Plasma Physics

1
Plasma Physics Engineering

• Lecture 7

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Electronically Excited Molecules, Metastable
Molecules.

• Properties of excited molecules their
  contribution into plasma kinetics depend if
  molecules are stable or not stable WRT radiative
  and collisional relaxation processes
• Major factor, defining the stability
  ----radiation
• Electronically excited particles --easily decay
  to a lower energy state by a photon emission if
  not forbidden
• Selection rules for electric dipole radiation of
  excited molecules require
• ?S0

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• For transitions between S-states, and for
  transitions in the case of homonuclear molecules,
  additional selection rules require
• or g ?u or u ?g
• If radiation is allowed, frequency ? 109 sec-1.
• lifetime of excited species ---short in this
  case.
• Some data on such lifetimes for diatomic
  molecules and radicals are given in the Table 3.4
  together with the excitation energy of
  corresponding state

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Table 3.4. Life times and energies of
electronically excited diatomic molecules and
radicals on the lowest vibrational levels.
Molecule or Radical Electronic State Energy of the State Radiative Lifetime
CO 7.9 eV 9.510-9 sec
C2 2.5 eV 1.210-7 sec
CN 1 eV 810-6 sec
CH 2.9 eV 510-7 sec
N2 7.2 eV 610-6 sec
NH 2.7 eV 210-2 sec
NO 5.6 eV 310-6 sec
O2 4.3 eV 210-5 sec
OH 4.0 eV 810-7 sec
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• In contrast to resonance states,, metastable
  molecules have very long lifetimes
• seconds, minutes and sometimes even hours.
• reactive species generated in a discharge zone ?
  transported to a quite distant reaction zone.

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Table 3.5. Life times and energies of the
metastable diatomic molecules (on the lowest
vibrational level).
Metastable Molecule Electronic State Energy of the State Radiative Lifetime
N2 6.2 eV 13 sec
N2 8.4 eV 0.7 sec
N2 8.55 eV 210-4 sec
N2 11.9 eV 300 sec
O2 0.98 eV 3103 sec
O2 1.6 eV 7 sec
NO 4.7 eV 0.2 sec
oxygen molecules have two very low laying (energy
about 1eV) metastable singlet-states with a
pretty long lifetimes
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• The energy of each electronic state depends on
  the instantaneous configuration of nuclei in a
  molecule. In the simplest case of diatomic
  molecules energy depends only on the distance
  between two atoms.
• This dependence can be clearly presented by
  potential energy curves. -- very convenient to
  illustrate different elementary molecular
  processes like ionization, excitation,
  dissociation, relaxation, chemical reactions etc.

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Potential energy diagram of a diatomic molecule
AB in the ground and electronically excited
state.
Electronic terms of different molecules
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VIBRATIONALLY AND ROTATIONALLY EXCITED MOLECULES.

• When the electronic state of a molecule is fixed,
  the potential energy curve is fixed and it
  determines the interaction between atoms of the
  molecules and their possible vibrations
• Vibrational excitation of molecules plays ?
  essential and extremely important role in plasma
  physics and chemistry of molecular gases.
• Largest portion of discharge energy transfers ?
  primarily to excitation of molecular vibrations,
• and only after that to different channels of
  relaxation and chemical rxns
• Several molecules, e.g. N2, CO, H2, CO2 , --
  maintain vibrational energy without relaxation
  for relatively long time ? accumulating large
  amounts of the energy which then can be
  selectively used in chemical rxns
• Such vibrationally excited molecules are the most
  energy effective in the stimulation of
  dissociation and other endothermic chemical
  reactions.
• Emphasizes the importance of vibrationally
  excited molecules in plasma chemistry the
  attention paid to the physics and kinetics of
  these active species.

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Potential Energy Curves for Diatomic Molecules,
Morse Potential

• The potential curve U(r) corresponds closely to
  the actual behavior of the diatomic molecule if
  it is represented by the Morse potential
• (3.4)
• r0, a, D0 ? Morse potential parameters
• The Morse parameter D0 is called
• the dissociation energy of a diatomic
• molecule WRT the minimum energy

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• real dissociation energy D lt Morse parameter
• difference between D and D0 is not large --often
  can be neglected
• important sometimes --isotopic effects in plasma
  kinetics
• r gt r0 -- attractive potential,
• r lt r0 -- repulsion between nuclei.
• Near U(r)min_at_r r0 , potential curve parabolic
  
• corresponds to harmonic oscillations of the
  molecule
• With energy growth,, the potential energy curve
  becomes quite asymmetric
• Central line -- the increase of an average
  distance between atoms and the molecular
  vibration becomes anharmonic.

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• For specific problems ---Morse potential
  describes the potential energy curve of diatomic
  molecules
• Especially important for molecular vibration
• permits analytical description of the energy
  levels of highly vibrationally excited molecules,
  when the harmonic approximation no longer
  applies

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Vibration of Diatomic Molecules, -- Harmonic
Oscillator

• Consider potential curve of interaction between
  atoms in a diatomic molecule as
  parabolic---harmonic oscillator approximation
• quite accurate for low amplitude molecular
  oscillations
• QM -- sequence of discrete vibrational energy
  levels
• (3.6)
• vibrational levels sequence -- equidistant in
  harmonic
• approximation, e.g. - the energy distance is
  constant
• and equals to the vibrational quantum h?.

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Compare frequencies and energies for electronic
excitation and vibrational excitation of molecules

• Electronic excitation -- f(electron mass m not
  mass of heavy particles M )
• vibrational excitation
• a vibrational quantum is typically lt
  characteristic electronic energy I10-20 eV
• typical value of a vibrational quantum
  0.1-0.2eV.

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• typical value of vibrational quantum (about
  0.1-0.2eV) occurs in a very interesting energy
  interval.
• From one hand this energy is relatively low WRT
  typical electron energies in electric discharges
  (1-3 eV) and for this reason vibrational
  excitation by electron impact is very effective.
• From another hand, the vibrational quantum
  energy is large enough to provide at relatively
  low gas temperatures, high values of the Massey
  parameter PMa ?E/hav ?/av gtgt 1 to make
  vibrational relaxation in collision of heavy
  particles a slow, adiabatic process
• As a result at least in non-thermal discharges,
  the molecular vibrations are easy to activate and
  difficult to deactivate,
• makes vibrationally excited molecules very
  special in different applications of plasma
  chemistry.

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Table 3.6. Vibrational quantum and coefficient of
anharmonicity for diatomic molecules in ground
electronic states
Molecule Vibrational Quantum Coefficient of Anharmonicity Molecule Vibrational Quantum Coefficient of Anharmonicity
CO 0.27 eV 610-3 Cl2 0.07 eV 510-3
F2 0.11 eV 1.210-2 H2 0.55 eV 2.710-2
HCl 0.37 eV 1.810-2 HF 0.51 eV 2.210-2
N2 0.29 eV 610-3 NO 0.24 eV 710-3
O2 0.20 eV 7.610-3 S2 0.09 eV 410-3
I2 0.03 eV 310-3 B2 0.13 eV 910-3
SO 0.14 eV 510-3 Li2 0.04 eV 510-3
the lightest molecule H2 has the highest
oscillation frequency and hence the highest value
of vibrational quantum h? 0.55 eV.
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Vibration of Diatomic Molecules, Model of
Anharmonic Oscillator

• parabolic potential and harmonic
  approximation for vibrational levels
• Valid for low vibrational quantum numbers, far
  from dissociation
• unable to explain the molecular dissociation
  itself
• Solution QM oscillations ? based on the Morse
  potential
• anharmonic oscillator approximation
• the discrete vibrational levels ?exact QM
  energies
• (3.7)
• xe , dimensionless coeff. of anharmonicity
• typical value of anharmonicity is xe 0.01

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Table 3.6. Vibrational quantum and coefficient of
anharmonicity for diatomic molecules in their
ground electronic states.
Molecule Vibrational Quantum Coefficient of Anharmonicity Molecule Vibrational Quantum Coefficient of Anharmonicity
CO 0.27 eV 610-3 Cl2 0.07 eV 510-3
F2 0.11 eV 1.210-2 H2 0.55 eV 2.710-2
HCl 0.37 eV 1.810-2 HF 0.51 eV 2.210-2
N2 0.29 eV 610-3 NO 0.24 eV 710-3
O2 0.20 eV 7.610-3 S2 0.09 eV 410-3
I2 0.03 eV 310-3 B2 0.13 eV 910-3
SO 0.14 eV 510-3 Li2 0.04 eV 510-3
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• Harmonic oscillators Equal Vibrational levels
  ?Ev h?
• Anharmonic oscillators, -- energy distance
  ?Ev(v,v1) decrease with increase of vibrational
  quantum number v
• (3.8)
• finite number of vibrational levels
• v vmax , corresponds ?Ev(v,v1) 0 e.g.
  dissociation
• distance between vibrational levels?vibrational
  quantum, f(v)
• smallest vibrational quantum v vmax - 1 and v
  vmax

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• Last vibrational quantum before dissociation --
  smallest one, -- typically 0.003 eV.
• Corresponding Massey parameter
• PMa ?E/hav ?/av ? low.
• Means transition between high vibrational levels
  during collision of heavy particles is a fast
  non-adiabatic process in contrast to adiabatic
  transitions between low vibrational levels
• Thus, relaxation of highly vibrationally excited
  molecules is much faster, than relaxation of
  molecules

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• Vibrationally excited molecules -- quite stable
  WRT collisional deactivation.
• Their lifetime WRT spontaneous radiation is also
  relatively long.
• The electric dipole radiation, corresponding to a
  transition between vibrational levels of the same
  electronic state, is permitted for molecules
  having permanent dipole moments pm.
• In the framework of the model of the harmonic
  oscillator,- the selection rule requires
• However, other transitions are also possible
  in the case of the anharmonic oscillator, though
  with a much lower probability.
• The transitions allowed by the selection rule
  provide spontaneous infrared (IR) radiation

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• The radiative lifetime of vibrationally excited
  molecule can then be found according to the
  classical formula for an electric dipole pm,
  oscillating with frequency ?
• (3.11)
• Radiative lifetime strongly depends on the
  oscillation frequency.
• Earlier showed the ratio of frequencies
  corresponding to vibrational excitation and
  electronic excitation
• Then, taking into account Eq.(3.11) the radiative
  lifetime of vibrationally excited molecules
  should be approximately times
  longer than that of electronically excited
  particles.
• Numerically, the radiative lifetime of
  vibrationally excited molecules is about
  10-3-10-2 sec, gtgt typical time of resonant
  vibrational energy exchange and some chemical
  reactions , stimulated by vibrational excitation.

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Vibrationally Excited Polyatomic Molecules, the
Case of Discrete Vibrational Levels

• Polyatomic molecules - more complicated, than
  that of diatomic molecules
• due to possible strong interactions between
  different vibrational modes inside of the
  multi-body systems
• Non-linear triatomic molecules have three
  vibrational modes with three frequencies ?1, ?2,
  ?3
• When the energy of vibrational excitation is
  relatively low, the interaction between the
  vibrational modes is not strong and the structure
  of vibrational levels is discrete
• The relation for vibrational energy of such
  triatomic molecules at the relatively low
  excitation levels is just a generalization of a
  diatomic, anharmonic oscillator

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• (2.12)
• The six coefficients of anharmonicity have energy
  units in contrast with those coefficients for
  diatomic molecules
• Table 3.7. list information about vibrations of
  some triatomic molecules, including their
  vibrational quanta, coefficients of anharmonicity
  as well as type of symmetry

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• Table 3.7. Parameters of oscillations of
  triatomic molecules.

Molecules Symmetry Molecules Symmetry Molecules Symmetry Normal Vibrations their Quanta, eV Normal Vibrations their Quanta, eV Normal Vibrations their Quanta, eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV Coefficients of Anharmonicity, 10-3 eV
Molec. Sym. ?1 ?1 ?2 ?3 ?3 x11 x22 x33 x12 x13 x23
NO2 C2v 0.17 0.17 0.09 0.21 0.21 -1.1 -0.06 -2.0 -1.2 -3.6 -0.33
H2S C2v 0.34 0.34 0.15 0.34 0.34 -3.1 -0.71 -3.0 -2.4 -11.7 -2.6
SO2 C2v 0.14 0.14 0.07 0.17 0.17 -0.49 -0.37 -0.64 -0.25 -1.7 -0.48
H2O C2v 0.48 0.48 0.20 0.49 0.49 -5.6 -2.1 -5.5 -1.9 -20.5 -2.5
D2O C2v 0.34 0.34 0.15 0.36 0.36 -2.7 -1.2 -3.1 -1.1 -10.6 -1.3
T2O C2v 0.285 0.285 0.13 0.30 0.30 -1.9 -0.83 -2.2 -0.76 -7.5 -0.90
HDO C1hCs 0.35 0.35 0.18 0.48 0.48 -5.1 -1.5 -10.2 -2.1 -1.6 -2.5
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• The types of molecular symmetry clarify
  peculiarities of vibrational modes of the
  triatomic molecules
• Three molecular symmetry groups Cnv Cnh Dnh
• Transformations of coordinates , rotations and
  reflections, which keeps the Schroedinger
  equation unchanged for a triatomic molecule.
• Read section in text

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• As an example, a linear CO2 molecule -- three
  normal vibrational modes, asymmetric valence
  vibration ?3 symmetric valence vibration ?1 and a
  doubly degenerate symmetric deformation vibration
  ?2.
• It should be noted, that there occurs a resonance
  in CO2 molecules ?1 2?2 between the two
  different types of symmetric vibrations (see
  Table 3.7). For this reason

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Rotationally Excited Molecules

• The rotational energy of a diatomic molecule with
  a fixed distance r0 between nuclei can be found
  from the Schroedinger equation as a function of
  the rotational quantum number
• (3.22)
• B is the rotational constant,
  is the momentum of inertia of the diatomic
  molecule with mass of nuclei M1 and M2
• the momentum of inertia and hence the correct
  rotational constant B are sensitive to a change
  of the distance r0 between nuclei during
  vibration of a molecule. As a result, the
  rotational constant B, for diatomic molecules,
  decreases with a growth of the vibrational
  quantum number
• (3.23)

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• Estimate value of rotational constant (
  rotational energy)
• Similar to discussion on vibrational quantum lt
  characteristic electronic energy, because a
  vibrational quantum is proportional to
• the values of rotational constant B and
  rotational energy are
• proportional to
• This means that the value of the rotational
  constant lt value of a vibrational quantum (which
  is about 0.1 eV) and numerically is about 10-3 eV
  (or even 10-4 eV).
• These rotational energies in temperature units
  (1eV11,600K) correspond to about 1-10K, is the
  reason why molecular rotation levels are already
  well populated even at room temperature (in
  contrast to molecular vibrations).
• Values of some rotational constants Be and ae
  Table3.8

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• Energy levels in the rotational spectrum of a
  molecule are not equidistant.
• For this reason, the rotational quantum, which
  is an energy distance between the consequent
  rotational levels, is not a constant.
• In contrast to the case of molecular vibrations,
  the rotational quantum is growing with the
  increase of quantum number J and hence with the
  growth of rotational energy of a molecule
• The value of the rotational quantum (in the
  simplest case of fixed distance between nuclei)
  can be easily found from Eq.(3.22)

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• Typical value of rotational constant --10-3-10-4
  eV,
• E.g at room temperatures, the quantum number J is
  about 10.
• Thus even the largest rotational quantum is
  relatively small, about 510-3eV.
• In contrast to the vibrational quantum, the
  rotational one corresponds to low values of the
  Massey parameter PMa ?E/hav ?/av even at low
  gas temperatures.
• e.g. the energy exchange between rotational and
  translational degrees of freedom is a fast
  non-adiabatic process
• therefore the rotational temperature of molecular
  gas in a plasma is usually very close to the
  translational temperature even in non-equilibrium
  discharges, while vibrational temperatures can
  be significantly higher.

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• Homework Assignment
• 3.1 3.4 3.5 3.6