CHAPTER 10
Molecules and Solids
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10.1 Molecular Bonding and Spectra
10.2 Stimulated Emission and Lasers
10.1: Molecular Bonding and Spectra
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The Coulomb force is the only one to bind atoms.
The combination of attractive and repulsive forces creates a
stable molecular structure.
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Force is related to potential energy F = −dV / dr, where r is the
distance separation.
it is useful to look at molecular binding using potential
energy V
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Negative slope (dV / dr < 0) with repulsive force
Positive slope (dV / dr > 0) with attractive force
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Molecular Bonding and Spectra
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An approximation of the potential of one atom in the vicinity of
another atom is
where A and B are positive constants.
Because of the complicated shielding effects of the various
electron shells, n and m are not equal to 1.

Eq. 10.1 provides a stable
equilibrium for total energy E < 0.
The shape of the curve depends on
the parameters A, B, n, and m.
Also n > m.
Molecular Bonding and Spectra
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Vibrations are excited thermally, so the exact
level of E depends on temperature.
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Once a pair of atoms is joined, then:

One would have to supply energy to raise the
total energy of the system to zero in order to
separate the molecule into two neutral atoms.

The corresponding value of r at the minimum
value is an equilibrium separation. The
amount of energy to separate the two atoms
completely is the binding energy which is
roughly equal to the depth of the potential
well.
Molecular Bonds
Ionic bonds:
 The simplest bonding mechanisms.
 Ex: Sodium (1s22s22p63s1) readily gives up its 3s electron to
become Na+, while chlorine (1s22s22p63s23p5) readily gains an
electron to become Cl−. That forms the NaCl molecule.
Covalent bonds:
 The atoms are not as easily ionized.
 Ex: Diatomic molecules (H2, N2, O2) formed by the combination
of two identical atoms tend to be covalent. These are referred to
as homopolar molecules.
 Larger molecules are formed with covalent bonds.
Molecular Bonds
Van der Waals bond:
 Weak bond found mostly in liquids and solids at low temperature
 Ex: In graphite, the van der Waals bond holds together adjacent
sheets of carbon atoms. As a result, one layer of atoms slides over
the next layer with little friction. The graphite in a pencil slides easily
over paper.
Hydrogen bond:
 Holds many organic molecules together
Metallic bond:
 Free valence electrons may be shared by a number of atoms.
Rotational States
Molecular spectroscopy:
 We can learn about molecules by studying how molecules
absorb, emit, and scatter electromagnetic radiation.

From the equipartition theorem, the N2 molecule may be thought
of as two N atoms held together with a massless, rigid rod (rigid
rotator model).

In a purely rotational system, the kinetic energy is expressed in
terms of the angular momentum L and rotational inertia I.
Rotational States

L is quantized.

The energy levels are

Erot varies only as a function of the
quantum number l.
Clicker - Questions
Vibrational States
There is the possibility that a vibrational energy mode will be excited.
 No thermal excitation of this mode in a diatomic gas at ordinary
temperature.
 It is possible to stimulate vibrations in molecules using
electromagnetic radiation.
Assume that the two atoms are point masses connected by a
massless spring with simple harmonic motion:
Vibrational States

The energy levels are those of a quantum-mechanical oscillator.

The frequency of a two-particle oscillator is


Where the reduced mass is μ = m1m2 / (m1 + m2) and the spring
constant is κ.
If it is a purely ionic bond, we can compute κ by assuming that the
force holding the masses together is Coulomb.
and
Clicker - Questions
Compare the fundamental vibrational frequencies of HCl
and NaCl and select the true statement
a) The fundamental vibrational frequencies are
equal
b) The fundamental vibrational frequency of HCl is
higher
c) The fundamental vibrational frequency of HCl is
lower
d) The fundamental vibrational frequency of HCl is
changing with temperature
Vibration and Rotation Combined
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It is possible to excite the rotational and vibrational modes
simultaneously.
Total energy of simple vibration-rotation system:
Vibrational energies are spaced at regular intervals.
emission features due to vibrational transitions appear at
regular intervals: ½ħω, 3/2ħω, etc.
Transition from l + 1 to l:
Photon will have an energy
Vibration and Rotation Combined
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An emission-spectrum spacing that varies with l
The higher the starting energy level, the greater the photon
energy.
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Vibrational energies are greater than rotational energies. This
energy difference results in the band spectrum.
Typical section of the emission spectrum of a diatomic molecule.
Equally spaced groups of lines correspond to the equal spacings
between vibrational levels. The structure within each group is due to
transitions between rotational levels.
Vibration and Rotation Combined
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The positions and intensities of the observed bands are ruled by
quantum mechanics. Note two features in particular:
1) The relative intensities of the bands are due to different transition
probabilities.
- The probabilities of transitions from an initial state to final state are not
necessarily the same.
2) Some transitions are forbidden by the selection rule that requires
Δℓ = ±1.
Absorption spectra:

Within Δℓ = ±1 rotational state changes, molecules can absorb
photons and make transitions to a higher vibrational state when
electromagnetic radiation is incident upon a collection of a
particular kind of molecule.
Vibration and Rotation Combined

ΔE increases linearly with l as
in Eq. (10.8).
A schematic diagram of the
absorptive transitions between
adjacent vibrational states ( n
= 0 to n = 1) in a diatomic
molecule.
Vibration and Rotation Combined
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In the absorption spectrum of HCl, the spacing between the
peaks can be used to compute the rotational inertia I. The
missing peak in the center corresponds to the forbidden Δℓ = 0
transition.
The central frequency
Vibration and Rotation Combined
Fourier transform infrared (FTIR) spectroscopy:
 Data reduction method for the sole purpose of studying
molecular spectra. It is based on the Michelson interferometer.
 A spectrum can be decomposed into an infinite series of sine and
cosine functions.
 With slow scanning random and instrumental noise can be
reduced in order to produce a “clean” spectrum. Typical
scanning time is tens of minutes/spectrum
Dual frequency comb spectroscopy
 A novel version of FTR without moving parts and with 10^-3
s/spectrum
 “seeing the heart beat of molecules”
Optical depth (L )
Dual comb spectroscopy in ambient air
8 Hitran simulation 580 m multipass
1.2 ppmv CH4 (1.8 ppmv CH4~ wiki)
0.0012 H2O
6
~ 4% relative humidity in room temp.
4
2
data acquisition time
< 100 ms
Optical depth (-ln (I/I 0))
0
0
2
4
6
MIR Dual comb spetroscopy
8
2900
2950
3000
3050
Wavenumber (cm-1)
3100
3150
Stimulated Emission and Lasers
Einstein’s analysis:
 Consider transitions between two molecular states with energies E1
and E2 (where E1 < E2).
 Eph is an energy of either emission or absorption.
 f is a frequency where Eph = hf = E2 − E1.
If stimulated emission occurs:
 The number of molecules in the higher state (N2)
 The energy density of the incoming radiation (u(f))
the rate at which stimulated transitions from E2 to E1 is
B21N2u(f) (where B21 is a proportional constant)
 The probability that a molecule at E1 will absorb a photon is B12N1u(f)
 The rate of spontaneous emission will occur is AN2 (where A is a
constant)
Stimulated Emission and Lasers
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Once the system has reached equilibrium with the incoming radiation,
the total number of downward and upward transitions must be equal.
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In the thermal equilibrium each of Ni are proportional to their
Boltzmann factor
.

In the classical time limit T → ∞. Then
becomes very large.
and u(f)
The probability of stimulated emission is approximately equal
to the probability of absorption.
Stimulated Emission and Lasers
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Solve for u(f),
or, use Eq. (10.12),

This closely resembles the Planck radiation law, but Planck law is
expressed in terms of frequency.
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Eqs.(10.13) and (10.14) are required:
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The probability of spontaneous emission (A) is proportional to the
probability of stimulated emission (B) in equilibrium.
Vibration and Rotation Combined
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A transition from l to l + 2
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Let hf be the Raman-scattered energy of an incoming photon and
hf ’ is the energy of the scattered photon. The frequency of the
scattered photon can be found in terms of the relevant rotational
variables:
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Raman spectroscopy is used to study the vibrational properties
of liquids and solids.
10.2: Stimulated Emission and Lasers
Spontaneous emission:
 A molecule in an excited state will decay to a lower energy
state and emit a photon, without any stimulus from the outside.
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The best we can do is calculate the probability that a
spontaneous transition will occur.
If a spectral line has a width ΔE, then a lower-bound estimate
of the lifetime is Δt = ħ / (2 ΔE).
Stimulated Emission and Lasers
Stimulated emission:
 A photon incident upon a molecule in an excited state causes the
unstable system to decay to a lower state.
 The photon emitted tends to have the same phase and direction as
the stimulated radiation.

If the incoming photon has the same energy as the emitted photon:
The result is two photons of the same
wavelength and phase traveling in the
same direction.
Because the incoming photon just
triggers emission of the second
photon.
Stimulated Emission and Lasers
Einstein’s analysis:
 Consider transitions between two molecular states with energies E1
and E2 (where E1 < E2).
 Eph is an energy of either emission or absorption.
 f is a frequency where Eph = hf = E2 − E1.
If stimulated emission occurs:
 The number of molecules in the higher state (N2)
 The energy density of the incoming radiation (u(f))
the rate at which stimulated transitions from E2 to E1 is
B21N2u(f) (where B21 is a proportional constant)
 The probability that a molecule at E1 will absorb a photon is B12N1u(f)
 The rate of spontaneous emission will occur is AN2 (where A is a
constant)
Stimulated Emission and Lasers

Once the system has reached equilibrium with the incoming radiation,
the total number of downward and upward transitions must be equal.

In the thermal equilibrium each of Ni are proportional to their
Boltzmann factor
.

In the classical time limit T → ∞. Then
becomes very large.
and u(f)
The probability of stimulated emission is approximately equal
to the probability of absorption.
Stimulated Emission and Lasers

Solve for u(f),
or, use Eq. (10.12),

This closely resembles the Planck radiation law, but Planck law is
expressed in terms of frequency.

Eqs.(10.13) and (10.14) are required:

The probability of spontaneous emission (A) is proportional to the
probability of stimulated emission (B) in equilibrium.
Stimulated Emission and Lasers
Laser:
 An acronym for “light amplification by the stimulated emission of
radiation”
Masers:
 Microwaves are used instead of visible light.

The first working laser by Theodore H. Maiman in 1960
helium-neon laser
Clicker - Questions
If laser=light amplification by stimulated emission of
radiation, then what is ‘maser’ stand for?
a) macrowave amplification by stimulated emission
of radiation
b) microwave amplification by stimulated emission
of radiation
c) milliwave amplification by stimulated emission
of radiation
Stimulated Emission and Lasers
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The body of the laser is a closed tube, filled with about a 9/1 ratio
of helium and neon.
Photons bouncing back and forth between two mirrors are used to
stimulate the transitions in neon.
Photons produced by stimulated emission will be coherent, and the
photons that escape through the silvered mirror will be a coherent
beam.
How are atoms put into the excited state?
We cannot rely on the photons in the tube; if we did:
1) Any photon produced by stimulated emission would have to be
“used up” to excite another atom.
2) There may be nothing to prevent spontaneous emission from
atoms in the excited state.
The beam would not be coherent.
Stimulated Emission and Lasers
Use a multilevel atomic system to see those problems.

Three-level system
1)
2)
3)
Atoms in the ground state are pumped to a higher state by some
external energy.
The atom decays quickly to E2.
The transition from E2 to E1 is forbidden by a Δℓ = ±1 selection rule.
E2 is said to be metastable.
Population inversion: more atoms are in the metastable than in the
ground state
Stimulated Emission and Lasers
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After an atom has been returned to the ground state from E2, we
want the external power supply to return it immediately to E3, but
it may take some time for this to happen.
A photon with energy E2 − E1 can be absorbed.
result would be a much weaker beam
This is undesirable because the absorbed photon is unavailable
for stimulating another transition.
Stimulated Emission and Lasers
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Four-level system
1)
Atoms are pumped from the ground state to E4.
They decay quickly to the metastable state E3.
The stimulated emission takes atoms from E3 to E2.
The spontaneous transition from E2 to E1 is not forbidden, so E2 will
not exist long enough for a photon to be kicked from E2 to E3.
 Lasing process can proceed efficiently.
2)
3)
4)
Stimulated Emission and Lasers

The red helium-neon laser uses transitions between energy
levels in both helium and neon.
The Nobel Prize in Chemistry 1999 was awarded to Ahmed Zewail"for his studies of
the transition states of chemical reactions using femtosecond spectroscopy".
http://www.lindau-repository.org/nobellabs360/gm_theodorhaensch/index.html
My groups scientific applications of lasers

Atto and femto second spectroscopy in strong laser fields
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Precision measurements of the Fundamental Constants
Sensing of Greenhouse gases in the atmosphere
Sniffing methane from natural seeps and petroleum reservoirs
Looking for exoplanets (Qatar b,Khalid Alsubai)
Breath analysis for monitoring stages of diabetes
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.
10.1) Consider again the rotational energy
states of the N2 molecule as described in
Example 10.1. Find the energy involved in a
transition (a) from the l=2 to l=1 state, and (b)
from the l=10 to l=9 state.
10.2) (a) Use the data in table 10.1 to find the
approximation spacing between vibrational
energy levels in CO. (b) What temperature
would be needed to excite this vibration
thermally?
10.7) If the energy of a vibrational transition
from n=0 state to the n=1 state in CO could be
absorbed in a rotational transition that begins in
the ground state ( l=0 ), what would be the
value of l for the final state? Explain why such a
rotational transition is impossible.
10.18) (a) How many photons are emitted each
second from a 5.0mW helium-neon laser (
λ=632.8 nm )? (b) If the laser contains 0.02
mole of neon gas, what fraction of the neon
atoms in the tube participate in the lasing
process during each second of operation? (c)
Comment on the relatively low numerical result
in (b).
10.19) A laser emits 5.50x1018 photons per
second, using a transition from an excited state
with energy 1.15 eV to a ground state with
energy 0 eV. (a) What is the laser’s power
output? (b) What is the wavelength?
Homework 9
Chap.10
#1,2,7,18,19