit cell, also known as a simple unit cell, includes exactly one lattice point and has the minimal volume required to represent the lattice translations.[10] In contrast, non-primitive unit cells, such as body-centered (with two lattice points) or face-centered (with four lattice points), contain additional lattice points at internal positions like the body center or face centers, resulting in larger volumes but often higher symmetry for practical description.[11] For example, the body-centered cubic structure features a lattice point at each corner and one at the cube's center, while the face-centered cubic adds points at the centers of each face.[10] The geometry of a unit cell is characterized by three lattice constants—a, b, and c, representing the lengths of the edges along the three crystallographic axes—and three interaxial angles—\alpha (between edges b and c), \beta (between a and c), and \gamma (between a and b).[10] These parameters fully define the shape and size of the unit cell, varying across crystal systems; for instance, in the cubic system, a = b = c and \alpha = \beta = \gamma = 90^\circ, forming a cube with equal edges and right angles.[8] In the hexagonal system, a = b \neq c, with \alpha = \beta = 90^\circ and \gamma = 120^\circ, resulting in a hexagonal prism.[12] Visualizations of these often depict the cubic unit cell as a symmetric cube with atoms at corners (and possibly centers for non-primitive types), and the hexagonal unit cell as a taller prism with three equivalent basal edges forming 120° angles.[8][12] Through translational symmetry, identical unit cells are repeated infinitely in three dimensions along the lattice vectors, generating the complete crystal lattice as an extended periodic array.[9] This repetition ensures that every point in the crystal can be reached by integer combinations of the unit cell's defining vectors, preserving the structural integrity across the material.[10]Crystal lattice
A crystal lattice is defined as an infinite, periodic array of discrete points in three-dimensional space, where each point represents the position of a unit cell that repeats translationally to fill the entire volume without gaps or overlaps.[13] This arrangement captures the long-range order inherent to crystalline solids, distinguishing them from amorphous materials by their repeating translational symmetry. The periodicity of the crystal lattice is mathematically described by three primitive lattice vectors, conventionally denoted as \mathbf{a}, \mathbf{b}, and \mathbf{c}, which are non-coplanar and connect a lattice point to its nearest neighbors along the three independent directions.[14] Any lattice point can then be reached by integer linear combinations of these vectors: \mathbf{R} = m\mathbf{a} + n\mathbf{b} + p\mathbf{c}, where m, n, and p are integers. These vectors define the fundamental translations that preserve the lattice structure, ensuring that the environment around every lattice point is identical.[15] The reciprocal lattice provides a dual representation in momentum space, constructed from basis vectors \mathbf{b}_1, \mathbf{b}_2, and \mathbf{b}_3 that satisfy \mathbf{a} \cdot \mathbf{b}_i = 2\pi \delta_{i,j} (with \delta as the Kronecker delta), ensuring orthogonality to pairs of direct lattice vectors. Reciprocal lattice points correspond to wavevectors where plane waves exhibit the same periodicity as the direct lattice, making it indispensable for analyzing diffraction phenomena, as scattering intensities peak at these points in experiments like X-ray diffraction. In qualitative terms, direct space describes the real-space positions and arrangements of atoms within the crystal, while reciprocal space captures the Fourier transform of this density, relating spatial frequencies to scattering angles and enabling the interpretation of diffraction patterns as a map of the lattice's periodicity.[16] For instance, in a simple cubic lattice with lattice constant a, the direct lattice vectors are \mathbf{a} = a\hat{x}, \mathbf{b} = a\hat{y}, \mathbf{c} = a\hat{z}, yielding lattice points at coordinates (ma, na, pa) for integers m, n, p. The corresponding reciprocal lattice is also simple cubic but scaled by $2\pi/a, with points at (2\pi h/a, 2\pi k/a, 2\pi l/a) for integers h, k, l.[17]Indexing and Geometry
Miller indices
Miller indices are a symbolic notation system used in crystallography to designate the orientation of planes and directions within a crystal lattice relative to the unit cell axes. This system was introduced in 1839 by the British mineralogist and crystallographer William Hallowes Miller in his work A Treatise on Crystallography, providing a concise way to describe lattice features using small integers derived from geometric intercepts. The notation facilitates the analysis of crystal symmetry and structure without requiring detailed coordinate descriptions, making it essential for identifying specific atomic arrangements in materials.[12][18] For crystal planes, the Miller indices are denoted as (hkl), where h, k, and l are integers representing the reciprocals of the fractional intercepts that the plane makes with the crystallographic axes a, b, and c, respectively, scaled to the smallest integers by clearing fractions. To determine the indices, one identifies the intercepts of the plane on the axes (in units of the lattice parameters); takes the reciprocals; and multiplies through by the least common multiple of the denominators to obtain whole numbers, with the lowest values preferred. If a plane is parallel to an axis, the intercept is infinite, resulting in a zero index for that component (e.g., a plane parallel to the b- and c-axes has k = 0 and l = 0). Planes with negative intercepts are indicated by placing a bar over the index (e.g., (\bar{1}00)). The notation {hkl} refers to a family of equivalent planes related by the crystal's symmetry operations. For instance, in a cubic lattice, the (100) plane corresponds to a face of the unit cell parallel to the yz-plane, intersecting the a-axis at one unit length while being parallel to the others.[12][18] Directions in the crystal lattice are specified using Miller indices in the form [uvw], where u, v, and w are the smallest integers proportional to the components of the direction vector along the a, b, and c axes, respectively. Unlike planes, direction indices are not based on reciprocals but directly on the lattice vector coordinates, often reduced to the lowest terms. Negative directions are denoted with bars (e.g., [\bar{1}10]). The notationCrystal planes and directions
Crystal planes in a crystal structure are defined as families of parallel planes that pass through the lattice points, representing sets of atomic layers stacked in a repeating manner. These planes are fundamental to understanding the geometric arrangement of atoms within the lattice, as they delineate the layers where atoms are densely packed or exhibit specific bonding characteristics. In face-centered cubic (FCC) lattices, for instance, the {111} family of planes consists of close-packed atomic layers that form equilateral triangular arrangements, contributing to the high density and stability of these structures.[20][21] Crystal directions, in contrast, refer to straight lines that connect lattice points along specific vectors within the crystal lattice, defining pathways for atomic alignment or movement. These directions often coincide with the shortest lattice vectors or high-symmetry axes, influencing processes such as atomic diffusion or dislocation motion. In plastic deformation, slip directions are particular crystallographic directions along which dislocations glide, typically the close-packed directions like <110> in FCC crystals, enabling shear without bond breaking in other orientations.[22][23][24] Due to the symmetry of the crystal lattice, multiple planes and directions that are equivalent under rotational or reflection operations form families, denoted by curly braces {} for planes and angle brackets <> for directions. In cubic crystals, the <100> family includes all directions equivalent to [25], such as and , which point along the principal axes and exhibit identical physical properties due to the lattice's isotropic symmetry in these orientations.[20][22][26] Crystal planes and directions play a critical role in determining material properties, such as cleavage, where crystals fracture preferentially along planes of weak atomic bonding, like the {100} planes in some ionic crystals, resulting in smooth, flat surfaces. Similarly, during crystal growth, facets often develop perpendicular to low-index directions or along specific planes with minimal surface energy, influencing the overall morphology of the crystal. In hexagonal close-packed (HCP) structures, the basal plane, denoted as (0001), serves as a prominent example of a close-packed layer that governs anisotropic growth and deformation behaviors in materials like magnesium or zinc.[27][20][28]Interplanar spacing
Interplanar spacing, denoted as d_{hkl}, represents the perpendicular distance between successive parallel crystal planes characterized by Miller indices (hkl). These planes are defined by their intercepts on the lattice axes, and the spacing provides a key geometric parameter for understanding crystal periodicity and diffraction behavior. The concept arises from the arrangement of atoms in the lattice, where parallel planes of atoms scatter waves constructively under specific conditions. In fractional coordinates (where positions are expressed relative to the lattice vectors), the equation for planes is h x + k y + l z = p (with p integer for lattice planes). The general formula for interplanar spacing is d_{hkl} = \frac{1}{|\vec{G}_{hkl}|}, where \vec{G}_{hkl} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3} is the reciprocal lattice vector, with reciprocal basis vectors \vec{b_i} defined such that \vec{a_i} \cdot \vec{b_j} = 2\pi \delta_{ij} (or 1 in some conventions). The magnitude |\vec{G}_{hkl}| is computed using the metric tensor of the reciprocal lattice, which accounts for the angles between direct lattice vectors. This reciprocal formulation is universal and simplifies calculations for diffraction, as \vec{G}_{hkl} is perpendicular to the (hkl) planes by construction.[14] For crystal systems with orthogonal axes (cubic, tetragonal, orthorhombic), where all angles are 90° but edge lengths may differ, the formula simplifies to d_{hkl} = \left( \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2} \right)^{-1/2}. In the cubic system, where a = b = c, it further simplifies to d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}. In the hexagonal system, accounting for the sixfold symmetry and c/a ratio with \gamma = 120^\circ, the expression is d_{hkl} = \left[ \frac{4(h^2 + hk + k^2)}{3a^2} + \frac{l^2}{c^2} \right]^{-1/2}. These formulas enable precise computation of spacings from known lattice dimensions.[29][30] A primary application of interplanar spacing lies in X-ray diffraction, where Bragg's law relates it to observable diffraction angles: n\lambda = 2d_{hkl} \sin\theta, with n as the diffraction order, \lambda the X-ray wavelength, and \theta the incidence angle. This equation allows experimental determination of d_{hkl} from measured peak positions, facilitating crystal structure analysis without deriving the full interference conditions here.[31] Interplanar spacing is influenced by external factors that modify lattice parameters. Lattice strain, arising from mechanical deformation or defects, can compress or expand planes, shifting d_{hkl} and thus diffraction peaks. Temperature effects occur via thermal expansion, where increasing heat causes lattice parameters to grow, enlarging d_{hkl} proportionally; for instance, coefficients of thermal expansion quantify this change per degree Kelvin.[32][33] As an example, consider sodium chloride (NaCl), which adopts a face-centered cubic structure with lattice parameter a = 5.64 Å. For the (111) planes, d_{111} = \frac{a}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{5.64}{\sqrt{3}} \approx 3.26 Å, illustrating how the formula yields atomic-scale distances relevant to ionic bonding and diffraction studies.[34]Symmetry Classification
Bravais lattices
A Bravais lattice is defined as an infinite array of discrete points in three-dimensional space where each point has an identical environment, generated solely by translational symmetry. These lattices represent the distinct ways to arrange points such that no two are equivalent except through pure translations, ensuring the lattice cannot be reduced to a simpler form by redefining the unit cell. In 1850, French physicist and crystallographer Auguste Bravais systematically enumerated these unique arrangements, identifying exactly 14 Bravais lattices in three dimensions.[35] The criteria for uniqueness among Bravais lattices emphasize that additional lattice points within the conventional unit cell must arise only from translations of the primitive vectors; any extraneous points would imply either a smaller primitive cell or a different lattice type, violating the minimal description. This leads to four primary centering types: primitive (P), where lattice points are only at the corners; base-centered (C), with additional points at the centers of two opposite faces; body-centered (I), with a point at the body center; and face-centered (F), with points at the centers of all six faces. These centering variations, combined with the geometric constraints of the lattice systems, yield the 14 distinct types.[35][36] The 14 Bravais lattices are classified within seven crystal systems, each defined by specific relationships among the unit cell parameters (lattice constants a, b, c and angles α, β, γ). For instance, the cubic system features equal lengths and right angles (a = b = c, α = β = γ = 90°), while the tetragonal system has a = b ≠ c and α = β = γ = 90°. Representative examples include the primitive cubic lattice in the cubic system and the body-centered tetragonal lattice in the tetragonal system. The full classification is summarized below:| Crystal System | Bravais Lattice Types | Description |
|---|---|---|
| Triclinic | Primitive (P) | No symmetry constraints; a ≠ b ≠ c, α ≠ β ≠ γ. |
| Monoclinic | Primitive (P), Base-centered (C) | One right angle; a ≠ b ≠ c, α = γ = 90°, β ≠ 90°. |
| Orthorhombic | Primitive (P), Base-centered (C), Body-centered (I), Face-centered (F) | Three right angles; a ≠ b ≠ c, α = β = γ = 90°. |
| Tetragonal | Primitive (P), Body-centered (I) | Two equal lengths with right angles; a = b ≠ c, α = β = γ = 90°. |
| Trigonal (Rhombohedral) | Primitive (R) | a = b = c, α = β = γ ≠ 90°. |
| Hexagonal | Primitive (P) | a = b ≠ c, α = β = 90°, γ = 120°. |
| Cubic | Primitive (P), Body-centered (I), Face-centered (F) | a = b = c, α = β = γ = 90°. |
Lattice systems
Lattice systems classify the possible geometries of crystal lattices into seven distinct categories, determined by the constraints on the unit cell's edge lengths a, b, c and the angles between them \alpha (between b and c), \beta (between a and c), and \gamma (between a and b). This classification arises from the requirement that the lattice must be periodic and translationally symmetric, with the systems reflecting increasing levels of metric symmetry from the lowest in triclinic to the highest in cubic. The grouping enables systematic analysis of crystal structures without considering full rotational symmetries, focusing solely on the metric relations that define distances and angles within the lattice./07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure) The seven lattice systems, along with their parameter constraints, are summarized in the following table:| Lattice System | a | b | c | \alpha | \beta | \gamma | Number of Bravais Lattices |
|---|---|---|---|---|---|---|---|
| Triclinic | \neq b | \neq c | arbitrary | \neq 90^\circ | \neq 90^\circ | \neq 90^\circ | 1 (primitive) |
| Monoclinic | \neq b | \neq c | arbitrary | $90^\circ | \neq 90^\circ | $90^\circ | 2 (primitive, base-centered) |
| Orthorhombic | \neq b | \neq c | arbitrary | $90^\circ | $90^\circ | $90^\circ | 4 (primitive, base-centered, body-centered, face-centered) |
| Tetragonal | = b | \neq c | arbitrary | $90^\circ | $90^\circ | $90^\circ | 2 (primitive, body-centered) |
| Trigonal (Rhombohedral) | = b = c | \neq 90^\circ | \neq 90^\circ | \neq 90^\circ | 1 (rhombohedral) | ||
| Hexagonal | = b | \neq c | arbitrary | $90^\circ | $90^\circ | $120^\circ | 1 (primitive) |
| Cubic | = b = c | $90^\circ | $90^\circ | $90^\circ | 3 (primitive, body-centered, face-centered) |
Crystal systems
Crystal systems represent a fundamental classification in crystallography, grouping the 32 point groups into seven categories based on the overall symmetry compatible with the underlying lattice geometry. These systems—triclinic, monoclinic, orthorhombic, tetragonal, trigonal (or rhombohedral), hexagonal, and cubic—define the possible macroscopic symmetries of crystals by integrating rotational and reflection symmetries from point groups with the metric constraints of the lattice. Unlike lattice systems, which focus solely on geometric parameters like axis lengths and angles, crystal systems emphasize the full symmetry repertoire, ensuring that only point groups whose operations preserve the lattice are assigned to each category.[39][40] Each crystal system corresponds directly to one of the seven lattice systems, but restricts inclusion to point groups that align with the lattice's metric symmetry, creating a mapping that excludes incompatible symmetries. For instance, the orthorhombic lattice system (with three mutually perpendicular axes of unequal length) maps to the orthorhombic crystal system, which accommodates point groups like 222, mm2, and mmm, all of which respect the 90° angles and distinct axis lengths. Similarly, the cubic lattice system aligns with the cubic crystal system, incorporating high-symmetry point groups such as 23, m3, 432, \bar{4}3m, and m\bar{3}m, where operations like threefold and fourfold rotations are feasible due to equal axes and right angles. This mapping ensures that the symmetry elements do not distort the lattice, with lower-symmetry point groups fitting into higher-symmetry systems if their operations are subgroups.[41][40] Holohedry refers to the point group exhibiting the maximal symmetry within each crystal system, representing the "complete" form that includes all possible symmetry operations allowed by the lattice. For the cubic system, the holohedral group is m\bar{3}m (also denoted 4/m \bar{3} 2/m), featuring inversion, mirror planes, and multiple rotation axes, as seen in structures like halite (NaCl). In the triclinic system, the holohedry is simply \bar{1}, limited to inversion without rotations or mirrors, reflecting the absence of higher symmetries. These holohedral forms serve as benchmarks, with other point groups in the system being hemihedral or merohedral subgroups that omit certain operations.[42][40] Illustrative examples highlight the diversity: the isometric (cubic) crystal system demonstrates maximal isotropy, with equal lattice parameters (a = b = c) and α = β = γ = 90°, enabling highly symmetric minerals like diamond (point group 4/m \bar{3} 2/m) or pyrite (point group \bar{4} 3 m). Conversely, the anorthic (triclinic) system lacks any symmetry constraints beyond the lattice, with arbitrary parameters (a ≠ b ≠ c, α ≠ β ≠ γ ≠ 90°), as in turquoise (point group 1) or microcline (point group \bar{1}), where even basic rotations are absent. These extremes underscore how crystal systems encapsulate both geometric and symmetric aspects./07%3A_Molecular_and_Solid_State_Structure/7.01%3A_Crystal_Structure)[41] Transitions between crystal systems arise from subtle variations in lattice parameters or symmetry-breaking during phase changes, reclassifying structures when parameters deviate from defining thresholds. For example, a cubic system (a = b = c, all angles 90°) may shift to tetragonal if one axis elongates slightly (a = b ≠ c, angles 90°), reducing the point group from m\bar{3}m to 4/mmm, as observed in some perovskites under pressure. Similarly, orthorhombic symmetry can distort to monoclinic by tilting one angle away from 90°, lowering the holohedry from mmm to 2/m; such changes often occur in temperature-driven phase transitions, where thermal expansion or atomic displacements break higher symmetries while preserving lower ones. These transitions illustrate the continuum of symmetry in crystals, governed by energetic stability.[43][44]Point groups
Point groups in crystallography refer to the finite collections of symmetry operations—rotations, reflections, and inversions—that map a crystal lattice onto itself when performed around a fixed point, preserving the overall periodicity of the structure. These groups describe the external symmetry of crystals without involving translations, focusing solely on operations that leave a central point invariant. The possible rotation axes are limited to 1-, 2-, 3-, 4-, and 6-fold due to compatibility with translational symmetry in three dimensions./02%3A_Rotational_Symmetry/2.04%3A_Crystallographic_Point_Groups) The fundamental symmetry elements comprising these point groups include the identity operation (which leaves the lattice unchanged), proper rotations about principal axes (denoted as n-fold, where n = 1, 2, 3, 4, or 6), mirror planes (perpendicular or parallel to axes), the inversion center (which maps each point to its opposite through the origin), and improper rotoinversions (combinations of rotation and inversion). For instance, a 2-fold rotation reverses direction by 180 degrees, while a mirror plane reflects across its surface. These elements combine in specific ways to form closed groups under composition, ensuring all operations are consistent with lattice invariance.[40][45] There are exactly 32 crystallographic point groups, arising from the permissible combinations of these elements that align with the seven crystal systems. They are denoted using two primary notations: the Schoenflies system (common in molecular spectroscopy, e.g., D_{4h} for a group with a 4-fold axis, horizontal mirrors, and dihedral planes) and the international (Hermann-Mauguin) system (standard in crystallography, e.g., 4/mmm for the same group, indicating a 4-fold axis with mirrors and dihedral planes). Examples include the trivial group 1 (or C_1, no symmetry beyond identity) in the triclinic system and the highly symmetric O_h (or m\bar{3}m) in the cubic system, which incorporates 48 operations including 3-fold, 4-fold, and 2-fold axes along multiple directions.[42][46] These 32 point groups are distributed across the crystal systems as follows: 2 in triclinic (1, \bar{1}), 3 in monoclinic (2, m, 2/m), 3 in orthorhombic (222, mm2, mmm), 7 in tetragonal (4, \bar{4}, 4/m, 422, 4mm, \bar{4}2m, 4/mmm), 5 in trigonal (3, \bar{3}, 32, 3m, \bar{3}m), 7 in hexagonal (6, \bar{6}, 6/m, 622, 6mm, \bar{6}m2, 6/mmm), and 5 in cubic (23, m\bar{3}, 432, \bar{4}3m, m\bar{3}m). This distribution reflects the increasing symmetry constraints from lower (triclinic) to higher (cubic) systems, with cubic hosting the highest symmetry groups. For clarity, the groups can be summarized in the following table, using international notation with representative Schoenflies equivalents:| Crystal System | Number of Point Groups | Examples (International / Schoenflies) |
|---|---|---|
| Triclinic | 2 | 1 / C_1, \bar{1} / C_i |
| Monoclinic | 3 | 2 / C_2, m / C_s, 2/m / C_{2h} |
| Orthorhombic | 3 | 222 / D_2, mm2 / C_{2v}, mmm / D_{2h} |
| Tetragonal | 7 | 4 / C_4, 4mm / C_{4v}, 4/mmm / D_{4h} |
| Trigonal | 5 | 3 / C_3, 3m / C_{3v}, \bar{3}m / D_{3d} |
| Hexagonal | 7 | 6 / C_6, 6mm / C_{6v}, 6/mmm / D_{6h} |
| Cubic | 5 | 23 / T, 432 / O, m\bar{3}m / O_h |
Space groups
Space groups represent the complete set of symmetries for periodic crystal structures in three dimensions, extending the 32 crystallographic point groups by incorporating lattice translations along with nonsymmorphic operations such as screw axes and glide planes. These elements allow the symmetry operations to fill space while maintaining the periodic arrangement of atoms. There are exactly 230 distinct space groups, enumerated and classified in the International Tables for Crystallography.[50] Of these, 73 are symmorphic space groups, which combine point group operations with pure lattice translations without fractional shifts, whereas the remaining 157 are nonsymmorphic, featuring screw axes (rotations combined with partial translations parallel to the axis) or glide planes (reflections combined with partial translations parallel to the plane)./03:_Space_Groups/3.04:_Group_Properties) Space groups are denoted using the Hermann–Mauguin symbol, as standardized in the International Tables for Crystallography, which specifies the lattice type, principal axes, and any nonsymmorphic elements.[50] For instance, the symbol P2₁/c describes a primitive (P) monoclinic lattice with a twofold screw axis (2₁) and a glide plane (c) perpendicular to the b-axis, reflecting combined rotational and translational symmetries. The distribution of the 230 space groups varies by crystal system, reflecting the increasing constraints on symmetry as metric parameters become more equal:| Crystal System | Number of Space Groups |
|---|---|
| Triclinic | 2 |
| Monoclinic | 13 |
| Orthorhombic | 59 |
| Tetragonal | 68 |
| Trigonal | 25 |
| Hexagonal | 27 |
| Cubic | 36 |