structures of matters

7/31/2019 Structures of Matters

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Crystal Structures

How do atoms arrange themselves to form solids?

Lecture Outline

Fundamental concepts and languageUnit cells

Crystal structuresClose packed crystal structuresDensity computations

Types of solids

Types of Solids

Polycrystalline material:comprised of many small

crystals or grains

Crystalline material: atoms self-organize in a periodic array

Single crystal :atoms are in a repeating or periodic array overthe entire extent of the material



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Crystal structure

To discuss crystalline structures it is useful toconsider atoms as being hard spheres with

well-defined radii. In this hard-sphere model,the shortest distance between two like atomsis one diameter.

We can also consider crystalline structure as

a lattice of points at atom/sphere centers.

Lattices

A Lattice is defined as an array of equivalent pointsin one, two or three dimensions (normally three forinorganic materials). The environment of an atomplaced on any one of these lattice points would be

identical to that placed on any other lattice point.Therefore the lattice locates equivalent positions andshows the translational symmetry. The actualpositions of atoms or molecules, however, is notprovided.



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Unit CellThe unit cell is the smallest structural unit or

building block that can describe the crystalstructure. Repetition of the unit cell generatesthe entire crystal.

This image shows a line of equally spacedpoints as a one dimensional lattice

Unit Cell

Example:2D honeycomb net can be representedby translation of two adjacent atoms that form aunit cell for this 2D crystalline structure



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Metallic Crystal Structures

Metals are usually polycrystalline; although formationof amorphous metals is possible by rapid cooling.

Amorphous: lacks a systematic atomic arrangement.

Metallic Crystal Structures

Two Dimensional Lattice

Is it right?

Is it real?

The atomic bonding in metals isnon-directional no restriction

on numbers or positions ofnearest-neighbor atoms large

number of nearest neighborsand dense atomic packing.



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The Two Dimensional Lattice

6R

12balls

inside

6R

12balls

inside

A A A A A

A A A A A

A A A A

B B B B

BBB

C C C C

C C C

A A A A

B B B B

BBB

C C C C

C C C

A A A A A Layer I - A

Layer II - B

Layer III A or C ?!

Ways to Pack Close-PackedLayers in 3D

A

Peaks

Valleys

B C

Valleys



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A A A A A

A A A A A

A A A A

B B BB

BBB

C C C C

C C C

A A A A

B B B B

BBB

C C C C

C C C

A A A A A

Layer I - A

Layer II - B

Layer III - A !!!

ABABABABABAB

Layer III = Layer I !!!

Close Package

A A A A A

A A A A A

A A A A

B B B B

BBB

C C C C

C C C

A A A A

B B B B

BBB

C C C C

C C C

A A A A A

Layer I - A

Layer II - B

Layer III - C !!!

ABCABCABCABC

Layer III = Layer I !!!

Close Package



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HolesTWO different types of HOLES (INTERSTITIAL sites) are leftOCTAHEDRAL (O) holes with 6 nearest sphere neighbours

TETRAHEDRAL (T ) holes with 4 nearest sphere neighbours

Bravais Lattices



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BCC Crystal Structure

a

a

b

Examples: Alkali metals (Li, Na, K, Rb), Cr,Mo, W, Mn, α-Fe (< °C), β-Ti (> °C).

Body centered cubic (BCC) crystal structure. (a) A BCC unit cellwith closely packed hard spheres representing the Fe atoms. (b)Areduced-sphere unit cell.

FCC Crystal Structure

R

a

a

a

a

(c)(b)

(a) The crystal structure of copper is Face Centered Cubic (FCC). The

atoms are positioned at well defined sites arranged periodically and there is

a long range order in the crystal.

(b) An FCC unit cell with closed packed spheres.

(c) Reduced sphere representation of the FCC unit cell. Examples: Ag, Al,

Au, Ca, Cu, γ -Fe (> C), Ni, Pd, Pt, Rh



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HCP Crystal Structure

Layer A

Layer B

Layer A

(a) (b)

Layer A

Layer B

Layer A

c

a

(c) (d)

Examples: Be, Mg, α-Ti ( < 882 C ), Cr, Co, Zn, Zr, Cd

The Hexagonal ClosePacked (HCP) CrystalStructure.(a) The HexagonalClose Packed (HCP)Structure. A collection ofmany Zn atoms. Colordifference distinguisheslayers (stacks).(b) The stacking

sequence of closelypacked layers is ABAB(c) A unit cell withreduced spheres(d) The smallest unit cellwith reduced spheres.

The Diamond Structure

a

C

a

a

The diamond unit cell is cubic. The cell has eight atoms. Grey Sn

(α-Sn) and the elemental semiconductors Ge and Si have this

crystal structure.



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The Zinc Blende Cubic CrystalStructure

S

Zn

a

a

a

The Zinc blende (ZnS) cubic crystal structure. Many importantcompound crystals have the zinc blende structure. Examples: AlAs,GaAs, GaP, GaSb, InAs, InP, InSb, ZnS, ZnTe.



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Unit Cell Characterization

Face-Centered Cubic Crystal Structure

The hard spheres or ion cores touch oneanother across a face diagonal the cube

edge length, a=2R 2



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The coordination number, CN =the number ofclosest neighbors to which an atom is bonded =numberof touching atoms, CN =12

Number of atoms per u nit cell, n =4 .(For an atomthat is shared with m adjacent unit cells, we only counta fraction of the atom, 1/m). In FCC unit cell we have:

6 face atoms shared by two cells:6 x 1/2 =38 corner atoms shared by eight cells: 8 x 1/8 =1

Atomic packing factor, APF = fraction of volumeoccupied by hard spheres =(Sum of atomicvolumes)/(Volume of cell)=0.74 (maximum possible)



Corner and face atoms in the unit cell are equivalentFCC crystal has APF of 0.74,the maximum packing for

a system equal-sized spheres FCC is a close-packedstructureFCC can be represented by a stack of close-packed

planes (planes with highest density of atoms)




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Unit Cell CharacterizationBody-Centered Cubic Crystal Structure

The hard spheres touch one another along cube diagonalthe cube edge length, a=4R/ 3The coordination number, CN =8Number of atoms per unit cell, n =2

Center atom (1) shared by no other cells:1 x 1 =18 corner atoms shared by eight cells: 8 x 1/8 =1

Atomic packing factor, APF =0.68

Corner and center atoms are equivalent

Unit Cell CharacterizationHexagonal Close-Packed Crystal Structure

Two lattice parameters a and c. Ideal ratio c/a =1.633The coordination number, CN =12 (same as in FCC)Number of atoms per unit cell, n =6.3 mid-plane atoms shared by no other cells:3 x 1 =312 hexagonal corner atoms shared by 6 cells:12 x 1/6 =22 top/bottom plane center atoms shared by 2 cells:2 x 1/2 =1Atomic packing factor, APF =0.74 (same as in FCC)

All atoms are equivalent



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Unit Cell CharacterizationClose-packed Structures (FCC and HCP)

Both FCC and HCP crystal structures have atomicpacking factors of 0.74 (maximum possible value)

Both FCC and HCP crystal structures may begenerated by the stacking of close-packed planes

The difference between t he two structures is

in the stacking sequence

HCP: ABABAB... FCC: ABCABCABC

Density Computations

Atoms in the unit cell, n = 2 (BCC); 4 (FCC); 6 (HCP)

NA =6.023 1023 atoms/molThe volume of the cell, Vc =a3 (FCC and BCC)a =2R2 (FCC); a =4R/ 3 (BCC)where R is the atomic radius

= the density of the unit cell, n = atoms in the unit cell,Vc = the volume of the unit cell, Mass of an atom A/NA =M = Atomic weight, A, in amu (or g/mol) is given in theperiodic table.

Thus, the formula for the density is:



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Polymorphism and Allotropy

Some materials may exist in more than one crystal

structure, this is called polymorphism. If the material is anelemental solid, it is called allotropy.An example of allotropy is carbon, which can exist asdiamond, graphite, and amorphous carbon.

Polymorphism



FullerenesTwo kinds of fullerenes are shown here:

carbonnanotube

buckminsterfullerene(buckyball)

structures of matters

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