材料科学基础 Fundamentals of Materials Science

1. 材料科学基础Fundamentals of Materials Science Chapter 2 Fundamentals of Crystallology

2. §2.1 Crystal Character & Space Lattice Chapter 2 Basic Crystallology

3. Ⅰ.Crystals versus non-crystals 1. Substance states

4. 气体: 高度无序，分子的空间位置完全是无规则的,自由运动 液体：短程有序、长程无序结构，有一定体积，无固定形状 固体 晶体/ crystal: 原子周期性排列/长程有序 The materials atoms are arranged periodically 非晶体/amorphous solid：原子无序排列 The materials atoms are arranged disorderly 准晶体/quasicrystal:介于晶体和非晶体之间, 具有完全有序的结构， 不具有晶体的平移对称性，具有5次和6次以上对称轴 .

6. 2. Classification of materials based on structure Regularity in atom arrangement —— periodic or not (amorphous) • Crystal: The materials atoms are arranged in a periodic fashion. Amorphous: The material’s atoms do not have a long-range order (0.1～1nm). • Single crystal: in the form of one crystal grains • Polycrystal: grain boundaries

9. Ⅱ. Space lattice 1. Definition: Space lattice consists of arrays of regularly arranged geometrical points, called lattice points. The (periodic) arrangement of these points describes the regularity of the arrangement of atoms in crystals. 2. Two basic features of lattice points • Periodicity: Arranged in a periodic pattern. • Identity: The surroundings of each point in the lattice are • identical.

11. b a (1) (2) (3) A lattice may be one , two, or three dimensional two dimensions Space lattice is a point array which represents the regularity of atom arrangements

12. Three dimensions Each lattice point has identical surrounding environment

14. Ⅲ. Unit cell and lattice constants • Unit cell is the smallest unit of the lattice. The whole lattice can be obtained by infinitive repetition of the unit cell along it’s three edges. • The space lattice is characterized by the size and shape of the unit cell.

17. How to distinguish the size and shape of the deferent unit cell ? The six variables , which are described by lattice constants ——a , b , c α, β, γ

18. c c β b α β α b a a γ γ lattice constants ——a , b , c α, β, γ Lattice Constants

19. §2.2 Crystal System & Lattice Types If a rotation around an axis passing through the crystal by an angle of 360o/n can bring the crystal into coincidence with itself, the crystal is said to have a n-fold rotation symmetry. And axis is said to be n-fold rotation axis. We identify 14 types of unit cells, or Bravais lattices, grouped in seven crystal systems.

20. Ⅰ.Seven crystal systems All possible structures reduce to a small number of basic unit cell geometries. There are only seven, unique unit cell shapes that can be stacked together to fill three-dimensional lattices. We must consider how atoms can be stacked together within a given unit cell.

21. Seven Crystal Systems

22. 七个晶系的划分 晶系名称点阵常数特征 (1)立方(等轴)晶系 a=b=c α=β=γ=90° P.I.F (2)四方(正交)晶系 a=b≠c α=β=γ=90° P.I (3)正交(斜面)晶系 a≠b≠c α=β=γ=90° P.I.C.F (4)三方(菱面)晶系 a=b=c α=β=γ≠90° R (5)六方(六角)晶系 a=b≠c α=β=90ºγ=120° P(C) (6)单斜晶系 a≠b≠c α=γ=90°≠β C.P (7)三斜晶系 a≠b≠c α≠β≠γ≠90° P代表简单格子 I代表体心格子 F代表面心格子 C底心格子

23. Ⅱ.14 types of Bravais lattices 1. Derivation of Bravais lattices Bravais lattices can be derived by adding points to the center of the body and/or external faces and deleting those lattices which are identical.

24. + + + P I C F 7×4＝28 Delete the 14 types which are identical 28－14＝14

25. 2. 14 types of Bravais lattice • Tricl: simple (P) • Monocl: simple (P). base-centered (C) • Orthor: simple (P). body-centered (I). base-centered (C). face-centered (F) • Tetr: simple (P). body-centered (I) • Cubic: simple (P). body-centered (I). face-centered (F) • Rhomb: simple (P). • Hexagonal: simple (P).

27. 常见的晶体结构 面心立方点阵 FCC 体心立方点阵 密排六方点阵 BCC HCP

28. Seven crystal systems and fourteen lattice types

29. 120o 120o 120o Ⅲ. Complex Lattice﹡ The example of complex lattice

30. Crystal structure is the real arrangement of atom in crystals Crystal structure ＝ Space lattice + Basis or structure unit

31. + =

32. Fe Al The difference between space lattice and crystal structure Fe : Al = 1 : 1

33. 2×3 atoms / cell

34. Ⅳ. Primitive Cell 1. For primitive cell, the volume is minimum Primitive cell only includes one lattice point 固体物理学原胞

35. (批评、判断等的)标准, 准则，尺度 [krai'tiəriən] 2. Criterion for choice of unit cell • Symmetry • As many right angle as possible • The size of unit cell should be as small as possible

36. P简单格子 I体心格子 F面心格子 C底心格子 P → C I → F But the volume is not minimum.

37. Examples and Discussions 1. Why are there only 14 space lattices? • Explain why there is no base centered and face centered tetragonal Bravais lattice.

38. P简单格子 I体心格子 F面心格子 C底心格子 P → C I → F But the volume is not minimum.

39. Exercise 1. Determine the number of lattice points per cell in the cubic crystal systems. If there is only one atom located at each lattice point, calculate the number of atoms per unit cell. 2. Determine the relationship between the atomic radius and the lattice parameter in SC, BCC, and FCC structures when one atom is located at each lattice point. 3. Determine the density of BCC iron, which has a lattice parameter of 0.2866nm.

40. 4. Prove that the A-face-centered hexagonal lattice is not a new type of lattice in addition to the 14 space lattices. 5. Draw a primitive cell for BCC lattice. Thanks for your attention !