Download chapter43

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
A B
U (r )   n  m
r
r




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
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
1
ψ1s (r ) 
e  r ao
πao3

There is little overlap
between the wave
functions of the two
atoms when they are
far away from each
other
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
PLAY
ACTIVE 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


Vibrational energy


Due to the rotation of the molecule about its center of
mass
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
Erot

1

I ω2
2
I is the moment of inertia of the molecule
 m1 m2  2
2
I 
r

μr

m

m
2 
 1

µ is called the reduced mass of the molecule
Rotational Motion of
Molecules, Angular Momentum


Classically, the value of the molecule’s
angular momentum can have any value
L = Iω
Quantum mechanics restricts the values of
the angular momentum to
L  J  J  1

J  0, 1, 2,
J is an integer called the rotational quantum
number
Rotational Kinetic Energy of
Molecules, Allowed Levels

The allowed values are
Erot  EJ 


2
2I
J  J  1
J  0 , 1, 2,
The rotational kinetic energy is quantized and
depends on its moment of inertia
As J increases, the states become farther
apart
Allowed Levels, cont.


For most molecules,
transitions result in
radiation that is in the
microwave region
Allowed transitions are
given by the condition
E photon  Erot 
h2

J
2
4π I

2
I
J
J  1, 2, 3 ,
J is the number of the
higher state
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
PLAY
ACTIVE 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
Evib


1

  v   hƒ
2

v  0 , 1, 2,
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
Vibrational Energy, Final

The allowed vibrational
energies can be expressed
as
1 h

E vib   v 

2  2π

v  0 , 1, 2,


k
μ
Selection rule for allowed
transitions is Δv = 1
The energy of an absorbed
photon is Ephoton = ΔEvib =
hƒ
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:
2
1

E   v   hƒ 
J  J  1
2
2I

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
PLAY
ACTIVE 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
n  noe

J (J 1)/(2 I kBT )
2
Intensity of Spectral Lines,
cont

Taking into account both factors by
multiplying them,
 2J (J 1)/(2 I kBT )
I   2J  1 e



The 2J + 1 term increases with J
The exponential term decreases
This is in good agreement with the observed
envelope of the spectral lines
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 nearestneighbor atoms
Ionic Bonds in Solids, 2

The net effect of all the interactions is a
negative electric potential energy
e2
Uattractive  αke
r
 α is a dimensionless number known as the
Madelung constant
 The value of α depends only on the crystalline
structure of the solid
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
Utotal

e2
B
 αke
 m
r
r
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
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 ) 


1
(E EF ) kBT
e
1
ƒ(E) is called the Fermi-Dirac distribution
function
EF is called the Fermi energy
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
PLAY
ACTIVE 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:
8 2πm
g (E )dE 
h3
32
e
12
E dE
Fermi Energy at T = 0 K

The Fermi energy at T = 0 K is
h2  3ne 
EF (0) 


2me  8π 


2 3
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
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:
ψs (r )   A ƒ(r )e  Zr
nao
ψs (r )   A ƒ(r )e  Zr
nao
Combined Wave Functions

The wave functions can
combine in the various
ways shown


 s- +  s- is equivalent to
 s+ +  s+
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 electronhole 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 hightemperature superconductivity