2015 Exercise 4-14 Bravais

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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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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


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


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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


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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


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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

Documents

2015 Exercise 4-14 Bravais