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Vector correlations in the O2(a1Δg, v = 1) fragment

formed in the 265 nm photodissociation of ozoneG. Hancock a , G. A.D. Ritchie a & T. R. Sharples ba Department of Chemistry , University of Oxford, Physical and Theoretical ChemistryLaboratory , Oxford , UKb School of Engineering and Physical Sciences , Heriot-Watt University , Edinburgh , UKAccepted author version posted online: 28 Feb 2013.Published online: 28 Mar 2013.

To cite this article: G. Hancock , G. A.D. Ritchie & T. R. Sharples (2013) Vector correlations in the O2(a1Δg, v = 1) fragment

formed in the 265 nm photodissociation of ozone, Molecular Physics: An International Journal at the Interface BetweenChemistry and Physics, 111:14-15, 2012-2019, DOI: 10.1080/00268976.2013.780104

To link to this article: http://dx.doi.org/10.1080/00268976.2013.780104

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Molecular Physics, 2013Vol. 111, Nos. 14–15, 2012–2019, http://dx.doi.org/10.1080/00268976.2013.780104

INVITED ARTICLE

Vector correlations in the O2(a1�g, v = 1) fragment formed in the 265 nmphotodissociation of ozone

G. Hancocka,∗, G.A.D. Ritchiea and T.R. Sharplesb

aDepartment of Chemistry, University of Oxford, Physical and Theoretical Chemistry Laboratory, Oxford, UK; bSchool of Engineeringand Physical Sciences, Heriot-Watt University, Edinburgh, UK

(Received 14 January 2013; final version received 21 February 2013)

Angular momentum polarisation and translational anisotropy have been measured for the J = 17 and 18 rotational levelsin the O2(a1�g, v = 1) fragment formed from the photodissociation of ozone at 265 nm. The results show similaritiesto those previously measured for the most populated vibrational level, v = 0, produced from photolysis in the Hartleyband. Translational anisotropy parameters are very similar for both fragments, consistent with prompt dissociation from abent intermediate. Angular momentum alignment and orientation, however, are found to be consistently lower for J = 17,compared with J = 18, and the magnitude of the depolarisation is consistent with previous results obtained for the oddand even J fragments in v = 0, produced with the same speed at longer photolysis wavelengths. The results indicate avibrational-state-independent mechanism for the difference in polarisation between even and odd rotational states observedat relatively low fragment speeds.

Keywords: ozone; photodissociation; alignment

1. Introduction

The photolysis of ozone in the Hartley band between 230and 300 nm has been intensively studied in recent years,partly because of the atmospheric importance of the pro-cesses involved [1–3], but also because of the wide rangeof quantum states in the products which are amenable tobeing probed with molecular beam and laser techniques[4–13]. The dominant dissociation channel near the peak ofthe Hartley band is the spin-allowed formation of electron-ically excited molecular and atomic oxygen fragments.

O3(X1A1) + hν → O2(a1�g) + O(1D2) (1)

In this paper we concentrate on one specific feature ofthe O2(a1�g) fragment, namely the information about theformation of its first excited level, v = 1, contained in thevarious vector correlations between the fragment’s angularmomentum and velocity, and the transition dipole momentin the parent molecule.

The vibrational and rotational distributions of theO2(a1�g) fragment formed via a singlet channel have beenmeasured at a range of wavelengths across the Hartley band.Valentini et al. [7] found that the molecular fragment isformed primarily in its vibrational ground state and ob-served little variation in the vibrational distribution over 17photolysis wavelengths between 230 and 311 nm. These ob-servations were found to broadly follow the predictions of a

∗Corresponding author. Email: gus.hancock@chem.ox.ac.uk

model for the dissociation in which the vibrational distribu-tion is determined by Franck–Condon overlap of the initialparent wavefunction and the continuum wavefunction inthe excited state [4,14,15]. The vibrational distribution isexpected to broadly follow the predictions of a Franck–Condon model since the impulse imparted to the molecularfragment acts predominantly perpendicular to the O2(a1�g)bond axis and hence, the vibrational and dissociation coor-dinates may be adiabatically separated; the impulse actingin the upper state does not significantly couple differentvibrational levels in the upper state and the distributionof final states reflects the initial excitation step. These vi-brational distributions have subsequently been reproducedwith good agreement by wavepacket calculations [16,17].

The rotational distribution for each vibrational level, onthe other hand, was peaked at high J and, fascinatingly, thepopulations of the odd J states were found to be significantlylower than for even J. The alternation in the population ofJ state was attributed to a selective depletion of the oddJ O2(a1�g) due to a curve crossing in the exit channel,leading to the formation of triplet molecular and atomicproducts. As the 16O nucleus has a nuclear spin quantumnumber, I, equal to zero, the Pauli principle dictates that theground state of 16O16O (3�−

g ) may only be formed in oddN states. Only odd J fragments undergo the crossing, lead-ing to greater numbers of even J (a1�g) fragments. It wasdemonstrated that the populations of even and odd J statesof 16O18O, for which such restrictions do not apply, did not

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alternate and that furthermore the populations of both statesdropped to the values for odd J 16O16O, strongly suggest-ing that the crossing mechanism accounts for the populationdifference. More recently, however, Baloitcha and Balint-Kurti [18] have suggested that symmetry restrictions on theparent ozone initial and dissociative wavefunction with re-spect to nuclear exchange may be the origin of the even Jpropensity. A complete understanding of the origin of thiseffect has clearly not been achieved.

As well as an alternation in populations of even and oddrotational states, an oscillation has been observed in the vec-tor correlations involving rotational angular momentum Jin the O2(a1�g) fragment in v = 0 [19–22]. Correlationsof J with the transition dipole moment μ, the fragment ve-locity v and the term describing the orientation of J withrespect to v are consistently weaker for odd J fragments thanfor even J. However, there does not appear to be the sameorigin for the two sets of even/odd oscillations with J: thevector correlation oscillations become less pronounced forfragments produced at short wavelengths, i.e. with higherfragment speeds [22]. For example, alternations in polarisa-tion are essentially zero for photolysis near 248 nm, wherethe population alternation observed by Valentini et al. [7]is strongly pronounced.

This paper reports the measurement of vector cor-relations for the vibrationally excited product O2(a1�g,v = 1) from the photolysis of ozone at a single wavelengthof 265 nm. For this minor fragment we are able to studytwo rotational levels, J = 17 and 18, in detail, and findthat the magnitude of the difference in vector correlationsmatches very closely the one previously observed for v =0 fragments of the same speed, obtained by photolysis atlonger wavelengths. The origin of the effect thus appears tobe independent of vibrational state, and more pronouncedat the low fragment speeds for both v = 0 and 1.

2. Experimental

The experimental details have been described in previouspublications [19–23], and are summarised here. The par-ent ozone molecule (in a 50% mix with molecular oxygen)was admitted to the interaction region of a modified Wiley–McLaren mass spectrometer [24] via a pulsed valve (Parkerseries 9, 0.51 mm orifice diameter) operated at 10 Hz. Mildrotational cooling to ca. 140 K (as measured for an equiv-alent beam of NO) is expected under these conditions; theweak signals observed in these experiments precluded sub-stantial cooling. After an adjustable delay of ∼2 ms, allow-ing the molecular beam to reach the source region of thespectrometer, the laser trigger sequence was initiated. Theexperiment used two counterpropagating Nd:YAG pumpeddye laser systems, one of which provided the photoly-sis light at 265 nm (LAS LDL 2051 dye laser pumpedby a Spectra Physics GCR 230-10 Nd:YAG laser) whilethe second (a Sirah CSTR-G-2400 dye laser pumped by a

Continuum Surelite II-10 Nd:YAG laser) probed the frag-ments using resonance enhanced multiphoton ionisation(REMPI) of the O2 (d1�g, v = 2) ←← (a1�g, v = 1) tran-sition at wavelengths near 328 nm. Spectra were also takenof the O2 (d1�g, v = 3) ←← (a1�g, v = 1) transition near318 nm. Ions were detected as a function of arrival timeafter the REMPI pulse using a pair of microchannel platesat the end of a time-of-flight (TOF) tube whose axis wasperpendicular to the laser propagation axis. When mea-surement of polarisation dependence of the TOF signalswas required, one laser polarisation was held fixed using aGlan-Taylor polariser preceded by a half wave plate whilethe other was switched either between linear polarisationsdiffering by 90◦ or between right and left circular polarisa-tions using a photoelastic modulator (PEM). The methodemployed to deal with signals produced by the probe laseralone are different from those employed previously and aredescribed in Section 2.2.

2.1. Extraction of vector correlationsand spectral identification

For a REMPI transition, which probes a single rotationallevel J in O2(a1�g, v = 1), the vector correlations extractedrelate the angular momentum J, the recoil velocity v and thetransition dipole moment in the parent ozone molecule μ.The correlations that we determine are described by bipolarmoments defined within the molecular frame [25,26]. Theβ2

0 (2, 0), β20 (0, 2) and β0

0 (2, 2) moments describe the align-ment of the μ–v, μ–J and v–J vector pairs, respectively,with their limiting values of + 1 and −1/2 corresponding topure parallel and perpendicular distributions, respectively.β2

0 (2, 0), the μ–v alignment, takes the value of one half ofthe more often used translational anisotropy parameter, β

[27]. The β20 (2, 1) moment describes the orientation of J

with respect to an axis perpendicular to the plane definedby μ and v and hence, describes the preferred sense of rota-tion of a diatomic fragment formed from a triatomic parent.Methods of extracting the moments, and of allowing for theeffects of core extraction, higher order vector correlations,laser pulse width and laser bandwidth, have been describedin detail previously [23,28,29].

2.2. Probing O2(a1�g, v = 1) nascent fragments

For dissociation of ozone at 265 nm, some 20% of thefragments are formed vibrationally excited in v = 1 [7,9],but we find that the REMPI signals are considerably lowerthan those from the dominant v = 0 product, indicating thatthe total ionisation probability of v = 1 is (significantly)less than that for v = 0. Overlap of the strong O2(d1�g,

v = n) ←← (a1�g, v = 0) band (hereafter termed as (n,0))and the weaker O2(d1�g, v = n + 1) ←← (a1�g, v = 1)

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Figure 1. REMPI spectrum of O2(a1�g) fragments measured following photolysis of ozone at 265 nm with the probe laser scannedbetween 318.0 and 319.3 nm, containing a number of transitions assigned to the O2(d1�g, v = 3) ←← O2(a1�g, v = 1) and O2(d1�g,v = 2) ←← O2(a1�g, v = 0) bands. All assignments are based on the work of Morrill et al. [30].

(n + 1,1) is an additional problem, and this can be seen inthe spectrum of Figure 1 near 318 nm, where n = 2.

Transitions in the (3,1) band were assigned from theterm values of the (d1�g, v = 3) and (a1�g, v = 1) levelsgiven by Morrill et al. [30]. It can be seen from the spec-trum that most of the lines are blended, and this presentsa problem for the extraction of rotational quantum stateresolved vector correlations. Difference profiles of TOFspectra taken with different polarisations are used to deter-mine several of the values of the bipolar moments, and boththe magnitude and sign of the difference depend upon thebranch being probed [31–33]. For example, in the deter-mination of β0

0 (2, 2), the v–J correlation, the appropriatedifference profile for an S-branch transition is of oppo-site sign to that for an R-branch transition. Blended linespredominate in Figure 1, and the spectral region has limiteduse.

A portion of the (2,1) band near 328 nm is shown inFigure 2. The weakness of the (2,1) band compared with the(1,0) is apparent in Figure 2a, but the expanded version inFigure 2b shows the appearance of several apparently iso-lated S-branch lines, identified as before from the data ofMorrill et al. [30]. As a consequence of the relative weak-ness of the ion signals obtained from these minor productsof the photolysis, it was found necessary to employ higherpump laser powers and a tighter focus than in previous ex-periments. A simple calculation from the product of thephoton fluence in the photolysis beam and the ozone ab-sorption cross section at 265 nm [34] indicates that thedepletion due to the pump laser is complete under the con-ditions of these experiments. Ion signals were found to beindependent of 265-nm photolysis power, consistent with

this conclusion. Such depletion does not seem to be accom-panied by saturation, as the vector correlations were foundto be independent of laser power in the regions probed;the ca. 40 fs upper state lifetime [35] following excitationin the Hartley band is short in comparison with the 10 nsphotolysis pulse, and this can result in depletion withoutsaturation. Structured probe-only signals were found in thiswavelength region, with TOF profiles indicating fragmentswith significantly higher velocities than those from the265-nm dissociation. The quantum state identity of thesefragments is unclear: at 328 nm, the quantum yield ofprocess (1) is low [36–38], and faster O2(a1�g) can beformed in a spin forbidden process, or highly excited groundstate O2 molecules could contribute from the spin-allowedchannel forming triplet products. There was, however, noevidence of these faster fragments (which contribute to thewings of the TOF profile) in the two laser experiments, againconsistent with substantial removal of the ozone sample by265-nm photolysis.

Vector correlations were therefore measured from theunsubtracted two-laser signals in these experiments. Clearlythe above justification for this approach assumes that theprobe-only signal has its origin in only one set of fragmentswhose contribution to the whole of the TOF profile can bededuced from a consideration of the behaviour at long andshort arrival times, and so as a further precaution any vari-ations of the measured vector correlations with photolysisor probe laser power were checked for. While correlationsmeasured for the S(17) and S(18) transitions showed nodependence on laser powers, a significant variation was ob-served for the S(19) transition and hence, only data for theS(17) and S(18) transitions are reported here.

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Figure 2. (a) REMPI spectrum of O2(a1�g) fragments measuredfollowing photolysis of ozone at 265 nm with the probe laserscanned between 327.5 and 330.0 nm containing a portion ofthe O2(d1�g, v = 1) ←← O2(a1�g, v = 0) (2 + 1) REMPIband. Assignments of a number of features to this band are madeon the basis of the work of Morrill et al. [30]. (b) Detail of theregion between 328 and 329 nm containing a number of transitionsassigned to the O2(d1�g, v = 2) ←← O2(a1�g, v = 1) transitionon the basis of term values given by Morrill et al. [30].

3. Results

3.1. Translational anisotropy (β = 2 β20 (2, 0))

Figure 3(a) shows an example of the raw data taken forthe determination of the translational anisotropy parameterβ, probed on the S(18) transition. Measurement of this pa-rameter involved holding the probe beam’s electric vectorat the magic angle (54.7◦) with respect to the TOF axis, andswitching the polarisation of the photolysis laser betweenhorizontal (parallel to the TOF axis) and vertical polari-sations [22]. The change in the shape of the TOF signalsas a function of photolysis laser polarisation is consistentwith the excitation of a parallel transition, as expected forprompt dissociation following B(1B2) ← X(1A2) excitationof ozone in the Hartley band [39].

In Figure 3(b), the normalised and symmetrised dataare presented and the fit shown returns a value of β =1.07 ± 0.08. Errors are determined by a Monte Carlo sam-pling method similar to that employed by Costen et al. [40]and are quoted at the 2σ level. Table 1 shows the datafor the two levels measured through the S(18) and S(17)transitions.

3.2. Angular momentum polarisation

Data from three different pairs of polarisation geometriesare used to determine the three major bipolar moments in-volving the rotational angular momentum [22]. For β0

0 (2, 2),the v–J correlation, the photolysis laser is held at themagic angle (to minimise the effect of the translationalanisotropy upon the shape of the profile) while the probelaser is switched from horizontal to vertical polarisation.For β2

0 (0, 2), the μ–J alignment, the photolysis laser is heldhorizontal, and the probe switched between horizontal andvertical. For the orientation momentβ2

0 (2, 1), the photoly-sis laser polarisation is held at 45◦ to the TOF axis, andthe probe laser switched between right and left circular po-larisation. β2

0 (0, 2) is extracted from the ratio of areas ofthe profiles for horizontal and vertical probe polarisations,whilst for the other two moments the difference profilesare fit to obtain the appropriate moment. Further details arepresented in previous publications [19–23]. Figure 4 showsthe difference profiles and appropriate fits for geometriesused to determine the β0

0 (2, 2) and β20 (2, 1) moments for the

two transitions, S(17) and S(18), and what is immediatelyobvious is the marked dissimilarity of the difference signalsin the two cases.

Table 1 lists the values of the moments extracted fromthe data on the S(17) and S(18) lines. Errors for the β0

0 (2, 2)and β2

0 (2, 1) moments were obtained by Monte Carlo sam-pling while those for the β2

0 (0, 2) moment were determinedfrom the standard deviation of 10 replicate measurementsof the same parameter.

4. Discussion

The stereodynamics observed for the O2(a1�g, v = 1; J =17,18) are now compared with those for the vibrationallycold fragments reported elsewhere [20,22]. Values of thetranslational anisotropy parameter are both within errorbounds of the value of β = 1.18, expected for dissocia-tion from the ozone with the same bent geometry as theground state equilibrium value [39]. As with the ground vi-brational state products [22], the values observed here aresmaller than that of β = 1.46 ± 0.15 measured for O(1D2)formed in coincidence with O2(a1�g, v = 1) by Dylewskiet al. [8] following photolysis at this wavelength due to dif-ferences in the rotational temperatures of the ozone sam-ples photolysed and the averaging of the O(1D2) resultsover the rotational distribution of the molecular cofragment.

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Figure 3. (a) TOF profiles measured as explained in the text to determine β. The photolysis laser power was 0.24 mJ/pulse, theprobe power is 2 mJ/pulse and profiles were averaged over 6000 sweeps. Profiles recorded with the photolysis laser polarised paralleland perpendicular to the TOF axis are shown in black and red, respectively, and the difference between the profiles is shown in blue.(b) Simulated fit (red line) to the difference between these profiles after symmetrisation and area normalisation (black circles) returning avalue of β = 1.07 ± 0.08. The error represents twice the standard deviation of values obtained through Monte Carlo simulation.

However, the fact that the values of β reported here are sosimilar for v = 0 and v = 1 fragments is in good agree-ment with the trend in the O(1D2) results, for which littlevariation in the translational anisotropy with cofragmentvibrational state was observed.

The absolute values of the angular polarisation mo-ments are similar to those observed for the production of thecorresponding v = 0 fragments [22], and in previous pub-lications we have commented on the magnitudes of theseparameters [19,20,22]. Two points, however, can be madehere. First, the overall consistency of the values (and par-ticularly of their signs) compared with those for v = 0 givesconfidence to the assignments of the lines in the weak (2,1)band near 328 nm to S(17) and S(18); assignment of linesto different branches of the two photon transition can beaided through measurements of the polarisation momentsbecause of the marked dependences of the signs of the dif-ference profiles on the magnitude and sign of �J. This useof ‘spectroscopy through dynamics’ has been noted pre-viously [22,23]. For the angular momentum polarisations,although the error bars shown in Table 1 are larger thanthose previously reported for fragments formed in v = 0,the reduction in alignment for J = 17 compared with that

for J = 18 is clear, and mirrors the alternation seen pre-viously for odd and even J. A more pertinent comparison,however, comes from observing the difference in polari-sation for even and odd J as a function of the fragmentspeed. Figure 5 shows that for v = 0 the differences forthe β0

0 (2, 2) and β20 (2, 1) terms tend to zero as the fragment

speed increases to values near 1000 ms−1. Also plotted onthe figure are the present limited data for v = 1, whichappear to follow the same trend – photolysis at 265 nm toproduce v = 1 in the observed rotational levels yields frag-ments of the same speed as would be found for photolysisnear 276 nm to form v = 0, and the differences in the po-larisation behaviour for fragments at 270 nm and 280 nm(the closest speeds in previous measurements to those of thev = 1 fragments measured here) are similar. We have chosento calculate error bars of the differences in the alignment pa-rameters as the maximum and minimum of the differencesbetween the moments as given by the reported 2σ error bars.These necessarily result in large error bars on the data, butthe trends in the values with fragment speed can be seen.

It is clear that the qualitative behaviour observed forformation of the v = 0 fragments is well reproduced in thev = 1 fragments suggesting that the dynamics leading to

Table 1. Values of the translational anisotropy parameter and bipolar moments describing the angular momentum polarisation measuredfor the O2(a1�g, v = 1) fragments. Errors represent twice the standard deviation of values obtained through Monte Carlo simulation orare determined from replicate measurements of the same parameter as described in the text.

Transition β ± 2σ β20 (0, 2) ± 2σ β0

0 (2, 2) ± 2σ β20 (2, 1) ± 2σ

S(17) 1.26 ± 0.10 −0.10 ± 0.10 −0.03 ± 0.12 0.48 ± 0.08S(18) 1.07 ± 0.12 −0.29 ± 0.10 −0.21 ± 0.07 0.65 ± 0.10

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Figure 4. Differences between symmetrised and area normalised profiles recorded to measure the β00 (2, 2) and β2

0 (2, 1) moments (blackcircles) together with the simulated fits to the data (red lines). (a) and (b) show the difference profiles recorded in the polarisation geometryappropriate for extraction of β0

0 (2, 2) for the S(18) and S(17) transitions respectively, and (c) and (d) show the appropriate differenceprofiles for the S(18) and S(17) transitions where the orientation moment β2

0 (2, 1) is to be extracted. Although the intensity scales are inarbitrary units, the relative values for each difference profile are comparable and differences in the magnitudes of the difference profilesreflect differences in the values of the angular momentum polarisation for the even and odd levels.

the formation of these two sets of fragments are very sim-ilar. It would appear, therefore, that the vibrational motionof the O2(a1�g) fragments is decoupled from the factorsinfluencing their translational and rotational anisotropy. Asmentioned above, Valentini and co-workers [7] have shownthat while the rotational distribution of the fragments isreasonably well reproduced by an impulsive model for thedissociation, the vibrational distribution may be treated asbeing determined by the Franck–Condon overlap of thecontinuum wavefunctions corresponding to different prod-uct vibrational states and the initial bound state of the par-ent. The translational anisotropy and angular momentumpolarisation of the O2(a1�g) fragments appear to be largelydetermined by such effects as the impulse imparted to thefragments and the parent internal motion, and hence, it is

not surprising that these properties do not vary with prod-uct vibrational state, since their origin is different to thatdetermining the vibrational distribution. However, somediscrepancies between the vibrationally adiabatic modeland the vibrational distribution were observed, and it ispossible that slight differences in the upper state trajecto-ries between the two sets of products may account for thedifferences in the vector correlations observed in this work.Finally, we note that we have previously commented [22]on a speed-dependent variation in the degree of orienta-tion of the O(1D) product of ozone photolysis as observedby Lee et al. [9]. For the atomic product it is the elec-tronic angular momentum orientation which is probed, anda speed-dependent value has been attributed to the transitionfrom adiabatic to diabatic dynamics [9]: in the present case

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Figure 5. Differences between the β20 (2, 1) and β0

0 (2, 2) mo-ments for adjacent even and odd J states normalised by the averageof the two moments as a function of average fragment speed. Thedata from this paper for the O2(a1�g, v = 0; J = 17, 18) fragmentsare denoted by the filled squares. These are compared with datafor O2(a1�g; v = 0) fragments from previous publications. Datataken probing the J = 18 and 19 fragments following photolysisat 248 nm, 260 nm and 265 nm, together with data taken probingthe J = 33 and 34 fragments following photolysis at 248 nm andfor probing the J = 32 and 33 fragments following photolysis at260 and 265 nm are denoted by the unfilled squares [22]. Datafor probing the J = 18 and 19 fragments following photolysis at270, 280 and 290 nm are denoted by the unfilled circles [20] andmeasurements of the orientation of the J = 18 and 19 fragmentsfollowing photolysis at 301 nm are denoted by the unfilled triangle[21].

the rotational angular momentum of the diatomic fragmentdominates, and it is the difference in orientation values foreven and odd J levels which appears to be speed dependent.However, as even and odd J levels result from different� doublets, a full treatment of electronic symmetry con-straints may elucidate the various speed dependencies nowobserved in the photodissociation of ozone.

We conclude by emphasising that although the datashow clear differences between the angular momentum po-larisation of the J = 18 and J = 17 states, the measurementsare extremely sensitive to these differences; for example,for β0

0 (2, 2) values of −0.2 and 0 the average value of theangle between the velocity and angular momentum vectorsis estimated to be 63.4◦ and 54.7◦, respectively. Measure-ments of the angular momentum polarisation for each ro-tational state as a function of the rotational temperature ofthe parent molecule should further help elucidate the originof these subtle differences. This could be possible for theground vibrational state fragment (where signals are larger)by using a less intense but cooler molecular beam, and thusremoving the effect of parent rotational angular momen-tum on reducing both the correlations involving J and oninfluencing � doublet formation [22]. A further intriguingpossibility is the photolysis of rotationally and vibrationallyselected ozone, now possible with the development of con-tinuous wave high power, single mode quantum cascadelaser operating around 10 μm.

AcknowledgementsT.R. Sharples is grateful to the EPSRC for a studentship. The workwas supported by the EPSRC programme grant EP/G00224X/1:New Horizons in Chemical and Photochemical Dynamics.

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