材料科学基础
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
§ 2.1 Space Lattice
? Ⅰ.Crystals versus non -crystals
1,Classification of functional materials
Chapter Ⅱ
Fundamentals of Crystallography
Lesson three
2,Classification of materials based on structure
Regularity in atom arrangement
—— periodic or not (amorphous)
? Crystalline,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
? Polycrystalline,
grain boundaries
? Ⅱ.Space lattice
1,Definition:
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,Two basic features of lattice points
① Periodicity,Arranged in a periodic pattern.
② Identity,The surroundings of each point in
the lattice are 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
bαβ γ
a
c
bαβ
γ
§ 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.
?Ⅰ.Seven crystal systems
All possible structure reduce to a small
number of basic unit cell geometries.
① There are only seven,unique unit cell
shapes that can be stacked together to fill
three-dimensional.
② We must consider how atoms can be
stacked together within a given unit cell.
Seven Crystal Systems
Triclinic a≠b≠c, α≠β≠γ≠90°
Monoclinic a≠b≠c, α＝ β＝ 90° ≠γα＝ γ＝ 90° ≠β
Orthorhombic a≠b≠c, α＝ β＝ γ＝ 90°
Tetragonal a＝ b≠c, α＝ β＝ γ＝ 90°
Cubic a＝ b＝ c, α＝ β＝ γ＝ 90°
Hexagonal a＝ b≠c, α＝ β＝ 90° γ＝ 120°
Rhombohedral a＝ b＝ c, α＝ β＝ γ≠90°
? 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,
7× 4＝ 28
Delete the 14 types which are identical
28－ 14＝ 14
+ + +
P I C F
2,14 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).
Crystal systems
(7)
Lattice types (14)
P C F IA 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
? Ⅲ.Primitive Cell
For primitive cell,the volume is minimum
Primitive cell
Only includes one
lattice point
? Ⅳ,Complex Lattice
The example of complex lattice
a
ab ?
c
120o
120o
120o
Examples and Discussions
1,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.
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
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.
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.
Thank you !
3
Fundamental of Materials
Prof,Tian Min Bo
Tel,62795426, 62772851
E-mail,tmb@mail.tsinghua.edu.cn
Department of Material Science and Engineering
Tsinghua University,Beijing 100084
§ 2.1 Space Lattice
? Ⅰ.Crystals versus non -crystals
1,Classification of functional materials
Chapter Ⅱ
Fundamentals of Crystallography
Lesson three
2,Classification of materials based on structure
Regularity in atom arrangement
—— periodic or not (amorphous)
? Crystalline,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
? Polycrystalline,
grain boundaries
? Ⅱ.Space lattice
1,Definition:
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,Two basic features of lattice points
① Periodicity,Arranged in a periodic pattern.
② Identity,The surroundings of each point in
the lattice are 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
bαβ γ
a
c
bαβ
γ
§ 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.
?Ⅰ.Seven crystal systems
All possible structure reduce to a small
number of basic unit cell geometries.
① There are only seven,unique unit cell
shapes that can be stacked together to fill
three-dimensional.
② We must consider how atoms can be
stacked together within a given unit cell.
Seven Crystal Systems
Triclinic a≠b≠c, α≠β≠γ≠90°
Monoclinic a≠b≠c, α＝ β＝ 90° ≠γα＝ γ＝ 90° ≠β
Orthorhombic a≠b≠c, α＝ β＝ γ＝ 90°
Tetragonal a＝ b≠c, α＝ β＝ γ＝ 90°
Cubic a＝ b＝ c, α＝ β＝ γ＝ 90°
Hexagonal a＝ b≠c, α＝ β＝ 90° γ＝ 120°
Rhombohedral a＝ b＝ c, α＝ β＝ γ≠90°
? 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,
7× 4＝ 28
Delete the 14 types which are identical
28－ 14＝ 14
+ + +
P I C F
2,14 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).
Crystal systems
(7)
Lattice types (14)
P C F IA 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
? Ⅲ.Primitive Cell
For primitive cell,the volume is minimum
Primitive cell
Only includes one
lattice point
? Ⅳ,Complex Lattice
The example of complex lattice
a
ab ?
c
120o
120o
120o
Examples and Discussions
1,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.
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
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.
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.
Thank you !
3
