1

Characteristics of Solids

Bonding

Electrons in Solids

Band Theory

Defects

Classification of Solids

There are several forms solid state materials can adapt

•Single Crystal

-Preferred for characterization of structure and properties.

•Polycrystalline Powder (Highly crystalline)

-Used for characterization when single crystal can not be easily obtained, preferred for industrial production and certain applications.

•Polycrystalline Powder (Large Surface Area)

-Desirable for further reactivity and certain applications such as catalysis and electrode materials

•Amorphous (Glass)

-No long range translational order.

•Thin Film

-Widespread used in microelectronics, telecommunications, optical applications, coatings, etc.

Solid State Strength

The strength of the solid depends on the molecular forces that hold the solid together.

Ionic interactions are strong because the opposite charges resist breaking of the intermolecular bonds. For example, the melting temperature of salts is very high, typically 400 to 800C.

Some molecular interactions are strong because of the 3-D arrangement of the atoms. For example, diamonds are exceptionally hard because the solid state forms, a 3-D network such that each carbon is held close to 4 other atoms. The bonds are molecular, not ionic.

Types of Crystals

(a)ion crystal(NaCl) (b)metallic crystal (c)covalent crystal(InSb) (d)molecule crystal (solid state Ar) (e)hydrogen bond crystal(H3BO3) (f)mixed bond crystals (graphite)

According to the bonding in solids

Bonding

Electrons transfer

between atoms

Between many

atoms

Electrons sharing

two atoms

Bonding

Primary bonds strong attractions between atoms

Ionic Metal ion (+) & Nonmetallic ion ()

Covalent local sharing of electrons between atoms

Metallic global sharing of electrons by all atoms

Secondary bonds attraction forces between molecules

2

Primary Bonding

e

+

Na+

e

e e

e

e e

e

e

e e

+ e

e e

e

e e

e

e

e

F-

Ionic

+ + e

e

Covalent

e

+ + +

+ + +

+ + +

e e

e e

e e

e e

Metallic

Types of Non-Bonded (intermolecular,

van der Waals) Interactions

Dipole-dipole Ion-dipole Induced dipole Dispersion or London

Hydrogen bonding

H Cl +

H +

Cl

Dipole-dipole interactions: when two polar molecules with net dipoles come closer, one end of the dipole in one molecule will be attracted to the opposite end in the second molecule. (a). These forces are strong, but are weaker than ion-ion interactions in ionic compounds. (b). These forces decrease with distance as

3r

1

Ion-dipole interactions: When polar molecules encounter ions, the positive end of the dipole is attracted to negative ions and vice versa. The ion-dipole interactions require two species to be present, one to provide ions and another to provide dipoles.

O

H H + +

Na+

Cl-

In non-polar molecules, electrons are distributed symmetrically. This symmetry can be distorted by an ion/dipole, by inducing dipole in non-polar molecule. These forces are weak and are of short range.

e e

e

e e e

e +

e e

e

e e

e +

e

Induced dipole interactions:

Noble gases are atomic gases and do not have dipole moments or net charges. The fact that they can be liquefied suggests that forces of attraction exist between atoms of a noble gas. These forces are called Dispersion or London forces.

The distribution of electrons in an atom/molecule fluctuate over time. These fluctuations set up temporary dipoles which induce dipoles in others. The attraction between temporary dipoles is responsible for dispersion forces. These forces exist between all atoms and molecules.

Strength of dispersion forces increases with number of electrons in atoms/molecules. “Dispersion force increases down a group”. Thus, for rare gases, dispersion forces increase as Xe > Kr > Ar > Ne > He.

Dispersion or London forces:

A hydrogen atom covalently bonded to N, O, or F is attracted to the lone pair of a different atom nearby forming a hydrogen bond. Hydrogen bonding is stronger than any other non-bonded interaction, yet weaker than covalent bonds/ionic bonds

O H H

..

Hydrogen bond

Hydrogen bonding

3

Types of Non-Bonded Interactions

Interaction Energy Ion-ion interaction ~250 kJ/mol H-bonding ~20 kJ/mol Ion-dipole Dipole-dipole ~2 kJ/mol Dispersion/London <2 kJ/mol Induced dipole

Examples:

Compound/element Type Dominant Interaction NaH ionic Ion-ion ClBr covalent Dipole-dipole Rn noble gas Dispersion NH3 covalent Hydrogen bonding NH4Cl ionic Ion-ion HBr(g) covalent Dipole-dipole HF(g) covalent Hydrogen

Ion Bond and Ion Crystal

Some properties of ion crystal

Lattice energy of ion crystals

Ionic radii in crystals

Pauling’s Rules

Ion bonds with part covalent bond

Some Properties of Ionic Crystals

Relative stable and hard crystals

Poor electrical conductors (lack of free electrons)

High melting and vaporization temperatures

Transparent to visible light, but absorb strongly infrared light

Soluble in water and polar liquids!

Lattice Enthalpy

Two definitions: The lattice enthalpy change is the standard

molar enthalpy change for the following process:

If entropy considerations are neglected, the most

stable crystal structure of a given compound is the one with the highest lattice enthalpy.

Lattice Energy (U) of ionic compounds: disassemble one mole of a crystalline ionic compound at 0K into free components

o

LH

o

LHU

LH( ) ( )gas gasM X MX 0 0 (solid)

Equilibrium Distance & Cohesive Energy

)n

11(

r

eZZ

EEE

0

2

repulsiveattractive

Ep r

nr

B

r

eZZ2

total

At equilibrium:

(Erepulsive)

(Eattractive)

0

2

attractiver

eZZE

Find B and n at equilibrium:

nrepulsiver

BE

n

2

repulsiveattractiveTotalr

B

r

eZZEEE

0dr

dE Total 0rr

1n

0

2

1n

0

2

0

2

rn

eZZB0

r

nB

r

eZZ

Total energy at r0: )

n

11(

r

eZZE

0

2

rr 0

What is n? Compressibility n 9

4

Bohr-Madelung equation

Lattice Energy of Ionic Compounds: Bohr-Madelung equation：

N = 6.02x1023 mol-1

The Madelung constant (A) is independent of the ionic charges and the lattice dimensions, but is only valid for one specific structure type

If know the crystal structure, you can choose a suitable Madelung constant, and the distance between the ions ro ,

you can estimate the lattice energy of ion compound.

)n

11(

r

eZANZU

0

2

Structure Type Madelung Constant

CsCl 1.763

NaCl 1.748

ZnS (Wurtzite) 1.641

ZnS (Zinc Blende) 1.638

thermal stabilities of ionic solids

stabilities of oxidation states of cations

solubility of salts in water

calculations of electron affinity data

stabilities of “non existent” compounds

Applications of Lattice Enthalpy Calculations

The Kapustinskii Equation

Kapustinskii noticed that A /, is almost constant for all structures

ν is the number of ions in the formula unit

r0 = r++r, unit: pm

Variation in A/ with structure is partially canceled by change in ionic radii with coordination number

)r

5.341(

r

ZZ125200U

00

Variation of Ionic Radii With Coordination Number

Linus Pauling: The radius of one ion has to be fixed to a reasonable value (r(O2-) = 1.40Å ) . That value is then used to compile a set of self consistent values for all other ions.

Ionic Radii

1. Ionic radii increase on going down a group.

(Lanthanide contraction restricts the increase of heavy

ions !!)

2. Radii of equal charge ions decrease across a period

3. Ionic radii increase with increasing coordination

number (the higher its CN the bigger the ions seems to

be !!)

4. The ionic radius of a given atom decreases with

increasing charge (r(Fe2+) > r(Fe3+))

5. Cations are usually the smaller ions in a cation/anion

combination (exceptions: r(Cs+) > r(F-) ...!!!)

6. Frequently used for rationalization of structures:

“radius ratio” r(cation)/r(anion) (< 1)

Some General Trends for Ionic Radii Bonds Coord. No. Length (Å)

C-O 3 1.32

Si-O 4 1.66

Si-O 6 1.80

Ge-O 4 1.79

Ge-O 6 1.94

SnIV-O 6 2.09

PbIV-O 6 2.18

PbII-O 6 2.59

Notes: Ion radii for given element increase with coordination

number (CN) Ion radii for given element decrease with increasing

oxidation state/positive charge Radii increase going down a group Anions often bigger than cations

5

not “in touch” in touch

Limiting and Optimal Radius Ratios for Specific Coordination

Rationalization for octahedral coordination: R= radius of large ion, r=radius of small ion

414.0R

r

rR)12(

rRR2

2

145cos

rR

R

Radius Ratio Rules

If r/R < 0.414, the cation is too small and can “rattle” inside the octahedral site

If r/R > 0.414, the anions are pushed apart

Coordination Minimum r/R Linear, 2 Trigonal, 3 0.155 Tetrahedral, 4 0.225 Octahedral, 6 0.414 Cubic, 8 0.732 Close packed, 12 1.000 A simple prediction tool, but beware it doesn’t always work!

If r/R or 0.414, coordination changes:

Limiting Radius Ratios anions in the coordination polyhedron of cation are in contact with the cation and with each other

Radius Ratio Coordination No. Binary (AB)

Structure-type

r+/r- = 1 12 none known

1 > r+/r- > 0.732 8 CsCl

0.732 > r+/r- > 0.414 6 NaCl

0.414 > r+/r- > 0.225 4 ZnS

The critical ratio determined by geometrical analysis

2 <0.155

3 0.1550225

4 0.225 0.414

6 0.414 0.732

8 0.732 1.0

C.N. rC/rA

Geometry

6

Coordination Number vs Geometrical Shapes

C.N. = 3

C.N. = 4

C.N. = 6

Cubic hole

Cuboctahedral hole

Cuboctahedral and

Anti-cuboctahedral Structure

Anti-cuboctahedron Hexagonal close packing

Cuboctahedron Cubic close packing

Common Coordination Polyhedra Ceramic Crystal Structures

The ratio of ionic radii (rcation/ranion ) dictates the coordination number of anions around each cation.

As the ratio gets larger (i.e. as rcation/ranion1), the coordination number gets larger and larger.

Separation of structure types is achieved on structure diagrams but, the boundaries are complex

Conclusion Size does matter, but not necessarily in any simple way!

Structure Maps Plots of rA versus

rB with structure-type indicated

晶格能U = -Ae2/r + Be-r/m + C2/r6

－正负离子比下降，晶格能升高，直至不满足正负离子相互接触。

－ZnS-NaCl转变的离子半径比为0.32

－CsCl结构一般难以形成

(配位数从6增至8, r0将增大3%, 这相应于晶格能减少3%,修正后的CsCl晶格能曲线用虚线表示,不管 取何值,CsCl型晶格能都将小于NaCl型.)

－CsCl, CsBr, CsI结构由于Van der Waals作用增强, 为CsCl结构。

7

Separation of

structure types is

achieved on structure

diagrams but, the

boundaries are

complex

Conclusion Size

does matter, but not

necessarily in any

simple way!

Structure Maps Plots of rA versus

rB with structure-type indicated Pauling’s Rules

for Ionic Crystals

Cation environment in a polyhedron (cation-anion distance and coordination number)

Relationship between bond valence and oxidation number

Corner, edge and face sharing polyhedra

Large valence and small coordination number cations tend not to share polyhedra elements

Rule of parsimony

Linus Pauling, J. Am. Chem. Soc. 1929, 51, 1010

Deal with the energy state of the crystal structure

1st Rule

A coordinated polyhedron of anions is formed about each cation, the cation-anion distance determined by the sum of ionic radii and the coordination number by the radius ratio.

The cation-anion distance = radii

Can use r+/r to determine the

coordination number of the cation

2nd Rule

First note that the strength of an electrostatic bond = valence / CN

Cl

Cl Cl

Cl Na

Na+ in NaCl is in VI coordination

For Na+, the strength = +1 divided by 6 = + 1/6

The bond valence of each ion should be approximately equal to its oxidation state.

2nd Rule the electrostatic valence principle

+ 1/6

+ 1/6

+ 1/6

+ 1/6

Na

Na

Na

Na Cl

-

An ionic structure will be stable to the extent that the sum of the strengths of electrostatic bonds that reach an anion from adjacent cations = the charge of that anion 6( + 1/6 ) = +1 (sum from Na’s) charge of Cl = 1 These charges are equal in magnitude, so the structure is stable

对于理想的CaTiO3结构，

Ca2+与12个O2-配位，

sCa= 2/12 = 1/6

Ti4+与6个O2-配位，

sTi = 4/6 = 2/3

O2-周围有4个Ca2+和2个Ti4+

zO = 4 sCa + 2 sTi = 2

8

In [SiO4], strengths of Si-O=1, Strength=2, stable

If 1 Al replace 1 Si, Strength = 1+3/4=1.75, unstable

If 2 Al replace 2 Si, Strength =3/4+3/4=1.5, very unstable

2nd Rule the electrostatic valence principle The presence of shared edges, and particularly shared faces decreases the stability of a structure. This is particularly true for cations with large valences and small CN.

3rd Rule

Polyhedral Linking

The stability of structures with different types of polyhedral linking is vertex-sharing > edge-sharing > face-sharing

effect is largest for cations with high charge and low coordination number

especially large when r+/r- approaches the lower limit of the polyhedral stability

4th Rule

In a crystal containing different cations those with large valence and small CN tend not to share polyhedron elements with each other.

An extension of Rule 3

Si4+ in IV coordination is very unlikely to share edges or faces

Olivine (Mg2SiO4)

2

1hcp, Oct. & Tetr. Hole filled) 8

1

Mg Coord.Octahedron

Si Coord. Tetrahedron

In a crystal containing

different cations those with

large valence and small CN

tend not to share polyhedron

elements with each other.

Si4+ in IV coordination is very unlikely to share edges or faces MgO6 octahedron shared edges with SiO4 tetrahedrons

5th Rule Rule of Parsimony

The number of chemically different coordination environments for a given ion in a crystal tends to be small.

5th Rule

9

Polarization of Ion

Polarization of an ion is the distortion of the electron cloud of the anion, due to the influence of the nearby cation.

+ –

+ – Perfect model of ionic compound

Electron cloud of anion is attracted towards the cation and result in higher electron density between the ions, stronger bond is resulted.

Ionic compound with polarization of ion Not a purely ionic compound

Polarizing Power of Cation

The ability to distort the electron distribution of adjacent ions or atoms.

The polarizing power of cation is favored by higher charge and smaller size to have a higher charge density

Al3+ > Mg2+ > Na+

Li+ > Na+

Polarizability of Anion

The ease of the electron cloud being distorted by the influence of adjacent ion or atom.

The more polarizable anion would have higher charge and larger size

S2- > O2-

S2- > Cl-

Polarization Effect on AB2 Structures (A = Transition Metal)

Same r+/r, larger size of anion induced more polarizable anion part of electrons are active in the whole crystal property of semiconductor and metal, such as FeS2

smaller cation has more polarizing power ion crystals to molecule crystals along vertical axis.

Polarization

Increasing Polarization in bonding

low-dimensionality layers/chains

Covalent Crystals Held Together by Covalent Bonds

Share electrons lead to strongest bonds

Some Properties:

- Very hard.

- High melting points.

- Insulators/semiconductors.

10

Covalent Bonding

Review some important features of covalent bonding:

•Basic Concepts of Molecular Orbital Theory

half-filled orbital

hybrid orbital

atomic orbital & molecular orbital

bonding (symmetric) molecular orbital

antibonding (antisymmetric) molecular orbital

What is the Largest Molecule in the world?

Not polymer aggregates The largest molecules that have ever been found are

Diamonds. The largest of all was the so-called Cullinan diamond, found on Jan. 25, 1905, in South Africa and sporting a weight of 621.6 g (3106 carats).

The largest man-made synthetic molecule is an artificial diamond of 38.4 carats, the growth of which required 25 days.

The hardest element with the highest thermal conductivity of >2000 W/mK (the best metallic thermal conductor is Ag (429 W/mK).

Molecular Crystals held together by van der Waals bonds

weak … but everywhere.

Polar molecules electric dipole moment

Polar molecules attract each other:

Van der Waals Attraction between Non-polar molecules:

On the average, non-polar molecules are symmetric distributions, but at any moment the distributions are asymmetric. The fluctuations in the charge distributions of nearby molecules lead to an attractive force, given also by:

6attractiver

1~U

C C Pradzynski et al. ，Science，2012，337，1529-1532

水结冰后，分子集群间的距离被拉大。

Photoionization scheme, origin of action

effect and (right) comparison of IR spectra

of Na doped with phenol-doped (H2O)19

clusters.

Evolution of size-selected (H2O)n IR

spectra from n = 85 to 475.

A Fully Size-Resolved Perspective on the Crystallization of Water Clusters

275个水分子，大小在1毫微米到3毫微米间的微小冰簇——碎冰晶的极限体积

制造一个最小的冰晶需要多少个水分子？ Metallic Bond and Metallic Crystal

Free Electrons Gas Model in Metals

The Nearly-Free-Electron Model in Metals

Distribution of Electrons in Metals

Band Theory

11

Like copper and gold, most of the known elements are metals.

Metals are solids which require extra discussion to explain their special properties:

Ductility Shiny surface High electrical conductivity High heat conductivity

Superductility of Nanoscaled Copper Metallic Crystals Held Together by Metallic Bonds

Some Properties: Weaker than covalent

or ionic crystals. High melting points. Electrical and thermal

conductors.

A gas of negatively charged free electrons holds metal ions together.

Free Electron Gas Model

Drude and Lorentz: The valence electrons in metals are loosely bound to their atoms and form a gas of particles that may wander through the crystal and conduct electricity.

Many properties of metals can be

understood by treating electrons in metals as free-electron gas.

Metallic Sea of Electrons

High electrical conductivity

High thermal conductivity

High reflectivity of visible light

High malleability and ductility

+ + + + + + + + + + + + + +

+ + + + + + + + + + + + + +

Valence electrons are not bonded to any particular atom and are free to move about in the solid.

12

Hall Effect If an electric current flows through a conductor in a magnetic field, the magnetic field exerts a transverse force on the moving charge carriers which tends to push them to one side of the conductor. This is most evident in a thin flat conductor as illustrated. A buildup of charge at the sides of the conductors will balance this magnetic influence, producing a measurable voltage between the two sides of the conductor. The presence of this measurable transverse voltage is called the Hall effect after E. H. Hall who discovered it in 1879.

When electrons flow without magnetic field...

t

d

conductor slice

+ _

I I

When the magnetic field is turned on ...

B-field

I qBv

As time goes by...

I

qE

qBv = qE low

potential

high potential

Finally...

B-field

I

V H

Hall Effect

Lorentz force likes to deflect Ix

However, E-field is set up which balances Lorentz force

Balance occurs when Ey = Bzvx

is defined as Hall coefficient Nq

1R H

xzHyxzyx

xxx IBREIBNq

1E

Nq

IvNqvI

xz

y

HIB

ER

研究霍尔场的意义在于可以通过测量霍尔场的方向来确定试样中载流子的符号和密度

13

the Nearly Free Electron Model

Arnold Sommerfeld modified free electron

gas model: the Nearly-Free-Electron Model

To better account for the electrical properties of different materials, we need to consider the interaction that arises between the electrons and the crystal structure. One model that attempts to do this in a simple way is the nearly-free electron model. To understand the key features of this, we need to recall the diffraction of waves that is generated by a crystal structure.

In the nearly-free electron model it is therefore considered that the main effect of the electron-crystal interaction is to diffract electrons whenever the Bragg condition is satisfied

Electrons can exhibit a variety of wave effects including diffraction，it accords with De Broglie relation: =h/m

If a particle is confined into a rectangular volume, the same kind of process can be applied to a three-dimensional "particle in a box", and the same kind of energy contribution is made from each dimension. The energies for a three-dimensional box are

Particle in a Box The idealized situation of

a particle in a box with infinitely high walls is an application of the Schrodinger equation which yields some insights into particle confinement. The wave function must be zero at the walls and the solution for the wave function yields just sine waves.

2

2

22

m8

hm

2

1E

0 L x

2

2

1mL8

hE

2

22

2mL8

h2E

2

22

3mL8

h3E

2

22

4mL8

h4E

2

2

2

8

m

hE

2m

2

1E

m

h

2

0 L x 2

2

1

2

22

2

2

22

3

2

22

4

mL8

hE

L

2L2

mL8

h2E

L

22L

mL8

h3E

L

32L

3

2

mL8

h4E

L

42L

2

1

De Broglie Relation

Fermi Energy (Fermi Level)

"Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature.

Fermi energy (or Fermi level): highest occupied energy level in the ground state (T=0K) of the N electron system. At T=0K, the N electron system is in the ground state: The electrons occupied all the energy states up to the Fermi level.

Enrico Fermi

(1901-1954)

Fermi Energy Distribution Function

The distribution function f(E) is the probability that a particle is in energy state E.

The Fermi function f(E) gives the probability that a given available electron energy state will be occupied at a given temperature. Some electrons excited above Fermi level at T > 0K, Fermi distribution.

The basic nature of this function dictates that at ordinary temperatures, most of the levels up to the Fermi level EF are filled, and relatively few electrons have energies above the Fermi level.

1e

1)E(f

kT/)EE( F

The possibility that a particle will have the energy E is:

Origin and Temperature Dependence of Electronic Conductivity in Metals

Electrons with energies close to the Fermi level can easily be promoted to nearby empty levels.

they are mobile and can easily move through the solid in an electric potential gradient (free electrons)

At elevated temperatures, thermal vibrations of the atoms reduce the mobility of the free electrons. As a consequence, the electrical conductivity decreases with increasing temperature

14

Basic Concepts of Band Theory

Electrons in crystals are arranged in energy bands. These bands are separated by regions in which no

electron states exist. These regions are called energy gaps or band gaps.

The lowest empty band is the conduction band.The highest band with occupied electron levels is the valence band.

The highest occupied band can be completely or partly filled depending on the ratio between the number of electron levels available in the band and the number of valence electrons given by each atom in the crystal.

The occupancy of the valence band and the size of the gap between the valence and the conduction band will determine the conductive, semiconductive or insulator character of a crystal.

Classification of Solids Depending on Their Structures

Metal: partly filled valence band Semiconductor: completely filled valence band, small

energy gap (Eg) between valence and conduction band Insulator: completely filled valence band, large energy

gap (Eg) between valence and conduction band Semi-metal: partly filled valence band due to

overlapping of conduction and valence band

Two different pictures of the same problem

Fermi Energy Distribution Function of Semiconductor

The illustration below shows the implications of the Fermi function for the electrical conductivity of a semiconductor. The band theory of solids gives the picture that there is a sizable gap between the Fermi level and the conduction band of the semiconductor. At higher temperatures, a larger fraction of the electrons can bridge this gap and participate in electrical conduction.

Bands in Metals Example: Aluminum

Consider Al(1s22s22p63s23p1) Core atomic orbitals:

contracted, poor overlap, narrow bands, filled

Valence atomic orbitals: diffuse, partially occupied, good overlap, wide bands

Electrons in the partially filled VB are responsible for electrical and thermal conductivity, optical (reflectivity) and magnetic properties

Metals: Free Electron Model

Many physical properties of metals can be explained in terms of a free electron model. It assumes a lattice of positive ion cores surrounded by a cloud of electrons that can move freely through the solid. In this approximation, we neglect the electron-electron and electron-core interaction.

The free electron model allows us to understand electrical and thermal conductivity.