g Factors and lifetimes in the B state of molecular iodine

HAL Id: jpa-00208248

https://hal.archives-ouvertes.fr/jpa-00208248

Submitted on 1 Jan 1975

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

g Factors and lifetimes in the B state of molecular iodine

M. Broyer, J.-C. Lehmann, J. Vigue

To cite this version:

M. Broyer, J.-C. Lehmann, J. Vigue. g Factors and lifetimes in the B state of molecular iodine.

Journal de Physique, 1975, 36 (3), pp.235-241. �10.1051/jphys:01975003603023500�. �jpa-00208248�

g

FACTORS AND LIFETIMES IN THE

B

STATE

OF MOLECULAR IODINE

M.

BROYER,

^{J.-C. LEHMANN} and J. VIGUE Laboratoire de

Spectroscopie

Hertzienne de l’ENS

Université de Paris

VI, 4, place Jussieu,

Tour 12-E1 75230 Paris Cedex

05,

^France

(Reçu

le 31 octobre

1974)

Résumé. ²⁰¹⁴ Nous avons mesuré les facteurs de Landé et les durées de vie des niveaux de vibration- rotation de la molécule I2 excités par les diverses raies des lasers à argon ionisé ^et à krypton ^ionisé.

Pour cela nous avons utilisé les méthodes de l’effet Hanle et des résonances en lumière modulée.

Les valeurs expérimentales obtenues pour gJ varient beaucoup ^{avec le nombre} quantique ^{de vibra-}

tion v’ de l’état B. La relation théorique que l’on peut ^{établir entre} gJ ^et CI ^{la constante de structure} hyperfine

magnétique

^{est assez} bien vérifiée.

Abstract. ²⁰¹⁴ We have measured

the g

^{factors and lifetimes of the} vibration-rotation levels of molecular iodine excited

by

the various lines of Ar+ and Kr+ lasers. To obtain these results we have used the Hanle effect and the method of resonances in a modulated

light

^{beam. The}

experimental

values of gJ vary very much with v’ the vibrational quantum ^{number of the B state. A} theoretical

relationship

between gJ and

CI,

^the

magnetic hyperfine

constant, ^{has been}

proved

^{to be valid to a} good

approximation.

Classification

Physics ^Abstracts

5.448

1. Introduction. ^- The first measurements of the

magnetism

^{of the B state of}

12

^were

performed

^{in 1914}

by

^{Wood and} Ribaud who studied the

Faraday

effect

[1].

In

1937, using

^the

theory

of Serber

[2],

^Thomas

Caroll

[3] interpreted

^the results of Wood and Ribaud : for the level v’ ⁼ 27 J’ ⁼ 109 excited

by

^the

Hg

line 5 461

Á, they

gave : gJ = 0.06. Until

recently [4, 5]

it was the

only

information about the

magnetism

of the B state.

In order to measure

the g

factors of several indi- vidual rotational-vibrational levels of the B state we

have used the Hanle effect

technique

^{which was}

developed

^{a few years ago for the}

study

of molecular excited states

[6].

When it was necessary ^{to measure} the _{g J} factor

[4]

directly,

^{we used the} method of resonances in a

modulated

light

beam introduced

by

^Series

[7].

By

^these

experiments

^we have measured the values of _{g J} and i for the

following

levels of the B state :

Independently

^{of our}

work,

Paisner and Wal- lenstein

(5) (7)

have used the method of quantum beats to measure g factors and lifetimes in the B state. We shall _compare their results with our own at the end of this _paper.

2. Hanle effect. ^- 2.1 ZEEMAN SPLITTING IN THE

B STATE OF

12.

^- The Hanle effect of molecular iodine is

complicated

since each rotational level J is

split

^into

a great ^{number of}

hyperfine

^sublevels.

The nuclear

spin

^{of the}

^12’I

^nucleus

being 5/2,

each level of the

B3Ilô

^{u state is}

split

^{as follows :}

e For even J the state is

ortho,

the total nuclear

spin

^{I =}

h

⁺

/2

takes the values

1,

^{3 or 5. Each} rotational level is

split

^{into 21}

hyperfine

^sublevels.

e For odd J the state is _{para, I} ⁼

0, 2, 4,

^each rotational level is

split

^{into 15}

hyperfine

^sublevels.

The

hyperfine

^{structure of some of} these levels has been measured

by

Levenson et al.

[8].

^{It is}

mainly

^due

to the

quadrupole

^{moment of the two} nuclei. There-

fore,

^{I =}

h

⁺

12

^{is not a}

good

^{quantum number and}

the

eigenstates

^are

given by :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003603023500

236

The Zeeman hamiltonian is written :

g J is the

rotational g

^factor

gl is the

nuclear g

factor of the 127 1

nucleus,

gi = 1. 12 [9].

^’

gl, should be the

magnetic shielding

^{tensor. In fact if}

J >

I, (i.

e. as soon as J >

10),

^it may be shown

[10]

that it is sufficient to

consider g1

^{as a number. For}

most

rnolecules g1

^is

negligible

gi z

10-4 [11].

^We

shall come back to this

point

^at the end of this _paper.

In the low field limit a Landé factor _{9 tE} ^can be defined for each

hyperfine

^{sublevel :}

where

The

intensity

of the fluorescence is obtained

using

irreducible tensor formalism

[20].

^{It is}

proportional

^to

A(E, F, k, J’)

^is the contribution of each

hyperfine

sublevel.

J’ is the rotational number of the excited state.

Tr

is the inverse of the radiative lifetime of the excited state and T, the inverse of the total relaxation time.

E is the excitation

operator, 5)

the detection

operator.

E:

^and

U)k

^{are the} components of î and 0 on the basis of irreducible tensor operators

Tq.

By

this method the H anle effect curves may be calculated. It must be noted that for a

given

^rota-

tional level the

shape

^{of the curves}

depends only

upon the relative values of _gj and

(g,

⁺

g,),

^r

being only important

for the width of the curves. Two cases

should then be considered :

almost

equal

^to gj. The Hanle effect curves are then lorentzian. From the width of the

experimental

^curves

at zero pressure the

product

g J z is deduced.

2) 1 gj 1 5 j (gi ^{+ gt).gü}

^{varies very much for the}

various

hyperfine

sublevels. Its

sign

may

change.

^The

Hanle effect curves are very different from lorentzian

ones. The relative size and width of the

dispersion

and

absorption

Hanle effect curves are very sensitive

to the relative values of _gj and

(gl

⁺

gl).

^{From the}

comparison

between the

shape

^of

experimental

^and

theoretical curves the _gi value is deduced as a

function of (g,

⁺

g).

^Il g, « gj, ^{one can}

neglect gl

^{and deduce}

the value of _gj. Then from the width of the curves the

lifetime

^is also obtained.

Both of these cases occurred in the set of rotational levels studied here.

2.2 EXPERIMENTAL sEr uP. ^-

Figure

^{1 shows the}

experimental

^set up. The

magnetic

^{field is}

produced by

^an

electromagnet.

The fluorescent

light

is observed in the direction of the

magnetic

^field

using

^{a mirror}

(this

^{mirror is not drawn on the}

figure).

^The

pola-

rization is detected

by

^a

rotating quarter

^wave

plate

followed

by

^{a fixed}

polarizer. By changing

^the

phase

of the lock-in detector

by

^{900 it is}

possible

^{to record a}

dispersion

^{or an}

absorption

Hanle effect curve. The monochromator is _necessary when a laser line excites

simultaneously

several rotational levels of the B state

[21].

. |

FIG. 1. ^- Expérimental set up for the Hanle effect experiment.

The laser beam ^was

expanded by

^a

lens,

^the power

density

of the laser

being

then about 20

mW/cm.

We have verified that in these conditions saturation effects are

negligible.

^Our observations

related

^to these saturation effects in

12

have been

published

elsewhere

[12] [13].

2.3 EXPERIMENTAL RESULTS. ^- From the observa- tion of the Hanle

dispersion

^curves recorded under suitable

conditions,

^the

sign

^of gj ^can be deduced.

For all levels studied _gj is found to be

negative,

^that

is the same

sign

^{as an electron.}

2. 3 .1 Case 1.

gj (g, ^{+ . -}

^{The levels excit-}

ed

by

the laser lines 5 145

A

and 5 309

A

^are most

representative

^{of this case.} For the levels excited

by

the laser line 5 682

A ;

^the determination of gj

by comparison

^between

experimental

^and theoretical

curves was also

performed

^{but the}

precision

^was very poor. The lifetimes of these levels ^are deduced from the width of the Hanle curves

extrapolated

^{to zero}

pressure. The whole ^set of results obtained is _pre- sented in table 1.

2. 3 .1.1 Excitation at 5 145

A. - Two

^rotational

levels J’ = ₁₂ and J’ ⁼ 16 of the set v’ ⁼ 43 are

TABLE 1

The whole set

of results.

The values

of gl

^{with * are estimated}

by

the method

explained

^{in 4. It must be}

remarked that we measure

a (2)

the cross section

of destruction of alignment.

^On the contrary, ^{Paisner et al.}

measure

0’(0)

the cross section

of destruction of

^the

population.

( 1 ) ^{On this} level, ^{we had} already reported [4] g J ^{= -} 0.31 ± 0.06 and ^{i =} 2.9 ± 0.05 s. These values were obtained by assuming

g, ⁼ 0 which is evidently wrong since we have measured gl ⁼ 3.4 for the laser line 5 017 A.

excited. The Hanle effect was

simultaneously

^observed

for both levels since in order to

separate

the fluores-

cence from the two levels the slits of the monochroma-

tor would need to be

20 Il

wide and would therefore

give

^{too low a}

signal

^to noise ratio for _any serious measurements. To calculate the theoretical curve

shapes

^we assumed that the two levels involved have the same g J factor and the same lifetime 1’. This

assumption

^{is in}

good

agreement ^{with the}

interpre-

tation of

the g

^factor

developed

in the last

paragraph

of this paper;

following

^this

interpretation

gj is

independent

of the rotational number J’ and

depends only

upon the vibrational quantum ^{number v’.}

The

dispersion

^and

absorption

curves are in this

case very different from each other both in their size and their width. Therefore the

precision

of the fit is rather

good :

²⁰

%.

Furthermore the Hanle effect

curves are so narrow that

magnetic predissociation [14]

can be

neglected.

^-

2.3.1.2 Excitation at 5 309

A.

^- The situation is _very similar. Two levels v’ ⁼ 32 J’ = 9 and v’ ⁼ 32 J’ ⁼ 14 are excited and

they

^are observed simul-

taneously.

^The gj factor of both levels is fitted with

a

precision

^{about 15}

%. Again

^the

magnetic predis-

sociation has no influence on the Hanle curves.

2. 3 .1. 3 Excitation at 5 682

Á.

^- Each level excited

by

this line has been studied

separately, using

the monochromator with narrow slits

(60

^{to 100}

03BC).

Furthermore, gj being

of the order

of gj,

^{the curves}

were not

drastically

different from lorentzian. At the

end,

the width of the Hanle ^curves

being

^{in this}

case about 2 000 to 5 000 _gauss the

magnetic predisso-

ciation was not

negligible.

Therefore the

precision

^of

the _gj determination was rather _{poor :} about 40

%

or

50 %.

The results are shown in table I. The levels v’ ⁼

21,

J’ ⁼ 11 b and J’ ⁼ 122 have been studied

separately

but no

meaningful

difference was noticed in their _gj factors.

As an illustration of our method for the determi- nation of _gj,

figure

2 shows the

agreement

^between the theoretical and

experimental

^{curves in the case}

of an excitation at 5 309

Á.

The

dispersion

^and

absorption

curves are recorded at the same

sensitivity.

It must be noticed that their sizes and their widths are

quite different.

FIG. 2. ^- Comparison between theoretical and expérimental ^Hanle

curves for the excitation at 5 309 A. The points ^are theoretical.

The 12 pressure ^was 3 mtorr.

2.3.2

Case 2. gj » (g,

gj ~

9r 9i) + gl). -

^{This case cor-}

responds

^{to an} excitation

by

the laser lines 5 017

Å

and 5 208

A.

^The values of the

product

gj i obtained

by extrapolation

of the width of the Hanle curves to

zero vapour pressure ^are also

presented

^{in table I.}

2.3 .2.1 Excitation at 5 017

Á.

^- One rotational level J’ ⁼

27,

^{v’ = 62 is}

mainly

excited. The

product

238

gi ⁱ is

large.

^The

extrapolation

of the line width of the curves to zero vapour pressure is _very difficult since it is _very sensitive

[15]

^{to small}

quantities

^of

gaseous

impurities

^{in our} cells. This is due to the

high

value of the lifetime r.

2.3.2.2 Excitation at 5 208

À.

^- One rotational level J’ ⁼

77,

^{v’ = 40 is}

mainly

excited. The

product

^

gj,r is much weaker and there is no

problem

^for

extrapolation

^{to zéro} vapour pressure.

Figure

³ shows the Hanle effect curves recorded for the laser line 5 208

Á.

Of course

they

^{are lorentzian}

and one can notice that the

dispersion

^and

absorption

curves have about the same

amplitude

^{and the same}

width.

FIG. 3. ^- Hanle effect curves for the excitation at 5 208 Á. The

points correspond ^{to the} best fit of lorentzian curves. The 12 pressure

was 3 mtorr.

2. 3. 3 Particular case

of

^the excitation at 6 328

Â.

^-

The excited level is v’ ⁼ 6 J’ ⁼ 32. The

interpretation

of the Hanle effect is _very difficult in this case since the various

hyperfine

^{sublevels are not}

equally

^excited

by

^the

laser,

^the laser line

being only

^coïncident

with the

wing

^{of the}

absorption

^{line of}

I2.

Furthermore the

magnetic predissociation

^{is so}

important [16]

that is

strongly

modifies the

shape

^{of the Hanle curves.}

However

preliminary experiments

and calculations show that : - 0.04 _{g J} 0.

The whole set of results for lifetimes shown in table 1 are in

good agreement

with those obtained

independently by

^{J. A.} Paisner and R. Wallenstein

[17]

by

^the

decay

^method.

On table 1 we also

give

the values of

0’(2), the

^cross section for destruction of the

alignment. ^0’(2)

^is

deduced from

the slope

^{of the curve AH=}

f ( p)

FIG. 4. ^- dHD is the distance of the two maxima of the dispersion

Hanle curves. p is ^the 12 pressure. Laser line ^{5 145} Á.

where AH is the width of the Hanle curves

and p

the

12

pressure.

Figure

⁴ shows such a curve for an

excitation at 5 145

Á.

Apart

from the levels v’ = 32 J’ ⁼ 14 and 9 excited at 5 309

Á,

^the

depolarisation

^of the fluores-

cence as a function of the

12

pressure is

generally

very weak and

(1(2)

is not very different from

(1(0)

the cross section for destruction of the

population (quenching

^cross

section).

3. Resonances in modulated

light.

^{- This}

technique permits

^{us to measure}

directly

the Landé factors g eF of the various

hyperfine

sublevels. The

principle

of this

experiment

^{is the}

following :

^{the laser}

light

^is

modulated at a fixed

frequency

^m and the modulation of the fluorescence

light

is detected at the same

frequency

^{co. If the}

magnetic

^{field is}

swept,

^resonances

are observed when hm is

equal

^{to the}

splitting

^between

two Zeeman sublevels of the excited states. The exci- tation and the detection must be coherent. This condition is obtained

using

^{linear 6}

polarisation

^both

for excitation and detection. Therefore at resonance,

we induce coherence between Zeeman sublevels such

as Am ⁼ + 2.

The

experimental

^{set up is very}

simple.

^{The laser}

light

is modulated at 800 kHz

by

^a Pockels cell or an

electro-optic

modulator. The modulation of the fluo-

rescence is monitored

by

^a

photomultiplier

^and

by

a brookdeal lock-in

amplifier.

This

experiment

^was

performed

^{on the levels}

excited

by

the laser lines 5 017

Á

and 5 208

À.

3.1 EXCITATION AT 5 017

A. - Figure

^{5 shows}

the

spectrum

^of resonances

corresponding

^{to the 15}

hyperfine

sublevels of the para ^state v’ ⁼ 62 J’ ⁼

27.

FIG. 5. ^- Resonances in modulated light ^for the laser line 5 017 À.

The same resonances are obtained with the direction of the

magnetic

field reversed.

The

position

of the central resonance

gives

g J ^{= -} 1.8 ± ^{0.2. The}

dispersion (or scattering)

^of

resonances

gives (gl

⁺

gl).

^We

obtain gi

^{= 3.4} ± ^0.5.

Figure

5 shows the theoretical

position

^{of the reso-}

nances calculated for these values

of gl

^and gj.

The agreement

^is

^very good.

^{In fact one resonance is not}

explained by

^this

theory.

^However it is known

[19]

that the laser line 5 017

A

also

significantly (10 to

20

%)

excites another level

(v’

^{= 71}

± 2

^{J = 55} ±

2)

of the B state. Most

probably

^this resonance corres-

ponds

^to the level v’ ⁼ 71.

Thé

^{value of} g J deduced from the

position

^of

this

^{resonance would be}

gj _ - 5.2 ±

0.8,

ⁱⁿ

good agreement

with the inter-

pretation of 9J developed

in the last

part

^{of this} paper.

The lifetime of the v’ ⁼ 62 J’ ⁼ 27

level

is deduced from the,

product

gi T

given previously.

^{We find}

1 ⁼ 14 ± ³ J.1s. This result was confirmed

by

^{a direct}

measurement of the

exponential decay [18]

^{of the}

fluorescence

following

^a

pulsed

excitation.

These values

of g J

^{and i for} the level v’ ⁼ 62 J’ ⁼ 27

are in

disagrqement

with those

reported by

^Paisner

and Wallenstein.

They

^find gj = - 2.6 andr ⁼ 8.8 _J.1s.

We believe that in the present ^{case of a} very

compli-

cated

spectrum

^{of 15} resonances the

technique

^of

of resonances in a modulated

light

^{beam is more}

accurate than the

quantum

beats method that

they

used.

Indeed,

Paisner and Wallenstein observed ^an almost unresolved

quantum

^beat

signal

^that

they

had to fit with a theoretical curve

involving

^two

parameters gl

^and gj. In ^our

experiment

^{we can}

observe well resolved resonances from which it is

extremely

easy ^to determine the values of these two

parameters ^{and even to} check the theoretical _expres- sions used for the Landé factors.

Furthermore,

^the lifetime is difficult to obtain with

precision

^{because it}

is _very

long

^and very sensitive to gaseous

impurities

in the cells. Therefore we have confidence in our

results.

3.2 EXCITATION AT 5 208

Â.

^-

Figure

^{6 shows}

the resonances obtained in this case. The width of each resonance

being

^too

large

^{it is not}

possible

^to

Fm. 6. ^- Resonances in modulated light for the laser line 5 208 A.

measure gl. We. can

only

^deduce gj = - 0.32 ± 0.04 in

agreement

with the results of Paisner and Wallenstein who find _{g J} ^{= -} 0.28.

4.

Interprétation

^{of the}

experimental

^{results. -}

4.1 SECOND ^ORDER PERTURBATION THEORY. - The gj factor of molecules without electronic

angular

momentum

(Q

⁼

0)

^is

generally interpreted

^{in terms}

of second order

perturbation theory [11].

The

perturbation

hamiltonian can be written :

where

L and S are the electronic

angular

^momenta,

J ⁼ L + S. 1 ⁼

Il

⁺

12

is the total nuclear

spin.

re is the

position

of the electron.

V,

^{is the off}

diagonal

part of the rotational

hamiltonian, V2

^{is the Zeeman}

hamiltonian, V3

^{is the}

magnetic hyperfine

hamiltonian.

A molecular

eigen

^state may be

written : 1 rQva >

where ^T and S2 are characteristic of the electronic state. v is the vibrational

quantum number,

^a represents

(8JF MF).

For the

perturbated state 1 r’,

^{Q =}

0, v’,

^a’

>,

^the

second order

perturbation

^{term is :}

The summation over v becomes an

integral

^when

the

perturbing

electronic states form a continuum.

The cross

perturbation

^term

240

gives CI,

^{the constant of the}

magnetic hyperfine

^struc-

ture term

(C, I.J). CI

has been measured for several levels

by

^Levenson and Schawlow

[8].

The cross term

gives

g J.

The cross term

gives gl.

These constants are

independent

^of

^J, they depend only

^on

v’,

the vibrational quantum number of the

perturbed

^level.

Suppose

^{that we assume that :}

Then,

^{for each} term,

expression (1)

^{has a factor :}

and the various constants gl, g J

and CI

^{are propor-}

tional to each other.

Indeed this

approximation

^{is rather}

rough,

^{but it}

is useful to comparç the measurements of _gj and

CI,

and to

estimate gl

for the levels

where gl

^{is not mea-}

sured.

For

instance,

^{it would}

give :

The value

of gl

^estimated

by

^this way for the level

(27, 62)

^excited

by

^{5 017}

^A

^is g, ⁼ 3 in

good agreement

^{with our}

experimental

^result.

For the other

levels,

the estimated values

of gl

have been

reported

^{in table 1.}

4.2 VARIATION

OF g J

^{wm v’. -} This variation is very similar to that

of CI

^with

^v’,

^observed

by

^Leven-

son and Schawlow

[8].

The authors have

given

the formula :

which

they

both derived

theoretically

and verified

experimentally.

E is the _energy of the studied level in the

potential

curve of the B state.

Ec is

^an

adjustable

parameter.

It is

possible

^to

verify experimentally

that such a

law is also valid for _{g J.} We find :

where

Ec

^{and E are taken in cm-1.}

Ec

^{= 4 360}

^cm-1

ⁱⁿ

good

agreement with the value of reference

[8] Ec

^{= 4 340 ± 40}

^cm-1.

Also Paisner and Wallenstein have

reported Ec=4 ^{391 ± 40}

^{cm -1.}

Figure

^{7 shows our}

expérimental

values for _{g J}

plotted against

^the theoretical curve obtained from eq.

(2).

The

reported

values for

Ec

^are very close to

Do

the dissociation _energy of the B state. This fact seems to prove, ^{as was}

already

^noticed

by Levenson,

^that the

perturbing

^state

(or states)

^{is a} dissociative state with the same

asymptote

^{as the B state.}

FIG. 7. ^- gj ^{as a} fonction of vibrational energy.

5. Conclusion. ^- In this _paper we have

reported

a set for measured values

for g J,

gl and 1 in the B3

II Q

^u

state of iodine.

Although

these results seem in

good agreement

^{with an}

oversimplified

theoretical inter-

pretation, they

should be considered as

preliminary

to a more

complete analysis.

^{We are in the} process of

measuring

^a series of such

parameters

^{over a}

great

number of vibrational and rotational levels of the B electronic state. We use for this _purpose several

types

of

dye

lasers. A careful

analysis

^of

these

^{data will} then

permit

^{us to}

give

^{a more detailed} theoretical

interpretation. Especially interesting

^{will be the para-}

meters measured for levels _very close to the dissocia- tion limit for which the Landé factors become very

high.

^Also

interesting

^will be the results obtained for low vibrational levels ^for which the Landé factors

are very small and become sensitive to first order _per- turbation terms

giving

^{rise to a}

^small,

pure ^rota-

tional,

Landé factor.

References [1] WOOD ^and RIBAUD, ^Phil. Mag. ²⁷ (1914) ^1009.

[2] SERBER, Phys. ^{Rev. 41} (1932) ^489.

[3] CAROLL, T., Phys. ^{Rev. 52} (1937) ^822.

[4] BROYER and LEHMANN, Phys. ^{Lett. 40A} (1972) ^43.

[5] WALLENSTEIN, R., PAISNER, ^{J. A. and} SCHAWLOW, ^A. L., Phys. ^{Rev. Lett. 32} (1974) ^1033.

[6] MCCLINTOCK, M., DEMTRÖDER, W. and ZARE, ^R. N., ^{J. Chem.}

Phys. ⁵¹ (1969) 5509 ;

MARECHAL, M. A., JOURDAN, A., NEDELEC, O. and PEBAY PEYROULA, J. C., C. R. Hebd. Séan. Acad. Sci. 268 (1969)

1428.

[7] CORNEY, A. and SERIES, ^G. W., ^Proc. Phys. ^{Soc. 83} (1964)

207.

[8] LEVENSON, M. D. and SCHAWLOW, ^A. L., Phys. Rev. A 6

(1972) 10.

[9] MURAKAWA, K., Phys. ^{Rev. 100} (1955) ^1369.

[10] BROYER, M., Thesis Paris (1973).

[11] ^{TOWNES, C. H. and} SCHAWLOW, ^A. L., Microwave Spectroscopy (McGraw Hill).

[12] BROYER, M., DALBY, F. W., VIGUÉ, ^{J. and} LEHMANN, J. C., Can. J. Phys. ⁵¹ (1973) ^227.

[13] DUCLOY, M., VIGUÉ, J., BROYER, M. Communication au

Colloque ^CNRS 217, p. 241 (Aussois, ^mai 1973).

[14] ^VAN VLECK, ^J. H., Phys. ^{Rev. 40} (1932) ^544.

[15] BROYER, M., VIGUÉ, J., Communication au Colloque ^CNRS 217, _p. 185 (Aussois, ^mai 1973).

[16] CHAPMAN, ^{G. D. and} BUNKER, ^P. R., ^{J. Chem.} Phys. ⁵⁷ (1972)

2951.

[17] PAISNER, J. A. and WALLENSTEIN, R., ^{to be} published ⁱⁿ

J. Chem. Phys.

[18] KELLER, J. C., BROYER, M., LEHMANN, ^J. C., C. R. Hebd. Séan Acad. Sci. B 277 (1973) ^369.

[19] LIEHN, J. C., BERJOT, ^{M. and} JACON, M., Optics ^Commun.

10 (1974) 341.

[20] FANO, U., ^{Rev. Mod.} Phys. ²⁹ (1957) ^74.

GOUEDARD, G., LEHMANN, ^J. C., J. Physique ³⁴ (1973) ^693.

[21] VIGUÉ, J., LEHMANN, J. C., ^Chem. Phys. ^{Lett. 16} (1972) ^385.