J. Phys. Chem. 1995,99, 16307-16314
16307
Ion Imaging of the Photodissociation of OCS Near 217 and 230 nm Yoshihiro Sato, Yutaka Matsumi,* and Masahiro Kawasaki Institute for Electronic Science, and Graduate School of Environmental Sciences, Hokkaido Universiv, Sapporo 060, Japan
Koichi Tsukiyama Department of Chemistry, Faculty of Science, Science University of Tokyo, Tokyo 162, Japan
Richard Bersohn Department of Chemistry, Columbia University, New York, New York 10027 Received: July 12, 1995; In Final Form: August 23, 1995@
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+
OCS was photodissociated near 217 and 230 nm, and the resulting CO was probed by 2 1 resonant X'Z+ transitions. The CO+ was detected by the multiphoton ionization (REMPI) using the B ' F , C'X+ ion imaging technique of Chandler and Houston. The nascent CO was found only in the v = 0 state, and its rotational state population distribution was bimodal. It is remarkable that this CO rotational distribution has no tail extending to states of lower J. /3 values rise monotonically with J . Recent potential energy surface calculations by Rokutan et al. provide reasonable explanations for the results: (1) dissociation can take place from near degenerate A" and A' states, each of which generates a different CO product distribution; (2) a molecule excited by the initial absorption to an A' state can cross over in linear geometry at two different C-S distances to dissociating states.
Introduction The dynamics of photodissociation can be most simply studied with triatomic molecules because their potential energies depend on only three coordinates. OCS is a particularly interesting molecule to investigate because there is an additional simplification in that the C-0 bond length seems to be only a spectator in the dissociation process. The C-0 bond length is 1.157 8, in OCS and 1.128 8, in CO. Moreover on photodissociating OCS the CO is liberated only in the v = 0 state. Strong forces act during the dissociation event but evidently not on the C-0 bond length, which therefore can be frozen without affecting the physics. The photodissociation of OCS has been investigated by Paul Houston's group in the first very weak absorption band (222, 235, and 248 n m ) ' s 2 and in an allowed band at 157 nm. At 222 nm there is approximately a 6% yield of S(3P), but the major channel is to CO(v=O) and S(1D).2The CO is strongly rotationally excited. At 222 nm the distribution of J-state populations has two peaks, the major one at J = 56 and a secondary weaker maximum at J = 67. At 235 and 248 nm the secondary peak is no longer seen and the major peak has decreased to lower Ss, that is, J = 45 (235 nm) and J = 31 (248 nm). The high rotational excitation implies that a strong torque is exerted on the CO fragment as it separates. In other words, when the OCS molecule is excited in its first absorption band from its linear X'X+ ground state, the upper state is bent. This result is entirely in accord with Walsh's rules which predict that for a triatomic, the energy of the ninth valence molecular orbital decreases as the molecule is bent.4 The opposite is true of the eighth molecular orbital. Therefore a low-energy excitation in a 16 valence electron triatomic is invariably to a bent upper state. OCS photodissociation is important as a paradigm of this process. The photofragment velocity anisotropy parameter measured by Sivakumar et al. was ,8 = 1.9 f 0.1 for the higher J @
Abstract published in Aabance ACS Abstracts, October 15, 1995.
0022-365419512099-16307$09.00/0
maximum and ,8 = 0 for the lower J maximum. This quite unprecedented result brings with it the need for further investigation of this linear-to-bent dissociation process. The present paper describes results of a study of the photodissociation dynamics of OCS using the ion-imaging technique of Chandler and Houston. A one-color laser experiment is employed for the photodissociation of OCS at 217 and 230 nm. A single wavelength is used both to excite the OCS and subsequently to ionize the CO using a two-photon resonant three-photon ionization process. The resulting CO+ is detected by an imaging technique:
+ hw - CO(X'X+) + S('D) photodissociation CO(X'X+) + 2 h ~ CO(B'X+ or C'X+) CO(B'X+ or C'X') + hw - CO+ (2+1)REMPI OCS
Experimental Section The experimental apparatus is essentially the same as reported by Chandler and Houston5 which consists of three differentially pumped chambers, as shown in Figure 1. The molecular beam and reaction chambers are pumped by separate 6-in. diffusion pumps. The detection chamber is pumped with a turbo molecular pump (150 W s ) . The three axes, molecular beam, laser beam, and detector, are orthogonal. The molecular beam is formed by a pulsed valve, a skimmer, and a collimator. The pulsed valve (General Valve) has a 0.8mm-diameter orifice and is typically driven with a pulse (220300 ps, 10 Hz). The molecular beam is skimmed twice using two skimmers. The molecular beam then travels another 150 mm to the center of the reaction chamber. The resulting molecular beam is less than 2 mm across in the interaction region, which is tested by observing one-photon UV REMPI signals of NO gas at 226.3 nm. 0 1995 American Chemical Society
Sat0 et al.
16308 J. Phys. Chem., Vol. 99, No. 44,1995 is given by7 Image Intensifier
MCP
To Pump
T n Diimn I W I Ulllr/
Figure 1. Schematic diagram of experimental apparatus. The 355nm output of a Nd3+:YAG laser (Quanta-Ray, GCR190) was used to pump a dye laser (Lambda Physik, SCANmate2, coumarin dyes). The visible radiation was doubled in a BBO crystal. The W outputs at 217 and 230 nm were typically -0.5 mJ/pulse and 0.1 cm-' bandwidth (fwhm). The light beam, the molecular beam, and the detector axes are mutually orthogonal in the interaction region. The laser beam was focused by a lens (f= 200 mm). The W laser pulse was used as the light souce for both the photodissociation of OCS and the REMPI of the fragment CO. The time-of-flight (TOF) mass spectrometer is based on the design of Wiley and McLaren.6 The electrodes are nickel etched meshes (90% transmission, 60 line&. supported by 120-mmdiameter stainless steel rings. The typical acceleration voltage is 3000 V. The length of the field-free region is 600 mm. The ions strike a multichannnel plate (MCP, Hamamatsu, F1217, 40-mm diameter). The MCP is equipped with a phosphor screen (P47, response time * 50 ns) at the end. A CCD camera (Hamamatsu, C4346) attached with a gated image intensifier (Hamamatsu, C2925, minimum gate width 5 ns), observes the image on the phospher screen through a lens. The image signal from the CCD camera is accumulated in a microcomputer over 10 000 laser pulses. A timing box divides the 30 Hz camera frame signal to produce a 10 Hz pulse signal used for the experiment. Subsequent delays are provided by a pulse generator (Stanford Research Systems) for the YAG laser trigger and the gate signal for the image intensifier. The gate position is adjusted to the CO+ ( d e = 28) or S+ ( d e = 32) peak signal separated by the TOF spectrometer. For the measurements of the MPI laser excitation spectra, the current on the phospher screen is monitored, which is fed into a gated integrator and processed. Angular Distribution and 2-DImage of Photofragments. If the dissociation is prompt, Le., fast compared to rotation of the parent molecule, the fragments are distributed anisotropically. The angular distribution of the photofragment velocity
whereflu) is the speed distribution of the fiagments, 8 is the angle between the polarization vector Ed of the photodissociation laser and the velocity vector of the fiagments, p is a spatial anisotropic parameter which ranges between -1 and 2, and &(x) is the second Legendre polynomial defined as P2(x) = '/2[3$ - 11. For recoil along the breaking bond, the two limiting cases are recoil parallel to p or p = 2, and recoil perpendicular to p or /3 = - 1. In the photofragment imaging experiment, the three-dimensional (3-D) distribution is projected onto a two-dimensional (2-D) detector. It is difficult to discern the speed and angular distribution of photofragments directly from the 2-D image. Therefore, some computational processing is essential to recover the 3-D velocity distribution. This processing is similar to that used in computerized tomography method.* Thus far in the studies on the 2-D photofragment imaging spectroscopy, direct calculations using the inverse Abel transform have been used? which contains a fast Fourier transform and Hankel transform. Instead, we used a filtered backprojection technique.I0 This method involves neither the Fourier transform nor the Hankel transform but only convolution and summation in the position coordinates. The calculation algor i t h m of the filtered back-projection method is described in Appendix A.
ReSUlts A. 230 nm Photodissociation. The CO fragments fiom the photodissociation of OCS were detected by the resonanceenhanced multiphoton ionization technique using the BIZ+ XIC+ transition in this study. Two-photon absorption of the CO(B-X) transition at 230 nm has been reported by Loge et al." The BIZ+ state for u = 0, J 1 38 and v = 1, J 2 20 predissociates. Since Rottke and Zacharias12reported that weak predissociation process of the CO(BIZ+,v=lJ 2 20) is independent of the rotational level J, it is reasonable to assume no J dependence of the weak predissociation process at CO(B,v'=OJ>38). Thus, the two-photon spectra reflect the J population of CO (XIC+,v=O,J). The line strength of the CO (B1C+J--X1C+,J) two-photon transition is given in eq B.l of Appendix B and approximated by (W 1) for high-J levels. Since there is a possible correlation between the velocity and rotational motion of a molecular photofragment, the J vector of the fragment can have the quadrupole and hexadecapole alignment factors, A:' and A:', along the polarization vector of the photodissociation laser.13-15 The line strength for twophoton excitation depends on the quadrupole and hexadecapole alignment factors of the J vector.I6-I8 However, the two-photon transition probability of the CO (BIZ+-XIC+) Q-branch is independent of the quadruple and hexadecapole factors. A detailed discussion about the alignment effect is given in Appendix B. Therefore the observed CO spectra are free from the J vector alignment of the CO fragments. The essential results of the experiment are the distribution of populations of CO rotational states, J and photofragment velocity anisotropy parameters, p(J). Figure 2 shows the REMPI spectrum for CO (XIC+,v=O,J)as well as for S(lD). One sees here the great advantage of probing in a Q-branch; the various rotational states are resolvable but still so close that all the transitions can be excited in a single short sweep. As the fundamental laser line is swept through only 20 cm-l the rotational energy is probed through 7000 cm-l. If the REMPI intensities were divided by the two-photon line strength, W
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+
J. Phys. Chem., Vol. 99, No. 44, 1995 16309
The Photodissociation of OCS I
50
60
I
I "
CO(v"=O, J")
"
Q
40
1 " " I
1
II
Wavelength I nm Figure 2. REMPI spectra of CO(v=O,J) and S(ID) from OCS near 230 nm. The two-photon transition is CO(B'Z+,v=O--X'Z+,v=O) Q-branch.
TABLE 1: Population and Velocity Anisotropy Factors of Product CO(v=OJ) from OCS Near 230 and 217 nm 230 nm 217 nm J population" B(J) population" B(J) 43 0.04 0.6 44 0.06 45 0.07 0.6 46 0.1 1 47 0.6 0.13 0.12 48 0.10 49 0.6 0.08 0.7 0.01 50 51 0.05 0.6 0.02 52 0.04 0.8 0.6 0.02 0.03 53 0.8 0.04 0.02 0.07 0.6 1.1 54 0.0 1 0.07 1.4 55 0.0 1 0.09 0.6 1.5 56 0.0 1 0.08 0.7 1.5 57 0.02 0.07 0.8 1.7 58 1.4 0.07 0.02 1.7 59 0.07 0.03 1.7 60 61 1.7 0.05 0.03 0.04 0.01 62 1.7 0.03 0.01 63 64 65 66 67 68 69 70 71 a
0.0 1
0.04
0.05
1.7
0.04 0.04
0.03 0.02 0.02 0.0 1
(2+1) REMPI signal intensity is divided by a factor of (W + 1).
1, one obtains the relative populations as shown in Table 1. This is a one-color experiment in that the probe and the dissociating photon have the same wavelength. Inasmuch as the absorption has no structure and the total sweep range is only about 30 cm-' as compared to a full width at half maximum of about 5500 cm-I, the variation in absorption wavelength is considered to be unimportant. The images of CO ( v = 0,J = 47,55, and 60) are shown in Figures 3 and 4. When the stagnation pressures of OCS were changed from 10 to 200 Torr, the images were essentially the same. In these figures, the laser light beam was polarized in the plane as shown by an mow. Figure 3 shows a 2-D image
Figure 3. Top: 2-D photofragment image of CO(v=OJ=60). Bottom: backprojected 3-D image of CO(v=OJ=60). Electric vector of the dissociation laser light lies on plane as shown by an arrow.
obtained experimentally and a 3-D backprojected image calculated for CO(v=O,J=60). To eliminate a striped shadow caused by the meshed electrodes, the 2-D images are first filtered in the frequency domain and then processed with the backprojection method. The 3-D backprojected images in Figure 4 are equatorial slices of the 3-D distribution. The speed distribution $flu) is obtained from the radial distribution of an image. The center images are caused by background signals and not real ones. The angular distribution, g(@, is obtained by integrating over all speeds at a constant angle. Figure 5 shows a similar image of the S(ID) REMPI signal which by chance is in the same wavelength region. The upper state of the two-photon excitation of the sulfur atom is unknown but the identification of its source as a S(ID) atom is shown by the facts that (1) the ionic mass is 32, (2) the kinetic energy corresponding to the most populated CO(J) level as shown in Figure 5 is consistent with S(lD) and not with S(3P), (3) exactly the same REMPI signal was obtained when photodissociating thiirane C2H4S with focused 230 nm light and (4) the photodissociation of CS;!at 230 nm is thermodynamically below the formation of S(lD), and did not give the REMPI signal. The /I value for S(lD) from OCS was 0.7, which is equal to the
Sat0 et al.
16310 J. Phys. Chem., Vol. 99, No. 44, 1995
J=47
t
J=55
J=60
Figure 4. 3-D backprojected images of CO(v=OJ=47,55,60) from photodissociationof OCS near 230 nm. The photographs on the left show the equatorial slice of the back projected 3-D angular distribution. The contour plots on the right convey the same information in more quantitative form. Note that as J increases, the images become smaller because less energy is available for translation. As J increases, the images become more anisotropic (Table 1). Electric vector of the dissociation laser light lies on plane as shown by an arrow.
The Photodissociation of OCS
J. Phys. Chem., Vol. 99, No. 44, 1995 16311
Figure 5. Photofragment backprojected image of S(lD) from OCS near 230 nm. The inner and outer circles indicate the energies available for S(’D) CO(v=OJ=47) and S(3P) CO(v=OJ=47), respectively. Electric vector of the dissociation laser light lies on plane as shown by an
+
+
Discussion J=47
B =0.6
.
Previous results on the photodissociation of OCS were obtained by measuring the Doppler broadened one-photon vacuum UV fluorescence excitation spectrum ( A I I I XiZ+).I In the present experiment (2 1)REMPI is used together with an ion-imaging system as a detector. The results on the CO rotational populations at 230 nm appear to interpolate reasonably well between previous data taken at 222 and 235 nm. However, as shown in Table 1 the /3 values at 217 and 230 nm are consistently larger than those previously measured at 222 nm. The rotational population distributions are remarkable in three ways: (1) no low-J states are occupied at all, (2) the velocity anisotropy factor increaes with increasing J, (3) at 217, 222 and 230 nm the distribution is bimodal. The fact that only high-J states are occupied is a result of a mechanical constraint; on the upper state the CO must rotate as it separates from the S atom. The increase of /3 with increasing J was explained by Sivakumar et al.I as a result of absorption to two upper states of A” and A’ symmetry with respect to the newly formed OCS plane. As the ground state has even (A’) symmetry with regard to reflection, an A” A‘ transition dipole moment must be perpendicular to the plane and hence /3 = - 1 for that A” surface. On the other hand the transition moment for an A’ A’ transition must lie in the plane; /3 could, in principle, lie between -1 and +2. The observed /3 was assumed to be an incoherent average of /3 values on the two surfaces. The electronic spectrum of OCS was measured by Rabalais et al.19 The first two bansitions *A, IZ- I F which overlap are both forbidden in the linear configuration but become allowed when the molecule bends. The assignments are based on a reasonable ordering of the molecular orbitals and are confirmed by their agreement with the intensities. Recently, Rokutan et al.*O have computed two dimensional potential surfaces by ab initio methods for the nIA’ states (n = 1-4) and the dA’’ states (n = 1-3). The C-0 bond distance was fixed at its equilibrium distance in OCS (1.157 A). Sivakumar et al.’ assumed that the two dissociating A’ and A” states were the Renner-Teller pair which is a degenerate IA state in linear geometry which splits in bent geometry. The 2’A‘ state falls in energy and the 2IA” state rises as the molecule bends away from the linear form. Figure 8 shows the potential energy of the different states as a function of angle as calculated by
-
+
20
40
30
50
60
70
Angle 8 /degrees Figure 6. Typical angular plot of a image signal of CO(v=OJ=47) from OCS near 230 nm as a function of polarization angle 8. The dashed line is a plot of 1 + 0.6f~(cose). 80 I
’ ’ ’ ’
70 I
’
60
“
’
50
Q
I ’ “ ‘ I ”
-
-
Wavelength I nm
Figure 7. REMPI spectra of CO(v=O,J)from OCS at 217 nm. Two-
photon transition is CO (C’C+,v=ocCX’X+,v=O)Q-branch. weighted average of all the /3(J)’s for the CO molecules which recoil from the sulfur atom. Figure 6 shows a typical plot of the angular distribution of CO,in this case for J = 47, which is fitted by the parameter /3 = 0.6. B. 217 nm Photodissociatioh. The excitation spectrum and image of the product CO from OCS at 217 nm were measured using the Q-branch two-photon transition of CO (CIZ+,d=O--XIZ+,t/’=O). Results are shown in Figure 7 and Table 1, which are similar to what are observed at 230 nm, that is, (a) the photodissociation process is CO S(ID), (b) the J distribution is bimodal, and (c) CO with low J ( 5 5 8 ) have low values and CO with high J (259) have high /3 values. It should be noted that the relative population of the high /3 component increases with respect to the low /3 component.
+
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16312 J. Phys. Chem., Vol. 99, No. 44, I995
2‘A“
I ‘A”
the Renner-Teller pair, 2IA’ and 2IA”, they are 2IA’ and 1IA”. At the shorter wavelength of 217 nm, since the excitation to the upper ‘A’ (‘A) state becomes more probable than at 230 nm, the relative contribution of the higher J’s increases with respect to that of the lower Ss as shown in Figure 7. /3 is always -1 for the A” surface, while we do not necessarily assume a fixed /3 value for the A’ surface which can be range from - 1 to 2. The excited potential surfaces of OCS exhibit strong forces which increase the C-S distance and decrease the OCS angle. The resultant asymptotic trajectories will produce CO molecules with different Ss and different dissociation angles, Le., p’s. Since our largest /?value is 1.7 which is close to the upper limit of 2, the transition moment of the 2IA’ XIA’ transition lies almost along the C-S bond of OCS. Up to now there has been no explanation of the bimodal rotational distribution. Figure 9 supplies a possible answer. The optical transition is from the ground 1‘A’ state to both the 1IA” and the 2’A’ states. In each case the molecule is held temporarily in a potential well but would then cross over to dissociating surfaces at the same value of the C-S bond distance, which is marked by an arrow (a) in Figure 9. Since the crossing points of the A’ and A” surfaces are at almost the same internuclear distance of C-S, the parent OCS would have similar structures and then the J distributions of the product CO become similar. Thus, the /3 values for lower J’s are the incoherent average of /3 values on the two surfaces and show a constant value of 0.6. In addition, on the 2IA’ surface, another dissociation pathway may be open. OCS can dissociate to S(lD) CO, passing over the energy hump shown by an arrow (b) in Figure 9. This dissociation path produces the higher Ss of CO with large p values (-1.7), thus explaining the bimodality. At the lower photon energies, 235 and 248 nm, the higher energy crossing region may no longer be accessible; these narrower population distributions have only a single peak. It is interesting to compare the photodissociation dynamics of OCS with that of CS2. When OCS is photodissociated the CO product is mainly or exclusively in the v = 0 state. On the other hand when CS;!is photodissociated the CS product vibrational excitation is so strong that the populations are inverted, Le., tr = 0 is not the most populated state. The difference is explicable on spectroscopic grounds. The electronic spectrum of CS;! begins at energies much lower than that of CO;!. Therefore in OCS the S atom is the exclusive’ chromophore and the CO bond is unaffected. In CS2 the nonbonding electrons of both S atoms contribute to the absorption and both CS bonds are excited and stretched. Ultimately, only one CS bond breaks but the survivor is strongly vibrational e ~ c i t e d . ~ ~ . ~ *
-
1
I
I
I
220
200
180
I
I
160
140
1 0
Bond angle / degrees Figure 8. Potential energies of the ground and three excited states of OCS as a function of bond angle with fixed bond distances (from Rokutan et al.*”).
+
R , S
Figure 9. Potential energies of the ground and several excited states of OCS as a function of C-S distance with fixed 180” bond angle and C - 0 distance (from Rokutan et al.?”).Solid curves refer to A’ surfaces and broken ones to A”. An arrow (a) is designated for surface crossing of IZ-(1 IA”) to a ‘A” state and also ‘A(2’A’) to a ‘A’ state, which are correlated with S(lD) CO(IZ+). An arrow (b) is designated for a direct dissociation pathway of ‘A(2’A’) to S(lD) CO(’Z+).
+
+
Rokutan et ~ 1 The . upper ~ ~ 2’A” state of the Renner-Teller pair has an energy minimum at the linear configuration. Dissociation on this surface would produce low J states of CO contrary to fact. Therefore, either the optical transition to this state from the ground state is very weak or crossing to a lower surface is extremely efficient. As shown in Figures 8 and 9 besides the 2IA’ and 2IA” pair there is also a 1‘A’‘ state which has a maximum at the collinear configuration. In linear geometry this state has IZ-symmetry and, as with the RennerTeller pair, a transition to it from the ground state becomes allowed in the planar geometry. The increase in p with increasing J was ascribed by Sivakumar et af.’ to different J state distributions and p(J) on the two excited potential surfaces. They assumed that /3 was always -1 on the A” surface and always 2 on the A’ surface. The latter value implies that the transition dipole remains parallel to the dissociation direction, i.e., the C-S bond. The assumption that higher Ss are produced mainly on the A’ surface is strengthened by the fact that, as shown in Figure 8, at 160” bond angle the falloff of potential with angle is much steeper on the A’ surface. Our explanation of the variation of the anisotropy parameter with J is essentially the same as that of Sivakumar et a1.I Two excited surfaces are assumed to contribute to the absorption but instead of being
~
Acknowledgment, This work was partly supported by a Grant-in-Aid in priority field of “Free Radical Science’’ from the Ministry of Education, Science and Culture of Japan (Y.M.), the Mitsubishi Foundation (M.K.), and the U. S. National Science Foundation (R.B.). The authors thank Prof. S. Iwata of Institute for Molecular Science for helpful discussions and allowing us to use their theoretical calculation data prior to publication. Appendix A
Transformation of 2-D Images to 3-D Structures. The 3-D velocity distribution is cylindrically symmetric along the z axis, where the z axis is defined along the polarization vector Ed of the photodissociation light. The x-axis is defined as it is perpendicular to the z axis and lying in the surface of the 2-D image. Let the intensity distribution of one row across the 2-D image along the x axis be p(x), and the intensity distribution of
J. Phys. Chem., Vol. 99, No. 44,1995 16313
The Photodissociation of OCS the row across the center slice image of the 3-D structure be (x2 y2)'I2and 4(r) is at the 4(r), where r is defined as r same z-axis position with the p ( x ) . Then the function p ( x ) is the projection of the rotated structure of the function 4(r) around the z-axis. The computational processing of the p ( x ) function to produce the q(r) function is called back-projection. The calculations of the back-projection row by row across the z axis result in the construction of the 3-D structure from the 2-D image. The function p ( x ) is the projection of the cylindrical symmetric 3-D structure 4(r), that is
+
eq A.8 is reduced to the following expression: s = [WIP
(A. 10)
which is the covolution procedure with the kemel [w]. The filter function W(6)= 161 is high-frequency intensification processing in the Fourier space. The use of this filter is not practical, because it amplifies high frequency noise in the data p ( x ) . Instead of W(8)= 161, Shepp and Logan2I proposed the SL filter to reduce the problems caused by the sharp cutoff at tcin the SL filter:
The sampled values of the SL filter are given by In polar coordinates this becomes the Abel transform of 4(r) given by
k"(n) = 2N2/d(1 - 4n2)
6412)
The convolution in eq A.10 is expressed as
c N
so') = (1/N)
P ( t ) = J:J(x)
e x p ( - W d.x
('43)
where 6 is the frequency coordinate in the Fourier space corresponding to the x axis in the position space. The original distribution q(r) can be recovered by the inverse Abel transform as follows: 40.) = 2xJWtJo(2xrt) P ( t ) d t
(-44)
where JO is the zeroth-order Bessel function of the first kind. The Bessel function, Jo, can be written as
p(lnl) ko' + n )
Then the slice of the 3-D structure q = {4(i), i = 0-N} is calculated by the following summation of s(i) according to eq A.8: M
40) = ( 1 / z ~ ) ~ scos f i (zm/2~)1
where P(&, the Fourier transform of p ( x ) , is an even function around the origin 8 = 0. Then, making a change of variables yields
For the resolution of the summation in eq A.14, we used M = 256. The calculations row by row for the 2-D image yields the 3-D velocity structure.
Appendix B Two-Photon Line Strength and Its Alignment Dependence for the CO B W , C W XIX+ Transition. Bray and I: two-photon Hochstrasser2* (BH) pointed out that a I: transition has rather special characteristics. The observables which depend on polarization are sensitive to the nature of the virtual intermediate states, that is, whether they have I: or l7 symmetry. For a given initial state J the I: I: Q-branch absorption intensity is proportional to22
+
('4.8) s = [UI- [WI [UIP where [W] is the matrix corresponding to the filter function W(8). The matrix [W] has only diagonal elements. If we know the matrix [WI=
~ul-'[wl[~l
(A.9)
-
-
and for two circularly polarized photons the corresponding expression is
cos 0) d 0 (A7) where the substitution x = r cos 8 is made. In practice, the p ( x ) function, which is the row data of the 2-D image obtained experimentally, consists of a number of intensity data sampled pixel by pixel. Now, we regard p ( x ) as the vector p = (Po),j = 0-N), where 2N 1 is the number of the pixels in one row of the 2-D image. The s(x) function is also regarded as a vector s = {so), j = 0-N). The Fourier and its inverse transformation are expressed by 2-dimensional transformation matrixes [VIand [V]-I, respectively. Using the matrixes, eq A.6 can be written as
0414)
m=O
-
We used a filtered back-projection methodlo instead of the direct calculation of the inverse Able transform (eq A.4) to get the 3-D structure 4(r) from the 2-D projection p ( x ) . Defining the inverse Fourier transform of l t l P ( [ ) as
(A13)
n=-N
The Fourier transform P(6) of the function p ( x ) is given by
+
+ +
(2J l)J(J 1) 30(2J - 1)(2J 3)"
2
The electronic matrix elements, called by BH the transition dipole factors are
Ps2 = 12PIP1,'
+ P+P-' + P-P+'I2
Pu,2 = IP,P,; - P#-' The transition moments for the Z = ('iIPzI')
(B3)
- P-P+'12 +
I:,
+
Z path are
and PI/ = (I'Pz.Ii)'
( z is the molecular axis). Those for the I: P* = W1l(1/fi)(Px
034)
- l7 -
035) Z path are
* iP,)IZ)
(B6) iP,)III) Here we used the standard conventionI6 for the tensor moments P,'
= WI(1/fi)(Px
Sato et al.
16314 J . P h y . Chem., Vol. 99, No. 44, 1995 ,piinstead
of BH’s definition. The only intermediate states that can occur are X and n states. For a Q-branch, the ratio of the absorption cross section for two linearly polarized photons to that for two circularly polarized photons is
TABLE 2: Moments of Alignment Factors in the Z-Z Two-Photon Transitiona a two-photon process of CO P$P; P$P:
x-rI-x
0 -1
,UI
0.510
= 0, ,us = 0
0 -0.1 15 0.1 13 0.713 -0.715
-1.4 -0.7 2
-
Exactly this ratio has been measured for the B X transition by Tjossem and S m ~ t h for ’ ~ J’s large enough that eq B7 is close to ?/j i”/.(u’&2~). They found a lower limit of 200 for the intensity ratio ull/ucc, implying a lower limit of 17 for the ratio
+
,U’IlU’S.
To understand the meaning of this ratio, we consider the various dipole moment matrix elements contained in p21. In the absence of optical rotation left and right circularly polarized light must have the same effect. Therefore p+,u-’ = p-,u+’. Consider first an extreme case in which only I: virtual states have nonzero matrix elements. In this case p+,u- = 0, ,u’s = 4,u’r and ullluCc= 4 for large J . In the other extreme case in which only ll intermediate states occur, puip~’ = 0, p’s = p21 and u11/u~~ = 14 for large J . Given that p 2 2 ~ 17&, using the equal sign eqs B.3 and B.4 can be solved for the ratio and one finds two roots, - 1.4 5 a 5 - 1 and - 1 5 a 4 -0.67. Recalling that only a lower limit has been measured for C J ~ I / O ~ ~ , one sees that as this ratio increases the two roots both approach -1. In this limit p1Ips = c-, that is, ps = 0. This limit is not nearly so extreme as the cases in which the intermediate states were assumed to be purely 2 or purely n. Indeed it has a physical basis. The limit of zero absorption of two circularly polarized photons simply means that a set of matrix elements with intermediate states makes the same overall contribution as either one of the two ll intermediate states. We infer that there are numerous intermediate states with the I l states being statistically twice as abundant as the I: states. Thus, on average, ,Ul$lI‘ = -,u+,u-’ = -u-p-’. This equality means that the alignment, whether finite or zero, will be unaffected by the absorption of two linearly polarized probe photons of the CO BIZt X’2+ Q-branch transition. Even if ps is not strictly zero, its small size means that the CO molecule will absorb two linearly polarized photons isotropically in this transition. This is also the case for the CO C’Z+ XII- Q-branch transition. In the photodissociation of OCS, the high rotational excitation of the CO fragment observed implies that a strong torque is exerted on the CO fragment as it separates. This motion can result in the strong correlation between the translational recoil velocity v and rotational angular momentum vector J of the CO fragment in the molecular frame. The v-J correlation is carried over to the alignments of the J vector in the laboratory frame, when the photodissociation is caused by a polarized laser beam. DixonIJ has given a general theory of the correlation between the velocities and angular momenta of photofragments. He showed that Doppler broadened line shapes depend on the anisotropies of both velocity and angular momentum. Actually, Sivakumar et nl.’ have indicated the vector corrletaion between v and J of the CO photofragments in the photodissociation of OCS from their measurements of the Doppler profiles in the laser-induced fluorescence detection of the CO AIIl--X’X* system. Kummel et al.lh have presented the method for determining the population A:’(Ji), the quadrupole alignment factor A:,’’(.Ji).and the hexadecapole alignment factor A:;’(J,) for a J ,
-
-
Ill
= 0, us
z0
x-1-2
co
0.123
0 0.007
0.006 2.572 0.429
“ S e e eqs B.3-B.8 for a and ,u and eq B.9 for P:, Pi,and Pi. The ratios are calculated for the Q-branch transition for J , = 50.
rotational level of a diatomic molecule probed by linearly polarized two-photon nonresonant excitation. They have given a general expression for the alignment dependence of the twophoton transition probability I:
where k = 0, 2, 4; q = 0, 1, 2 , 3, 4; q 5 k, C(det) is the overall detection-sesitivity constant, n(J;) is the population in a given rotational state J,, and
