Square lattice
The square lattice is a two-dimensional lattice in the Euclidean plane consisting of all points with integer coordinates, denoted as \mathbb{Z}^2 = \{(m, n) \mid m, n \in \mathbb{Z}\}.[1] It is generated by the basis vectors \mathbf{a}_1 = (1, 0) and \mathbf{a}_2 = (0, 1), which are orthogonal and of equal length, forming the simplest primitive Bravais lattice in two dimensions among the five possible 2D Bravais lattice types.[2] This lattice possesses four-fold rotational symmetry, with its point group being the dihedral group D_4 of order 8, encompassing 90-degree rotations, reflections across horizontal, vertical, and diagonal axes, and inversion through the origin.[3] The full symmetry group, when including translations, corresponds to wallpaper group p4m, which also incorporates glide reflections, making the square lattice a foundational structure for analyzing periodic patterns in the plane.[4] In mathematics, the square lattice underpins discrete geometry, number theory, and graph theory, where it models the infinite 4-regular grid graph used in problems like percolation and random walks.[5] In physics and materials science, it is central to theoretical models such as the two-dimensional Ising model, which exhibits a ferromagnetic phase transition at a critical temperature and serves as a paradigm for studying critical phenomena and statistical mechanics.[6] The structure also appears in crystallography for describing atomic arrangements in square-based unit cells of thin films, surfaces, and quasi-2D materials.[7]Definition and Basic Properties
Mathematical Definition
The square lattice \Lambda in the Euclidean plane \mathbb{R}^2 is defined as the discrete set of points \Lambda = \{ (m a, n a) \mid m, n \in \mathbb{Z} \}, where a > 0 is the lattice constant denoting the distance between nearest-neighbor points.[8] This construction forms a regular grid aligned with the coordinate axes, with each point reachable by integer linear combinations of the basis vectors.[5] The basis vectors generating the square lattice are \mathbf{e}_1 = (a, 0) and \mathbf{e}_2 = (0, a), so \Lambda = \{ m \mathbf{e}_1 + n \mathbf{e}_2 \mid m, n \in \mathbb{Z} \}.[8] Under uniform scaling by a and arbitrary translation, the square lattice is equivalent to the integer lattice \mathbb{Z}^2 = \{ (m, n) \mid m, n \in \mathbb{Z} \}, which corresponds to the case a = 1.[5] As a Bravais lattice, the square lattice is distinct among the five two-dimensional Bravais lattices; unlike the rectangular lattice (generated by (a, 0) and (0, b) with a \neq b) or the hexagonal lattice (generated by vectors of equal length at a 60° angle), it features equal basis vector lengths and a 90° angle, yielding fourfold rotational symmetry.[2] The primitive cell of the square lattice is the square parallelogram spanned by \mathbf{e}_1 and \mathbf{e}_2, with side length a and area a^2.[5]Geometric Characteristics
The square lattice is characterized by its points arranged in a regular grid pattern, defined by two orthogonal basis vectors of equal length a, forming right angles of 90 degrees between them. This orthogonal arrangement ensures that the lattice exhibits rectangular symmetry in its primary directions. The nearest-neighbor distance, which connects adjacent points along the horizontal and vertical axes, is precisely a, establishing the fundamental scale of the structure. Each lattice point thus has four nearest neighbors at this distance, forming the edges of the unit square.[9][10] Beyond the nearest neighbors, the next-nearest neighbors lie along the diagonals of the unit squares, separated by a distance of a\sqrt{2}. This diagonal spacing arises directly from the Pythagorean theorem applied to the right-angled geometry of the lattice. The overall point density of the square lattice is $1/a^2 points per unit area, as the primitive unit cell—a square of side a—encloses an area of a^2 and contains exactly one lattice point. This density metric highlights the efficient packing of points in the plane, with implications for applications in materials science and computational modeling.[11][10] The square lattice possesses a bipartite structure, allowing it to be partitioned into two interpenetrating square sublattices distinguished by the even or odd parity of the sum of their integer coordinates, akin to a checkerboard pattern where one sublattice occupies black squares and the other white. This division underscores the lattice's ability to support alternating site properties, such as in antiferromagnetic models. Additionally, a diagonal orientation variant of the square lattice, obtained by rotating the standard configuration by 45 degrees, aligns the principal axes with the diagonals; in this orientation, the effective spacing between points along these new axes is a/\sqrt{2}, while preserving the overall unit cell area and point density.[12][13]Symmetry
Point Group Symmetry
The point group symmetry of the square lattice refers to the finite set of rotational and reflectional symmetries that fix a lattice point, excluding translations. This symmetry is captured by the dihedral group D_4, which consists of eight elements: the identity, rotations by $90^\circ, $180^\circ, and $270^\circ about the axis perpendicular to the plane through the lattice point, and four reflections across the horizontal, vertical, and two diagonal axes aligned with the lattice directions.[14][15] The group D_4 has order 8 and represents the holohedry of the square lattice, meaning it is the maximal point group compatible with the lattice's periodicity in two dimensions. This holohedry aligns with the tetragonal crystal system in its 2D manifestation, where the square lattice exhibits the highest possible symmetry among orthorhombic-like 2D Bravais lattices. In comparison, the rectangular lattice possesses a lower-symmetry point group D_2 (order 4, with $180^\circ rotations and reflections along the principal axes), while the square lattice's D_4 reflects its equal side lengths and right angles, distinguishing it as a special case of rectangular geometry with enhanced rotational invariance.[16] In terms of reflection group structure, the point symmetries of the square lattice correspond to the finite Coxeter group of type B_2, often denoted in simplified Coxeter notation as [17], generated by reflections at $90^\circ angles. The full symmetry group of the square lattice tiling, incorporating the infinite wallpaper group, is described by the affine Coxeter group [4,4], which extends D_4 through translational elements but isolates the point group as its rotational-reflective core.[18]Space Group and Wallpaper Groups
The symmetry group of the square lattice extends the point group symmetries by incorporating translational invariances, forming a wallpaper group that describes the full set of isometries preserving the infinite periodic structure. The translational subgroup is generated by the two basis vectors of the lattice and is isomorphic to ℤ², an abelian group that captures the discrete shifts in two independent directions. This subgroup is normal in the full wallpaper group, and the overall structure arises as a semi-direct product of the translations with the finite point group, ensuring compatibility with the square geometry.[19][20] For the square lattice, three wallpaper groups are compatible, all featuring 4-fold rotational symmetry: p4, p4m, and p4g. The group p4m represents the maximal symmetry case, primitive with 4-fold rotations and mirror reflections across axes aligned with the lattice vectors and their diagonals; it has 8 symmetries per unit cell, including four rotations (orders 1, 2, and 4) and four reflections. In orbifold notation, p4m is denoted 442, reflecting two 4-fold rotation points, two 2-fold rotation points, and mirror lines. The group p4 lacks reflections, relying solely on rotations (90° and 180°), resulting in 4 symmetries per unit cell, with orbifold notation 442. Finally, p4g incorporates glide reflections along the diagonals and perpendicular mirror reflections at 45° to the lattice axes, also yielding 8 symmetries per unit cell and orbifold notation 42.[20][21][22]Crystallography
Bravais Lattice Classification
In two dimensions, Bravais lattices are classified into five distinct types based on their symmetry and geometric constraints: oblique (or parallelogram), rectangular, centered rectangular (also known as rhombic), square, and hexagonal.[23] These lattices represent the unique ways to tile the plane with identical unit cells without gaps or overlaps, where the square lattice is one of the primitive types distinguished by its high rotational symmetry. The square lattice is specifically a primitive lattice, denoted as type P, with lattice vectors of equal length (a = b) and a right angle between them (\gamma = 90^\circ) in the conventional cell parameters. Unlike some other 2D lattices, such as the centered rectangular, there is no base-centered variant of the square lattice, as introducing centering in a square arrangement would reduce to an equivalent primitive rectangular lattice under a change of basis vectors.[23] This primitive nature ensures that each unit cell contains exactly one lattice point, emphasizing the square lattice's simplicity and efficiency in space filling. Among 2D lattices with orthogonal axes—such as the rectangular and square types—the square lattice possesses the highest symmetry due to its fourfold rotational invariance, setting it apart from the lower-symmetry orthorhombic-like rectangular lattice where a \neq b. This elevated symmetry arises directly from the metric constraints a = b and \gamma = 90^\circ, making the square lattice a special case within the rectangular family.[23] Extending to three dimensions, the square lattice forms the basis for the tetragonal Bravais lattice system, where the basal (ab) plane is square (a = b, \alpha = \beta = \gamma = 90^\circ) and the c-axis length may differ. The tetragonal system includes two Bravais types: primitive tetragonal (P), which directly stacks square layers without additional centering, and body-centered tetragonal (I), featuring an extra lattice point at the body center. This 3D classification maintains the square lattice's core geometry while accommodating variations along the unique c-direction, linking 2D and 3D crystallographic hierarchies.Crystal Systems and Classes
The square lattice arrangement in three-dimensional crystals is associated with the tetragonal crystal system, where the basal plane is square due to equal lattice parameters a = b and right angles between all axes, with the unique c-axis perpendicular to this plane.[24] This system is distinguished by a principal 4-fold rotation axis aligned with the c-direction, enabling square symmetry in the base.[24] The tetragonal system includes 7 crystal classes, each corresponding to a specific point group that governs the external symmetry of crystals with this lattice.[24] These classes are defined using Hermann-Mauguin symbols and incorporate variations of the 4-fold axis, combined with possible mirror planes, 2-fold axes, or inversion centers.[24] Key examples include the cyclic group C_4 (Hermann-Mauguin 4), featuring only a 4-fold rotation axis, the dihedral group D_4 (422), with a 4-fold axis and four 2-fold axes, and C_ {4v} (4mm), with vertical mirror planes. Other symbols in the system include \bar{4}, 4/m, \bar{4}2m, and 4/mmm, each describing distinct combinations of symmetry elements aligned with the square basal plane.[24] For centrosymmetric structures in the tetragonal system, the relevant Laue groups are 4/m and 4/mmm, which include an inversion center and determine the symmetry of diffraction patterns.[25] These classes build upon the primitive tetragonal Bravais lattice, emphasizing the geometric foundation of the square base.[24]| Crystal Class | Hermann-Mauguin Symbol | Point Group (Schoenflies) | Key Symmetry Elements |
|---|---|---|---|
| Tetragonal Pyramidal | 4 | C_4 | 4-fold rotation axis |
| Tetragonal Sphenoidal | \bar{4} | S_4 | 4-fold rotoinversion axis |
| Tetragonal Dipyramidal | 4/m | C_{4h} | 4-fold axis + horizontal mirror |
| Tetragonal Trapezoidal | 422 | D_4 | 4-fold + four 2-fold axes |
| Ditetragonal Pyramidal | 4mm | C_{4v} | 4-fold + four vertical mirrors |
| Tetragonal Scalenohedral | \bar{4}2m | D_{2d} | 4-fold rotoinversion + 2-fold axes + diagonal mirrors |
| Ditetragonal Dipyramidal | 4/mmm | D_{4h} | 4-fold + 2-fold axes + mirrors + inversion |