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Chapter 3
物理冶金
魏茂國
Crystal Binding
 The internal energy of a crystal
 Ionic crystals
 The Born theory of ionic crystals
 Van Der Waals crystals
 Dipoles
 Inert cases
 Induced dipoles
 The lattice energy of an inert-gas solid
 The Debye frequency
 The zero-point energy
 Dipole-quadrupole and quadrupole-quadrupole terms
 Molecular crystals
 Refinements to the Born theory of ionic crystals
 Covalent and metallic bonding
1
物理冶金
Crystal Binding
魏茂國
 Introduction
- Crystalline solids are empirically grouped into 4 classifications: (1) ionic,
(2) van der Waals, (c) covalent, and (d) metallic.
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Internal Energy of a Crystal
物理冶金
魏茂國
 Internal energy (內能)
- The internal energy of a crystal is considered to be composed of 2 parts.
1. There is the lattice energy U (晶格能) that is defined as the potential energy
(位能) due to the electrostatic attractions and repulsions that atoms exert on one
another.
2. There is the thermal energy (熱能) of the crystal, associated with the vibrations
of the atoms about their equilibrium lattice positions.
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物理冶金
Ionic Crystals
魏茂國
 Ionic crystal (離子晶體)
- Fig. 3.1 shows the lattice structure of sodium chloride crystal, which is simple cubic
with alternating lattice positions occupied by positive and negative ions. This lattice
can be pictured as two interpenetrating face-centered cubic structures made up of
positive and negative ions respectively.
In the sodium chloride lattice, the corresponding coordinate number is 6.
4
Fig. 3.1 The sodium chloride lattice
Ionic Crystals
物理冶金
魏茂國
- Fig. 3.2 shows the lattice of cesium chloride. Here
each ion of a given sign is surrounded by 8 neighbors
of the opposite sign.
- Fig. 3.3 shows the lattice of zincblende (閃鋅礦).
The coordination number is 4.
- Ideally, ionic crystals are formed by combining
a highly electropositive metallic element with a
highly electronegative element.
Fig. 3.2 The cesium chloride lattice.
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Fig. 3.3 The zincblende lattice, ZnS.
物理冶金
Born Theory of Ionic Crystals
魏茂國
 Born theory of ionic crystals
- Born and Medelung assumed that the ions are electrical charges with spherical
symmetry and that they interact with each other according to simple central-force
laws.
- In ionic crystals, these interactions take 2 basic forms, one long range and the other
short range.
- The first is the electrostatic, or coulomb, force that varies inversely with the square
of the distance between a pair ions,
ke1e2
f 
r12 2
(3.1)
where e1 and e2 are the charges on the ions, r12 is the center-to-center distance
between the ions, and k is a constant.
If cgs units are used, k = 1 dyne (cm2)/(statcoulombs)2, and if mks units are
employed, k = 9109 Nm2/C2.
e1
r12
e2
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物理冶金
Born Theory of Ionic Crystals
魏茂國
The corresponding coulomb potential energy for a pair of ions is
ke1e2

r12
(3.2)
- The other type of interaction is a short-range repulsion that occurs when ions are
brought so close together that their outer electron shells begin to overlap. When this
happens, very large forces are brought into play that force the ions away from each
other. In a typical ionic crystal, such as NaCl, both the positive and negative ions
have filled electron shells characteristic of inert gases.
- According to the Born theory, the total potential energy of a single ion in an ionic
crystal of the NaCl type, due to the presence of all the other ions, may be expressed
in the form
  M   R
(3.3)
where  is the total potential energy of the ion, M is its energy due to coulomb
interactions with all the other ions, and R is the repulsive energy.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
If one uses the cgs system of units where, in Eqs. 3.1 and 3.2, k = 1 this expression
may be written
Az 2 e 2 Be 2
 
 n
r
r
(3.4)
where e is the electron charge, z is the number of electronic charges on the ions, r is
the distance between centers of an adjacent pair of negative and positive ions (Fig. 3.4),
n is a large exponent, usually of the order of 9, and A and B are constants.
Fig. 3.4 Interionic distances in the sodium chloride lattice.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
If we now think in terms of the potential energy of a crystal containing one mole of
NaCl, the preceding equation becomes,
ANz 2 e 2 NBe 2
 n
U 
r
r
(3.5)
where N is Avogadro’s number (6.0221023) and U is the total lattice potential
energy. The first term on the right-hand side of this equation represents the
electrostatic energy due to simple coulomb forces between ions, whereas the second
term is that due to the repulsive interactions that arise when ions closely approach
each other.
It is a basic assumption of the Born theory that the repulsive energy can be expressed
as a simple inverse power of the interionic distance.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
- The cohesive energy U (內聚能) is obtained as a function of r by summing the
curves of the individual terms. It is done in Fig. 3.5 for an assumed value of 9 for the
exponent n. Note that the repulsive term, because of the large value of the exponent,
determines the shape of the total energy curve at small distances, whereas the
coulomb energy, with its smaller dependence on r, is the controlling factor at large
value of r.
The cohesive energy shows a minimum,
U0, at the interionic distance r0, where r0 is
the equilibrium separation between ions
at 0 K.
Fig. 3.5 Variation of the lattice energy of an ionic crystal
with the spacing between ions.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
- Let us consider the coulomb energy term of the Born equation, which for a single ion
is
z 2e 2 A
M  
r
(3.6)
Assumed that we are specially interested in a sodium chloride crystal where there is a
unit charge on each ion and z2 = 1.
e2 A
M  
r
(3.7)
Because the coulomb energy varies inversely as the first power of the distance
between charged ions, coulomb interactions act over large distances, and it is not
sufficient to consider only the coulomb energy between a given ion and its immediate
neighbors.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
In Fig. 3.4, the coulomb energy of a single ion equals a series of terms of the form
6e 2 12e 2 8e 2 6e 2 24e 2
M  




...
1r
2r
3r
4r
5r
(3.8)
Ae2
e2
M  
  6  8.45  4.62  3.00  10.7...
r
r
(3.9)
For sodium chloride, the Madelung number (A) is
1.7476 and the coulomb, or Madelung energy, for
one ion in the crystal is, accordingly,
1.7476e 2
M  
r
(3.10)
12
Fig. 3.4
Born Theory of Ionic Crystals
物理冶金
魏茂國
- For a single ion, the repulsive energy term is
Be 2
R   n
r
(3.11)
In this term the 2 quantities, B and n, must be evaluated. This can be accomplished
with the aid of 2 experimentally determined quantities: r0, the equilibrium interionic
separation at 0 K; and K0, the compressibility of the solid at 0 K.
At r0 the net force on an ion due to the other ions is zero, so that the first derivative of
the total potential energy with respect to the distance, which equals the force on the
ion, is also zero or
d  Ae 2 Be 2 
 d 
 n   0
    
r 
 dr  r  r0 dr  r
(3.12)
Since the quantity A is already known, the preceding expression produces an equation
relating n and B.
The equilibrium separation of ions r0 can be experimentally obtained from X-ray
diffraction measurements of the lattice constant extrapolated to 0 K, In the NaCl
crystal, this quantity equals 0.282 nm.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
- The compressibility is a function of the second derivative of the cohesive energy
(d2/dr2)r=r0 at r = r0. The compressibility is defined by the expression
1  V
K 0  
V  p


T
(3.13)
where K0 is the compressibility, V is the volume of the crystal, and (V/P)T is the
rate of change of the volume of the crystal with respect to pressure at constant
temperature. The compressibility is a quantity capable of experimental evaluation and
extrapolation to 0 K.
When the calculations outlined above are made, it is found that the Born exponent for
the sodium chloride lattice is 8.0. The computed cohesive energy is 180.4 Kcal
(7.56105 J) per mole. The latter is actually the energy of formation of a mole of solid
NaCl from a mole of Na+ ions, in the vapor form, and a mole of gaseous Cl- ions.
The experimental value of U for the NaCl lattice turns out to be 188 Kcal
(7.88105 J) per mole.
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物理冶金
Born Theory of Ionic Crystals
魏茂國
- The general equation for the force between electric charges is f  ke1e2 / r122 , where k is
a constant with units force  distance2  charge2. In the electrostatic or cgs system of
units, k = 1 dynecm2/(statcoulombs)2. Prove that k = 9109 Nm2/C2 in the
international (mks) system of units. (1 dyne = 10-5 N, 1 statcoulomb = 10-9 C/3)
2
1dyne  cm
1  10 N  10 m 
N

m
9
k


9

10
statcoulombs 2
C2
109 C / 32
2
5
2
2
15
物理冶金
Van der Waals Crystals
魏茂國
 Van der Waals (凡得瓦) crystals
- We shall consider another type of bonding that makes possible the formation of
crystals from atoms or even molecules that are electrically neutral and possess
electron configurations characteristic of inert gases.
- The forces that hold this type of solid together are usually quite small and of short
range. They are called van der Waals forces and arise from nonsymmetrical charge
distributions.
The most important component of these forces can be ascribed to the interactions
of electrical dipoles.
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Dipoles
物理冶金
魏茂國
 Dipole (偶極)
- An electrical dipole is formed by a pair of oppositely charged particles (+e1 and –e1)
separated by a small distance (a). Because the charges are not concentric, they
produce an electrostatic field that is capable of exerting a force on other electrical
charges.
- In Fig. 3.6, let l1 and l2 be the respective distances from the 2 charges of the dipole
to a point in space at a distance r from the midpoint of the dipole. If r is large
compared to a, then the potential at the point p, using cgs units, is given by
V 
e1 e2
 
l1 l2
e1
e1

(3.14)
a
a
 
 
r    cos  r    cos 
2
2


a 
a 
  1    cos   1    cos  
e 

 2r 
 2r 
V  1

2
r 
 a 


1    cos  


 2 r 



(3.15)
17
Fig. 3.6 An electrical dipole.
Dipoles
物理冶金


a


  cos 
e1 

r
V 
2
r   a 

1    cos   
  2 r 
 

魏茂國
(3.16)
Since (a/2r) <1, Eq. 3.16 simplifies to
V
e1 a cos  e1a cos 


r
r
r2
(3.17)
The components of the electric field intensity in the radial and transverse directions are
then given by
V 2e1a cos 

r3
r
1 V e1a sin 
E   

r 
r3
Er  
(3.18)
where r is the distance from the dipole to the point p, and  is the angle between the
axis of the dipole and the direction r. It is customary to call the quantity e1a, the
product of one of the dipole charges and the distance between the dipole charges, the
dipole moment () (偶極矩).
18
Dipoles
物理冶金
魏茂國
2e1a cos  2  cos 

r3
r3
e a sin   sin 
E  1 3

r
r3
- The component of the force, which would act on a charge of magnitude e if placed at
Er 
point p, are accordingly
2 e cos 
r3
e sin 
F  eE 
r3
Fr  eEr 
(3.19)
or using mks units with k = 9109 Nm2/C2
2ke cos 
r3
ke sin 
F 
r3
Fr 
(3.20)
Note that the electric field intensity due to a dipole varies as the inverse cube of the
distance from the dipole, whereas the field of a single charge varies as the inverse
square of the distance.
19
物理冶金
Inert Cases
魏茂國
 Inert cases
- It is interesting that the inert-gas atoms crystallize (at low temperature) in the facecentered cubic system.
- Because of their closed-shell structures, we can consider that over a period of time
the negative charges of the electrons are distributed about the nucleus with complete
spherical symmetry. The center of the negative charge on a time-average basis
therefore coincides with the center of the positive charge on the nucleus, which
means that the inert gas atoms have no average dipole moment.
- They do, however, have an instantaneous dipole moment because their electrons, in
moving around the nuclei, do not have centers of gravity that instantaneous coincide
with the nuclei.
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物理冶金
Induced Dipoles
魏茂國
 Induced dipoles (感應偶極，誘導偶極)
- When an atom is placed in an external electrical field, its electrons are displaced
from their normal positions relative to the nucleus. This charge redistribution may
be considered equivalent to the formation of a dipole inside the atom. Within limits,
the size of the induced dipole is proportional to the applied field,
 I  E
(3.21)
where I is the induced dipole moment, E is the electric field intensity, and  is a
constant known as the polarizability (極化率).
- Fig. 3.7 represents 2 inner-gas atoms of the same kind separated by the distance r.
Let it be assumed that the atom on the left possesses an instantaneous dipole
moment  due to the movement of the electrons around the nucleus. This moment
will produce a field E at the position of the second atom, which, in turn, induces a
dipole moment in the latter as given by Eq. 3.21.
21
Induced Dipoles
物理冶金
魏茂國
- In Fig. 3.7, consider the force exerted by the left dipole on the right dipole. The
induced dipole in the right-hand atom is equivalent to the pair of charges –e’ and +e’
separated by the distance a’. According to this, the induced dipole moment is e’a’.
Let E be the field intensity due to the instantaneous dipole on the left atom at the
negative charge (-e’) of the induced dipole. The corresponding field at the position of
the positive charge (+e’) of the induced dipole is E+a’dE/dr. The total force on the
induced dipole to the field of the other dipole is
dE 
dE
dE


'
'




f  e' E  e'  E  a'
e
a

I
dr
dr
dr


dE
dE
  I  E  f   I 
 E 
dr
dr
Fig. 3.7 Dipole-dipole interaction in
a pair of inert-gas atoms.
(3.22)
(3.23)
22
Induced Dipoles
物理冶金
魏茂國
In general the field of a dipole is proportional to the inverse cube of the distance
E

2e1a cos  2  cos  




E


r
3
3
r
r


(3.24)
r3
dE
 d 
   3   3 2
2
f  E
 7
 3
 3    3  4  
7
dr
r dr  r 
r  r 
r
r
(3.25)
The energy of a pair of inert-gas atoms due to the dipole interactions can be evaluate
as follows
r
  
 2
 2
2
2  1 6 
dr    r    6   6
7
r
6r
r
 6 
r
(3.26)
The van der Waals energy between a pair of inner-gas atoms due to dipole interactions
varies as the square of the dipole moment and the inverse sixth power of their distance
of separation.
Over a period of time the average dipole moment for an inert-gas atom must be zero.
The square of this quantity does not equal zero, and it is on this basis that inert-gas
atoms can interact.
23
物理冶金
Lattice Energy of an Inert-Gas Solid
魏茂國
 Lattice energy of an inert-gas solid
- When the atoms of a rare-gas solid have their equilibrium separation, the van der
Waals attraction is countered by a repulsive force. The latter is due to the
interaction that occurs when closed shells of electrons start to overlap.
- The cohesive energy of an inert-gas solid may be expressed in the form
A B
U  6  n
(3.27)
r
r
where A, B, and n are constants.
24
物理冶金
Debye Frequency
魏茂國
 Debye frequency
- In a crystalline solid, an atom is free to vibrate independently in 3 orthogonal
directions. A crystal of N atoms is considered as equivalent to 3N oscillators of
various frequencies v.
- Debye assumed that the forces of interaction between a neighboring pair of atoms
were roughly equivalent to a linear spring.
Pushing the atoms together would have the effect of compressing the spring, and in
so doing, a restoring force would be developed that would act to return the atoms to
their rest positions.
Pulling the atoms apart would produce an equivalent opposite result.
On this basis, Debye concluded that the entire lattice might be considered to be a
3-dimensional array of masses interconnected by spring.
In fact, assuming a simple cubic crystal, each atom would be held in space by a set
of 3 pairs of springs, as indicated in Fig. 3.8B. A one-dimensional crystal will be
considered, as indicated in Fig. 3.8B.
25
物理冶金
(A)
Debye Frequency
魏茂國
(B)
Fig. 3.8 (A) Debye model of a simple cubic crystal pictures an atom
as a mass joined to its neighbors by springs. (B) A one-dimensional
crystal model.
- The vibration modes of such an array are analogous to the standing waves that can be
set up in a string. According to Debye, as shown in Fig. 3.9, the minimum wavelength
or the mode of maximum frequency is obtained when neighboring atoms vibrate
against each other.
Fig. 3.9 The highest frequency vibration mode
for an array of 4 masses.
26
Debye Frequency
物理冶金
魏茂國
- The minimum wavelength corresponds to twice the spacing between the atoms, or
min = 2a, where a is the interatomic spacing.
The (maximum) vibrational frequency associated with this wavelength is
vm 
v
(3.28)

where v is the velocity of the shortest sound waves. This latter is normally of the
order of 5103 m/s.
At the same time, the interatomic spacing in metals is ~0.25 nm, so that
min  2a  2  0.25nm  0.50nm
5  103 m / s
13 1
13
vm  

10
s

10
Hz
9
 0.50  10 m
v
(3.29)
The value, vm = 1013 Hz, is often used in simple calculations to represent the vibration
frequency of an atom in a crystal.
27
物理冶金
Zero-Point Energy
魏茂國
 Zero-point energy
- The zero-point energy of a crystal is its thermal energy when the atoms are
vibrating in their lowest energy states.
- In a 3-dimensional crystal of N atoms, each atom inside the
crystal can undergo transverse vibrations in 3 independent
directions (Fig. 3.8A), it is possible to show that there are
3N independent transverse modes of vibrations.
- In a linear crystal (Fig. 3.8B), the density of vibrational
Fig. 3.8(A)
modes is the same in any frequency interval dv.
However, in a 3-dimensional array or crystal the
Fig. 3.8(B)
vibrational modes are 3-dimensional, and the multiplicity of the standing wave
patterns increases with increasing frequency. As a result, in the 3-dimensional case
the number of modes possessing frequencies in the range v to v+dv is given by
9N
f ( v )dv  3 v 2dv (3.30)
vm
where f(v) is a density function, N is the number of atoms in the crystal, v is the
28
vibrational frequency of an oscilllator, and vm is maximum vibrational frequency.
Zero-Point Energy
物理冶金
魏茂國
- Fig. 3.10 is a schematic plot of the Debye density function f(v) as a function of v. The
area under this curve from v = 0 to v = vm equals 3N, the total number of oscillators.
According to the quantum theory, the zero-point energy of a simple oscillator is hv/2.
The total vibrational energy of the crystal at absolute zero is, accordingly,
vm
 9 Nhv 4 
9 N 2 hv
9 Nhv
9
hv

EZ  0 f ( v ) dv  0 3 v  dv  0
dv

Nhvm (3.31)
 8v 3 
2
2
2vm3
8
vm
m 0

- The correction due to the zero-point energy is ~31% or ~0.5 kJ/mol in the case of
vm
vm
vm
3
neon, so that the lattice energy U0 should be ~2.47 kJ/mol rather than 1.88 kJ/mol, as
shown in Table 3.1.
Fig. 3.10 Frequency spectrum of a crystal according to Debye. The
maximum lattice frequency is vm.
29
物理冶金
Dipole-Quadrupole and Quadrupole-Quadrupole Terms
魏茂國
 Dipole-quadrupole and quadrupole-quadrupole terms
- Modern quantum mechanical treatments generally use an expression for the van der
Waals attractive energy (expressed in terms of a single ion) of the type
 c1 c2 c3 
 8  10 
6
r
r
r 

where c1, c2, and c3 are constants.
( r )  
(3.32)
- The first of this expression is the dipole-dipole interaction already considered.
- The second term is the inverse eighth power of the distance is called the dipolequadrupole term because the interaction between a dipole on one atom and a
quadrupole on another will lead to an energy that varies as the inverse eighth power
of the distance. The dipole-quadrupole term equals ~16% of the total attractive
energy.
- The last term, varying as the inverse tenth power of the distance, is called the
quadrupole-quadrupole term. It is, in general, small and amounts to less than 1.3%
of the total van der Waals attractive energy for all of the inert-gas solids.
30
物理冶金
Dipole-Quadrupole and Quadrupole-Quadrupole Terms
魏茂國
 Electric quadrupole (電四極)
- A general distribution of electric charge may be characterized by its net charge, by
its dipole moment, its quadrupole moment and higher order moments.
- An elementary quadrupole can be represented as two dipoles oriented antiparallel.
31
Source: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elequad.html
物理冶金
Molecular Crystals
魏茂國
 Molecular crystals
- Many molecules form crystals which are held together by van der Waals forces.
Among these are N2, H2, and CH4; these are typical covalent molecules in which the
atoms share valence electrons to effectively obtain closed shells for each atom in
the molecule. These molecules are nonpolar molecules; they do not have permanent
dipole moments.
- There are also polar molecules, such as water (H2O), which possess permanent
dipoles. The interaction between a pair of permanent dipoles is, in general, much
stronger than that between induced dipoles.
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物理冶金
Refinements to the Born Theory of Ionic Crystals
魏茂國
 Refinements to the Born theory of ionic crystals
- The forces of van der Waals exist in other solids, but when the binding due to other
causes is strong they may contribute only a small fraction of the total binding
energy. This is generally true in ionic crystals, although some types, like the silver
halides, may have van der Waals contribution of more than 10%.
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物理冶金
Covalent and Metallic Bonding
魏茂國
 Introduction
- In Fig. 3.11, the coordination number of the diamond structure is 4.
- Pauli exclusion principle: two electrons can occupy the same quantum state only if
the spins are oppositely directed.
- In general, covalent crystals follow what is known as the (8-N) rule, where N is the
number of valence electrons, and the factor (8-N) gives the number of nearest
neighbors in the structure.
Fig. 3.11 The diamond structure. Each carbon atom is surrounded by
4 nearest neighbors. Note: this structure is the same as that of
zincblende (ZnS). Fig. 3.3, except that this lattice contains one kind
of atom, instead of two.
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物理冶金
Covalent and Metallic Bonding
魏茂國
 Covalent bonding (共架鍵結)
- Suppose that 2 hydrogen atoms are made to approach other. Then there are 2 cases
to be considered: when the spins of the electrons on the 2 atoms are parallel, and
when the spins are opposed.
- Spins are opposed
As the atoms come closer and closer together, the electron on either atom begins to
find itself in the field of the charge on the nucleus of the other atom. Since the spins
of the electrons are opposed, each nucleus is capable of containing both electrons in
the 1s ground state. Under these conditions, there is a strong probability that the
electrons will spend more time in the neighborhood of one nucleus than the other,
and the hydrogen molecule becomes a pair of charged ions-one positive and the
other negative. This structure is unstable, as may be estimated from the energy
required to form a positive and a negative pair of hydrogen ions (-1237 kJ/mol). At
the normal distance of separation of the atoms in a hydrogen molecule, the ionic
structure exists for limited periods of time and contributes ~5% of the total binding
energy.
35
物理冶金
Covalent and Metallic Bonding
魏茂國
A much more important type of electron interchange occurs when both electrons
simultaneously exchange nuclei. The resulting shifting of the electrons back and forth
between the nuclei, which occurs at a very rapid rate, is commonly known as a
resonance effect, and ~80% of the binding energy of the hydrogen molecule is
attributed to it.
In addition to the ionic and the exchange, or resonance energy, there are other more
complicated electrostatic interactions between the two electrons and the two protons.
These contribute the remaining 15% of the binding energy of the hydrogen molecule.
Quantum mechanics shows that, on the average, the electrons spend more of their time
in the region between the two protons than they do on the far sides of the protons.
From a very elementary point of view, we may consider that the binding of the
hydrogen molecule results from the attraction of the positively charged hydrogen
nuclei to the negative charge which exists between them.
36
物理冶金
Covalent and Metallic Bonding
魏茂國
Attention is called to the interrelation between space and time implied in this
discussion concerning the time-average position of the electrons.
This is a coordination system that includes both the positions of the particles and their
momenta (that is, velocities). Thus, for a single particle free to move along a single
direction (the x-axis), there will be 2 dimensions in phase space: its position along the
axis, and its momentum.
For n particles capable of moving in a single direction, there will be 2n linear
dimensions in the phase space associated with these particles. These are x1, x2, … xn,
the positions of the particles; and p1, p2, … pn, the momenta of the particles.
For particles capable of moving in 3 dimensions, there will be 6n degrees of freedom
and therefore a corresponding number of dimensions in phase space.
An important theorem in statistical mechanics that bears on the subject of phase space
states that the position and momentum average in phase space coincide with the same
average over an infinite time.
37
物理冶金
Covalent and Metallic Bonding
魏茂國
- Spins are parallel
Now consider the case where 2 hydrogen atoms with parallel spins are made to
approach each other. Here, as the electron on either of the atoms comes within the
range of effectiveness of the field on the nucleus of the other atom, it is found that
the energy level it would normally occupy is already filled.
The normal electronic orbits become badly distorted,
or else the second electron moves into a higher
energy state, such as 2s1. In either case, bringing
together 2 hydrogen atoms with electrons that have
parallel spins increases the energy of the system.
A stable molecule cannot be formed in this
fashion. This is shown in Fig. 3.12, where the
uppermost curve represents the hydrogen
molecule with parallel spins and both electrons
in 1s orbits.
Fig. 3.12 Interaction energy of two
38
hydrogen atoms.
物理冶金
Covalent and Metallic Bonding
魏茂國
The lower curve is for opposed electron spins, and it can be observed that this curve
has a pronounced minimum, indicating that in this case a stable molecule can be
formed.
Fig. 3.12
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物理冶金
Covalent and Metallic Bonding
魏茂國
 Metallic bonding (金屬鍵結)
- In a metal the valence electrons are able to move at will through the lattice, while in
a covalent crystal the electrons form directed bonds between neighboring atoms.
- Metals tend to crystallize in close-packed lattices (face-centered cubic and closepacked hexagonal structures) in which the directionality of the bonds between
atoms is of secondary importance, while covalent crystals form complicated
structures so that the bonds between neighboring atoms will give each atom the
effect of having a closed-shell configuration of electrons.
- Inside a metal the valence electrons are free.
- A metal consists of an ordered array of positively charged ions between which the
valence electrons move in all directions with high velocities. Over a period of time,
this movement of electrons is equivalent to a more or less uniform distribution of
negative electricity which might be thought of as an electron gas that holds the
assembly together.
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物理冶金
Covalent and Metallic Bonding
魏茂國
The cooperative interaction between the electron gas and the positively charged nuclei
forms a structure that is stable.
The binding forces that hold a metallic crystal together can be assumed to come from
the attraction of the positively charged ions for the cloud of negative charge that lies
between them
As the distance between nuclei is made smaller (as the volume in metal is decreased),
the velocities of the free electrons increase with a corresponding rise in their kinetic
energy. This leads to a repulsive term which becomes large when a metal is
compressed.
41
物理冶金
Covalent and Metallic Bonding
魏茂國
 Born-Haber cycle
- There are several important concept to understand before the Born-Haber Cycle can
be applied to determine the lattice energy of an ionic solid; ionization energy,
electron affinity, dissociation energy, sublimation energy, heat of formation, and
Hess's Law.
- Ionization energy is the energy required to remove an electron from a neutral atom
or an ion. This process always requires an input of energy, and thus will always
have a positive value. In general, ionization energy increases across the periodic
table from left to right, and decreases from top to bottom. There are some excepts,
usually due to the stability of half-filled and completely filled orbitals.
- Electron affinity is the energy released when an electron is added to a neutral atom
or an ion. Usually, energy released would have a negative value, but due to the
definition of electron affinity, it is written as a positive value in most tables.
Therefore, when used in calculating the lattice energy, we must remember to
subtract the electron affinity, not add it. In general, electron affinity increases from
left to right across the periodic table and decreases from top to bottom.
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物理冶金
Covalent and Metallic Bonding
魏茂國
- Dissociation energy is the energy required to break apart a compound. The
dissociation of a compound is always an endothermic process, meaning it will always
require an input of energy. Therefore, the change in energy is always positive. The
magnitude of the dissociation energy depends on the electronegativity of the atoms
involved.
- Sublimation energy is the energy required to cause a change of phase from solid to
gas, bypassing the liquid phase. This is an input of energy, and thus has a positive
value. It may also be referred to as the energy of atomization.
- The heat of formation is the change in energy when forming a compound from its
elements. This may be positive or negative, depending on the atoms involved and
how they interact.
- Hess’s law states that the overall change in energy of a process can be determined by
breaking the process down into steps, then adding the changes in energy of each step.
The Born-Haber Cycle is essentially Hess's Law applied to an ionic solid.
43
Source: http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Lattices/Thermodynamics_of_Lattices/Lattice_Energy%3A
_The_Born-Haber_cycle
Covalent and Metallic Bonding
物理冶金
魏茂國
 Using the Born-Haber cycle
- The values used in the Born-Haber Cycle are all predetermined changes in enthalpy
for the processes described in the section above. Hess' Law allows us to add or
subtract these values, which allows us to determine the lattice energy.
Na+(g) + Cl(g)
IE
EA step 4
step 4
Na+(g) + Cl-(g)
Na(g) + Cl(g)
Energy
1
H d0 step 3
2
Na(g) +1/2 Cl2(g)
H a0 step 2
0
step 5
H lattice
Na(s) +1/2 Cl2(g)
H 0f
step 1
NaCl(s)
44
Source: http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Lattices/Thermodynamics_of_Lattices/Lattice_Energy%3A
_The_Born-Haber_cycle
物理冶金
Covalent and Metallic Bonding
魏茂國
- Step 1: Determine the energy of the metal and nonmetal in their elemental forms.
(Elements in their natural state have an energy level of zero.) Subtract from this the
heat of formation of the ionic solid that would be formed from combining these
elements in the appropriate ration. This is the energy of the ionic solid, and will be
used at the end of the process to determine the lattice energy.
- Step 2: The Born-Haber Cycle requires that the elements involved in the reaction are
in their gaseous forms. Add the changes in enthalpy to turn one of the elements into
its gaseous state, and then do the same for the other element.
- Step 3: Metals exist in nature as single atoms and thus no dissociation energy needs to
be added for this element. However, many nonmetals will exist as polyatomic species.
For example, Cl exists as Cl2 in its elemental state. The energy required to change Cl2
into 2Cl atoms must be added to the value obtained in Step 2.
45
Source: http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Lattices/Thermodynamics_of_Lattices/Lattice_Energy%3A
_The_Born-Haber_cycle
物理冶金
Covalent and Metallic Bonding
魏茂國
- Step 4: Both the metal and nonmetal now need to be changed into their ionic forms,
as they would exist in the ionic solid. To do this, the ionization energy of the metal
will be added to the value from Step 3. Next, the electron affinity of the nonmetal will
be subtracted from the previous value. It is subtracted because it is a release of energy
associated with the addition of an electron.
This is a common error due to confusion caused by the definition of electron affinity,
so be careful when doing this calculation.
- Step 5: Now the metal and nonmetal will be combined to form the ionic solid. This
will cause a release of energy, which is called the lattice energy. The value for the
lattice energy is the difference between the value from Step 1 and the value from
Step 4.
46
Source: http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Lattices/Thermodynamics_of_Lattices/Lattice_Energy%3A
_The_Born-Haber_cycle
Covalent and Metallic Bonding
物理冶金
魏茂國
- The diagram below is another representation of the Born-Haber Cycle.
H a0
1
H d0
2
Na(s) +1/2 Cl2(g)
IE  EA
H 0f
NaCl(s)
Na(g) + Cl(g)
H
0
lattice
Na+(g) + Cl-(g)
47
Source: http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Crystal_Lattices/Thermodynamics_of_Lattices/Lattice_Energy%3A
_The_Born-Haber_cycle