材料科学基础 Fundamentals of Materials Fundamentals of Materials ScienceScience

Chapter 2Fundamentals of Crystallology

gas state气态（ ）substance l i qui d state物质（ ）液态（ ）

crystal晶体（ ）sol i d state固态（ ）amorphous sol i d非晶体（ ）

Chapter 2 Basic Crystallology§2.1 Crystal Character & Space Lattice

ⅠⅠ.. Crystals versus non-crystalsCrystals versus non-crystals

1. Substance states1. Substance states

.

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

O

O

1O

ClA

C

1A

*AB

）面示意图晶体结构（100NaCl

22. Classification of materials based on structure. 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:Single crystal: in the form of one crystal grains Polycrystal: Polycrystal: grain boundaries

ⅡⅡ. Space lattice. Space lattice 1. Definition: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 points2. Two basic features of lattice points①① Periodicity: Arranged in a periodic pattern.Periodicity: Arranged in a periodic pattern.②② Identity: The surroundings of each point in the lattice are Identity: The surroundings of each point in the lattice are

identical.identical.

A lattice may be one , two, or three dimensional

two dimensions

Space lattice is a point array which represents the regularity of atom arrangements

(1) (2) (3)

a

b

Three dimensions

Each lattice point has identical surrounding environment

Ⅲ. Unit cell and lattice constants

1. 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.

2. The space lattice is characterized by the size and shape of the unit cell.

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

Lattice Constants

a

c

b

αβγ

a

c

b

αβγ

lattice constants —— a , b , c α, β, γ

§2.2 Crystal System & Lattice §2.2 Crystal System & Lattice TypesTypes

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.

Ⅰ.Seven crystal systems

All possible structures reduce to a small All possible structures reduce to a small number of basic unit cell geometries.number of basic unit cell geometries.

① There are only seven, unique unit cell shapes that can be stacked together to fill three-dimensional lattices.

② We must consider how atoms can be stacked together within a given unit cell.

Seven Crystal SystemsSeven Crystal Systems

Triclinic 三斜 a≠b≠c ， α≠β≠γ≠90°

Monoclinic 单斜 a≠b≠c ， α＝ β＝ 90°≠γ α＝ γ＝ 90°≠β

Orthorhombic 正交 a≠b≠c ， α＝ β＝ γ＝ 90°

Tetragonal 四方 a＝ b≠c ， α＝ β＝ γ＝ 90°

Cubic 立方 a＝ b＝ c ， α＝ β＝ γ＝ 90°

Hexagonal 六方 a＝ b≠c ， α＝ β＝ 90°γ＝ 120°

Rhombohedral 菱方Trigonal 三角 a＝ b＝ c ， α＝ β＝ γ≠90°

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

ⅡⅡ.. 14 types of Bravais lattices14 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.

7×47×4＝＝ 2828

Delete the 14 types which are identicalDelete the 14 types which are identical

2828－－ 1414＝＝ 1414

+ + +

P I C F

2. 14 types of Bravais lattice2. 14 types of Bravais lattice

① Tricl: simple (P)② Monocl: simple (P). base-centered (C)③ Orthor: simple (P). body-centered (I). base-centered (C). face-centered (F)④ Tetr: simple (P). body-centered (I)⑤ Cubic: simple (P). body-centered (I). face-centered (F)⑥ Rhomb: simple (P). ⑦ Hexagonal: simple (P).

1

2

3

A face- centred cubi c l atti ce面心立方结构（ ）

A body- centred cubi c l at常见金属晶体结构 体心立方结构（ ） ti ce A hexagonal cl ose- packed l atti ce密排立方结构（ ）

体心立方点阵 密排六方点阵BCC HCP

面心立方点阵FCC

常见的晶体结构

Crystal systems(7)

Lattice types (14)

PC

F I A B C

1 Triclinic √2 Monoclinic √ √ or √

(γ≠90°or β ≠ 90° )

3 Orthorhombic √ √or √ or√ √ √4 Tetragonal √ √5 Cubic √ √ √6 Hexagonal √7 Rhombohedral √

Seven crystal systems and fourteen lattice types

ⅢⅢ. Complex Lattice. Complex Lattice﹡﹡ The example of complex lattice

a ab

c

120o

120o

120o

Crystal structure is the real arrangement of atom in crystals

Crystal structure ＝ Space lattice + Basis or structure unit

++ ==

Fe : Al = 1 : 1FeFe

AlAl

The difference between space lattice and crystal structure

2×3 atoms / cell

ⅣⅣ. Primitive Cell. Primitive Cell1. For primitive cell, the volume is minimum

Primitive cell only includes one lattice point固体物理学原胞

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

[krai'tiəriən] ( 批评、判断等的 ) 标准 , 准则，尺度

P → C I → F

But the volume is not minimum.

P 简单格子 I 体心格子 F 面心格子 C 底心格子

Examples and DiscussionsExamples and Discussions1. Why are there only 14 space lattices?

Explain why there is no base centered and face centered tetragonal Bravais lattice.

P → C I → F

But the volume is not minimum.

P 简单格子 I 体心格子 F 面心格子 C 底心格子

Exercise1. 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.

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 !