10.1Molecular Bonding and Spectra

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CHAPTER 10Molecules and Solids

• 10.1 Molecular Bonding and Spectra
• 10.2 Stimulated Emission and Lasers

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10.1 Molecular Bonding and Spectra

• The Coulomb force is the only one to bind atoms.
• The combination of attractive and repulsive
  forces creates a stable molecular structure.
• 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
• Negative slope (dV / dr lt 0) with repulsive force
• Positive slope (dV / dr gt 0) with attractive
  force

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Molecular Bonding and Spectra

• 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 lt 0. The shape of the curve depends on
  the parameters A, B, n, and m. Also n gt m.

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Molecular Bonding and Spectra

• Vibrations are excited thermally, so the exact
  level of E depends on temperature.
• 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.

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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.

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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.

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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.

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Rotational States

• L is quantized.
• The energy levels are
• Erot varies only as a function of the quantum
  number l.

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Clicker - Questions
 
10
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

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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 ?.
• If it is a purely ionic bond, we can compute ? by
  assuming that the force holding the masses
  together is Coulomb.
• and

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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
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Vibration and Rotation Combined

• 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 ½h?,
  3/2h?, etc.
• Transition from l 1 to l
• Photon will have an energy

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Vibration and Rotation Combined

• An emission-spectrum spacing that varies with l
• The higher the starting energy level, the
  greater the photon energy.
• 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.

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Vibration and Rotation Combined

• 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 ?l 1.
• Absorption spectra
• Within ?l 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.

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Vibration and Rotation Combined

• ?E 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.

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Vibration and Rotation Combined

• 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 ?l 0
  transition.
• The central frequency

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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

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Dual comb spectroscopy in ambient air
 
data acquisition time lt 100 ms
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Stimulated Emission and Lasers

• Einsteins analysis
• Consider transitions between two molecular states
  with energies E1 and E2 (where E1 lt 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)

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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 ? 8. Then
  and u(f) becomes very large.
• The probability of stimulated emission is
  approximately equal to the probability of
  absorption.

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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.

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Vibration and Rotation Combined

• A transition from l to l 2
• 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
• Raman spectroscopy is used to study the
  vibrational properties of liquids and solids.

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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.
• The best we can do is calculate the probability
  that a spontaneous transition will occur.
• If a spectral line has a width ?E, then a
  lower-bound estimate of the lifetime is ?t h /
  (2 ?E).

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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.

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Stimulated Emission and Lasers

• Einsteins analysis
• Consider transitions between two molecular states
  with energies E1 and E2 (where E1 lt 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)

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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 ? 8. Then
  and u(f) becomes very large.
• The probability of stimulated emission is
  approximately equal to the probability of
  absorption.

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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.

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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
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Clicker - Questions
If laserlight 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
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Stimulated Emission and Lasers

• 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
• Any photon produced by stimulated emission would
  have to be used up to excite another atom.
• There may be nothing to prevent spontaneous
  emission from atoms in the excited state.
• The beam would not be coherent.

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Stimulated Emission and Lasers

• Use a multilevel atomic system to see those
  problems.
• Three-level system
• 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 ?l 1 selection
  rule.E2 is said to be metastable.
• Population inversion more atoms are in the
  metastable than in the ground state

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Stimulated Emission and Lasers

• 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.

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Stimulated Emission and Lasers

• Four-level system
• 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.

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Stimulated Emission and Lasers

• The red helium-neon laser uses transitions
  between energy levels in both helium and neon.

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The Nobel Prize in Chemistry 1999 was awarded to
Ahmed Zewail"for his studies of the transition
states of chemical reactions using femtosecond
spectroscopy".
37
http//www.lindau-repository.org/nobellabs360/gm_t
heodorhaensch/index.html
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My groups scientific applications of lasers

• Atto and femto second spectroscopy in strong
  laser fields
• 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
• .