APPENDIX 2: DERIVATION OF SPEED DEPENDENT RAUTIAN PROFILE FORMULA

Keywords: DNO 3, Infrared spectroscopy, Positions, Intensities, Coriolis resonance, ν 5 band, ν 7 + ν 9 band, CH 3 F, Microwave spectroscopy, Line shapes, Rautian profile, Speed dependent profile, Collisional broadening, Stark effect, Dipole moment, Line-Mixing, IOS

The work presented in this thesis belongs to the domain of theoretical and experimental gas phase molecular spectroscopy. It consists of two parts dedicated to two relatively different aspects in this field.

The first part presents a high resolution analysis of the ν 5 fundamental band (NO 2 in plane bending mode) positions and intensities of D 14 NO 3 (deuterated nitric acid) in the 11 µm spectral region. For this study, we used an infrared spectrum of D 14 NO 3 recorded in the 700-1400 cm -1 region on a Fourier transform spectrometer at Bergische Universität in Wuppertal (Germany). Our analysis demonstrates that the fundamental ν 5 band centered at 887.657 cm -1 is strongly perturbed. Indeed, it proves that 5 1 and 7 1 +9 1 energy levels of DNO 3 are coupled through A and B type Coriolis resonances. The resonance scheme for the isotopologue D 14 NO 3 therefore differs substantially from the schemes of H 14 NO 3 and H 15 NO 3 that feature dominantly Fermi type resonances.

The second theme treated in this work is devoted to the lineshapes of pure rotational transitions of CH 3 F with the study of collisional broadening (collisions CH 3 F-CH 3 F and CH 3 F-He) of optical transitions and their Stark components. The microwave measurements were realised at ICT in Prague. Their analysis enabled to provide collisional parameters using various line profiles (Voigt, Rautian, Speed dependent) for the J, K → J + 1, K (K = 0, …, J) transitions with J = 1 and J = 3 as well as for their various Stark components J, K, M → J + 1, K, M' (|M| = 0, …, J ; |M -M'| = 0, 1). Moreover, a correct use of model based on Infinite Order Sudden approximation led to very satisfactory results of the observed line-mixing effects. The retrieved experimental results complete and extend the previous studies and provide the first successful demonstration of the ability of the IOS approximation to model line-mixing effects among Stark transitions.

Résumé

Le travail qui est présenté dans ce mémoire de thèse s'inscrit dans le domaine de la Spectroscopie moléculaire en phase gazeuse théorique et expérimentale. Il comprend deux parties dédiées à des aspects relativement différents de cette discipline.

La première partie présente l'analyse à haute résolution des positions et intensité des raies de la bande ν 5 (ouverture angulaire du groupe NO 2 dans le plan de la molécule) de DNO 3 dans la région spectrale de 11 µm. Pour ce faire, nous avons utilisé un spectre infrarouge enregistré dans la région 700-1400 cm -1 à l'aide d'un spectromètre à transformée de Fourier de l'université « Bergische Universität » à Wuppertal, en Allemagne. Ce travail nous a permis de montrer que la bande ν 5 , centrée à 887.657 cm -1 , est fortement perturbée. L'analyse prouve en effet que les niveaux d'énergie 5 1 et 7 1 +9 1 sont couplés par des résonances de Coriolis de types A et B. Le schéma de résonances pour la variété isotopique DNO 3 diffère donc fortement de ceux observés pour les états 5 1 et 9 2 de H 14 NO 3 et H 15 NO 3 qui sont majoritairement de type Fermi.

Le deuxième problème abordé dans ce travail est celui des profils de raies rotationnelles pures de CH 3 F avec l'étude des élargissements collisionnels (collisions CH 3 F-CH 3 F et CH 3 F-He) des transitions optiques et de leurs composantes Stark. Des mesures microondes ont été réalisées à l'ESTC à Prague. Leur analyse a permis d'en extraire des paramètres collisionnels à l'aide de divers profils de raies (Voigt, Rautian, dépendantes de vitesse) pour les transitions J, K → J + 1, K (K = 0, …, J) avec J = 1 et J = 3 et leurs diverses composantes Stark J, K, M → J + 1, K, M' (|M| = 0, …, J ; |M -M'| = 0, 1). De plus, un modèle fondé sur l'approximation Infinite Order Sudden (IOS) a été construit pour décrire les effets d'interférences collisionnelles observés. Les résultats expérimentaux obtenus complètent et étendent des travaux précédents et sont la première démonstration de la capacité de l'approximation IOS à modéliser les couplages collisionnels entre composantes Stark.

Mots clefs: DNO 3 , spectroscopie infrarouge, positions, intensités, résonance Coriolis, bande ν 5 , bande ν 7 + ν 9 , CH 3 F, spectroscopie microonde, profils de raies, profil de Rautian, profil avec dépendance de vitesse, élargissement collisionel, effet Stark, moment dipolaire, couplage des raies, approximation IOS.

Shrnutí

Předkládaná dizertační práce tematicky zahrnuje jak teoretickou, tak i experimentální molekulovou spektroskopii v plynné fázi. Předmětem práce jsou dvě relativně nezávislé studie, které spojuje právě zaměření na vysoce rozlišenou spektroskopii plynů.

V první části je vypracována analýza poloh a intenzit linií vysoce rozlišeného spektra základního vibračního pásu ν 5 (mód kyvadlové rovinné vibrace skupiny NO 2 ) DNO 3 v 11 µm spektrální oblasti. V této studii bylo použito infračervené spektrum naměřené v rozsahu 700-1400 cm -1 na FT spektrometru v Bergische Universität ve Wuppertalu (Německo).

Tato práce ukazuje, že základní pás ν 5 o poloze 887.657 cm -1 je silně porušen. Podrobná analýza naměřených spekter dokázala, že energetické hladiny 5 1 a 7 1 +9 1 jsou spřaženy prostřednictvím Coriolisových rezonancí typu A a B. Rezonanční schéma isotopologu DNO 3 se tedy podstatně liší od rezonancí pozorovaných u H 14 NO 3 a H 15

Thesis aims (form as required by ICT Prague)

The thesis projects aim at three tasks:

1. Analysis of rovibrational spectra of the deuterated nitric acid (DNO 3 ) fundamental band ν 5 with respect to the interactions between states 5 1 and 7 1

2. Measurement of the methyl fluoride (diluted or not in helium) pure rotational microwave spectra with and without Stark effect.

3. Analysis of the collisional effects observed in the measured spectra of CH 3 F, comparison of the Stark lines' experimental collisional broadening values with those calculated using Infinite Order Sudden (IOS) approximation method.

I Introduction

The work presented in this thesis had been done in the frame of co-tutelle thesis program organized by Université Paris-Est (UPE) and by Institute of Chemical Technology, Prague (ICT Prague). Some studies have been carried out in the Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA, CNRS UMR 7583) under the supervision of Dr. Agnès Perrin and at the Department of Analytical Chemistry (DAC-402) of ESCT Prague in the laboratory LMSVR under direction of Prof. Štěpán Urban. Another topic of the thesis, suggested at the end of the first year of the co-tutelle, was studied afterwards and continued further on, in cooperation with Dr. Jean-Michel Hartmann (LISA) and Prof. Christian Boulet (ISMO). The text describes therefore two completely independent spectroscopic studies, as the thesis name suggests: an analysis of the infrared spectra of deuterated nitric acid (DNO 3 ) and an analysis of collisional effects in the Stark and non Stark microwave spectra of the methyl fluoride (CH 3 F). These two subjects are introduced in the following two sections.

The lay-out of the thesis text is presented afterwards.

I.1 Rovibrational study of DNO 3 ν 5 band

Nitric acid (HNO 3 ) plays an important role as a ''reservoir'' molecule for both the NO x (nitrogen oxides) and HO x (hydrogen oxides) species in the stratosphere [START_REF]Scientific Assessment of Ozone Depletion: 1998[END_REF][START_REF]Scientific Assessment of Ozone Depletion: 2006[END_REF]. These radicals are potentially active contributors to the ozone destruction in the stratosphere through catalytic reactions. The 11 µm region corresponds to one of the strongest absorption bands of nitric acid and coincides with a rather clear atmospheric window of the atmosphere. This spectral region is therefore commonly used for the retrieval of nitric acid in the atmosphere, not only for the most abundant isotopic species (H 14 NO 3 , Refs. [START_REF] Mencaraglia | [END_REF]4) and references therein) but also for H 15 NO 3 (isotopic concentration of about α ~ .00365 (7)) which could be detected by MIPAS (Michelson Interferometer for Passive Atmospheric Sounding) aboard the ENVISAT satellite (5). For this reason, numerous spectroscopic studies were devoted to the improvement of line position, line intensities ((6-9) and references therein) and line broadening parameters (10) for nitric acid at 11 µm for both the H 14 NO 3 and H 15 [START_REF] Mencaraglia | [END_REF] species.

On the other hand, the chances to detect DNO 3 in atmospheric spectra at 11 µm are rather weak. DNO 3 exists with rather weak isotopic abundance in a natural sample of nitric acid (α ~1. 55 10 -4 ) and the ν 5 band of DNO 3 is located at 887.7 cm -1 in the middle of the 11 µm absorption region for H 14 [START_REF] Mencaraglia | [END_REF] , where strong absorptions from the ν 5 , 2ν 9 and ν 5 +ν 9 -ν 9 Q-branches are clearly observable in atmospheric spectra at 879.109, 896.448, 885.424 cm -1 , respectively. However studying DNO 3 is interesting since it provides input for the validation of recent and future ab initio studies on nitric acid (11). Indeed the resonance scheme for the interacting DNO 3 vibrational states differs as compared to H 14 NO 3 and H 15 NO 3 .

I.2 Collisional effect studies of CH 3 F microwave spectra with and without Stark effect

Line-mixing (LM) is a process manifesting through transfers of intensity among overlapping optical transitions and resulting from rotational population exchanges induced by intermolecular collisions. It has been the subject of numerous experimental and theoretical studies for various molecular systems and types of spectra, as reviewed in Chapter IV of Ref. [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF]. However, few of these studies have considered the case of collisional exchanges among Stark resolved molecular lines, i.e. when an applied electric field removes the spatial degeneracy of the rotational levels. As first shown by Bréchignac (12) and afterwards measured (13)(14)(15) for CH 3 F in the IR region, the "zero field widths" are much smaller than the widths of the various Stark resolved lines. Various theoretical treatments have been proposed to describe collisional couplings between M-components. Let us first mention the pioneering work of Buffa, Tarrini and co-workers (13), based on a generalization of the well known Anderson-Tsao-Curnutte (ATC) model (16). An alternative approach to take LM into account is to build the associated relaxation matrix using the Infinite Order Sudden (IOS) approximation [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF][START_REF] Belli | [END_REF] or its improved Energy Corrected Sudden (ECS) extension [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF]19,20). IOS/ECS models have enabled precise modeling of LM effects for many systems and types of spectra [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF], including hyperfine components due to nuclear spin (21,22) and symmetric-top spectra (23,24). However, quite surprisingly, as shown in Ref. (14), the IOS model developed in (12) apparently failed to reproduce the details of coupling processes between the Stark components of CH 3 F lines. With the aim of clarifying this result we have re-examined the problem and the present work shows that this failure was only due to an oversimplification of the IOS formalism.

During the analysis of the measured CH 3 F pure rotational spectra using Voigt line shapes, a "W signature", characteristic of a narrowing, was observed in the fits residuals. The line shapes were therefore also studied with more refined models that involve a more thorough description of collisional effects by taking Dicke and speed dependence effects into account, in order to attempt to diminish the residuals and to understand the contributions of various collisional processes.

* * *

The remained of the thesis text is divided into the following parts: After this introductory chapter, a chapter dedicated to infrared spectra of deuterated nitric acid is presented, in which the general theory is first recalled. Then the model used for the rovibrational spectra of asymmetric top molecules like DNO 3 is discussed following the approach established by J.K.G. Watson for cases with vibrationaly isolated states and generalised further by J.-M. Flaud and C. Camy-Peyret for cases with interacting vibrational states. The final section of the chapter presents the results of the analysis of ν 5 band of DNO 3 using the previously mentioned formalism.

The next three chapters are devoted to collisional effects on rotational spectra of CH 3 F.

The third chapter describes the experimental apparatus used for in this study as well as some experimental details on the measurements made using the Stark effect.

The fourth chapter is devoted to the analysis of the shapes of isolated lines in the pure rotational CH 3 F spectra. Some general theory is first mentioned before some selected models describing isolated line profiles are discussed. The results obtained CH 3 F J = 1 and J = 3 pure rotational transitions are then presented and discussed.

The fifth chapter is devoted to the analysis of the Stark pure rotational CH 3 F spectra in situations when lines overlap and collisional line mixing effect get involved. The modelling of this effect using the Infinite Order Sudden (IOS) approximation is detailed before the analysis of Stark and non Stark CH 3 F (pure or diluted in helium) J = 1 and J = 3 pure rotational transitions is presented.

The next and final parts cover a general conclusion, references and appendices.

II Vibration -rotation study for a molecule of atmospheric and theoretical interest: the case of the DNO 3 isotopomer of nitric acid

The present chapter is organized in the following way. Paragraph II.1 will describe the overall approximation which led to establish the vibration rotational Hamiltonian written in the form given by Darling and Dennison. Assuming that the nuclei vibrations are small compared to the interatomic distances, the Hamiltonian can be developed in series according to the vibrational quanta. The structure of the DNO 3 , its vibrational modes, its symmetry properties will be presented in Paragraph II.2.1 together with the rigid rotor rotational wavefunctions. The selection rules for an electric dipole moment transition will be deduced in Paragraph II.2.3. The "contact transformations" which lead to a partial block diagonalization of the vibration rotation Hamiltonian will be presented in Paragraph II.2.4. Finally the effective Hamiltonian used in this work will be described shortly in Paragraph II.3.4.

II.1 General considerations, Rovibrational Hamiltonian

A molecule can be considered as a rather stable aggregate of atomic nuclei and electrons.

The time dependence of Φ(t) is defined by the Schrödinger equation:

) ( ) ( t H t t i Φ = ∂ Φ ∂ (II.1)
where Φ(t) is the molecule wavefunction of the molecule which is function of its electronic and nuclear coordinates. In Eq. (II.1) H is the Hamiltonian of the molecule which is assumed, in the following, to be time independent. In this way the Φ(t) eigenfunctions for the different stationary states are solutions of the equation:

) ( ) ( t E t H Φ = Φ (II.2)
Solving Eq. (II.2) for a molecular system consisting in moving particles (several atoms and numerous electrons) is usually impossible. Several more or less severe approximations must be formulated.

¤ The Born Oppenheimer approximations (25):

The fact that the nuclei are much heavier than electrons (neutron-electron mass ratio m n /m e = 1838.683 6605 (11) -2006 CODATA values (26), proton-electron mass ratio m p /m e = 1836.152 672 47 (80)) led Born and Oppenheimer (25) to formulate approximation that fast movements of electrons are separated from the relatively slow movements of nuclei.

The rovibronic (electronic vibration rotational) Hamiltonian then takes the form:

evr e vr H H H = + , (II.3)
where H e is electronic Hamiltonian describing the movements of electrons considered for a given fixed nuclear geometry, and H vr is the rovibrational (vibration rotational) Hamiltonian describing the movement of the nuclei for the corresponding electronic state.

In the following of this manuscript, the Born Oppenheimer approximation is assumed to be appropriate. The studies presented here are for molecules (nitric acid and methyl fluoride) in their ground electronic state.

¤ The molecules under study are assumed to be isolated and moving in an isotropic laboratory space. By using a reference system of axes centered at the molecular center of gravity, O, the translation of the center of inertia is completely separated from the internal coordinates (vibration and rotation) of the molecule. The so-called vibration-rotational Hamiltonian involve 3N -3 degrees of freedom, for a molecule with N nuclei. We are dealing with non linear polyatomic molecules, and the molecular rotation and the nuclear vibrations around their equilibrium positions are associated to 3 and 3N -6 degrees of freedom, respectively.

¤ It is assumed that the amplitudes of the nuclei vibrations are small compared to the interatomic distances. This is relevant for the present DNO 3 study. Let us mention, however, that this approximation fails when considering the OH torsional motion (ν 9 mode) for the H isotopomers of nitric acid (H 14 NO 3 and H 15 NO 3 ).

Even in such conditions, it is not possible to separate completely rotation from the vibration motion. At this level we need to define two orthogonal systems of axes centered at O, the molecular center of gravity: a floating O xyz -reference system of molecular axis -bound to a reference molecular configuration (i.e. equilibrium configuration in case of rigid or semirigid molecules) and the O XYZ "laboratory fixed" reference system that is in translational motion relative to the laboratory system. In order to fix the origin, O, of both systems to the center of gravity of the molecule, the following relationship applies:

N i i i 1 m = = ∑ r 0 (II.4)
where r i is instantaneous radius vector of nucleus i with mass m i .

In order to minimize the interactions between rotation and vibration, Eckart wrote the following constrains (27) which set the orientation of O xyz relative to O XYZ :

N 0 i i i i 1 m = ∧ = ∑ r r 0, (II.5)
where r i 0 is radius vector of nucleus i at its reference position.

In this way, the vibration-rotation Hamiltonian, as developed by Darling-Dennison (28) takes the following form, presented here as it was simplified later by Watson (29):

( ) ( ) 2 2 , 1 2 2 k k H J J P U V α α α β β β α β π μ π = - - + + + ∑ ∑ (II.6)
In this equation, α and β refer to the molecular axes with values x, y or z, k refers to the 3N -6 normal coordinates Q k . The other symbols are defined as follows:

-J α are components of the molecular axes of the total angular momentum in units α π are components on the molecular axes of the vibrational moment defined as:

, , 1 k l k l k l Q P α α π ς = ∑ (II.7)
αβ μ is matrix element (α, β) of the inverse matrix of the effective inertia tensor referring to the O xyz molecular system of axis k P is conjugate moment to the normal coordinate Q k , and V is the vibrational potential energy function whose quadratic part can be diagonalized through the 3N -6 vibrational coordinates:

(3 6) 2 1 2 1 ... N k k k V Q λ - = = + ∑ (II.8)
and:

-

U αα α μ = ∑ , (II.9)
as shown by J.K.G. Watson (29) In the following of the text, we will use , k k q p which are the dimensionless normal coordinates and their conjugate momenta defined by:

1/4 1/2 , k k k k k q Q p i q λ ∂ = =-∂ (II.10)
In case of small vibrations (in comparison to interatomic distances), V and αβ μ can be expanded as series in the normal coordinates q k : 0 1 2

H H H H

= + + +… (II.11)
with:

( )

2 2 2 0 1 2 k k k k H p q BJ α α α ω = + + ∑ ∑ (II.12) 1 2 , 1 , , 2 l k k k l kl klm k l m k k l k l m k H B J J q q p B J K qqq α β α α β α α α β α ω ς ω < < ⎛ ⎞ = - + ⎜ ⎟ ⎝ ⎠ ∑∑ ∑ ∑ ∑ (II.13) ( ) ( ) 1 2 , , , , 2 3 8 1
4 k l k l k l k k k k l k klmn k l m n k l m n H qq B B B B B J J B q q J B B K qqq q γα γβ γβ γα αβ γ α β α α β α β γ α β α α α α α π π π - < < < = + - + + - + ∑ ∑ ∑∑ ∑ ∑ ∑ (II.14)
In these formulas, k ω is the harmonic vibrational wavenumber of the kth normal mode:

1/4 2 k k c λ ω π = (II.15)
-B α is the equilibrium rotational constants (in cm -1 ):

2 8 B cI α α π = (II.16) - , k B α β are rotational derivatives: ( ) 2 , 1 0 0 , k k l k l k k l B B B B B B q qq αβ αβ α β γα γβ γβ γα γ μ μ - ⎛ ⎞ ∂ ∂ ⎛ ⎞ = + = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂ ∂∂ ⎝ ⎠ ⎝ ⎠ (II.17) -kl α ς are Coriolis coupling constants - α π
are components on the molecular axes of the vibrational momentum:

1 2 , , 1 l k l k l k l k q p α α ω π ς ω ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ ∑ (II.18)
klm K , (resp. klmn K ) are cubic, (resp. quartic) potential constants.

We can rewrite the Eq. (II.12) as:

0 0 0 vib rot H H H = + , (II.19)
where ( )

3 2 2 0 1 1 2 vib k k k k H p q ω = = + ∑ , (II.20) 2 0 rot H B J α α α = ∑ , (II.21) with 0 0 , , , vib rot k B H H α ω in cm -1 .
The zeroth order Hamiltonian is a rough approximation. At this level, the vibration and rotation are separated, and it is easy to label the vibrorotational energy levels. We will detail the description of the eigenfunctions and eigenvalues of the rotational and vibrational zeroth order Hamiltonian. Let us remind, however, that at higher orders (H 1 , H 2 and over …) the vibration and rotational motions are coupled. Recently, a semi-experimental equilibrium structure has been derived from experimental ground state rotational constants achieved for several isotopomers of nitric acid ((30), (31) and Refs. therein) and of rovibrational interaction parameters calculated from the ab initio force field. In this way, the best equilibrium structure was found to be A (cm -1 ) B (cm -1 ) C (cm -1 ) Ref.

II.2 Application to

H 14 Therefore nitric acid may be considered as a "near oblate rotor".

For a near-symmetric top molecule, it is useful to define the (x,y,z) reference axes, choosing the z axis as the symmetry axis of the rotor. In this way, the standard base J k formed by eigenvectors common to 2 J and z J is the bases of all rotational wavefunctions. is the z-reference axis. However, the most recent and accurate determinations of the rotational constants for various isotopomers of nitric acid by millimeter wave techniques (30,31) were satisfactorily performed using a rotational Hamiltonian written in the I r representation (x = b, y = c, z = a), and the so called A-type reduction which will be defined further in the text. Therefore the I r convention was also adopted in this study. Under these conditions, the zeroth order rotational Hamiltonian has following form: v v v and of the rotational eigenstate of 0 rot H , denoted as

0 0 ( ) 0 0 a c rot rot a c J K K a c H J K K E J K K = (II.
0 a c J K K : 0 0 0 a c a c J K K J K K = … … 1 2 9 1 2 9 v v v v v v (II.25)
The zeroth-order vibrational Hamiltonian, 0 vib H , is a tensorial product of nine non degenerated harmonic oscillators. Each corresponding vibrational eigenstate,

0 … 1 2 9 v v v
is the product of nine i v harmonic oscillators eigenstates. Each associated eigenvalue is the sum of the nine associated eigenvalues: The equilibrium structure of nitric acid has two elements of symmetry: the identity E and the reflection in the molecular plane σ xz . Therefore the appropriate symmetry point group is

0 1 2 9 1 1 1 2 2 2 1 2 9 v v v vib E ω ω ω ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = + + + + + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ … (II.
C s with { } , s C E σ = .
This point group is isomorph to the G 2 = {E, E*}, where E* is the inversion of the positions of all particles (electron and atomic nuclei) in the center of mass.

Table II.3 presents the characters of the C s group together with the symmetry species of the molecular (μ x ,μ y ,μ z ) and space-fixed (μ Z ) electric dipole moment components, of the total angular moment J molecular components, of the normal coordinates, and the rotational levels with regards to the K c parity.

C s E σ xz [J K a K c ] G 2 E E* μ J Vibrations parity of K c A' 1 1 μ x μ y J z q 1 , ..., q 7 even A" 1 -1 μ z , μ Z J x J y q 8 , q 9 odd
G 2 or C s .
For DNO 3 such large amplitude motions are expected to be significantly weaker than for HNO 3 because of the mass ratio D/H~2. For example the tunneling splitting in the v 9 = 2 vibrational state of DNO 3 is expected to be only of 2 MHz, instead of 60 MHz for HNO 3 .

Therefore, the D-O tunneling around the O O N < moiety will not be considered for this DNO 3 study.

II.2.2.2 Symmetry properties of H 0 eigenstates

The eigenstates of H 0 can also be classified using the two irreducible representations of C s group, their type of symmetry Γ (Γ = A' or A'') depends on the parity of the rotational and vibrational quantum numbers and is obtained by multiplication of the types of symmetry of the vibrational state and of the rotational level:

( ) ( ) ( ) 0 0 0 a c a c J K K J K K Γ = Γ ⊗ Γ … … 1 2 9 1 2 9 v v v v v v (II.27)
Afterwards, it is convenient to calculate the matrix elements of the operators involved in the Hamiltonian using a symmetry adapted basis defined from the J k and 0

… 1 2 9 v v v .

II.2.2.2.1 Symmetry of vibrational states

The C s symmetry group is an Abelian group and the type of symmetry of a particular vibrational state …

1 2 9
v v v is therefore given by the direct product of the types of symmetry of its each vibrational state i v :

( ) ( ) ( ) ( )

1 2 9 1 2 9 v v v v v v Γ = Γ ⊗ Γ ⊗ ⊗ Γ … … (II.28)
The eigenfunctions of an one dimension harmonic oscillator i v , are Hermite's polynomes, H v (q i ). During the q i → -q i change of sign, the i v wavefunction changes as:

q i → -q i ( ) i i i i v v i v i v i ( ) ( ) 1 ( ) H q H q H q → -= - . (II.29)
The symmetry type of the i v vibrational wave function according to C s depends on the i and v i values. For i = 1 to 7, the q i normal mode does not change during any C s symmetry operation (E or σ xz ). Consequently, ( ) '

Γ = i v A for i =
Γ = + = + = … 1 2 9 8 9 8 9 v v v A' (resp. A'') if v v even (resp. v v odd ) (II.30) ( ) ( ) ( )

Symmetry of rotational levels

The inertia ellipsoid has at least three planes of symmetry and three binary axes, whatever the molecular symmetry group is. The molecular rotation Hamiltonian is therefore invariant under any operation of the V symmetry group (Vierergruppe, noted also D 2 ): identity, three rotations by π around the respective axes x, y and z. The symmetry types of the rotational levels in this group are hence to be determined. The characters of the V symmetry group are given in Table II.4.

V (D 2 ) E C x 2 C y 2 C z 2 A 1 1 1 1 B x 1 1 -1 -1 B y 1 -1 1 -1 B z 1 -1 -1 1

Table II.4 Character table of the V group of symmetry

As was already mentioned, the eigenstates of 0 rot H are determined through diagonalization of this Hamiltonian in a standard basis J k , and the wavefunctions 0 a c J K K are given as linear combinations of J k basis sets. In order to determine the symmetry of the 0 a c J K K levels it is necessary to determine the symmetry of the J k functions.

An appropriate symmetric basis of rotational wavefunctions is the Wang type basis defined as:

( )

1 2 J K J k J k γ γ = + - (II.31) with K = |k|, γ = ±1 and 0 1 0 J J γ = + = (II.32)
The J K γ are grouped in four classes named usually E + , E -, O + and O -with regards to the K parity (K even or odd) and the γ sign (γ = ±1) .

The following useful relations were demonstrated (40):

( )

exp z R J k i k J k β β = (II.33) ( ) ( )
1 exp 2 J R J k i k J k π α α = - - - (II.34)
In this expression z R β is the rotation of a β angle around z of the x and y axes. Also, R π α is the π rotation of the x, y and z axes around an "α " axis. This "α " axis is located in the (xy) plane with an α angle with regards to the x axis.

The three rotation operations of the group

V (C x 2 , C y 2 , C z 2 ) are equivalent to the 0 / 2 , , z R R R π π π π
operations, respectively. The equations (II. [33][34] give the J K γ symmetry types under the V operations. The irreducible representation of this group under the I r convention (x = b,

y = c, z = a) is: , , A A, B B B B B B x b y c z a → → → → (II.35)
Table II.5 gives the symmetry of the basis functions J K γ under the V symmetry group following the I r convention.

J even J odd E + A B a E - B a A O + B b B c O - B c B b
Table II.5 Symmetry of fonctions J K γ under the V group of symmetry

Under the under the I r convention, the 0 a c J K K symmetry types according to V operation are deduced from those of J K γ in the two limit cases, K → K a and K → K c . This is possible since the levels sharing the same J do not cross each other and the symmetry properties are conserved. As a final result, the symmetry type of 0 a c J K K depends only on the K a and K c parity, as reported in the Once the symmetry types of the 0 a c J K K and J K γ wavefunction relative to V are established, it is easy to set the symmetry properties of these wavefunctions relative to the C s point group. Indeed V is homomorph onto G 2 or C s .

Therefore, the correspondence between the irreductible representations of the V and C s group is obtained:

; A, B A'; B , B A'' s y a b V C → → → (II.36)
The symmetry of the 0 a c J K K levels and of the basis functions J K γ under the C s symmetry group is given in Tables II.7-8. parities, as is shown in Table II.9.

J even J odd E+ A' A'' E- A'' A' O+ A'' A' O- A' A''
J K a K c K c even A' K c odd A''
K c even K c odd (v 8 +v 9 ) = even A' A'' ( ) 0 a c J K K Γ … 1 2 9 v v v (v 8 +v 9 ) = odd A'' A'
Table II.9 Symmetry of rovibrational levels of DNO 3 under C s symmetry group

II.2.3 Selection rules

The intensity of an electric dipole moment transition between two levels

' ' a c J K K … 1 2 9 v ' v ' v ' ' and '' '' a c J K K … 1 2 9 v '' v '' v '' '' is proportional to the square of the transition moment matrix element: | ' ' '' '' a c Z a c J K K J K K μ … … 1 2 9 1 2 9 v ' v ' v ' ' v '' v '' v '' '' | 2 (II.37)
where Z μ is the component on the laboratory axis Z of the molecule electric dipole moment μ . In order this matrix element be non-zero, the following conditions must be satisfied:

-For an electric dipolar transition, one has:

0, 1 J Δ = ± (II.38)
-The transition moment matrix element must belong to the totally symmetric representation of the molecular symmetry group. For a C s type molecule this is stated by:

( ) ( ) ( ) ' ' '' '' ' a c Z a c J K K J K K A μ Γ ⊗ Γ ⊗ Γ = … … 1 2 9 1 2 9 v ' v ' v ' ' v '' v '' v '' '' (II.39)
Following Table II

( ) ( ) ' ' '' '' '' a c a c J K K J K K A Γ ⊗ Γ = … … 1 2 9 1 2 9 v ' v ' v ' ' v '' v '' v '' '' (II.40)
The symmetry types of the vibration -rotational wavefunctions are gathered in Table II.9.

Using these results, it is easy to deduce the ' ''

a a a K K K Δ = - and ' '' c c c K K K Δ = - selection
rules as a function of the Δ(v 8 +v 9 ) variation. These results are collected in Table II.10. Let us remind that the ν 5 and ν 7 +ν 9 bands of DNO 3 bands correspond to Δ(v 8 +v 9 ) = 0 and Δ(v 8 +v 9 ) = 1, respectively.

odd K a = Δ even K c = Δ even K a = Δ odd K c = Δ odd K a = Δ odd K c = Δ Table II.

Selection rules for an electric dipole moment transition

For vibrational transitions with Δ(v 8 +v 9 ) = even, the band is in principle hybrid, with both Aand B-type characters. This is the case, in principle for the ν 2 to ν 7 fundamental bands of H 14 NO 3 or DNO 3 . For example, during the analysis of the ν 4 band of H 14 NO 3 , both A and B type vibro-rotational transitions were observed [START_REF] Perrin | [END_REF].

It is important to underline that for a hybrid band the A to B character depends on the relative values of the a to b components of the transition moment operator of the considered band.

Indeed, for H 14 NO 3 , H 15 NO 3 and DNO 3 , the ν 2 bands are almost pure B-type bands, while ν 3 , ν 6 and ν 7 are mostly A-type bands. This is obvious when comparing the general structure of the ν 6 to ν 2 bands of nitric acid (Figs II.2 and II.3, respectively). Finally the ν 5 band is a pure A-type band for H 14 NO 3 and H 15 NO 3 , and is hybrid for DNO 3 with a significantly stronger Atype component as compared to its B-type counterpart.

II.2.4 Contact transformation and energy levels calculation

It is possible, in principle, to compute the rotation-vibration energy levels of a given molecule together with its associated transition intensities. For this it is necessary to build a potential energy surface generated through ab initio calculations and subsequently fitted against experimental data. Such studies are presently performed for water (43), methane (44), ozone (45), phosphine (46). These calculations necessitate important computation capacities and a large set of accurate experimental data to validate the results of the calculations. Even in this case, the accuracy of the calculation is often far from the experiment. Such theoretical methods are far for being applicable for nitric acid at the present time.

Nitric acid with five atoms, several N-O bonds and the existence of the large amplitude OH bonds is quite a challenge for ab initio calculations (11). In addition the experimental data are still scarce. To give an example, the ν 1 fundamental band at 3 µm was not yet investigated at high resolution. Finally, in the literature, the infrared studies pointed out the existence of numerous complicated resonances.

For this reason it is not possible to perform a full diagonalisation of the vibration-rotation Hamiltonian. To perform the analyses presented here, it is necessary to use effective Hamiltonians which were generated through the so called contact transformations which are described here.

II.2.4.1 Vibrational Hamiltonian

A widely used perturbation method in molecular spectroscopy is the unitary transformation method first proposed by Van Vleck (47) and developed further in Refs. (48)(49)[START_REF] Amat | Rotation-vibration of polyatomic molecules: Higher order energies and frequencies of spectral transitions[END_REF].

This method leads to a partial vibrational diagonalization of the Hamiltonian matrix in vibrational blocks and is based on the physical properties of the molecule. Each vibrational block groups together vibrational states with close enough vibrational energies. These states are presumed to interact together, while interactions between states belonging to different blocks are assumed to be much weaker. The goal of the contact transformation method is to eliminate, at a given order of approximation, the matrix elements coupling the states that do not belong to the same polyad.

For instance, let the Hamiltonian H be developed:

0 1 2 p H H H H H = + + + + + … … (II.41)
The first contact transformation T 1 leads to: v v v following the scheme of resonance.

The contact transformation may be repeated, in principle, until the elimination, in sufficient order of approximation, of all operators coupling states belonging to different polyads.

The Hamiltonian matrix appears then in a bloc diagonal form following a chosen resonance scheme. This simplifies the Hamiltonian matrix diagonalization because it is sufficient to calculate individually each block related to the particular polyad of interacting states.

However, as it is detailed in Ref. [START_REF] Papousek | Molecular Vibrational--Rotational Spectra[END_REF], it is important to underline that this method is suitable only if several conditions are fulfilled.

The T i (i = 1, 2, …) unitary operators are usually written as T i = exp(S i ). The vibration (and or) rotational operators, S i , must obey the following conditions.

-as it is the case for H i , the S i operators must fulfill strict symmetry conditions for all the terms appearing at a given order in the expansion of the T expansion.

-There exist also severe criteria of convergence, both for the H i and S i operators appearing in the expansion of the Hamiltonian and of T, respectively.

More or less, and without going into details, one must have:

η ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
where m e and m p are the electron and proton masses, respectively, the other one adopted by Nielsen ( 52) is ( )

1/ 2 B η ω = ,
H H = = vv T T v v v v v v H (II.45) In the vibrational state v , rot H E H = + vv v v
contains only the operators acting on rotational functions, i.e. the powers of operators ( , , ) J

x y z α α = and can be written as:

rot H E H = + vv v v , (II.46)
Where E v is vibrational energy dependent on v and coinciding with the band center, in this case of non resonating state. rot H v is the rotational Hamiltonian of the state v which can be written in general form proposed by J.K.G. Watson (29,[START_REF] Watson | Vibrational Spectra and Structure[END_REF][START_REF] Watson | [END_REF](55)56 ) as:

( ) , , 2 v v v rot pqr p q r r q p pqr x y z z y x pqr H h J J J J J J h p q r = + ∈ + + ∈ ∑ (II.47)
This expansion of rot H v is written with strict conditions for the existing non zero pqr h v parameters and for the p, q and r exponents values which can appear at a given n order (p + q + r = n). Indeed, rot H v must be fully symmetrical in the molecular symmetry group, and remain unchanged during other symmetry operations (time reversal, exchange of the particules etc…).

The direct diagonalization of H vv gives the energy levels if the v pqr h coefficients are known.

The inverse problem consists in the determination of the v pqr h coefficients from the observed eigenvalues which are the experimental energy levels. As J.K.G. Watson showed all the v pqr h coefficients cannot be determined simultaneously from the experimental energy levels.

The method proposed by J.K.G. Watson consists in performing a series of rotational contact transformations. These unitary transformations do not change the eigenvalues of a given rotational Hamiltonian and eliminate the undeterminable parameters at a given order.

The used unitary operator is v iS e where S v is a rotational operator that can be expanded as a series with respect to the components of J :

( ) , , 2 v v v pqr p q r r q p pqr x y z z y x pqr S s J J J J J J s p q r = + ∈ + + ∈ ∑ (II.48)
There are multiple ways of reduction to go from v rot H to )).

iS iS red rot H e H e - = v v v v (J.K.G. Watson ( 53

Case of an orthorhombic molecule:

Orthorhombic molecule are molecules for which the symmetry point group is isomorph to V (C 2v , D 2 etc..) or homomorph into V (D 2V etc..). For this type of molecules several types of reductions exist, among which Watson described in detail the A and S type reductions.

We will only consider here the A reduction, where v red H has non zero matrix elements only for 0, 2 K Δ = ± . In this way, the rotational Hamiltonian is written as:

(

2) (4) (6) v red H H H H = + + +… , (II.49) with ( ) ( ) ( ) 2 2 2 (2) 1 1 1 2 2 2 v v v v v v v z x y H A B C J B C B C J ⎛ ⎞ = - + + + + - ⎜ ⎟ ⎝ ⎠ J (II.50) ( ) { } 2 4 2 2 2 (4) 2 2 2 2 , 2 v v v v v K z JK z J K z x y J x y H J J J J J δ δ = -Δ -Δ -Δ - - J J J (II.51) ( ) ( ) { } { } ( ) 2 3 6 4 2 2 2 2 (6) 2 4 2 2 2 2 2 2 , , 2
v v v v v v v K z KJ z JK z J K z x y K J z x y J x y H H J H J H J H h J J h J J h J = + + + + + + J J J J J , (II.52)
where

{ } 2 2
2 , ,

xy x y J J J A B AB BA = - = + . (II.53)
Case of a C s type molecule:

We are dealing here with DNO 3 which is not an orthorhombic molecule. At the second order its rotational Hamiltonian takes a form which is more complicated than for a V molecule:

{ } z x v xz 2 z v 2 z v 2 z v , J J h J C J B J A H Red Non (2) + + + = (II.54)
In this expression the {J x ,J z } is a non-orthorhombic term. However, Watson has demonstrated that, in principle, it is possible to remove all the non-orthorhombic terms (in

{ } z x v xz , J J h
for example) by a suitable contact transformation. For this reason the form of the reduced rotational Hamiltonian written in Eqs. (II.50-53) can be used for a C s type molecule for DNO 3 .

In the present study, the rotational Hamiltonian that was used for DNO 3 was written in the A reduction and with a I r representation. Indeed the same model was successfully used recently during the ground state studies performed for HNO 3 and DNO 3 ((30, 31) and Refs. therein).

II.2.4.3 Resonance conditions

As was suggested previously, interactions can couple energy levels belonging to different vibrational states with rather close energy. These resonances must be taken into account explicitly.

Let us consider two states ,

1 2 v v with energies 1 2 0 0 , v v E E . If an interaction X is
considered, the second order corrections to the energies are: 0

0 E E - 2 1 1 2 v v v X v
. If this correction is small when compared to 1, the problem can be solved by perturbation methods.

If 0 0 E E - 2 1 v v is
of the same order as 1 2 v X v , the two states in question are in resonance.

In order that the 1 2 v X v matrix element differs from zero, several conditions must be fulfilled:

( ) ( ) ( ) 1 2 v v ' X A Γ ⊗Γ ⊗Γ = (II.55) 2 , 0 H ⎡ ⎤= ⎣ ⎦ J
, and therefore

2 , 0 X ⎡ ⎤= ⎣ ⎦ J . (II.56)
The resonance conditions for states therefore are:

-same value of J quantum number -same type of symmetry -close energy levels

II.2.4.4 Rotational operators appearing in the expansion of the v-off diagonal operators in case of resonances

When several vibrational state have to be considered in the same block of interacting states, the relevant part T H of the transformed Hamiltonian T H has the following general form:

H H = = ∑ ∑ vv' T T v v' v v' v v v' v' v v' H (II.57) with H H = vv' T v v' (II.58)
where v and v' represent vibrational states belonging to the same polyad.

¤ v diagonal blocks

When v = v', it is possible, in principle, to apply the results of the previous discussion.

By performing rotational contact transformation the v-diagonal H vv Hamiltonian is reduced to the Hamiltonian red H vv that can be written as:

red red H E H = + v vv v (II.59)
where the operator, red H v takes the form given in Eqs. (II. [START_REF] Amat | Rotation-vibration of polyatomic molecules: Higher order energies and frequencies of spectral transitions[END_REF][START_REF] Papousek | Molecular Vibrational--Rotational Spectra[END_REF][START_REF] Nielsen | [END_REF]. One defines the vibrational band center ν ~ as the (fictive) position of the [J = 0, K a = 0, K c = 0] energy levels.

The vibrational level energy E v does not always coincide with ν ~ especially when anharmonic interaction operators exist. This is particularly the case for the {5 1 ,9 2 } system of interacting states for H 14 NO 3 and H 15 NO 3 , as it can be seen in Table II.2.

¤ v ≠v' off diagonal blocks.

When strong resonances couple energy levels from different vibrational states, it is necessary to account explicitly these interactions. The theoretical approach used to carry out the calculation of the resonating energy levels is described by J.-M. Flaud and C. Camy-Peyret [START_REF] Flaud | Water Vapour Line Parameters from Microwave to Medium Infrared[END_REF]. In this approach, the expansion of the rotational operators in the off diagonal blocks is written in an expansion based on symmetry considerations. The rotational terms are, in principle, expanded according to their J power.

Indeed for a molecule with symmetry C s like DNO 3 , the

' v v

H rotational operator belongs to

A' the totally symmetric representation of the group. Following expressions apply for the non diagonal operators of H vv' :

( ) ( ) ( ) ' H Γ ⊗Γ ⊗Γ = v'v'' v v A' (II.60)
The form of the ' v v

H

rotational operator for v and ' v interacting vibrational states depends on the Δ(v 8 +v 9 ) variation, more explicitly:

¤ Δ(v 8 +v 9 ) = even, ' v v

H

= anharmonic and Fermi-type rotational operators together with C-type Coriolis operators

¤ Δ(v 8 +v 9 ) = odd, ' v v

H

= A-and B-type Coriolis operators.

Table II.11 gives the type of rotational operators which appears in the expansion of the off diagonal in v rotational operators (up to n = 2).

Δ(v 8 +v 9 ) = even Fermi or anharmonic C-type Coriolis n = 0 n = 2 n = 1 n = 2 A' rotational operators: Cst ( ) 2 z 2 x 2 z 2 , , J J J - J y iJ { } z x , J J Δ(v 8 +v 9 ) = odd A-type Coriolis B-type Coriolis n = 1 n = 2 n = 1 n = 2 A'' rotational operators: z J { } y x i , J J x J { } y z i
, J J

Table II.11 Type of rotational operators appearing in the expansion of the v-off diagonal resonating blocks as a function of Δ(v 8 +v 9 ).

If we consider the dyad {5 1 , 7 1 +9 1 } of DNO 3 , with 5 1 and 7 1 +9 1 of A' and A'' symmetry, respectively, all operators appearing in the expansion of the H vv' off diagonal interaction block are of A-or B-Coriolis types and must be of A'' symmetry.

On the other hand, for the dyad {5 1 , 9 2 } of HNO 3 , Fermi and C-type Coriolis resonances couple together the 5 1 , and 9 2 energy levels of H 14 NO 3 (7) and H 15 NO 3 (8).

Finally, let us mention that in the expansion of the interacting operators, the terms in J 1 (J x , iJ y and J z for an B-type, C-type or A-type Coriolis resonances), are usually, but not always weaker but than those in J 2 ({iJ y ,J z }, {J x ,J z }, or {J x ,iJ y }, respectively). This was discussed in details by Perevalov and Tyuterev in 1982 (58), considering the rotational constants values, the strength of the resonance, and the distance between the interacting vibrational states.

II.2.4.5 Symmetrized basis for energy level calculation

The expansion of

a c J K K … 1 2 9 v v v wavefunctions is performed on J K γ ⊗ v
cross product of zero order vibration and rotation wavefunctions:

a c K K J K K C J K γ Γ Γ Γ = ∑∑ … v r o t v' 1 2 9 v' v v v v' (II.61) with Γ ⊗Γ = Γ v r o t
symmetry conditions. In this expression the sum over v' runs only through the states belonging to the same polyad of interacting states, B'. For the study performed during this thesis for DNO 3 , B' = {5 1 , 7 1 +9 1 }.

II.2.5 Intensities

The intensity of a line (57) for a "pure" isotopic sample of DNO 3 is given (in cm -1 /(molecule.cm -2 )) by:

U L L N ~R kT hcE exp kT hc exp ) T ( Z hc k ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ν - - πε ν π = ν 1 3 4 8 0 3 (II.62)
where L and U are respectively the lower and upper levels of the transition,

(

) ( )

U L E E hc ν =
is the wavenumber of the transition in cm -1 , Z(T) is the total partition function and U L R is line strength.

The total partition function can be written as:

vib rot ( ) ( ) ( ) Z T Z T Z T = ⋅ (II.63)
and includes a vibrational contribution Z vib (T) and a rotational contribution Z rot (T).

Finally, U L R is the square of the matrix element:

2 U L l u R l u = ∑∑ μ , (II.64)
where l and u are the states corresponding to the degeneracy in quantum number M of the lower and upper level, and μ is the transition moment operator.

In case of zero external field, the expression (II.64) simplifies in to

2 3 U L Z l u R l u μ = ∑∑ , (II.65)
where Z μ is the projection of electric dipole moment on axis OZ of the laboratory system of axis.

Intensities are often given other units, and the following correspondence applies:

N k N k ν ν = (in cm -1 /(cm -2 )) (II.66) 0 P N k T k L k P T ν ν ν = = (in cm -2 .atm -1 ), (II.67)
where L is Loschmidt constant (L = 2.686754 10 19 cm -3 at 0 °C and 1 atm), N is number of molecules and P is the pressure.

In Eq. (II.62) the quantity difficult to evaluate is U L R . Also, the initial upper and lower state

Hamiltonian were transformed by rotational and vibrational contact transformations in order to enable the energy levels calculation. As a consequence, these contact transformations also affect the transition moment operator. Therefore, the transformed dipole moment,

' Z μ , is used for the calculation of U L R : 2 ' ' ' " " " , ' , v' v'' U L a c Z a c R J K K J K K μ = (II.68)
It is, in principle possible to calculate explicitly the transformed dipole moment using the expression of the vibrational and rotational contact transformations. This was indeed performed numerically for several C 2v triatomic molecules (59). However, for molecules with five atoms, like DNO 3 , this task is impossible. Therefore it is necessary to use an a priori expansion of the transformed transition moment operator based on the symmetry properties of the molecule.

II.2.5.1 Transformed dipolar moment

In the present study we are dealing only on cold bands, and the lower vibrational state is the ground state ( ) 0 . The operator of the transformed dipole moment may be written as:

' ' 0 Z Z μ μ = ∑ v,0 v v , (II.69)
where v runs through all members of the polyad B', and

' Z μ v,0
is the transition moment between the ground state and the state v that can be written as:

' ' Z j j j A μ μ = ∑ v,0 v,0 , (II.70)
where

' j μ v,0
are the coefficients to be determined from the experimental intensities, and j A are the rotational operators of the form { } , p q r r q p x y z z y x J J J J J J ¤ For Δ(v 8 +v 9 ) = even, both A-and B-type transitions are observable, and, up to the second order, the j A operators appearing in the right hand side of Eq. (II.70) are in ϕ z , {ϕ x , iJ y }, {iϕ y , J x }, and in ϕ x , {ϕ z , iJ y }, {iϕ y , J z }, for the A-and B-type expansion, respectively.

¤ For Δ(v 8 +v 9 ) = odd, only C-type transitions are observable. In this case Eq. (II. 70) involves only terms in iϕ y , {ϕ z ,J x }, {ϕ x ,J z }.

II.2.5.2 Calculation of the line strengths:

The eigenvectors are written on the symmetrized basis functions as: , 0 ,

a c K K J K K C J K γ γ γ = ∑∑ … v 1 2 9 v v v v v .
(II.71)

The expression (II.68) is now written in the following way: II.2, the vibrational energies between 5 1 and 9 2 differ in Δ 99-5 = E 99 -E 5 = 14.448 and Δ 99-5 = 3.098 cm -1 for H 14 NO 3 or H 15 NO 3 , respectively. This value has to be compared with the value, F 0 = 8.53 cm -1 , where F 0 is the zeroth order term in the expansion of the 5 1 9 2 Fermi operator for both H 14 NO 3 (7) or H 15 NO 3 (8). By diagonalizing the J = 0 Hamiltonian matrix using the Δ 99-5 and F 0 values it is possible to estimate the mean

2 ' ' ' ' ' ' ' ' ' ' " " " U L KK j j K K j R C C J K A J K μ γ γ = ∑∑∑∑ v'* v" v,
a c K K K K J K K C J K C J K γ γ γ γ γ γ = + ∑ ∑ (II.74)
The bracket is for the mean mixing value on the overall 9 2 [J K a K c ] energy levels.

By using the numerical values of Δ 99-5 and F 0 and diagonalizing the J = 0 matrix, this 5 99

R

mixing is estimated to be about 41 %, 18 % and 0.2 % for H 14 NO 3 (7) and H 15 NO 3 (8) and DNO 3 , respectively. Therefore, the very weak resonances coupling the 5 1 and 9 2 energy levels of DNO 3 were not considered explicitly.

On the other hand, when examining Table II.2, it is clear that 7 1 9 1 vibrational state is located ~160 cm -1 above the 5 1 state for H 14 NO 3 and H 15 NO 3 . On the contrary, for DNO 3 , 7 1 9 1 is only ~5.4 cm -1 below 5 1 . Therefore, due to symmetry considerations it is clear that A-type and Btype Coriolis resonances coupling together the 5 1 and 7 1 9 1 energy levels have to be explicitly taken into account for DNO 3 . In a 100 mL glass bulb equipped with a magnetic stirring bar and a 10 mm valve with PTFE stem (Young) 40 g of KNO 3 (0.4 mol) was placed. After the KNO 3 was dried in vacuum for one hour, 25 mL (0.5 mol) of D 2 SO 4 was introduced into the bulb. Subsequently the bulb was connected to the vacuum line and the volatile products were distilled through a series of traps held at -20, -100, and -196 °C. The content in the bulb was stirred and heated in a water bath until the gas evolution was terminated. About 20 g of pure DNO 3 was collected in the trap held at -100 °C.

The IR gas cell (glass body, AgBr windows sealed with Kel-F wax, optical path length of 28 cm) was treated with DNO 3 vapour several times, until no proton exchange with the surface could be detected any more. For the final measurements sample pressures of 1.3, 0.4 and 0.1 mbar at cca 23°C (296 K) were used.

The infrared spectra of DNO 3 were recorded with an instrumental resolution of 0.0022 cm -1 in the 700-1400 cm -1 region on the Bruker IFS 120 HR Fourier transform spectrometer at the University of Wuppertal, equipped with a Globar source, a KBr beam splitter, and a MCT 800 detector. An optical filter was used to eliminate radiation > 1400 cm -1 . 60, 121 and 169 scans were co-added for the spectra recorded at pressures of 0.1, 0.4 and 1.3 mbar, respectively.

The calibration of both FT spectra recorded at two different pressures (1 Torr and < 0.1 Torr) in the region 730 -1400 cm -1 was performed via model ( ) A spectral resolution of 0.0027 cm -1 was determined from the FWHM value of isolated lines.

II.3.3 Analysis

Because of the particular values of the A, B and C rotational constants for nitric acid (A ~ B ~ 2C) there exists a K a degeneracy (resp. a K c degeneracy) for the rather high K c values (resp.

the very high K a values): this means that the [J,

K a = J -K c , K c ] and [J, K a = J -K c + 1, K c ] (resp. [J, K a , K c = J -K a ] and [J, K a , K c = J -K a + 1]
) energy levels are degenerate.

As already mentioned in part II. In addition, in the wings of the P and R branches, series of regular weaker transitions involving high K a values were also observed. These series were assigned as "pure" B-type transitions, as it is exemplified on Fig. II.9. Searching for these B-type transitions, as for example the [J, The details of the line intensity calculations will be given in Section II.3.4.3.

K a = J, K c = 0 or 1] -[J ± 1, K " a = J ± 1, K " c = 1 or 0] type transitions, helped us to confirm the A-type assignments in the narrow Q-branch ([J, K a = J, K c = 0 or1] - [J, K " a = J, K " c = 1 or 0] type-transitions).
Some low and medium J and K a transitions in the ν 5 band were first assigned. Then, using the ground state parameters of Ref. (31), the lower state energy levels were calculated and added to the newly observed line positions to get a list of experimental upper state energy levels.

These upper state levels were inserted in a least squares fit to get an improved set of upper states parameters allowing better predictions and hence more assignments to be made.

This process was repeated iteratively until it appeared that several series were significantly perturbed, because of resonances involving the energy levels belonging to the 7 1 9 1 dark state located at 882.211 cm -1 . A model accounting for these resonances was then set up, which allowed more accurate predictions, and therefore new assignments. This iterative process was carried out until the ν 5 band was satisfactorily assigned. Table II.12 gives an overview of the results of the present analysis. As the resonances coupling the 5 1 and 7 1 +9 1 energy levels are rather strong, it was possible to assign several transitions belonging to the dark ν 7 +ν 9 band.

E E δ = - in 10 -3 cm -1 )
In addition, it has to be mentioned that lines from several hot bands were observed in the spectrum during the analysis. The form of the Hamiltonian matrix used for the {5 1 ,7

( ) ( ) ( ) ( ) { } { } v v v 2 v v 2 v v 2 v,v v 2 v 4 v 2 v v 2 2 v 2 2 2 2 1 1 1 2 2 2 , 2
, ,

z x y K z JK z J K z xy J xy xy x y H E A B C J B C B C J J J J J J A B AB BA J J J δ δ ⎡ ⎤ = + - + + + + - ⎣ ⎦ -Δ -Δ -Δ - - + = + = - 2 2 2 J J J J … (II.78)
Due to the relative symmetry of the 5 1 ↔ 7 1 9 1 interacting states (A' ↔ A"), both A-type and B-type Coriolis operators are considered in the 5 1 7 1 9 1 off diagonal blocks:

H 5,79 = C A + C B (A-type Coriolis, B-type Coriolis) { } 1 2 , A A z A y x C C J C iJ J = + (II.79) { } ( ) 2 2 3 3 1 2 3 4 5 , B B x B y z B x B x z B C C J C iJ J C J C J J C J J + - = + + + + + J (II.80)
The final energy levels calculation was performed for the {5 1 ,7 1 9 1 } resonating states.

The experimental 5 1 and 7 1 9 1 energy levels obtained in this work were introduced in a least squares fit using this Hamiltonian model. Tables II. 15 and II.16 list the Hamiltonian parameters (band centers, rotational and interacting constants, in cm -1 units) resulting from the fit, together with their associated statistical uncertainties (for one standard deviation).

Several rotational or centrifugal distortion constants were maintained fixed at their ground state values (31). One has to remind that since the informations available on the rotational structure of the 7 1 9 1 dark state concern a small sample of perturbed levels (75 as compared to 1078 for the 5 1 bright state). For this reason the rotational and centrifugal distortion constants for the 7 1 9 1 resonating state are effective ones, with poor physical meaning.

The results of the energy levels calculations proved to be rather satisfactory, as can be seen from the standard deviation and statistical analysis given in the Table II.13. However, several resonances could not be completely accounted for satisfactorily. Indeed the most significant disagreements are observed starting from J~30 for the levels of the 5 1 state with K c = 1, 2 and 3 (with K a = J -K c and J -K c + 1). As it will be discussed in the next paragraph, this is because these resonances are local, and difficult to predict. To the writer's knowledge, there exists no reported band intensity for the ν 5 band of DNO 3 .

0
-0.6856x10 -11 # # H JK 1.7855x10 -12 # # H J -0.8156x10 -13 # # h K 0.7103x10 -12 # # h JK 0.4954x10 -12 # # h J -0.3070x10 -13 # # L K -1.420x10 -16 # # L KKJ 1.2329x10 -16 # # l KKJ 1.90x10 -17 # # l JJK -4.420x10 -18 # #
γ γ γ γ γ γ k J C ' ' k J C K K J , , k , k ' , ' k ' , ' k c a ∑ ∑ + =
Consequently as for H 15 NO 3 (8), an estimation of the "absolute" line intensities was performed using the transition moment operator of the ν 5 band of H 14 NO 3 (7).

For the (almost) pure A-type ν 5 band of H 14 NO 3 the first order in the expansion of the transition moment operator is written Finally, the synthetic spectrum of the ν 5 band of DNO 3 was calculated in the region 846.5 -929.6 cm -1 . The line positions were computed using the band centers and rotational and coupling constants which are given in Tables II. 15-16 for the upper {5 1 ,7 1 9 1 } resonating states and in Ref. (31) for the vibrational ground state.

The intensities were computed using the method described above and the A-and B-type components of the ν 5 transition moment operator given in Eq. (II.86).

The calculations were performed for a "pure" isotopic sample of DNO 3 using an intensity cutoff of 0.5 x10 -22 cm -1 /(molecule.cm -2 ) at 296 K, maximum values of 70 for J' and 50 for K' a , maximum upper and lower state energies of 2000 cm -1 and 1200 cm -1 , respectively.

In this way, about 28000 lines were generated and the values ( )

5 -17 -1 -2 (296 K) =1.37x10 cm / molecule.cm S ν (II.87) ( ) 7 9
-17 -1 -2

(296 K) =0.12x10 cm / molecule.cm S ν ν + (II.88) at 296 K were achieved for the ν 5 and ν 7 +ν 9 band intensities of DNO 3 , showing that the ν 7 +ν 9 dark band borrows about 8 % of the ν 5 band's intensity. This final list of line positions and intensities generated for the 11 µm bands of DNO 3 has been deposited as supplementary data at the Journal of Quantitative Spectroscopy and Radiative Transfer and is available online (34). Some comparisons between the observed and calculated spectra are given through Figs. II.6-10, the agreement between the observed and calculated spectra is in general reasonably good when one considers the difficult treatment of the interaction between 5 1 and 7 1 9 1 states.

II.3.5 Conclusion

Using new high resolution Fourier transform infrared spectra recorded at Bergische Universitat in Wuppertal, the first analysis of the ν 5 bright band and a partial identification of the ν 7 +ν 9 dark band of DNO 3 have been performed. The theoretical models which were used to calculate the line positions and intensities account for very strong Coriolis resonance (Atype and B-type) which perturb the 5 1 and 7 1 +9 1 rotational energy levels. Finally a synthetic spectrum of the ν 5 band of DNO 3 was generated, using a realistic estimation of its transition moment operator.

III CH 3 F collisional effect studies -experimental part III.1 Experimental set-up

Pure rotational spectra and Stark split pure rotational spectra of 12

III.2 Pure rotational transition measurements

In the present study, microwave spectra were recorded for total pressures ranging from 3 to 250 μbar in the case of pure CH 3 F and from 100 to 500 μbar in the case of CH 3 F diluted in He (x CH3F around 0.1). All measurements were made at room temperature and the sample pressures were measured by a capacitance gauge from Leybold CTR 91 in the range from 5 to 130 μbar and Leybold CTR 100 in the range from 5 to 500 μbar with relative uncertainty of about 5 %. Methyl fluoride (with natural isotopolog abundances) and helium were provided by Linde Gas with stated purities of 99.5 %, and 99.994 %, respectively.

First, a check of optimal absorption conditions was carried out in order to avoid the higher energy levels' saturation that would lead to nonlinear absorption. The power of generated probing microwaves at 102 (resp. 204) GHz was held at -22 (resp. -7) dBm for pressures above 7 (resp. 5) μbar. At lower sample pressures, the respective radiation input powers led to the significant saturation of CH 3 F spectra and lower powers detered the signal to noise ratio due to high noise, as is shown in Fig. III.4-5. The indicated saturation of a line for given pressure was calculated as proportion of its intensity to the intensity of the same line measured with least intensive power. This calculation gives only approximative values since they are overrated -the signal to noise ratio is particularily low for low input microwave powers and so the maximal intensity may be overestimated. for all the pressures in each iteration, this approach enables to fit also the characteristics of the spectral lines as pressure shift [MHz/Torr] and collisional broadening [MHz/Torr] over multiple pressures, including the higher pressures under which some or all treated lines are no more isolated due to pressure broadening. The pressure values are on the other hand with respect to the calculated absorption coefficients. This approach assumes the identity of the pressure value for all lines at given pressure level. This is not granted and care had to be taken during experiment to keep the pressure for such different measurements as close as possible since the dosing of the gas in the cell was carried out manually in this work. This applies for low pressures and higher J (e.g. for J″=3 lines, different K lines were measured separetely at the same pressure level).

III.3 Specificities of the measurements in the significant external electric field

Stark spectra of 12 CH 3 F (whether pure or in mixture with helium) were recorded for J: For precise measurements of these separations, the intensity of the static electric field must be as homogenous and stable in time as possible. The better the precision, homogeneity and stability of the electric field intensity makes the better the precision of the electric dipole moment determined possible. In optimal case, the 1σ uncertainty shall not disturb at least four significant digits of the involved quantities. In the laboratory of Prof. Urban at ICT, it is possible to measure using both the parallel and orthogonal Stark effect arrangements.

1 → 2 ( 102
As alredy mentionned, these two designations refer to orientation of the intensity of the electric field between Stark plates towards the orientation of the electric component of the probing beam. By rotating the transmitter parts (multiplier, amplifier) together with detecting part (detector, bias) around the beam axis, it is possible to switch from one regime to the other one. The two different orientations have the different selection rules for the Stark components they enable to generate.

The symmetry of this class of molecules (symmetrical top) makes that the frequencies of the pure rotational lines are multiples of J"+1, the transitions follow rules ΔJ = 1, ΔK = 0 and for parallel orientation Stark effect there are 2J"+1 Stark components. Roughly, ν rot (J",K") = 2B(J"+1)-2D J" (J"+1) 2 (J"+2)-2D J"K" (J"+1)K 2 , where B is a molecular rotational constant and D J" , D J"K" are centrifugal distortion constants (inherent to a vibronical state).

Note that the individual transition is described by the quantum numbers of the lower (e.g. J", K", M", …) and upper (e.g.

i i i i i E J K M UF S E E J K M μ = = ∑ , (III.1)
where the electric field intensity E is in V/cm, dipole moment μ is in D, UF is a unity factor to pass from Debye unit to C.m via statcoulomb.cm for the dipole moment and to pass from V/cm to V/m for the intensity of the electric field: While evaluating some of the constants, the Eq. (III.1) simplifies to: ( )

( )( )
( ) ( ) ( ) 2 Stark 1 coef ,1 2 coef ,2 3 3 c o e f , 3 2 ( , , ) ( , , ) ( , , ) ( , , ) E E J K M X E E J K M X E J K M B E X E J K M B μ μ μ = + + , (III.
coef ,1 ( , , ) 1
KM E J K M J J = - + (III.5) coef ,1 (0, 0, 0) 0 E = (III.6) ( )( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 coef ,2 3 3 1 1 ( , , ) 2 1 2 1 1 2 1 2 3 J K J M J K J M E J K M J J J J J J + - + - - - = - - + + + + (III.7) coef ,2 (0, 0, 0) 1 3 E = - (III.8) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( ) ( ) ( )( )( ) 2 2 2 2 2 5 coef ,3 2 2 2 2 5 1 1 2 1 2 1 ( , , ) 1 1 1 2 2 1 2 3 J K J M J J J J J E J K M MK J K J M J J J J J ⎛ ⎞ - - ⎜ ⎟ ⎜ ⎟ - + - + ⎜ ⎟ = - ⎜ ⎟ + - + - ⎜ ⎟ - ⎜ ⎟ + + + + ⎝ ⎠ (III.9) 2 2 coef ,3 coef ,3 coef ,3 1 4 (0, 0, 0) (1, 0, 0) 0, (1,1, ) 3 480 M M E E E M M ⎛ ⎞ - - = = =- - ⎜ ⎟ ⎝ ⎠ (III.10)
Relative intensities of the Stark components J, K, M → J +1, K, M +ΔM (ΔM = -1, 0, 1) are given by Eqs. (III.11-12) :

( ) ( ) ( )( ) 2 2 Stark, rel Stark 2 2 1 ( , , , 0) ( , , , 0) 1 2 2 1 1 1 3 M J M I J K M M I J K M M J M J J J + - Δ = = Δ = + - = ⎛ ⎞ + + + ⎜ ⎟ ⎝ ⎠ ∑ (III.11) ( )( ) ( )( ) ( ) Stark, rel Stark Stark . 1 . 2 ( , , , 1) ( , , , 1) . 1 . 2 2
( , , ,

.

M M M J M M J M M I J K M M I J K M M J M M J M M I J K M M J M M J M Δ + Δ + + Δ + Δ = ± = Δ = ± + Δ + + Δ + = Δ = + + Δ + + Δ = ∑∑ ∑ ( ) ( )( ) 2 4 2 2 1 1 2 3 M J J J + ⎛ ⎞ + + + ⎜ ⎟ ⎝ ⎠ (III.12)
We can also remark that the fastest Stark components (the M components with the highest absolute value of coef , ( , , )

U = 0 V U = 250 V U = 700 V U = 150 V U = 900 V U = 85 V U = 500 V
F" = 10.5, 13.5, 11.5, 12.5 J" = 12, K" = 6

F" = 10.5, 13.5, 11.5, 12.5 J" = 12, K" = 7

F" = 10.5, 13.5, 11.5, 12.5 J" = 12, K" = 8

FIG. III.9 Methyl bromide rotational spectrum and Stark spectrum for orthogonal orientation

Calibration of Stark cell

Though the distance that separates the Stark plates may be measured rougly using Vernier caliper, the spectroscopic calibration leads to a more precise effective value. By measuring precisely Stark shifts of Stark components of rotational lines belonging to a species which dipole moment is known precisely and knowing the precise value of electric tension applied to electrodes (i.e. using a precise and stable Voltage generator, e.g. Fluke 5440B featuring 10 ppm output accuracy), we can determine precisely the distance separating the electrodes. The usual calibration molecules are: OCS, CH 3 F and CH 3 CN. Their electric dipole moments determined by different methods and groups are given in Tables III (12) least squares analysis

Table III.3 Review table of CH 3 CN electric dipole moment value determinations

The

f [MHz] f [MHz] f [MHz] f [MHz] f [MHz] f [MHz] U =
f [MHz] f [MHz] U = 40 V U = 45 V U = 50 V U = 55 V U = 60 V U = 65 V CH 3 F, parallel

IV Line profiles of pure rotational CH 3 F lines IV.1 General considerations

The various types of interactions of electromagnetic (e.m.) radiations with gaseous molecules manifest through specifics shapes of the resulting (absorption, scattering, etc.) spectra.

Under sufficient gas density (i.e. pressure/temperature) conditions, these spectra can be significantly influenced by collisional processes occurring in the system.

The early studies dealing with spectrally isolated optical transitions, overlapping and collisionally coupled lines either in a practical way or through the development of formal theories are reviewed in chapter II. of Ref. [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF] and thus are not recalled here. Some other models, approximate but amenable to practical calculations and/or useful for analyses and fits of experimental spectra will be presented directly (see also chapter III of Ref. [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF], rather than the fully general formalism underlying them. From now on, we shall consider molecular gases with the following restrictions, all valid for our studies:

-Neutral molecular species -Absorption in the spectral domain -Weak radiation fields (no saturation, no alignment, no high order effects, etc) and non reactive collisions (no chemical reactions) -Local thermodynamic equilibrium (LTE) so that all populations of energy levels are given by the Maxwell-Boltzmann statistics with equal translational, rotational, and vibrational temperatures.

-The inter-molecular collisions are binary: this approximation, which assumes that a molecule interacts with a single other one at a given time is valid except at very high pressures not considered in this work. The spectral shape then results from the interaction of intrinsic properties of one molecule (called the radiator or absorber) with the e.m. field and with the other particles (perturbers) through binary collisions.

-The "impact approximation" which assumes very short collisions and limits the modeling of the line-shape to the spectral region within a few cm -1 (a few ten GHz) around the center of the resonant optical transition.

-In cases where the transitions in the spectrum are spectrally separated (negligible overlapping), the isolated lines approximation is used which neglects the influence of linemixing studied in chapter V. The absorption then can be computed by simply adding up the individual contributions of the transitions with proper line shapes.

-We assume the dipolar approximation which assumes that the variation of the e.m. field amplitude due to its spatial modulation over the radiator size is negligible and only take into account the contribution of the permanent electric dipole.

IV.1.1 Absorption

In classical electrodynamics a neutral gas (characterized by the scalar relative complex permittivity ε) is polarized (linear macroscopic electric polarization P ) by the electromagnetic field (of frequency ν, polarization vector E and wave vector k ) through

Maxwell equations:

0 ( 1) , with ' " i ε ε ε ε ε = - = + P E , (IV.1)
ε 0 being the vacuum dielectric constant.

For electromagnetic waves travelling through a gas in the z direction, the Beer-Lambert law gives the light intensity I(ν,z) at frequency ν and depth z:

( , ) ( , 0) exp[ ( ) ] I z I z z ν ν αν = = × - (IV.2)
where α(ν) is the absorption coefficient which can be related to the imaginary part of the permittivity through: C t . Note that, in the following, we shall use frequencies ν instead of angular frequencies ω or wavenumber σ but that conversions are easy to make using the relations ω = 2πν = 2πcσ. One can then show that the absorption coefficient is given by ( 17):

2 4 ( ) 1 exp ( ) 3 a B n I c kT π ν ν α ν ν ⎡ ⎤ ⎛ ⎞ = - - × ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ , (IV.4) ( ) 0 1 ( ) Im[ F( )] Im ( ) d i t I i i e Ct t ν ν ν π ∞ = = ∫ , (IV.5)
where F( ) ν is the complex spectral density and ( ) I ν is a line profile function.

IV.2 Introduction to isolated lines profiles

As mentioned before, within the isolated lines approximation, the contributions of the various transitions to the spectrum are additive. This disregards the influence of line-mixing treated in following chapter.

Let us consider an optical transition from the initial internal state i to the final state f (i → f) with a corresponding frequency at zero pressure ν if [Hz] (or equivalently wavenumber

σ if [m -1 ] or angular frequency ω if [rad.s -1 ] with ν if = cσ if = ω if /2π
). The basic parameters characterizing the line shape are the frequency of its maximum ν MAX thus defining the line shift ν MAX -ν if and its full width at half maximum (FWHM). For spectrally asymmetric lines, when their peak is skewed toward higher or lower frequencies while the center of gravity remains the same for both asymmetrical and symmetrical profiles, an additional and convenient asymmetry parameter can be defined, for example by:

A = [ν HF +ν LF -2ν Max ]/[( ν HF -ν LF )/2] (IV.6)
where ν HF and ν LF are the high and low frequencies at half maximum.

Two general types of line broadening may occur in a gas in thermal equilibrium for given temperature and pressure conditions: homogenous broadening and inhomogeneous broadening that, respectively, affect the line identically or not at all frequencies.

One example of inhomogeneous broadening is that due to the Doppler effect through the Maxwell-Boltzmann distribution of absorber velocity v given for the volume element

[ 3 , d + v v v ]: 3 3 / 2 2 3 MB B ( ) d ( / 2 ) exp[ (v / v) ] d f M kT π = - v v v , (IV.7)
where M is the absorber mass, v = v the absorber speed, and

1/2 v (2 / ) B k T M =
the most probable speed. Doppler broadening is dominant when collisional processes influencing the line shape are negligible, i.e. at sufficiently low pressures. It is a symmetric broadening.

However, the velocity equilibrium distribution may be altered by the radiator velocity changes due to collisions. Since the collisions favor changes from large velocities to smaller ones, this so-called Dicke effect leads to a narrowing of the line.

IV.2.1 Doppler broadening, Dicke narrowing

We consider here the Doppler effect associated with the velocity v and position at time t

( ) t r ( ( ) , (0) t t = ⋅ = r v r
( ) ( ) ( ) 2 ext D ( ) exp [ ( ) (0)] exp ( . ) exp (2 / 2) C t i t i t t π ν = - - = - = -Δ k r r k v , (IV.8)
where ... stands for mean value, averaged over all velocities through Maxwell-Boltzmann

distribution MB ( ) f v , D i f B v v / , v 2 2 k c k T M ν ν π Δ = = =
, v being the most probable speed.

The Doppler profile may be calculated from (IV.8) by using Laplace transform:

( )

2 D e x t D 0 D 1 1 ( ) Im ( ) d exp ( / ) 2 i t I i e C t t ν ν νν π π π ν ∞ Δ ⎡ ⎤ Δ = = -Δ Δ ⎢ ⎥ ⎣ ⎦ Δ ∫ , (IV.9) where if ν ν ν Δ = -
The Doppler (Gaussian) profile is usually characterized by its half width at half maximum

(HWHM) D γ : if B D D 7 1 D i f 2 ln (2) ln(2)
[MHz] 3.5812 10

[MHz] [K] [g.mol ] k T c M T M ν γ ν γ ν - - = Δ = = × (IV.
VC MB VC MB ( ) ( ) ( ) ( ) f f f f → = → v' v v' v v' v , one has: ( ) 2 2 VC z z VC z z B (v v' ) (v ' v ) exp v v' 2 z z M f f k T ⎛ ⎞ → = → - ⎜ ⎟ ⎝ ⎠ (IV.11)
From this relation one can conclude that collisions decreasing |v z | are more probable than those increasing it. This is in favor of the decrease of the absolute value of the Doppler shift if z v c ν so that transfers are more probable from the wing of the line to the center rather than vice versa. Hence the signature of this Dicke effect is a reduction of the line width explaining the largely used term of "Dicke narrowing".

The Dicke narrowed profile can be modeled within the diffusional approximation of the thermal motion of the absorber that travels a distance z(t) along the e.m. beam between times 0 and t:

( )

2 2 ext ( ) exp ( ) exp[ ( ) /2] C t ikz t k z t = - = -< > (IV.12)
The diffusional approximation assumes that <z(t) 2 >=2Dt (D is the diffusion coefficient,

[D] = m 2 .s -1 ) which shall apply in cases when the square root of the mean square displacement is small enough when compared to the wavelength λ. where

0 z z 0 0 1 v (0)v ( ) d (0) ( ) d 3 D D P t t t t ∞ ∞ ≡ = = ∫ ∫ v v , (IV.14)
where D 0 ([D 0 ] = N) is the pressure independent value for the diffusional coefficient.

The diffusional profile may be calculated from (IV.13-14) by using Laplace transform:

( )

2 0 Diff 2 2 2 0 / 1 ( ) ( / ) k D P I k D P ν π ν Δ = Δ + (IV.15)
This profile has a Lorentzian shape with 2 0 / k D P as HWHM that decreases with increasing pressure. This conforms the assumption of many collisions to occur to the absorbing molecule before it travels a wavelength.

IV.2.2 Lorentz, Dicke, Voigt, Galatry and Rautian profiles

The Lorentz profile If the thermal motion, and thus the Doppler effect, is disregarded, the dipole autocorrelation function int ( ) C t bound to the absorber internal (rotation, vibration) degrees of freedom reads

( ) int ( ) exp ( ) C t i t φ = - (IV.16)
where ( ) t i φ = Δ + Γ is the averaged collisional phase shift of the radiator dipole for the optical transition, Δ being the frequency shift of the oscillating radiator dipole and Γ its decay rate, hence:

( ) int ( ) exp ( ) C t i t = -Γ+ Δ (IV.17)
Frequency shift and the decay rate are pressure dependent:

, P P δ γ Δ = Γ = , (IV.18) 3/2 2 Voigt D i f D 1 ( ) exp( ) Im d ( )/ I t t t i π ν ν ν ν ν - ∞ -∞ ⎧ ⎫ = - ⎨ ⎬ - + -Δ-Γ ⎩ ⎭ ∫ (IV.24)
Using the dimensionless parameters andx x y

if if D D D ,
ν ν ν ν ν ν ν - --Δ Γ = = = Δ Δ Δ the Voigt
profile can be also written as:

[ ] Voigt D L 1 ( , ) ( ') ( ', ) d ' Im ( , ) I x y I x I x x y x i w x y π +∞ -∞ = - = ∫ , (IV.25)
where w(x, y) is a complex probability function: Note that, using the same dimensionless parameters, the Doppler and Lorentz profiles take the form:

2 D L 2 2 1 1 ( ) exp( ) , ( , ) y I x x I x y x y π π = - = + (IV.27)
The Galatry profile

The Galatry profile ( 95) is an extension of the Dicke model to the Doppler model that assumes very small velocity changes of the absorber caused by collisions. It is adapted to system involving heavy radiators and light perturbers. The correlation function then takes the form:

S VC 2 S S D ext VC S VC 1 ( ) exp 1 2 t C t t e ν ν ν ν - ⎧ ⎫ ⎛ ⎞ Δ ⎪ ⎪ ⎡ ⎤ = - -+ ⎨ ⎬ ⎜ ⎟ ⎣ ⎦ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ , (IV.28)
where S VC ν is the speed independent VC collision rate in the soft collision model (also called dynamical friction coefficient) that relates to the "optical" diffusion coefficient D by

S B VC Diff k T MD ν ν = ≡ .
Assuming the independence of VC and dephasing (broadening/shifting) collisions, the Laplace transform of S int ext ( ) ( ) C t C t gives the Galatry profile:

( )

G 2 0 1 1 ( , , ) Im exp ( ) 1 d , 2 zt I x y z i ix y t zt e t z π ∞ - ⎧ ⎫ ⎡ ⎤ = -+ + -- ⎨ ⎬ ⎢ ⎥ ⎣ ⎦ ⎩ ⎭ ∫ (IV.29) S VC D D
with and z t t

ν ν ν = Δ =Δ (IV.30)
The Rautian profile

The Rautian profile (96,97), also called Nelkin-Ghatak profile or Rautian-Sobelman profile, is a model, which, as the Galatry profile, describes velocity changes. It is a hard collision model, thus conceptually opposite to the soft collision Galatry approach, which assumes that each collision completely thermalyses the velocity. The VC probability density for a → v' v collision is thus given by:

H H VC VC MB ( ) ( ) f f ν → = v' v v ,
( ) ( ) H H 3 if VC VC ( , ) ( , ) ( ) ( , )d f t i f t f f t t ν ν ∂ ⎡ ⎤ = + +Δ -Γ+ + → ⎣ ⎦ ∂ ∫∫∫ v k.v v v' v v' v' (IV.32)
The solution after Laplace transform gives the expression:

3 MB Raut H 3 MB VC ( ) d 1 ( ) ( ) Im ( ) d ( ) f i A I f i A ν ν π ν ν ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ - ⎪ ⎪ ⎩ ⎭ ∫∫∫ ∫∫∫ v v v v , (IV.33)
where

H if VC ( ) ( ) A i ν ν ν ν = ---Δ-Γ+ k.v .
Calculations can thus be performed using the complex probability function introduced for the Voigt profile: )] , I

{ } Raut 1 ( , , ) Im ( , ) [1 ( ,
x y iw x y w x y

ς ς π ς ς π = + - + (IV.34) with H VC D / ς ν ν = Δ (IV.35)
The Rautian profile reduces to the Doppler profile when Δ = Γ = ς = 0, to the Lorentz profile when y → infinity, and to Voigt profile when 0 ς = .

IV.2.3 Speed dependence

All the models presented above disregard the dependences of the broadening and shifting coefficients on the radiator speed v ( v = v ) (98). Nevertheless, these dependences exist and generally lead to a narrowing of the line shape, as do the velocity changes modeled in the Rautian and Galatry profiles.

The speed dependent version of the Voigt profile is easily derived and takes the form:

3 MB sdVoigt ( ) 1 ( ) Im d Δ(v) . Γ(v) f I i ν π ν = Δ - - - ∫∫∫ v v k v , (IV.36)
which can be developed, as shown in Appendix, in

3/2 2 sdVoigt D 1 '( ) ( ) exp( ) Im d ( ) z t I t t x t z t π ν ν - ∞ -∞ ⎧ ⎫ - = - ⎨ ⎬ + - ⎩ ⎭ ∫
, where (IV.37)

0 D D D D D Δ (v ) (v ) ( ) i Δ(v ) (v ) ( ) i x t t z t t t z t ν ν ν ν ν ν ν - = ′ ′ Γ ′ = + Γ = +
Note at this step that the quadratic law of Rohart ( 99) is convenient for fits of measured spectra and has been widely used for this purpose. Its expression is 2

(v ) 2 t t t t Γ =Γ + - Γ ′ Γ = Γ (IV.38) 0 2 2 (v ) ( 1.5)
The speed dependent generalization of the Rautian profile has the form:

3 MB sdRaut H 3 MB VC ( ) d 1 ( , ) ( ) Im ( ) d ( , ) f i A I f i A ν ν π ν ν ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ - ⎪ ⎪ ⎩ ⎭ ∫∫∫ ∫∫∫ v v v v v v , ( IV.39)
where

H if VC ( , ) (v) ( (v) ) A i ν ν ν ν = ---Δ -Γ + v k . v .
Note that, for H VC 0 ν = , the speed dependent Voigt profile is, of course, obtained. Indeed

H 3 3 M M VC sdRaut sdV ( ) ( ) if 0, then I ( ) Im d Im d ( ) A( ,ν) A ( ν) f f I ν ν ν ⎧ ⎫ ⎧ ⎫ = = = = ⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭ ∫∫∫ ∫∫∫ v v v v v (IV.40)
One can easily show that the Dicke effect (due to velocity changes) vanishes at elevated pressures, when the collisional width gets much larger than the Doppler one. For instance, in the absence of speed dependence, the Rautian and Galatry profiles tend toward a Lorentz shape. On the opposite, speed dependence effects remain whatever the pressure.

Hence, although the speed dependence and Dicke narrowing have similar spectral effects, it is, in principle, possible to differentiate them experimentally provided that the investigated pressure range goes from the nearly Doppler to the collisional regime, i.e. from conditions for which the Doppler broadening is dominant to negligible.

IV.3 Case of CH 3 F

IV.3.1 Introduction

While treating the measured pure rotational spectra for the line-mixing study presented in the next chapter, the detailed analysis of the fit residuals showed that the observed profiles of the lines, when isolated, are slightly narrower than their Voigt fits, as demonstrated by the W shaped residuals in Figs. IV.1 and IV.2. As recalled above, this narrowing, which has been observed for many molecular systems [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF], may results from velocity changes (Dicke effect), from the speed dependence, or from simultaneous contributions of these two processes. Note that, for CH 3 F in the microwave, line narrowing has been studied, to the best of our knowledge, in Ref. (99) only. In this work, the J = 1 K = 1 CH 3 F pure rotational transition has been studied in a time domain experiment and its narrowing has been attributed to the speed dependence of the broadening coefficient only, thus a priori disregarding any Dicke effect.

IV.3.2 Analysis

It shall be reminded that the presented analysis was carried out as a first trial study of narrowing processes in pure rotational absorbance spectra of CH 3 F. The measured spectra were fitted line by line, first using the Rautian profile and the speed dependent Voigt profile presented in Sec. IV.2. As shown by Figs. IV.3-4 this leads to very significant improvements of the fits qualities with respect to what is obtained when a purely (speed independent) Voigt profile is used. Furthermore, the fits residuals obtained with these two more refined models are very good and comparable thus not permitting, at this step, any conclusion on the mechanism(s) truly involved. In fact, recall that the narrowing obtained with the Rautian and SDV profiles originate from completely different processes, i.e. the velocity changes (Rautian) and the speed dependence of the broadening coefficient (SDV). Although measured spectra were available in the range from about 5 to 250 μbar, only those for the lowest pressures (below about 30 μbar) were used since, for higher pressures, the fitting procedure either failed to converge or converged to physically meaningless values.

This could be expected for the Rautian profile since it tends to a Voigt profile at elevated pressure forbidding a fit of the narrowing parameter. It is more unexpected for the speed dependent Voigt shape and the reason for this problem remains unclear but may be due to uncertainties and noise on the measurements. Further studies are needed to clarify this point.

Nevertheless, Table IV.1 gives the values of the parameters obtained from these fits. As can be seen, the only possible comparison with previous values is reasonable. ( ) . After integration over the orientation, Eq. (IV.41) transforms to:

2 3 p 3 a p r p p v (v ) v exp (v ) d v π σ - Γ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ Γ = -⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫∫∫ v , (IV.
2 2 a r a r a r r r 2 2 p p 0 a p v v 2v v 2 (v ) v exp (v ) dv v v v v sh σ π +∞ Γ ⎛ ⎞ ⎛ ⎞ + Γ = - ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ∫ (IV.42)
The calculation of

K = 0 K = 1 p [mTorr] K = 0 K = 1 K = 2 K = 3

IV.3.3 Conclusion

The pure rotational lines of CH 3 F were studied using Voigt profile and other more subtle Energy In summary, collisional line-mixing leads to exchanges of populations between energy levels of the considered absorption transitions and this manifests, in the spectrum, through the fact that the absorption lines partially exchange their intensities.

transition ν i'f' E f' E i' E i E f J +1, K, M' J, K, M' J +1, K, M transition ν if J, K, M
For this effect to occur and to significantly affect absorption by pure rotational transitions (vibrational ground state), the following conditions must be fulfilled. First, collisional transfers between levels i i ', f f ' ↔ ↔ must be permitted: levels must thus belong to the same isotopologue and be "connectable" considering their symmetries and the inter-molecular potential. For instance, within the model presented thereafter, the ΔM = -1 and ΔM = + 1 optical transitions in Stark orthogonal orientation spectra are not coupled by collisions.

Secondly, radiator-perturber interactions inducing i i ', f f ' ↔ ↔ transfers must be efficient.

One may thus disregard transfers between levels belonging to different vibrational states (or having different nuclear spins); indeed the collisional transfers between such levels are very slow and several orders of magnitude less efficient that pure rotational transfers. Finally, the populations' exchanges must have a significant effect on the spectrum itself. Recalling that the pressure broadening is some kind of measure of the effect of molecular interactions, a criterion is thus that the collisional widths of the interacting lines should be comparable or greater than the distance between line centers ( if

if i 'f ' ν ν Γ ≈ > - ).
In fact, Γ is generally smaller (17) than WA Γ because of a usually negative coupling term

{ } 1 ' ' ' Re W S dd ρ - ≠ ∑∑
that must be taken into account. Note that WA Γ is the width of the "effective" line obtained from Eq. (V.4) in the strongly overlapping case and that it is thus the value associated with the neglect of LM effects.

In the present study we consider individual Stark transitions associated with given values of the rotational quantum numbers J and K, i.e.: , , 1, , ' J K M J K M M M → + = + Δ . The sums over and ' in Eq. (V.2) thus become sums over M and M'. As shown in ( 13) and ( 14), and as can also be established from the IOS formalism of ( 12) and Sec. V.

V.1.2 Construction of the relaxation matrix

The models available to build collisional relaxation matrices within the impact and binary collision approximation range from simple empirical approaches for complex molecular systems to fully quantum calculations starting from the intermolecular potential energy surface (PES) for simple systems. With the exception of direct calculation from the PES, the models provide calculations only of the real part of the relaxation matrix.

Note at this step that one can show that the relaxation matrix W verifies the following properties:

1) The diagonal elements of W are the individual pressure-induced widths and shifts of the optical transitions (Eq. (V.3))

2) W satisfies the detailed balance principle which ensures the overall conservation of the populations ρ of the levels ' ' '

ρ ρ = W W (V.8)
3) For rigid rotors and intermolecular interactions only dependent on the rotational states (i.e. no dependence on vibration), W verifies the following sum rule:

' ' ' ' 1 ' 0 ' ( ) ( ) d d T i T d ≠ = ⇔ - =Γ -Δ ∑ ∑ W W (V.9)
The numerous models for the construction of W are detailed in [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF] and are (very) briefly recalled below.

-The simplest is a partly empirical approach based on the hard collision model where the off diagonal real elements are proportional to the relative population of the initial level:

' ' , ' ρ γ = -× ≠ W (V.10)
It has been largely used in the past when computer power was limited and more sophisticated models were not available, which is not the case anymore.

-

i i i i K i i f E E E E = ← = - ≥ W , (V.11)
where EG ( )

f E Δ
is usually a power or exponential function ( -The dynamically based scaling laws, which were originally introduced to model collisions with atoms, have more robust physical bases since they explicitly take into account the couplings of the various (spectroscopic and collisional) angular momenta. The first of these models is based on the so-called Infinite Order Sudden (IOS) approximation (104) and assumes instantaneous collisions to disregard the molecules rotation during collisions.

Its main interest is that this model separates spectroscopic effects dependent on the internal states of the absorber (described by angular momenta coupling terms) and dynamical effects dependent on absorber-perturber interactions (described by a set of collisional quantities).

Nevertheless, the IOS neglecting of the energy spacing between rotational levels leads to the breakdown of the detailed balance relation of Eq. (V.8) and to inaccurate results when collisions are "slow" (for relatively heavy molecular pairs). To correct for these limitations, the Energy corrected Sudden (ECS, (19) model was proposed later on, which improves the IOS by imposing the detailed balance condition and including the collisional effects of the finite duration of collisions. The resulting approach has enabled very satisfactory predictions of line-mixing in a large variety of spectra for many molecules including linear and symmetric-top species (see review in ( 17)). Note that, for Stark components in which all M sub-levels have the same rotational energy and population, the IOS model, used in the following, is identical to the ECS approach.

Among approaches based on knowledge of the intermolecular potential, the fully quantum ones are the most accurate and they are considered as essentially exact. Nevertheless, in spite of the considerable increase of computer power, they are still hardly applicable to complex molecular pairs and limited to simple system and/or low temperatures. The alternative is to use a semi-classical treatment in which the translation is treated classically while the internal states are treated quantum mechanically. Among these approaches, the Neilsen and Gordon model ( 105) is considered most sophisticated but limited to interactions between linear molecules and atoms. The Anderson-Tsao-Curnutte (ATC, (16, 106) model enables to treat more complex collisional pairs and the Robert-Bonamy (100) model extends the ATC approach with a better description of close collisions. Note that a model derived from the ATC approach was applied previously to line-mixing among Stark components as discussed below.

V.2 Case of Stark components of CH 3 F and CH 3 F in He

V.2.1 Introduction

As reviewed in chapter IV of Ref. [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF], LM has been the subject of numerous experimental and theoretical studies for various molecular systems and types of spectra. However, only very few studies have considered the case of collisional exchanges among hyperfine components and/or Stark resolved molecular lines, i.e. when an applied electric field removes the spatial degeneracy of the rotational levels. As first shown by Bréchignac (12) for CH 3 F in the IR region, the "zero field widths" are significantly smaller than the widths of the various Stark resolved lines. Later on, accurate measurements were performed on pure rotational transitions of CH 3 F (13-15), confirming the expected M-dependence of the widths of the Mcomponents as well as the importance of LM effects.

Various theoretical treatments have been proposed to describe collisional couplings between M-components. Let us first mention the pioneering work of Buffa, Tarrini and co-workers (13), based on a generalization of the well known Anderson-Tsao-Curnutte (ATC) model (16). This approach has enabled a clear understanding of the collisional processes between hyperfine and Stark components of molecular spectral lines, for molecules with large dipole moments (13)(14)(15)[START_REF] Belli | [END_REF]. An alternative approach to take LM into account is to build the associated relaxation matrix using the Infinite Order Sudden (IOS) approximation [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF]107) or its improved Energy Corrected Sudden (ECS) extension [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF]19,20). IOS/ECS models have enabled precise modeling of LM effects for many systems and types of spectra [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF], including hyperfine components due to the nuclear spin (21,22) and symmetric-top spectra (23, 24). However, quite surprisingly, as shown in ( 14), the IOS model developed in (12) apparently failed to reproduce the details of coupling processes between the Stark components of CH 3 F lines. With the aim of clarifying this result we have re-examined the problem and the presented results suggest that this failure was only due to an oversimplification of the IOS formalism. Indeed, when applied properly, and using basic cross-sections previously (and independently) determined from double resonance experiments and He-broadening data, the IOS formalism leads to an agreement with experimental data comparable to that obtained in ( 14) and ( 15) within a different (ATC-like) approach.

V.2.2 The IOS relaxation matrix

The basic equations used here have been given in (12). In the IOS limit, the relaxation matrix elements are expressed in terms of spectroscopic factors (containing the coupling of the various angular momenta of the system) and dynamical factors Q(L,M a ,M b ). Note that M diagonal values Q(L,M,M) can be identified (20,107) with the state to state rotational crosssections:

( , , ) ( 0, 0 , ) Q L M M J K J L K M σ ≡ = = → = = . (V.12)
Starting from Ref. (12), we first assume that:

,0 ,0 ( , , ) ( )

a b a b M M Q L M M Q L δ δ = , (V.13) where: B B ( ) ( 0, 0 , 0) ( , 0) ( , 0 0, 0)(2 1) exp ( , 0) '( )(2 1) exp . Q L J K J L K E J L K J L K J K L k T E J L K Q L L k T σ σ ≡ = = → = = ⎛ ⎞ = = = = = → = = + - ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = = ≡ + - ⎜ ⎟ ⎝ ⎠ (V.14)
Q(L) and Q'(L), as the corresponding cross sections σ, are thus related by the detailed balance relation. Equation (V.13) generalizes the approach of (12) where it was assumed that a predominant dipole-dipole interaction implies that only Q'( 1) is non zero. Even if the dipole-dipole interaction (with its 1; 0 J K Δ = ± Δ = selection rule) is the dominant contribution to intermolecular forces that connect rotational states through collisions, it can induce | | 1 J Δ > rotational changes through higher order terms in the expansion of the S matrix. The 1 J Δ = ± processes are likely the fastest but they are not the only ones authorized by this interaction. In our opinion, this is the origin of the relative failure; observed in (14), of the IOS model of Bréchignac (12), in reproducing the experimental data, as will be shown later on.

For CH 3 F-He, the validity of Eq; (V.13) approximation has been assumed and discussed in (108) where the basic Q'(L) cross-sections, that will be used here, were deduced from an IOS analysis of He-broadening data .

Inserting Eq. (V.13) into the IOS formalism of (12), one obtains the following expression for the elements of the relaxation matrix:

( ) W( , , ; 1, , , , '; 1, , ' ) ( 1)
(

)

1 1 W ( , ,; 1, , ; 1, ) ' ' M M k k J K M J K M M J K M J K M M k J J k J J k J K J K J K J K M M M M M M M M + + +Δ → + +Δ = - + + + ⎛ ⎞ ⎛ ⎞ × + → + ⎜ ⎟ ⎜ ⎟ + Δ - -Δ -Δ - -Δ ⎝ ⎠ ⎝ ⎠ ∑ (V.15) with: ( ) W ( , ; 1, , ; 1, ) ( , , 1
, ; , , 1
, ; ) ( ) k k L J K J K J K J K F J K J K J K J K L Q L + → + =- + + ∑ (V.16)
where:

1 ( , , 1, ; , , 1, ; ) ( 1) (2 1)(2 3) 0 1 k L k J L J J J k F J K J K J K J K L J J K K J J L + + ⎛ ⎞ ⎧ ⎫ + + = - + + ⎨ ⎬ ⎜ ⎟ - + ⎝ ⎠ ⎩ ⎭ (V.17)
where ( ) ::: and { } ::: are 3-j and 6-j coefficients, respectively. Note that: W( , , ; 1, , , , ; 1, , )

M JKM J K M J K M M J K M J K M M Δ + +Δ → + +Δ =Γ , (V.18)
and that the symmetry relations verified by the elements of W given in (14,15,[START_REF] Belli | [END_REF] can be easily obtained within the IOS formalism. They will not be detailed here but one has, for instance:

0 0 M M JKM JK M Δ = Δ = - Γ =Γ . (V.19)
Using the following expression of the dipole reduced matrix element ( 12):

1 1 1 3 ( 1) 2 1 J M M JKM M J J d M M M M J + + +Δ Δ + ⎛ ⎞ = - ⎜ ⎟ --Δ Δ + ⎝ ⎠ , (V.20)
and neglecting the influence of the Stark field on the rotational distribution, it can be easily verified that , provided that the relative populations are defined by 2 1 J ρ = + . This last equality, which is consistent with the IOS approximation which neglects the spacing between rotational levels, is a good approximation for the low J lines considered in this paper. Then, one can show that, within the IOS limit:

{ } (1) ' ' , ' Re W W ( , ; 1, , ; 1,
) JK d d J K J K J K J K ρ Γ = ≡ + → + ∑ . (V.21)
This result is of course expected since (1) W ( , ; 1, , ;

, ) J K J K J K J K + → + is 1

V.2.3 Data

The experimental details on the recording of the Stark microwave spectra used for the analysis of line-mixing effects were given in chapter III. Let us only recall here that Stark spectra of 12 CH 3 F (either pure or in mixtures with helium) were recorded for the transitions J: Finally, in order to estimate the uncertainty, the spectra have also been fitted one by one each giving a value of the broadening coefficient. From the scatter of the results for a given transition and various total pressures, the error on our determinations was estimated to be of less than 5 % for pure CH 3 F and about twice for CH 3 F-He (due to the fact that the broadening coefficients by He are about one order of magnitude smaller than those for pure CH 3 F).

1 → 2 ( 102

V.2.4 Analysis

According to Eqs. (V.14-17), the information required for the calculation of the relaxation matrix are the dynamical factors

Q(L) [or Q'(L)].
Pure CH 3 F: Over the past decades, rotational energy transfer in pure CH 3 F samples have been widely investigated, in particular, by a time resolved millimetre/sub-millimeterinfrared double resonance experiment (110). Rotational state-to-state rates were measured and it was shown that they can be modeled using the IOS theory and the following law for the basic cross-sections:

[ ]

'( 0) ( , 0 0, 0) ( 1) Q L L A L L α σ - ≠ ≡ → = × + , (V.22) [with 0 ( 0) ( ) L Q L Q L ≠ = = -∑
]. The parameters A and α deduced from the measurements of (110) are given in Table V. Finally, let us recall that for CH 3 F-He mixtures, the W matrix is written according to the binary collision approximation as: The experimental result is 0.572, the semi-classical result of ( 14) is 0.527, while the present IOS prediction is 0.63 and not 0.306 as obtained by assuming that only Q'( 1) is non zero.

This quality of the IOS approach, when properly applied, is confirmed by the similarity of the IOS relaxation matrix for the case J = 5, K = 5 → J = 6, K = 5; ΔM = 0 with that obtained within the semi-classical approach (compare +16.0 -1.94 -0.08 0.06 0.04 0.02 0.01 0.00 0.00 0.00 0.00 -4 -1.94 +18.3 -2.76 -0.23 0.05 0.05 0.03 0.01 0.00 0.00 0.00 -3 -0.08 -2.76 +19.7 -3.22 -0.34 0.03 0.05 0.03 0.01 0.00 0.00 -2 0.06 -0.23 -3.22 +20.6 -3.49 -0.40 0.02 0.05 0.03 0.01 0.00 -1 0.04 0.05 -0.34 -3. J i ,K i → J f ,K f M i = M f Meas. IOS

V.2.5 Conclusion

The measurements presented in this study complete previous results of (14,15)

VI Conclusion

Two independent studies in experimental and theoretic gas phase molecular spectroscopy are presented in this thesis: an analysis of the infrared spectra of deuterated nitric acid (D 14 The first part of the thesis presents the first high resolution analysis of the ν 5 fundamental band (NO 2 in plane bending mode) of DNO 3 (deuterated nitric acid) in the 11 µm spectral region. In order to carry out the project, we used an infrared spectrum of D 14 NO 3 recorded with a Fourier transform spectrometer in the 700-1400 cm -1 region by Profs. Helmut Beckers and Helge Willner at Bergische Universität in Wuppertal. The analysis of the ν 5 band of DNO 3 centred at 887.657 cm -1 which is mostly an A-type band, was performed at LISA.

Our study shows that the fundamental ν 5 band is very perturbed. In the case of DNO 3 , there is the ν 7 +ν 9 "dark" combination band situated at 882.211 cm -1 . The 5 1 and 7 1

  NO 3 , kde mezi stavy 5 1 a 9 2 dominují Fermiho rezonance. Druhá část práce se věnuje profilům čistě rotačních linií CH 3 F. Studováno je kolizní rozšíření (kolize CH 3 F s CH 3 F a kolize CH 3 F s He) rotačních přechodů a jejich Starkových komponent. Mikrovlnná spektra byla naměřena na VŠCHT Praha. Jejich analýza vedla k určení kolizních parametrů vybraných profilů (Voigtův, Rautianův, rychlostně závislé profily) u přechodů J, K → J + 1, K (K = 0, …, J) s J = 1 a J = 3 a jejich Starkových komponent J, K, M → J + 1, K, M' (|M| = 0, …, J ; |M -M'| = 0, 1). Navíc byl pro popis pozorovaných kolizních interferenčních jevů sestaven model založený na Infinite Order Sudden (IOS) aproximaci. Získané výsledky doplňují a rozšiřují předchozí práce a jsou zároveň první úspěšnou aplikací aproximace IOS pro modelování kolizního spřažení (linemixing) mezi Starkovými komponentami. Klíčová slova: DNO 3 , infračervená spektroskopie, pozice, intenzita, Coriolisovy rezonance, pás ν 5 , pás ν 7 + ν 9 , CH 3 F, mikrovlnná spektroskopie, profily linií, Rautianův profil, rychlostně závislý profil, kolizní rozšíření, Starkův jev, dipólový moment, Line-Mixing, model IOS Declaration (form as required by ICT Prague) The thesis work had been carried out at the Department of Chemical Analysis of the Institute of Chemical Technology Prague from September 2007 to September 2011 and in the period from October 2008 to September 2011 also at the Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA, CNRS, UMR 7583) at Université Paris-Est Créteil in the frame of the program "thèse en co-tutelle". "I hereby declare that I have worked out the thesis independently while noting all the resources employed as well as co-authors. I consent to the publication of the thesis under Act No. 111/1998, Coll., on universities, as amended by subsequent regulations. " Prague, June 7th, 2011

the nitric acid molecule II. 2 . 1

 21 Figure II.1 gives the structure of nitric acid molecule which is planar in its equilibrium configuration.

Fig. II. 1

 1 Fig. II.1 Scheme of DNO 3 structure.

r

  Figure II.1 gives also the system of principal axes for the DNO 3 isotopomer. As compared to H 14 NO 3 , the inertia system of axes rotates of about 9° clockwise from H 14 NO 3 to D 14 NO 3 .

  The eigenstates and eigenvalues of the asymmetric Hamiltonian rotor, 0 rot H , are retrieved by diagonalization of matrix associated to the standard base J k . Different conventions can be chosen for the orientation of the (x, y, z) reference system for the symmetric rotor relative to the (a,b,c) inertia axes of the actual molecule. Naturally the III r representation, with z = c, x = a, y = b seems the most appropriate for a near oblate molecule like nitric acid, because c

1 J

 1 , a K and c K being associated, respectively, to the components of the total kinetic moment J on molecular axes a and c. Though these two last labels are not well behaved quantum numbers, nevertheless the triplet + ) enables labeling of the molecular rotational energy levels in a single way.Equation of the eigenvalues of 0 rot H has the form:

Fig. II. 2

 2 Fig. II.2 Calculated and observed spectrum of the ν 6 Q branch of HNO 3 (42). The upper curve is the calculated spectrum for only b-type transitions; the middle curve is the observed spectrum; and the lower curve is the calculated spectrum for only A-type transitions. The spectra are offset for clarity.

Fig. II. 3

 3 Fig. II.3 Overview of the ν 2 band of DNO 3 and HNO 3 for an D-enriched sample of nitric acid. The absence of sharp Q structure indicates a B-type character of the ν 2 band. From Ref. (33).

Fig. II. 4

 4 Fig. II.4 Overview of the C-type ν 8 band of DNO 3 . From Ref. (37).

Finally

  , Fig. II.4 gives an overview of the C-type ν 8 band of DNO 3 : it is clear that C-type bands differ in its structure from A-or B-type bands (see Figs. II.2-3).

  the authors, different definitions for η. One is based on the Born-

  expression, α ϕ is direction cosine between the molecular axes α and the laboratory axis Z. The operator ' Z μ has the A'' symmetry as Z μ , therefore the j A operators depends on symmetry properties of the vibrational operator 0 v .

  v" are obtained from the diagonalization of the upper and lower state Hamiltonian matrix (see Eq. II.71). Then the matrix elements of the j A operators on the J K γ basis wavefunctions are easily computable as it was established byFlaud et al 1981 (57). Therefore, for the computation of U L R the only missing entities are the ' j μ v,0 constants which are to be determined through a least squares fit performed on the experimental intensities, or estimated from those of other isotopomers, or predicted by ab initio calculations.II.3 First high resolution analysis of the ν 5 band of DNO 3II.3.1 Existing spectroscopic studies for the first vibrational states of DNO 3For the ground vibrational state of several isotopic species of nitric acid, accurate rotation constants were achieved from high resolution studies performed in the microwave submillimeter and centimeter region (Drouin et al. (31) and Chou et al. (60) and references therein). In the infrared, the vibrational spectrum of nitric acid was measured at low resolution by McGraw et al. (32) for several isotopic species (H 14 NO 3 , D 14 NO 3 , H 15 NO 3 , D 15 NO 3 ). As for DNO 3 , only the ν 2 (NO 2 a-stretch) (33), ν 6 (O-NO 2 stretch) and ν 7 (O-NO 2 bend) (35), ν 8 (out of plane NO 2 bend) (37), and ν 9 (O-D torsion) (38) fundamental bands were the subject of extensive high resolution infrared studies. Except for the ν 6 band for which a Fermi type resonance coupling the 6 1 9 2 interacting energy levels was noticed, all other investigated bands (ν 2 , ν 7 , ν 8 , and ν 9 ) appeared to be unperturbed. The present analysis describes the first high resolution study of the ν 5 fundamental band (NO 2 in plane bending mode) of DNO 3 in the 11 µm spectral region. Fig. II.5 compares the positions of the first vibrational states of H 14 NO 3 and DNO 3 . For H 14 NO 3 or H 15 NO 3 the energy levels calculation had to account the strong Fermi and C-type resonances coupling the 5 1 and 9 2 energy levels. This is because, as stated in Table

Fig. II. 5

 5 Fig. II.5 Resonance scheme for DNO 3 and HNO 3

  2.2-3, DNO 3 is a planar asymmetric molecule (C S symmetry) and as it is expected by symmetry considerations, the ν 5 band is a hybrid band, with both Aand B-type transitions, with (Δ|K a | = even, Δ|K c | = odd) and (Δ|K a | = odd, Δ|K c | = odd) selection rules, respectively.

Fig. II. 6

 6 Fig. II.6 shows the overview of the measured spectrum in the 11 µm region. The lower trace corresponds to the experimental spectrum (T = 296 K and P = 0.1 Torr). The upper trace is the line by line calculation for DNO 3 . The Q branch of ν 5 band is centered at 887.65 cm -1 . The A-type character of the ν 5 is predominant according to the general aspect of the narrow Q branch for which the strongest lines are A-type transitions involving very high K a values (K a ~ J).

Fig. II. 6

 6 Fig. II.6 Overview of the 855-920 cm -1 spectral region with the ν 5 band of deuterated nitric acid

Fig

  Fig. II.7 shows some lines of the P branch grouped in stacks corresponding to the same (2J -K c ) values. The quoted assignments are given for the upper K' c values, with (2J' -K' c ) = 20 (solid triangles, ) and (2J' -K' c ) = 21 (open triangles, ). Due to a resonance which occurs for K' c = 14 and 15, several ν 7 +ν 9 band transitions are also observed and some assignments are given (diamonds ♦). The "d" letter stands for degenerate transitions with K a = J -K c and K a = J -K c + 1. The lower trace corresponds to the experimental spectrum, the medium and the upper traces are line by line calculations performed for the ν 5 and ν 7 +ν 9 interacting bands and for the ν 7 +ν 9 dark band, respectively.

Fig. II. 7

 7 Fig. II.7 Part of P branch of the ν 5 band of DNO 3 near 878.6 cm -1 (with some upper K c assignements)

Fig. II. 9 (

 9 Fig. II.9 Part of R branch of the ν 5 band of DNO 3 near 903 cm -1

  Fig. II.10 Portion of the P branch of the ν 5 band of DNO 3 near 883 cm -1 : Identification of the ν 5 +ν 6 -ν 6 , ν 5 +ν 7 -ν 7 and ν 5 +ν 9 -ν 9 hot bands

4 . 2 C

 42 Resonances Due to the close proximity (Fig. II.5) of the 5 1 and 7 1 9 1 energy levels of DNO 3 , strong perturbations are observed during the analysis of the ν 5 band. Through A-type and B-type Coriolis resonances the wavefunctions of the 7 1 9 1 "dark" state are mixed with those from the "bright" 5 1 state, and therefore many ν 7 +ν 9 transitions become observable. In order to characterize these mixings and get an idea of the strength of the resonances we plot on Fig. II.11 the percentage of mixing coefficients of the 5 1 energy levels on the 1 γ are the coefficients appearing in the expansion of the v = 5 1 wavefunctions resulting from the diagonalization of the Hamiltonian matrix:

  Fig. II.11 shows that rather regular resonances couple together the 1 5 ,

Fig. II. 11 1 μ and 5 'B 1 μ

 1111 Fig. II.11 Mixing (in %) of the 5 1 [J, K a , K c ] energy levels on the 7 1 9 1 state versus J for different K c values.

  CH 3 F used in this study in Chapters IV and V were recorded for the transitions J: 1 → 2 (102 GHz) and J: 3 → 4 (204 GHz) with semiconductor millimeter wave high resolution spectrometer (described in Refs. (65-67)) and the 1.5 m long Stark cell (see Fig. III.1) at ICT Prague.

Fig. III. 1

 1 Fig. III.1 Scheme of the Prague MW spectrometer with the Stark module

Fig. III. 2

 2 Fig. III.2 Scheme of the Prague MW spectrometer with free path cell only

Figures III. 6 -

 6 Figures III.6-7 display experimental absorbancies for the J: 1 → 2 and J: 3 → 4 pure rotational transitions of methyl fluoride used further in the analysis.

3 F

 3 J: 1 -> 2, K = 0, p = 3 μbar (left) / 1 μbar (right), 5-15 scans mw source in dBm: from -16 to -30 A f [MHz]

Fig. III. 4 Saturation and signal to noise ratios of a CH 3 F J″ = 1 38 FIG. III. 5 35 CH 3 FCH 3 F

 4313853533 Fig. III.4 Saturation and signal to noise ratios of a CH 3 F J″ = 1 line for different pressures and input microwave powers

  GHz) and J: 3 → 4 (204 GHz) using the Stark module (see Fig. III.1) and the semiconductor millimeter wave high resolution spectrometer at the ICT in Prague. The probing electromagnetic radiation (microwave, millimeterwave, submm wave) travels between and along the Stark electrodes and transmits its power to the rotational modes of absorbing species. When the electric tension is applied on one electrode while the other is grounded, the spatial isotropy is withdrawn and the intensity of electric field between Stark plates separates the energy levels of the space orientation sensitive components (characterised by quantum number M) as shown in Fig. III.8. The selection rules for the Stark transitions depend on the orientation of the external electric intensity vector towards electric component of the probing beam. ΔM = 0 rule applies in case of parallel orientation, ΔM = ±1 characterizes othogonal orientation.In these measurements, the orientation of the electric field has been set parallel or orthogonal to the electric vector of the polarized microwave probing beam. Thus the Stark transitions were obtained with both the selection rules ΔM = 0 and ΔM = ±1. Finally, the electric field intensities used in our experiments ranged from 15 to 35 V/cm. The pressures of the pure methyl fluoride ranged from 7 to10 μbar and in the experiments with the methyl fluoride diluted in the helium (x He = 80-95 %), the total pressure range was 100-500 μbar.

Fig. III. 8

 8 Fig. III.8 Methyl fluoride rotational spectrum and Stark spectrum for orthogonal and parallel orientation

  c = 299792458 m/s, h = 6.62606896 10 -34 J.s(26), B in MHz..

  J, K, M energy level coefficients are given by Eqs. (III.5-10):

  Fig. III.8) and observe the fastest Stark components as well isolated (|M J "| = 1 for ΔM J = 0, M J " = -1 for ΔM J = +1 and M J " = +1 for ΔM J = -1) the situation becomes cumbersome for higher values of the rotational quantum number. Moreover, when the pure rotational lines are split significantly also by other effects (fine, hyperfine) as in J" = 12 Stark lines of CH 3 Br (see Fig. III.9) where each rotational (J", K") line is also split by nuclear quadrupole hyperfine effect to four components (described by quantum number F").

  of electric Stark field: E = 0 -cca 900 V/cm, p = 12 μbar, selection rules: rotation ΔJ = +1, ΔK = 0; hyperfine structure ΔF = +1; orthogonal Stark: ΔM F = +/-1 A f [MHz]

5 )

 5 ΔJ = 0, ΔM J = ±1, (J,K): (1of ν3: R(0,0)-R(5,5), Q(1,1)-Q(12,[START_REF]Scientific Assessment of Ozone Depletion: 2006[END_REF], P(1,0)-P(5,4) weighted mean value, CO 2 laser P(6)-P(30) lines in the 9.4 μm band,

  Stark module used consists of a stable DC voltage generator Fluke 5440B (0 -1100 V, claimed stability 10 ppm) and of two 150 cm × 8 cm × 1 cm finely polished stainless steel plates placed in a free path glass absorption cell. The separation d of the electrodes by quartz blocks was deduced from measured Stark shifts of theJ = 1 → 2, K = 1 → 1, M = -1 → -1transition at low pressure using an accurate value μ = 1.85840(7) D (70) of the CH 3 F permanent electric dipole moment. This was done using a relatively low voltage of 40 V creating a (small) shift of -12.741 MHz, leading to d = 0.9790(9) cm, a determination by far accurate enough for the low field measurements used further. At higher intensities of the Stark electrical field, the experimental spectra of calibration molecules CH 3 F and CH 3 CN suffered anomalies when confronted to calculations resulting from perturbation method. For both the molecules, it seemed like hyperfine structure is responsive, in term of the frequency position of its components, also to the higher electric field's intensity. Figs. III.10-11 resume the anomalies measured for CH 3 CN and CH 3 F Stark spectra recorded using in the amplitude and/or frequency modulations in the parallel orientation. The negative and positive M components sharing the same J, K numbers are given for the same voltage in sake of comparison. The pure rotational spectra are shown in the upper parts of the Figs. III.10-11. A consideration of inhomogeneity of the electric field between the Stark plates would explain this observed effect partly only since inhomogeneity would just continuously broaden the Stark lines (most obviously the Stark lines that shift easily) with increasing intensities, diminishing the absorbance peak of the transition for high intensities. Though the effect of the electric field heterogeneity may be observed in high voltage Stark amplitude modulation spectra of CH 3 CN and CH 3 F as given in Figs. III.10-11, the frequency modulation spectra suggest that the hyperfine structure components are separating with the external electrical field. These puzzling details of the spectra are not yet explained since apart what seems to be Stark field dependent hyperfine structure, there are also many lines appearing in the spectrum which are not accounted for by perturbation theory calculation. Stark spectra ΔM J = 0: J'' = 4, K'' = 3, M J '' = -3 (on left) and 3 (on right) M I = 1, calc. Stark f = 92284.023 MHz M I =-1, calc. Stark f = 92284.051 MHz M I = 0, calc. Stark f = 92284.124 MHz M I = 1, calc. Stark f = 91660.857 MHz M I =-1, calc. Stark f = 91660.885 MHz M I = 0, calc. Stark f = 91660.958 MHz A M I = 1, calc. Stark f = 92201.786 MHz M I =-1, calc. Stark f = 92201.814 MHz M I = 0, calc. Stark f = 92201.887 MHz M I = 1, calc. Stark f = 91741.974 MHz M I =-1, calc. Stark f = 91742.002 MHz M I = 0, calc. Stark f = 91742.075 MHz A

5 U

 5 Fig. III.10 CH 3 CN pure rotational and higher voltage Stark AM and FM spectra (identic pressure colour code)

Fig. III. 11 CH 3 F

 113 Fig. III.11 CH 3 F Stark parallel AM /FM spectra of one M component at lower and higher voltage Stark

3 )

 3 Within the dipolar approximation, "( ) ε ν is the Laplace transform in time of the classical dipole autocorrelation function ( )

  0 ) of the radiator. It results in a Doppler shift if z v / c ν for the absorbed photon frequency (v z is the radiator velocity component along the e.m. wave propagation vector k : z v . probable, the Maxwell-Boltzmann distribution f M (v z ) leads to an inhomogenously broadened symmetric profile centered at if ν . The dipole autocorrelation function ( ) ext C t resulting from the absorber translational motion (external degrees of freedom) then reads

10 )

 10 In the Dicke narrowing model, one assumes that the collisions have only a velocity changing (VC) effect. Let VC ( ) f → v' v be the probability per unit time of velocity changing (from to ) v' v collisions. These collisions change the Doppler shift from if z .v' c ν to if z .v c ν and therefore change the line shape. Assuming the velocity isotropy of gas and the constancy of the velocity equilibrium distribution and thus the detailed balance principle:

Fig. IV. 1

 1 Fig. IV.1 Voigt profile residuum, CH 3 F J: 1→2, K = 0-1, pressure 7 μbar.

Fig. IV. 2

 2 Fig. IV.2 Voigt profile residuum, CH 3 F J = 3→4, K = 2, pressure 7 μbar.

Fig. IV. 3 Fig. IV. 4

 34 Fig. IV.3 Residuals of Voigt, Rautian and their speed dependent profiles, CH 3 F, J: 1→2, K = 0-1, p =7 μbar.

Γ 2 ,Γ

 2 unc Γ multifit Γ unc Γ 0 unc Γ 2 unc Γ 0 unc Γ 2 unc Γ unc H VC ν unc Γ calc H VC ν unc In order to go further and to try to differentiate between the contributions of Dicke (through H VC ν ) and speed dependent (through the 0 Γ Γ parameters) effects, a speed dependent Rautian profile was set up. Since attempts to simultaneously derive (fit) from experiments the velocity changing and speed dependence parameters failed due to the strong correlation between these two unknown quantities an alternative procedure was used. In fact, in order to constrain the model, the speed dependent broadening coefficients a were fixed to values calculated with a semi-classical model adapted to CH 3 F from the method proposed for linear molecules by Bonamy et al. (100). These calculations, made with a program kindly communicated by Pr. Christian Boulet, provide the broadening cross section r (v ) σ Γ as a function of the the relative speed r r v = v with r p a =v v v being the relative velocity between the perturber and the absorber. Assuming a Maxwell-Boltzmann distribution for the perturber velocity, the values of a (v ) Γ , thus now versus the speed of the absorbed only, were calculated using the following relation (see Eqs. III.54-55 in (17)):

  Fig. IV.5 for the transitions under study in this work.

Fig. IV. 5 Γ

 5 Fig. IV.5 Speed dependent broadening coefficients (T = 296 K): calculated values for J = 1 and 3 CH 3 F lines

Fig. IV. 6

 6 Fig. IV.6 Rautian narrowing parameter H VC ν in [MHz/Torr] in the Rautian speed dependent profile with calculated (v) Γ for J = 1 lines (left) and J = 3 lines (right).

V. 1

 1 photon of frequency ν i'f' , thus contributing to the absorption line at frequency ν i'f' and then it can relax from level f ' 1, , 'J K M = + to f 1, , J K M = +by collisions (blue arrows on Fig. V.1). Through this process, a molecule initially on level i can contribute to the absorption line at frequency ν i'f' . The reverse path is also possible (collisional transfer from i' , , ' J K M = to i , , J K M = , absorption at ν if and collisional relaxation from f 1 , , J K M = + to f ' 1, , ' J K M = + ) as showed on Fig. V.1 by red arrows.

Fig. V. 1

 1 Fig. V.1 Scheme of line mixing process among two ΔM=0 Stark lines

  2.2, collisionalcoupling can arise only between lines having the same value of ΔM. The problem thus reduces to the three subspaces corresponding to the three possible values ( 0, 1± ) of M Δ . is the width of an isolated Stark component while JK Γ → Γ is the width of the , 1 , J K J K → +rotational line when unresolved, i.e. in the zero field limit (recall that the distance between M-components is driven by a first order Stark effect with a constant spacing -for given values of J and K -linear in the field). Finally, the weighted average of the widths of the Stark components [see Eq. (V.7)] will be noted WA JK Γ .

  which parameters are obtained through fits of experimental data. Such models have the advantage of easy implementation but have limited extrapolation possibilities since they rely on experimental data.

'

Fig. V. 2

 2 Fig. V.2 Experimental spectra of pure rotational CH 3 F J: 1→2, K = 0-1 transitions (on right) and the parallel Stark components M = -1, 0 ,1 at E = 23.5 V/cm (on left). Total pressure 0.1-0.5 mbar, mixture CH 3 F (x = 0.05-0.2) and He (x = 0.80-0.95).Other typical examples, for J = 3 → 4 transitions of methyl fluoride diluted in helium, are displayed in Figs. V.3-4.

40 CH 3 FFig. V. 3 CH 3 FFig. V. 4

 403334 Fig. V.3 Experimental (red line), Voigt fit (black line) and residuals (blue line) of CH 3 F J: 3→4, K = 0-3, M = -3, … , 3 parallel Stark lines at E = 27 V/cm. Total pressure 0.1 mbar, mixture CH 3 F (x = 0.05) and He (x = 0.95).

8 Table V. 1

 81 I. They have been used in the present work to construct the relaxation matrix, after a slight optimization (well within the experimental uncertainty) of the parameters, based on the experimental results (14) for the J = 5, K = 5 → J = 6, K = 5; ΔM = 0 line broadening coefficients. CH 3 F-He: Line mixing among usual (i.e. zero field limit) rotational components have been studied in the 3 ν band of CH 3 F perturbed by He in (108). The observed modifications with respect to the addition of Lorentzian contributions were analyzed in terms of models respectively based on the IOS and ECS approximations. Both lead to a satisfactory agreement with measurements even if the latter gives better results, particularly for high J values. The basic CH 3 F-He cross sections were modeled (108) through the simple analytical law given by Eq. (V.22). The two parameters deduced in this previous study are given in Table V.1 and will be used in the present calculations. Parameters of the law of Eq. (V.22) for the basic cross sections (A is given in MHz/Torr).

  P is the total pressure while x CH3F and x He are the mole fractions of CH 3 F and He, respectively. V.2.4.1 Results -Pure CH 3 F Experimental values of the broadening coefficients of transitions with and without a Stark effect from the present study and from (14) are compared with the IOS predictions in Fig. V.5 and in Tables V.2-3.

Fig. V. 5

 5 Fig. V.5 Measured and calculated (IOS) self-broadening coefficients (in MHz/Torr) of CH 3 F pure rotational lines

  49 +21.1 -3.61 -0.42 0.02 0.05 0.03 0) coupling M components of the J = 5, K = 5 → J = 6, K = 5, ΔM = 0 transitions in pure CH 3 F.

J: 5 → 6 , K = 5 , M = -5 to 5 ,•Fig. V. 7 1 C H 3 FFig. V. 8 Fig. V. 9 CH 3 F

 5655713893 Fig. V.6 CH 3 F self-broadening coefficientsM M JKM JK -M = γ γ Δ Δ

3 → 4 , K = 1 to 3 , M = -3 to 3 ,

 3433 Fig. V.10 CH 3 F-He broadening coefficientsM M JKM JK -M = γ γ Δ Δ

1 M

 1 Δ = ± case). When two lines ΔM = +1 and ΔM = -1 are superimposed and since they are not coupled, the calculated value is the weighted average of their individual widths.

  Figures V.11-12 give a good overall view of the present study. In these plots, the spectra have been directly calculated starting from the IOS relaxation matrix W, either using the full W or restricting it to its diagonal part. The first calculation thus takes line mixing into account while the second does not (uncoupled lines) and corresponds to the addition of individual Lorentzian/Voigt line contributions. As can be seen in Fig. V.11, in the "high-field/low pressures" regime where the Stark components are well resolved, LM effects are small, as expected (17), and both approaches give a good prediction of the various Stark components profiles.

3 CH 3 FFig. V. 11 CH 3 F 3 FFig. V. 12 CH 3 F

 331133123 Fig. V.11 CH 3 F Stark spectrum (Helium broadened): J: 3→4, K = 3, ΔM = 0, M = -3,…,3; E = 27 V/cm, p tot = 0.1mbar, x CH 3 F = 5 %If the static field is removed (and also in the "low-field/high pressures" regime), all these components collapse (Fig.V.12) into a single line whose width is significantly smaller than those of the individual transitions. Assuming a diagonal relaxation matrix then significantly overestimates the observed broadening while using a full matrix leads to very satisfactory results.

NO 3 )

 3 and an analysis of collisional effects in the Stark and non Stark microwave spectra of methyl fluoride (CH 3 F). These studies were carried out in the frame of co-tutelle thesis program organized by Université Paris-Est (UPE) and by Institute of Chemical Technology, Prague (ICT Prague).

  scheme for D14 NO 3 thus differs a lot from those involve in H14 NO 3 and H 15 NO 3 , since, for these two isotopologues, the 5 1 and 9 2 energy levels are coupled through Fermi and C type Coriolis resonances. Finally, the absolute line intensities for the ν 5 band of DNO 3 were calculated using the ν 5 transition operator from H 14 NO 3 . The hot bands ν 5 +ν 6 -ν 6 , ν 5 +ν 7 -ν 7 and ν 5 +ν 9 -ν 9 located at 881.03, 882.61 and 884.45 cm -1 , respectively, were also identified in the spectrum.

Table II .1 Ground state rotational constants for the H 14 NO 3 , H 15 NO 3 and D 14 NO 3 isotopomers of nitric a c i d .

 II

Table II .

 II 1 gives the ground state rotational constants for the H 14 NO 3 , H 15 NO 3 and D 14 NO 3 isotopomers. The following approximation holds:

	A~B ~2C	(II.22)

Table II .2 Vibrational modes of several nitric acid isotopologues. The E v and ν ~are the vibrational energy and

 II

the band center, respectively, expressed in cm -1 . ν ~ is the calculated position for the [J=0, K a =0, K c =0] energy level.

Table II .

 II 2 gives the description, the frequency, and the symmetry type according to C s of the nine vibrational modes of the D 14 NO 3 , H 14 NO 3 and H 15 NO 3 isotopologs. The C s point group will be defined later in the text.

	At zeroth-order, the rotation and vibration are well separated as shown previously
	( 0 H H =	0 vib	+	H	0 rot	). The eigenstates of H 0 are the product of vibrational eigenstate of 0 vib H
	denoted as		1	2	…	9	0

Table II .3 Symmetry properties of DNO 3 molecule Let

 II us remind that for the H 14 N 16 O 3 and H 15 N 16 O 3 isotopomers, the ν 9 mode (H-ONO torsion

	around the	-	N <	O O	group) is a large amplitude motion. This means that for excited energy
	levels in the v 9 vibrational quantum numbers, splittings of the energy levels can be observed,
	whose values are of about 2 MHz for v 9 = 1, 60 MHz for v 9 = 2 and 1800 MHz for v 9 = 3.
	This is because the ν 9 large amplitude motion enables the	O O	>	N -	O	H	O O	>	N -	O	H
	tunneling effects. In this operation, the two 16 O in the	-	N <	O O	radical are exchanged through
	a (12) permutation. The C s or G 2 groups are no more suitable, and it is necessary to use the
	G 4 = {E, (12), E*, (12)*} permutation-inversion group to characterize the symmetry
	properties of the vibration-torsion-rotation wavefunctions. Of course G 4 is homomorph onto

Table II .6 Symmetry of the rotational levels under the V symmetry group

 II Table II.6.

	K a	K c	type
	e	e	A
	e	o	B a
	o	o	B b
	o	e	B c

Table II .7 Symmetry of the levels J K γ under C s symmetry group

 II

Table II .8 Symmetry of the levels

 II

	0 J K K under C s symmetry group a c
	II.2.2.2.2 Symmetry of the rovibrational levels
	The symmetry of the H 0 eigenstates can be determined using the equation (II.29) from the
	known symmetry of the 0 vib H and 0 rot H eigenstates and depends only on the K c and (v 8 +v 9 )

  FT values of 1.68(30) 10 -6 , resp. 1.58(25) 10 -6 which gave the weighted mean value 1.62(27) 10 -6 .

	σ	Ref	=	σ	FT	1 C + FT ,	(II.75)
	using HITRAN 2004 (61) collection of H 2 O (62), HDO (62), N 2 O (63) reference line
	positions. These species were present in the measured gas mixtures impurities. In case of D 2 O
	the line position values issued directly from an article by Toth (64).
	13 lines of H 2 O, 22 lines of HDO, 50 lines of D 2 O and 15 lines of N 2 O were localised in the
	higher pressure spectrum, respectively in the regions 1187 -1387 cm -1 , 1354 -1386 cm -1 ,
	1127 -1388 cm -1 , and 1258 -1273 cm -1 . This led to determination of the C FT constants
	respectively 1.70(15) 10 -6 , 1.71(20) 10 -6 , 1.54(23) 10 -6 and 1.65(23) 10 -6 . These constants
	were weighted by square of inversed standard uncertainties to form the weighted mean value
	of 1.66(19) 10 -6 .

In the lower pressure spectrum, just H 2 O and D 2 O lines with sufficient intensity were traced.

The calibration through some 16, resp. 24, lines from region 1174 -1386 cm -1 , resp. 1042 -1271 cm -1 , led to C

Table II .

 II 13 provides a statistical analysis of the energy levels calculations. The standard deviation of the fit is 0.70 10 -3 cm -1 .

		ν 5	ν 7 + ν 9
	Nb of lines	2938	137
	J Max	68	40
	Maximum K a values	44	33
	Nb of levels	1078	75

Table II .12 Range of the observed energy levels

 II

						5 1	7 1 9 1
	Nb of levels	1078	75
	0	δ ≤ <	1.0	94.0 %	81.3 %
	1.0	δ ≤ <	2.0	3.6 %	4.0 %
	2.0	δ ≤ <	4.0	1.7 %	10.7 %
	4.0	δ ≤ <	10.0	0.7 %	4.0 %

Table II .13 Statistical analysis (

 II

	obs	calc

  1 9 1 } interacting states of DNO 3 is described in Table

		5 1	7 1 9 1
	5 1	H 5,5	Complex conjugate
	7 1 9 1	C A +C B	H 79,79

Table II .14 Hamiltonian matrix -The {5 1 ,7 1 9 1 } resonating states

 II As it is the case usually for planar C s type molecules, the v-diagonal parts of this model are A-reduced Watson-type Hamiltonians written in the I r representation:

Table II .15 Vibrational energies and rotational constants for the dyad {5 1 ,7 1 9 1 } ( # Fixed to the ground state values [Drouin et al. (31)])

 II

	Operator	Parameter	Value in cm -1
	J z	C A1	3.3410(25)x10 -2
	{i J y , J x }	C A2	4.622(110)x10 -4
	J x	C B1	6.94941(560)x10 -2
	{i J y , J z }	C B2	1.25664(370)x10 -3
	J x J 2	C B3	-1.044(100)x10 -6
	J x J z 2	C B4	6.3721(860)x10 -6
	J + 3 + J -3	C B5	1.1546(160)x10 -6

Table II .16 Coriolis interaction constants

 II

II.3.

  J', K', M', …) state involved. The intensity of the initial rotational line is least share of it. The higher rotational number J" the lesser the maximal Stark shift of

	Stark	( , , )	coef,	coef ,	( , , )
		1,3

the Stark components (see formula in Fig. III.8) and the lesser their intensities (see Eqs. III.11-12).

In the case of the strong Stark field when compared to fine or hyperfine fields, a perturbation method can be used to calculate the Stark splitting. Eq. (III.1) gives the energies of Stark levels (in MHz) up to third order of the perturbation development, the coefficients being specified further:

( )

Transition (No Stark Freq. [GHz] / wavenumber [cm -1 ] / wavelenght [μm])

  .1-3, the actually used ones are formatted in bold letters.

	Experimental technique	μ [D]		Comments	Authors	Year	Ref.
	OCS
	mw absorption	0.710	(4)		Shulman, Dailey,	1950	(71)
					Townes
		0.7124	(2) J=1→2, ΔM=0		Marshall, Weber 1957	(72)
	MBER	0.71521 (20) J: 1→1, M J : 0→±1	mean value	Muenter	1968	(73)
	MW absorption	0.7149	(3) (J, M) = (1,0)→ (2,1)	least square	Dijkerman,	1969	(74)
	resonant cavity				Ruitenberg
		0.71512 (3) 1.00031 (2) J: 1→1, M J : 0→±1, 16 O 12 C 34 S	μ relative to 16 O 12 C 32 S	Dymanus De Leeuw,	1970	(75)
		1.00017 (2) J: 1→1, M J : 0→±1, 16 O 13 C 32 S
		0.70423 (3) (01 1 0) vibrational state	least square	Reinartz, Meerts,	1972	(76)
	MBER	0.98473 (1) μ(OCS (01 1 0))/ μ (OCS (000))		Dymanus
	(Molecular	0.71519 (3)	Revision of Ref. (75) value based
	beam electric			on change of recommended value
	resonance)	0.70433 (3) (01 1 0) vibrational state	of Planck constant: 6.62559(16).10 -34 J.s [(77) used in (75)], 6.626196(50).10 -34 J.s (78)	Reinartz, Dymanus	1974	(79)
		0.98482 (2) μ (OCS (01 1 0))/ μ (OCS (000))	used in (79); 2011 recommendation CODATA2006:
				6.626 068 96(33) x 10 -34 J s (26)

Table III .1 Review table of OCS electric dipole moment value determinations

 III

	Experimental technique	μ [D]	Transition (No Stark Freq. [GHz] / wavenumber [cm -1 ] /	Comments	Authors	Year	Ref.
			wavelenght [μm])
	CH 3 F
	polarization 1.808	-		Smyth,

Table III .2 Review table of CH 3 F electric dipole moment value determinations

 III

	Experimental technique	μ [D]	Transition (No Stark Freq. [GHz] / wavenumber [cm -1 ] /	Comments		Authors	Year	Ref.
			wavelenght [μm])
	CH 3 CN
		3.97	J: 1→2, K = 0, ΔM J = 0	mean value		Coles, Good, Hughes 1950	(87)
			(36.79 GHz)
	mw absorption	3.92	(6) J=1→2 (36.79 GHz)	mean value		Ghosh, Trambaluro,	1953	(81)
						Gordy
		3.913	(2) J: 1→2, K = 0, 1 ΔM J = 0, ±1	mean value		Steiner, Gordy	1966	(83)
			(36.79 GHz)
		3.925191 (48)	least squares , ν 4	12 CO 2
				P(40)-P(50) and N 2 O P(18)-
		3.929493 (92) ν 7	P(29) around 10.6 μm, ν 7 12 CO 2 , P(34)-P(8), and R(8)-R(26) at 9.4 μm, and 13 CO 2	Rackley, Butcher, Römheld, Freund, Oka	1982	(88)
	laser-Stark spectroscopy	3.935513 (55) ν 4	P(14)-P(26), R(10)-R(24) at 9.8 μm, 550 CH 3 F Stark lines
				ΔM = ±1,0; P = 20-23 mTorr
		3.9256	(7) CH 3 C 15 N	least squares analysis, CO 2
				and N 2 O lasers, 650 Stark
		3.9354	(7) ν 4 , CH 3 C 15 N	vibrational transitions with lines for cca 140 ro-	Sakai, Katayama	1984	(89)
				J< 34 and K < 11,
				P = few mTorr
		3.9219	(15)	mean value
		3.92196

  Another type of phenomenological models uses statistically based energy gap fitting laws (103). In energy gap models, the real elements of W are expressed by state-to-state collisional transfer quantities between initial levels K(i'←i) that are represented as functions of energy gap ΔE i'i =|E i -E i' | in the statistical fitting law approach:

	'	( '	)	EG	(	'	) for	'

  other types (J or K manifolds, Q branches, hyperfine components) of closely spaced lines where LM reduces the broadening with increasing pressure[START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF].

					nothing but the
	IOS width of an "ordinary' rotational line , J K	J → +	1,	K	of a symmetric top molecule
	(107), in the absence of any static field.
	Coming back to the collisional line-mixing process, and according to Eqs. (V.6-7), one can
	see that if the off diagonal elements coupling the various Stark components M	M → +Δ	M
	are significant,

JK Γ will be significantly smaller than the weighted average WA JK Γ of the widths M JKM Δ Γ of these components. Stark manifolds of various M transitions thus behave similarly to clusters of

  Table V.4 with Table III of (15)). Consequently the IOS formalism will give, in the intermediate regime where the Stark components are neither well resolved nor completely degenerate, results very similar to those obtained in (15).

	M'/M	-5	-4	-3	-2	-1	0	1	2	3	4	5
	-5

Table V .5 Measured and calculated (IOS) He-broadening coefficients (in MHz/Torr) in the zero field limit and for isolated Stark components (ΔM = 0 case). J i ,K i → J f ,K f M i → M f Meas. IOS

 V

	1,1→2,1	Zero field	2.8	2.7
		-1→0	2 . 3	2 . 8
		1→0	2 . 3	2 . 8
		0 →1 and -1→-2	2.0	2.75
		0 →-1 and 1→2	2.0	2.75
	3,1→4,1	Zero field	2.6	2.7
		-3→-4 and 0→1	3.0	2.75
		-2→-3 and 1→2	3 . 6	2 . 8
		-1→-2 and 2→3	3 . 6	2 . 8
		0 →-1 and 3→4	3.0	2.75
	3,2→4,2	Zero field	2.6	2.6
		-3→-4 and 0→1	3.2	2.75
		-2→-3 and 1→2	3 . 0	2 . 8
		-1→-2 and 2→3	3 . 0	2 . 8
		0 →-1 and 3→4	3.2	2.75
	3,3→4,3	Zero field	2.4	2.5
		-3→-4 and 0→1	2.8	2.75
		-2→-3 and 1→2	2 . 9	2 . 7
		-1→-2 and 2→3	2 . 9	2 . 7
		0 →-1 and 3→4	2.8	2.75
	1,1→2,1	Zero field	2.8	2.7
		0	2.4	2.8
		1	3.0	2.8
	3,1→4,1	Zero field	2.6	2.7
		0	2.7	2.8
		1	2.8	2.8
		2	2.9	2.8
		3	3.1	2.8
	3,2→4,2	Zero field	2.6	2.6
		0	2.6	2.8
		1	2.7	2.8
		2	2.7	2.7
		3	2.7	2.7
	3,3→4,3	Zero field	2.4	2.5
		0	2.5	2.8
		1	2.6	2.8
		2	2.6	2.7
		3	2.6	2.6

Table V .6 Measured and calculated (IOS) He-broadening coefficients (in MHz/Torr) in the zero field limit and for isolated Stark components (

 V

Gamma calc (v a ) [MHz/Torr] v a [m/s] Γ calc (v a , J = 1,K = 0) Γ calc (v a , J = 1,K = 1) Γ calc (v a , J = 3,K = 0) Γ calc (v a , J = 3,K = 1) Γ calc (v a , J = 3,K = 2) Γ calc (v a , J = 3,K = 3)

Acknowledgements

I would like to thank my ancestors, especially mother Věra (in memoriam), father Jindřich and grand mother Marta. I thank Lucie who had accompanied me abroad for co-tutelle stays, delaying thus her own carrier and projects for a couple of years. I thank my supervisors Dr. Agnès Perrin and Prof. Štěpán Urban for the interesting yet manageable thesis themes they supported, for attentive care they provided and for their unwavering patience I experienced all along my doctoral studies. I thank Dr. Jean-Michel Hartmann for his original proposal, for his generous help with coding and for his kind

The work presented in this thesis was published in two articles: J. Koubek, C. Boulet, A. Perrin, Š. Urban and J.-M. Hartmann, Line-mixing between rotational Stark components of CH 3 F self-perturbed and perturbed by helium: Experimental results and IOS analysis, Journal of Molecular Spectroscopy, Journal of Molecular Spectroscopy, 266, 2011, 12-20 J. Koubek, A. Perrin, H. Beckers, H. Willner and Š. Urban, First analysis of the ν 5 band of DNO 3 (deuterated nitric acid) in the 11 μm region, Journal of Quantitative Spectroscopy & Radiative Transfer, 111, 2010, 1184-1192 Three more articles co-signed by Jindřich Koubek were published during his thesis studies: S. Zvánovec, P. Černý, P. Piksa, T. Kořínek, P. Pechač, M. Mazánek, J. Varga, J. Koubek, Š. Urban, The Use of the Fabry-Perot Interferometer for High Resolution Microwave Spectroscopy, Journal of Molecular Spectroscopy, 256, 2009, 141-145 L. Nová Stříteská, M. Šimečková, P. Kania, P. Musil, L. Kolesniková, J. Koubek and Š. Urban, Precise ground state molecular parameters of chloromethane, Journal of Molecular Structure, 919, 2009, 89-93 P. Kania, L. Nová Stříteská, M. Šimečková, P. Musil, L. Kolesniková, J. Koubek and Š. Urban, Rotational spectrum of 13 C chloromethanes, Journal of Molecular Spectroscopy, 252 , 2008, 90-92

radiation through its wires into the cell but mirrors the returning beam (characterized by orthogonal orientation of its components in comparison to the initial orientation) into the detector.

Fig. III.3 Scheme of a set-up with doubled absorption path

The microwave radiation is synthesized using a swept signal generator 83650 B (Agilent) operating from 10 MHz to 50 GHz. The signal generator is phase-locked to a rubidium atomic clock that is controlled by the cesium frequency standard via the GPS system.

The frequencies above 50 GHz are generated using a set of frequency multipliers. For this study a home made tripler by Dr. Vowinkel (Köln University) and a WR 4.3x6 sextupler produced by Virginia Diodes were used together with the high power semiconductor amplifier AHP 3404 by Wisewave Technologies that ensured a sufficient radiation power before the frequency multiplication. The millimeter set-up uses a set of Schottky non-cooled diodes as detectors, in this case WR 8 ZBD and WR 5.1 ZBD detectors by Virginia Diodes.

The rotational lines were measured using amplitude modulation since this modulation makes possible a more subtle numerical evaluation of the spectral background and overlapping lines than frequency modulation. An accuracy of the well developed lines measured by the amplitude modulation and subjected to the numerical treatment is estimated to be close to 10 kHz.

where δ is the frequency shifting coefficient and γ is the Lorentz broadening coefficient (per unit pressure).

After a Laplace transform of Eq. (IV.17) we obtain the Lorentz profile (91):

The Dicke profile Provided that velocity changing collisions do not correlate with dephasing collisions, the correlation function in the diffusional limit stands as:

The Dicke profile ( 93) is then obtained from the Laplace transform of Eq. (IV.21) using Eqs.

(IV.13) and (IV.17):

The expression of the Voigt profile can be obtained from the general expression (IV.5) of normalized absorption profile for isolated line, neglecting the velocity changing and speed dependence of Δ and Γ:

, that leads to: Since the equilibrium Maxwell-Boltzmann population distribution applies in collisions, the detailed balance principle relates the exchange terms:

where ( )

P → is probability of transition from i to i' and ( ) i ρ is the equilibrium relative population of the initial level of line ν if . The detailed balance indicates that more probable is the process starting from the less populated level. Therefore line-mixing promotes the transfer of intensity from the weak lines to the strong lines. More generally this leads to transfers from the weak to the strong absorption regions. In comparison with shapes of isolated lines, line-mixing leads to a narrowing of profile.

V.1.1 Absorption coefficient and widths

In the following, we assume binary collisions (valid at the pressures considered in this work), we disregard the Doppler effect (which can be easily introduced through a convolution by the proper Gaussian profile), and work within the frame of the "impact approximation" (valid since we consider regions close to the line centers). With these assumptions, the normalized spectral profile of a band composed of rotational components broadened and coupled by molecular collisions is given by Refs. [START_REF] Hartmann | Collisional Effects on Molecular Spectra[END_REF]101):

For each optical transition , d is its reduced dipole transition matrix elements and ρ is the relative population of its initial level. L 0 is a diagonal matrix, in the line (Liouville) space, containing the unperturbed line frequencies ν . W is the relaxation matrix, which is, within the binary collision approximation, proportional to the gas density (or total pressure P).

Its diagonal elements are related to the pressure induced width Γ and spectral shift Δ of the individual lines:

The off diagonal elements of W describe the couplings between the various transitions and they are responsible for the LM process. From now on, the imaginary part of W will be neglected. This is required in the absence of any model to construct them and this approximation is likely valid in view of the small pressure induced shifts. Knowing W, the collisional profile is obtained from the diagonalization of L 0 +iW (17, 102) and it can be convolved by a Gaussian function in order to take the Doppler effect into account, if necessary.

Two asymptotic behaviors will be of primary importance for the analysis of the results in section V.2. The first is the (very) weak overlapping regime when the widths of the pressure broadened transitions are negligible when compared with the spectral distances between

]. The collisional profile of Eq. (V.2) then becomes the sum of "isolated" Lorentzian contributions:

In the opposite case, i.e. for a group of completely overlapping lines

], the profile corresponds to that of a single "effective line" with a Lorentzian shape:

where the intensity, frequency and width of this "effective" line are given by :

Equation (V.6) shows that the line width of a bunch of completely overlapping collisionally coupled transitions is different from the weighted average of the widths of the isolated components, the latter being given by: For the J = 5 → 6, K = 5, and |M| values from 0 to 5, the IOS matrix is given in Table V. 4 and results are displayed in Fig. V.6 which also shows previous predictions (14). They show that, when properly applied, the IOS model does indeed lead to an agreement with the experimental data very similar to that obtained in (14) Consider, as in ( 14), the J = 5, K = 5 → J = 6, K = 5; line and more particularly the ratio of the Further similar studies of CH 3 F-He mixtures for more transitions and higher rotational quantum numbers would be of great interest. These would enable to test the limits of the IOS model and to evaluate the improvements brought by the adiabaticity corrections (20) of the Energy Corrected Sudden model. Furthermore, when available, extended experimental data could be compared with the results of direct calculations starting from the intermolecular potential. This could be done within the close-coupling scheme using the sophisticated CH 3 F-He potential of Ref. (113).

VIII Appendices VIII.1 Appendix 1: Derivation of speed dependent Voigt profile formula

The speed dependent Voigt profile is derived for an isolated line centered at frequency if ν , disregarding thus the line-mixing effects. The Doppler effect is taken in account as well as the dependences of collisional broadening and shifting parameters on the speed v of the absorbing molecule. The Dicke narrowing is not considered. The normalized absorption profile is that of equation (IV.36) stated in IV.2.3.:

Remark, that following equation is equivalent to VIII.1.

The wave propagation vector can be expressed, assuming that it is along the z axis, as:

. The normalised Maxwell-Boltzmann distribution of the radiator velocity expression using Euler angles is:

and gives after integration over φ and θ :

( )

Remarking the equivalence:

and after simplifications of the equation (VIII.1.7) we obtain:

After integrating per partes of the equation (VIII.1.9) we obtain:

( )

where ( )

Hard collision model sets:

we get:

then we can write also:

(VIII.2.9)

After integration over velocities and realizing that: