2015 Exercise 4-14 Bravais
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Transcript of 2015 Exercise 4-14 Bravais
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11inh Quang Sang
5/04/2015Tinh th hc
TP ON DU KH VIT NAMTRNG I HC DU KH VIT NAM
MNG CU TRC TINH TH(CRYSTAL STRUCTURES & BRAVAIS LATTICE)
Bi 4
CBGD : inh Quang SangEmail : [email protected] : www.pvu.edu.vn/sangdq
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Contents
1. Unit Cell ( mng c s)
2. Type of Bravais lattices in 3-D
3. The 14 Bravais lattices
4. Examples Cc v d
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1. Unit Cells
3
Six lattice parameters: 3 distances: a, b & c 3 angles: , &
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2. Type of Bravais lattices in 3-D
4
1. Th nguyn thy - Primitive (P): lattice points on the cell corners only (sometimes called simple)
Y
X
Z
ba
c
4
8
7
32
6
5
1
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Coordination number for Body Centered Cubic (BCC) Structure
Total 8 nearest neighbor atomsCoordination number = 8
Coordination # = 8(# nearest neighbors)
2.Th Tm khi - Body-Centered (I): lattice points on the cell corners with one additional point at the center of the cell
2. Type of Bravais lattices in 3-D
4
8
7
32
6
5
1
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4
8
7
32
6
5
1
3. Th Tm mt - Face-Centered (F): lattice points on the cell corners with one additional point at the center of each of the faces of the cell
2. Type of Bravais lattices in 3-D
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Coordination number for Face Centered Cubic (FCC)
Structure
1 4
32
8
7
6
5
12 11
109
Total 12 nearest neighbor atoms
Coordination # = 12(# nearest neighbors)
7
3. Th Tm mt - Face-Centered (F): lattice points on the cell corners with one additional point at the center of each of the faces of the cell
2. Type of Bravais lattices in 3-D
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4
8
7
32
6
5
1
4. Th Tm y - Base-Centered (A, B, or C): lattice points on the cell corners with one additional point at the centerof each face of one pair of parallel faces of the cell
2. Type of Bravais lattices in 3-D
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3. The 14 Bravais lattices
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3.1 Cubic (H Lp phng) P I F C
Lattice point
4 23m m
Symmetry of Cubic lattices
P
I
F
a = b = c; = = = 90
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3.2 Tetragonal (H 4 phng)
Symmetry of Tetragonal lattices
4 2 2m m m
P I F C
a = b c; = = = 90
P I
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3.3 Orthorhombic (H Trc thoi)
Symmetry of Orthorhombic lattices
2 2 2m m m
Note the position of a and b
a b c One convention
P I F C
a b c; = = = 90
PI
FC
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3.4 Hexagonal (H 6 phng)
A single unit cell (marked in blue) along with a 3-unit cells forming a
hexagonal prism
Symmetry of Hexagonal lattices
6 2 2m m m
P I F C
a = b c; = = 90, = 120
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3.5a. Trigonal (H 3 phng)
Symmetry of Trigonal lattices
Rhombohedral
23m
Note the position of the origin and of a, b & c
P I F C
a = b = c; = = 90
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A trigonal cell can be produced from a cubic cell by pulling along [111] (the body diagonal) (keeping the edge length of the cube constant)
Rhombohedral3.5b. Trigonal (H 3 phng)
P I F C
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Some times an alternate hexagonal cell is used instead of the Trigonal Cell
Rhombohedral3.5c. Trigonal (H 3 phng)
P I F C
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3.6 Monoclinic (H 1 nghing)
Symmetry of Monoclinic lattices
2m
Note the position of a, b & c
P I F C
a < b < ca b c; = = 90
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3.7 Triclinic (H 3 nghing)
Symmetry of Triclinic lattices
1
P I F C
a b c; 90
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Summary 3-D SymmetryThe 32 3-D Point Groups Regrouped by Crystal System
(more later when we consider translations)
Crystal System No Center CenterTriclinic 1 1
Monoclinic 2, 2 (= m) 2/m
Orthorhombic 222, 2mm 2/m 2/m 2/m
Tetragonal 4, 4, 422, 4mm, 42m 4/m, 4/m 2/m 2/m
Hexagonal 3, 32, 3m 3, 3 2/m
6, 6, 622, 6mm, 62m 6/m, 6/m 2/m 2/m
Isometric 23, 432, 43m 2/m 3, 4/m 3 2/m
Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons
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4. Examples 1- Diamond Structure
The diamond-cubic crystal of silicon, with atoms on the cube corners and faces in red, and those inside in blue; each blue atom has four neighbors, and each lies in six-fold rings
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Can be viewed as a face-centered cubic array of the anions, with the cations in all of the octahedral holes, or A face-centered cubic array of the cations with anions in all of the octahedral holes.
The coordination number is 6 for both ions.
4. Examples 2- Rock Salt (NaCl) structure
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Chloride ions occupy the corners of a cube, with a Cesium ion in the center (called a cubic hole) or vice versa. Both ions have a coordination number of 8, with the two ions fairly similar in size.
4. Examples 3- CsCl structure
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The Zinc-blende or Sphalerite structure:Anions (S2-) ions are in a face-centered cubic
arrangement, with cations (Zn2+) in half of the tetrahedral holes.
4. Examples 4- Sphalerite (ZnS) structure
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Pyramids and DipyramidsPyramids and Dipyramids
PrismsPrisms
Low Symmetry Forms
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Trapezohedron, Scalehedron, Rhombehedron, Disphenoid
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Isometric Forms