DOPPLER MEASUREMENT OF DIFFERENTIAL CROSS SECTIONS IN CROSSED BEAM EXPERIMENTS : DEPENDENCE OF THE DETECTION SENSITIVITY WITH THE PRODUCT VELOCITY

Differential cross sections can be measured as a function of the internal state of a reaction product thanks 
to the analysis of the Doppler profile of the laser induced fluorescence detection line. This analysis is 
complicated by two effects: first, the LIF signal intensity depends on the interaction time of the molecule 
with the laser, and this time depends on the scattering angle, second, the angular and velocity distributions 
of the beams have non negligible widths. We present here a treatment of these effects in the case of the 
F


I. INTRODUCTION
Doppler measurement of the velocity of a collision product can be used to measure the differential cross section of a reaction as a function of the internal state of the detected product.This very powerful technique has been used many times in photodissociation experiments z and only in a few collision experiments. 3We have performed such an experiment for the F + Iz reaction and in order to analyse these data, it appears necessary to model carefully the relation between the differential cross section and the Doppler profile of the laser induced fluorescence signal.Two main effects must be taken into account: as soon as the product recoil velocity in the center of mass frame is not small with respect to the velocity of this frame, the detection sensitivity depends noticeably on the direction of the recoil velocity; rathe experiment is not an ideal one, in the sense that the beams have non negligible velocity and angular spreads, and this induces various averaging effects.
-Associated with Ecole Normale Sup6rieure, Universtit6 P. et M. Curie and C.N.R.S. (UA 18).The authors are also belong to GRECO 87 (C.N.R.S.) "Dynamique des R6actions Mol6culaires." The goal of the present paper is to set-up the mathematical formalism necessary to represent as exactly as possible these two effects in the analysis of the experimental data.This formalism is used to establish the relation between the differential cross section and the Doppler profile as well as the total intensity of the LIF signals.

II. DOPPLER PROFILE OF THE LASER INDUCED FLUORESCENCE SIGNAL IN AN IDEAL EXPERIMENT
We analyse here the LIF signal given by a perfect crossed beam experiment, perfect in the sense that we do not consider here velocity distribution or angular spread of the beams.So we only deal with the detection sensitivity versus the direction of the recoil velocity.Previously a special attention is given to the nature of the measured signal.

II.1 Principles
We will particularize immediately to the case of the F + 12 reaction but, except where noted, this will be done without loss of generality, as long as one considers only reaction between an atom and a diatomic molecule.So we consider the following reaction" F+

IF+ I
(1) which is endothermic: AH0 -1.23 eV/molecule.The geometrical arrangement is represented on Figure 1" two molecular beams cross at right angle and we assume Jre I Oeomeldcal aaemet: the I beam (aIo the Ox axis) and the Fbeam coss at dbt aJe.The OP axis is alo the eladve veIocty , and OQ is the decdo peedJcla to OP n the Oxy plae.I ADDS eomety, the lase s colinea to the eIadve velocity, .e. aloB OP ad, i PADDS eomety, I s pependcla o the o beams, Le., aIon Oz.I the centre o mass ame, the I podct velocity is described by its pbedcal coodinate , 9, @ elafive to the am OzQP.
here that their velocity distributions as well as their angular distributions have negligible widths.Accordingly, the centre of mass velocity vG and the relative velocity v do not present any dispersion" VG-" (mI2vI + mFVF)/M (2) with obvious notations (M me + rn).
The available energy for the products, E,,v, is the sum of the reaction endothermi- city and of the collision kinetic energy: where/x is the reduced mass of the reactants (/z mxmM).We have neglected here any internal energy of the reactants.Conservation of energy is not sufficient, by itself, to define the relative velocity of the products, as both products can be excited (the iodine atom can be produced in its 2p1/2 excited fine structure state, but this is a very minor branch of the reaction4).However for IF levels with sufficient internal energy (in our conditions v > 6) the iodine atom is necessarily produced in its ground state and the final relative velocity v has a modulus given by: where E(v, J) is the internal energy of IF in its v, J rovibrational level and/z' is the reduced mass of the products (/z' rnxrn/M).Then the velocity u.F of IF in the level v,J, in the centre of mass frame, has a modulus given by: The collision problem, in the centre of mass frame, presents a cylindrical symmetry around the reactant relative velocity vector yr.Therefore, the velocity Ule, which is distributed on a sphere, is equally probable on each circle of axis OP (OP is parallel to v, and is taken as the polar axis).To take advantage of this symmetry, one must shine the laser along OP.If o0 is the angular frequency at resonance for the LIF transition, the Doppler shifted resonance frequency is given by: Here VGp is the projection of vG on the OP axis, and UtF, 0, tp are the spherical coordinates of the vector uie.The sign appears when the laser propagates in a direction opposite to vr, as in Figure 1.Because tot depends only on 0 (and not on tp), this arrangement, named ADDS, 5 allows a direct measurement of the differential cross section [dtr/dfl] (0) in the centre of mass frame.But because 0 appears only through its cosine, the angular resolution of this measurement is very poor near 0 0 and r.This can be remedied by using a geometrical arrangement in which the laser beam is along the axis Oz, perpendicular to the plane of the molecular beams.In this case, the resonance frequency is given by: w= c0011 +uesin 0 cos ] ( c The angular resolution near 0 0 and r is then good, but in this arrangement (named PADDS), the signal at frequency cot corresponds to a range of 0 values (and not to a single one, as for ADDS).Moreover the signal is only sensitive to the sum [da/dfl] (0) + [da/df] (z 0), i.e., one cannot distinguish backward and forward scattering. 5

II.2 The Measured Signal
Let us now consider the laser induced fluorescence signal.We assume that the laser is a c.w. single frequency laser with a long coherence time.We have already discussed in details the saturation effects in this case. 6e LIF signal, i.e., the number of photons emitted per unit time for a laser frequency WL is: where x(wz o) is the number of fluorescence photons emitted by one molecule, for a laser detuning OL W, and d2/ represents the number of molecules produced per unit time in the (0, 99) direction in the centre of mass flame.Obviously we have: d2/ nlnFVr "--V sin 0 dO dq9 (10)   Here n and ne are the densities of the two beams, v is the collision volume and (do/df) is the reaction differential cross section in the centre of mass frame.
Starting from Eq. ( 9), we discuss now two problems: first, the ultimate velocity resolution of such an experiment and second, the variation of the unimolecular signal with the interaction time between the IF product and the probe laser.

Velocity Resolution
The X(OL C0r) function has been studied in Ref. (6), especially under resonant conditions.This quantity is resonant for 0L 0. The resonance width c%o is proportional to the ultimate velocity resolution of the Doppler profile.In the absence of saturation, it is equal to the natural width F of the excited state, and this corresponds to a velocity resolution 6v (F/wo)C (of the order of 0.02 m/s for the IF B-X transition).Saturation is due to power broadening of the resonance.We consider here only the saturation case which applies to our experiment: where to1 is the Rabi frequency of the transition, and r the interaction time with the laser field (supposed to be constant inside the beam and zero outside).In that case, the saturation width is" &o=eol for Fr.l (12a) &0=w for Fv>>I (12b) In our experimental conditions, we have Fv---1 and wl is in the range lOClO s s-1.The corresponding velocity resolution 3v (wl/Wo) c remains very good (3v 10 m/s).
Variation of the Individual Signal with the Interaction Time Using the same assumption as in Ref. (6), i.e., the resonance width is smaller than any structure of (dcr/df2), expressed in frequency unit, we get: with I x(w) dw.I appears to be proportional to the mean LIF signal produced by one molecule at resonance.If olv -> 1, I is given by: FradVexp (---'-)II0(--)+ II(F-v2)] ( 14) Here Frad is the radiative decay rate of the excited state (different from F if this state suffers predissociation), and I0 and Ix are modified Bessel functions. 8I behaves asymptotically like: I=-o91F for Fv'I (15a) I= -1 for rv->l (15b) (with Frad F).If we consider that the experiment is in the first case, the signal is a linear function of the interaction time v (and this would be also the case for an incoherent laser in the absence of saturation).In any case, I depends on the direction (0, p) of the IF velocity, via the interaction time 3, and the LIF signal is: II. 3 The Doppler Profile We first consider here the ADDS geometry.The laser beam is assumed to be centered on the OP axis, and has a radius rL.For a very small scattering volume centered in O, the interaction time is given by: rL (0, 99) [va + uz sin e 0 + 2VaQ u sin 0 cos q]/e where VGQ is the component of v on the axis OQ.In the two following limiting cases, r is dependent of uv "> Vaa r rL (18a) uiF sin 0 rL uIF ' VaQ r (18b) VGQ and the calculation of the q9 average of r is trivial.We see that in general, the q9 average of r will present a strong dependence on 0, and the detection sensitivity will have the same dependence.For example, in the limiting case U,F>> Vaa (and for Fr, 1, i.e., I r), the Doppler profile s(aL 00) is proportional to: where cos 0 aL o0 .__c0)0

UlF
The same effect acts also in the PADDS geometry.Now the laser beam is assumed to be centered on the OZ axis.The same kind of hypothesis leads, in the limiting case uv-> va, to a Doppler profile proportional to: In addition to the effect discussed above, we are going to take into account the averaging effects introduced by the finite widths of the angular and velocity distri- butions of the molecular beams, as well as the internal state distribution of the Iodine beam.
We represented each molecular beam as a perfect molecular flow emitted by a point source S. The density at a point M near the scattering center O is given by: n(M, v) dv no SM (v) where no is the total density at O. The velocity vector v is assumed to be parallel to the vector SM and the distribution of its modulus v is given by the formula usually accepted for supersonic molecular beams: 9 (C is the normalization constant, u and cr are defined in Ref. ( 9) and can be measured by the time of flight technique).Each beam is collimated (by a circular orifice for the Fluorine beam, by a long slit for the Iodine beam) and we assume that the density vanishes out of the volume defined by S and the collimator.We neglect the possible angular dependence 1 of the density inside this volume; this is a good approximation as the angular width of the collimator of the beam is small.The internal state distribution of the Iodine molecules is characterized by two temperatures, namely a vibrational one Tv, and a rotational one Tr.This non equilibrium distribution is well known in supersonic beams and can easily be characterized by spectroscopic techniques for Iodine. 11The probability of occurrence of the (v", J") rovibrational level is then proportional to: htOV" P(v", J") (2J" + 1) exp v + l) ] (

23) kTr
Ortho-para alternation is here neglected.Hereafter rotation is treated through the continuous variable x J"(J" + 1).
The fluorine atom is also distributed on its two fine structure states (2p3/2 E 0, 2p1/2 E 404 cm-), probably with an equilibrium distribution at the oven temperature (T 1000 K).Because the potential energy surfaces which correlate to these two states are different, it is not realistic to assume that the two types of collision lead to the same differential cross section.We have chosen her to treat only the collision involving the 2p3/2 state.Neglecting the 2p/2 contribution to describe our data is supported by the fact that this state represents at most 20% of the reactants and may be a very good assumption if the corresponding potential energy surface is repulsive or presents a high barrier.
Taking into account these averages, the LIF signal for a laser frequency becomes: n do X P(v", J ) 12(M, vi2 ) dvi2nF(M, VF) dVF Vr (0) d3V sin0 dO dqo (24) It appears as a mean over all the trajectories.r is the interaction time calculated exactly for each trajectory, defined by the point M where the collision occurs, and the IF velocity va + uxF.The initial conditions fix the available energy, and consequently the uxF modulus.The function expresses the condition on the IF velocity along the laser axis which gives the right Doppler shift (as in Eq. 16).
It should be noticed that the time r is not a simple function of the parameters.
Namely three different cases occur: some trajectories begin out of the laser beam and do not go through it (then r 0), some other trajectories, beginning outside the laser beam, go through it, --and finally some trajectories begin inside the laser beam.For the last two cases, the algebraical expressions of the time are given in the appendix.We consider here that the observation zone is large enough to allow the detection of the fluorescence from each molecule along the whole part of its trajectory inside the laser field.
In fact what we are looking for is not to calculate the LIF signal s(oz o0) from (da/dg2), but to invert this relation.This can be done practically by a least square fit for the following reasons: --The signal at frequency o/ appears to be a convolution of the differential cross section by a complicated apparatus function defined by Eq. ( 24).
Din our case, IF B-X transitions present an hyperfine structure 12 which introduces further complications.The observed signal can be written as: S(ooL) [ok s(ooL wk) k where c0, is the frequency of the k th hyperfine component of the transition, and a its weight factor.The differential cross section, and consequently the signal s, must be independent of k if we assume that the hyperfine Hamiltonian is too small to be efficiently coupled during the collision, i.e., if: 14FS" ton " h (26) This is a well known property in collision theory 13 and condition (26) should be fulfilled as the estimated duration of the reactive collision ton is a few picoseconds 4 and the order of magnitude of the hyperfine Hamiltonian due to the Iodine nucleus is ,HFS/h 101 S-1.15

III.2 Signal Simulation
In order to fit the experimental data, we develop the differential cross section (da/dg2) on a polynomial basis in cos0; we have chosen to use Legendre polynomials: do t latp,(co s 0) (27) and we calculate the functions sl(w/ w0) obtained by replacing (da/df2) by Pl(cos 0) in Eq. (24).
These integrals are too intricate to be calculated by means other than the Monte-Carlo method. 6Figure 2 shows different Sl(WZ o0) functions for a IF internal energy Ele 5000 cm -, which have been computed both for ADDS and PADDS geometries.All the input data are collected in Table 1.Each calculation Figure 2 The SI(toL tOo) functions with different values (0 < < 4) obtained for an internal energy EIF 5000 cm-1, in the ADDS (part a) and PADDS (part b) geometries.Sl corresponds to the experimental Doppler profile for a differential cross section proportional to Pl(cos 0).They are plotted here as a function of the dimensionless parameter D (tOz tOo)/(tOo) x (c/(uxF)) where (uxF) is the quadratic mean of the IF velocity in the centre of mass frame, i.e., the IF velocity corresponding to the mean recoil energy.In PADDS geometry, the Sl functions with odd values average to zero, and have not been plotted here.For ADDS geometry, as expected, st looks like PI. However the experiment has a low sensitivity to the molecules whose velocity is perpendicular to the laser: this induces distortion for coz coo, i.e. at the center of the Doppler profile.On the contrary, the internal energy distribution and angular averages smooth the two ends of the profile.In PADDS geometry, the st functions with odd are quite zero, according to the idea that this arrangement cannot distinguish backward and forward scattering.Because in this geometry, the Doppler shift coz COo is not correlated to a single value of 0, even in a perfect experiment, the other curves st (with > 0) are not easy to interpret qualitatively, but they are also affected by the effects described above.This can be clearly seen for l 0 (corresponding to the case of an isotropic differential cross section).
Figure 3 shows the evolution of the Sl--O curves as a function of the IF internal energy, for the two geometries.We verify here that So, which is almost flat for low recoil energy (case 18b), becomes more and more sharply peaked near its extreme values when the recoil energy increases (case 18a).These results show clearly that an analysis of the Doppler profile of the line not taking into account the variation of the interaction time with the velocity direction would give a strongly incorrect differen- tial cross section.

IV. THE INTEGRATED LINE INTENSITY
The integrated line intensity S is naturally used as a measurement of the total cross section.However the effects discussed in Part III affect also this measurement.Some of their consequences have already been discussed in several experimental condi- tions (17 and references therein).Let S be given by: S s(CO/too)dcoz, =/I() d2/ Figure 3 The st(toz.tOo) functions, with 0, obtained for different values of the IF internal energy EzF (curves to 5 correspond to Ew 0, 5000, 7500, 9000 and 10,000 cm-1), both in the ADDS (part a) and PADDS (part b) geometries.So is plotted as a function of the dimensionless parameter D (tOz tOo)/(tOo) x (c)/((ulv)) where (ulv) is the quadratic mean of the IF velocity in the centre of mass frame.
Each calculation involves 2.5 10 6 trajectories with a non zero interaction time and the D axis has been divided in 100 boxes.
10000 EiF(crn ') 1oobo E IF em ") Figure 4 St coeefficients, measuring the sensitivity of the total LIF line intensity to the part of the differential cross section proportional to Pl(COSO), plotted for several values as a function of the IF internal energy, both for ADDS geometry (part a) and for PADDS geometry (part b).The mean total available energy is 10,886 cm-1.
with a non zero interaction time : and 100 boxes have been used on the co axis.

Table 1
Beams and laser parameters used as input data in the calculations presented here.