材料科学基础 第3课

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.
grains
Polycrystalline:
grain boundaries
Amorphous: The material’s atoms do not have a long-range order (0.1～1nm).
2.1.2 Space lattice
1. Definition:
① There are only seven, unique unit cell
shapes that can be stacked together to fill three-dimensionally.
② We must consider how atoms can be
stacked together within a given unit cell.
√ √ √ √ √ √ √ √
or
√
or√
( ≠ 90°or β ≠ 90°)
√or √
√
√ √ √
7
Rhombohedral
√
2.2.3 Primitive cell
For primitive cell, the volume is minimum
Primitive cell Only includes one lattice point
Space lattice consists of an array of regularly arranged geometrical points, called lattice points. The (periodic) arrangement of these points describes the regularity of the arrangement of atoms in crystals.
2.2.4 Complex lattice
The example of complex lattice
Examples and Discussions 1. Why are there only 14 space lattices?
Explain why there is no base centered and face
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.
Regularity in atom arrangement
—— periodic or not (amorphous)
Crystalline: The material’s atoms are arranged in a periodic fashion.
Single crystal: in the form of one crystal
a＝b＝c ，α＝β＝γ＝90° a＝b≠c ，α＝β＝90°γ＝120°
n=4*3 n=6
n=3
Rhombohedral a＝b＝c ，α＝β＝γ≠90°
2.2.2 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.
3
Thank you !
3
Seven Crystal Systems
Triclinic
a≠b≠c ，α≠β≠γ≠90°
a≠b≠c ， α＝β＝90°≠γ α＝γ＝90°≠β a≠b≠c ，α＝β＝γ＝90° a＝b≠c ，α＝β＝γ＝90°
n=1பைடு நூலகம்
n=2 n=2*3 n=4， 2*2
Monoclinic
Orthorhombic Tetragonal Cubic Hexagonal
材料科学基础
Fundamental of Materials Science
Prof: Tian Min-Bo, Sun Xiao-Dan
Tel: 62772851 ，62772977 E-mail: tmb@mail.tsinghua.edu.cn sunxiaodan@mail.tsinghua.edu.cn Department of Material Science and Engineering Tsinghua University. Beijing 100084
centered tetragonal Bravais lattice.
P→C
I→F
But the volume is not minimum.
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
2.1.3 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. The space lattice is characterized by the size and shape of the unit cell.
An infinite periodic array of geometric points in space defines a space lattice or simply a lattice.
A lattice may be one , two, or three dimensional. two dimensions
+
P I
+
C
+
F
7×4＝28
Delete the 14 types which are identical
28－14＝14
2. 14 types of Bravais lattice
① Tricl: ② ③ ④ ⑤ ⑥ ⑦
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) Hexagonal: simple (P). Rhomb: simple (P).
b a
(1) (2) (3)
Space lattice is a point array which represents the regularity of atom arrangements
Three dimensions
Each lattice point has identical surrounding environment
Lattice Constants
c c
β α b a γ β α a γ
b
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 nfold rotation symmetry. And axis is said to be n-fold rotation axis.
Lesson three
Chapter 2
Fundamentals of Crystallography
2.1 Space Lattice
2.1.1 Crystals versus non-crystals
1. Classification of functional materials
2. Classification of materials based on structure
Seven crystal systems and fourteen lattice types
Crystal systems (7) Lattice types (14) P C A B C F I
1 2 3 4 5 6
Triclinic Monoclinic Orthorhombic Tetragonal Cubic Hexagonal
2.
How to distinguish the size and shape of different unit cells?
The six variables , which are described by lattice constants —— a , b , c ; α, β, γ
2. Two basic features of lattice points
① ②
Periodicity: Arranged in a periodic pattern. Identity: The surroundings of each point in the lattice are identical.
We identify 14 types of unit cells, or Bravais lattices, grouped in seven crystal systems based on the symmetry of crystal.
2.2.1 Seven crystal systems All possible structures reduce to a small number of basic unit cell geometries.