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The secret of the unit cell revealed: Why is the primitive unit cell the smallest unit of crystal structure?
In geometry, biology, mineralogy, and solid-state physics, a unit cell is a repeating unit formed by the vectors that describe the points on the lattice. Despite the very suggestive name, a unit cell does not necessarily have a unit size, or even any particular size. In contrast, the primitive unit cell is arguably the closest concept to a unit vector, since it has a definite size for a given lattice and is the basic unit from which larger unit cells are constructed.
The geometric characteristics of the unit cell not only affect the planning of the structure, but also affect the physical properties of the crystal.
The concept of a unit cell is particularly useful for describing crystal structures in two and three dimensions, although it can be understood in all dimensions. A lattice can be characterized by the geometry of its unit cell, a part that generates a whole tiling, usually a parallelogram or parallelepiped, which is generated only by translations.
There are two special cases of unit cells: primitive cells and regular cells. The primitive unit cell corresponds to a single lattice point and is the smallest possible unit cell. In some cases, the full symmetry of the crystal structure may not emerge from the primitive unit cell, in which case a conventional unit cell can be used. A regular unit cell (which may or may not be a primitive unit cell) is a unit cell with complete symmetry of the lattice, and may contain more than one lattice point.
The definition of the primitive unit cell is closely linked to the primitive axes (vectors), which are the smallest volume unit of the lattice.
The primitive unit cell contains exactly one lattice point, so for a normal unit cell, the lattice points belonging to n units are treated in the calculation as if each unit cell contained 1/n of the lattice points. Grid. This means that in three-dimensional space, if a primitive unit cell has lattice points at all eight vertices, then the primitive unit cell actually only contains 1/8 of each lattice point. This calculation method allows the primitive unit cell to accurately represent the basic repeating form of the lattice structure.
For every Bravais lattice, there is another primitive unit cell, called the Wiegand–Seitz cell. The lattice point of the Wiegand–Seitz unit cell is located at the center of the unit cell and is usually not a parallelogram or parallelepiped. This unit cell is a Voronoi-type partitioning of space, and the reciprocal lattice of the Wiegand–Seitz unit cell in momentum space is called a Brillouin zone.
In crystallography, for each specific lattice a conventional unit cell is chosen based on computational convenience. These regular unit cells can have additional lattice sites added to the faces or volume of the unit cell, where the number of such sites and the volume of the regular unit cell are integer multiples of the original unit cell (e.g., 1, 2, 3, or 4).
For any two-dimensional lattice, the unit cell is usually a parallelogram, although in some special cases its internal angles can be right angles, its sides can be equal in length, or both. All four and five two-dimensional Bravais lattices can be represented using conventional primitive cells, while the concentrated rectangular lattice also has a primitive cell similar to a rhombus. In order to distinguish them based on symmetry, they are usually represented using a primitive cell containing two Conventional unit cell representation of lattice points.
For any three-dimensional lattice, the conventional unit cell is usually a parallelepiped, and in special cases may have right angles, or sides of equal length, or both. There are seven three-dimensional Bravais lattices represented using the regular primitive cell, and another seven (called concentrated lattices) are also represented using the parallelepiped primitive cell but are represented using the regular cell because this allows These units are distinguished by their symmetry by having more than one lattice point in the unit cell.
Scientists' long-standing understanding of crystal structure has enabled many technological advances, so in the future, can we use this knowledge to unlock more mysteries of nature?
