Chapter 43

Molecules and Solids

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

Features of Molecular Bonds

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

The force between the atoms is attractive at larger distances

Potential Energy Function

The potential energy for a system of two atoms can be expressed in the form

r is 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 r

r r

Potential Energy Function, Graph At large separations, the

slope of the curve is positive Corresponds to a net

attractive force At the equilibrium

separation distance, the attractive and repulsive forces just balance At this point the potential

energy is a minimum The slope is zero

Molecular Bonds – Types

Simplified models of molecular bonding include Ionic Covalent van der Waals Hydrogen

Ionic Bonding

Ionic bonding occurs when two atoms combine in such a way that one or more outer electrons are transferred from one atom to the other

Ionic bonds are fundamentally caused by the Coulomb attraction between oppositely charged ions

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 amount of

energy needed to break the molecular bonds and produce neutral atoms

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

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

Covalent Bonding

A covalent bond between two atoms is one in which electrons supplied by either one or both atoms are shared by the two atoms

Covalent bonds can be described in terms of atomic wave functions

The example will be two hydrogen atoms forming H2

Wave Function – Two Atoms Far Apart

Each atom has a wave function

There is little overlap between the wave functions of the two atoms when they are far away from each other

1 3

1( ) or a

s

o

ψ r eπa

Wave Function – Molecule

The two atoms are brought close together

The wave functions overlap and form the compound wave shown

The probability amplitude is larger between the atoms than on either side

Active Figure 43.3

Use the active figure to move the individual wave functions

Observe the composite wave function

PLAYACTIVE FIGURE

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

Van der Waals Bonding

Two neutral molecules are attracted to each other by weak electrostatic forces called van der Waals forces Atoms that do not form ionic or covalent bonds

are also attracted to each other by van der Waals forces

The van der Waals force is due to the fact that the molecule has a charge distribution with positive and negative centers at different positions in the molecule

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

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 in a nonpolar molecule

Types of Van der Waals Forces, cont.

Dispersion force An attractive force occurs between two nonpolar

molecules The interaction results from the fact that, although

the average dipole moment of a nonpolar molecule is zero, the average of the square of the dipole moment is nonzero because of charge fluctuations

The two nonpolar molecules tend to have dipole moments that are correlated in time so as to produce van der Waals forces

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 the molecule The electrons from the hydrogen atoms are more

likely to be found near the oxygen atom than the hydrogen atoms

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

Hydrogen Bonding, Final

The hydrogen bond is relatively weak compared with other electrical bonds

Hydrogen bonding is a critical mechanism for the linking of biological molecules and polymers

DNA is an example

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 molecule’s electrons and nuclei

Translational energy Due to the motion of the molecule’s center of mass

through space

Energy States of Molecules, 2

Categories, cont. Rotational energy

Due to the rotation of the molecule about its center of mass

Vibrational energy Due to the vibration of the molecule’s constituent

atoms The total energy of the molecule is the sum of

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

Spectra of Molecules

The translational energy is unrelated to internal structure and therefore unimportant to the interpretation of the molecule’s spectrum

By analyzing its rotational and vibrational energy states, significant information about molecular spectra can be found

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 an x axis has only two rotational degrees of freedom Corresponding to rotations

about the y and x axes

Rotational Motion of Molecules, Energy

The rotational energy is given by

I is the moment of inertia of the molecule

µ is called the reduced mass of the molecule

2rot

1

2IE ω

2 21 2

1 2

Im m

r μrm m

Rotational Motion of Molecules, Angular Momentum

Classically, the value of the molecule’s angular momentum can have any valueL = Iω

Quantum mechanics restricts the values of the angular momentum to

J is an integer called the rotational quantum number

1 0 1 2, , ,L J J J

Rotational Kinetic Energy of Molecules, Allowed Levels

The allowed values are

The rotational kinetic energy is quantized and depends on its moment of inertia

As J increases, the states become farther apart

2

rot 1 0 1 22

, , ,IJE E J J J

Allowed Levels, cont. For most molecules,

transitions result in radiation that is in the microwave region

Allowed transitions are given by the condition

J is the number of the higher state

2

rot

2

21 2 3

4

I

, , ,I

photonE E J

hJ J

π

Active Figure 43.5

Use the active figure to adjust the distance between the atoms

Choose the initial rotational energy state of the molecule

Observe the transition of the molecule to lower energy states

PLAYACTIVE FIGURE

Vibrational Motion of Molecules

A molecule can be considered to be a flexible structure where the atoms are bonded by “effective springs”

Therefore, the molecule can be modeled as a simple harmonic oscillator

Vibrational Motion of Molecules, Potential Energy

A plot of the potential energy function

ro is the equilibrium atomic separation

For separations close to ro, the shape closely resembles a parabola

Vibrational Energy

Classical mechanics describes the frequency of vibration of a simple harmonic oscillator

Quantum mechanics predicts that a molecule will 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

Vibrational Energy, cont.

The allowed vibrational energies are

v is an integer called the vibrational quantum number

When v = 0, the molecule’s ground state energy is ½hƒ The accompanying vibration is always present,

even if the molecule is not excited

vib

10 1 2

2ƒ , , ,E v h v

Vibrational Energy, Final

The allowed vibrational energies can be expressed as

Selection rule for allowed transitions is Δv = 1

The energy of an absorbed photon is Ephoton = ΔEvib = hƒ

vib

1

2 2

0 1 2, , ,

h kE v

π μ

v

Molecular Spectra

In general, a molecule vibrates and rotates simultaneously

To a first approximation, these motions are independent of each other

The total energy is the sum of the energies for these two motions:

21

12 2

ƒI

E v h J J

Molecular Energy-Level Diagram

For each allowed state of v, there is a complete set of levels corresponding to the allowed values of J

The energy separation between successive rotational levels is much smaller than between successive vibrational levels

Most molecules at ordinary temperatures vibrate at v = 0 level

Molecular Absorption Spectrum

The spectrum consists of two groups of lines One group to the right of center satisfying the selection rules ΔJ

= +1 and Δv = +1 The other group to the left of center satisfying the selection rules

ΔJ = -1 and Δv = +1 Adjacent lines are separated by h/2πI

Active Figure 43.8

Use the active figure to adjust the spring constant and the moment of inertia of the molecule

Observe the effect on the energy levels and the spectral lines

PLAYACTIVE FIGURE

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 I’s are present

in the sample

Intensity of Spectral Lines

The intensity is determined by the product of two functions of J The first function is the number of available states

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

The second function is the Boltzmann factor

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

on n e

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 the observed envelope of the spectral lines

2B( 1)/(2 )2 1 II J J k TJ e

Bonding in Solids

Bonds in solids can be of the following types Ionic Covalent Metallic

Ionic Bonds in Solids

The dominant interaction between ions is through the Coulomb force

Many crystals are formed by ionic bonding Multiple interactions occur among nearest-

neighbor atoms

Ionic Bonds in Solids, 2

The net effect of all the interactions is a negative electric potential energy

α is a dimensionless number known as the Madelung constant

The value of α depends only on the crystalline structure of the solid

2

attractive e

eU αk

r

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) α = 1.747 6 for the NaCl structure

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 of electrostatic

forces and the exclusion principle

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

2

total e m

e BU αk

r r

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

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 tightly

bound 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

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

Cohesive Energies for Some Covalent Solids

Covalent Bond Example – Diamond

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

This forms a tetrahedral structure

Another Carbon Example -- Buckyballs

Carbon can form many different structures

The large hollow structure is called buckminsterfullerene Also known as a

“buckyball”

Metallic Solids

Metallic bonds are generally weaker than ionic or covalent bonds

The outer electrons in the atoms of a metal are relatively free to move through the material

The number of such mobile electrons in a metal is large

Metallic Solids, cont.

The metallic structure can be viewed as a “sea” or “gas” of nearly free electrons surrounding a lattice of positive ions

The bonding mechanism is the attractive force between the entire collection of positive ions and the electron gas

Properties of Metallic Solids

Light interacts strongly with the free electrons in metals Visible light is absorbed and re-emitted quite

close to the surface This accounts for the shiny nature of metal

surfaces High electrical conductivity

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 in varying amounts The resulting solid solutions, or alloys, may be

designed to have particular properties Metals tend to bend when stressed

Due to the bonding being between all of the electrons and all of the positive ions

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

Fermi-Dirac Distribution Function

Applying statistical physics to a collection of particles can relate microscopic properties to macroscopic properties

For electrons, quantum statistics requires that each state of the system can be occupied by only two electrons

Fermi-Dirac Distribution Function, cont.

The probability that a particular state having energy E is occupied by one of the electrons in a solid is given by

ƒ(E) is called the Fermi-Dirac distribution function

EF is called the Fermi energy

B( )

1( )

1ƒ

FE E k TEe

Fermi-Dirac Distribution Function at T = 0

At T = 0, all states having energies less than the Fermi energy are occupied

All states having energies greater than the Fermi energy are vacant

Fermi-Dirac Distribution Function at T > 0

As T increases, the distribution rounds off slightly

States near and below EF lose population

States near and above EF gain population

Active Figure 43.15

Use the active figure to adjust the temperature

Observe the effect on the Fermi-Dirac distribution function

PLAYACTIVE FIGURE

Electrons as a Particle in a Three-Dimensional Box

The energy levels for the electrons are very close together

The density-of-states function gives the number of allowed states per unit volume that have energies between E and E + dE:

3 21 2

3

8 2( ) eπm

g E dE E dEh

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 Eavg = (3/5) EF

2 32

F

3(0)

2 8e

e

h nE

m π

Fermi Energies for Some Metals

Wave Functions of Solids

To make the model of a metal more complete, the contributions of the parent atoms that form the crystal must be incorporated

Two wave functions are valid for an atom with atomic number Z and a single s electron outside a closed shell:

( ) ( ) ( ) ( )ƒ ƒo oZr na Zr nas sψ r A r e ψ r A r e

Combined Wave Functions

The wave functions can combine in the various ways shown ψs

- + ψs- is equivalent to

ψs+ + ψs

+ These two possible

combinations of wave functions represent two possible states of the two-atom system

Splitting of Energy Levels The states are split into two

energy levels due to the two ways of combining the wave functions

The energy difference is relatively small, so the two states are close together on an energy scale

For large values of r, the electron clouds do not overlap and there is no splitting of the energy level

Splitting of Energy Levels, cont.

As the number of atoms increases, the number of combinations in which the wave functions combine increases

Each combination corresponds to a different energy level

Splitting of Energy Levels, final

When this splitting is extended to the large number of atoms present in a solid, there is a large number of levels of varying energy

These levels are so closely spaced they can be thought of as a band of energy levels

Energy Bands in a Crystal In general, a crystalline

solid will have a large number of allowed energy bands

The white areas represent energy gaps, corresponding to forbidden energies

Some bands exhibit an overlap

Blue represents filled bands and gold represents empty bands in this example of sodium

Electrical Conduction – Classes of Materials

Good electrical conductors contain a high density of free charge carriers

The density of free charge carriers in an insulator is nearly zero

Semiconductors are materials with a charge density between those of insulators and conductors

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

Metals

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

The motion of electrons in response to an electric field represents an increase in the energy of the system

When an electric field is applied to a conductor, the electrons move up to an available higher energy state

Metals – Energy Bands At T = 0, the Fermi energy

lies 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 energies near EF require only a small amount of additional energy from the applied field to reach nearby empty states above the Fermi energy

Metals As Good Conductors

The electrons in a metal experiencing only a weak applied electric field are free to move because there are many empty levels available close to the occupied energy level

This shows that metals are excellent electrical conductors

Insulators

There are no available states that lie close in energy into which electrons can move upward in response to an electric field

Although an insulator has many vacant states in the conduction band, these states are separated from the filled band by a large energy gap

Only a few electrons can occupy the higher states, so the overall electrical conductivity is very small

Insulator – Energy Bands The valence band is filled

and the conduction band is empty at T = 0

The Fermi energy lies somewhere in the energy gap

At room temperature, very few electrons would be thermally excited into the conduction band

Semiconductors

The band structure of a semiconductor is like that of an insulator with a smaller energy gap

Typical energy gap values are shown in the table

Semiconductors – Energy Bands

Appreciable numbers of electrons are thermally excited into the conduction band

A small applied potential difference can easily raise the energy of the electrons into the conduction band

Semiconductors – Movement of Charges

Charge carriers in a semiconductor can be positive, negative, or both

When an electron moves into the conduction band, it leaves behind a vacant site, called a hole

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 the holes

migrating through the material in the direction opposite the direction of the electrons The hole behaves as if it were a particle with

charge +e

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 called electron-

hole pairs

Doped Semiconductors

Impurities can be added to a semiconductor This process is called doping Doping

Modifies the band structure of the semiconductor Modifies its resistivity Can be used to control the conductivity of the

semiconductor

n-Type Semiconductors

An impurity can add an electron to the structure

This impurity would be referred to as a donor atom

Semiconductors doped with donor atoms are called n-type semiconductors

n-Type Semiconductors, Energy Levels

The energy level of the extra electron is just below the conduction band

The electron of the donor atom can move into the conduction band as a result of a small amount of energy

p-Type Semiconductors An impurity can add a hole

to the structure This is an electron

deficiency This impurity would be

referred to as a acceptor atom

Semiconductors doped with acceptor atoms are called p-type semiconductors

p-Type Semiconductors, Energy Levels The energy level of the hole

is just above the valence band

An electron from the valence band can fill the hole with an addition of a small amount of energy

A hole is left behind in the valance band

This hole can carry current in the presence of an electric field

Extrinsic Semiconductors

When conduction in a semiconductor is the result of acceptor or donor impurities, the material is called an extrinsic semiconductor

Doping densities range from 1013 to 1019 cm-3

Semiconductor Devices

Many electronic devices are based on semiconductors

These devices include Junction diode Light-emitting and light-absorbing diodes Transistor Integrated Circuit

The Junction Diode

A p-type semiconductor is joined to an n-type This forms a p-n junction A junction diode is a device based on a

single p-n junction The role of the diode is to pass current in one

direction, but not the other

The Junction Diode, 2

The junction has three distinct regions a p 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

The Junction Diode, 3

Because the two sides of the depletion region each carry a net charge, an internal electric field exists in the depletion region

This internal field creates an internal potential difference that prevents further diffusion and ensures zero current in the junction when no potential difference is applied

Junction Diode, Biasing

A diode is forward biased when the p side is connected to the positive terminal of a battery This decreases the internal potential difference

which results in a current that increases exponentially

A diode is reverse biased when the n side is connected to the positive terminal of a battery This increases the internal potential difference

and results in a very small current that quickly reaches a saturation value

Junction Diode: I-V Characteristics

LEDs and Light Absorption

Light emission and absorption in semiconductors is similar to that in gaseous atoms, with the energy bands of the semiconductor taken into account

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

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

Transistors

A junction transistor is formed from two p-n junctions A narrow n region sandwiched between two p

regions or a narrow p region between two n regions

The transistor can be used as An amplifier A switch

Integrated Circuits

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

Integrated circuits Solved the interconnectedness problem posed by

transistors Possess the advantages of miniaturization and

fast response

Superconductivity

A superconductor expels magnetic fields from its interior by forming surface currents

Surface currents induced on the superconductor’s surface produce a magnetic field that exactly cancels the externally applied field

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 a metal can be described by a single wave function

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 the

Cooper pairs because of the energy gap

Superconductivity - Critical Temperatures

The critical temperature is the temperature at which the electrical resistance of the material decreases to virtually zero

A new family of compounds was found that was superconducting at “high” temperatures First discovered in 1986 Found materials that are superconductive up to

temperatures of 150 K Currently no widely accepted theory for high-

temperature superconductivity