School of Physics and Electronics 双语教学示范课程 Solid State Physics Yuanxu Wang School of Physics and Electronics Henan University wyxhenu@gmail.com

§3 Bravais lattice, Primitive and Conventional cell (a)、(b)、(c) are 2D diagram for crystal structures What are the differences and similarities among them?

1. Crystal periodicity An ideal crystal is composed of identical structure unit which is periodically arranged in space. All crystal structures can be depicted with crystal lattice, there are a group of atoms in every lattice point for this crystal lattice, this atomic group is called basis. Crystal structure is composed of periodically repeated arrangement of basis. (1) Basis Properly select some atoms as basic structural unit in crystal, this basic structural unit is called basis. Basis is the smallest repeated unit in structural unit, and basis forms crystal structure with periodically repeated arrangement in space.

(2) Crystal lattice (c) (a) (b) The crystal inner structure could be regarded as some same points regularly distributed infinitely in the space, consist some net lattices by drawing 3 groups of non-coplanar parallel lines through these points, which is called crystal lattice (or crystal lattice is the framework formed by the points that are periodically arranged in space.)

Vector represents arrangement of lattice. Crystal lattice is arithmetic abstract for periodic structure, which ignored concrete content and maintained periodicity for crystal structure

(3) Lattice point (a) (b) A point in crystal represents same position of crystal structure, which is called lattice point. A lattice point represents a basis, which could be centrobaric position of basis or represents arbitrary point in basis. (emphasize) (b) (a) Lattice + basis=crystal structure

Lattice + basis=crystal structure 奥古斯特·布拉菲（Auguste Bravais，又译布拉伐、布喇菲，1811年－1863年），法国物理学家，于1845年得出了三维晶体元祖排列的所有14种布拉菲点阵结构，首次将群的概念应用到物理学，为固体物理学做出了奠基性的贡献。除此之外，布拉菲还对磁性、极光、气象、植物地理学、天文学和水文学等方面进行过研究。

2. Bravais lattice, simple crystal lattice and complex crystal lattice Lattice point collectivity is called Bravais lattice. The characteristic of this lattice is that the ambient situation for every point is identical. (2) simple and complex lattice Simple crystal lattice: if the crystal is composed of one type atoms, and ambient situation for every atom is identical, then, this net lattice is called simple crystal lattice. Complex crystal lattice: if crystal is composed of two or more than two type of atoms, and each same atoms consist net lattice same as lattice point. The net lattices relatively translate to forms a complex lattice.

Simple lattice Complex lattice

3. Primitive cell In crystal lattice, selecting one lattice point as vertex, then taking periodicities along three non-coplanar directions as side length, the periodicities consist parallelepiped which is taken as a repeated unit, this parallelepiped is “primitive cell”. The primitive cell is repeated unit with the smallest cell volume. Vectors representing three sides of the primitive cell are called the basic translation vector, or called “basis vector”.

(1) Primitive cell Characteristic: lattice points only occupy vertices of the parallelepiped, no lattice point in the surface and internal. Each primitive cell contains only one lattice point on an average, which shows periodicity of crystal structure. Basis vector: primitive cell is often represented by in solid state physics. Volume is： Position vector of arbitrary point in the primitive cell is:

(2) Crystallographic cell Construct: directions of the three vectors are as far as possible along the symmetry axes. The cell shows obvious symmetry and periodicity. Characteristic：lattice points of crystallographic cell are distributed not only on the vertexes of parallelepiped, but also on the surface and inner. Its volume is the integer times of the primitive cell one. Basis vector: basis vector of crystallographic cell usually represents with . Cell Volume is:

(a) Simple cubic Every conversional cell contains one lattice point. Cell volume of a unit cell is

(b) Face-centered cubic Each conversional cell contains 4 lattice points. Volume of primitive unit cell:

(c) Body-centered cubic Each conversional cell contains two lattice points. The volume of a primitive cell:

(d) Diamond structure c Diamond structure is composed of two face-centered cubic sub-lattices which translate 1/4 along the diagonal. Diamond structure belongs to face-centered structure, every unit cell contains 4 lattice points.

(e) Sodium chloride structure 氯化钠结构由两个面心立方子晶格沿体对角线位移1/2的长度嵌套而成。 Cl- and Na+ form the face-centered cubic structure, respectively Each primitive cell contains one lattice point, and each conversional cell does 4 ones.

(f) CsCl structure 氯化铯结构是由两个简立方子晶格沿体对角线位移1/2的长度嵌套而成。 Cl-和Cs+分别组成简立方格子。 Cl- and Cs+ form simple cubic structure, respectively Each primitive cell contains one lattice point, and each conversional cell does 1 point.

(g) Perovskite structure Oxide octahedron in perovskite structure O Ti The Ba, Ti, O atoms form simple cubic lattice, respectively. BaTiO3 is formed by five simple cubic lattices. Perovskite structures are always described as “ABO”

Key concepts: Bravais lattice Primitive cell and conversional cell Simple lattice and complex lattice Close-packing