2.1 THE PLANETARY MODEL OF AN ATOM
We know that atoms consist of a small, positively charged nucleus that is
surrounded by negatively charged electrons. The nucleus is composed of protons and
neutrons. Each proton carries an electric charge of + 1.6x10-19 C, and the neutron has no
charge. The atomic number refers to the number of protons in a given element, and this
number is fixed.
Each electron carries a negative charge of - 1.6x10-19 C. A neutral
atom has the same number of protons and electrons. If this number is not equal, then the
atom carries a net positive or negative charge, and it is called an ion. Protons and
neutrons are both about 1840 times heavier than an electron, so the electron mass is
negligible in comparison. The number of protons and neutrons a given element
possesses thus determines its atomic mass. All atoms of a given element have the same
number of protons but not all have the same number of neutrons. Atoms of a particular
element which have different numbers of neutrons are referred to as isotopes, and,
because different isotopes have slightly different atomic masses, the atomic mass
characterizing a given element is not necessarilty an integer.
Iron, for instance, has
atomic number 26 and its atomic mass is 55.85. A mole of iron is the number of atoms
that weighs 55.85 grams. This is Avogadro’s number, N, which equals 6.023x1023.
A simple way to picture an atom employs the so-called planetary model.
According to the planetary model, electrons circle the nucleus in well-defined orbits in
close analogy with how the planets
orbit the sun. A schematic illustration
of the planetary model of a sodium
(Na) atom is presented in figure 2-1.
At its center is the nucleus consisting
of 11 protons and 11 neutrons with 11
electrons orbiting this nucleus. The
radius of each orbit is determined by
Coulombic interaction between the
electrons and the positively charged
nucleus. In the simplest atom,
hydrogen which consists of a single
electron orbiting a single proton, the
radius of the electron orbit is referred
to as the Bohr radius and is equal to
0.059 nm. This radius is of order 104
times greater than the nuclear radius,
and we see that the size of the
hydrogen is determined almost
entirely by the size of the electron
orbital. This is true for all atoms.
Appendix A lists atomic radii for
most elements, and we see that these
range between 0.06 nm for oxygen (O) to 0.263 nm for cesium (Cs). Most elements have
atomic radii between about 0.1 and 0.15 nm despite the fact that different elements
1
contain very different numbers of electrons. The fact that many-electron atoms are all
roughly the same size because the electrons close to the nucleus feel the entire Coulombic
potential of the nucleus and the radii of their orbitals are small whereas the outermost
electrons feel only a fraction of the nuclear charge due to the partial shielding of the
nuclear by the inner electrons.
2.2
THE WAVE MODEL OF ELECTRONS IN AN ATOM
The planetary model of the atom provides an easy way to conceptualize what an
atom looks like, but this model is fundamentally flawed. In the early 1900’s Neils Bohr
and others recognized that the planetary model is unphysical for the reason that a charged
particle must suffer an acceleration in order to travel in a circular path. In the case of an
atom, this acceleration would cause an orbiting electron to spiral into the nucleus. The
fact that atoms are very stable tells us that such spiraling does not occur. To solve this
dilemma, Bohr proposed that atomic electrons are better thought of as waves rather than
as particles. In the Bohr model, the circumferential path that an electron would travel in
its orbit would have to equal an integral number of wavelengths, so the electron behaves
like a standing wave which closes on itself (Figure 2-2).
Figure 2-2: Model of a hydrogen atom showing an electron executing a
circular orbit around a proton. (B) De Broglie standing waves in a hydrogen
atom for an elkectron orbit corresponding to n=4.
The Bohr model very quickly led to a very rich period in the history of physics
during the 1920’s and early 1930’s when quantum mechanics was developed, which,
among many other things, developed in great detail the nature of electrons in atoms.
Central to quantum mechanics is the Schroedinger equation, and solutions to this
equation describe the wave nature of atomic electrons. For our purposes, we do not need
to know the details of these solutions but we can take advantage of the fact that each
solution is characterized by four quantum numbers n, l, m and s. These four quantum
numbers give us a tool to describe systematically the nature of the electron wave in
atomic orbitals.
2
The principal quantum number, n, can only assume integer values 1,2,3,... , and
this quantum number organizes the electrons into shells or groups of orbitals. When n =1
we speak of the K electron shell, while for the L and M shells, n =2 and n =3 ,
respectively, etc. As will become evident after introduction of the other quantum
numbers there are 2 electrons in the K shell, 8 in the L shell, 18 in the M shell, etc.
The angular momentum arising from the rotational motion of orbiting electrons is
also quantized, i.e. forced to assume specific whole-number values. This gives rise to the
second quantum number called the orbital quantum number, l, which can assume
values of 0, 1, 2, ....n -1. The shape of electron orbitals is essentially determined by the l
quantum number. When l =0 we speak of s electron states. These electrons have no net
angular momentum, and since they move in all directions with equal probability, the
charge distribution is spherically symmetric about the nucleus. For l = 1, 2, 3, ... we have
corresponding p, d, f, ... states.
A third quantum number, m, specifies the orientation of the angular momentum
along a specific direction in space. Known as the magnetic quantum number, m takes
on integer values between + l , i.e., -l , -l+1,... to ...+l -1, +l.
Lastly there is the spin quantum number, ms, in recognition of the fact that
electrons behave as if they simultaneously spin while they orbit the nucleus. Because
there are only two orientations of spin angular momentum, up or down, ms assumes - 1/2
and + 1/2 values.
These four quantum numbers are useful to us, because they provide a means with
which we can label and identify the various electrons in an individual atom or, as we
shall see later, in a large assembly of atoms that make up a solid. Furthermore, these
quantum numbers provide information both about the shape of the electron orbitals
around the nucleus as well as the energy a particular electron has in a given orbital. Note
that the shapes of the orbitals are not the simple spherical shells that we used in the
planetary model of the atom. Instead, the shape is largely determined by the orbital
quantum number l. When l=0, the orbital has a spherical shape. As the orbital angular
momentum increases and l increases, the shape of the orbital deviates from being
perfectly spherical. The shapes of these orbitals are described by figure 2-3. When the
orbital shapes are non-spherical, the third quantum number m provides additional
information regarding the orientation of the orbital relative to a set of Cartesian
coordinates x, y, and z. While much can be said about the physics of these electron
orbitals, most important from the point of view of engineering materials is that the shape
and orientation of the orbitals in which the outermost electrons in an atom are found
plays a significant role in how atoms form bonds with each. Atoms that are bonded
together by electrons in p orbitals, for example, usually have dramatically different
properties than those bonded together, for example, by electrons in s orbitals.
3
Figure 2-3: Pictorial representation of the charge distribution in hydrogen-like
s, p, and d wavefunctions. s orbitals are spherically symmetric, whereas p
orbitals have two lobes of high electron density extending along the x, y, and z
coordinate axes. d orbitals have four such lobes.
To systematically and accurately label the electrons in multielectron atoms we
take advantage of the Pauli exclusion principle. In order to account for many of the
properties observed in the periodic table, Wolfgang Pauli postulated that no two electrons
4
in an atom can have the same four quantum numbers. This simple statement immediately
enables us to uniquely describe each of the electrons in an atom. In the case of sodium
where Z =11 we have to specify the four quantum numbers (n, l, m, s) for each of the 11
electrons. These are summarized in Table 2.1.
Table 2.1 - The quantum numbers for each of the 11 electrons in atomic sodium (Na)
Electron
Energy
State
1s states
2s states
2p
states
3s state
Quantum Numbers (n, l, m, ms)
Shell
(1,0,0,+1/2), (1,0,0,-1/2)
(2,0,0,+1/2), (2,0,0,-1/2)
(2,1,0,+1/2), (2,1,0,-1/2), (2,1,1,+1/2),
(2,1,1,-1/2), (2,1,-1,+1/2), (2,1,-1,1/2)
(3, 0, 0, +1/2)
K
L
L
Another way to identify the
electron distribution in sodium uses
the spdf notation: 1s2 2s2 2p6 3s1. In
this shorthand notation the integers
and letters are the principal and orbital
quantum numbers, respectively, and
the superscript number tells how many
electrons have the same n and l values.
The order in which the orbitals are
filled does so to minimize the total
energy of the atom and follows a
specific pattern (figure 2-4) which
gives the periodic table its particular
shape.
2.3
THE ENERGIES OF
ELECTRONS IN ATOMS
M
6s
5s
4s
3s
2s
1s
6p
5p
4p
3p
2p
6d 6f
5d 5f
4d 4f
3d
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d…
Figure 2.4 - The filling order of
electron orbitals in the spdf notation.
We have seen that electronic structure of an atom can be described in terms of a
real-space picture where electrons, whether we choose to envision them as point particles
or as waves, surround the nucleus at various very small distances around its center. An
alternate way to describe electronic structure is to use energy space rather than real space.
In energy space, the important coordinate is energy as opposed to x, y, z directions in real
space. As we discuss various types of materials and the origins of many of their
properties, we will find very useful the ability to think about atoms and their electrons in
both a real-space picture and in an energy-space picture.
We create an energy-space representation of an atom plotting the energies
5
characteristic of each electron in an atom. A very important consequence of the Bohr
model is that only a particular wavelength will satisfy the requirement that the path length
of an electron orbital must equal an integral number of wavelengths (figure 2-2). Since
energy is related to wavelength – recall E=h - a particular electron orbital will have
associated with it a very specific energy. Furthermore, because the various orbitals in a
given atom all have their own characteristic sizes – e.g. some orbitals are very close to
the nucleus while others are much farther away - the electron wavelengths associated
with different orbitals are different. Thus, not only must the energy of an electron in a
particular orbital assume a very specific value, electrons in different orbitals must have
different and very well defined energies.
The energy of an electron in any given orbital is largely determined by the
principal quantum number n. For the simple case of the electron orbitals in the hydrogen
atom, for which the Schroedinger equation can be solved exactly, the energy E of each
shell is:
E=-
2 me q4/ h2n2 = - 13.6 / n2 (eV)
…[2.1]
Energy (eV)
where n = 1, 2, 3, …, h is Planck’s constant, and the electron mass and charge are me and
q, respectively. With this we can draw a diagram describing the electron energy levels in
hydrogen (figure 2-5). When the
hydrogen atom is in its ground
state, or, in other words, at its
0
lowest energy, its one electron
n=3
-1.5
occupies the n=1 energy level
with an energy of –13.6 eV.
n=2
-3.4
Notice, however, that we can
still calculate the energies for
Forbidden
n=2 and n=3 despite the fact that
energies
these are unfilled electron energy
levels. Recognizing that empty
levels exist is important, because
these empty levels are energy
n=1
-13.6
states into which electrons can
be excited from lower-energy
filled states. Very importantly,
Figure 2-5: A one-dimensional energy-level
diagram for atomic hydrogen .
most of the energies on the
diagram do not correspond to
allowed electron energy states in
hydrogen. For example, no electron in hydrogen can have an energy in the range –13.6
eV < E < -3.4 eV. Only the specific energies of –13.6 eV, -3.4 eV, -1.5 eV, etc. are
allowed. We label the other energies as forbidden energies.
The electron energy levels for multielectron atoms can be calculated using exact
or approximated solutions to the Schroedinger equation. A very rough estimate of the
electron energies is given by:
6
…[2.2]
E = -13.6 Z2/n2 (eV)
Energy (eV)
where Z is the atomic number. Many of the electron energies have been measured
experimentally, and Table 2.2 lists these for several different elements. Figure 2-6
illustrates the
electronic
0
structure of the
3p
sodium atom in
3s
-1
energy space
-31
using an energy2p
level diagram.
2s
-63
Note in Table 2-2
that the energies
of the inner-shell
electrons become
increasingly more
negative as the
-1072
1s
atomic number
increases. This is
because the inner
shell electrons
Figure 2-6: Energy-level diagram for atomic sodium (Na).
feel the entire
Coulombic potential of the increasingly positive nucleus and are bound tightly to it.
These electrons are referred to as core electrons, because, together with the nucleus, they
make up the core of the atom and are not very much affected when brought in close
proximity to other atoms. The outermost electrons, on the other hand, have energies of
just a few eV, because the Coulombic attraction of the nucleus is in great measure
shielded by the intervening core electrons. These outer-shell electrons are referred to as
valence electrons. They are very much involved in bonding, and their energies as well
as the nature of their orbitals can be dramatically affected when brought in close
proximity to other atoms.
Table 2.2 - Binding energies of Electrons in selected Elements.
From American Institute of Physics Handbook, pp. 7-98 and 158-165
Carbon
Atomic 6
Number
1s
2183.8
2s
6.4
2p
6.4
3s
3p
Aluminum
13
Silicon
14
Iron
26
Gold
79
1,559.6
117.7
73.3
72.9
2.2
2.2
1,838.9
148.7
99.5
98.9
3.0
3.0
7,112.0
846.1
721.1
708.1
100.7
54.0
80,724.9 K shell
14,352.8 L shell
13,733.6 “
11,918.7 “
3,424.9 M shell
3,147.8 “
2,743.0 “
2,291.1 “
2,205.7
“
758.8
N
3d
3.6
4s
shell
4p
3.6
643.7
545.4
“
“
7
2.4 ATTRACTIVE AND REPULSIVE INTERACTIONS BETWEEN ATOMS
Bonds form between atoms because of the balance between attractive forces and
repulsive forces between them. These act simultaneously but do so over different length
scales. When the atoms or molecules are far apart, the attractive force dominates. When
they are very close together, the repulsive force dominates.
When two atoms are relatively far apart, say several atomic diameters away from
each other, there is a small attractive force due to the electrostatic attraction between the
outer-shell electrons in one atom and the partially shielded nucleus of the other atom.
The strength of this attraction increases as the two atoms grow closer together, because
Coulomb’s Law tells us that the magnitude of the force is inversely proportional to r2:
…[2.3]
Potential Energy (eV)
F= -kq1q2/r2
Starting first with the
potential energy of the interaction
between the two atoms, U, the most
general way to describe the
attraction and repulsion is:
U r 
A B

rm rn
Force (eV/nm)
The positively charged nuclei repel
each other only weakly at these large
separation distances, because they
are shielded from each other by
valence and core electrons in both
atoms. When the atoms get so close
together that the valence-electron
orbitals begin to overlap, the
magnitude of the shielding
decreases, and the repulsive force
grows and does so very rapidly as
the atoms overlap more and more.
At a critical separation, the
magnitude of the attractive force is
equal to the magnitude of the
repulsive force and this condition
defines the equilibrium length of the
bond between the atoms.
3
Repulsive
2
Net
1
0
-1
Attractive
-2
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Interatomic Distance (nm)
10
Attractive
5
0
-5
Net
Repulsive
-10
-15
0.15
…[2.7]
0.2
0.25
0.3
0.35
0.4
0.45
Interatomic Distance (nm)
where r is the distance between the
centers of the two atoms. The first
 term represents the repulsive part of
Figure 2-7 - The repulsive and attractive
forces in a Lennard-Jones (12-6) model of a
Cu-Cu bond.
8
the potential, and the second term represents the attractive part of the potential. The force
between the two atoms, F(r), can be determined by evaluating F = -dU/dr. Expressions
such as these for either the potential energy or the force with these are useful in
characterizing interactions not only between atoms, but also between molecules and even
between macroscopically sized particles like dust. Different values of A, B, m, and n
would characterize these different situations, but in all cases there is a balance between
an attractive force which acts over long distances and a repulsive force which acts over
very short distances.
The so-called Lennard-Jones potential is a common and useful model for
simulating the interaction between different atoms based on equation [2.7]. In it, m=12
and n=6. Here we illustrate the balance between attractive and repulsive forces for the
specific case of the bond between two copper atoms, and we rewrite equation [2.7] more
explicitly as:
 12  6
U r  4LJ      
r  r  


…[2.8]
where for copper LJ  0.583 eV , giving U units of energy (eV) as it should have, and
  0.227 nm . This function, as well as F(r), can be easily calculated using a spreadsheet
(figure 2-7). With such a model of the interatomic energy, one can determine the
equilibrium distance between the two atoms by evaluating dU/dr = 0. This identifies the
 distance, ro, at which U(r) is a minimum. For the case of copper, we find that
interatomic
ro = 0.255 nm. This is, as it should be, almost exactly double the atomic radius listed for
copper in Appendix I.
When the interatomic distance deviates from the equilibrium spacing, ro, the bond
produces a restoring force. When the atoms move apart, the attractive part of the
potential dominates, and the two atoms are pulled back towards each other. Likewise,
when they move closer together, the repulsive component of the potential dominates and
the two atoms are pushed apart. This behavior is very similar to how two balls connected
by a spring would behave where the restoring force is related to the spring constant, ks, in
Hooke’s law (F= - ksx). Thinking of atoms as being connected to each other by springs is
a useful qualitative picture to have of atomic bonds in solids.
2.5 BONDING BETWEEN ATOMS AND MOLECULAR ORBITALS
When two atoms are separated by a distance greater than several atomic
diameters, then, for all intents and purposes, they behave like isolated atoms. The
attractive and repulsive interactions between them are weak, and their electron orbitals
and their electron energy levels follow the well-defined rules we have outlined in the
preceding sections. This situation changes, however, when they are sufficiently close
together that their outermost electron orbitals begin to overlap. This is perhaps easiest to
visualize with hydrogen. The hydrogen atom has only one electron with spherically
symmetrical 1s orbital. When two hydrogen atoms approach each other and the electron
orbitals overlap, each electron feels the attraction of both nuclei. As a consequence, each
9
electron begins to not just follow an atomic orbital around its parent nucleus but instead it
follows a molecular orbital that extends around both nuclei. Two different molecular
orbitals are possible (figure 2-8) which satisfy the standing-wave requirements of the
Bohr-type model. One concentrates the two electrons between the two nuclei, and this is
called the bonding orbital. For hydrogen,
the energy of the bonding orbital is lower
than the energy of the two separate atomic
orbitals. The difference is the chemical
bond energy.
The second possible
molecular orbital concentrates the two
electrons at opposite extremes from each
other. In this case, the energy of the
molecular orbital is higher than that of the
two atomic orbitals, so this is called an
antibonding orbital. According to the
Figure 2-8 - The contours of electron density in
Pauli exclusion principle, the bonding
(a) bonding and (b) antibonding orbitals of the
orbital and the antibonding orbital can
hydrogen molecule. The energy of the two
each accommodate two electrons, and,
atoms is lowered when a bonding orbital forms
since each hydrogen atom only contributes
when the nuclei are separated by a specific
one electron, both can be accommodated
distance (c).
in the bonding orbital and the H2 molecule
is thus stable.
On
an
energy-level
diagram (figure 2-9), the
electron energy levels of
molecular orbitals are different
from atomic levels in two ways.
First, there are now two distinct
energy levels rather than just
one. This is necessary because
the Pauli principle tells us that
only two electrons occupy a
given energy level and these
must have opposite spin. The
energy of the bonding molecular
orbital is slightly lower than that
of the atomic orbital, and,
similarly, the energy of the
antibonding orbital is slightly
higher. Second, the molecular
energy levels traverse two
atoms. In contrast, the core
electron orbitals from the two
atoms do not overlap since they
are close to their respective
10
nuclei. The size, shape, and energies of the core orbitals do not change much when
bonds form between the outer shell valence electrons.
2.6 ELECTRONIC STRUCTURE OF MANY-ATOM SOLIDS: ENERGY BANDS
What happens when 3 atoms get together? Just as in the case of two hydrogen
atoms, the overlapping atomic orbitals split to form molecular orbitals. The orbitals of
the three atomic orbitals of the valence electrons combine to form 3 molecular orbitals of
distinct energies. Again, these orbitals or energy levels extend over the entire molecule.
When 10 atoms combine, 10 orbitals of different shapes and energies are formed from the
atomic orbitals. Figure
2-10
illustrates
E=0
schematically
this
Split molecular
schematically for the
orbitals that form from
case of two atoms and
overlapping unfilled
four
atoms.
atomic orbitals.
Importantly,
the
Split molecular
splitting occurs over a
orbitals that form

from overlapping
relatively small energy
filled or partially
range. The difference
filled atomic
between the top energy
valence orbitalsl
level and the bottom
Figure 2-10: When two atoms bond together, both the valence atomic
energy level, d in figure
orbitals (red) and outer unfilled atomic orbitals split into molecular orbitals
which extend across all of the atoms in the assembly. The number of
2-10, is on the order of
molecular energy levels is exactly identical to the number of atoms in the
10
eV
or
less.
assembly. As the number of levels increases, the energy difference between
Consequently,
when
different molecular energy levels decreases.
more atoms participate
in the bonding, the number of orbitals increases and the energy difference between them
decreases. Also shown in figure 2-10 is the orbital splitting associated with the first
unfilled atomic orbitals (green). These were included in figure 2-9a and subsequently
omitted for clarity. Despite the fact that they are empty, these outer orbitals also overlap,
and empty molecular orbitals must form.
A typical solid material contains something on the order of Avogadro’s number of
atoms. If we extend the picture in figure 2-10 to describe N atoms where N is of order
1023 or 1024 then the atomic orbitals must split into N orbitals. Exactly as many orbitals
are formed as there are atoms in the solid. These electron orbitals are distinct, but the
energy difference from one to the next is equal to /N and is thus immeasurably small.
Because the molecular energy levels are so close together, they form a quasi-continuous
energy band, a few electron volts in width, that contains exactly one orbital for every
atom in the solid. Two energy bands for an N-atom solid are illustrated schematically in
figure 2-11. The red band is called the valence band, and it contains the valence
electrons from the various atoms in the solid. The green band is called the conduction
band. It is empty, but it contains N molecular energy levels corresponding to the
corresponding N empty atomic orbitals one shell above the valence electron shell. Notice
that for each of the N atoms, the nucleus is at a well defined position and with it are the
11
core electrons which remain in their atomic orbitals. In contrast to the valence electrons
which, because of the molecular orbitals can move anywhere in the entire solid, the
nucleus and its core electrons remain at a specific position in the solid. Together they
form an ion core, which has a net positive charge. In the case of sodium, for example,
the 11 protons in the nucleus and 10 core electrons (1s22s22p6) for an ion core with a net
charge of +1 while the 3s electron participates in the bonding and has an energy in the
valence band.
E=0
Conduction band
+Z
+Z
+Z
+Z
Position (r))
Increasing Energy
Valence band
+Z
+Z
Figure 2-11: The valence band, the conduction band, and the ion cores conduction
band for a solid of N atoms.
As in the case of hydrogen, the average energy of the electrons in the valence
band is lower than the energy of the atomic orbital. This lowering of the electron
energies is responsible for the cohesion, or bond energy, of the solid. Chemical bonds
formed by the such sharing of electrons among the atoms are primary bonds, and the
three main types of primary bonds – metallic, covalent, and ionic – can all be discussed in
the context of energy band diagrams.
We have presented a highly simplified representation of energy-band diagrams to
understand the electronic structure of many-atom solids. While the energy-band
diagrams characteristic of most real solids can be quite complex, we already have
sufficient tools to use them to categorize materials into classes and begin to understand
how and why different types of materials have the characteristic properties that empower
them for use in a broad variety of engineering applications. Based on four general types
of energy-band diagrams, we can quickly understand why some materials are good
electrical conductors while others are insulators, why some materials are transparent to
visible light while others are opaque, and why some materials are ductile and easily
deformed while others are very brittle.
12
A
Figure 2-12
summarizes the four
main types of energyband diagrams. For
simplicity, we show
only the valence band
(VB)
and
the
conduction
band
(CB).
One must
remember that these
diagrams are plots of
energy
versus
distance, and they
show the allowed
energy levels for the
bonding electrons in a
solid. The ion cores
are
not
shown.
Critical to using these
diagrams is simply
understanding
the
relative position and
filling of the two
bands
B
CB
CB
VB
VB
D
C
CB
CB
Eg
Eg
VB
VB
Figure 2-12: The four principal categories of energy-band diagrams.
(A) a partially filled valence band (conductor); (B) an overlapping
valence band and conduction band (conductor); (C) a filled valence
band separated from an empty conduction band by a large gap energy,
Eg (insulator); (D) a filled valence band separated from an empty
conduction band by a small gap energy (semiconductor).
2.7 ELECTRICAL CONDUCTORS (METALS AND ALLOYS)
When the atoms of a solid have an odd number of electrons, the band with the
highest energy is not full. Sodium is a good example of this situation, despite the fact
that pure sodium metal is not useful for any engineering application. The valence
electrons of the solid occupy the valence energy levels with the lowest energies available,
so two electrons occupy the lowest orbital, then two occupy the next higher level, and so
on, until the band is half full. The highest filled level is referred to as the Fermi level.
Above the Fermi level, a continuum of empty energy levels is available to the electrons
(figure 2-12A). Electrons in the lower-energy levels can accept energy, for example from
some sort of applied electrical, thermal, or mechanical field, and, very importantly, there
are empty energy states above the Fermi level into which the electron can be excited. In
other wordsw, these electrons can accept the energy being offered to them. In the
particular case of an electric field, electrons can be accelerated by the field – their energy
is slightly increased – and they then move to form an electric current. A material with a
partially filled valence band is thus a good electrical conductor. Similarly, with little
expense in energy, electrons can occupy different orbitals and take different shapes when
one atom slides past another, and the material plastically deforms. Furthermore, when
visible light of energy strikes the material, electrons can always absorb the photon and be
13
excited to higher energies. This means the material absorbs light and is opaque. These
are all properties of metals.
In most elements, the width of the highest lying bands is larger than the separation
of the atomic energy levels. The bands overlap in energy as shown in figure 2-12B.
Even when the atom has an even number of electrons such as in magnesium, neither of
the overlapping bands is completely occupied and the solid is again a good electrical
conductor. There are empty electron energy levels easily accessible to electrons in the
valence band, so they can accept energy in response to an applied field. For this reason,
most elements in the periodic table are considered as metals.
2.7 ELECTRICAL INSULATORS (MOSTLY CERAMICS)
Diamond (i.e. carbon), silicon and germanium, have 4 valence electrons. Their
valence band is completely filled. An energy gap separates the filled valence band from a
higher, empty band of electron orbitals. In diamond, this energy gap is 8.5 eV, which is
too large for thermal excitation of electrons at any practicable temperature. (See figure 212C). By virtue of Pauli’s exclusion principle, none of the electrons can change orbitals
since two electrons already occupy any possible orbital. An applied electric field cannot
accelerate the electrons in this material: no current can flow. The orbitals (shapes) of the
four valence electrons dictate the positions of the four atoms to which any atom is
bonded. To change the position of any atom would require the excitation of electrons
into the higher, empty, band. This, in diamond, with a gap of 8.5 electrons, is impossible.
No plastic deformation is possible and diamond is the hardest material known.
Now to the optical properties: light has photon energy between 2 and 3 eV. No
electron can be excited by this amount: diamond is transparent to all light except
ultraviolet with h larger than 8.5 eV. These are the properties of ceramics we have
described in section 1.1. Chemical compounds, such as SiO2 (silica, quartz), Al2O3
(Alumina, sapphire), SiC, TiO2 (rutile), NaCl, etc. have full bands and are ceramics.
Silicon and germanium are ceramics with a relatively small energy gap between
the filled and the empty energy band as illustrated in figure 2-12D. This gap is 1.15 eV
in silicon and 0.76 eV in germanium. The III-V compounds, such as GaAs, GaP, etc.
(see the periodic table), have similar band gaps. This allows the promotion of electrons
from the valence band to the empty band, called conduction band, and the conduction of
electrons. These are the semiconductors we will examine later.
In ceramics, the energy bands are completely filled or empty.
The chemical bond of ceramics can be covalent or ionic, depending on the shape
of the valence orbitals.
1.2.3.3. Covalent, ionic and mixed bonds.
14
In solid diamond, silicon and germanium, all atoms being the same, the sharing of
the electron orbitals between neighboring atoms is symmetrical as shown in figure 1.9 A.
All atoms remain neutral. This constitutes the covalent bond.
A
B
Figure 1.9. (A) Covalent bond, (B) Mixed, covalent-ionic bond, the circles are the
atomic orbitals before bonding: the thicker line indicates the molecular orbital; the
positive ion decreases in size, the negative ion increases.
In a covalent solid, the shapes of the molecular orbitals govern the positions of the
atoms. This is especially important in the covalently bonded materials such as diamond,
silicon and their compounds SiC, Si3N4, SiO2 because the orbitals of carbon and silicon
are formed from sp3 and sp2 hybrids. The sp3 hybrid is a new atomic orbital that is
formed by the combination of the s and the three p orbitals shown in figure 1.4. Four such
linear combinations can be formed; these four hybrids extend from the atom in the four
directions shown in figure 1.11.
Figure 1.11. The four directions of the sp3 hybrids.
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These hybrid orbitals are responsible for the structure of diamond and silicon, as shown
in figure 1.12
Figure 1.12. Left: structure of diamond and silicon. Right: Structure of SiO2; small black
atom is silicon; the larger gray atoms are oxygen.
In compound ceramics, the energy with which atoms bind electrons differs from
one element to the other. The shared valence electrons are more strongly attracted by one
element than by the other. The power to attract electrons in a chemical bond, the
electronegativity, has been measured and is shown in figure1.10
The electronegativity of the elements is smallest at the left of the periodic table,
where the atoms have one electron in addition to the completed shells; they give up this
extra electron relatively easily. The electronegativity increases as we go to the right of
the table and is largest for the halogens, which attract electrons more strongly in order to
complete their shell. The shape of the molecular orbital in a compound is sketched in
figure 1.9B. The valence electrons move towards the atom on the right, with higher
electronegativity. As a result, the atom on the left carries a positive charge and
Figure 1.10 Electronegativities of the elements.
diminishes in size; the atom on the right carries a negative charge and is enlarged. The
result is an ionic bond. The ionicity of the bond, that is, the fraction of an electronic
16
charge that is transferred from the positive to the negative ion is approximated by the
equation
% ionicity = 1 – exp{(–0.25)*(XA-XB)2}
where XA and XB are the electronegativities of the two elements in the bond.
The most ionic bond is that of cesium fluoride, with an ionicity of 95%. In this
compound, the cesium atom retains only 5% of its original electronic charge. Sodium
chloride has an ionicity of 68 %. At the other extreme, SiC possesses 12 % ionicity and
GaAs 3.9%. A pure ionic bond, in which the electron is totally transferred from one atom
to the other, does not exist. Only diamond, silicon and germanium have purely covalent
bonds in which no charge transfer takes place.
The degree of ionicity of the chemical bonds has practical implications for the
properties of the solids. In a purely ionic solid, where the valence electrons are no longer
shared but transferred totally to the more electronegative ion, the positions of the atoms
in the solid are governed by the neutrality of the material (a negative ion must be
surrounded by positive ions and vice versa) and the relative sizes of the ions. In practice,
one considers as ionic the solids whose structure and chemical properties are determined
by their ionic character. Any solid with ionicity larger than 50% is considered ionic.
Similarly, compounds with small ionicity, such as SiC, the compound semiconductors
such as GaAs, GaP, Si3N4, are considered covalent. Covalent materials are usually
harder and more brittle than the ionic. The intermediate compounds are called mixed or
polar covalent bonds. A useful example of a polar covalent bond is that of water with an
ionicity of 40 %.
Polymers and Secondary Bonds:
Polyethylene is an organic material consisting of long chains of carbon atoms to
each of which two hydrogen atoms are attached. This is shown in figure 1.13. The C-C
and C-H bonds are covalent sp3 and the valence band is completely filled.
Figure 1.13 Portion of a polyethylene molecule. The white atoms are carbon; the dark
atoms are hydrogen.
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When two polyethylene molecules approach, there is no sharing of valence
electrons between them. The electric charges of the valence electrons in the two
molecules repel each other so that small electric dipoles are induced in the two
approaching molecules. These dipoles attract each other weakly and form the van der
Waals bond. Similar bonds attract oxygen or nitrogen molecules and are responsible for
the formation of liquid gases at very low temperatures.
Polyvinyl chloride (PVC) has a similar structure to that of polyethylene, except
that some hydrogen atoms are replaced by chlorine. The latter is more electronegative
than carbon and hydrogen and attracts valence electron charge to itself, forming a polar
covalent bond. The negative charge on the chlorine and the positive charge in the carbon
and hydrogen form a permanent dipole. The permanent dipoles of neighboring
molecules attract each other and form a bond that is stronger than the van der Waals
bond.
When the positive charge of a polar bond is hydrogen that is attracted to the
negative charge of the neighboring molecule, the permanent dipole bond is called a
hydrogen bond. This is, in particular, the bond that forms water and ice.
Van der Waals, permanent dipole and hydrogen bonds are secondary bonds.
They are much weaker than the primary bonds of metals and ceramics and account for
the characteristic properties of polymers. We shall see later that primary bonds, called
cross-links exist between some polymers. These materials are stronger and can be used at
higher temperatures.
Table 1.2. Chemical Bond energy of some materials.
Material
Bonding Type
Chemical Bond Energy
Melting Temperature
o
KJ/mol eV/atom, Molecule
C
Hg
Al
Cu
Fe
W
Diamond
Si
WC
NaCl
MgO
SiO2
Ar
Polyethylene
H2O
PVC
Metallic
68
334
338
406
849
713
450
0.7
3.4
3.5
4.2
8.8
7.4
4.7
640
1000
879
7.7
3.3
5.2
51
0.52
Covalent
Polar Covalent
Ionic
van der Waals
Hydrogen
Permanent Dipole
0.08
-39
660
1083
1538
3410
4350
1410
2776
801
2800
1710
-189
N.A.*
0
NA.*
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* These polymers are amorphous and do not have a melting temperature.
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