II. Depolarization cross-section of the (1s 3p) 3II u (N =

1) level H*2 due to electrostatic long range interactions

M.-A. Melieres-Marechal, M. Lombardi

To cite this version:

M.-A. Melieres-Marechal, M. Lombardi. II. Depolarization cross-section of the (1s 3p) 3II u (N= 1) level H*2 due to electrostatic long range interactions. Journal de Physique, 1977, 38 (6),pp.547-552. <10.1051/jphys:01977003806054700>. <jpa-00208615>

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Submitted on 1 Jan 1977

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547

II. DEPOLARIZATION CROSS-SECTION OF THE (1s 3p) 3IIu (N = 1)LEVEL OF H*2 DUE TO ELECTROSTATIC LONG RANGE INTERACTIONS

M.-A. MÉLIÈRES-MARÉCHAL and M. LOMBARDI

Laboratoire de Spectrométrie Physique, Université Scientifique et Médicale de Grenoble, B.P. 53,38041 Grenoble Cedex, France

(Reçu le 5 octobre 1976, accepté le 10 fevrier 1977)

Résumé. 2014 La section efficace de dépolarisation du niveau (1s 3p) 3IIu (v = 0, N = 1, I = 0)de H*2 dans les collisions H*2-H2, calculée en ne considérant que les interactions électrostatiques àlongue distance, est de 160 Å2. L’influence du recouplage du spin électronique après la collisionest faible (03C3 = 175 Å2 lorsqu’on néglige le spin). Le résultat trouvé est en bon agrément avec l’expé-rience; ceci montre que l’interaction quadrupôle-quadrupôle peut être responsable de la collision.

Abstract. 2014 The depolarization cross-section of the (Is 3p) 3IIu (v = 0, N = 1, I = 0) level ofH*2 in H*2-H2 collision is found to be 160 Å2, when only the long range electrostatic interactionsare considered. The influence of electronic spin recoupling is weak (03C3 = 175 Å2 when neglectingspin). Agreement with experimental value is satisfactory and shows that the quadrupole-quadrupoleinteraction may be responsible for the collision.

LE JOURNAL DE PHYSIQUE TOME 38, JUIN 1977, ’

Classification

Physics Abstracts5.480

1. Introduction. - In this paper we calculate the

depolarization cross-section, Qk-2, of the moleculartriplet (S = 1) excited level (Is 3p) 3nu (v = 0, N = 1,I = 0) of H* due to collisions H*-H2- We use thetheory developed in paper I [1], based on the predo-minance of the long range electrostatic forces, whichare usually responsible for depolarization. We thencompare the calculated value of al=2 with previouslypublished experimental results [2]. In appendix Awe briefly recall how the observed polarization signalis related to ak=2 in this type of magnetic depolari-zation experiment [2].The interactions considered here are the quadru-

pole-quadrupole interaction at first order of pertur-bation, and the dipole-dipole interaction at secondorder of perturbation. We shall give their exact formsin § 2. The collision cross-section ak (k = 0, 1 or 2

correspond to the population, orientation, or align-ment, destruction) resulting from the relation (1.48)when transfers (No - N) are neglected is given by

The analytical expression of the AF’ matrix, whenneglecting the electronic spin recoupling, is given in§ 3. In § 4, we present the calculated values of a’when neglecting the spin recoupling. In § 5, we study

the influence of the spin recoupling, and in § 6 wecompare the theoretical and experimental values ofthe depolarization cross-section.

2. Interactions. - The electrostatic long rangeinteraction potential, V, is given by (I.22, 23). Atfirst order of perturbation, the first non zero term ofinteraction, CO’lKx corresponds in the present case tothe interaction of the permanent quadrupole momentsin the excited (K = 2) and in the fundamental (X -= 2)states. This term which varies as R-5 (R being theintermolecular distance) is given by (I.24) where

The following term, which varies as R -’, corres-ponds to (K = 2, x = 4). Note that resonant colli-sions do not exist as one state is a triplet, and theother one a singlet.At second order of perturbation, the first non zero

term of interaction, ’BJ 2K lK2X lX2 corresponds to theinteraction of the induced dipole-dipole moments inthe excited (Kl = K2 =1) and fundamental (Xl = X2 = 1)states. This term, which varies as R- 6, is given by(I . 32, 33) where K, x and J are obtained by couplingrespectively Kl and K2, Xl and x2, K and x. K, Xand J can then have only the following sets of values :

i) K = 2, x=0 and J = 2.ii) K = 2, X = 2 and J = 0, 2, 4.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003806054700

548

Cases where K = 0 are not considered as theycorrespond to a zero value of the AT matrix. Thetensorial ’6 operators given by (I . 30) and the y(KX, J)coefficients given by (I.34) involved in the presentcalculation are :

where the summation concerns all the charged par-ticules of the molecule (in fact, in the present caseof H2, homonuclear diatomic molecule, the contri-bution of the nuclei is zero and the summationsconcern the electrons only).

Here AE is an average excitation energy betweenthe initial state of the molecular ensemble

and the various intermediate states : e.g.

composed of a triplet state connected by a dipolarmatrix element to (ls 3p) 3 llu and a singlet state

connected to (1s2) 1 Eg . Due to the peculiar form ofthe energy level diagram of the molecule (the funda-mental state is well separated (~ 100 000 cm-’) fromthe bunch of the excited levels ( ± 20 000 cm -1 aroundthe mean)), we have taken AE = 100 000 cm -’(i.e. 0.5 Rydberg). Since as we shall see this interactionis not the dominant one, this rough approximationis sufficient.

3. The Ar matrix. - The dTk matrix expressesthe evolution of the tensorial density matrix compo-nent, p’, of the excited molecule during collisions(1.6, 9). When coherences are neglected (which isthe case in the experiments under consideration [2]),the AF’ matrix is given by (I.42) which can be writtenin the present case in the form (1.45) with threedifferent terms :- ð.rk(’tJ lKx’ U1Kx) with K = x = 2 which will

be written ð.r:uad-qUad. This term expressed by (1.43)varies as b-8 (b being the collision impact parameter)and depends on the cut-off function defined by (I . 41),45,5-

which will be written as 4Td;p-a;p. This term givenexplicitly by (1.44) varies as b - 10 and depends onal, with J = 0, 2, 4. The two corresponding differentcases i) and ii) mentioned in § 1 will be written as

Arb;p-d;p (K = 2, X = 0) and Arb;p- d;p (K = 2, X = 2).Note that the last case must be considered only whena permanent quadrupolar moment exits in the fun-damental state (x = 2), and is therefore not consideredin atomic collisions.- 2 Ark(’BJ lKx, ’BJ2KIK2XIX:J or 2 Ar:uad-dip. This

term is given by the general expression (1.42) withs = 1 KX (K = X = 2) and s’ = 2 K1 K2 X1 X2(Kl = K2 = xi = X2 = 1) where the b functions

TABLE I

Mean values of the electronic part of the UKQ operators

549

imply Ks’ = ki = 2, xs, = Ki = 2, Js’ = 4. This termvaries as b-9 and depends on a 4The analytical expressions of the cut-off functions

are given in the appendix B. The reduced matrixelements of the ’B1 operators are calculated usingexpressions (1.67) for electronic and (1.68) fornuclear contributions. In these calculations we haveused the electronic wavefunctions of Rothenberg andDavidson [3] relative to the (Is 3p) 3llu, v = 0, stateand the mean values of the operators z2, r2, ZI z2,XI X2 in the fundamental state given by Kolos andWolniewicz [4]. The corresponding results are givenin table I.The nuclear contribution has to be calculated to

the first order of perturbation only (see § 1). ThehK(P) (see § 1, appendix D) reduce then to

where p is the intemuclear distance. The mean valueof this function has been calculated using the resultsof Matcha [5]. We found that p &#x3E;2 = 4.147(a2) inthe excited state and ( p &#x3E;2 = 2.074(aõ) in the fun-damental.

4. Results. - The cross-section ak relevant to theexcited state (Is 3p) 3llu (v = 0, N = 1, I = 0) is

calculated using (I.48). The

states involved in the calculation are related to N

by the selection rules N’ = N ± (2,0). The sameselection rules connect v and vo in the fundamentalstate. The level energy values are given by Dieke [6]and the first 11 fundamental rotational states are

included. The following approximations are made :the relative velocity v is replaced by its mean valuev(T), Ilrk is calculated for b varying from 5 ao to25 ao (ao being the Bohr radius). We have also

neglected the spin recoupling. In table II, and figure 1,the results correspond to T = 600 K; figure 2 showsthe variation of the depolarization cross-section ak= 2with temperature.

FIG. 1.

- I1r2 total.--------- (1) AF 2 dip.-dip. (K = 2, x = 2).--------- (2) I1r 2 dip.-quad.--------- (3) AF 2 dip.-dip. (K = 2, x = 0).--------- (4) I1r 2 quad.-quad.

FIG. 2.

In order to show the respective contributions ofthe different interactions we have presented in thefirst column, of table II, the value of a’, when thequadrupole-quadrupole and the dipole-dipole interac-tions are simultaneously considered and in the secondcolumn, the value of a’ when only part of årk isincluded. We indicate the value of the cut-off para-meter, bo, in, each case.We have plotted in figure 1 the function årk=2(b)

and its different components.

TABLE II

Values of ak corresponding to T = 600 K

550

5. Spin influence on the fIc=2. - The recoupling ofthe electronic spin (S = 1) of the excited moleculewith the orbital momentum N after collision reducesthe anisotropy of this state and therefore reduces thepolarization of the emitted light. The depolarizationcross-section will therefore be smaller when the spinrecoupling is included. Calculations concerning thespin recoupling are given in (I, § 4). We have calculatedthe quantities NJ NJ NJoNJ°ð.rk using the expression(I . 54) ; the corresponding values of ak (nvak = AF)are reported in table III.

TABLE III

Values Of NJ NJ NJo NJOak in (Á 2) for T = 600 K

In the present case the fine structure of the levelconsidered is not optically resolved and we experi-mentally [2] observe the superposition of the threeoptical lines emitted by the three fine structure levels.The cross-section Uk = 2 experimentally determined isthe slope of the curve dH( p) where p is the gas-pressure and AH the width at half-height of thepolarization, P(H), as a function of the magneticfield, H (Hanle or depolarization effect [7]).The polarization P is calculated using the D’Yako-

nov [8] expressions for the intensity of the emittedlight, and is proportional to

The p’s are solutions of a system of coupled equations :

The excitation density matrix has been previouslycalculated [2] with the Percival and Seaton [9] hypo-thesis of fixed spin excitation (it was shown that allthe NJ NJ’pqxc. k depend on one NNpq parameter only).

When k = 2, this system reduces to two coupledequations. We have here supposed as usual that theradiation lifetime T is independent of J, and havetaken for 9 [2] :

The calculated quantity AH(p) shows a linear varia-tion with pressure, and the slope gives the cross-

section value of Qk - 2 = 160 A2.

6. Discussion of results. Comparison between expe-riment and theory. - 6.1. COMMENTS ON RESULTS. -From table II and figure 1 we can see that the qua-drupole-quadrupole interaction is mainly responsiblefor the collision. We notice that, in the dipole-dipoleinteraction, the main contribution comes from theterm Ar k=2 ip (K = 2, x = 0), the contribution ofthe other term, Ar k=2 ip (K = 2, x = 2), whichmakes the calculation somewhat more difficult isalmost negligible. In the same way the contributionof the cross terms Ar k=2 is quite negligible.Figure 2 shows a relatively small variation of Uk=2with temperature (180 A to 155 A when T vary from500 to 1 000 K) (1). However this variation has to beconsidered because, in the present electron impactexcitation type of experiment [2], the temperature isnot very homogeneous in the observation zone andthe mean value of T has not actually been preciselydetermined ( ± 100 K) (see § 6.2).From the results obtained in § 5, we can see that

the spin recoupling does not strongly affect the cross-section value (minus 10 %).

6.2 COMPARISON BETWEEN EXPERIMENTAL AND

THEORETICAL RESULTS. - The published experimen-tal [2] value of ak=2 associated with the temperature800 K was 230 ± 30 A2. Later temperature measu-rements have shown that, in that early experiment thetemperature was overestimated, and the uncertaintyin T underestimated; the new estimation isT = 600 ± 100 K. The corresponding experimentalvalue of Uk=2, relative to the slope AH(p), is afterthis correction Uk=2 = 200 + 40 A2. Independentmeasurements have been performed by Baltayan [10]using a different apparatus and have given similarresults (167 ± 25 A2 for T = 800 ± 100 K). Thetheoretical value of 160 A2 agrees with the experimen-tal value. This agreement suggests that the quadru-pole-quadrupole interaction is responsible for thecollision.

7. Conclusion. - In this article, we have calculatedthe depolarization cross-section Qk-2 of the (Is 3p)3nu (v = 0, N = 1, 1 = 0) of H2, in a collision

H2-H2. This calculation was based on the hypothesisof predominant long range electrostatic interactions,which has been theoretically described in a precedingarticle [1]. Agreement between experiment and theory

(1) When neglecting the spin recoupling.

551

allows us only to conclude that if long range forcesare predominant, the quadrupole-quadrupole interac-tion is responsible of the collision. It does not excludethe possibility of the existence of other mechanisms

(short range forces), if these mechanisms give thesame value for Uk. This could be verified by an expe-rimental study of the dependence of cross-section ontemperature.

Appendix A. - In the magnetic depolarization experiment [2], the excited molecular state is aligned anda static magnetic field, Hz, is applied perpendicularly to this alignment. The polarization of the emitted lightis studied as a function of Hz at different pressures. This polarization is calculated as follow.

The components of the intensity of the emitted light (and therefore the polarization) are expressed as afunction of the tensorial density matrix components, NN’pk, using the D’Yakonov expression [8] when no spincoupling exist (N = J, S = I = 0). The NN’pq are obtained by resolving equation (I.5); neglecting coherencesand transfers from No to N and using (I.6) one obtains

where g is the Lande factor

The type of excitation imposes

The dependence of the polarization on Hz is then given by

The full width at half maximum of P(Hz) varies linearly with pressure, the slope being related to (J’k=2.

Appendix B. - Analytical expression of the cut-off function. - l. a 45,5 CAN BE EXPRESSED AS FOLLOWS

where J. is given by

These integrals can be expressed in terms of other integrals in which are related to Bessel functions ofthe second kind, Kn(1 a 1), with a = wblv. If in is given by

we finally obtain :

- These functions can be expressed using Jm integrals where Jm is given by

We obtain :

552

The a. integrals can be expressed in terms of other jn integrals, these being evaluated in the complex planeand given by

Thus we finally obtain :

with

3. a 45,6 IN THE SAME WAY WE HAVE : :

with

References

[1] MÉLIÈRES-MARÉCHAL, M. A. and LOMBARDI, M., J. Physique38 (1977) 527.

[2] MARÉCHAL, M. A., JOST, R., LOMBARDI, M., Phys. Rev. A 5(1972) 740.

[3] ROTHENBERG, S. and DAVIDSON, E. F., J. Chem. Phys. 45 (1966)2560.

[4] KOLOS, W., WOLNIEWICZ, L., J. Chem. Phys. 43 (1965) 2429.[5] MATCHA, Private communication.

[6] DIEKE, G. H., Wavelength Tables of the Hydrogene Molecule(Interscience, New York) 1971.

[7] HANLE, W., Z. Phys. 30 (1924) 93.[8] D’YAKONOV, M. I., J.E.T.P. 20 (1965) 1484.[9] PERCIVAL, J. C. and SEATON, M. J’, Phil. Trans. Roy. Soc.

251 (1958) 113.[10] BALTAYAN, P., Thesis, Grenoble (1973).