Transcript Molecules & Condensed Matter

Chapter 42
Molecules and
Condensed Matter
PowerPoint® Lectures for
University Physics, Thirteenth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by Wayne Anderson
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Goals for Chapter 42
• To understand the bonds holding atoms together
• To see how rotation and vibration of molecules affect
spectra
• To learn how and why atoms form crystalline structures
• To apply the energy-band concept to solids
• To develop a model for the physical properties of metals
• To learn how impurities affect semiconductors and to
see applications for semiconductors
• To investigate superconductivity
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Ionic bonds
• There are several ways in which atoms can bind
together to form more complex arrangements,
each using a somewhat different type of
electrostatic attraction. Two strong bond types
are ionic bonds and covalent bonds (1-5 eV),
while weaker bonds (0.5 eV or less) include van
der Waals bonds and hydrogen bonds.
• An ionic bond is an interaction between
oppositely charged ionized atoms, such as Na+
and Cl-., with a binding energy of 4.2 eV.
• Figure 37.1 (right) shows a graph of the
potential energy of two oppositely charged ions.
• Example 42.1: What is the electrostatic potential
energy of a pair of Na+ and Cl- ions separated
by their equilibrium distance of 0.24 nm? The
actual energy is -5.7 eV.
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Consider the Na+ and Clions as +e and –e charges.
U -
e2
4 0 r
 -6.0 eV
Covalent bonds
• In a covalent bond, the electrons are more
shared among the atoms of the molecule.
The wave functions of the outer shells are
distorted and become more concentrated in
certain places.
• Figure 42.2 (right) shows the hydrogen
covalent bond (binding energy -4.8 eV,
and Figure 42.3 (below) shows the
methane molecule.
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Rotational energy levels
• Once two or more atoms are bound into a molecule, they can undergo various
rotations and vibrations. Because of their small sizes, these various motions are
quantized in certain allowed energy states.
• Rotation of a 2-atom molecule might be modeled as below. As we saw earlier, the
relevant mass of both rotational and vibrational motions is the reduced mass,

m1m2
m1  m2
• There is a nice analogy with classical rotation:
K  12 I  2
•
and
L  I
L2
For an isolated atom U = 0, so K  E 
2I
I   r02
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Rotational energy levels
• The angular solutions to the Schrödinger equation are the same as those for the
hydrogen molecule (both have no angular dependence of U), hence as before
L  l (l  1)
• Combining this with the “classical” energy dependence on L, we have quantized
energies:
2
2
El  l (l  1)
2I
 l (l  1)
2 r02
(rotational energy levels for diatomic molecule)
• Example 42.2: Carbon monoxide atoms are 0.1128 nm apart. The atomic masses are
mC = 1.993 x 10-26 kg, mO = 2.656 x 10-26 kg.
(a) Find energies of the lowest three rotational energy levels of CO.
(1.993)(2.656)
10-26 kg  1.139 10 -26 kg
1.993  2.656
I   r02  (1.139 10-26 kg)(0.1128 10-9 m)2  1.449  10-46 kg m2

E0  0;
E1  l (l  1)
2
2I
1.055 10 
-34 2
2
2(1.449 10-46 )
1.055 10 

6
 0.479 meV
-34 2
E2
2(1.449 10-46 )
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 1.437 meV
(b) Find wavelength of photon
emitted in transition from l = 2
to l = 1 rotational level.

hc
 1.29 mm
E2 - E1
Vibrational energy levels
• In a similar manner, a diatomic molecule can vibrate, as in the classical analog
below, two masses on a spring. Both atoms vibrate about their center of mass,
so once again the relevant mass to use is the reduced mass.
• This system has the same energy levels as the quantum-mechanical harmonic
oscillator, which was in Chapter 40, although we did not spend time on it then.
The energy levels are given by
En  (n  12 )   (n  12 )
k

(k  is the spring constant)
• Figure 42.7 shows some vibrational energy levels of a diatomic molecule.
Combining both rotation and
vibration, one has:
Enl  l (l  1)
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2
2I
 (n  12 )
k

Rotation and vibration combined
• Combining both rotation and vibration, one has:
Enl  l (l  1)
2
2I
 (n  12 )
k

• Figure 42.8 (right) shows an energy-level diagram
for rotational and vibrational energy levels of a
diatomic molecule. Figure 42.9 (below) shows a
typical molecular band spectrum.
• Quantum-mechanical rules require Dl = ±1 and
Dn = ±1 (plus for absorbing a photon, and minus
for emitting a photon).
• The arrows in the figure show allowed transitions
between from n = 2 levels.
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Complex Molecules
• More complex molecules like CO2 have additional
vibration modes as shown schematically at the
right.
• Each of these motions follows its own
quantization rules, with separations in energy less
than 1 eV, so they produce infrared photons with
wavelength longer than 1 m.
• This fact makes CO2 a very effective
“greenhouse” gas, that absorbs heat from the
ground and traps it in the atmosphere.
• Venus’ atmosphere is nearly entirely CO2, hence
the planet’s surface temperature is near 800 K.
• Methane (CH4) is an even more effective
greenhouse gas.
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Crystal lattices
• The next simplest multi-atom structures to consider are crystalline materials. A
crystal lattice is a repeating pattern of mathematical points. Figure 42.11
(below) shows some common types of lattices.
• These are “simple” because of their repeating structures, although actual
crystals have imperfections that can strongly influence their bulk behavior.
• Ionic or covalent bonds can occur in crystals, which are called either ionic or
covalent crystals. An example of an ionic crystal is salt (NaCl). Examples of
covalent crystals are carbon, silicon, germanium, tin. They have four electrons
in their outermost shell, and form tetrahedral (hcp) bonds.
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Crystal lattices and structures
• Example 42.4: It is an interesting exercise to calculate the potential energy of an
ionic crystal. Consider such a crystal with alternating +e, –e charges with equal
spacing a along a line. Show that the total interaction potential energy is
negative, which means the crystal is stable.
e2 1
e2 1
e2 1
U  - 4 a  4 2a - 4 3a 
0
0
0
e2 1
1 - 12  13 - 14 
4 0 a

• The quantity in parentheses is actually the expansion of the function ln(2),
which is clearly a positive constant, so the overall potential energy is negative.
• In addition to ionic crystals and covalent crystals, a common type is metallic
crystals. In this structure, one or more of the outermost electrons in each atom
become detached from the parent atom, and are free to move through the
crystal. The corresponding wave functions extend over many atoms.
• The freely moving electrons have many of the properties of a gas, and can be
considered in the context of the electron-gas model (simplest of which is called
the free-electron model).
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Types of crystals
• The figure below-left is indicative of the free-electron model—fixed ions and
mobile electrons.
• Real crystals have defects, like the edge dislocation seen at right in two
dimensions. Many materials are polycrystalline, having regions of uniform
crystal structure in different orientations bonded together at grain boundaries.
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Energy bands
• The energy band concept, introduced in 1928, looks at
how the outer energy levels of states in an atom vary with
distance. When atoms are close enough together, the
bands from one atom can join with another to permit lots
of states in a closely spaced band of energy.
• Some bands are filled, some are empty, and in some
materials some are partially filled. In an insulator, there
is a large gap between occupied states and the
“conduction” band where electrons can move in a
metallic state. In a conductor, some electrons are in the
conduction band, and therefore can conduct electricity.
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Free-electron model of metals
• The free-electron model assumes that electrons are completely free inside the
metal, but that there are infinite potential-energy barriers at the surface. The 3D
potential box from the previous chapter becomes the size of the entire lump of
metal! For a cubic box, the energies are as we saw before:
E   nX  nY  nZ
2
2
2
2
2
 2mL
2
• When L is large, these energies can be very small, and very close together.
• The density of states, dn/dE, is the number of states per unit energy range.
Consider a sphere in “quantum-number-space” of radius n  n 2  n 2  n 2 .
rs
X
Y
Z
The number of states in the sphere is just the
volume
3
n  43  nrs
• Because we are only dealing with positive numbers, though, we only have 1/8th
of a sphere. Also, each n-combination can have electrons with spin ±½, so 3
twice as many. Finally, the number of states in the sphere-quadrant is n  13  nrs
• The energy at the surface of the sphere is
E  nrs
2
2
2
2mL2
• So writing L3 = V, we have
 2m 
n
3/2
3 2
VE 3/2
3
 2m  V E1/2
dn
 g (E) 
dE
2 2 3
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3/2
Fermi-Dirac distribution
• The Fermi-Dirac distribution f(E) is the
probability that a state with energy E is
occupied by an electron.
• The fraction of available states that are
occupied at some temperature is denoted
f(E). At absolute zero, all states up to
some energy are filled (f(E) = 1), and that
energy is called the Fermi energy EF0. As
the temperature is increased, some
electrons gain energy and go to higher
states. The distribution of states is given
by the Fermi-Dirac distribution:
f (E) 
1
e( E - EF )/ kT  1
• This function is graphed for increasing
temperature as shown at the right. It is a
bit tricky, because for some materials (like
semiconductors) EF itself varies with
temperature, but for solid conductors we
can make the assumption EF = EF0.
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Electron concentration and free-electron energy
• We now have two quantities: the density of states between E and dE given by
 2m 
g (E) 
2
3/2
V
E1/2
2 3
and the fraction of states filled as a function of temperature:
f (E) 
1
e( E - EF )/ kT  1
• We can combine these to find out the number dN of electrons with energies in the
range dE as a function of temperature
 2m 
dN  g ( E ) f ( E )dE 
• From our earlier expression
2
 2m 
n
3/2
V
2 3
3/2
3 2
E1/2
1
e
( E - EF )/ kT
1
dE
VE 3/2
3
at absolute zero n = N (the total number of electrons) and E = EF0 (the Fermi
energy at absolute zero, from which we can solve for EF0 as
EF 0
32/3  4/3

2m
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2
N
 
V 
2/3
N
 electron concentration
V
Semiconductors
• A semiconductor has an electrical resistivity that
is intermediate between those of good conductors
and good insulators.
• Follow Example 42.9 using Figure 42.24 below.
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Holes
• A hole is a vacancy in a
semiconductor.
• A hole in the valence band
behaves like a positively
charged particle.
• Figure 42.25 at the right
shows the motions of
electrons in the conduction
band and holes in the
valence band with an
applied electric field.
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Impurities
• Doping is the deliberate addition of impurity
elements.
• In an n-type semiconductor, the conductivity is
due mostly to negative charge (electron) motion.
• In a p-type semiconductor, the conductivity is due
mostly to positive charge (hole) motion.
• Follow the text analysis of impurities.
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n-type and p-type semiconductors
• Figure 42.26 (left) shows an n-type semiconductor, and
Figure 42.27 (right) shows a p-type semiconductor.
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Photocell
• A photocell is a simple semiconductor device.
• See Figure 42.28 below.
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The p-n junction
• A p-n junction is the boundary in a semiconductor
between a region containing p-type impurities and
another region containing n-type impurities.
• Figure 42.29 below shows the behavior of a
semiconductor p-n junction in a circuit.
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Currents through a p-n junction
• Follow the text analysis of currents through a p-n
junction.
• Figure 42.30 below shows a p-n junction in equilibrium.
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Forward and reverse bias at a p-n junction
• Figure 42.31 (left) shows a p-n junction under forwardbias conditions.
• Figure 42.32 (right) shows a p-n junction under reversebias conditions.
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Transistors
• Follow the analysis of transistors in the text.
• Figure 42.33 (left) shows a p-n-p transistor in a circuit.
• Figure 42.34 (right) shows a common-emitter circuit.
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Integrated circuits
• Follow the text discussion of integrated circuits.
• Figure 42.35 (left) shows a field-effect transistor.
• Figure 42.36 (right) shows an actual integrated
circuit chip.
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Superconductivity
• Follow the text summary of BCS theory
and superconductivity.
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