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Orthorhombic crystal system

The orthorhombic crystal system is one of the seven fundamental crystal systems in crystallography, defined by a lattice with three mutually perpendicular axes of unequal lengths (a ≠ b ≠ c) and all interaxial angles equal to 90° (α = β = γ = 90°).^[1] This system encompasses crystals where the unit cell forms a rectangular parallelepiped, distinguishing it from higher-symmetry systems like cubic or tetragonal by the lack of equal axis lengths.^[2] The orthorhombic system features four distinct Bravais lattices: primitive (P), base-centered (C), body-centered (I), and face-centered (F), each with varying numbers of lattice points per unit cell—1 for P, 2 for C and I, and 4 for F.^[1]^[2] Symmetry in this system is governed by three point groups: 222 (rhombic-disphenoidal, with three perpendicular 2-fold rotation axes), mm2 (rhombic-pyramidal, with one 2-fold axis and two mirror planes), and mmm (rhombic-dipyramidal, with three 2-fold axes, three mirror planes, and an inversion center).^[3] These point groups combine with the Bravais lattices to yield 59 space groups, representing a significant portion of the 230 total space groups in three-dimensional crystallography.^[4] Crystals in the orthorhombic system often exhibit prismatic, tabular, or blocky habits, with the b-axis conventionally the longest, followed by a and then c (b > a > c).^[5] Notable examples include minerals such as barite (BaSO₄, mmm class), topaz (Al₂SiO₄(F,OH)₂, mmm class), and epsomite (MgSO₄·7H₂O, 222 class), which demonstrate the system's prevalence in natural and synthetic materials.^[3]^[5] The orthorhombic structure is common in ionic, molecular, and metallic compounds, influencing properties like anisotropy in thermal expansion and mechanical strength due to the unequal axes.^[1]

Fundamentals

Definition and Unit Cell

The orthorhombic crystal system is one of the seven fundamental crystal systems in crystallography, distinguished by its geometric constraints on the unit cell. It features three mutually perpendicular axes of unequal lengths, denoted as a \neq b \neq c, with all interaxial angles exactly 90 degrees (\alpha = \beta = \gamma = 90^\circ).^[6]^[7]^[8] The unit cell in this system forms a rectangular prism, also known as a rectangular parallelepiped, which serves as the smallest repeating unit that tiles the crystal lattice. This shape arises conceptually from stretching a cubic lattice along two orthogonal directions by different amounts, resulting in the elongation of two axes while preserving right angles.^[6]^[8]^[9] The basis vectors align with the coordinate axes, where the vector \mathbf{a} extends along the a-direction, \mathbf{b} along the b-direction, and \mathbf{c} along the c-direction, each of distinct magnitude and oriented perpendicularly to define the cell's boundaries.^[10]^[7] This configuration provides a foundational symmetry lower than that of higher systems like the tetragonal, where two axes are equal (a = b \neq c) but angles remain 90 degrees, making the orthorhombic system a more general case for crystals lacking such axial equality.^[8]^[6]

Key Parameters and Metrics

The orthorhombic crystal system is characterized by a rectangular parallelepiped unit cell with three orthogonal axes of unequal lengths, defined by the lattice parameters a, b, and c, where a ≠ b ≠ c, and the interaxial angles α = β = γ = 90°. These parameters, typically expressed in angstroms (Å), fully describe the geometry of the unit cell and are determined experimentally through techniques such as X-ray diffraction.^[6]^[1] The volume V of the orthorhombic unit cell is calculated as the product of these parameters:

V = a \times b \times c

This formula provides a direct measure of the space occupied by one unit cell, essential for scaling structural models.^[1] The theoretical density ρ of an orthorhombic crystal is derived from the mass of its contents divided by the unit cell volume, given by:

\rho = \frac{Z \times M}{N_A \times V}

where Z is the number of formula units per unit cell, M is the molar mass of the formula unit, and N_A is Avogadro's number (6.022 × 10²³ mol⁻¹). This metric helps validate crystal structures by comparing calculated densities to experimental values, such as pycnometry measurements.^[11]^[12] Identification of the orthorhombic system relies on the distinct inequality of the lattice parameters, for instance, a ratio a/b ≠ 1 differentiates it from the tetragonal system where a = b ≠ c, while all angles remain 90° to exclude monoclinic or triclinic symmetries. These criteria are applied during refinement of diffraction data to confirm the system.^[1]^[13] In X-ray diffraction analysis, the orthorhombic lattice parameters enable indexing of powder or single-crystal patterns by solving for Miller indices (hkl) using the interplanar spacing equation:

\frac{1}{d^2} = \frac{h^2}{a^2} + \frac{k^2}{b^2} + \frac{l^2}{c^2}

where d is the spacing of planes reflecting at a given Bragg angle, allowing reconstruction of the unit cell from observed peak positions.^[14] The parameters also underpin selections among the four orthorhombic Bravais lattices by influencing Z values in density calculations.^[1]

Bravais Lattices

Primitive Orthorhombic (oP)

The primitive orthorhombic Bravais lattice, denoted as oP, represents the simplest arrangement within the orthorhombic crystal system, featuring lattice points exclusively at the eight corners of the unit cell, which collectively contribute one full lattice point due to shared corners.^[2] This configuration adheres to the defining orthorhombic parameters of three unequal edge lengths (a ≠ b ≠ c) and mutual orthogonality (α = β = γ = 90°).^[2] The basis vectors of the oP lattice are aligned directly along the principal crystallographic axes: \vec{a} = a \hat{x}, \vec{b} = b \hat{y}, and \vec{c} = c \hat{z}, where \hat{x}, \hat{y}, and \hat{z} are the Cartesian unit vectors.^[15] In this lattice, the conventional unit cell is identical to the primitive cell, with no internal lattice points or centering translations, resulting in a unit cell volume of V = abc.^[15] This direct correspondence simplifies the description of the lattice without the need for multiple primitive cells to fill a conventional one. Space groups constructed on the oP lattice incorporate the primitive translations with additional symmetry operations compatible with orthorhombic constraints. A representative example is P222 (space group No. 16), which includes three mutually perpendicular twofold rotation axes intersecting at the origin, providing a foundational symmetry for orthorhombic structures without glide planes or mirrors.^[16] Other primitive orthorhombic space groups, such as Pmm2 (No. 25), build upon this lattice by adding mirror planes, but all maintain the corner-only point distribution.^[15] The oP lattice's minimalistic structure offers advantages in visualization and indexing for crystals exhibiting low symmetry, where the absence of centering allows straightforward assignment of Miller indices to diffraction patterns without complications from additional lattice points.^[2] Unlike centered orthorhombic lattices, it prioritizes conceptual simplicity over enhanced atomic packing density.^[15]

Base-Centered Orthorhombic (oC)

The base-centered orthorhombic Bravais lattice, also known as C-centered orthorhombic, consists of lattice points located at the eight corners of a rectangular parallelepiped unit cell and additional points at the centers of two opposite faces, conventionally the faces normal to the c-axis (the ab-planes).^[2] This arrangement yields a total of two lattice points per conventional unit cell, as the face-centered points contribute one full point each due to shared faces with adjacent cells.^[2] The lattice vectors are mutually orthogonal, with unequal lengths a ≠ b ≠ c, all angles at 90°. Although the conventional cell is non-primitive, an equivalent primitive description can be obtained by redefining the basis vectors within the ab-plane while keeping the c-direction unchanged:

\mathbf{a}_p = \frac{1}{2} (\mathbf{a} + \mathbf{b}), \quad \mathbf{b}_p = \frac{1}{2} (-\mathbf{a} + \mathbf{b}), \quad \mathbf{c}_p = \mathbf{c}

^[17] These transformations generate a primitive cell with one lattice point and a volume half that of the conventional cell, V_p = V_\text{conv}/2 = abc/2, reflecting the two points in the larger cell.^[1] The Pearson symbol for this lattice type is oC, where "o" denotes orthorhombic symmetry and "C" indicates base (C-type) centering. This lattice is observed in various compounds and minerals, such as stromeyerite (AgCuS), which adopts a C-centered orthorhombic structure in its low-temperature form.^[18] In X-ray or electron diffraction studies, the base-centering imposes systematic absences on certain hkl reflections: those with h + k odd are forbidden, while reflections with h + k even are allowed, providing a diagnostic signature for identifying this lattice type.^[19] This lattice extends the primitive orthorhombic arrangement by incorporating centering on the base faces to enable more efficient packing in structures requiring such symmetry.^[20]

Body-Centered Orthorhombic (oI)

The body-centered orthorhombic (oI) lattice is characterized by lattice points positioned at the eight corners of the conventional unit cell, each contributing 1/8 to the total, along with a full lattice point at the body center at coordinates (½, ½, ½), yielding two lattice points per unit cell overall.^[2] This arrangement preserves the orthorhombic symmetry with three unequal axes (a ≠ b ≠ c) and all angles at 90°, distinguishing it from more symmetric centered lattices like body-centered cubic.^[21] The body centering introduces translational symmetry that halves the effective volume compared to a primitive equivalent, facilitating denser atomic packing in certain anisotropic materials. The primitive unit cell of the body-centered orthorhombic lattice is derived by selecting basis vectors that connect a corner to the body center and adjacent corners, resulting in a parallelepiped with volume V_p = V_conv / 2, where V_conv = abc is the conventional cell volume.^[1] This primitive cell exhibits a rhombohedral-like geometry adapted to orthorhombic constraints, with unequal edge lengths reflecting the distinct a, b, and c parameters, unlike the equilateral rhombohedral primitive cell in cubic systems.^[21] The Pearson symbol for this Bravais lattice is oI, indicating its orthorhombic (o) body-centered (I) nature.^[22] Representative examples of the oI lattice occur in intermetallic compounds such as MoPt₂, which adopts the Immm space group (No. 71) with lattice parameters approximately a = 3.20 Å, b = 9.70 Å, c = 4.10 Å at ambient conditions.^[23] Another instance is the high-pressure phase III of GaAs, which adopts an oI configuration (space group Imm2) above approximately 24 GPa, enabling metallic properties under compression.^[24] In X-ray crystallography, the body centering imposes systematic absences, permitting reflections only when the Miller indices satisfy h + k + l = even.^[25] This lattice type supports higher coordination numbers in metallic structures, with the central atom bonded to eight nearest neighbors along the body diagonals, promoting efficient space filling and mechanical stability in alloys despite the reduced symmetry compared to cubic counterparts.^[21]

Face-Centered Orthorhombic (oF)

The face-centered orthorhombic (oF) Bravais lattice is characterized by lattice points located at the eight corners of the conventional unit cell and at the centers of all six faces, resulting in four lattice points per conventional cell.^[1] This configuration arises from adding centering translations of (1/2, 1/2, 0), (1/2, 0, 1/2), and (0, 1/2, 1/2) to the primitive orthorhombic lattice, enhancing symmetry while maintaining the orthorhombic metric with unequal edge lengths a \neq b \neq c and right angles \alpha = \beta = \gamma = 90^\circ.^[1] The conventional unit cell is defined by orthogonal basis vectors \mathbf{a}, \mathbf{b}, and \mathbf{c} along the three axes, with volume V_\text{conv} = abc. A primitive unit cell, which contains only one lattice point, can be constructed using the vectors:

\mathbf{v_1} = \frac{a}{2} \hat{x} + \frac{b}{2} \hat{y}, \quad \mathbf{v_2} = \frac{a}{2} \hat{x} + \frac{c}{2} \hat{z}, \quad \mathbf{v_3} = \frac{b}{2} \hat{y} + \frac{c}{2} \hat{z},

yielding a primitive volume V_p = V_\text{conv}/4 = abc/4.^[1] The Pearson symbol for this lattice is oF, where "o" denotes orthorhombic and "F" indicates face-centering.^[26] Although relatively rare in nature, the oF lattice occurs in certain close-packed orthorhombic structures, often as distortions of more symmetric cubic arrangements. An example is α-sulfur (S₈), which adopts the Fddd space group (No. 70).^[27] In X-ray diffraction, the face-centering imposes systematic absences on the structure factor F_{hkl}, which vanishes unless h + k, h + l, and k + l are all even—equivalent to h, k, and l being either all even or all odd.^[28] Reflections with mixed parity indices (e.g., two even and one odd) are thus forbidden, aiding in the identification of this lattice type from diffraction patterns. Among the orthorhombic Bravais lattices, the oF structure exhibits the highest packing efficiency for spheres of equal radius, reaching up to \pi / (3\sqrt{2}) \approx 74\% when a = b = c, akin to the face-centered cubic limit, though typically lower for unequal axes. This maximum efficiency surpasses the maxima for primitive (up to 52%), base-centered (up to 68%), and body-centered (up to 68%) orthorhombic lattices in their limiting cases approaching higher symmetries, making oF suitable for dense atomic packings in materials with orthorhombic distortion.^[29]

Crystal Classes and Point Groups

Rhombic Disphenoidal (222)

The rhombic disphenoidal point group, also known as D₂ or class 222, represents the lowest symmetry variant within the orthorhombic crystal system, characterized exclusively by rotational symmetries without reflection or inversion elements. It features three mutually perpendicular twofold rotation axes aligned along the crystallographic a, b, and c directions, which impose orthorhombic metric symmetry on the lattice while preserving a minimal set of symmetry operations. This configuration results in a total of four symmetry operations: the identity and the three 180° rotations about each axis.^[3]^[30] The Hermann-Mauguin symbol for this point group is 222, reflecting the three perpendicular twofold axes, while the Schoenflies notation designates it as D₂, emphasizing its dihedral structure derived from cyclic rotations. Absent mirror planes or an inversion center, the group exhibits no improper rotations, rendering it chiral and capable of supporting enantiomorphic crystal forms that are non-superimposable mirror images. This chirality distinguishes it from higher-symmetry orthorhombic classes, such as those incorporating mirrors, and allows for potential optical activity in suitable materials.^[3]^[30] In the orthorhombic system, the 222 point group corresponds to nine space groups, which combine the rotational symmetries with translations compatible to the primitive (P), base-centered (C), face-centered (F), and body-centered (I) Bravais lattices: P222, P222₁, P₂₁₂₁₂, P₂₁₂₁₂₁, C222₁, C222, F222, I222, and I₂₁₂₁₂₁. The characteristic crystal form is the rhombic disphenoid, a closed polyhedron resembling four tetrahedral wedges joined at their edges, with eight triangular faces grouped into four pairs of identical, opposite faces parallel to the twofold axes. This form arises from the general position in the point group, where no faces are equivalent under reflection, leading to a wedge-like morphology often described as shoebox-shaped in idealized habits.^[31]^[3] Representative examples of minerals crystallizing in this point group include epsomite (MgSO₄·7H₂O), a hydrated magnesium sulfate that commonly forms prismatic to disphenoidal crystals, illustrating the chiral potential and low symmetry in natural orthorhombic structures. Other instances, such as certain synthetic organic compounds, further demonstrate the group's utility in accommodating asymmetric molecular arrangements within an orthorhombic framework.^[3]

Rhombic Pyramidal (mm2)

The rhombic pyramidal class corresponds to the point group C_{2v}, which is characterized by a twofold rotation axis aligned with the unique c-axis of the orthorhombic system and two mutually perpendicular mirror planes containing the a- and b-axes, respectively.^[32] This configuration results in four symmetry operations: the identity, the 180° rotation about the c-axis, reflection across the plane normal to the b-axis (ac-plane), and reflection across the plane normal to the a-axis (bc-plane).^[33] The rhombic pyramidal class builds upon the rhombic disphenoidal class (222) by incorporating these mirror planes, introducing reflection symmetry while retaining the twofold axis.^[3] In the Hermann-Mauguin notation, this point group is denoted as mm2, reflecting the two mirror planes (m) and the twofold axis (2) along the c-direction.^[3] There are 22 space groups associated with the mm2 point group in the orthorhombic system, accommodating primitive (oP), base-centered (oC), body-centered (oI), and face-centered (oF) Bravais lattices; representative examples include Pmm2 (no. 25) and Cmm2 (no. 35).^[34] This point group is unique to the orthorhombic crystal system among the non-centrosymmetric classes of low symmetry, as higher-symmetry systems like tetragonal or hexagonal incorporate equivalent elements into more isotropic arrangements.^[3] The absence of an inversion center in mm2 renders crystals polar along the c-axis, permitting a spontaneous electric dipole that cannot be reversed by symmetry operations.^[35] This polarity enables hemimorphic crystal development, where the top and bottom faces along the c-axis differ morphologically, often manifesting as distinct terminations. Common crystal forms include rhombic pyramids, consisting of four scalene triangular faces converging along the c-axis, and domes, which are paired wedge-shaped faces related by the mirror planes.^[36] Representative examples include hemimorphite (Zn₄Si₂O₇(OH)₂·H₂O), which forms hemimorphic crystals illustrating the polar nature.^[37] Due to its non-centrosymmetric nature, the mm2 point group supports piezoelectricity, where mechanical stress induces electric polarization, and conversely, applied electric fields generate strain; this property is absent in centrosymmetric orthorhombic classes.^[35] Such characteristics make mm2 crystals valuable in applications requiring electromechanical coupling, though specific quantitative coefficients vary by material.^[38]

Rhombic Dipyramidal (mmm)

The rhombic dipyramidal class corresponds to the point group D_{2h}, also known as mmm in Hermann-Mauguin notation and representing the holosymmetry of the orthorhombic system.^[3] This point group is characterized by three mutually perpendicular 2-fold rotation axes aligned with the crystallographic a, b, and c directions, three perpendicular mirror planes parallel to the principal faces (xy, xz, and yz), and a center of inversion at the origin.^[3] These eight symmetry elements collectively ensure that the class exhibits the maximum symmetry compatible with orthorhombic lattice constraints, where all angles are 90° and the lattice parameters a, b, and c are unequal.^[39] As a centrosymmetric point group, D_{2h} lacks polarity, meaning it possesses no net dipole moment and exhibits inversion symmetry that relates each point in the crystal to its antipodal counterpart.^[3] This centrosymmetry is a defining feature, distinguishing it from lower-symmetry orthorhombic classes and making it prevalent in structures of ionic and metallic crystals that favor balanced charge distributions.^[16] The class encompasses the symmetries of the subordinate orthorhombic point groups, incorporating their rotational and reflectional elements while adding the inversion center.^[3] Of the 59 total orthorhombic space groups, 28 belong to the mmm class, integrating the D_{2h} point group symmetries with translational elements such as screw axes and glide planes.^[39] Notable examples include Pnma (No. 62) and Imma (No. 74), which are among the most frequently observed due to their versatility in accommodating diverse atomic arrangements.^[39] These space groups support a variety of crystal forms, including dipyramids (eight-faced polyhedra formed by two square pyramids joined at their bases), prisms (six rectangular faces parallel to the axes), and pinacoids (pairs of parallel faces perpendicular to each axis), all related by the group's operations. Representative examples include barite (BaSO₄), which commonly forms tabular crystals demonstrating the high symmetry.^[3]

Two-Dimensional Analogy

Primitive Rectangular Lattice

The primitive rectangular lattice represents the simplest two-dimensional Bravais lattice analogous to the orthorhombic system, featuring a rectangular unit cell with unequal side lengths a \neq b and a right angle \gamma = 90^\circ between the basis vectors.^[40] This configuration arises from the rectangular crystal symmetry in two dimensions, where the lattice points are positioned exclusively at the corners of the rectangle, and the basis vectors \mathbf{a} and \mathbf{b} align perpendicularly along the x- and y-axes, respectively.^[40] The area A of this primitive unit cell is calculated as the product of the side lengths, A = a \times b, reflecting the absence of additional lattice points within the cell boundaries.^[40] In relation to three-dimensional structures, the primitive rectangular lattice corresponds to a planar projection of the primitive orthorhombic (oP) lattice by disregarding the extension along the c-axis, which is particularly relevant in surface crystallography for modeling the atomic arrangement on crystal surfaces derived from orthorhombic bulk materials.^[41] For diffraction analysis in two dimensions, such as in techniques like low-energy electron diffraction (LEED), the reciprocal lattice points are indexed using integers h and k, with all combinations allowed due to the primitive nature of the lattice, resulting in no systematic absences from centering. This simplicity facilitates the interpretation of diffraction patterns for surface reconstructions on orthorhombic crystals.^[41]

Centered Rectangular Lattice

The centered rectangular lattice in two dimensions features lattice points located at the corners and at the center (a/2, b/2) of a conventional rectangular unit cell, with the centering translation vector (a/2, b/2). This arrangement results in two lattice points per conventional unit cell, with parameters defined by unequal side lengths a and b, and a right angle γ = 90° between them.^[40]^[42] The primitive cell of this lattice takes the form of a rhombus, constructed from primitive vectors such as \vec{v_1} = (a/2, b/2) and \vec{v_2} = (a/2, -b/2), yielding equal side lengths of \sqrt{(a/2)^2 + (b/2)^2} = \frac{1}{2} \sqrt{a^2 + b^2} for each side. The angle between these vectors depends on the ratio a/b, given by \cos \theta = \frac{a^2 - b^2}{a^2 + b^2}, which deviates from 90° unless a = b (reducing to a square lattice). The area of the primitive cell A_p relates to the conventional cell area A_{\text{conv}} = ab by A_p = A_{\text{conv}} / 2 = ab/2, reflecting the doubling due to the additional centering point.^[40]^[42]^[43] In diffraction studies, this lattice imposes reflection conditions such as h + k even, arising from the translational symmetry that causes systematic absences for reflections where h + k is odd.^[44] Unlike the oblique lattice, which lacks right angles and has lower symmetry (γ ≠ 90°), the centered rectangular maintains orthogonal axes, enabling higher symmetry elements like 2-fold rotation axes and mirror planes perpendicular to the sides. This configuration is analogous to the base-centered orthorhombic lattice in three dimensions, providing a planar projection for understanding layered structures.^[40]^[43] Such lattices appear in layered materials, including soft matter systems where they form modulation patterns in smectic phases, as observed in diffraction peaks indexing to a centered rectangular array.^[45]

Examples and Applications

Natural and Synthetic Materials

The orthorhombic crystal system encompasses a variety of natural minerals that showcase its structural diversity. Olivine, a nesosilicate mineral with the composition (Mg,Fe)₂SiO₄, crystallizes in an orthorhombic structure belonging to the rhombic dipyramidal point group (mmm). Topaz, another silicate mineral with formula Al₂SiO₄(F,OH)₂, adopts an orthorhombic lattice in the rhombic dipyramidal point group (mmm). Elemental sulfur occurs naturally in an orthorhombic form consisting of S₈ rings, corresponding to the rhombic disphenoidal point group (222). Synthetic materials in the orthorhombic system include distorted perovskites such as GdFeO₃, which exhibits an orthorhombic structure in the Pbnm space group. Zeolites like natrolite, a hydrated aluminosilicate, also form orthorhombic frameworks, often in the Fdd2 space group, enabling their porous structure for ion exchange and adsorption. In pharmaceuticals, cyclosporine A, an immunosuppressant cyclic peptide, has an orthorhombic polymorph characterized by a distinct crystal packing that influences its solubility and stability, as identified in structural studies of its solid forms. Alloys provide further examples, with alpha-uranium displaying a body-centered orthorhombic (oI) structure in the Cmcm space group. Certain high-entropy alloys, such as those in the Gd-Er-Ho-Co-Cr system, stabilize orthorhombic phases due to compositional complexity and entropy effects. Polymorphism in the orthorhombic system is illustrated by calcium carbonate, where aragonite adopts a primitive orthorhombic (oP) structure as the high-pressure polymorph, contrasting with the stable trigonal (rhombohedral) calcite form under ambient conditions. This difference arises from aragonite's denser packing, making it metastable at surface pressures but prevalent in biogenic structures like shells.

Technological and Scientific Uses

In materials science, orthorhombic perovskites, such as variants of methylammonium lead iodide (MAPbI₃), play a crucial role in photovoltaic applications due to their phase stability and enhanced charge transport properties. The orthorhombic phase of MAPbI₃, achieved through controlled phase transitions from tetragonal structures, has been studied for its structural properties. In the pharmaceutical industry, controlling the orthorhombic polymorph of aspirin (acetylsalicylic acid, form II) is essential for optimizing drug stability and bioavailability. This polymorph, characterized by its orthorhombic P2₁2₁2₁ space group, exhibits distinct solubility and dissolution rates compared to the more stable monoclinic form I, influencing formulation strategies to prevent unwanted phase transformations during storage. Studies on crystallization techniques have shown that orthorhombic aspirin form II can be isolated under specific solvent conditions, aiding in the development of stable dosage forms with predictable therapeutic performance. Recent advancements from 2020 to 2025 have leveraged orthorhombic crystal phases in spintronics for efficient spin-orbit torque (SOT) devices. Orthorhombic iridate films, such as SrIrO₃, integrated at interfaces with ferromagnetic layers, enhance SOT efficiency through Rashba-Edelstein effects, enabling low-power magnetization switching without external fields. This has been demonstrated in heterostructures where the orthorhombic lattice distortion amplifies spin Hall conductivity, paving the way for next-generation memory devices.^[46] Additionally, progress in 3D imaging techniques, including dark-field X-ray microscopy with structured illumination, has advanced the characterization of ordered orthorhombic crystals, achieving sub-micrometer resolution for defect mapping in complex materials.^[47] Orthorhombic phases also contribute to catalysis, particularly in TiO₂-based composites for photocatalysis. The brookite polymorph of TiO₂, which adopts an orthorhombic structure, outperforms anatase and rutile in photoreforming reactions due to its unique bandgap and surface reactivity, achieving higher hydrogen evolution rates under UV irradiation. Composites incorporating orthorhombic TiO₂ additives with other metal oxides further boost pollutant degradation efficiency by facilitating charge separation.^[48] In geophysics, orthorhombic MgSiO₃ phases, such as the perovskite (bridgmanite), are studied under high-pressure conditions to model Earth's mantle dynamics. These phases provide insights into seismic wave propagation and phase transitions at depths exceeding 400 km, with thermoelastic properties revealing stability up to 40 GPa and 2000 K, informing models of mantle convection and convection.^[49]

References

[1]
Orthorhombic Crystal System - Aflow
In the orthorhombic system, the conventional unit cell is a parallelepiped defined by three mutually orthogonal vectors of unequal length: ...
[2]
nglos324 - orthorhombic
The Orthorhombic Bravais lattice has four unit cells. The primitive cell (P) has fractional lattice points at each corner for a total of one lattice point ...Missing: crystal system
[3]
External Symmetry of Crystals, 32 Crystal Classes - Tulane University
Aug 20, 2013 · The Orthorhombic System has only two fold axes or a 2-fold axis and 2 mirror planes. The Tetragonal System has either a single 4-fold or 4-fold ...
[4]
[PDF] CHEMISTRY 583 (Part II): Symmetry in Crystalline Solids 37
Space groups describe rotational and translational symmetry of crystalline structures, combining 32 point groups and 14 Bravais lattices. In 3D, there are 230 ...
[5]
Crystallography: The Orthorhombic System
### Summary of Orthorhombic Crystal System
[6]
https://chem.libretexts.org/Bookshelves/Analytical_Chemistry/Physical_Methods_in_Chemistry_and_Nano_Science_(Barron)/07:_Molecular_and_Solid_State_Structure/7.01:_Crystal_Structure
[7]
Unit Cells
A unit cell is the simplest repeating unit in a crystal, defined by lattice points, where opposite faces are parallel and edges connect equivalent points.
[8]
11 Crystallography – Mineralogy - OpenGeology
Crystallography involves unit cells, which are basic building blocks of crystals, and their 7 fundamental shapes. Unit cells fit together in 14 ways.
[9]
[PDF] Equation of State for Solid Benzene Valid for Temperatures up to ...
Orthorhombic lattices stem from stretching a cubic lattice along two ... (left) Orthorhombic unit cell from the crystal structure of solid phase I benzene.<|separator|>
[10]
https://pd.chem.ucl.ac.uk/pdnn/symm3/sgortho.htm
[11]
[PDF] v2 - A Guide to the Calculation of Theoretical Densities of Crystal ...
. Density is simply mass divided by volume. Because of the orthorhombic structure, there are four formula units in each unit cell, which requires the ...Missing: system | Show results with:system
[12]
Density of Unit Cell - BYJU'S
Jul 7, 2021 · What is the formula for the density of any crystal? ... Density of a unit cell is given as the ratio of mass and volume of the unit cell. The mass ...
[13]
Tetragonal Crystal System - Aflow
In the tetragonal system, like the orthorhombic system, the conventional unit cell is a parallelepiped, but two sides are equal, so that a=b and c =a, ...
[14]
Powder diffraction/Indexing powder patterns - CLASSE (Cornell)
Indexing the pattern is a process where each peak is identified by the (hkl) Miller-index of the lattice plane that it results from.
[15]
Orthorhombic Crystal System - Aflow
There are four Bravais lattices in the orthorhombic system. Lattice 4: Simple Orthorhombic ./PICS/simple_orthorhombic icon. The simple orthorhombic Bravais ...
[16]
Orthorhombic Space Groups
Orthorhombic space groups have 59 groups, divided into classes 222, mm2, and mmm. Their unit cell axes are orthogonal, and they are named by lattice centering ...<|control11|><|separator|>
[17]
Symmetry Line Files - Questaal
C-centered Orthorhombic (ORCC) Lattice. The ORCC lattice has real- and reciprocal lattice vectors (units of a and 2π/a, respectively) Plat Qlat a/2 -b/2 0 1 ...
[18]
[PDF] The Crystal Structure of Stromeyerite, AgCuS - RRuff
Abstract. Stromeyerite, below 93° C., forms orthorhombic crystals having space group symmetry. Cmcm, with a.Missing: example | Show results with:example
[19]
[PDF] Assignment 1 Due Tuesday, February 6, 2018 (1) Consider a two ...
(5) There are two ways to identify systematic absences for C-centered orthorhombic lattice. Clearly show that these two approaches give the same results. (a) ...<|control11|><|separator|>
[20]
Base-Centered Orthorhombic Unit Cell - MSE Student
The base-centered orthorhombic unit cell can be imagined as a box with 3 different side lengths, with an atom on each corner and at the center of the top ...
[21]
Body-Centered Orthorhombic Unit Cell - MSE Student
The body-centered orthorhombic unit cell looks like a box with 3 different edge lengths, with an atom on each corner and another in the center.
[22]
Body-centred orthorhombic (ORCI) — rad-tools 0.7.16
Body-centred orthorhombic (ORCI)#. Pearson symbol: oI. Body-centered orthorombic lattice is described by the class ORCI . It is defined by three parameter: ...
[23]
mp-952: MoPt2 (Orthorhombic, Immm, 71) - Materials Project
Pt₂Mo crystallizes in the orthorhombic Immm space group. Mo is bonded to two equivalent Mo and ten equivalent Pt atoms to form MoMo₂Pt₁₀ cuboctahedra that ...
[24]
Structural and mechanical properties of GaAs under pressure up to ...
In previous studies of GaAs high-pressure phases, it has been suggested that GaAs–ZB transforms to an orthorhombic phase near 17 GPa [2], [3]. By studying the X ...
[25]
Diffraction Selection Rules
Body-centered, H+K+L = 2n (even), H+K+L = 2n+1 (odd) ; Face-centered, H,K,L all odd or all even, H,K,L mixed odd, even ; FCC Diamond, H,K,L all odd. H,K,L all ...
[26]
Orthorhombic Structures - Aflow
Pearson Symbol · Space Group ... oF28. AB2C4_oF28_69_a_g_cg-001. Face-Centered Orthorhombic La2CuO4 AB2C4_oF28_69_a_g_cg-001 ...
[27]
Face-Centered Orthorhombic Unit Cell - MSE Student
Face-centered orthorhombic has 4 atoms per unit cell, lattice constants a, b ... orthorhombic crystal with math, you would describe the cell with the primitive ...Missing: systematic absences
[28]
[PDF] Systematic Absences and Space Group determination
May 1, 2014 · Direct lattice represented with an F-centered cell. Reciprocal space lattice represented with an I-centered cell. Page 24. Systematic absences.
[29]
https://chem.libretexts.org/Courses/UW-Whitewater/Chem_260%3A_Inorganic_Chemistry_(Girard)/10%3A_Metals_and_Alloys-_Structure_Bonding_Electronic_and_Magnetic_Properties/10.02%3A_Unit_Cells_and_Crystal_Structures
[30]
https://geo.libretexts.org/Bookshelves/Geology/Mineralogy_(Perkins_et_al.)/10%3A_Crystal_Morphology_and_Symmetry/10.05%3A_Point_Groups_and_Crystal_Systems
[31]
the C2v point group
The C2v Point Group. This point group contains four symmetry operations: E the identity operation. C2 a twofold symmetry axis σv the first mirror plane (xz)
[32]
Orthorhombic: C 2v (mm2)
Orthorhombic: C2v (mm2) ; sm_C2v stereograph · C2v solid · H2O ; Stereograph · 3D object · Molecule: H2 O (Water) ...
[33]
Orthorhombic Space Groups
Orthorhombic Space Groups ; 40. A m a 2 · B b m 2 ; 41. A b a 2 · B b a 2 ; 42. F m m 2 ; 43. F d d 2 · F d 2 d ; 44. I m m 2.
[34]
Piezoelectricity and Crystal Space Groups - Actuate Lab
Table 3. Orthorhombic crystal system. Number, Point Group, Symbol, Piezoelectric Coefficient(s). 16, 222, P222, Shear ...
[35]
Crystal Form, Zones, & Habit - Tulane University
Jan 10, 2011 · Rhombic pyramid: 4-faced form where the faces are related by mirror planes. In the drawing shown here the faces labeled "p" are the four faces ...Missing: hemimorphic | Show results with:hemimorphic
[36]
Piezoelectric anisotropy of orthorhombic ferroelectric single crystals
Jun 11, 2013 · This fact tempts to think that negative values of longitudinal PR ν 13 may be inherent to ferroelectric crystals of orthorhombic symmetry (mm2), ...
[37]
High-Resolution Space Group Diagrams and Tables
Orthorhombic. (For a fuller list with alternative axes and origins click ... P n a 21, 34. P n n 2, 35. C m m 2. 36. C m c 21, 37. C c c 2, 38. A m m 2, 39. A ...
[38]
https://pubs.aip.org/aip/jap/article/113/22/224105/151744/Piezoelectric-anisotropy-of-orthorhombic
[39]
[PDF] Surface crystallography - physics.muni
"p" or "c" denotes primitive or centered surface lattice, and X is the chemical symbol of an adsorbed species. The substrate net is therefore denoted: "(1 x ...
[40]
[PDF] Handout 4 Lattices in 1D, 2D, and 3D Bravais Lattice
The Wigner-Seitz (WS) primitive cell of a Bravais lattice is a special kind of a primitive cell and consists of region in space around a lattice point that ...
[41]
[PDF] Crystal Structure
This plane (call it the base plane) is a 2D Bravais lattice for which we can choose—in accordance with the above-discussed 2D rules—a primitive vector a2 such ...
[42]
[PDF] Three-dimensional textures and defects of soft material layering ...
Nov 26, 2013 · At small angles, this produces a lattice of peaks indexing to a centered rectangular layer/modulation lattice. With cooling, additional ...<|control11|><|separator|>
[43]
Phase Transition Kinetics of MAPbI 3 for Tetragonal-to-Orthorhombic ...
As a result, planar perovskite solar cells with a certificated efficiency of 23.32% (quasi-steady state) are obtained. In addn., a VOC as high as 1.18 V is ...
[44]
http://yclept.ucdavis.edu/course/240A.F17/supple/Lattices_SpaceGroups.pdf
[45]
Dark-field X-ray microscopy with structured illumination for ... - Nature
Jan 22, 2025 · We introduce a structured illumination technique optimized for 3D imaging of ordered materials at sub-micrometer length scales.
[46]
Brookite TiO 2 as an active photocatalyst for photoconversion of ...
This work introduces the brookite polymorph of TiO 2 as an active photocatalyst for photoreforming with an activity higher than anatase and rutile polymorphs.
[47]
majorite at high temperatures and pressures: A first principles study
In this study, we have carried out extensive first principles calculations to determine the thermoelastic properties of MgSiO3-majorite up to 2000 K and 40 GPa.