ν_3 vibrational ladder of SF_6

January 1981 / Vol. 6, No. 1 / OPTICS LETTERS V3 39 vibrational ladder of SF6 Chris W. Patterson and Burton J. Krohn Theoretical Division, Los Alomos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545 A. S. Pine Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Massachusetts 02173 Received September 12, 1980 From an analysis of the high-resolution spectrum of the 3v3 manifold of SF 6 , we have determined the anharmonicity constants to be (in reciprocal centimeters) X3 3 = -1.7426, G 3 3 = 0.9188, and T33 = -0.24635. By using these constants, one can for the first time to our knowledge accurately predict the energies of the lower-vibrational sub- levels of the v3 ladder and determine the pathways for multiple-photon absorption. Currently the multiple-photon excitation of the v3 vibrational mode of SF6 is thought to occur in two stages before dissociation, as discussed by Isenor et al. I In the first stage, the laser pump frequency must be resonant with the first few discrete vibrational substates that compose the v3 ladder. Since the anharmonic terms in the vibrational potential split the harmonic nv3 levels into complex patterns of vibrational sublevels, the v3 ladder is quite complicated. As one proceeds up the v3 ladder, the background density of levels, consisting mainly of low-frequency bending-mode combinations, forms a quasi-continuum into which population from the nV3 sublevels can leak by means of high-order vi- brational or Coriolis interactions. Once substantial leakage has occurred, there will always be resonant transitions within the vibrational quasi-continuum up to dissociation. Any modeling of the discrete multiple-photon pathways in the first stage of excitation depends on the exact positions of the vibrational sublevels in the i' 3 ladder. These positions are determined primarily by the effective harmonic frequency co30and the second-order anharmonic constants X 33 , T 3 3 , and G33, as shown by Hecht 2 and, more recently, Akulin et al. 3 The higher-order anharmonic constants become more important as n increases, but we shall not consider them in this Letter. One may determine the anharmonic constants by T3 3. These constants are compared with earlier esti- mates in Table 1. Some of these estimates5 -8 "16'1 7 led to the correct ordering of vibrational sublevels up to 3V3, whereas others did not, largely because of misinterpretations of the low-resolution data. 1 4"15' 1 8- 20 detailed analysis of all the 3v3 high-resolution data will be published separately. The 3v3 manifold of SF6 has two Fl, vibrational sublevels to which electric-dipole transitions from ground state are allowed. One can either consider that the vibrational states form a Cartesian basis (n,,nyn,), in which the V3 x, y, and z components are uncoupled, Table 1. Anharmonic Constants for the v3 Ladder of SF6 (in cm-') Akulin et aJ.,4'a Double Resonance 3V3 Spectroscopy 0146-9592/81/010039-03$0.50/0 -0.85 Cantrell and Galbraith5 -2.4 Jensen et al. 6 Cantrell and Galbraith7 -2.8 Heenan and Robiette8 -1.36 techniques. There are several spectroscopic means available to determine these constants. First, one can analyze the V3 - 2v3 hot-band spectrum in the i'3 the double-resonance data.9 -' 0 Finally, the nV3 overtone spectra can be used to assess these constants. So far, only the 3v3 overtone band has been observed, initially at low resolutionl-' 6 and more recently at Doppler-limited resolution by Pine and Robiette."7 We report in this Letter the results of an interacting band analysis of this 3V3 high-resolution spectrum, which yields definitive anharmonic constants X 3 3 , G33, and G:313 X:33 T 33 Force-Field Models means of force-field modeling 4 - 8 or by spectroscopic high-resolution data. However, because of the weakness of the lines, this has not been accomplished to date. One can also determine the anharmonic constants from Our results establish the vibrational energies in the v3 ladder of SF6 and allow us to predict the multiple-photon pathways up the ladder for CO2 laser excitation. A v3 - Bulaninet al. 11,a McDowellet al. 12 Fox' -1.2 Cantrell and Fox' 9 Ackerhalt and -0.4 -0.397 -0.44 -0.16 +0.53 +0.27 +0.24 0.0 20 -2.54 -2.56 +0.303 +0.24 -0.6 -2.08 -1.83 -1.7426 +0.24 -0.25 -1.05 +0.74 +1.00 +0.9188 Ackerhalt et al. 14 Alimpiev et al. 15,Q Fuss' 6 Pine and Robiette"7 This work " +0.4 +0.8 +1.214 +1.34 +0.98 -3.1 8 Galbraith 2v:, -1.55 0.0 -0.01 +0.525 -0.12 -0.21 -0.24635 The constants in these works are related to ours by X / + 2,y)/5,G33 = (-2CY +fl - 8y)/10,and T,33 © 1981, Optical Society of America = 33 (2a - fl - = (3a + 2,y)/20. OPTICS LETTERS / Vol. 6, No. 1 / January 1981 40 or one can make use of a strongly coupled basis with good total-vibrational angular momentum 1,depending on the values of the anharmonic constants. In the first case, the two Fl, states correspond to (300) and (210) type states, whereas in the latter case they correspond to I = 1 and 1 = 3 type states. One may expect the overtone (n,0,0) to have a much-larger dipole moment I basis is a good representation, we would expect to see both the 1 = 1 and 1 = 3 Fla subbands. In Figs. 1 and 2 we show the Q branches of the two dipole-allowed subbands taken with near-Doppler resolution (-0.002 cm-'). We see that, since the (300) Fl0 Q branch is about 50 times more intense than the associated with it than does a combination like (n - 1, (210) Fl, (AR = 0) Q branch, the Cartesian basis is valid. The I = 1 Q branch at 2827.5 cm-' was analyzed 1,0).21 Indeed, explicit calculations for SF6 have predicted the (300) dipole moment to be considerably larger than the (210) dipole moment.2 2 Thus, if the previously by Pine and Robiette17 by using a noninteracting fundamental-type model Hamiltonian, and they correctly attributed the weak structure seen at Cartesian basis is a good representation, 2838-2841 cm-' to an 1 = 3 Q branch. we would ex- pect to see only the (300) Fl, subband; whereas, if the This Q branch is shown here for the first time to our knowledge, and its rovibrational structure has now been assigned by using a model Hamiltonian in which the 1 = 1 and 1 = 3 subbands interact. For the line assignments, we have 'V QB 10.0 _ used a concise cluster notation.2 3 The relative weak- a ness of the (210) or I = 3 subband is primarily responsible for the misinterpretations of the low-resolution 3v3 F data mentioned above. 's~~~~~~~~~~~o From a fit of over 700 lines from 10 branches of 3V3, 27 S75 - A - achieved by using the Hecht Hamiltonian,2 we have determined the three anharmonic constants X3 3, T3 3, T11nT 27or and G:33 shown in Table 1, along with seven other rovi- 0~~~~~~~~C\ I- 5.0 - n II brational constants. The details of the assignmentsand VI 1111111111' 1 1111 TI O C W Z2.5_A f1 II fit will be given in a later work. However, it is impor- tant to note that all the second-order rovibrational constants of 3V3 are within -1% of their V3values.2 4' 25 This implies that neither the V3nor the 3v3 band interacts strongly with other bands, including the numerous bending-mode combinations. In fact, since the lowpressure 2828.0 28275 2827.0 WAVE NUMBER (cm- 1 ) Fig. 1. The (300) or I = 1 Q branch of the 3v:3manifold of SF6 recorded at 160 K with a fill pressure of 1.00 Torr and a path length of 64 m. 3V3 linewidths are Doppler limited, leakage into the quasi-continuum certainly is not important at 3v3. Thus we feel justified in using the constants derived from the 3v3 analysis to predict the vibrational structure further up the ladder. The vibrational part of the Hecht Hamiltonian for nv3 may be written as26 At = I'(W 3o) + (X 33 - 6T 3 3 )N 2 + (G3 3 + 2T 3 3)12 + (1O[ - 8N)T3 3 , (1) 0 where W3 is the effective harmonic frequency, (N) = n = n. + ny + nz, and (M) = nx2 + ny2+ n, 2 in the Cartesian basis. The Y3 band origin may be written 020 _ as m = w3 0 + X 33 + 2G 33 -2(B3), , 015 (2) where we use the accurate V3values m = 947.976 cm-' and (Bt,3 ) = 0.0631 cm-' in Ref. 25. This gives 0 3 0 = H 948.008 cm-'. We now use Eq. (1) with the anharmonic constants derived from our 3V3analysis to predict the vibrational -0 10 z energy levels for nV 3 , as shown in Table 2. Although the vibrational energies are given to hundredths of reciprocal centimeters, we do not wish to imply that such 005 accuracy is attainable for high n, where the higher-order anharmonic constants will certainly contribute or where 0 2840.0 2839.0 WAVE NUMBER (cm-l) Fig. 2. The (210) or I = 3 (AR = 0) Q branch of the 3i'a manifold of SF 6 recorded at 160 K with a fill pressure of 1.86 Torr and a path length of 80 m. interband perturbations become important. Still, the grouping of eigenvalues at high n is interesting. The energies in Table 2 do not change for one standard de- viation of the anharmonic constants. Before we can use Table 2 to discuss multiple-photon pathways, it is necessary to consider first the selection January 1981 / Vol. 6, No. 1 / OPTICS LETTERS Table 2. Predicted Vibrational Energy Levels for nv 3 of SF6 (in cm-') n =0 (000) 0.0 Al n =1 E 948.10 n =2 1889.05 F2 Al (100) F. (200) Al n =6 5606.24 F, F, 7431.46 7471.03 7471.40 Al 5646.70 Al 7495.69 1891.60 E 1896.53 F2 E E 7496.95 7501.21 A2 7502.26 F2 F2 F2 7505.10 7510.94 7525.60 7527.09 7528.70 7528.94 7529.25 E (300) (210) F, F, F2 2827.55 2839.04 2840.35 A2 5658.98 (111) A2 2845.28 F, (110) F2 n =3 n =4 (400) (310) (220) (500) (410) (320) (311) (320) (311) (221) 5660.47 F, 5670.48 5673.10 F2 A, 5677.01 n=7 F2 F, Al F, Al 3758.89 E 3759.49 F2 3774.15 F, F, F, 3779.57 F2 Al 3781.87 F, 6521.60 6555.54 6555.62 6576.36 E 3783.82 F2 6577.91 E A2 6581.75 F2 (211) F 2 3788.87 n =5 F, F, F2 F1 A2 F, E F, I Parity is (-1) 7431.45 5606.32 5633.62 5635.48 5648.43 5656.48 5658.05 E n =8 A, E 4685.52 4707.54 4708.09 4721.15 4721.29 4722.46 4728.96 4732.78 E A, F2 F, 7538.70 7544.02 7547.06 7553.26 7556.23 6584.89 6594.52 6595.28 6597.48 E F, F2 F. F2 n. rules for electric-dipole An = 1 transitions. The nV3 vibrational states are best represented by a Cartesian basis only for n < 5 as a result of the particular values of the anharmonic constants. The Cartesian selection rules for transitions are 26 Ani = 1, Anj = 0, (3) where i = x,y,z and j F#i. These selection rules differ from the standard selection rules for vibrational angular momentum Al = +1 except for 0 : 13 and 13 : 21v3transitions, where the two descriptions are equivalent. That is, for n > 2, it is possible to have Al 5d ±1 transitions in 13 of SF 6 . For n > 5 even the Cartesian selection rules can be vio- lated. We now can determine the multiple-photon pathways near the 13 band origin. From Table 2 we see that the highest-frequency transitions will be resonant near 948 cm-1 up to (400), where a vibrational bottleneck occurs. The rotational splittings of the vibrational levels20 may compensate somewhat for this bottleneck. From Table 2 wesee that two-photon transitions will be resonant with the 21'3A1g,Eg, and F2g vibrational levels at frequencies 944.5, 945.8, and 948.3 cm-', respectively. The two-photon resonance to the 2V3A1g level is responsible for the strong absorption of the 11P(20) line of CO2 at 944.2 cm-' by SF6 . Further- more, this CO2 line is mostly resonant with anharmonically split sublevels of the v3 the most anharmonically depressed levels in their respective manifold and could lead to population bottlenecks if pumped by a CO2 laser. It followsfrom Table 2 that there is a three-photon transition at 942.5cm-, which is resonant with the (300) F,0 sublevel and close to the 11P(22) CO2 line at 942.4 cm- 1 . Similarily, there are four-photon transitions at 939.4cm-1 , resonant with the (400) Alg and Eg levels, and close to the 11P(24) CO2 line at 940.6cm-'. Note that there are no resonant pathways from (300) or (400) higher up the V3ladder at 942.4 or 940.6 cm-l, respectively, so population will tend to collect at these levels. It might be possible to learn more about the higher-vibrational states by probing the (n,0,0) bottlenecks that have been produced by intense CO2 radiation tuned to the n-photon resonances. We are grateful to H. Galbraith, J. Ackerhalt, and A. Robiette for helpful comments and discussion. This work was performed under the auspices of the U.S. Department of Energy and the U.S. Department of the Air Force. 1. 2. 3. 4. 5. 6612.21 6616.58 E F, (n,0,0) for n > 2 are of particular interest since they are References 6607.69 6608.46 A2 41 ladder up to 8v3. The n-photon transitions to vibrational sublevels N. Isenor et al., Can. J. Phys. 51, 1281 (1973). K. Hecht, J. Mol. Spectrosc. 5, 355 (1960). V. M. Akulin et al., Sov. Phys. JETP 45, 47 (1977). V. M. Akulin et al., JETP 74, 490 (1968). C. D. Cantrell and H. W. Galbraith, Opt. Commun. 18,513 (1976). 6. C. C. Jensen et al., Opt. Commun. 20, 275 (1977). 7. C. D. Cantrell and H. W. Galbraith, Opt. Commun. 21,374 (1977). 8. R. Heenan and A. Robiette, Department of Chemistry, University of Reading, Whiteknights, Reading RG6 2AD, U.K., personal communication. 9. P. F. Moulton et al., Opt. Lett. 1, 51 (1977). 10. C. C. Jensen et al., J. Chem. Phys. 71, 3648 (1979). 11. M. 0. Bulanin et al., JETP Lett. 29, 43 (1979). 12. R. McDowell, J. Aldridge, and R. Holland, J. Phys. Chem. 80, 1203 (1976). 13. H. Kildal, J. Chem. Phys. 67, 1287 (1977). 14. J. Ackerhalt et al., J. Chem. Phys. 69, 1461(1978). 15. S. S. Alimpiev et al., Sov. J. Quantum Electron. 9, 699 (1979). 16. W. Fuss, Chem. Phys. Lett. 71, 77 (1980). 17. A. S. Pine and A. G. Robiette, J. Mol. Spectrosc. 80, 388 (1980). 18. K. Fox, J. Chem. Phys. 68, 2512 (1978). 19. C. Cantrell and K. Fox, Opt. Lett. 2, 151 (1978). 20. J. Ackerhalt and H. Galbraith, J. Chem. Phys. 69, 1200 (1978). 21. D. F. Heller, Chem. Phys. Lett. 61, 583 (1979). 22. C. Marcott, W. G. Golden, and J. Overend, Spectrochim. Acta 34A, 661 (1978). 23. E. G. Brock et al., J. Mol. Spectrosc. 76, 301 (1979). 24. M. Loete et al., C. R. Acad. Sci. Ser. B 285, 175 (1977). 25. Ch. J. Bord6 et al., in Laser Spectroscopy IV, H. Walther and K. W. Rothe, eds. (Springer-Verlag, New York, 1979). 26. H. W. Galbraith and C. D. Cantrell, in The Significance of Nonlinearity in the Natural Sciences, B. Kursunoglu, A. Perlmutter, 1977). and L. Scott, eds. (Plenum, New York,