PDF Solid State Chemistry

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Solid State Chemistry

2nd week

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1. Crystal Structures and Crystal Chemistry

2. Crystal Defects, Non-stoichiometry and Solid Solutions 3. Synthesis, Processing and Fabrication Methods

4. Characterization Technique (Thermal Analysis) 5. Bonding in Solids

Table of Contents

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Periodic Arrays of Atoms

• Crystal : periodic arrangement of atoms or groups of atoms - Ideal crystal: infinite repetition of identical structure units (atom, atoms, molecules) in space

- In the simplest crystal, the structure unit is a single atom (e.g. Cu, Ag, Au, Fe, …)

• Crystal structure = Lattice + Basis

- Lattice: a periodic array of point in space, identical surroundings. (Symmetry) - Basis: atom or atoms “attached” to each lattice point. (Composition)

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Periodic Arrays of Atoms

A

B

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Periodic Arrays of Atoms

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Lattice Translational Vectors

• How to define the lattice?

- Lattice can be defined translational vectors.

- In three dimension, we need three translational vectors (a1, a2, a3)

r

j

= x

j

a

1

+ y

j

a

2

+ z

j

a

3

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Lattice Translational Vectors

• Primitive translation vectors, ai

- The lattice is said to be primitive if any two points from which the atomic arrangement looks like the same always satisfy the below equation with a suitable choice of the integers ui.

r′ = r + x 1 a 1 + y 2 a 2 + z 3 a 3

^{(In a 3-D)}

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Lattice Translational Vectors

• Primitive lattice cell

- The parallelepiped defined by primitive axes a1, a2, a3 is called a primitive cell.

primitive cell

- A cell will fill all space by the repetition of suitable crystal translation operations.

- A primitive cell is a minimum-volume cell.

- There are many ways of choosing the primitive axes and primitive cells.

volume, Vc = |a1 ･ a2 × a3 |

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Lattice Translational Vectors

• Primitive lattice cell - Wigner-Seitz cell

1) draw lines connecting central point to all other lattice points

2) Construct planes that are bisector of lines in (1)

3) Minimum volume

* Brillouin zone = Wigner-Seitz cell of reciprocal lattice

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Fundamental Types of Lattices

• Symmetry operation

- typical symmetry operation: rotation about an axis that passes through a lattice point

1.Rotation 1-fold: 2π 2-fold: 2π/2 3-fold: 2π/3 4-fold: 2π/4 6-fold: 2π/6

2. Reflection at a plane (mirror reflection) 3. Inversion operation

4. Rotary-inversion

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Fundamental Types of Lattices

- if a rotation through 2π/n about an axis brings a figure into congruent position, the axis is called an n-fold symmetry.

2 fold 3 fold 4 fold

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- 2, 3, 4, and 6 fold symmetry are possible but 5, 7, and 8 fold are not possible.

On the non-existence of symmetry axes other than 1,2,3,4 and 6-fold in crystal. Notice that in (e), (g) and (h) space cannot be filled without leaving gaps which are shown shaded. 5, 7 and 8- fold axes, therefore, cannot exist in crystals.

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• Two-Dimensional Lattice Types - 5 Bravais lattice

Fundamental Types of Lattices

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• As with the case of the 2-D, the choice of the unit cell in a lattice is not unique, that is, there are many possible unit cells.

• The one that is “standard” for a given array of lattice points is the one which displays the most symmetry.

• In three dimensions, there are only 14 distinct lattices. There are called Bravais Lattices after Auguste Bravais (1811-1863)

• We will now develop the 14 Bravais Lattices by starting with the most general 2-D net.

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Fundamental Types of Lattices

• A square net with the second layer directly over the first and a = b = c. This is called cubic.

10

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11

• A square net with the second layer directly over the center of the first and a = b = c. This is called cubic.

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12

• A square net with the second layer directly over the center of the first and a = b = c. This is called cubic.

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Fundamental Types of Lattices

• A square net with the second layer directly over the first, but c ≠ a = b.

This is called Tetragonal.

4

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• A square net with the second layer placed over the centre of the first net, but c ≠ a = b. This is called Tetragonal.

5

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• A rectangular net with the second layer directly over the first, and a ≠ b ≠ c. This is called orthorhombic.

6

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Fundamental Types of Lattices

8

• A rectangular net with the second layer directly over the center of the first, and a ≠ b ≠ c. This is called orthorhombic.

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• A centered rectangular net with the second layer directly over the first, and a ≠ b ≠ c. This is called orthorhombic.

7

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Fundamental Types of Lattices

9

• A centered rectangular net with the second layer directly over the side center of the first, and a ≠ b ≠ c. This is called orthorhombic.

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• The second layer could be placed directly over the first layer, producing a 3-D lattice with right angles. This is called monoclinic.

2

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Fundamental Types of Lattices

• The second layer could be placed directly over the position half way between either of the vectors a or c, producing another 3-D lattice with right angles. This is called side-centered monoclinic.

3

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• If the next layer up is placed over a general position the least symmetric 3- D lattice is obtained. This is called triclinic.

1

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Fundamental Types of Lattices

• Triangular nets stacked directly over each other, and a = b ≠ c. α = β = 90°

and γ = 120°. This is called hexagonal.

13

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14

• Triangular nets stacked directly over center of triangles, and a = b = c. α = β = γ < 120°, ≠ 90°. This is called trigonal.

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14

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• Two-Dimensional Lattice Types - 14 Bravais lattice

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Fundamental Types of Lattices

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• 230 Space groups

- In mathematics and physics, a space group is the symmetry group of a configuration in 3-D space.

- The space group of the crystal structure is composed of the translational symmetry operations in addition to the operations of the point group:

translation, screw axes, glide planes

Ref.) http://en.wikipedia.org/wiki/Space_group

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Quasicrystals

• Quasicrystal

• rotational symmetry (n = 5, 10, 12)

• A regular crystal lattice

exhibiting fivefold rotational symmetry cannot exist.

• discovered in a wide range of alloy, organic polymer, and

liquid crystal systems.

(e.g. Al-Cu-Fe alloy:

icosahedrite)

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Quasicrystals

* Ordered but aperiodic (“quasi-periodic”) crystals first discovered in 1982

2011 Nobel Prize Chemistry D. Shechtman

10-fold symmetry !!

Al

0.86

Mn

0.14

alloy

Shechtman et al. PRL 53, 1951 (1984).

YMgCd