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Solid state structurehttp://web.chemistry.gatech.edu/~barefield/1311/handouts.html

click on Wilkinson’s Class for the version used in classclick on handout #5 for an alternative view of material

The URL for the webpage on unit cells ishttp://www.cwru.edu/artsci/chem/chime/solids.html

The structure of metalsThe structure of most metals can be described in terms of packing hard spheres (representing atoms) in a regular array– this type of description is also used for ionic solids– these metals are crystalline

Several different simple packing arrangements are found in elemental metals– Packing arrangements of atoms in metals play an important

role in determining their mechanical properties> Ductility of copper is directly related to its packing

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The unit cell

• The atoms in a crystal are in a regular repeating pattern called the crystalline lattice.

• The crystalline lattice can be reproduced by translation of the unit cell in three dimensions. The unit cell is that unique part of the crystal structure such that when translated along parallel lines, generates the entire crystal structure.

• For a substance containing more than one kind of atom, the ratio of atoms in the unit cell must be exactly the same as in the entire crystal.

Simple cubic

body-centered cubic

face-centered cubic

Contribution of atoms to unit cell

Cubic

Body centered cubic

bcc

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Close packing

Close packed structures are built up from hexagonal arrays of atomsClose packing is the most efficient way of filling space

HCP and CCP (or FCC)Hexagonal and cubic close packing are very common. – Face Centered Cubic packing is the same as cubic close packing

FCC unit cell has four(whole) atoms in it

HCP hasABAB.. repeat

FCC hasABCABC.. repeat

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Other views of HCP and FCC

ccp hcp

Note that face centered (FCC) cubic packing and cubic close packing (CCP) are the same thing!

Not all unit cells are cubic!

There are seven unique crystal systems

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Packing efficiency

Determination of density from structural data

Aluminum has the face-centered cubic structure (to right) with a unit cell dimension of 4.041 D.

volume of unit cell = (4.041 x 10-8 cm)3 = 6.599 x 10-23 cm3

mass of atoms in the cell = 4 atoms x 1 mol/6.022 x 1023 atoms x 26.98 g mol-1= 17.922 x 10-23 g

d = mass/volume = 17.922 x 10-23 g/6.599 x 10-23 cm3 = 2.715 g cm-3

d = mass of contents of cell/volume of cell

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Ionic bonding

Many compounds can be thought of as a collection of ions (Mn+, Xn-) held together electrostaticallyThis idea arose out of experiments by Arrhenius looking at the conductivity of solutions prepared by dissolving “ionic compounds” in water– Not believed at first, but got the 1903 Nobel

prize. http://www.nobel.se/chemistry/laureates/1903/

Evidence for presence of ions in solids

•Formed from elements with )O>2

•Exist as nonvolatile solid with poor conductance at RT

•Good conductance when molten

•Process a 3-dimensional lattice with regular array of cationsand anions where nearest neighbors are of opposite charge

Electron density map of a portion of the face of NaCl unit cell

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Metals and ionic compounds: structure, bonding and energetics

• Metals adopt three basic structures; cubic closest packed (ccp), hexagonal closest packed (hcp) or body centered cubic (bcc).

• Many simple salts can be viewed as having a structure in which one ion forms a closest packed array with the oppositely charged ion occupying voids (holes) in the array.

• Properties of metals, salts and non-metallic elements/molecular compounds differ greatly.

ccp hcp

bcc NaCl

ZnS

Comparison of properties of metals, simple salts and non-metallic elements

Non-metallic elementsSaltsMetalsStructures

Loose packed, low connectivityMolecular, chain, layer or infinite

structures

Close packed, C.N. = 4-8Infinite structures

Close packed, C.N. = 8-12Infinite structures

PropertiesPoor conductorsBrittle (can be hard)Non-reflective, low tensile

strengthLarge range of densities and

melting points

Insulators as solids, good conductors molten

BrittleNon-reflective, low tensile

strengthLarge range of densities; high

melting points

Good conductorsMalleable, ductile, elasticLustrous, hard, high tensile

strengthLarge range of densities and

melting points

Chemical PropertiesTend to form anionsOxidizing agentsNever liberate H2 from waterForm salts with metalsReact with O2 to form covalent

(mostly acidic) oxides

Combination of ionsSimple salts are neither

oxidizing or reducingNever liberate H2 from waterDo not react with metalsDo not react with oxygen

Tend to form cationsReducing agentsMore electropositive liberate H2

from waterForm alloys with other metalsElectropositive react with O2 to

form (mostly ionic, basic) oxides

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Structures of ionic compoundsIt is often convenient to think about the cationslying in holes (interstices) between arrays of anionsTypically, assume ions are hard spheresUsually, a compound will adopt a structure that maximizes the number of anions around each cation without causing the anions to touch

Salts based on simple cubic packingA simple cubic array of ions contains holes that are eight coordinate– These holes can be filled by ions of opposite charge to make

salts» structures include CsCl and CaF2

Note this is not BCC packing

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More unit cells

NaCl CsCl

ZnS Li2Se

Salts based on octahedral cationcoordination

Close packed arrays of anions have both octahedral and tetrahedral interstices– filling octahedral holes in a cubic close packed

array gives the NaCl structure– filling octahedral holes in a hexagonal close

packed array gives the NiAs structure

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Holes in close packed arrays

There are one octahedral and two tetrahedral holes for every atom in a close packed array

x Marks octahedral holes x Marks tetrahedral holes

The NaCl structure

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Tetrahedral coordination

Structures based on filling tetrahedral holes in close packed anion arrays are commonly found– fill all tetrahedral sites in cubic close packed array -

ZnS zinc blende– fill all tetrahedral sites in a hexagonal close packed

array - ZnS Wurtzite

ZnS structures

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More unit cells

NaCl CsCl

ZnS Li2Se

Structure prediction from ion radius ratios

The maximum electrostatic interaction between ions occurs when they are surrounded by the maximum number of ions of opposite charge separated by the shortest distance.

Assuming a model in which ions are considered as hard spheres with the smaller ion occupying holes in an array of larger ions allows calculation of the ideal size for a sphere in each type of hole.

Cubic hole

r+/r- = 0.15 r+/r- = 0.41

r+/r- = 0.73

r+/r- = 0.22

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Radius ratio rules

It is possible to predict the type of ion coordination that you will get if you know the ratio of the cation to anion size

r+/r- values Preferred coordination number

> 0.732 8 – cubic coordination

0.414 – 0.732 6 – octahedral coordination

0.225 – 0.414 4 – tetrahedral coordination

How the limiting values were calculated

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Violations of the radius ratio rulesRadius ratio rules only work for ~2/3 known compounds– ions are not really hard spheres– covalent contribution to bonding can mess things up– ionic radius varies with coordination number

There are empirical methods that can be used to reliably predict structure– structure maps

What do the following have in common?● Push-button igniters (BBQ grills, cigarette lighters) ● Electronic beepers ● Tweeters in stereo speakers ● Sound-generating arrays for sonar, fish finders, etc. ● Crystal microphones● Nanoscale positioners (Å precision)● Functional element in time keepers (clocks, watches,

etc.)

Crystals that are piezoelectric● Piezo (Greek) means pressure; such crystals are pressure

sensitive● Piezoelectric crystals generate current when deformed

and deform when a potential is applied● Only materials whose crystal structures lack a center of

symmetry can exhibit piezoelectric properties

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Pb(Zr,Ti)O3 (PZT) phase diagram

Only structures without a center of symmetry show piezoelectric properties

0

100

200

300

400

500

0 20 40 60 80 100Percent Titanium

Tem

pera

ture

(ο C)

tetragonal

rhombohedral

cubic

The Piezoelectric Effect

Crystal

Current Meter= 0

+ - + - + -

+ - + - + -Charges canceleach other, sono current flows

Crystal material at rest: No forces applied, so net current flow is 0

Adapted from copyrighted ValpeyFisher Corp. materials

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The Piezoelectric Effect

Crystal

Current meter deflects

- - - - -

+ + + + +

Charges are net + on one end and net - on the opposite end: crystal gets thinner and longer; changing direction of applied force will shorten and fatten crystal and reverse the charges (and current).

Crystal material with forces applied in direction of arrows………..

Force

Adapted from copyrighted ValpeyFisher Corp. materials

The electromechanical effect

Crystal

…. the crystal should get longer and skinnier (shown) or shorter and fatter depending upon polarity.

Now, replace the current meter with a power source capableof supplying a current meter….

power source(battery)

- side

+ side

- - - - -

+ + + + +

Adapted from copyrighted ValpeyFisher Corp. materials

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The electromechanical effect

Crystal

With the switch open, the crystal material is now at rest again:the positive charges cancel the negative charges.

+ - + - + -

+ - + - + -

power source

switch

charges cancel

Adapted from copyrighted ValpeyFisher Corp. materials

Summary of the Piezoelectric & Electromechanical Effect

● A deformation of the crystal structure (eg: squeezing it) will result in an electrical current.

● Changing the direction of deformation (eg: pulling it) will reverse the direction of the current.

● If the crystal structure is placed into an electrical field, it will deform by an amount proportional to the strength of the field.

● If the same structure is placed into an electrical field with the direction of the field reversed, the deformation will be opposite.

Adapted from copyrighted ValpeyFisher Corp. materials

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Ionic sizeCations are always smaller than the parent atom and anions are always larger than their parent atoms– outermost electrons in a cations experience a higher effective

charge than the outer electron in the neutral atom would» Na 186 pm but Na+ 116 pm

– outermost electrons in a anions experience a lower effective charge than the outer electron in the neutral atom would

Determining ionic radiiMany different ways to do this. Each gives slightly different answers. Be consistent with the source of you data when doing calculations– A good way involves measuring electron density in

crystals. Minimum in density between ions is the boundary between ions

Electron density map for NaCl

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Effect of ion chargeIsoelectronic ions get smaller as the nuclear charge goes up

Ion Radius / pm

Νa+ 116

Mg2+ 86

Al3+ 68

Ion Radius / pm

Ν3- 132

O2- 124

F- 117

Periodic trends in size

Similar to those found for atomsIncrease down a group decrease across a period assuming the ion has the same charge

Ion Radius / pm

F- 117

Cl- 167

Br- 182

I- 206

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Trends in physical propertiesDecreasing ion size and increasing ion charge favor better binding of the solid (higher lattice energy)– this tends to give increased melting and boiling points

Compound Melting point / ºC

ΚF 857

KCl 772

KBr 735

KI 685

Compound Melting point / ºC

NaF 988

MgF2 1266

AlF3 1291 sublimes

The ionic to covalent continuumIn practice, no solid compound is fully ionicCompounds containing elements with very different electronegativities tend to be more ionic

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Polarization and covalency

“Ionic” compounds tend have a considerable covalent contribution to their bonding when they contain polarizing cations– polarizing cations are cations capable of distorting

the anion’s electron cloud towards the cation

Fajan’s rules

Fajan’s rules help us determine which species will be polarizing or polarizable– Small highly charged cations are more polarizing– Large highly charged anions are more polarizable– Polarization is favored for cations that do not have

a noble gas electron configuration» Ag+, Cu+, Zn2+, Cd2+, Hg2+, Tl+ etc.

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Charge density

The charge density of a cation is a good indication of its ability to polarize an anion– Charge density = (Charge on proton x valency)/volume

For Na+, ionic radius 1.16 Å– Charge density = (1 x 1.6 x 10-19 C)/((4/3) x π x (1.16 x 10-7mm)3) = 24

C.mm-3 (this is a low charge density)

High charge density ions form compounds displaying covalent bonding

Cation chargeHigh charge ions tend to be small. So high charge ions usually have high charge densities and form compounds with covalent bonding– +1 and +2 ions often form predominantly ionic

compounds– +3 ions more covalent except for compounds with

unpolarizable anions such as F-

– +4 charge and above, bonding is usually highly covalent

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Physical effect of covalency“Ionic” solids with a significant covalent contribution to bonding show “anomalous” physical properties– may not be water soluble AgCl, CuI etc.– AlF3 MP 1290 oC, AlI3 MP 190 oC– Often compounds with significant covalency in

their bonding have low melting points» As melting does not involved breaking the compound up

into a collection of mobile individual ions

Manganese oxides example

The effect of charge density/covalency can be illustrated by looking at manganese oxidesMnO – Mn2+, charge density 84 C/mm3

– Ionic, melting point 1785 º CMn2O7 – Mn7+, charge density 1240 C/mm3

– Covalent, liquid at room temperature

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Silver halides

Non-noble gas configuration of Ag+ makes the species polarizing and many of its compounds are at least partly covalent

AgF AgCl AgBr AgIMP/ºC 435 455 430 558

All lower melting than alkali halides due to covalent contribution to bondingAgCl, AgBr and AgI are insoluble in water. AgF dissolves in water. AgF has an hard to polarize anion so it is more ionic and hence more soluble

Structure maps

The structure of a compound can be predicted based on the difference in electronegativity between the elements and the average principle

quantum number of the valence orbitals

Structure mapfor AB compounds