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

Transcription

1 In La LE JOURNAL DE PHYSIQUE TOME 38, JUIN 1977, 547 Classification Physics Abstracts II. DEPOLARIZATION CROSSSECTION OF THE (1s 3p) 3IIu (N 1) LEVEL OF H*2 DUE TO ELECTROSTATIC LONG RANGE INTERACTIONS M.A. MÉLIÈRESMARÉCHAL and M. LOMBARDI Laboratoire de Spectrométrie Physique, Université Scientifique et Médicale de Grenoble, B.P. 53, Grenoble Cedex, France (Reçu le 5 octobre 1976, accepté le 10 fevrier 1977) 2014 Résumé. section efficace de dépolarisation du niveau (1s 3p) 3IIu (v 0, N 1, I 0) de H*2 dans les collisions H*2H2, calculée en ne considérant que les interactions électrostatiques à longue distance, est de 160 Å2. L influence du recouplage du spin électronique après la collision est faible (03C3 175 Å2 lorsqu on néglige le spin). Le résultat trouvé est en bon agrément avec l expérience; ceci montre que l interaction quadrupôlequadrupôle peut être responsable de la collision Abstract. depolarization crosssection of the (Is 3p) 3IIu (v 0, N 1, I 0) level of H*2 in H*2H2 collision is found to be 160 Å2, when only the long range electrostatic interactions are considered. influence of electronic spin recoupling is weak (03C3 175 Å2 when neglecting spin). Agreement with experimental value is satisfactory and shows that the quadrupolequadrupole interaction may be responsible for the collision. 1. Introduction. this paper we calculate the depolarization crosssection, Qk2, of the molecular triplet (S 1) excited level (Is 3p) 3nu (v 0, N 1, I 0) of H* due to collisions H*H2 We use the theory developed in paper I [1], based on the predominance of the long range electrostatic forces, which are usually responsible for depolarization. We then compare the calculated value of al2 with previously published experimental results [2]. In appendix A we briefly recall how the observed polarization signal is related to ak2 in this type of magnetic depolarization experiment [2]. interactions considered here are the quadrupolequadrupole interaction at first order of perturbation, and the dipoledipole interaction at second order of perturbation. We shall give their exact forms in 2. collision crosssection ak (k 0, 1 or 2 correspond to the population, orientation, or alignment, destruction) resulting from the relation (1.48) when transfers (No N) are neglected is given by analytical expression of the AF matrix, when neglecting 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 we compare the theoretical and experimental values of the depolarization crosssection. 2. Interactions. electrostatic long range interaction potential, V, is given by (I.22, 23). At first order of perturbation, the first non zero term of interaction, CO lkx corresponds in the present case to the interaction of the permanent quadrupole moments in the excited (K 2) and in the fundamental (X 2) states. This term which varies as R5 (R being the intermolecular distance) is given by (I.24) where following term, which varies as R, corresponds to (K 2, x 4). Note that resonant collisions do not exist as one state is a triplet, and the other one a singlet. At second order of perturbation, the first non zero term of interaction, BJ 2K lk2x lx2 corresponds to the interaction of the induced dipoledipole moments in the 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 coupling respectively Kl and K2, Xl and x2, K and x. K, X and J can then have only the following sets of values : i) K 2, x0 and J 2. ii) K 2, X 2 and J 0, 2, 4. Article published online by EDP Sciences and available at

2 ð.rk( tj Cases where K 0 are not considered as they correspond to a zero value of the AT matrix. tensorial 6 operators given by (I. 30) and the y(kx, J) coefficients given by (I.34) involved in the present calculation are : where the summation concerns all the charged particules of the molecule (in fact, in the present case of H2, homonuclear diatomic molecule, the contribution of the nuclei is zero and the summations concern the electrons only). connected to (1s2) 1 Eg. Due to the peculiar form of the energy level diagram of the molecule (the fundamental state is well separated (~ cm ) from the bunch of the excited levels ( ± cm 1 around the mean)), we have taken AE cm (i.e. 0.5 Rydberg). Since as we shall see this interaction is not the dominant one, this rough approximation is sufficient. 3. Ar matrix. dtk matrix expresses the evolution of the tensorial density matrix component, p, of the excited molecule during collisions (1.6, 9). When coherences are neglected (which is the case in the experiments under consideration [2]), the AF matrix is given by (I.42) which can be written in the present case in the form (1.45) with three different terms : lkx U1Kx) with K x 2 which will be written ð.r:uadquad. This term expressed by (1.43) varies as b8 (b being the collision impact parameter) and depends on the cutoff function defined by (I. 41), 45,5 Here AE is an average excitation energy between the initial state of the molecular ensemble and the various intermediate states : e.g. composed of a triplet state connected by a dipolar matrix element to (ls 3p) 3 llu and a singlet state which will be written as 4Td;pa;p. This term given explicitly by (1.44) varies as b 10 and depends on al, with J 0, 2, 4. two corresponding different cases i) and ii) mentioned in 1 will be written as Arb;pd;p (K 2, X 0) and Arb;p d;p (K 2, X 2). Note that the last case must be considered only when a permanent quadrupolar moment exits in the fundamental state (x 2), and is therefore not considered in atomic collisions. Ark( BJ lkx, BJ2KIK2XIX:J or 2 Ar:uaddip. This term is given by the general expression (1.42) with s 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

3 I1r2 (2) (3) (4) 549 imply Ks ki 2, xs, Ki 2, Js 4. This term varies as b9 and depends on a 4 analytical expressions of the cutoff functions are given in the appendix B. reduced matrix elements of the B1 operators are calculated using expressions (1.67) for electronic and (1.68) for nuclear contributions. In these calculations we have used the electronic wavefunctions of Rothenberg and Davidson [3] relative to the (Is 3p) 3llu, v 0, state and the mean values of the operators z2, r2, ZI z2, XI X2 in the fundamental state given by Kolos and Wolniewicz [4]. corresponding results are given in table I. nuclear contribution has to be calculated to the first order of perturbation only (see 1). hk(p) (see 1, appendix D) reduce then to where p is the intemuclear distance. mean value of this function has been calculated using the results of Matcha [5]. We found that p > (a2) in the excited state and ( p > (aõ) in the fundamental. (1) FIG. 1. total. AF 2 dip.dip. (K 2, x 2). I1r 2 dip.quad. AF 2 dip.dip. (K 2, x 0). I1r 2 quad.quad. 4. Results. crosssection ak relevant to the excited state (Is 3p) 3llu (v 0, N 1, calculated using (I.48). I 0) is states involved in the calculation are related to N by the selection rules N N ± (2,0). same selection rules connect v and vo in the fundamental state. level energy values are given by Dieke [6] and the first 11 fundamental rotational states are included. following approximations are made : the relative velocity v is replaced by its mean value v(t), Ilrk is calculated for b varying from 5 ao to 25 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 shows the variation of the depolarization crosssection ak 2 with temperature. FIG. 2. In order to show the respective contributions of the different interactions we have presented in the first column, of table II, the value of a, when the quadrupolequadrupole and the dipoledipole interactions are simultaneously considered and in the second column, the value of a when only part of årk is included. We indicate the value of the cutoff parameter, bo, in, each case. We have plotted in figure 1 the function årk2(b) and its different components. TABLE II Values of ak corresponding to T 600 K

4 6.1. In Spin influence on the fic2. recoupling of the electronic spin (S 1) of the excited molecule with the orbital momentum N after collision reduces the anisotropy of this state and therefore reduces the polarization of the emitted light. depolarization crosssection will therefore be smaller when the spin recoupling is included. Calculations concerning the spin recoupling are given in (I, 4). We have calculated the 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 level considered is not optically resolved and we experimentally [2] observe the superposition of the three optical lines emitted by the three fine structure levels. crosssection Uk 2 experimentally determined is the slope of the curve dh( p) where p is the gaspressure and AH the width at halfheight of the polarization, P(H), as a function of the magnetic field, H (Hanle or depolarization effect [7]). polarization P is calculated using the D Yakonov [8] expressions for the intensity of the emitted light, and is proportional to p s are solutions of a system of coupled equations : excitation density matrix has been previously calculated [2] with the Percival and Seaton [9] hypothesis of fixed spin excitation (it was shown that all the NJ NJ pqxc. k depend on one NNpq parameter only). When k 2, this system reduces to two coupled as usual that the equations. We have here supposed radiation lifetime T is independent of J, and have taken for 9 [2] : calculated quantity AH(p) shows a linear variation with pressure, and the slope gives the crosssection value of Qk A2. 6. Discussion of results. Comparison between experiment and theory. COMMENTS ON RESULTS. From table II and figure 1 we can see that the quadrupolequadrupole interaction is mainly responsible for the collision. We notice that, in the dipoledipole interaction, the main contribution comes from the term Ar k2 ip (K 2, x 0), the contribution of the other term, Ar k2 ip (K 2, x 2), which makes the calculation somewhat more difficult is almost negligible. In the same way the contribution of the cross terms Ar k2 is quite negligible. Figure 2 shows a relatively small variation of Uk2 with temperature (180 A to 155 A when T vary from 500 to K) (1). However this variation has to be considered because, in the present electron impact excitation type of experiment [2], the temperature is not very homogeneous in the observation zone and the mean value of T has not actually been precisely determined ( ± 100 K) (see 6.2). From the results obtained in 5, we can see that the spin recoupling does not strongly affect the crosssection value (minus 10 %). 6.2 COMPARISON BETWEEN EXPERIMENTAL AND THEORETICAL RESULTS. published experimental [2] value of ak2 associated with the temperature 800 K was 230 ± 30 A2. Later temperature measurements have shown that, in that early experiment the temperature was overestimated, and the uncertainty in T underestimated; the new estimation is T 600 ± 100 K. corresponding experimental value of Uk2, relative to the slope AH(p), is after this correction Uk A2. Independent measurements have been performed by Baltayan [10] using a different apparatus and have given similar results (167 ± 25 A2 for T 800 ± 100 K). theoretical value of 160 A2 agrees with the experimental value. This agreement suggests that the quadrupolequadrupole interaction is responsible for the collision. 7. Conclusion. this article, we have calculated the depolarization crosssection Qk2 of the (Is 3p) 3nu (v 0, N 1, 1 0) of H2, in a collision H2H2. This calculation was based on the hypothesis of predominant long range electrostatic interactions, which has been theoretically described in a preceding article [1]. Agreement between experiment and theory (1) When neglecting the spin recoupling.

5 In Analytical 551 allows us only to conclude that if long range forces are predominant, the quadrupolequadrupole interaction is responsible of the collision. It does not exclude the possibility of the existence of other mechanisms (short range forces), if these mechanisms give the same value for Uk. This could be verified by an experimental study of the dependence of crosssection on temperature. Appendix A. the magnetic depolarization experiment [2], the excited molecular state is aligned and a static magnetic field, Hz, is applied perpendicularly to this alignment. polarization of the emitted light is studied as a function of Hz at different pressures. This polarization is calculated as follow. components of the intensity of the emitted light (and therefore the polarization) are expressed as a function of the tensorial density matrix components, NN pk, using the D Yakonov expression [8] when no spin coupling exist (N J, S I 0). NN pq are obtained by resolving equation (I.5); neglecting coherences and transfers from No to N and using (I.6) one obtains where g is the Lande factor type of excitation imposes dependence of the polarization on Hz is then given by full width at half maximum of P(Hz) varies linearly with pressure, the slope being related to (J k2. Appendix B. expression of the cutoff function. l. a 45,5 CAN BE EXPRESSED AS FOLLOWS where J. is given by se integrals can be expressed in terms of other integrals in which are related to Bessel functions of the second kind, Kn(1 a 1), with a wblv. If in is given by we finally obtain : se functions can be expressed using Jm integrals where Jm is given by We obtain :

6 552 a. integrals can be expressed in terms of other jn integrals, these being evaluated in the complex plane and given by Thus we finally obtain : with 3. a 45,6 IN THE SAME WAY WE HAVE : with References [1] MÉLIÈRESMARÉCHAL, M. A. and LOMBARDI, M., J. Physique 38 (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) [4] KOLOS, W., WOLNIEWICZ, L., J. Chem. Phys. 43 (1965) [5] MATCHA, Private communication. [6] DIEKE, G. H., Wavelength Tables of the Hydrogene Molecule (Interscience, New York) [7] HANLE, W., Z. Phys. 30 (1924) 93. [8] D YAKONOV, M. I., J.E.T.P. 20 (1965) [9] PERCIVAL, J. C. and SEATON, M. J, Phil. Trans. Roy. Soc. 251 (1958) 113. [10] BALTAYAN, P., sis, Grenoble (1973).