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Chapter 43

Molecules and Solids

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Molecular Bonds

Introduction The bonding mechanisms in a molecule

are fundamentally due to electric forces

The forces are related to a potential

energy function

A stable molecule would be expected at

a configuration for which the potential

energy function has its minimum value

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Molecular Bonds Feature 1 The force between atoms is repulsive at

very small separation distances This repulsion is partially electrostatic and

partially due to the exclusion principle Due to the exclusion principle, some

electrons in overlapping shells are forced

into higher energy states The energy of the system increases as if a

repulsive force existed between the atoms

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Molecular Bonds Feature 2 The force between the atoms is

attractive at larger distances The attractive force (for many molecules) is

due to the dipole-dipole interactionbetween charge distributions within theatoms of the molecules

The electric fields of two dipoles willinteract, resulting in a force between thedipoles

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Potential Energy Function The potential energy for a system of two

atoms can be expressed in the form

ris the internuclear separation distance

m and n are small integers A is associated with the attractive force B is associated with the repulsive force

( )n m

A BU rr r

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Potential Energy Function,

Graph At large separations,

the slope of the curve ispositive Corresponds to a net

attractive force

At the equilibriumseparation distance, theattractive and repulsive

forces just balance At this point the potential

energy is a minimum The slope is zero

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Molecular Bonds Types Simplified models of molecular bonding

include Ionic

Covalent

van der Waals

Hydrogen

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Ionic Bonding Ionic bonding occurs when two atoms

combine in such a way that one or more

outer electrons are transferred from oneatom to the other

Ionic bonds are fundamentally caused

by the Coulomb attraction betweenoppositely charged ions

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Ionic Bonding, cont. When an electron makes a transition

from the E= 0 to a negative energy

state, energy is released The amount of this energy is called the

electron affinity of the atom

The dissociation energy is the amountof energy needed to break themolecular bonds and produce neutralatoms

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Ionic Bonding, NaCl Example

The graph shows the total energy of the molecule

vs the internuclear distance

The minimum energy is at the equilibrium

separation distance

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Ionic Bonding,final The energy of the molecule is lower

than the energy of the system of two

neutral atoms It is said that it is energetically

favorable for the molecule to form The system of two atoms can reduce its

energy by transferring energy out of the

system and forming a molecule

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Covalent Bonding A covalent bond between two atoms is

one in which electrons supplied by

either one or both atoms are shared bythe two atoms

Covalent bonds can be described interms of atomic wave functions

The example will be two hydrogenatoms forming H2

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Wave Function Two Atoms

Far Apart Each atom has a

wave function

There is little overlap

between the wavefunctions of the two

atoms

1 31( ) or as

o

r ea

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Wave Function Molecule The two atoms are

brought close together

The wave functionsoverlap and form the

compound wave shown

The probability

amplitude is larger

between the atoms than

on either side

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Active Figure 43.3

(SLIDESHOW MODE ONLY)

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Covalent Bonding, Final The probability is higher that the electrons

associated with the atoms will be located

between them This can be modeled as if there were a fixed

negative charge between the atoms, exerting

attractive Coulomb forces on both nuclei

The result is an overall attractive force

between the atoms, resulting in the covalent

bond

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Van der Waals Bonding Two neutral molecules are attracted to each

other by weak electrostatic forces called vander Waals forces Atoms that do not form ionic or covalent bonds are

also attracted to each other by van der Waalsforces

The van der Waals force is due to the fact

that the molecule has a charge distributionwith positive and negative centers at differentpositions in the molecule

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Van der Waals Bonding, cont. As a result of this charge distribution,

the molecule may act as an electric

dipole Because of the dipole electric fields, two

molecules can interact such that there

is an attractive force between them Remember, this occurs even though the

molecules are electrically neutral

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Types of Van der Waals

Forces Dipole-dipole force

An interaction between two molecules each

having a permanent electric dipole moment Dipole-induced dipole force

A polar molecule having a permanent

dipole moment induces a dipole moment ina nonpolar molecule

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Types of Van der Waals

Forces, cont. Dispersion force

An attractive force occurs between two nonpolarmolecules

The interaction results from the fact that, althoughthe average dipole moment of a nonpolarmolecule is zero, the average of the square of thedipole moment is nonzero because of charge

fluctuations The two nonpolar molecules tend to have dipole

moments that are correlated in time so as toproduce van der Waals forces

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Hydrogen Bonding In addition to covalent bonds, a

hydrogen atom in a molecule can also

form a hydrogen bond Using water (H2O) as an example

There are two covalent bonds in themolecule

The electrons from the hydrogen atoms aremore likely to be found near the oxygenatom than the hydrogen atoms

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Hydrogen Bonding

H2O Example, cont.

This leaves essentially bare protons at

the positions of the hydrogen atoms

The negative end of another molecule

can come very close to the proton

This bond is strong enough to form a

solid crystalline structure

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Hydrogen Bonding, Final The hydrogen bond

is relatively weakcompared with otherelectrical bonds

Hydrogen bonding isa critical mechanismfor the linking ofbiological moleculesand polymers

DNA is an example

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Energy States of Molecules The energy of a molecule (assume one

in a gaseous phase) can be divided into

four categories Electronic energy

Due to the interactions between the moleculeselectrons and nuclei

Translational energy Due to the motion of the molecules center of

mass through space

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Energy States of Molecules, 2 Categories, cont.

Rotational energy

Due to the rotation of the molecule about itscenter of mass

Vibrational energy Due to the vibration of the molecules constituent

atoms The total energy of the molecule is the

sum of the energies in these categories: E= Eel + Etrans + Erot + Evib

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Spectra of Molecules The translational energy is unrelated to

internal structure and therefore

unimportant to the interpretation of themolecules spectrum

By analyzing its rotational and

vibrational energy states, significantinformation about molecular spectra

can be found

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Rotational Motion of

Molecules A diatomic model will be

used, but the same

ideas can be extended

to polyatomic molecules A diatomic molecule

aligned along anxaxis

has only two rotational

degrees of freedom Corresponding to

rotations about the yand

xaxes

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Rotational Motion of

Molecules, Energy The rotational energy is given by

I is the moment of inertia of the molecule

is called the reduced mass of the molecule

2

rot

1

2

IE

2 21 2

1 2

Im m

rr

m m

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Rotational Motion of Molecules,

Angular Momentum Classically, the value of the molecules

angular momentum can have any value

L = I Quantum mechanics restricts the values

of the angular momentum to

J is an integer called the rotationalquantum number

1 0 1 2 , , ,L J J J h K

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Rotational Kinetic Energy of

Molecules, Allowed Levels The allowed values are

The rotational kinetic energy isquantized and depends on its moment

of inertia As Jincreases, the states become

farther apart

2

rot 1 0 1 22 , , ,IE J J J

hK

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Allowed Levels, cont. For most molecules,

transitions result in radiationthat is in the microwave

region Allowed transitions are given

by the condition

Jis the number of the higher

state

2 2

rot 24

1 2 3

I I

, , ,

h

E J J

J

h

K

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Active Figure 43.5


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Sample Transitions

CO Example

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Vibrational Motion of

Molecules A molecule can be

considered to be aflexible structurewhere the atoms arebonded by effectivesprings

Therefore, themolecule can bemodeled as a simpleharmonic oscillator

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Vibrational Motion of

Molecules, Potential Energy A plot of the

potential energyfunction

ro is the equilibrium

atomic separation For separations

close to ro, theshape closelyresembles aparabola

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Vibrational Energy Classical mechanics describes the frequency

of vibration of a simple harmonic oscillator

Quantum mechanics predicts that a moleculewill vibrate in quantized states

The vibrational and quantized vibrational

energy can be altered if the molecule

acquires energy of the proper value to cause

a transition between quantized states

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Vibrational Energy, cont. The allowed vibrational energies are

vis an integer called the vibrational quantum

number

When v= 0, the molecules ground state

energy is h The accompanying vibration is always present,

even if the molecule is not excited

vib

10 1 2

2 , , ,E v h v

K

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Vibrational Energy, Final The allowed vibrational

energies can be expressedas

Allowed transitions arev = 1

The energy betweenstates is Evib = h

vib

1

2 2

0 1 2 , , ,

h kE v

v

K

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Some Values for Diatomic

Molecules

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Molecular Spectra In general, a molecule vibrates and

rotates simultaneously

To a first approximation, these motionsare independent of each other

The total energy is the sum of the

energies for these two motions:

21

12 2

I

E v h J J

h

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Molecular Energy-Level

Diagram For each allowed state ofv,

there is a complete set of

levels corresponding to the

allowed values ofJ The energy separation

between successive

rotational levels is much

smaller than betweensuccessive vibrational levels

Most molecules at ordinary

temperatures vibrate at v= 0

level

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Molecular Absorption

Spectrum

The spectrum consists of two groups of lines

One group to the right of center satisfying the selection rulesJ= +1 and v= +1

The other group to the left of center satisfying the selection

rules J= -1 and v= +1

Adjacent lines are separated by /2I

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Active Figure 43.8


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Absorption Spectrum of HCl

It fits the predicted pattern very well

A peculiarity shows, each line is split into a doublet Two chlorine isotopes were present in the same

sample

Because of their different masses, different Is are

present in the sample

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Intensity of Spectral Lines The intensity is determined by the

product of two functions ofJ The first function is the number of available

states for a given value ofJ There are 2J+ 1 states available

The second function is the Boltzmannfactor

2B( 1)/(2 )IJ J k T

on n e

h

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Intensity of Spectral Lines,

cont Taking into account both factors by

multiplying them,

The 2J+ 1 term increases with J

The exponential term decreases

This is in good agreement with theobserved envelope of the spectral lines

2

B( 1)/(2 )2 1 II J J k T J e h

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Bonding in Solids Bonds in solids can be of the following

types Ionic Covalent

Metallic

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Ionic Bonds in Solids The dominant interaction between ions

is through the Coulomb force

Many crystals are formed by ionicbonding

Multiple interactions occur among

nearest-neighbor atoms

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Ionic Bonds in Solids, 2 The net effect of all the interactions is a

negative electric potential energy

is a dimensionless number known as the

Madelung constant The value of depends only on the

crystalline structure of the solid

2

attractive e

eUk

r

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Ionic Bonds, NaCl Example

The crystalline structure is shown (a) Each positive sodium ion is surrounded by six negative

chlorine ions (b) Each chlorine ion is surrounded by six sodium ions (c) = 1.747 6 for the NaCl structure

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Total Energy in a

Crystalline Solid As the constituent ions of a crystal are

brought close together, a repulsive

force exists The potential energy term B/rm accounts

for this repulsive force

This repulsive force is a result ofelectrostatic forces and the exclusion

principle

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Total Energy in a Crystalline

Solid, cont The total potential energy

of the crystal is

The minimum value, Uo, is

called the ionic cohesive

energy of the solid It represents the energy

needed to separate the solid

into a collection of isolated

positive and negative ions

m

2

etotalr

B

r

ekU +=

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Properties of Ionic Crystals They form relatively stable, hard

crystals

They are poor electrical conductors They contain no free electrons

Each electron is bound tightly to one of the

ions They have high melting points

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More Properties of Ionic

Crystals They are transparent to visible radiation, but

absorb strongly in the infrared region

The shells formed by the electrons are so tightlybound that visible light does not possess sufficient

energy to promote electrons to the next allowed

shell

Infrared is absorbed strongly because the

vibrations of the ions have natural resonant

frequencies in the low-energy infrared region

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Final Properties of Ionic

Crystals Many are quite soluble in polar liquids

Water is an example of a polar liquid

The polar solvent molecules exert anattractive electric force on the charged ions

This breaks the ionic bonds and dissolves

the solid

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Properties of Solids with

Covalent Bonds Properties include

Usually very hard

Due to the large atomic cohesive energies High bond energies

High melting points

Good electrical conductors

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Cohesive Energies for Some

Covalent Solids

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Covalent Bond Example

Diamond

Each carbon atom in a diamond crystal iscovalently bonded to four other carbon atoms

This forms a tetrahedral structure

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Another Carbon Example --

Buckyballs

Carbon can form

many different

structures The large hollow

structure is called

buckminsterfullerene Also known as a

buckyball

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Metallic Solids

Metallic bonds are generally weaker

than ionic or covalent bonds

The outer electrons in the atoms of ametal are relatively free to move

through the material

The number of such mobile electrons ina metal is large

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Metallic Solids, cont.

The metallic structure can

be viewed as a sea or

gas of nearly free

electrons surrounding alattice of positive ions

The bonding mechanism

is the attractive force

between the entirecollection of positive ions

and the electron gas

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Properties of Metallic Solids

Light interacts strongly with the free

electrons in metals

Visible light is absorbed and re-emittedquite close to the surface

This accounts for the shiny nature of metal

surfaces

High electrical conductivity

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More Properties of Metallic

Solids

The metallic bond is nondirectional This allows many different types of metal

atoms to be dissolved in a host metal invarying amounts The resulting solid solutions, oralloys, may

be designed to have particular properties

Metals tend to bend when stretched Due to the bonding being between all of

the electrons and all of the positive ions

f

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Free-Electron Theory of

Metals

The quantum-based free-electron theory of

electrical conduction in metals takes into

account the wave nature of the electrons The model is that the outer-shell electrons

are free to move through the metal, but are

trapped within a three-dimensional box

formed by the metal surfaces Each electron can be represented as a

particle in a box

F i Di Di ib i

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Fermi-Dirac Distribution

Function

Applying statistical physics to a

collection of particles can relate

microscopic properties to macroscopicproperties

For electrons, quantum statistics

requires that each state of the systemcan be occupied by only two electrons

F i Di Di t ib ti

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Function, cont.

The probability that a particular statehaving energy Eis occupied by one of

the electrons in a solid is given by

(E) is called the Fermi-Diracdistribution function EF is called the Fermi energy

B( )

1( )

1

FE E k T E

e

F i Di Di t ib ti

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Function at T= 0

At T= 0, all states

having energies less

than the Fermienergy are occupied

All states having

energies greater

than the Fermienergy are vacant

F i Di Di t ib ti

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Function at T> 0

As Tincreases, the

distribution rounds

off slightly States near and

below EF lose

population

States near and

above EF gain

population

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Active Figure 43.15


El t P ti l i

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Electrons as a Particle in a

Three-Dimensional Box

The energy levels for the electrons are

very close together

The density-of-states function givesthe number of allowed states per unit

volume that have energies between E

and dE:

F B

3 2 1 2

( )3

8 2( )

1e

E E k T

m E dE g E dE

h e

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Fermi Energy at T= 0 K

The Fermi energy at T= 0 K is

The order of magnitude of the Fermi

energy for metals is about 5 eV

The average energy of a free electron in

a metal at 0 K is Eav = (3/5) EF

2 32

F

3

(0) 2 8e

e

h n

E m

F i E i f S

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Fermi Energies for Some

Metals

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Wave Functions of Solids

To make the model of a metal more

complete, the contributions of the

parent atoms that form the crystal mustbe incorporated

Two wave functions are valid for an

atom with atomic numberA and asingle s electron outside a closed shell:

( ) ( ) ( ) ( ) o oZr na Zr nas s r A r e r A r e

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Combined Wave Functions

The wave functionscan combine in thevarious ways shown

s- + s

- is equivalent to

s+ + s

+

These two possiblecombinations of wavefunctions representtwo possible states ofthe two-atom system

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Splitting of Energy Levels The states are split into

two energy levels due tothe two ways of combiningthe wave functions

The energy difference isrelatively small, so the twostates are close togetheron an energy scale

For large values ofr, theelectron clouds do notoverlap and there is nosplitting of the energylevel

S litti f E

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Splitting of Energy

Levels, cont.

As the number ofatoms increases,the number of

combinations inwhich the wavefunctions combineincreases

Each combinationcorresponds to adifferent energylevel

Splitting of Energy

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Splitting of Energy

Levels, final

When this splitting is

extended to the large

number of atoms

present in a solid, thereis a large number of

levels of varying energy

These levels are so

closely spaced they canbe thought of as a band

of energy levels

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Energy Bands in a Crystal

In general, a crystalline solidwill have a large number ofallowed energy bands

The white areas representenergy gaps, correspondingto forbidden energies

Some bands exhibit anoverlap

Blue represents filled bandsand gold represents emptybands in this example ofsodium

Electrical Conduction

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Electrical Conduction

Classes of Materials

Good electrical conductors contain a highdensity of free charge carriers

The density of charge carriers in an insulatoris nearly zero

Semiconductors are materials with a chargedensity between those of insulators andconductors

These classes can be discussed in terms of amodel based on energy bands

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Metals

To be a good conductor, the charge carriersin a material must be free to move inresponse to an electric field We will consider electrons as the charge carriers

The motion of electrons in response to anelectric field represents an increase in theenergy of the system

When an electric field is applied to aconductor, the electrons move up to anavailable higher energy state

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Metals Energy Bands

At T= 0, the Fermi energylies in the middle of the band All levels below EF are filled

and those above are empty If a potential difference is

applied to the metal,electrons having energiesnearEF require only a small

amount of additional energyfrom the applied field toreach nearby higher energylevels

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Metals As Good Conductors

The electrons in a metal experiencing

only a small applied electric field are

free to move because there are manyempty levels available close to the

occupied energy level

This shows that metals are goodconductors

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Insulators

There are no available states that lie close inenergy into which electrons can move upwardin response to an electric field

Although an insulator has many vacant statesin the conduction band, these states areseparated from the filled band by a largeenergy gap

Only a few electrons can occupy the higherstates, so the overall electrical conductivity isvery small

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Insulator Energy Bands

The valence band isfilled and the conductionband is empty at T= 0

The Fermi energy liessomewhere in theenergy gap

At room temperature,very few electrons

would be thermallyexcited into theconduction band

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Semiconductors

The band structure

of a semiconductor

is like that of an

insulator with a

smaller energy gap

Typical energy gap

values are shown inthe table

Semiconductors

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Semiconductors

Energy Bands

Appreciable numbers

of electrons are

thermally excited into

the conduction band

A small applied

potential difference can

easily raise the energyof the electrons into the

conduction band

Semiconductors

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Semiconductors

Movement of Charges

Charge carriers in asemiconductor canbe positive,

negative, or both When an electron

moves into theconduction band, it

leaves behind avacant site, called ahole

Semiconductors

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Semiconductors

Movement of Charges, cont.

The holes act as charge carriers Electrons can transfer into a hole, leaving

another hole at its original site

The net effect can be viewed as theholes migrating through the material inthe direction opposite the direction of

the electrons The hole behaves as if it were a particle

with charge +e

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Intrinsic Semiconductors

A pure semiconductor material

containing only one element is called an

intrinsic semiconductor It will have equal numbers of conduction

electrons and holes

Such combinations of charges are calledelectron-hole pairs

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Doped Semiconductors

Impurities can be added to asemiconductor

This process is called doping Doping

Modifies the band structure of thesemiconductor

Modifies its resistivity Can be used to control the conductivity of

the semiconductor

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n-Type Semiconductors

An impurity can addan electron to thestructure

This impurity would bereferred to as a donoratom

Semiconductorsdoped with donoratoms are called n-typesemiconductors

n-Type Semiconductors

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n-Type Semiconductors,

Energy Levels

The energy level ofthe extra electron is

just below the

conduction band The electron of the

donor atom canmove into the

conduction band asa result of a smallamount of energy

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p-Type Semiconductors

An impurity can add a holeto the structure This is an electron

deficiency

This impurity would bereferred to as a acceptoratom

Semiconductors doped

with acceptor atoms arecalled p-typesemiconductors

p-Type Semiconductors

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p-Type Semiconductors,

Energy Levels

The energy level of thehole is just above thevalence band

An electron from thevalence band can fill thehole with an addition of asmall amount of energy

A hole is left behind in

the valance band This hole can carry

current in the presence ofan electric field

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Extrinsic Semiconductors

When conduction in a

semiconductor is the result of

acceptor or donor impurities, thematerial is called an extrinsic

semiconductor

Doping densities range from 1013

to1019 cm-3

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Semiconductor Devices

Many electronic devices are based on

semiconductors

These devices include Junction diode

Light-emitting and light-absorbing diodes

Transistor Integrated Circuit

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The Junction Diode

Ap-type semiconductor is joined to an

n-type

This forms a p-n junction Ajunction diode is a device based on

a singlep-n junction

The role of the diode is to pass currentin one direction, but not the other

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The Junction Diode, 2

The junction has three

distinct regions ap region

an n region a depletion region

The depletion region is

caused by the diffusion of

electrons to fill holes This can be modeled as if

the holes being filled were

diffusing to the n region

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Because the two sides of the depletionregion each carry a net charge, aninternal electric field exists in thedepletion region

This internal field creates an internalpotential difference that prevents further

diffusion and ensures zero current inthe junction when no potentialdifference is applied

The Junction Diode, 3

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Junction Diode, Biasing

A diode is forward biasedwhen thep side isconnected to the positive terminal of a battery This decreases the internal potential difference

which results in a current that increasesexponentially

A diode is reverse biasedwhen the n side isconnected to the positive terminal of a battery

This increases the internal potential difference andresults in a very small current that quickly reachesa saturation value

Junction Diode:

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Junction Diode:

I-VCharacteristics

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LEDs and Light Absorption

Light emission and absorption in semiconductors issimilar to that in gaseous atoms, with the energy bands ofthe semiconductor taken into account

An electron in the conduction band can recombine with ahole in the valance band and emit a photon

An electron in the valance band can absorb a photon andbe promoted to the conduction band, leaving behind a

hole

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Transistors

A transistor is formed from twop-n

junctions

A narrow n region sandwiched betweentwop regions or a narrowp region between

two n regions

The transistor can be used as An amplifier

A switch

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Integrated Circuits

An integrated circuit is a collection ofinterconnected transistors, diodes,resistors and capacitors fabricated on asingle piece of silicon known as a chip

Integrated circuits Solved the interconnectedness problem

posed by transistors Possess the advantages of miniaturization

and fast response

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Superconductivity

A superconductor expels magnetic

fields from its interior by forming surface

currents Surface currents induced on the

superconductors surface produce a

magnetic field that exactly cancels theexternally applied field

Superconductivity and

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Superconductivity and

Cooper Pairs

Two electrons are bound into a Cooper pair

when they interact via distortions in the array

of lattice atoms so that there is a net

attractive force between them

Cooper pairs act like bosons and do not obey

the exclusion principle

The entire collection of Cooper pairs in ametal can be described by a single wave

function

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Superconductivity, cont.

Under the action of an applied electric field,

the Cooper pairs experience an electric force

and move through the metal

There is no resistance to the movement of

the Cooper pairs They are in the lowest possible energy state

There are no energy states above that of theCooper pairs because of the energy gap

Superconductivity -

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Superconductivity

Critical Temperatures

A new family ofcompounds was found thatwas superconducting at

high temperatures The critical temperature is

the temperature at whichthe electrical resistance of

the material decreases tovirtually zero

Critical temperatures for

pc chapter 43

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