Crystal Systems and Bravais Lattices PDF

## Les Sept Systèmes Cristallographiques en Trois Dimensions et Leurs Mailles Correspondantes | Système | Symétrie Minimale | Maille | Orientation des Axes | |---|---|---|---| | triclinique (anorthique) | 1 (ou 1) | a, b, c, α, β, γ | n'est pas spécifiée | | monoclinique | Un 2 (ou 2 m) | a, b, c,...

## Les Sept Systèmes Cristallographiques en Trois Dimensions et Leurs Mailles Correspondantes | Système | Symétrie Minimale | Maille | Orientation des Axes | |---|---|---|---| | triclinique (anorthique) | 1 (ou 1) | a, b, c, α, β, γ | n'est pas spécifiée | | monoclinique | Un 2 (ou 2 m) | a, b, c, α=γ= 90°, β | b parallèle à 2 xe d'ordre 2: rotahonde<br>n=2<br>(quelquefoisa + β ≠ 90°, γ) | | orthorhombique | Trois 2 (ou 2) | a, b, c, α = β = γ = 90° | a, b, c parallèles aux trois axes<br>d'ordre 2 | | tétragonal (quadratique) | Un 4 (ou 4) | a = b, c, α = β = γ = 90° | c parallèle à 4 | | trigonal (rhomboédrique) | Un 3 (ou 3) | 1) maille rhomboédrique<br>a = b = c, α = β = γ ≠ 90° | a également inclinés par rapport à l'axe 3 | | |- |- | 2) maille hexagonale<br>a = b, c, α - β = 90°, γ ≠ 120° | c parallèle à 3 | | hexagonal | Un 6 (ou 6) | a = b, c, α - β = 90°, γ ≠ 120° | c parallèle à 6 | | cubique (isométrique) | Quatre 3 (ou 3) | a = b = c, α = β = γ = 90° | 1, 2, 3 doivent être parallèles aux 2 (ou 4)<br>Les quatre 3 sont dans les 4 diagonales du cube | ## Systèmes Cristallins et Réseaux de Bravais | Systèmes Cristallins (7) | a | b | c | α | β | γ | Réseaux de Bravais (14) | |---|---|---|---|---|---|---|---| | Triclinique | quelconques | quelconques | quelconques | quelconques | quelconques | P | | Monoclinique | quelconques | 90° | 90° | P, C | | Orthorhombique | quelconques | 90° | 90° | 90° | P, C, I, F | | hexagonal | a = b ≠ c | 90° | 90° | 120° | P | | Rhomboédrique | a = b = c | α = β = γ ≠ 90° | P | | Quadratique | a = b ≠ c | 90° | 90° | 90° | P, I | | Cubique | a=b=c | 90° | 90° | 90° | P, I, F | ## Réseaux de Bravais **Image:** A diagram depicting the 14 Bravais Lattices. It shows the different types of unit cells that can be found in crystals. Each unit cell is represented by a box with the crystallographic axes labelled. The angles α, β, γ are also indicated. - **Triclinique P**: A simple unit cell with no symmetry elements other than the identity. - **Monoclinique P**: A unit cell with one unique direction (the b-axis) perpendicular to the other two directions. - **Monoclinique C**: A unit cell with one unique direction (the b-axis) perpendicular to the other two directions, and a lattice point centered on the c-axis. - **Orthorhombic P**: A unit cell with three unique directions (a-axis, b-axis, c-axis) and all angles being 90 degrees. - **Orthorhombic C**: A unit cell with three unique directions (a-axis, b-axis, c-axis), all angles being 90 degrees, and a lattice point centered on the c-axis. - **Orthorhombic F**: A unit cell with three unique directions (a-axis, b-axis, c-axis), all angles being 90 degrees, and a lattice point centered at the face of the unit cell. - **Orthorhombic I**: A unit cell with three unique directions (a-axis, b-axis, c-axis), all angles being 90 degrees, and a lattice point centered in the body of the unit cell. - **Tetragonal P**: A unit cell with two equal directions (a-axis and b-axis) and a distinct c-axis. All angles are 90 degrees. - **Tetragonal I**: A unit cell with two equal directions (a-axis and b-axis) and a distinct c-axis. All angles are 90 degrees, and a lattice point centered in the body of the unit cell. - **Hexagonal P**: A unit cell with two equal directions (a-axis and b-axis) at an angle of 120 degrees, and a distinct c-axis. The c-axis is perpendicular to the a-axis and b-axis. - **Trigonal R**: A unit cell with three equal directions (a-axis, b-axis, c-axis), and all angles equal to 120 degrees. - **Cubic P**: A unit cell with three equal directions (a-axis, b-axis, c-axis), and all angles being 90 degrees. - **Cubic I**: A unit cell with three equal directions (a-axis, b-axis, c-axis), and all angels being 90 degrees, and a lattice point centered in the body of the unit cell. - **Cubic F**: A unit cell with three equal directions (a-axis, b-axis, c-axis), and all angels being 90 degrees, and a lattice point centered at the face of the unit cell. ## Body-Centered Cubic Lattice, Showing a Primitive Cell **Image:** A drawing of a body-centered cubic lattice showing its primitive cell. The central lattice point has a coordinate of (½, ½, ½) relative to the corner lattice points of the primitive cell. - The primitive cell shown is a rhombohedron of edge √3 a. - The angle between adjacent edges is 109°28′. ## Primitive Translation Vectors of the Body-Centered Cubic Lattice **Image:** A drawing of a body-centered cubic lattice showing its primitive translation vectors. The primitive translation vectors are denoted by a', b', and c'. - The three primitive translation vectors are: - a' = (½, ½, ½) - b' = (½, -½, ½) - c' = (-½, ½, ½) - The primitive cell is obtained by completing the rhombohedron. ## Rhombohedral Primitive Cell of the Face-Centered Cubic Crystal **Image:** A drawing of the rhombohedral primitive cell of the face-centered cubic crystal. The primitive translation vectors connect the lattice point at the origin to lattice points at the face centers. The corners of the primitive cell are at the face centers of the cubic lattice. - The primitive translation vectors are: - a' = (½, ½, 0) - b' = (0, ½, ½) - c' = (½, 0, ½) - The angles between the axes are 60°. ## Relation of the Primitive Cell in the Hexagonal System to a Prism of Hexagonal Symmetry **Image:** A drawing of the primitive cell in the hexagonal system (heavy lines) in relation to a prism of hexagonal symmetry (light lines). The primitive cell is a rhombohedron, while the prism is a hexagonal cell with equal a's and b's and a c-axis different from the a's and b's. - The primitive cell in the hexagonal system is obtained by completing the rhombohedron. - The prism has hexagonal symmetry and can be divided into 3 rhombohedra. ## Positions Atomiques dans la Maille Cubique du Diamant **Image:** A drawing of the positions of atoms in the diamond cubic lattice. The drawing shows a projection of the atoms onto a face of the unit cell. The fractions indicate the positions of the atoms along the edges of the unit cell. - The points at 000 and 111 are on the fcc lattice. - This lattice can be described as two fcc lattices that occupy the center of the unit cell of the other. ## Structure Cristalline du Diamant Montrant les Liaisons Tétraédriques **Image:** A drawing of the diamond cubic lattice showing the tetrahedral bonds between the atoms. - Each carbon atom is bonded to four other carbon atoms. - The bonds form a tetrahedral angle. ## Structure Cubique du Sulfure de Zinc **Image:** A drawing of the zinc blende structure. This structure can be seen as two fcc lattices that are offset by a quarter of the diagonal of the cube. - The zinc atoms occupy one of the fcc lattices. - The sulfur atoms occupy the other fcc lattice. - Each zinc atom is tetrahedrally coordinated to four sulfur atoms. - Each sulfur atom is tetrahedrally coordinated to four zinc atoms. ## Structure Cristalline du Sulfure de Zinc Cubique **Image:** A drawing of the zinc blende structure showing the tetrahedral bonds between the atoms. - The structure is cubic and is described as a face-centered lattice. - There are four molecules of ZnS per primitive cell. ## Structure Hexagonale Compacte **Image:** Two diagrams: - **(a)** Diagram of a hexagonal close-packed (hcp) lattice. The circles represent atoms. The atoms form layers, and the layers are stacked in an ABABAB pattern. - **(b)** Diagram of a hexagonal primitive cell. The circles represent atoms. The cell is a prism with a triangular base. - The positions of the atoms in the hcp structure do not form a lattice. - The hexagonal primitive cell has two atoms in its base. - The coordinates of the two atoms in the base of the hexagonal primitive cell are "000" and "a+b+c". - The angle between the sides of the base is 120°. - The c-axis is perpendicular to the a and b-axis. - The ideal ratio is c = 1.633a.