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Polarisation effects in electronically inelastic collisions: SiF C2Δ + H2→SiF B 2Σ+ + H2
Chemical Physics 233 Ž1998. 45–55
Polarisation effects in electronically inelastic collisions: SiF C 2D q H 2 ™ SiF B 2 Sq q H 2 Neil A. Jackson, Colin J. Randall, Kenneth G. McKendrick
)
Department of Chemistry, The UniÕersity of Edinburgh, King’s Buildings, Edinburgh EH9 3JJ, UK Received 6 March 1998
Abstract Linearly polarised laser excitation was used to prepare aligned, electronically excited samples of SiF molecules. Selected, spectroscopically isolated bandheads in the B–X and C–X systems were excited, and appropriate bands resolved in the dispersed fluorescence. The observed degree of polarisation anisotropy, R, of B–X Ž3,2. fluorescence from the directly excited SiF B 2 Sq, Õ s 3 state agreed, within experimental error, with the theoretical prediction. The value of R for C–X Ž0,0. fluorescence from the SiF C 2D, Õ s 0 level excited in the presence of H 2 Žand Ar carrier. was reduced from the theoretical prediction, consistent with previous observations of partial rotational redistribution under these conditions. As also previously established, collisions with H 2 induce SiF C ™ B transfer. The SiF B 2 Sq, ÕX s 0 product was found to be significantly less polarised than the SiF C 2D, Õ s 0 population from which it was produced. Two simple classical models were developed for the degree of depolarisation based on alternative assumptions about the spatial distribution of the additional angular momentum, D J, imparted during the collision process, relative to the initial J. Both predict a degree of depolarisation consistent with the experimental observations in this case. We note that there exists in principle an impulsive dynamical limit in which the product state will remain partially aligned regardless of the relative magnitudes of J and D J. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Electronically inelastic collisions play an important role in many gas phase processes of practical and fundamental interest. It has been known for a considerable time that non-reactive collisions may efficiently remove molecules from electronically excited states. Only relatively recently, however, have detailed investigations been possible of the fate of the quenched molecules, including the distributions over rovibrational levels of the product electronic states. Current progress in such state-to-state studies
)
Corresponding author. E-mail: [email protected]
of diatomic molecules has been reviewed recently by Dagdigian w1x. In another somewhat distinct research area, there is a quite extensive literature on the use of optical polarisation methods to determine the extent to which the plane of rotation Žor, quantum mechanically, the M quantum number. is conserved during rotationally inelastic collisions within single electronic states w2– 15x. There has also been very substantial parallel effort in the development of polarisation methods in the field of ‘dynamical stereochemistry’, broadly encompassing molecular photodissociation dynamics and reactive bimolecular collisions w16–20x. However, very little work has been done in the overlap between these fields, the retention of polari-
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 5 3 - 0
46
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
sation during collisional transfer between different electronic states. As far as we are aware, the only significant report of this kind is from McCaffery’s group w21x on electronically inelastic collisions of Na 2 . They observed that the collisionally produced 1 q emission on the A 1 Sq u –X Sg band was negatively polarised relative to the plane of polarisation of the laser used to excite the B 1 P u –X 1 Sq transition. g They concluded that this was the result of an indirect mechanism in which molecules were initially collisionally transferred to the Ž2. 1 Sq g state and then radiatively transferred to the A 1 Sq u state. Because of the overall combination of parallel and perpendicular electronic transitions involved, it was inferred from the sign and magnitude of the observed polarisation that there was a high propensity to conserve the plane of rotation Žor M . during P ™ S collisional transfer. We suggest that polarisation techniques have so far been under-exploited in this context, and potentially provide an additional diagnostic of the mechanisms. Our aim in this paper is to apply these methods to collision-induced SiF C 2D ™ B 2 Sq transfer, for which we have previously examined various aspects of the scalar partitioning of energy in the products w22–24x. The SiF C 2D and B 2 Sq excited states are subject to a zero point energy gap of ; 5000 cmy1 . Rather surprisingly, we have discovered that C ™ B transfer is nevertheless dominated by DÕ s 0 transitions, essentially independent of the collision partner, with the detailed vibrational propensities closely matching the C–B Franck–Condon overlap w22x. Subsequently, we established w23,24x, that a considerable fraction of the large energy defect was converted to SiF B 2 Sq product rotation, which is also relatively unusual for a process of this type w1x. We have argued that the energy partitioning is consistent with a sudden, impulsive mechanism, which may be associated with the valence–Rydberg nature of the C 2D–B 2 Sq collisioninduced transition w22–24x. We therefore set out to determine, in three stages, whether the degree of retention of polarisation during this process is consistent with our previous conclusions about the mechanism. The first stage, which was not strictly necessary from a theoretical point of view, was to use a linearly polarised laser to excite SiF directly to the B 2 Sq state and to disperse the
laser-induced fluorescence ŽLIF. on returning B–X bands. The measured polarisation ratios on known combinations of branches in absorption and emission could be compared with well-established theoretical predictions w25x and consequently were used as a preliminary test of the correct functioning of the apparatus. In the second stage, SiF molecules were excited to the C 2D state and the polarisation of the returning C–X emission determined. As explained more fully below, we anticipated that there might be complications due to some collisional depolarisation of the C 2D state and partial optical saturation of the strong C–X Ž0,0. transition w26x. It would be necessary to know the extent of these combined effects to interpret the final stage, where population was excited initially to the C 2D state and the polarisation of the emission from the B 2 Sq state populated in collisions with H 2 measured.
2. Theoretical basis of polarisation measurements The polarisation dependence of optical absorption and emission by isolated molecules has been thoroughly treated by Zare w25x. In essence, in the lowpower limit, the probability of a molecule being excited from an initial state, < i :, to an excited state, < e :, depends on the square of the projection of the linear polarisation vector, Eˆa , onto the transition dipole moment vector, m ei . Similarly, the probability of detecting a fluorescence photon emitted on a transition from < e : to a final state, < f :, depends on the square of the projection of the corresponding transition dipole, m e f , onto the detected polarisation vector, Eˆd . The extent to which the linear polarisations of the absorbed and emitted photons are correlated is normally measured by observing the relative intensities of the fluorescence, I 5 and I H , which correspond to Eˆd parallel to and perpendicular to Eˆa , respectively. Of the possible alternative measures of the fluorescence anisotropy w25x, we shall use R, defined by Rs
I5 y IH I5 q 2 IH
.
Ž 1.
R corresponds to the average degree of alignment relative to an isotropic sample. It is directly related
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
47
to the average angle between absorption and emission dipole moment vectors, through
3. Experimental
R s 25 P2 Ž m ˆ ei P mˆ e f .
The apparatus was similar to that used in our previous studies, w22–24,28–30x with the additional capability of polarisation-sensitive detection of fluorescence signals. Ground state SiF X 2 P radicals were generated in a flow system by microwave discharge of a dilute mixture of SiF4 in Ar. A second independent flow of the collision partner, which in this study was confined to H 2 , was added downstream of the discharge and upstream of the observation zone. The probe laser beam passed through a calcite linear polariser immediately prior to entering the observation zone Žvertically. via a Brewster’s angle window. We denote the laser propagation vector by k a . The polarisation of the beam, Eˆa , was chosen to be parallel to the flow tube axis. The SiF molecules were selectively excited on either the B 2 Sq–X 2 P or C 2D– X 2 P transition as appropriate to the desired measurement. Fluorescence mutually perpendicular to k a and to Eˆa was collected and passed through a photoelastic modulator ŽPEM. ŽHinds International PEM-80.. The principal axis of the PEM was set at 458 to the vertical, bisecting k a and Eˆa . The timing of the laser pulse relative to the phase of the PEM was used to control whether it acted either as a transparent optic or as a half-waveplate, rotating the plane of linear polarisation of all components of the fluorescence by 908. The light subsequently passed through a second, analysing calcite polariser with its polarisation axis set horizontally and therefore fixed parallel to Eˆa . Depending on the phase of the PEM this polariser would therefore transmit the parallel, I 5 , or perpendicular, I H , components of the fluorescence. The transmitted component was focused onto the entrance slit of a monochromator ŽHilger and Watts, Monospek 1000, 1 m focal length. and the dispersed intensity detected by a photomultiplier tube ŽEMI 9558QB or EMI 9789QB.. This method eliminated any potential problems which might otherwise have been associated with the polarisation dependence of the transmission efficiency of the monochromator or with physical realignment of mechanically rotated passive polarising optics. Signals were preamplified ŽEMI A2 or Stanford Research Systems SR240. and captured by a tran-
¦
;
Ž 2.
where the function P2 Ž x . is, as usual, the secondorder Legendre polynomial in x. Exact expressions for R for electronic transitions of linear molecules have been derived w25x. The results are J-dependent, and determined by the type of transition Žparallel or perpendicular. and the combination of branch types ŽP, Q, or R. in absorption and emission. They converge on various limiting high-J values. An insightful way of understanding these is to appeal w17x to the azimuthally averaged addition theorem ŽAAAT. for two otherwise uncorrelated distributions sharing a common axis of cylindrical symmetry, allowing R to be expressed as R s 25 P2 Ž m ˆ ei P Jˆ.
;¦P ž JˆP mˆ /;.
¦
2
Ž 3.
ef
Geometric considerations allow the values of
² P2 Ž mˆ P Jˆ. : to be deduced for different branch types w25x. The extension of this treatment to include collisions or other dynamical processes which might reorient the plane of rotation of the excited state, < e :, prior to re-emission of the fluorescence, was first presented by Gordon w27x. Equivalent results to Gordon’s treatment may be derived by a further application of the AAAT, fitting in with the framework above: R s 25 P2 Ž m ˆ ei P Jˆ.
;¦P Ž JˆP Jˆ .;¦P ž Jˆ P mˆ /;.
¦
X
2
X
2
ef
Ž 4. The quantity Ž JˆP Jˆ . is the cosine of the ‘tipping angle’, a , between initial and final rotational vectors. It is often convenient to isolate the dynamical effects on R from those purely to do with the spectroscopy of the isolated molecule w27x. This can be done by defining R 0 as the value of R that would have been obtained via Eq. Ž3. in the absence of collisions. The simple relationship R P2 Ž JˆP JˆX . s Ž 5. R0 X
¦
;
may then be used to extract the value of P2 Ž JˆP JˆX . and hence an average value of the tipping angle, ² a :, if desired w2,27x.
¦
;
48
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
sient digitiser ŽDSP Technologies 2001A, 100 MHz. incorporated in a CAMAC ŽIEEE 583. modular data collection system. Other experimental variables, such as scanning the laser or monochromator wavelengths and timing the firing of the laser, were controlled via appropriate CAMAC modules. This allowed, for example, spectra of the parallel and perpendicular components of the fluorescence to be collected on an alternate shot basis, so that long term drifts in the conditions would affect both components equally. Gases were obtained from the following manufacturers, and used without further purification: SiF4 ŽUnion Carbide, 99.99% and Fluorochem, 99.5%., Ar ŽBOC, 99.998%., H 2 ŽBOC, 99.99%..
4. Results 4.1. Directly excited B–X emission In principle, polarisation measurements could be made using a combination of any one of three branch types in absorption ŽP≠, Q≠, or R≠. and emission ŽPx, Qx, or Rx.. However, details of the spectroscopy of the B 2 Sq–X 2 P transition combined with the finite resolution of the exciting laser and, more severely, the detecting monochromator greatly limited the number of viable combinations in practice. It proved most convenient to select the B 2 Sq– X 2 P Ž3,0. band for these measurements because it was a sufficiently strong transition lying within the wavelength region accessible with the same laser dye as the C–X Ž0,0. band. A typical LIF excitation spectrum of the B–X Ž3,0. band is shown in Fig. 1. There are altogether 12 distinct spectroscopic branches for a 2 Sq– 2 P transition w31x. ŽWe shall use the contracted notation B–X 1r2 and B–X 3r2 to label branches connected to the well-separated 2 P 1r2 and 2 P 3r2 components of the ground state.. At the cost of not preparing a single J level in the upper state, there is a considerable advantage in exciting at a bandhead to maximise the overall signal in subsequent dispersed fluorescence measurements. This turns out not to be a severe limitation for the SiF B–X Žand C–X. bands because the relevant heads form for levels where the polarisation properties are approaching the limiting high J values and are relatively independent of J. Only the P1 and O P12 band-
Fig. 1. SiF B–X Ž3,0. LIF excitation spectrum. The locations of all 12 branches for 28 Si19 F are indicated. The features marked Ž). are due to minority isotopes of Si.
heads in the respective B–X 1r2 and B–X 3r2 subbands are suitable because the other features in the spectrum are blends of branches of more than one type. Of these two, the P1 head is preferable because of the larger thermal population of the lower-lying F1 ground state levels. In this case, the bandhead forms at J ; 20.5. Fig. 2 shows typical dispersed spectra of the corresponding components of the fluorescence returning on the B 2 Sq–X 2 P 1r2 Ž3,2. sub-band. The Ž3,2. band was chosen because it has the largest Franck–Condon factor w30x of bands originating in B 2 Sq, Õ s 3. The P≠Px and P≠Rx combinations show an enhancement of I 5 over I H , whereas the reverse is true for P≠Qx, consistent with simple physical arguments for a perpendicular transition. Because of the very short B 2 Sq state lifetime Ž- 10
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
Fig. 2. Parallel Ž- -`- -. and perpendicular Ž —v — . components of dispersed SiF B–X 1r 2 Ž3,2. fluorescence generated by pumping the B–X Ž3,0. P1 bandhead. In the absence of collisions, only the three branches indicated are present, as discussed in the text.
ns. we did not anticipate collisional depolarisation to compete with directly returning B–X emission. Therefore, only the three branches indicated in Fig. 2 contribute to the spectra because only the F1 levels in the B 2 Sq state are populated by pumping on the P1 branch.
49
Statistical analysis of repeated measurements of the type shown in Fig. 2 yielded polarisation ratios, R, for the different branch combinations. Signals were integrated over the wavelength range spanning each branch, starting from the minimum point between partially resolved branches. We were careful to avoid significant optical saturation of the absorption step, an aim which was helped by the relatively weak Franck–Condon factor for the Ž3,0. band w30x. As will be explained below, we were even more severely limited in the combinations of branches which could be used in the second and third stages of the measurements, so that we concentrate throughout on the numerical values of R for the P≠Px combination. The results are included in Table 1, together with the theoretical prediction. The agreement is seen to be good, which we regarded as confirmation of the validity of the experimental approach. In a less extensive analysis, qualitatively correct signs but quantitatively slightly less satisfactory magnitudes for the P≠Qx and P≠Rx combinations were obtained: we believe this may be due to the less sharply defined resolution of the R 1 and Q1 branches in the fluorescence spectrum. 4.2. Directly excited C–X emission The branch structure of the C 2D–X2 P bands is qualitatively similar to that of the B 2 Sq–X 2 P bands w32x, disregarding the unresolved splitting of all bands due to L-doubling in the C 2D state. Typical C–X LIF excitation spectra may be found in several of our previous reports w23,24,30x. In summary, it is again found that a P1 head is the most suitable for pumping
Table 1 Observed and predicted polarisation ratios, R, for P≠Px combinations of SiF bands P-type transition excited B–X 1r 2 Ž3,0. C–X 1r 2 Ž0,0. C–X 1r 2 Ž0,0. a
Polarisation ratio, R observed B–X 1r2 Ž3,2. C–X 3r2 Ž0,0. B–X 3r2 Ž0,0.
high J limit 1r10 1r10 y
predicteda b
0.0861 0.0918 c y
Predicted using the expressions in Ref. w25x. Assuming J ; 20.5 excited at the P1 bandhead of the B–X Ž3,0. band. c Assuming J ; 34.5 excited at the P1 bandhead of the C–X Ž0,0. band. d Error limits correspond to 1s uncertainties based on statistical analysis of repeated measurements. e In the presence of H 2 Ž1.5 Torr. and Ar Ž1 Torr.. b
observedd 0.089 " 0.033 0.060 " 0.010 e 0.020 " 0.016 e
50
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
a substantial population of molecules to the C 2D state on a branch of a pure spectroscopic type. The P1 head of the C–X Ž0,0. band forms at J ; 34.5. In a similar way to that described for the B–X transition, we then investigated the polarisation of fluorescence on the directly returning C 2D– X2 P 3r2 Ž0,0. sub-band following excitation at the P1 head in the C–X 1r2 sub-band. The emission and excitation wavelengths are well separated because of the relatively large Ž A ; 160 cmy1 . spin–orbit splitting in the X 2 P state. The situation is complicated, however, by the presence of sufficient H 2 to induce a substantial fraction of collisional transfer to the B 2 Sq state during the longer C 2D state lifetime Ž101.8 " 1.4 ns. w24x. We have shown previously that inelastic rotational redistribution within the C 2D Õ s 0 level is competitive with the inter-electronic state process w23,24x. This work also revealed that collisional transfer between F1 and F2 levels competes with rotational redistribution within the F1 levels. Consequently, the returning C–X 3r2 sub-band has some intensity on all six branches, and the only one which remains spectroscopically isolated is the O P12 branch to longest wavelength. As a result the only meaningful polarisation measurements we were able to make were for the P≠Px combination. The potential effects of optical saturation were again a further concern w26x. The C–X Ž0,0. band has a relatively high Franck–Condon factor, enhancing the likelihood of saturation, although this is mitigated by the weaker C–X electronic transition probability w30x. Operationally, we varied the laser pulse energy and simultaneously monitored the linearity of the total signal and any trends in the observed polarisation. Despite some significant curvature in the total signal, the polarisation ratios were statistically invariant for pulse energies up to ; 500 mJ, although there was increasing scatter about the mean polarisation at low pulse energies. The results reported in Table 1 represent averages over the pulse energy range ; 50–500 mJ. The value of R s 0.060 " 0.010 is significantly reduced from the prediction Ž R s 0.0918. for the P≠Px combination at the average J of the P1 head. We believe this reduction is primarily the result of collisional depolarisation, with a possible contribution from partial optical saturation. Nevertheless, we concluded that we had prepared a significantly polarised C 2D state sample and
could therefore proceed with the inter-electronic state measurements in Section 4.3. 4.3. Collisionally produced B–X emission Finally, we turn to the ultimate aim of this work which was to determine the extent of any retention of the plane of polarisation of the SiF B 2 Sq state produced from the C 2D state in collisions with H 2 . As described in Section 1, we have demonstrated previously w22x that the major product channel starting from SiF C 2D, Õ s 0 is B 2 Sq, ÕX s 0, despite the relatively large vibronic energy gap involved. The product B–X emission is therefore best monitored on the Ž0,0. band. Both F1 and F2 levels of the B state are populated in the collisional transfer process w24x and so only the P-type branches to longest wavelength in each sub-band have the potential to be used
Fig. 3. Collisionally produced parallel Ž- -`- -. and perpendicular Ž —v — . components of dispersed SiF B–X 3r 2 Ž0,0. fluorescence generated by pumping the C–X Ž0,0. P1 bandhead in the presence of H 2 Ž1.5 Torr. and Ar Ž1 Torr.. The head-forming branches are indicated Žfor a full description of the branch structure see Fig. 1..
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
for polarisation measurements. Furthermore, the rotational populations in the B 2 Sq, ÕX s 0 state are substantially hotter w23,24x than in the C 2D, Õ s 0 state from which they are produced. Therefore, there is a component in the emission spectrum from high-J rotational lines in the B–X 3r2 sub-band which partially underlies the P1 head of the B–X 1r2 sub-band. Consequently, only the O P12 branch at the red end of the B–X 3r2 sub-band remains viable for the determination of the B state polarisation. The corresponding parallel and perpendicular components of the collisionally produced B–X fluorescence are shown in Fig. 3. Interestingly, I 5 slightly exceeds I H in the region of the O P12 head, although the magnitude of any effect is admittedly not large relative to the signal-to-noise ratio. Statistical analysis of repeated scans of this region, under the same experimental conditions as used for the directly returning C–X fluorescence above, yielded the R value included in Table 1. At 0.020 " 0.016, R is statistically significantly reduced from the value for the directly returning C–X fluorescence, but still Žmore marginally. greater than zero.
interesting, although statistically less certain Ž; 80% confidence., that the non-zero value of RrR 0 in this work suggests there may be some partial retention of the SiF polarisation. At a qualitative level, the significant depolarisation of the SiF B 2 Sq state is intuitively consistent with our previous conclusions, based on the rotationally hot product state distributions w23,24x, that a large D J is imparted impulsively in this process. Put simply, if D J adds randomly to the initial J, it will tend to degrade the alignment of the product J X to an extent that depends on the relative magnitudes J and D J. This idea can be pursued more quantitatively through models based on different dynamical approximations. We start with probably the simplest and least constrained model. It is assumed that the initial J vector is aligned and of fixed magnitude. The D J vector is also of fixed magnitude, and isotropically distributed in space in the usual, spherical sense that the normalised distribution function of D J relative to J is w Ž J, D J . defined by w Ž J, D J . d V s
5. Discussion In the discussion which follows we take R 0 to be the observed polarisation of the C–X fluorescence returning from the laser-excited C 2D state, and R to be that of the collisionally produced B–X emission. The B 2 Sq–X 2 P and C 2D–X2 P bands are both perpendicular electronic transitions with the same polarisation properties, and therefore a positive value of RrR 0 corresponds to some retention of the original alignment during the collisional C ™ B transfer process. The experimental value of RrR 0 is deduced, using the values in Table 1, to be 0.33 " 0.27. A principal conclusion from this study is therefore that with a high statistical probability Ž) 98% confidence. RrR 0 differs from unity. This means that the SiF B 2 Sq, ÕX s 0 molecules are not aligned to the same extent as the C 2D, Õ s 0 molecules from which they are produced in collisions with H 2 . We note that this is rather a unique result because in the one previous related study of this type on the Na 2 system, w21x discussed in Section 1, a large degree of retention of the polarisation was inferred. It is also
51
1 4p
dVs
1 4p
sin u d u d f
s 12 sin u d u s 12 d Ž JˆP D Jˆ. .
Ž 6.
The polar angle, u , and cylindrically symmetric azimuthal angle, f , have their usual significance, as indicated in Fig. 4. The value of RrR 0 is then
Fig. 4. Geometrical relationship between the vectors J, D J and X J , defining the polar angles a and u , and the azimuthal angle f .
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
52
obtained from the expectation value of P2 Ž JˆP JˆX . averaged over all angles, R s P2 Ž JˆP JˆX . s P2 Ž JˆP JˆX . w Ž J, D J . d V R0 u ,f Ž 7.
¦
; H
which, using the trigonometric relationship between the angles u and a , becomes
¦P Ž JˆP Jˆ .; X
2
s
1 4
q1
Hy1
3 J q D J Ž JˆP D Jˆ.
ž
2
/
J 2 q D J 2 q 2 JD J Ž JˆP D Jˆ.
y1
0
=d Ž JˆP D Jˆ. . Ž 8. This integral may be manipulated into a form which has the analytical solution w33x
¦P Ž JˆP Jˆ .; X
2
2
s
1 8
ž
2
5 y 3 Ž D JrJ . q
=
Ž D JrJ . q 1 Ž D JrJ . y 1
/
3 Ž Ž D JrJ . y 1 .
2
2 Ž D JrJ .
ln
.
Ž 9.
As expected, this result only depends on the relative magnitudes of D J and J. The degree of depolarisation is plotted as a function of D JrJ in Fig. 5a. The behaviour is intuitively reasonable, with relatively minor degradation of the alignment when D J is significantly less than J, which becomes quite substantial when D J and J are comparable. The alignment asymptotically approaches zero for large D JrJ when the contribution to J X from the isotropic D J vector becomes totally dominant. To compare the predicted extent of depolarisation with experiment it is necessary to assess an appropriate value of the ratio D JrJ. This may be done, at least approximately, by comparing the average magnitude of product rotation, ² J X :, with J, because this relationship is also determined by D JrJ. Hence we also require from the model the value of ² J X :: X
² J :s
Hu ,f J w Ž J,D J . d V q1
Hy1 (ž J
2
q D J 2 q 2 JD J Ž JˆP D Jˆ.
=d Ž JˆP D Jˆ.
²
:
where the trigonometric relationships between J, D J and J X have again been used. Analytical integration Ženforcing a physical condition on the signs of roots. yields the result w33x ² JX:
1 s
J
6 J 2D J
ž Ž JqD J .
3
y Ž JyD J .
3
/ Ž 11 .
which simplifies to ² JX: J
X
s 12
Fig. 5. Predictions of the ‘spherical’ Žsolid lines. and ‘circular’ Ždashed lines. models of collisional depolarisation described in the text, as functions of the dimensionless ratio D Jr J. Ža. Expectaˆ ˆX . averaged over tion value of the depolarisation ratio, P2 Ž JPJ all angles. Žb. Expectation value of the ratio of magnitudes of final X and initial vectors, ² J :r J.
s Ž 1 q 13 Ž D JrJ .
2
.
for D J ( J
Ž 12a.
and ² JX:
/
J
Ž 10 .
s Ž Ž D JrJ . q 13 Ž JrD J . . for D J 0 J .
Ž 12b.
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
These results also, as expected, only depend on relative quantities. The variation with D JrJ is shown in Fig. 5b. In the work in this paper, we excited the SiF C 2D, Õ s 0 molecules at the P1 head in the presence of H 2 Žand some Ar.. We have previously characterised w23x the partially collisionally modified rotational distribution which this creates in the initial state, and found it to have an average rotational quantum number, ² J :, of 25.5. The average of the resulting B 2 Sq, Õ s 0 product rotational state distribution, ² J X :, was measured to be 35.5. Neglecting the fact that the initial rotational state distribution is in reality quite broad, the ratio of final to initial magnitudes is characterised by the ratio ² J X :r² J : ; 1.4. Within this approximation, the connection with the isotropic model may then be made by equating the predicted ² J X :rJ ratio to the observed ² J X :r² J : ratio. Interpolation on Fig. 5b yields a corresponding predicted D JrJ value of 1.08, and hence finally, according to Fig. 5a, a predicted depolarisation ratio P2 Ž JˆP JˆX . of 0.20. This now does fall within the experimental observed range of depolarisation, RrR 0 s 0.33 " 0.27, and lends support to the basic idea that the observed average change in rotational state is consistent with the observed degree of depolarisation. Care must be taken not to over-interpret this comparison, not only because of the relatively broad error limits but also because no attempt has been made to average over the initial distribution of the magnitude of J Žwhich is at least conceivable. or, much more problematically, the unknown distribution of magnitudes D J for a fixed J. It is therefore not reasonable to infer anything significant from the remaining discrepancy between the mean experimental RrR 0 value and the predictions of the spherically isotropic model. However, we note there may well be a valid dynamical reason in general why certain systems will exhibit a larger retention of alignment than this model predicts. In the construction above, it was assumed that the direction of the randomly imparted D J vector would have no special relationship to the initial J vector. However, as has been discussed previously w10x, for a system in which D J is imparted impulsively in a sudden collision between the partner and a linear rotor, the direction of D J is
¦
;
53
dynamically constrained to be perpendicular to the rotor axis, r. The point of impact will be cylindrically symmetrically distributed about r, and therefore D J is evenly distributed on a circle in a plane perpendicular to r. Averaging over all orientations of r in the plane perpendicular to J results in an overall D J distribution which is peaked along Žand against. the direction of J. This situation is mathematically analogous to the well-known w34x forward–backward peaking of product translational vectors in the decomposition of long-lived complexes in molecular beam scattering. In this alternative ‘circularly isotropic’ model, the appropriate distribution function analogous to Eq. Ž6. is w Ž J, D J . d V 1 1 1 s dVs du d f s du 2 2 p 2 p sin u 2p 1 s d Ž JˆP D Jˆ. . 2 p 1 y Ž JˆP D Jˆ.
Ž 13 .
(
Substitution in Eq. Ž7. yields a corresponding analogue of Eq. Ž8. for P2 Ž JˆP JˆX . , for which we have not been able to find a general closed form solution. Nevertheless, this integral may be evaluated numerically 1, with the results added to Fig. 5a. For D JrJ - 1, there is slightly less depolarisation than in the spherical model. This is consistent with the D J vectors being preferentially weighted along the direction of J in the circular model. The curves converge at D JrJ s 1, where P2 Ž JˆP JˆX . s 1r4 Žexactly.. Above D JrJ s 1, the degree of alignment in the circular model remains constant, which might at first sight appear rather remarkable, in contrast to the continued decline in the spherical model. However, this behaviour may be rationalised by considering that even in the limit of very large D J the product J X vectors in the circular model are still evenly distributed perpendicular to r. It is therefore the memory of the anisotropic distribution of r, rather than J, which causes the product state to be aligned. The correct limiting value is simply de-
¦
;
¦
;
1 Numerical integrations were done using the software package Maple V, Release 4, produced by Waterloo Maple.
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
54
duced 2 from the fact that the average Žacute. angle, a , between fixed J and random J X vectors distributed on a circle is pr4, and so cos 2a s 1r2 and P2 Žcos a . s 1r4. This value is achieved whenever the magnitude D J exceeds J, and therefore J X explores all directions in the plane containing J. The value of ² J X : may in turn be obtained for the circular model by substituting the distribution function from Eq. Ž12. in Eq. Ž10.. We have again not found a closed form analytical solution for the resulting integral, but it may also be evaluated numerically to produce the results plotted in Fig. 5b. The dependence of ² J X :rJ on D JrJ is only slightly different from that of the spherical model. Returning briefly to the comparison with our SiF experimental results, the exact value of D JrJ required to reproduce the observed ² J X :r² J : ratio would be less critical in the circular case, because all values of D JrJ ) 1 result in the limiting value of the alignment. Our mean experimental alignment does happen to exceed 1r4, while the interpolated value of D JrJ required to reproduce the substantial product rotation would be well above 1, but we reiterate that we do not wish to infer too much from this because of the significant experimental error bounds. However, an interesting conclusion may be drawn that a system which conforms strictly to this extreme impulsive limit, even when carefully averaged over all contributing combinations of J and D J, must always retain an alignment RrR 0 ) 1r4. The extent to which it will exceed 1r4 will depend on the contribution to the average from collisions with D J - J. If such a system were probed by pumping a low J level on a suitable transition Žpreferably an Rbranch line to maximise the initial alignment w25x. the degree of alignment of the product levels with D J much larger than J would indeed be a sensitive diagnostic of the impulsiveness of the mechanism. More generally for larger J, there should be a correlation between the alignment of J X and its magnitude, because, as we have argued previously w23,24x, and is inherent in the models in the current work, the
2
Compare similar arguments in the discussion of the limiting high-J polarisation of different spectroscopic branch types in Ref. w24x.
highest J X values require D J to be parallel to J. Although we have not yet achieved this level of rotational state specificity in the present study, we hope to have demonstrated that the results of polarisation measurements can usefully complement scalar properties of electronically inelastic collisions.
6. Principal conclusions Rotationally aligned SiF C 2D, Õ s 0 molecules have successfully been produced by linearly polarised laser excitation. The degree of retention of the alignment during transfer to the B 2 Sq, ÕX s 0 level in collisions with H 2 has been measured. The value of RrR 0 s 0.33 " 0.27 implies significant depolarisation during this process, which is qualitatively consistent with the relatively large changes in J which have previously been measured. Two alternative models for the extent of depolarisation have been developed. A dynamically unconstrained, spherically isotropic distribution of D J relative to J leads to the expected monotonic decline in product alignment as the ratio D JrJ increases. However, in an alternative dynamical limit where D J is constrained to be perpendicular to the molecular axis r, the degree of alignment never declines below 1r4 of that of the initial state. This observation has general implications for processes of this type where D J is imparted in a sudden, impulsive fashion, although the results of the present experiments are not able to distinguish between these cases.
Acknowledgements We are grateful to the EPSRC for an equipment grant and for studentships for NJ and CR, and to Shell Research Ltd. and The Nuffield Foundation for additional financial support.
References w1x P.J. Dagdigian, Annu. Rev. Phys. Chem. 48 Ž1997. 95. w2x R.B. Kurzel, J.I. Steinfeld, J. Chem. Phys. 56 Ž1972. 5188. w3x H. Kato, ˆ S.R. Jeyes, A.J. McCaffery, M.D. Rowe, Chem. Phys. Lett. 39 Ž1976. 573.
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55 w4x M.D. Rowe, A.J. McCaffery, Chem. Phys. 34 Ž1978. 81. w5x M.D. Rowe, A.J. McCaffery, Chem. Phys. 43 Ž1979. 35. w6x J. Husain, J.R. Wiesenfeld, R.N. Zare, J. Chem. Phys. 72 Ž1980. 2479. w7x G.W. Loge, J.R. Wiesenfeld, Chem. Phys. Lett. 78 Ž1981. 32. w8x S.J. Silvers, R.A. Gottsho, R.W. Field, J. Chem. Phys. 74 Ž1981. 6000. w9x P.M. Doherty, D.R. Crosley, Appl. Opt. 23 Ž1984. 173. w10x A.J. McCaffery, M.J. Proctor, B.J. Whitaker, Annu. Rev. Phys. Chem. 37 Ž1986. 223. w11x A.J. McCaffery, J. Phys. Chem. 91 Ž1987. 5451. w12x L.D. Snow, R.N. Compton, J.C. Miller, J. Chem. Phys. 88 Ž1988. 1652. w13x C.D. Pibel, C.B. Moore, J. Chem. Phys. 93 Ž1990. 4804. w14x G.H. Sha, W.Z. Sun, J. Bo, E. Hintsa, C.H. Zhang, J. Chem. Phys. 98 Ž1993. 9487. w15x G.O. Sitz, R.L. Farrow, J. Chem. Phys. 101 Ž1994. 4682. w16x R.B. Bernstein, D.R. Herschbach, R.D. Levine, J. Phys. Chem. 91 Ž1987. 5365. w17x J.P. Simons, J. Phys. Chem. 91 Ž1987. 5378. w18x P.L. Houston, J. Phys. Chem. 91 Ž1987. 5388. w19x M. Brouard, J.P. Simons, in: K. Liu, A. Wagner ŽEds.., The Chemical Dynamics and Kinetics of Small Radicals, Part II ŽWorld Scientific, Singapore, 1995. p. 795. w20x A.J. Orr-Ewing, R.N. Zare, in: K. Liu, A. Wagner ŽEds.., The Chemical Dynamics and Kinetics of Small Radicals, Part II ŽWorld Scientific, Singapore, 1995. p. 936. w21x A.G. Astill, A.J. McCaffery, S.C. Taylor, B.J. Whitaker, M.J. Wynn, J. Chem. Phys. 89 Ž1988. 184.
55
w22x C.W. Watson, K.G. McKendrick, Chem. Phys. 187 Ž1994. 79. w23x N.A. Jackson, C.W. Watson, K.G. McKendrick, Chem. Phys. Lett. 243 Ž1995. 564. w24x N.A. Jackson, C.J. Randall, K.G. McKendrick, J. Phys. Chem. 102 Ž1998. 51. w25x R.N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics ŽWiley, New York, 1988.. w26x R. Altkorn, R.N. Zare, Annu. Rev. Phys. Chem. 35 Ž1984. 265. w27x R.G. Gordon, J. Chem. Phys. 45 Ž1966. 1643. w28x S. Singleton, K.G. McKendrick, R.A. Copeland, J.B. Jeffries, J. Phys. Chem. 96 Ž1992. 9703. w29x S. Singleton, K.G. McKendrick, J. Phys. Chem. 97 Ž1993. 1389. w30x C.W. Watson, K.G. McKendrick, Chem. Phys. 187 Ž1994. 87. w31x G. Herzberg, Molecular Spectra and Molecular Structure, Vol. I, Spectra of Diatomic Molecules ŽVan Nostrand Reinhold, 1950; reprinted by Krieger Publ., Florida, 1989.. w32x Y. Houbrechts, I. Dubois, H. Bredohl, J. Phys. B 13 Ž1980. 3369. w33x M. Abramowitz, I. Stegun ŽEds.., Handbook of Mathematical Functions ŽDover, New York, 1965.. w34x R.D. Levine, R.B. Bernstein, Molecular Reaction Dynamics and Chemical Reactivity ŽOxford University Press, New York, 1987..
Polarisation effects in electronically inelastic collisions: SiF C 2D q H 2 ™ SiF B 2 Sq q H 2 Neil A. Jackson, Colin J. Randall, Kenneth G. McKendrick
)
Department of Chemistry, The UniÕersity of Edinburgh, King’s Buildings, Edinburgh EH9 3JJ, UK Received 6 March 1998
Abstract Linearly polarised laser excitation was used to prepare aligned, electronically excited samples of SiF molecules. Selected, spectroscopically isolated bandheads in the B–X and C–X systems were excited, and appropriate bands resolved in the dispersed fluorescence. The observed degree of polarisation anisotropy, R, of B–X Ž3,2. fluorescence from the directly excited SiF B 2 Sq, Õ s 3 state agreed, within experimental error, with the theoretical prediction. The value of R for C–X Ž0,0. fluorescence from the SiF C 2D, Õ s 0 level excited in the presence of H 2 Žand Ar carrier. was reduced from the theoretical prediction, consistent with previous observations of partial rotational redistribution under these conditions. As also previously established, collisions with H 2 induce SiF C ™ B transfer. The SiF B 2 Sq, ÕX s 0 product was found to be significantly less polarised than the SiF C 2D, Õ s 0 population from which it was produced. Two simple classical models were developed for the degree of depolarisation based on alternative assumptions about the spatial distribution of the additional angular momentum, D J, imparted during the collision process, relative to the initial J. Both predict a degree of depolarisation consistent with the experimental observations in this case. We note that there exists in principle an impulsive dynamical limit in which the product state will remain partially aligned regardless of the relative magnitudes of J and D J. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Electronically inelastic collisions play an important role in many gas phase processes of practical and fundamental interest. It has been known for a considerable time that non-reactive collisions may efficiently remove molecules from electronically excited states. Only relatively recently, however, have detailed investigations been possible of the fate of the quenched molecules, including the distributions over rovibrational levels of the product electronic states. Current progress in such state-to-state studies
)
Corresponding author. E-mail: [email protected]
of diatomic molecules has been reviewed recently by Dagdigian w1x. In another somewhat distinct research area, there is a quite extensive literature on the use of optical polarisation methods to determine the extent to which the plane of rotation Žor, quantum mechanically, the M quantum number. is conserved during rotationally inelastic collisions within single electronic states w2– 15x. There has also been very substantial parallel effort in the development of polarisation methods in the field of ‘dynamical stereochemistry’, broadly encompassing molecular photodissociation dynamics and reactive bimolecular collisions w16–20x. However, very little work has been done in the overlap between these fields, the retention of polari-
0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 0 1 0 4 Ž 9 8 . 0 0 1 5 3 - 0
46
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
sation during collisional transfer between different electronic states. As far as we are aware, the only significant report of this kind is from McCaffery’s group w21x on electronically inelastic collisions of Na 2 . They observed that the collisionally produced 1 q emission on the A 1 Sq u –X Sg band was negatively polarised relative to the plane of polarisation of the laser used to excite the B 1 P u –X 1 Sq transition. g They concluded that this was the result of an indirect mechanism in which molecules were initially collisionally transferred to the Ž2. 1 Sq g state and then radiatively transferred to the A 1 Sq u state. Because of the overall combination of parallel and perpendicular electronic transitions involved, it was inferred from the sign and magnitude of the observed polarisation that there was a high propensity to conserve the plane of rotation Žor M . during P ™ S collisional transfer. We suggest that polarisation techniques have so far been under-exploited in this context, and potentially provide an additional diagnostic of the mechanisms. Our aim in this paper is to apply these methods to collision-induced SiF C 2D ™ B 2 Sq transfer, for which we have previously examined various aspects of the scalar partitioning of energy in the products w22–24x. The SiF C 2D and B 2 Sq excited states are subject to a zero point energy gap of ; 5000 cmy1 . Rather surprisingly, we have discovered that C ™ B transfer is nevertheless dominated by DÕ s 0 transitions, essentially independent of the collision partner, with the detailed vibrational propensities closely matching the C–B Franck–Condon overlap w22x. Subsequently, we established w23,24x, that a considerable fraction of the large energy defect was converted to SiF B 2 Sq product rotation, which is also relatively unusual for a process of this type w1x. We have argued that the energy partitioning is consistent with a sudden, impulsive mechanism, which may be associated with the valence–Rydberg nature of the C 2D–B 2 Sq collisioninduced transition w22–24x. We therefore set out to determine, in three stages, whether the degree of retention of polarisation during this process is consistent with our previous conclusions about the mechanism. The first stage, which was not strictly necessary from a theoretical point of view, was to use a linearly polarised laser to excite SiF directly to the B 2 Sq state and to disperse the
laser-induced fluorescence ŽLIF. on returning B–X bands. The measured polarisation ratios on known combinations of branches in absorption and emission could be compared with well-established theoretical predictions w25x and consequently were used as a preliminary test of the correct functioning of the apparatus. In the second stage, SiF molecules were excited to the C 2D state and the polarisation of the returning C–X emission determined. As explained more fully below, we anticipated that there might be complications due to some collisional depolarisation of the C 2D state and partial optical saturation of the strong C–X Ž0,0. transition w26x. It would be necessary to know the extent of these combined effects to interpret the final stage, where population was excited initially to the C 2D state and the polarisation of the emission from the B 2 Sq state populated in collisions with H 2 measured.
2. Theoretical basis of polarisation measurements The polarisation dependence of optical absorption and emission by isolated molecules has been thoroughly treated by Zare w25x. In essence, in the lowpower limit, the probability of a molecule being excited from an initial state, < i :, to an excited state, < e :, depends on the square of the projection of the linear polarisation vector, Eˆa , onto the transition dipole moment vector, m ei . Similarly, the probability of detecting a fluorescence photon emitted on a transition from < e : to a final state, < f :, depends on the square of the projection of the corresponding transition dipole, m e f , onto the detected polarisation vector, Eˆd . The extent to which the linear polarisations of the absorbed and emitted photons are correlated is normally measured by observing the relative intensities of the fluorescence, I 5 and I H , which correspond to Eˆd parallel to and perpendicular to Eˆa , respectively. Of the possible alternative measures of the fluorescence anisotropy w25x, we shall use R, defined by Rs
I5 y IH I5 q 2 IH
.
Ž 1.
R corresponds to the average degree of alignment relative to an isotropic sample. It is directly related
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
47
to the average angle between absorption and emission dipole moment vectors, through
3. Experimental
R s 25 P2 Ž m ˆ ei P mˆ e f .
The apparatus was similar to that used in our previous studies, w22–24,28–30x with the additional capability of polarisation-sensitive detection of fluorescence signals. Ground state SiF X 2 P radicals were generated in a flow system by microwave discharge of a dilute mixture of SiF4 in Ar. A second independent flow of the collision partner, which in this study was confined to H 2 , was added downstream of the discharge and upstream of the observation zone. The probe laser beam passed through a calcite linear polariser immediately prior to entering the observation zone Žvertically. via a Brewster’s angle window. We denote the laser propagation vector by k a . The polarisation of the beam, Eˆa , was chosen to be parallel to the flow tube axis. The SiF molecules were selectively excited on either the B 2 Sq–X 2 P or C 2D– X 2 P transition as appropriate to the desired measurement. Fluorescence mutually perpendicular to k a and to Eˆa was collected and passed through a photoelastic modulator ŽPEM. ŽHinds International PEM-80.. The principal axis of the PEM was set at 458 to the vertical, bisecting k a and Eˆa . The timing of the laser pulse relative to the phase of the PEM was used to control whether it acted either as a transparent optic or as a half-waveplate, rotating the plane of linear polarisation of all components of the fluorescence by 908. The light subsequently passed through a second, analysing calcite polariser with its polarisation axis set horizontally and therefore fixed parallel to Eˆa . Depending on the phase of the PEM this polariser would therefore transmit the parallel, I 5 , or perpendicular, I H , components of the fluorescence. The transmitted component was focused onto the entrance slit of a monochromator ŽHilger and Watts, Monospek 1000, 1 m focal length. and the dispersed intensity detected by a photomultiplier tube ŽEMI 9558QB or EMI 9789QB.. This method eliminated any potential problems which might otherwise have been associated with the polarisation dependence of the transmission efficiency of the monochromator or with physical realignment of mechanically rotated passive polarising optics. Signals were preamplified ŽEMI A2 or Stanford Research Systems SR240. and captured by a tran-
¦
;
Ž 2.
where the function P2 Ž x . is, as usual, the secondorder Legendre polynomial in x. Exact expressions for R for electronic transitions of linear molecules have been derived w25x. The results are J-dependent, and determined by the type of transition Žparallel or perpendicular. and the combination of branch types ŽP, Q, or R. in absorption and emission. They converge on various limiting high-J values. An insightful way of understanding these is to appeal w17x to the azimuthally averaged addition theorem ŽAAAT. for two otherwise uncorrelated distributions sharing a common axis of cylindrical symmetry, allowing R to be expressed as R s 25 P2 Ž m ˆ ei P Jˆ.
;¦P ž JˆP mˆ /;.
¦
2
Ž 3.
ef
Geometric considerations allow the values of
² P2 Ž mˆ P Jˆ. : to be deduced for different branch types w25x. The extension of this treatment to include collisions or other dynamical processes which might reorient the plane of rotation of the excited state, < e :, prior to re-emission of the fluorescence, was first presented by Gordon w27x. Equivalent results to Gordon’s treatment may be derived by a further application of the AAAT, fitting in with the framework above: R s 25 P2 Ž m ˆ ei P Jˆ.
;¦P Ž JˆP Jˆ .;¦P ž Jˆ P mˆ /;.
¦
X
2
X
2
ef
Ž 4. The quantity Ž JˆP Jˆ . is the cosine of the ‘tipping angle’, a , between initial and final rotational vectors. It is often convenient to isolate the dynamical effects on R from those purely to do with the spectroscopy of the isolated molecule w27x. This can be done by defining R 0 as the value of R that would have been obtained via Eq. Ž3. in the absence of collisions. The simple relationship R P2 Ž JˆP JˆX . s Ž 5. R0 X
¦
;
may then be used to extract the value of P2 Ž JˆP JˆX . and hence an average value of the tipping angle, ² a :, if desired w2,27x.
¦
;
48
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
sient digitiser ŽDSP Technologies 2001A, 100 MHz. incorporated in a CAMAC ŽIEEE 583. modular data collection system. Other experimental variables, such as scanning the laser or monochromator wavelengths and timing the firing of the laser, were controlled via appropriate CAMAC modules. This allowed, for example, spectra of the parallel and perpendicular components of the fluorescence to be collected on an alternate shot basis, so that long term drifts in the conditions would affect both components equally. Gases were obtained from the following manufacturers, and used without further purification: SiF4 ŽUnion Carbide, 99.99% and Fluorochem, 99.5%., Ar ŽBOC, 99.998%., H 2 ŽBOC, 99.99%..
4. Results 4.1. Directly excited B–X emission In principle, polarisation measurements could be made using a combination of any one of three branch types in absorption ŽP≠, Q≠, or R≠. and emission ŽPx, Qx, or Rx.. However, details of the spectroscopy of the B 2 Sq–X 2 P transition combined with the finite resolution of the exciting laser and, more severely, the detecting monochromator greatly limited the number of viable combinations in practice. It proved most convenient to select the B 2 Sq– X 2 P Ž3,0. band for these measurements because it was a sufficiently strong transition lying within the wavelength region accessible with the same laser dye as the C–X Ž0,0. band. A typical LIF excitation spectrum of the B–X Ž3,0. band is shown in Fig. 1. There are altogether 12 distinct spectroscopic branches for a 2 Sq– 2 P transition w31x. ŽWe shall use the contracted notation B–X 1r2 and B–X 3r2 to label branches connected to the well-separated 2 P 1r2 and 2 P 3r2 components of the ground state.. At the cost of not preparing a single J level in the upper state, there is a considerable advantage in exciting at a bandhead to maximise the overall signal in subsequent dispersed fluorescence measurements. This turns out not to be a severe limitation for the SiF B–X Žand C–X. bands because the relevant heads form for levels where the polarisation properties are approaching the limiting high J values and are relatively independent of J. Only the P1 and O P12 band-
Fig. 1. SiF B–X Ž3,0. LIF excitation spectrum. The locations of all 12 branches for 28 Si19 F are indicated. The features marked Ž). are due to minority isotopes of Si.
heads in the respective B–X 1r2 and B–X 3r2 subbands are suitable because the other features in the spectrum are blends of branches of more than one type. Of these two, the P1 head is preferable because of the larger thermal population of the lower-lying F1 ground state levels. In this case, the bandhead forms at J ; 20.5. Fig. 2 shows typical dispersed spectra of the corresponding components of the fluorescence returning on the B 2 Sq–X 2 P 1r2 Ž3,2. sub-band. The Ž3,2. band was chosen because it has the largest Franck–Condon factor w30x of bands originating in B 2 Sq, Õ s 3. The P≠Px and P≠Rx combinations show an enhancement of I 5 over I H , whereas the reverse is true for P≠Qx, consistent with simple physical arguments for a perpendicular transition. Because of the very short B 2 Sq state lifetime Ž- 10
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
Fig. 2. Parallel Ž- -`- -. and perpendicular Ž —v — . components of dispersed SiF B–X 1r 2 Ž3,2. fluorescence generated by pumping the B–X Ž3,0. P1 bandhead. In the absence of collisions, only the three branches indicated are present, as discussed in the text.
ns. we did not anticipate collisional depolarisation to compete with directly returning B–X emission. Therefore, only the three branches indicated in Fig. 2 contribute to the spectra because only the F1 levels in the B 2 Sq state are populated by pumping on the P1 branch.
49
Statistical analysis of repeated measurements of the type shown in Fig. 2 yielded polarisation ratios, R, for the different branch combinations. Signals were integrated over the wavelength range spanning each branch, starting from the minimum point between partially resolved branches. We were careful to avoid significant optical saturation of the absorption step, an aim which was helped by the relatively weak Franck–Condon factor for the Ž3,0. band w30x. As will be explained below, we were even more severely limited in the combinations of branches which could be used in the second and third stages of the measurements, so that we concentrate throughout on the numerical values of R for the P≠Px combination. The results are included in Table 1, together with the theoretical prediction. The agreement is seen to be good, which we regarded as confirmation of the validity of the experimental approach. In a less extensive analysis, qualitatively correct signs but quantitatively slightly less satisfactory magnitudes for the P≠Qx and P≠Rx combinations were obtained: we believe this may be due to the less sharply defined resolution of the R 1 and Q1 branches in the fluorescence spectrum. 4.2. Directly excited C–X emission The branch structure of the C 2D–X2 P bands is qualitatively similar to that of the B 2 Sq–X 2 P bands w32x, disregarding the unresolved splitting of all bands due to L-doubling in the C 2D state. Typical C–X LIF excitation spectra may be found in several of our previous reports w23,24,30x. In summary, it is again found that a P1 head is the most suitable for pumping
Table 1 Observed and predicted polarisation ratios, R, for P≠Px combinations of SiF bands P-type transition excited B–X 1r 2 Ž3,0. C–X 1r 2 Ž0,0. C–X 1r 2 Ž0,0. a
Polarisation ratio, R observed B–X 1r2 Ž3,2. C–X 3r2 Ž0,0. B–X 3r2 Ž0,0.
high J limit 1r10 1r10 y
predicteda b
0.0861 0.0918 c y
Predicted using the expressions in Ref. w25x. Assuming J ; 20.5 excited at the P1 bandhead of the B–X Ž3,0. band. c Assuming J ; 34.5 excited at the P1 bandhead of the C–X Ž0,0. band. d Error limits correspond to 1s uncertainties based on statistical analysis of repeated measurements. e In the presence of H 2 Ž1.5 Torr. and Ar Ž1 Torr.. b
observedd 0.089 " 0.033 0.060 " 0.010 e 0.020 " 0.016 e
50
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
a substantial population of molecules to the C 2D state on a branch of a pure spectroscopic type. The P1 head of the C–X Ž0,0. band forms at J ; 34.5. In a similar way to that described for the B–X transition, we then investigated the polarisation of fluorescence on the directly returning C 2D– X2 P 3r2 Ž0,0. sub-band following excitation at the P1 head in the C–X 1r2 sub-band. The emission and excitation wavelengths are well separated because of the relatively large Ž A ; 160 cmy1 . spin–orbit splitting in the X 2 P state. The situation is complicated, however, by the presence of sufficient H 2 to induce a substantial fraction of collisional transfer to the B 2 Sq state during the longer C 2D state lifetime Ž101.8 " 1.4 ns. w24x. We have shown previously that inelastic rotational redistribution within the C 2D Õ s 0 level is competitive with the inter-electronic state process w23,24x. This work also revealed that collisional transfer between F1 and F2 levels competes with rotational redistribution within the F1 levels. Consequently, the returning C–X 3r2 sub-band has some intensity on all six branches, and the only one which remains spectroscopically isolated is the O P12 branch to longest wavelength. As a result the only meaningful polarisation measurements we were able to make were for the P≠Px combination. The potential effects of optical saturation were again a further concern w26x. The C–X Ž0,0. band has a relatively high Franck–Condon factor, enhancing the likelihood of saturation, although this is mitigated by the weaker C–X electronic transition probability w30x. Operationally, we varied the laser pulse energy and simultaneously monitored the linearity of the total signal and any trends in the observed polarisation. Despite some significant curvature in the total signal, the polarisation ratios were statistically invariant for pulse energies up to ; 500 mJ, although there was increasing scatter about the mean polarisation at low pulse energies. The results reported in Table 1 represent averages over the pulse energy range ; 50–500 mJ. The value of R s 0.060 " 0.010 is significantly reduced from the prediction Ž R s 0.0918. for the P≠Px combination at the average J of the P1 head. We believe this reduction is primarily the result of collisional depolarisation, with a possible contribution from partial optical saturation. Nevertheless, we concluded that we had prepared a significantly polarised C 2D state sample and
could therefore proceed with the inter-electronic state measurements in Section 4.3. 4.3. Collisionally produced B–X emission Finally, we turn to the ultimate aim of this work which was to determine the extent of any retention of the plane of polarisation of the SiF B 2 Sq state produced from the C 2D state in collisions with H 2 . As described in Section 1, we have demonstrated previously w22x that the major product channel starting from SiF C 2D, Õ s 0 is B 2 Sq, ÕX s 0, despite the relatively large vibronic energy gap involved. The product B–X emission is therefore best monitored on the Ž0,0. band. Both F1 and F2 levels of the B state are populated in the collisional transfer process w24x and so only the P-type branches to longest wavelength in each sub-band have the potential to be used
Fig. 3. Collisionally produced parallel Ž- -`- -. and perpendicular Ž —v — . components of dispersed SiF B–X 3r 2 Ž0,0. fluorescence generated by pumping the C–X Ž0,0. P1 bandhead in the presence of H 2 Ž1.5 Torr. and Ar Ž1 Torr.. The head-forming branches are indicated Žfor a full description of the branch structure see Fig. 1..
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
for polarisation measurements. Furthermore, the rotational populations in the B 2 Sq, ÕX s 0 state are substantially hotter w23,24x than in the C 2D, Õ s 0 state from which they are produced. Therefore, there is a component in the emission spectrum from high-J rotational lines in the B–X 3r2 sub-band which partially underlies the P1 head of the B–X 1r2 sub-band. Consequently, only the O P12 branch at the red end of the B–X 3r2 sub-band remains viable for the determination of the B state polarisation. The corresponding parallel and perpendicular components of the collisionally produced B–X fluorescence are shown in Fig. 3. Interestingly, I 5 slightly exceeds I H in the region of the O P12 head, although the magnitude of any effect is admittedly not large relative to the signal-to-noise ratio. Statistical analysis of repeated scans of this region, under the same experimental conditions as used for the directly returning C–X fluorescence above, yielded the R value included in Table 1. At 0.020 " 0.016, R is statistically significantly reduced from the value for the directly returning C–X fluorescence, but still Žmore marginally. greater than zero.
interesting, although statistically less certain Ž; 80% confidence., that the non-zero value of RrR 0 in this work suggests there may be some partial retention of the SiF polarisation. At a qualitative level, the significant depolarisation of the SiF B 2 Sq state is intuitively consistent with our previous conclusions, based on the rotationally hot product state distributions w23,24x, that a large D J is imparted impulsively in this process. Put simply, if D J adds randomly to the initial J, it will tend to degrade the alignment of the product J X to an extent that depends on the relative magnitudes J and D J. This idea can be pursued more quantitatively through models based on different dynamical approximations. We start with probably the simplest and least constrained model. It is assumed that the initial J vector is aligned and of fixed magnitude. The D J vector is also of fixed magnitude, and isotropically distributed in space in the usual, spherical sense that the normalised distribution function of D J relative to J is w Ž J, D J . defined by w Ž J, D J . d V s
5. Discussion In the discussion which follows we take R 0 to be the observed polarisation of the C–X fluorescence returning from the laser-excited C 2D state, and R to be that of the collisionally produced B–X emission. The B 2 Sq–X 2 P and C 2D–X2 P bands are both perpendicular electronic transitions with the same polarisation properties, and therefore a positive value of RrR 0 corresponds to some retention of the original alignment during the collisional C ™ B transfer process. The experimental value of RrR 0 is deduced, using the values in Table 1, to be 0.33 " 0.27. A principal conclusion from this study is therefore that with a high statistical probability Ž) 98% confidence. RrR 0 differs from unity. This means that the SiF B 2 Sq, ÕX s 0 molecules are not aligned to the same extent as the C 2D, Õ s 0 molecules from which they are produced in collisions with H 2 . We note that this is rather a unique result because in the one previous related study of this type on the Na 2 system, w21x discussed in Section 1, a large degree of retention of the polarisation was inferred. It is also
51
1 4p
dVs
1 4p
sin u d u d f
s 12 sin u d u s 12 d Ž JˆP D Jˆ. .
Ž 6.
The polar angle, u , and cylindrically symmetric azimuthal angle, f , have their usual significance, as indicated in Fig. 4. The value of RrR 0 is then
Fig. 4. Geometrical relationship between the vectors J, D J and X J , defining the polar angles a and u , and the azimuthal angle f .
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
52
obtained from the expectation value of P2 Ž JˆP JˆX . averaged over all angles, R s P2 Ž JˆP JˆX . s P2 Ž JˆP JˆX . w Ž J, D J . d V R0 u ,f Ž 7.
¦
; H
which, using the trigonometric relationship between the angles u and a , becomes
¦P Ž JˆP Jˆ .; X
2
s
1 4
q1
Hy1
3 J q D J Ž JˆP D Jˆ.
ž
2
/
J 2 q D J 2 q 2 JD J Ž JˆP D Jˆ.
y1
0
=d Ž JˆP D Jˆ. . Ž 8. This integral may be manipulated into a form which has the analytical solution w33x
¦P Ž JˆP Jˆ .; X
2
2
s
1 8
ž
2
5 y 3 Ž D JrJ . q
=
Ž D JrJ . q 1 Ž D JrJ . y 1
/
3 Ž Ž D JrJ . y 1 .
2
2 Ž D JrJ .
ln
.
Ž 9.
As expected, this result only depends on the relative magnitudes of D J and J. The degree of depolarisation is plotted as a function of D JrJ in Fig. 5a. The behaviour is intuitively reasonable, with relatively minor degradation of the alignment when D J is significantly less than J, which becomes quite substantial when D J and J are comparable. The alignment asymptotically approaches zero for large D JrJ when the contribution to J X from the isotropic D J vector becomes totally dominant. To compare the predicted extent of depolarisation with experiment it is necessary to assess an appropriate value of the ratio D JrJ. This may be done, at least approximately, by comparing the average magnitude of product rotation, ² J X :, with J, because this relationship is also determined by D JrJ. Hence we also require from the model the value of ² J X :: X
² J :s
Hu ,f J w Ž J,D J . d V q1
Hy1 (ž J
2
q D J 2 q 2 JD J Ž JˆP D Jˆ.
=d Ž JˆP D Jˆ.
²
:
where the trigonometric relationships between J, D J and J X have again been used. Analytical integration Ženforcing a physical condition on the signs of roots. yields the result w33x ² JX:
1 s
J
6 J 2D J
ž Ž JqD J .
3
y Ž JyD J .
3
/ Ž 11 .
which simplifies to ² JX: J
X
s 12
Fig. 5. Predictions of the ‘spherical’ Žsolid lines. and ‘circular’ Ždashed lines. models of collisional depolarisation described in the text, as functions of the dimensionless ratio D Jr J. Ža. Expectaˆ ˆX . averaged over tion value of the depolarisation ratio, P2 Ž JPJ all angles. Žb. Expectation value of the ratio of magnitudes of final X and initial vectors, ² J :r J.
s Ž 1 q 13 Ž D JrJ .
2
.
for D J ( J
Ž 12a.
and ² JX:
/
J
Ž 10 .
s Ž Ž D JrJ . q 13 Ž JrD J . . for D J 0 J .
Ž 12b.
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
These results also, as expected, only depend on relative quantities. The variation with D JrJ is shown in Fig. 5b. In the work in this paper, we excited the SiF C 2D, Õ s 0 molecules at the P1 head in the presence of H 2 Žand some Ar.. We have previously characterised w23x the partially collisionally modified rotational distribution which this creates in the initial state, and found it to have an average rotational quantum number, ² J :, of 25.5. The average of the resulting B 2 Sq, Õ s 0 product rotational state distribution, ² J X :, was measured to be 35.5. Neglecting the fact that the initial rotational state distribution is in reality quite broad, the ratio of final to initial magnitudes is characterised by the ratio ² J X :r² J : ; 1.4. Within this approximation, the connection with the isotropic model may then be made by equating the predicted ² J X :rJ ratio to the observed ² J X :r² J : ratio. Interpolation on Fig. 5b yields a corresponding predicted D JrJ value of 1.08, and hence finally, according to Fig. 5a, a predicted depolarisation ratio P2 Ž JˆP JˆX . of 0.20. This now does fall within the experimental observed range of depolarisation, RrR 0 s 0.33 " 0.27, and lends support to the basic idea that the observed average change in rotational state is consistent with the observed degree of depolarisation. Care must be taken not to over-interpret this comparison, not only because of the relatively broad error limits but also because no attempt has been made to average over the initial distribution of the magnitude of J Žwhich is at least conceivable. or, much more problematically, the unknown distribution of magnitudes D J for a fixed J. It is therefore not reasonable to infer anything significant from the remaining discrepancy between the mean experimental RrR 0 value and the predictions of the spherically isotropic model. However, we note there may well be a valid dynamical reason in general why certain systems will exhibit a larger retention of alignment than this model predicts. In the construction above, it was assumed that the direction of the randomly imparted D J vector would have no special relationship to the initial J vector. However, as has been discussed previously w10x, for a system in which D J is imparted impulsively in a sudden collision between the partner and a linear rotor, the direction of D J is
¦
;
53
dynamically constrained to be perpendicular to the rotor axis, r. The point of impact will be cylindrically symmetrically distributed about r, and therefore D J is evenly distributed on a circle in a plane perpendicular to r. Averaging over all orientations of r in the plane perpendicular to J results in an overall D J distribution which is peaked along Žand against. the direction of J. This situation is mathematically analogous to the well-known w34x forward–backward peaking of product translational vectors in the decomposition of long-lived complexes in molecular beam scattering. In this alternative ‘circularly isotropic’ model, the appropriate distribution function analogous to Eq. Ž6. is w Ž J, D J . d V 1 1 1 s dVs du d f s du 2 2 p 2 p sin u 2p 1 s d Ž JˆP D Jˆ. . 2 p 1 y Ž JˆP D Jˆ.
Ž 13 .
(
Substitution in Eq. Ž7. yields a corresponding analogue of Eq. Ž8. for P2 Ž JˆP JˆX . , for which we have not been able to find a general closed form solution. Nevertheless, this integral may be evaluated numerically 1, with the results added to Fig. 5a. For D JrJ - 1, there is slightly less depolarisation than in the spherical model. This is consistent with the D J vectors being preferentially weighted along the direction of J in the circular model. The curves converge at D JrJ s 1, where P2 Ž JˆP JˆX . s 1r4 Žexactly.. Above D JrJ s 1, the degree of alignment in the circular model remains constant, which might at first sight appear rather remarkable, in contrast to the continued decline in the spherical model. However, this behaviour may be rationalised by considering that even in the limit of very large D J the product J X vectors in the circular model are still evenly distributed perpendicular to r. It is therefore the memory of the anisotropic distribution of r, rather than J, which causes the product state to be aligned. The correct limiting value is simply de-
¦
;
¦
;
1 Numerical integrations were done using the software package Maple V, Release 4, produced by Waterloo Maple.
N.A. Jackson et al.r Chemical Physics 233 (1998) 45–55
54
duced 2 from the fact that the average Žacute. angle, a , between fixed J and random J X vectors distributed on a circle is pr4, and so cos 2a s 1r2 and P2 Žcos a . s 1r4. This value is achieved whenever the magnitude D J exceeds J, and therefore J X explores all directions in the plane containing J. The value of ² J X : may in turn be obtained for the circular model by substituting the distribution function from Eq. Ž12. in Eq. Ž10.. We have again not found a closed form analytical solution for the resulting integral, but it may also be evaluated numerically to produce the results plotted in Fig. 5b. The dependence of ² J X :rJ on D JrJ is only slightly different from that of the spherical model. Returning briefly to the comparison with our SiF experimental results, the exact value of D JrJ required to reproduce the observed ² J X :r² J : ratio would be less critical in the circular case, because all values of D JrJ ) 1 result in the limiting value of the alignment. Our mean experimental alignment does happen to exceed 1r4, while the interpolated value of D JrJ required to reproduce the substantial product rotation would be well above 1, but we reiterate that we do not wish to infer too much from this because of the significant experimental error bounds. However, an interesting conclusion may be drawn that a system which conforms strictly to this extreme impulsive limit, even when carefully averaged over all contributing combinations of J and D J, must always retain an alignment RrR 0 ) 1r4. The extent to which it will exceed 1r4 will depend on the contribution to the average from collisions with D J - J. If such a system were probed by pumping a low J level on a suitable transition Žpreferably an Rbranch line to maximise the initial alignment w25x. the degree of alignment of the product levels with D J much larger than J would indeed be a sensitive diagnostic of the impulsiveness of the mechanism. More generally for larger J, there should be a correlation between the alignment of J X and its magnitude, because, as we have argued previously w23,24x, and is inherent in the models in the current work, the
2
Compare similar arguments in the discussion of the limiting high-J polarisation of different spectroscopic branch types in Ref. w24x.
highest J X values require D J to be parallel to J. Although we have not yet achieved this level of rotational state specificity in the present study, we hope to have demonstrated that the results of polarisation measurements can usefully complement scalar properties of electronically inelastic collisions.
6. Principal conclusions Rotationally aligned SiF C 2D, Õ s 0 molecules have successfully been produced by linearly polarised laser excitation. The degree of retention of the alignment during transfer to the B 2 Sq, ÕX s 0 level in collisions with H 2 has been measured. The value of RrR 0 s 0.33 " 0.27 implies significant depolarisation during this process, which is qualitatively consistent with the relatively large changes in J which have previously been measured. Two alternative models for the extent of depolarisation have been developed. A dynamically unconstrained, spherically isotropic distribution of D J relative to J leads to the expected monotonic decline in product alignment as the ratio D JrJ increases. However, in an alternative dynamical limit where D J is constrained to be perpendicular to the molecular axis r, the degree of alignment never declines below 1r4 of that of the initial state. This observation has general implications for processes of this type where D J is imparted in a sudden, impulsive fashion, although the results of the present experiments are not able to distinguish between these cases.
Acknowledgements We are grateful to the EPSRC for an equipment grant and for studentships for NJ and CR, and to Shell Research Ltd. and The Nuffield Foundation for additional financial support.
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