Transcript ME 533 Lecture 7 Pla..

Plasma Physics & Engineering
Lecture 7
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:
  0,1
ΔS=0
• For transitions between Σ-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
bdA132312



ugg

Table 3.4. Life times and energies of electronically excited diatomic molecules and radicals on the lowest vibrational levels.
Electronic State
Energy of the State
Radiative Lifetime
7.9 eV
9.5*10-9 sec
Molecule or Radical
CO
A
1
C2
d 3 g
2.5 eV
1.2*10-7 sec
CN
A
1 eV
8*10-6 sec
CH
A
2.9 eV
5*10-7 sec
7.2 eV
6*10-6 sec
2.7 eV
2*10-2 sec
3*10-6 sec
N2
2
2
b g
3

NH
b
NO
b 
5.6 eV
O2
A

u
4.3 eV
2*10-5 sec
4.0 eV
8*10-7 sec
OH
1
2
3
A
2

• 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.
Table 3.5. Life times and energies of the metastable diatomic molecules (on the lowest vibrational level).
Electronic State
Energy of the State
Radiative Lifetime
6.2 eV
13 sec
8.4 eV
0.7 sec
8.55 eV
2*10-4 sec
11.9 eV
300 sec
3*103 sec
Metastable Molecule
A
baE


14331 
ggugu
N2
N2
A
3

u
a 1  u
N2
a g
N2
E 
O2
a g
0.98 eV
O2
b

g
1.6 eV
7 sec
NO
a 
4.7 eV
0.2 sec
1
3
1
1
4

g
oxygen molecules have two very low laying (energy about 1eV)
metastable singlet-states with a pretty long lifetimes
• 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.
Electronic terms of different
molecules
Potential energy
diagram of a diatomic
molecule AB in the
ground and
electronically excited
state.
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.
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
U (r )  D0 [1  exp(  (r  r0 ))] 2
(3.4)
•
r0, α, D0 → Morse potential parameters
•
The Morse parameter D0 is called
the dissociation energy of a diatomic
molecule WRT the minimum energy
• real dissociation energy D < Morse parameter
1
D  D0  
2
• difference between D and D0 is not large -often can be neglected
• important sometimes --isotopic effects in
plasma kinetics
• r > r0 -- attractive potential,
• r < r0 -- repulsion between nuclei.
• Near U(r)min@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.
• 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
Vibration of Diatomic Molecules, -- Harmonic Oscillator
•
Consider potential curve of interaction between atoms in a diatomic
molecule as parabolic---harmonic oscillator approximation
U  D0 2 (r  r0 ) 2
•
quite accurate for low amplitude molecular oscillations
•
QM -- sequence of discrete vibrational energy levels
2 D0
1
E v   ( v  )
  r0
(3.6)
2
I
•
vibrational levels sequence
--
equidistant in harmonic
approximation, e.g. -– the energy distance is constant
and equals to the vibrational quantum ħω.
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
•
~
1
1
1
( 

)
M
I
M
 a vibrational quantum is typically

electronic energy I~10-20 eV
m
M
 100 < characteristic
• typical value of a vibrational quantum ~ 0.1-0.2eV.
• 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
= ΔE/ħαv = /αv >> 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.
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
•6*10-3
•Cl2
•0.07 eV
•5*10-3
•F2
•0.11 eV
•1.2*10-2
•H2
•0.55 eV
•2.7*10-2
•HCl
•0.37 eV
•1.8*10-2
•HF
•0.51 eV
•2.2*10-2
•N2
•0.29 eV
•6*10-3
•NO
•0.24 eV
•7*10-3
•O2
•0.20 eV
•7.6*10-3
•S2
•0.09 eV
•4*10-3
•I2
•0.03 eV
•3*10-3
•B2
•0.13 eV
•9*10-3
•SO
•0.14 eV
•5*10-3
•Li2
•0.04 eV
•5*10-3
the lightest molecule H2 has the highest oscillation
frequency and hence the highest value of vibrational
quantum ħω = 0.55 eV.
Vibration of Diatomic Molecules, Model of Anharmonic
Oscillator
• parabolic potential
U  D0 2 (r  r0 ) 2
harmonic approximation for vibrational levels
and
1
E v   ( v  )
2
  r0
2 D0
I
– 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
1
1
Ev   (v  )  xe (v  ) 2
2
2
xe 

4D0
(3.7)
– xe , dimensionless coeff. of anharmonicity
– typical value of anharmonicity is xe ~ 0.01
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
•6*10-3
•Cl2
•0.07 eV
•5*10-3
•F2
•0.11 eV
•1.2*10-2
•H2
•0.55 eV
•2.7*10-2
•HCl
•0.37 eV
•1.8*10-2
•HF
•0.51 eV
•2.2*10-2
•N2
•0.29 eV
•6*10-3
•NO
•0.24 eV
•7*10-3
•O2
•0.20 eV
•7.6*10-3
•S2
•0.09 eV
•4*10-3
•I2
•0.03 eV
•3*10-3
•B2
•0.13 eV
•9*10-3
•SO
•0.14 eV
•5*10-3
•Li2
•0.04 eV
•5*10-3
• Harmonic oscillators – Equal Vibrational levels Ev = ħω
• Anharmonic oscillators, -- energy distance Ev(v,v+1)
decrease with increase of vibrational quantum number v
(3.8)
Ev  Ev1  Ev    2 xe  (v  1)
– finite number of vibrational levels
• v= vmax , corresponds Ev(v,v+1) = 0 e.g. dissociation
vmax 
1
1
2 xe
• distance between vibrational levels→“vibrational quantum”,
f(v)
• smallest “vibrational quantum” v = vmax - 1 and v = vmax
 min  Ev (vmax  1, vmax )  2 xe  
•
• Last vibrational quantum before dissociation -smallest one, -- typically ~0.003 eV.
• Corresponding Massey parameter
– PMa = ΔE/ħαv = /αv → low.
• Means transition between high vibrational levels
during collision of heavy particles is a fast nonadiabatic process in contrast to adiabatic
transitions between low vibrational levels
• Thus, relaxation of highly vibrationally excited
molecules is much faster, than relaxation of
molecules
• 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 v  1
• However, other transitions v  2,3,4... 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
• The radiative lifetime of vibrationally excited molecule can then be
found according to the classical formula for an electric dipole pm,
oscillating with frequency 
c 3 1
 R  120 2 3
pm 
(3.11)
• Radiative lifetime strongly depends on the oscillation frequency.
• Earlier showed the ratio of frequencies corresponding to vibrational
m  100
excitation and electronic excitation ~
M
• Then, taking into account Eq.(3.11) the radiative lifetime of 3
vibrationally excited molecules should be approximately ( M ) 2  10 6
m
times longer than that of electronically excited particles.
• Numerically, the radiative lifetime of vibrationally excited molecules
is about 10-3-10-2 sec, >> typical time of resonant vibrational energy
exchange and some chemical reactions , stimulated by vibrational
excitation.
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:
1
1
1
1
1
Ev (v1 , v2 , v3 )  1 (v1  )   2 (v2  )   3 (v3  )  x11 (v1  ) 2  x22 (v2  ) 2 
2
2
2
2
2
1
1
1
1
1
1
1
 x33 (v3  ) 2  x12 (v1  )(v2  )  x13 (v1  )(v3  )  x23 (v2  )(v3  )
2
2
2
2
2
2
2
(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
•Table 3.7. Parameters of oscillations of
triatomic molecules.
•Molecules &
Symmetry
•Mole
c.
•Sym.
•Normal Vibrations &
their Quanta, eV
•ν1
•ν2
•Coefficients of Anharmonicity,
•10-3 eV
•ν3
•x
•x22
•x33
•x12
•x13
•x23
1
1
•NO2
•C2v
•0.
17
•0.09
•0.
21
•1
.
1
•-0.06
•-2.0
•-1.2
•-3.6
•-0.33
•H2S
•C2v
•0.
34
•0.15
•0.
34
•3
.
1
•-0.71
•-3.0
•-2.4
•-11.7
•-2.6
•SO2
•C2v
•0.
14
•0.07
•0.
17
•0
.
4
9
•-0.37
•-0.64
•-0.25
•-1.7
•-0.48
•H2O
•C2v
•0.
48
•0.20
•0.
49
•5
.
6
•-2.1
•-5.5
•-1.9
•-20.5
•-2.5
•D2O
•C2v
•0.
34
•0.15
•0.
36
•2
.
7
•-1.2
•-3.1
•-1.1
•-10.6
•-1.3
•T2O
•C2v
•0.
28
5
•0.13
•0.
30
•1
.
9
•-0.83
•-2.2
•-0.76
•-7.5
•-0.90
•HDO
•C1h=
Cs
•0.
35
•0.18
•0.
48
•5
.
1
•-1.5
•-10.2
•-2.1
•-1.6
•-2.5
• 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
• As an example, a linear CO2 molecule
-- three normal vibrational modes,
asymmetric valence vibration ν3
symmetric valence vibration ν1 and a
doubly degenerate symmetric
deformation vibration ν2.
• It should be noted, that there occurs
a resonance in CO2 molecules ν1 ≈
2ν2 between the two different types
of symmetric vibrations (see Table
3.7). For this reason
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)
2
Er 
J ( J  1)  BJ ( J  1)
2I
M M
– B is the rotational constant, I  1 2 r02 is the momentum of
1  M2
inertia of the diatomic moleculeMwith
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
1
B  Be   e (v  )
2
(3.23)
• Estimate value of rotational constant (& rotational energy)
• Similar to discussion on vibrational quantum < characteristic electronic
energy, because a vibrational quantum is proportional to
m  100
M
• the values of rotational constant B and rotational energy are
proportional to 1 ( B  1  1 )
M
I
M
– This means that the value of the rotational constant < 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 (1eV=11,600K) 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 αe –Table3.8
• 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)
2
Er ( J  1)  Er ( J ) 
2( J  1)  2 B( J  1)
2I
• 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 5*10-3eV.
• In contrast to the vibrational quantum, the rotational one
corresponds to low values of the Massey parameter PMa
= ΔE/ħαv = /αv 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.
• Homework Assignment
3.1; 3.4; 3.5; & 3.6