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:

ΔS=01,0

• 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

Table 3.4. Life times and energies of electronically excited diatomic molecules and radicals on the lowest vibrational levels.1A gd 3 2A 2A gb 3 1b 2b uA3 2A

Molecule or RadicalElectronic State Energy of the State Radiative Lifetime

CO 7.9 eV 9.5*10-9 sec

C2 2.5 eV 1.2*10-7 sec

CN 1 eV 8*10-6 sec

CH 2.9 eV 5*10-7 sec

N2 7.2 eV 6*10-6 sec

NH 2.7 eV 2*10-2 sec

NO 5.6 eV 3*10-6 sec

O2 4.3 eV 2*10-5 sec

OH 4.0 eV 8*10-7 sec

1A

gd 3

2A

2A

gb 3

1b

2b

uA3

2A

• 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).

uA3 ua 1ga 1 gE 3ga 1 gb1 4a

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 2*10-4 sec

N2 11.9 eV 300 sec

O2 0.98 eV 3*103 sec

O2 1.6 eV 7 sec

NO 4.7 eV 0.2 sec

uA3

ua 1

ga 1

gE 3

ga 1

gb1

4a

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.

Potential energy diagram of a diatomic molecule AB in the ground and electronically excited state.

Electronic terms of different molecules

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

(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

200 ))](exp(1[)( rrDrU

• real dissociation energy D < Morse parameter

• 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.

2

10 DD

• 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

• 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 ħω.

20

20 )( rrDU

)2

1( vEv

I

Dr 0

0

2

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 < characteristic electronic energy I~10-20 eV

• typical value of a vibrational quantum ~ 0.1-0.2eV.

)11

(1

MIM

100Mm

• 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 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

)2

1( vEv

20

20 )( rrDU

I

Dr 0

0

2

2)2

1()

2

1( vxvE ev

04Dxe

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)

– finite number of vibrational levels

• v= vmax , corresponds Ev(v,v+1) = 0 e.g. dissociation

• distance between vibrational levels→“vibrational quantum”, f(v) • smallest “vibrational quantum” v = vmax - 1 and v = vmax

•

)1(21 vxEEE evvv

12

1max

exv

ev xvvE 2),1( maxmaxmin

• 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 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

• 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

1v...4,3,2 v

• 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, >> typical time of resonant vibrational energy exchange and some chemical reactions , stimulated by vibrational excitation.

32

3

0

112

mR p

c

100Mm

62

3

10)( mM

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:

(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

2222

2111332211321 )

2

1()

2

1()

2

1()

2

1()

2

1(),,( vxvxvvvvvvEv

)2

1)(

2

1()

2

1)(

2

1()

2

1)(

2

1()

2

1( 322331132112

2333 vvxvvxvvxvx

•Table 3.7. Parameters of oscillations of triatomic molecules.

•Molecules & Symmetry

•Normal Vibrations & their Quanta, eV

•Coefficients of Anharmonicity,•10-3 eV

•Molec.

•Sym. •ν1 •ν2 •ν3•x

1

1

•x22 •x33 •x12 •x13 •x23

•NO2 •C2v•0.1

7•0.09 •0.

21•-1.1

•-0.06 •-2.0 •-1.2 •-3.6 •-0.33

•H2S •C2v•0.3

4•0.15 •0.

34•-3.1

•-0.71 •-3.0 •-2.4 •-11.7 •-2.6

•SO2 •C2v•0.1

4•0.07 •0.

17•-0.49

•-0.37 •-0.64 •-0.25 •-1.7 •-0.48

•H2O •C2v•0.4

8•0.20 •0.

49•-5.6

•-2.1 •-5.5 •-1.9 •-20.5 •-2.5

•D2O •C2v•0.3

4•0.15 •0.

36•-2.7

•-1.2 •-3.1 •-1.1 •-10.6 •-1.3

•T2O •C2v•0.285

•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)

– 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)

)1()1(2

2

JBJJJI

Er

20

21

21 rMM

MMI

)2

1( vBB ee

• Estimate value of rotational constant (& rotational energy)

• Similar to discussion on vibrational quantum < 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 < 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

100Mm

)11

(1

MIB

M

• 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)

)1(2)1(22

)()1(2

JBJI

JEJE rr

• 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