43. Molecules and Solids
43.1 Molecular Bonds
43.2 The Energy and Spectra of Molecules
43.3 Bonding in Solids
43.4 Band Theory of Solids
43.5 Free-Electron Theory of Metals
43.6 Electrical Conduction in Metals,
Insulators, and Semiconductors
Suggested HW problems:
Chp43: 2,3,5-7,44-46
Molecular Bonds
molecule = cluster of atoms held together by chemical bonds
Effective Potential Energy
between two atoms
a molecule is energetically
stable if its energy is
smaller than the sum of
the energies of the
constituent atoms
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Ionic Bonding
In order to form a stable octet (nobel gas) electronic
configuration, one atom loses n<4 electrons (becoming a positive
ion) while the other atom gains these n electrons (becoming a
negative ion); the electrostatic attraction between the two ions
leads to the formation of an ionic bond.
Na
Cl
Na+
Cl
-
Na ionization energy = 5.1 eV
Cl electron affinity = 3.7 eV
Covalent Bonding
In order to form a stable octet (nobel gas) electronic configuration,
two or more atoms may share valence electrons leading to the
formation of one or several (single or double) covalent bonds.
Cl
H2
Cl
Cl Cl
CH4
2
Hydrogen Bonding
H
Two negative ions or polarized atoms
are bound together by a positively
charged proton
H
O
water
molecule
H
O
H
H2O ↔ H − O − H
Van der Waals Bonding
are due Van der Waals forces = weak electrostatic
attraction between atoms/molecules
Van der Waals forces: 1. dipole-dipole force
2. dipole-induced force
3. dispersion (London) force
F (r ) ~ r −6
3
Molecular Spectra
Electronic
transition
(in optical
or UV)
Energy
Excited electronic
state
dissociation
energy
Ground state
Rotational
transition
(in microwave)
Vibrational
transition
(in IR)
Internuclear separation
Rotational Spectra
E!
rot
L2
"2
!(! + 1),
=
=
2I 2I
! = 0,1, 2, ...
# #
I = ∑ mi (ri −rCM )2
i
momentum
of inertia
Transition energy:
∆E ! = E ! − E ! −1
E1 = ∆E1
"2
=
!
2I
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Vibrational Spectra
U (r ) =
Molecule
Frequency
(x1013 Hz)
K (N/m)
HF
8.72
970
HCl
8.66
480
HBr
7.68
410
HI
6.69
320
CO
6.42
1860
NO
5.63
1530
k # # 2 k 2
(r1 − r2 ) = r
2
2


1
Envib = "ω  n +  , n = 0,1,2,...
2

Classification of solids
Phases of matter
• solid (well defined shape and volume)
• liquid (only well defined volume)
• gas (no defined shape or volume)
• plasma (an overall neutral collection of charged and
neutral particles)
Solids
• crystalline (atoms form a regular periodic structure)
• amorphous (atoms have irregular spatial distribution)
Solids
• metals (good electrical/heat conductors)
• insulators (poor electrical/heat conductors)
• semiconductors
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Bonding in solids:
Ionic solids
Ionic solid crystals (e.g. NaCl) are
held together by the Coulomb
attractive interaction between ions
with opposite sign (ionic bonding)
U = −αk
e2 b
+
r rm
(m ~ 10)
k = 1 / 4πε 0
Madelung constant
(α = 1.7476 for Na +Cl − )
Ionic cohesive energy:
U 0 = min U (r )  = −αk  1 −

 mb 
r0 = 

 αk 
Bonding in solids:
11
m  r0
1
m −1
Ionic solids
Properties of ionic solid crystals
• relatively stable and hard
• poor electrical/heat conductors
• high melting/boiling temperatures
• transparent to visible light
• strong IR absorption
• soluble in polar solvents (e.g., water)
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Bonding in solids:
Covalent solids
Atoms in the crystal are held
together by covalent bonding
E.g., C atoms in diamond form a
tetragonal crystal structure
Properties of covalent crystals
• very hard and stable
• high melting point
• good insulators
• do not absorb light
• larger cohesive energies (~10 eV)
than in ionic crystals
Bonding in solids: Metallic solids
Atoms in a metallic crystal are
held together by the effective
attractive electrostatic
interaction mediated by the
conduction (valence) electron
gas (metallic bonding)
Metal
ion
Conduction
electron gas
Properties of metallic crystals
• smaller cohesive energies
(~1 eV) than in covalent/ionic
crystals
• sufficiently hard and stable
• good electrical/heat
conductors
• strong interaction with light
• form solid solutions
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Bonding in solids: Molecular crystals
Molecules in the crystal are held together by:
• weak Van der Waals bonds
exp: solid methane (Ec=0.10 eV/molecule)
solid argon (Ec=0.076 eV/molecule)
• relatively strong hydrogen bonds
exp: solid ice (Ec=0.53 eV/molecule)
Band Theory of Solids
Splitting of the 3s
level when 2 Na
atoms are brought
together
Splitting of the 3s
level when 6 Na
atoms are brought
together
Formation of a 3s
energy band when
Na atoms form a
crystalline solid
Energy bands
in sodium
For N atoms,
the capacity
of each band
is 2(2ℓ+1)N
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Energy Bands in Metals
nd
ba
n
io
t
c
du
n
co
(Fermi energy)
• partially filled conduction band (E<EF)
• states with E>EF are empty and are available for
electrons near the Fermi surface
⇒ conduction electrons can move freely in a perfect
metallic crystal
Energy Bands in Insulators
• The valence (conduction) band
is completely filled (empty)
• The conduction and valence
bands are separated by and
energy gap Eg~10eV
• The Fermi energy (chemical
potential) falls inside the
energy gap
• A valence electron requires
∆E>Eg to become a conduction
electron, i.e., the density of
conduction electrons is
n~ exp(-Eg/kBT)
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Energy bands in semiconductors
• similar band structure to
insulators but with much
smaller energy gap (Eg~1eV)
• poor (good) conductor
(insulator) at T=0
• conductivity increases
rapidly with temperature
There are two types of
charge carriers in an
intrinsic semiconductor
(i.e., #electrons = #holes)
Extrinsic (doped, impure) semiconductors
n-type
donor atom (P, As, Sb)
majority charge carriers = electrons
minority charge carriers = holes
p-type
acceptor atom (Al, Ga, In)
majority charge carriers = holes
minority charge carriers = electrons
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p-n Junction (Semiconducting Diode)
V =0
V ≠0
Diodes are used for rectifying AC currents
Next Lecture:
Problem Solving
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