Semi-empirical evaluation of Sr-Ar difference potentials

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Semi-empirical evaluation of Sr-Ar difference potentials

Le Quang Rang, D. Voslamber

To cite this version:

Le Quang Rang, D. Voslamber. Semi-empirical evaluation of Sr-Ar difference potentials. Journal de

Physique, 1986, 47 (7), pp.1149-1154. �10.1051/jphys:019860047070114900�. �jpa-00210303�

Semi-empirical evaluation of Sr-Ar difference potentials

Le Quang Rang

Département de Recherches Physiques, ^Tour 22, Université Paris VI, 75230 Paris Cedex 05, ^France and D. Voslamber

Association EURATOM-CEA, Département de Recherches

la Fusion Contrôlée,

B.P. N° 6, ⁹²²⁶⁵ Fontenay-aux-Roses, ^France (Reçu ^{le 27 dgcembre} 1985, accepté ^{le 6}

1986)

Résumé. 2014 En essayant ^{de faire coïncider}

profil d’absorption théorique

profil expérimental,

avons

tenté, pour le

de la raie de résonance de strontium perturbé par argon, d’évaluer les deux premières

différences de potentiels prises

la forme de Lennard-Jones. Le profil expérimental ^{utilisé est celui obtenu}

dans

expérience ^de Carlsten et al. Pour le profil théorique

employé

expression

d’aile unifiée » basée

l’approximation ^des trajectoires rectilignes pour les noyaux ^et

l’approximation adiabatique pour les états électroniques.

Abstract.

By fitting

theoretical to

experimental absorption profile ^of the strontium

line per- turbed by argon,

have attempted

evaluation in the Lennard-Jones form of the first two Sr-Ar difference potentials. ^The experimental absorption profile used in the procedure is taken from Carlsten et al’s redistribution

experiment. For the theoretical profile

^unified wing formula is used which is based

the classical straight path approximation ^for the nuclei and

the adiabatic approximation for the electron states.

Classification

Physique ^Abstracts

32.70J - 34.15

1. Introduction.

The amount and _accuracy of information about inter- atomic interactions and collision dynamics ^{have been}

increased substantially by ^recent light-scattering expe- riments using ^tunable dye ^lasers [1-4]. ^In particular,

the redistribution of radiation close to

atomic

resonance has proved ^{to be an effective tool for stu-} dying ^{such collision} complexes ^{in which an} optically

active atom in a selected state interacts with an

optically passive perturber. ^{To the extent that the}

collision occurs adiabatically, ^the scattering process is significantly determined by ^the corresponding

interatomic potentials ^and dipole strengths. ^These

are

therefore essential for the characteristics of the

spectroscopic ^data (such ^as e.g. line shapes ^and frequency-dependent polarization degrees) ^and may, in turn, be determined from these data

the basis of suitable theories.

In practice, ^the determination of potential ^and dipole ^curves from measured line shapes ^or polariza-

tion degrees, is difficult because the functional depen-

dence between the observed spectral data and the

quantities ^to be determined is rather involved. Com-

plications may arise from the dynamics of the combined collisional and radiative _process, from the multiplicity

of the transitions associated with a given frequency interval, and from the statistics of the problem.

Therefore, many of the previous investigations ^have

not strictly proceeded ^to determine the interatomic parameters from the observed spectral ^{data. Instead} they ^{were confined to} comparing these data with

corresponding theoretical results which ^were based

on

interatomic parameters ^obtained by independent

methods. The extent of agreement ^between experiment

and theory ^then provided ^some information about the quality of the interatomic parameters used in the

theory. ^On the other hand, in those papers where interatomic parameters

^extracted directly ^from

observed data, ^the complexity ^{of the} problem ^{had to}

be reduced by simplifying assumptions ^such

^the impact theory ^{for the line} body, ^the quasistatic theory for the far line wings and, ⁱⁿ nearly ^{all cases,}

the existence of only ^one transition (i.e. ^{one lower}

and one upper potential) contributing ^{to the line.}

Thus if

quasistatic ^line wing ^{is due to a} single ^tran-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047070114900

1150

sition and if one disregards ^the complications arising

from multiple ^Condon points, ^{it can be} directly

converted into the corresponding potential

[5-7]. ^{To extract}

difference potential ^{from the} impact-broadened ^line body,

usually employs

the method introduced by Behmenhurg [8] : ^the potential ^{is assumed to be of the} Lennard-Jones form whose two constants

then be determined from the width and the shift of the line. Either

both of these methods have been used in nearly ^all papers

dealing ^with e.g. alcaline-earth lines perturbed by

noble _gases. For the Sr-Ar case, which will be studied in the present paper, ^see e.g. references [1] ^and [9-13].

In our present investigation ^{we have} attempted ^to

go beyond ^the procedures mentioned above by using

a

dynamical (as opposed ^to quasistatic) theory ^{for the} semi-empirical evaluation of a pair ^{of difference} potentials corresponding to a E - E and a.E - 17 transi- tion. We assume that both potentials ^{can be} approxi-

mated in the Lennard-Jones form, ^so that the task is to determine the four parameters AC6-, ðC;2’

AC6’, OC 2. ^We have chosen the resonance transition of strontium perturbed by argon,

for this case

detailed experimental ^{data are available} (see ^references quoted above and in particular Carlsten et al. [1],

henceforth referred to as CSR). ^{In section 2 we will}

present

brief survey of the line broadening theory employed ⁱⁿ

investigation. Section 3 will summa-

rize the results for the Sr-Ar difference potentials

which we have obtained from

best-fit with the redistribution experiment of CSR. Some concluding

comments on these results will be given in section 4.

2. Theory.

It has been emphasized ^{in a} previous publication [14]

that results obtained from light scattering experiments

are closely ^{related to those obtained} in emission or

absorption spectroscopy. ^In particular, ^{the total} integrated fluorescence signal ^{as a function of the}

incident detuning from the atomic resonance is to a

very good approximation proportional ^{to the} absorp-

tion profile ^{of the resonance} transition considered. We therefore have chosen to take as point ^of departure ^a previously established formalism for the wings ^of

spectral line broadened by heavy perturbers (Le Quang Rang and Voslamber [15], henceforth referred to as LV). ^{In the} spectral range where the broadening

occurs adiabatically, this formalism provides ^a

^uni-

fied » wing expression ^{in the sense that} it describes the transition from the close (dynamic) ^{to the far} (possibly quasistatic) parts ^{of the line} wings. ^The

formalism is based on the following ^three assumptions : (a) ^the one-perturber approximation,

(b) the classical straight-path approximation ^{for the}

two nuclei involved in the collision complex,

(c) ^{the adiabatic} approximation for the electronic states.

As opposed ^to the unified Franck-Condon theory

of Szudy ^and Baylis [16]

treatment, though ^less sophisticated ^with regard ^{to the} dynamics ^{of the} nuclei,

has the advantage ^of including the adiabatic rotation of the dipole associated with the radiative transition.

It might ^be argued ^that

collision is

adiabatic in all its stages; ⁱⁿ particular, after radiative excitation of

collision complex ^{at small} internuclear separation,

the two particles fly apart ^to spend ^a long ^time (of ^the

order of the natural life time) ^at large distances before

reemitting ^the fluorescence photon. During ^this time, the dipole decorrelates from the nuclear motion, ^and

the present ^treatment might ^seem unsatisfactory. ^The

answer to such an objection ^{is that the} absorption profile ^is only ^{sensitive to what} happens during ^the

radiative excitation _process, i.e. during the _relatively

short instant when the interatomic potential ^{is near-}

resonant with the incident radiation. During ^this

instant the rotation of the dipole ^is important. ^Whether

later

the dipole ^{continues to rotate or not is irrele-}

vant to the total integrated fluorescence which is

uniquely determined by ^the strength of the transition

dipole, ^not by its direction.

While assumptions (a)4c) suffice for deriving ^the general line-wing expression, equation (1) presented below,

have made three further approximations

for numerical convenience :

(d) ^{the use of}

representative ^«thermal velocity »

instead of

average ^over velocities,

(e) ^{the use of the} dipole strengths ^{of the} unperturbed optically ^{active atom for all} atom-perturber distances, (f) ^the neglect ^{of the} Boltzmann factor (pa ⁼¹

in Eqs. (2), (3), (7) below).

As for the quality ^of approximations (a)-(f) ^{in the}

case we will be considering, ^{we first note that there}

should be

problem ^with approximation (a) ^because

the frequency ^domain investigated lies well beyond

the halfwidth of the spectral line. The validity ^of approximations (b) ^and (c), however, depends ^{on the} potentials ^{to be} determined and must be confirmed

a

posteriori. Since the order of magnitude ^{of the} potentials ^{was known in} advance, ^the approximate

domain of adiabaticity could be delimited from the outset; ^{this led us to} exclude the experimental point ^{at d}

^{5 cm-1} (see ^Sect 3) ^{from our best-fit} procedure. ^{The final} results then confirmed that the adiabatic approximation breaks down for I.J (

7 cm - 1 but holds well in the remaining part ^{of the} wing. The overall validity ^{of the} straight-path approxi-

mation was also confirmed, except perhaps ^{in the} region ^{of the E -II} potential ^well (mainly ^made up

by ^{the excited II} potential) ^whose depth is about 30 %

of the thermal _energy of a particle. ^{Since on the line} profile ^this corresponds ^{to the} far-wing ^{satellite at}

=- 278 cm-1, ^the shape of this satellite _may suffer

some influence from approximation (b). ^The bearing

of approximation (d) ^can hardly ^be judged ^without

performing ^the average ^over velocities; however,

there is an indication from the work of Caby-Eyraud

et al. [17] ^{on Stark} broadening by electrons that the

error connected with using ^{a thermal} velocity ^should

be less than 10 %. (It is in fact of the order of 5 % ^{in the}

case considered by ^these authors.) Approximation (e) ^is currently ^{used in} line-shape calculations because

as

matter of experience ^the dipole strength ^{of an}

allowed line usually depends ^{little on the atom-} perturber separation. Approximation (f) ^{is of a similar}

nature as approximation (b) but refers uniquely ^to

the groundstate potential ^{which is not} determined in the present paper. The dependence ^{of this} potential

on the atom-perturber distance is expected ^{to be}

weaker than that of the excited potentials ^{so that} approximation (f) should be at least as good ^as approximation (b). An ultimate judgement ^{of this} approximation, ^{however, can} only ^be given ^once

the groundstate potential ^{has been} determined, ^too.

Let us now turn to the line-shape ^{formalism derived} by ^{LV. We define} 4ro) ^{to be the} pressure-broadened absorption profile (without Doppler broadening)

normalized to the total sum of dipole strengths contributing ^{to the} spectral line under consideration.

As compared ^to equations (1) ^and (9) ^of LV, ^{we omit}

the factor (colcoo)P (p =1) because the absorption profile occurring in the redistribution function is not defined to be the shape ^{of the} absorption coefficient

K(ro) (absorbed intensity per unit length ^{and unit} frequency) but is the shape ^of K(ro)/C1iro (absorbed

number of photons per unit length ^{and unit} frequency).

As shown in LV, ^{the above} assumptions (a)-(c) ^then

lead to the following expression ^{for the} wings ^{of the} profile L(w) :

Here, n denotes the number density of perturbers ^and fM(v) is the Maxwell distribution of relative atom-

perturber velocities

(involving the reduced atom-

perturber mass). ^The subscripts

and fl ^{number the}

lower (a) ^and upper (P) molecular electronic _energy

eigenstates corresponding ^to given positions ^{of the}

nuclei in the collision complex. ^{In order to} specify

the quantities A’ , a# B.’ .8 ^{(k = 1, 2, 3), we have to dis-}

tinguish ^{between two} types ^{of radiative} transitions

a H

p, namely between those involving ^no change ^of

the quantum ^{number A} associated with the electronic

angular ^{momentum about the} internuclear axis

(AA 0,

parallel transitions ») and those for which AA

+ ¹ (o perpendicular transitions »).

2.1 PARALLEL TRANSITIONS.

For transitions a +-+ p

with AA

0, ^the A’ ^{are given by (see Eqs. (16) of LV)}

These expressions involve the Boltzmann factor

and the phase integral

where ga ^{is the} multiplicity of the lower states a, and

Ey(r) ^{is the energy} eigenvalue ^{of state y} (for ^inter-

nuclear distance r) plus ^the repulsion energy of the

two nuclei in the collision complex. Further, ^the quantity dflfJV bZ + r) ^{is defined} from the relation

where Dczp(b, ^{cp, z) is the} transition dipole ^{for a relative} (perturber ^to atom) position characterized by ^the cylindrical coordinates b, (p,

(the ^z-axis pointing parallel ^to the relative velocity v), ^and eb and ez ^are unit vectors in the collision plane pointing respectively perpendicular ^and parallel ^to the z-axis. As ^can be

seen from the unit vector (beb ⁺ zez)IJb2 ^{+ r, the}

transition dipole Dczp points parallel ^to the internuclear axis.

2.2 PERPENDICULAR TRANSITIONS.

For transitions

a

- fl with AA = ± 1, the formal expressions ^for

A 1 , B,,’P; A2and B 2 differ from those given in equa- tions (2) ^and (3) only by ^{a factor of} 1/J2. ^{If, as is}

usually done, ^the transitions A --+ A’(A’ ^{= A ±} 1)

and - A - - A’ (which yield ^{the same} contribution

to the line intensity) ^are considered as one transition

(I A ^{I -+} I A’ I) the factor of 1/,/2- ^{does not occur and the} A;p (k = 1., 2) ^are formally ^{the same as in} equations (2)

and (3).

1152

The quantities A2* ^and 82*

^{given by}

Again the factor 1/g/2 ^{in this} expression ^{is to be} dropped ^if

distinction is made between the tran- sitions A -+ A’ and - A - - A’. The functions _pa and 4JafJ ^{occurring in} A’ ^(k

^{1, 2, 3) are formally}

the same as in the ^case of parallel transitions (see Eqs. (4) ^and (5)), while the function dafJ ^{is now defined}

from the relation

where etp

e,,

eb. As ^can be seen from this equation, D,,,.8 ^{has only} components perpendicular ^{to the inter-}

nuclear axis.

3. Semi-empirical determination of Lennard-Jones constants.

We have applied the formalism of section 2 to the

resonance line 51 So - 5 1 Plat ^460.73

^{of Sr} per- turbed by Ar,

measured in the light scattering experiment of CSR. This line is characterized by ^two

transitions occurring from the E: ground potential ^to

the excited E and H potentials of the Sr-Ar collision

complex. ^{We assume that the} corresponding ^difference potentials ^{can be} approximated ^{in the Lennard-}

Jones form, ^i.e. (wither being ^{the resonance} frequency)

where

denotes the lower (E) potential and P ^either

of _{the upper} potentials ^{E and H. For} convenience,

we also introduce the dimensionless constants C6, Cf2 ^{defined from}

Inserting the rhs of equation (8) ^into equation (5) yields

with

Using this result and choosing ^the physical para- meters in accord with the experiment of CSR,

^have

evaluated the wing ^formula (1) numerically

^the

basis of approximations (d)-(f), and have converted it to the

section for collision-induced fluores-

cence.

We have performed this calculation ^{for various}

sets of constants C6, C 2, C6 , C 2 ^so

^to get optimal agreement ^{with the} measurements

presented ⁱⁿ figure ¹¹ of CSR. This

best-fit procedure » ^{had to be}

restricted to the red wing of the line. The

why

any attempts ^to also include the blue wing ^were

without _success, is probably ^{that this} part ^{of the} profile

is determined,

^{the one} hand, by very small inter- nuclear distances where the Lennard-Jones approxi-

mation becomes _poor, and on the other hand, by

transitions from the virtual level (laser detuning ^above resonance) ^{to the real} potential ^{curves below reso-}

nance ; it _appears that such transitions are not satis-

factorily accounted for by the adiabatic formalism of section 2. All calculations ^we carried out using

realistic Lennard-Jones parameters ^{led to blue} wing

intensities which ^were about five times smaller than the experimental ^ones.

The best-fit procedure ^{on the red side was} facilitated

by the fact that the far red line wing (detunings

I L1 > ⁶⁰ cm-1) is anti-static for the E - E transition and is thus uniquely determined by the E - ^{17 tran-}

sition, ^i.e. by ^{the constants} C6 ^and C 2. ^{In this}

spectral range the procedure could therefore be

limited ^to a two-parameter fit which led to the result

C6

^4.7, C 2

^{0.10. For} these values the agreement with the three experimental points existing ^{in that}

range (without regarding ^{their error} bars)

^better

than 3 %. The corresponding profile ^{shows a satellite}

near A - 250 cm-1 (situated between the last two

experimental points ^{of the red} wing). This satellite is much closer to line centre than that conjectured by

Lewis et al. [18] ^{and is also} contrary ^{to the} way in which CSR have drawn the curve through ^their experimental points. Inclusion of the error bars would open the possibility ^of drawing ^{a somewhat} patholo- gical ^curve which, ^if touching ^{the extreme bar} limits, could have negative ^{curvature in the} frequency

domain considered. Such a curve, which seems rather

unlikely, might ^be compatible ^{with a} satellite far out in the wing ^as considered by Lewis et al.

In contrast to the far line wing, ^{the close} part ^{of the} wing ^{is not} determined by ^a single transition, ^{but is}

made _up by contributions from both the E - E and ^the Z - 17 transition with comparable ^{order of} magnitude.

We therefore subtracted from the experimental profile

the contribution of the Z - H transition determined above and took the remaining intensity contribution

as a

basis for a best-fit procedure with calculated f - E ^curves. Small detunings (I L1 ( ⁷ cm-1)

excluded from the procedure ^{in order to} avoid falsi- fication arising from non-adiabatic effects. The two-

parameter ^{fit in} the remaininlspectral ^{domain then}

led to the result C6

^7.3, C;2

2.46. For these

values, ^the agreement ^{with the} experimental points

was better than 8 %.

The degree ^to which the theoretical intensities can

be brought ^to coincide with the experimental ^ones (3 % ^{for the Z - H} transition, ⁸ % for the L - L tran-

sition) ^{is not} necessarily

^{measure for the} precision

of the Lennard-Jones constants obtained or for the

precision with which Lennard-Jones potentials ^can approximate ^{the real} potentials. Indeed, ⁱⁿ the best-fit

procedure ^described above, any ^errors of the theore- tical and experimental intensities have been ignored.

While it seems hard to determine the theoretical uncertainties connected with approximations (a)-(f)

in section 2, ^the experimental uncertainties ^are known

as they

^marked by ^{error bars in} figure ^{11 of CSR.}

On the far wing, ^{which has served to determine the}

L - H transition, ^the experimental errors are about 15 %. ^{As can be}

from the various trial calculations

we

have performed, this corresponds ^to

uncertainty

of about 25 ^{% for the} Cf-constant and of about 50 %

for the C 2-constant Note, however, ^{that the uncer-}

tainties of the two constants are not independent ^of

each other since the ratio (C6 )2/C 2 ^{was found}

empirically ^to be invariant to better than 5 % (which

can be interpreted ^{as an} invariance ^{of the potential}

well depth). ^Thus, multiplying C6 ^=4.7 by ¹ + e(l ^{e 1;$}

0.25) brings the theoretical intensity ^{out of the} experi-

mental 15 % ^{error bars if} Cf2 ^{is not} multiplied by (1 ⁺ S)2 ^{± 0.05.}

In the close wing region which has served to deter- mine the E - E transition, ^the experimental ^errors

attain about 20 % ^{as referred to the total} intensity.

This amounts to more than 40 % after subtraction of the L - 17 transition. Note that only small additional uncertainties arise from the errors of the subtracted L - n transition. Indeed, for the close line wing,

this transition occurs in the van der Waals region

where the intensity ^is proportional ^to (C6 ) 1l2 (in the quasistatic approximation). ^The 25 % ^{error in} C6

stated above would thus entail a 12 % ^{error in the}

Z - H intensity, and the relative error of 0.40 mentioned above would be increased to only (0.42 + 0.122)1/2

^0.42. Unfortunately, ^a 40 % ^error

in the intensity of the E - Z transition implies ^a much larger error ^{in the} determination ^of the ^constants C6 and C12’ Calculations ^for C6

^4, C12 ^{= 0.56 and} Cl ^{= 14,} Ci2

^6.9 showed that these values are

still compatible ^{with the outermost} points ^{of the} experimental ^{error bars. A} pessimistic ^{error estimation}

therefore yields the result C6 ^{= 7.3 only up to}

factor of 2, and the result C 2

^2.46 only up ^{to a} factor of 3 or 4. Again, ^{the two} errors are not inde-

pendent of each other but linked by ^the approximate

invariance of the ratio (Cl)2/Cf2 which in the case of the Z - Z transition holds to about 24 %.

It is worth noting ^{that the error} limits stated above appear much more tolerable if they

mapped ^into

those of the depth Eo

(åcg)2/4 åcf2 ^{and distance} rfl

(2 ACO fl /6 ^{of the} potential well. These

constants may be taken as Lennard-Jones parameters instead of ACf ^and Acf2- As mentioned above, ^for

the Z - II transition the error of Eo ^is only ⁵ %.

Since ra

(ACf 12 Eff)1/6, ^{the 25 % error for} AC6

reduces to an error of about 4 % ^for ro’. ^{In the case of}

the Z - Z transition, ^{the error of} E0 ^{was stated to be}

24 %. ^{The error} of r§

(AC6/2 Et)1/6 ^is composed ^of

a 12 % ^error arising from the factor of 2 uncertainty ⁱⁿ ACI ^{and a 4} % ^error arising ^{from the 24} % uncertainty

of E5. The total relative error of r5 may therefore be estimated as (0.122 +0.04 2)1/2 ^0.13.

In terms of the depths ^and distances of the potential wells, ^{our results} may finally ^be expressed

^{follows :}

In figure ^{1 we have} plotted ^{the red} wing ^{of the cross}

section

for collision-induced fluorescence, ^calculated

with the above values for Eo, r-, Eo and ro (solid line). ^The experimental ^results (circles ^{with error} bars)

of CSR from which these values have been determined

by

^best-fit procedure, ^{are shown for} comparison.

In order to display ^{the effects} of nuclear motion,

we

have confronted

dynamical profile ^{with the} quasistatic ^{one, based on the same} Lennard-Jones parameters (dotted line). Also shown (dashed line)

is the result of a dynamical calculation using Eo ^{= 1 000} cm-’, r’ = ^3.4 A, Eo ^{= 20} cm -1,

Fig. ^1.

^Red wing ^{of the}

^section

for collision- induced fluorescence. Circles with

bars : experiment

of CSR. Solid

theoretical result obtained from

best-fit with CSR (corresponding ^{to the} Lennard-Jones

constants given ⁱⁿ Eqs. (9)). ^Dashed

analogous

theoretical result based

the Lennard-Jones constants

conjectured in reference [18]. ^Dotted

quasistatic

result based

the Lennard-Jones constants given ^{in equa-}

tions (9).

1154

r’

^{6.6 A.} These values had been conjectured by

Lewis et al. [18] (by ^also considering ^the experiment

of CSR) ^{for a} calculation of the polarization ^{of redis-}

tributed light. While their constants for the E - f transition are remarkably ^{close to ours, there is}

substantial difference for the E - H transition, espe-

cially ^with regard ^{to the} depth ^{of the} potential ^well

which determines the position of the second satellite

on the far wing.

For I A I 7 ^{cm-1 the} dynamical ^{curves deviate} significantly ^from experiment. ^{As mentioned} already,

this domain has been excluded from the best-fit

procedure because it is dominated by non-adiabatic effects which ^are not included in ^our formalism.

The main non-adiabatic effect is that for large ^atom- perturber distances the radiating dipole ^{does no} longer follow the rotation of the internuclear axis.

Therefore, ^{one of the main causes for} dynamical broadening (amplitude modulation due to the rotation of the dipole) disappears ^when approaching ^{the line}

centre. This is the reason why ^the experimental ^near wing intensity ⁱⁿ figure ¹ drops ^{below the calculated}

adiabatic intensity.

4. Concluding ^remarks.

A best-fit procedure ^{based on} Carlsten et al’s experi-

ment [1] ] ^{and on an} adiabatic line wing ^formalism developed previously (Le Quang Rang ^{and Voslamber} [15]) ^{has led us to a new} determination of Lennard- Jones constants for the first two Sr-Ar difference

potentials. Agreement ^{with the} measurements on the far red wing ^could only be obtained by assuming ^a

satellite located between the last two experimental points. ^We suggest that this satellite be confirmed

(or refuted) by measurements at intermediate detun-

ings. ^{In case it cannot} be confirmed experimentally,

we

would conclude that the Lennard-Jones approxi-

mation is probably ^not suitable for representing ^the

E - n difference potential.

It may be of interest to notice that the ratio tlC6 /4C6

of the two effective van der Waals constants equals

1.55 which is not far from the value 7/4 postulated by

e.g. Light and Szoke [19]

theoretical grounds.

This may be ^an indication that although ^{our results}

for the Lennard-Jones potentials ^{have been cons-}

tructed mainly from the far line wing, they possibly provide ^a good description ^{even for} large internuclear distances.

An error estimation based uniquely ^{on the} experi-

mental (not theoretical) ^error bars of the line wing intensity ^{leads to} relatively large uncertainties for the Lennard-Jones parameters ^of the r - r transition.

A more accurate determination of the difference

potentials by ^{the same method is} likely ^to require ^the improvement ^of both, experiment ^and theory. ^{On the} experimental ^{side, more} precise measurements cover-

ing ^a greater ^number of detunings ^{would be desirable.}

On the theoretical side, ^{some of the} approximations (a)-(f) (especially ^{the adiabatic} approximation (c))

should be refined, ^{and errors connected with the} approximations should be estimated. It might ^also

be _necessary to include additional constants to para- metrize the potentials, ^{such as the} dipole-quadrupole

constant C8 considered by e.g. Hindmarsh et al. [20].

Acknowledgments.

We are indebted to Dr. Cohen-Ganouna for his help

in some of the numerical calculations.

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