Limitations of the axial recoil approximation in measurements of moleculardissociationR. M. Wood, Q. Zheng, A. K. Edwards, and M. A. Mangan Citation: Review of Scientific Instruments 68, 1382 (1997); doi: 10.1063/1.1147588 View online: http://dx.doi.org/10.1063/1.1147588 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/68/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Electron ionization and dissociation of aliphatic amino acids J. Chem. Phys. 137, 105101 (2012); 10.1063/1.4749244 Dissociative Electron Attachment to Thymine: Bond and Site Selectivity in Different Molecular Environments AIP Conf. Proc. 901, 137 (2007); 10.1063/1.2727364 Dissociative recombination of dications J. Chem. Phys. 119, 839 (2003); 10.1063/1.1579470 Dissociative recombination and excitation of O 2 + : Cross sections, product yields and implications for studies ofionospheric airglows J. Chem. Phys. 114, 6679 (2001); 10.1063/1.1349079 Measurements of electron-impact-dissociation cross section for neutral products AIP Conf. Proc. 500, 349 (2000); 10.1063/1.1302668

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Limitations of the axial recoil approximation in measurements of moleculardissociation

R. M. Wood, Q. Zheng, A. K. Edwards, and M. A. ManganDepartment of Physics and Astronomy, The University of Georgia, Athens, Georgia 30602

~Received 21 October 1996; accepted for publication 9 December 1996!

The axial recoil approximation holds that when a diatomic molecular ion is formed in a dissociativestate, the atoms produced in the dissociation process will move outward along the straight linedefined by the internuclear axis of the molecule. Analysis of experiments measuring the angulardistribution of Auger electrons emitted by N2 following K-shell ionization of N2 molecules showsthat the axial recoil approximation is not strictly true. Significant corrections must be made for therotation of the molecule during the time of dissociation. Smaller corrections must be made for thethermal distribution of the translational velocities of the target molecules, and for instrumentaleffects. In the analysis of the N2 data, the corrections have the effect of smoothing the predictedangular distribution functions. The amount of the smoothing depends primarily on the temperatureof the target gas and the shape of the potential-energy curve for the N2

21 final state involved in theAuger transition. ©1997 American Institute of Physics.@S0034-6748~97!05103-4#

I. INTRODUCTION

Recent papers from this laboratory1–3 reported measure-ments and analysis of the angular distribution of Auger elec-trons emitted following theK-shell ionization of N2 by fastelectrons. The angular distributions were measured with re-spect to the internuclear axis of the target molecule. In theworks cited, the Auger decays of the N2

1 ions formed in theinitial ionizing collision produced dissociative states of N2

21 .The analysis of the measurements requires some knowledgeof the degree to which the dissociating molecules will rotateduring the dissociation process. This article shows that theaxial recoil approximation,4 which is commonly used in pho-todissociation and collision studies, asserts that the mol-ecules will not rotate at all is not sufficiently accurate toaccount for the data. The rotational corrections depend onthe N2

21 final states, the dissociation time of those states, andthe rotational temperature of the target molecules. Two othersmaller corrections are also discussed.

A thorough description of the experiment is to be foundin Refs. 1 and 3. What is presented here is just what isneeded to explain the rotational effects. The heart of theexperiment is a coincidence measurement of the simulta-neous occurrence of an Auger electron and the N1 fragmentwhich results from the dissociation of the N2

21 produced bythe Auger decay. If the axial recoil approximation werestrictly true, detecting the N1 fragment would determine theorientation of the target molecule at the time of the initialionizing collision. The coincidence measurement wouldtherefore allow measurement of the Auger electron yield at aparticular angle with respect to the internuclear axis of thetarget molecule. Figure 1 illustrates the scattering geometry.The N1 detector is set at angleuB , and the Auger electrondetector is set at angleuA with respect to the incident beamdirection. During measurements of the angular distributions,uB is held fixed anduA is varied so that angleu, the anglebetween the direction taken by the Auger electron and theinternuclear axis, is varied.

II. ROTATIONAL CORRECTION

The rotational correction is necessary because the disso-ciating molecular ions have time to rotate during the disso-ciation process. The physical reason for the rotation is thatthe molecules populating the target gas possess rotationalkinetic energy due to the nonzero rotational temperature ofthe target. The first step in the calculation of the correction isto determine a distribution function which describes theprobability that a molecular ion will rotate through an angleas a function of that angle. The model involves the followingassumptions, the first of which is made less restrictive in thelater stages of the calculation:

~1! The rotating target molecule is in its vibrational groundstate with angular momentumJ57, with its internuclearseparationR being its equilibrium value, and with theradial velocity of the constituent atoms equal to zero.

~2! TheK-shell ionizing collision and the subsequent Augerdecay take place so quickly thatR does not change andthere is a vertical transition to the N2

21 final state.~3! The N2

21 dissociates along the potential-energy curve forthe N2

21 state.~4! As the two N1 ions separate they share equally the ki-

netic energy derived from the decreasing potential en-ergy.

~5! Finally, during separation the angular velocity of the ro-tating molecular ion decreases as governed by the con-servation of angular momentum and the increasing inter-nuclear separation.

A computer code was written to follow the moleculethrough the process just described and calculate the totalangle through which the molecule rotates. The molecular ionis treated as a rotating dumbbell composed of two pointmasses a distanceR apart. When the N2

21 state is firstformed, its rotational energy is given by

1

2I 0v0

25J~J11!\2

2I 0, ~1!

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wherev0 is the initial angular velocity of the molecular ion,and I 0 is its moment of inertia at the instant of formation.The calculation proceeds by allowing a small increase inseparation to take place and then calculating the radial ve-locity of the constituent atoms at the midpoint of the step, thetime taken for the step, the angular velocity at the midpointof the step, and the angle turned through during the step.This process is continued untilR is very large and rotationhas essentially ceased. The total angle turned through forJ57 is the sum of all of the incremental angles.

In the full calculation described below, values ofJ otherthanJ57 are included. The calculation makes use of the factthat the total angle turned through scales approximately likethe square root ofJ(J11). In the experimental apparatusused, the target gas enters the interaction region throughtubes which are 6.35 mm in diameter. Taking the rotationaltemperature of the gas to be 300 K, and using the Boltzmanndistribution function, the probability of having molecularions with any givenJ can be calculated. Folding the prob-ability distribution into the model calculation allows deter-mination of a distribution function which indicates the prob-ability of finding target molecular ions which rotate throughan angle as a function of that angle. In our apparatus themost probable value ofJ is seven, which is why that value isused in the calculation just described.

The model is further improved by calculating the anglesturned through for different values of the initial internuclearseparation and performing a weighted average of thoseangles. The weighting factor is the square of the vibrationalground state wave function evaluated at the chosen internu-clear separation. In the worst case studied so far, the differ-ence between the weighted average value and the value cal-culated for the center of the Franck–Condon region was4.3%.

For example, the Auger transitions discussed in Ref. 1involve the dissociation of three N2

21 final states: the~1pu22!

1Sg1, the ~1pu

22! 1Dg , and the (2su21,1pu

21) 3Pg states. Weused the potential energy curves calculated by O’Neil.5 Ac-cording to the model described in the previous paragraphs, ittakes 396 fs for the dissociating N2

21 3Pg to reach an inter-nuclear separation of 100 bohr. During that time an N2

21

molecular ion havingJ57 rotates through an angle of 6.7°.Figure 2 shows the angle turned through as a function of theinternuclear separation for the N2

21 3Pg example just dis-cussed. As one might expect, nearly all the rotation takesplace while the separation is less than 10 bohr. For the1Sg

1

state the corresponding numbers are 381 fs and 4.6°. For the1Dg state they are 377 fs and 3.9°. The results for the threestates differ because of differing shapes of the potential-energy curves in and slightly beyond the Franck–Condonregion.

If the potential-energy curve possesses a shallow mini-mum which must be traversed during the dissociation, thenthe N1 ions will slow in the vicinity of the local potential

FIG. 3. The distribution function describing the relative probability of theinternuclear axis of the target molecule rotating through the angled duringthe dissociation process. The angle depends on the shape of the potential-energy curve for the final state. The curve shown applies to the N2

21

(2su21,1pu

21).

FIG. 1. A schematic diagram of the scattering geometry~not to scale!. Thedashed line represents the path of the incident electron beam. The N1 de-tector is set at angleuB , and the Auger electron detector atuA with respectto the incident beam direction. Both detectors can be rotated in the planedefined by the projectile beam and detectors.

FIG. 2. The angle turned through as a function of the internuclear separationfor the N2

21 (2su21,1pu

21) 3Pg state. The dissociating molecular ion is as-sumed to haveJ57, and to start at the center of the Franck–Condon regionwith zero radial velocity.

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maximum. As a consequence, the molecular ion has moretime to rotate. Indeed, if the N1 ions have barely enoughenergy to surmount the barrier, or are turned back by thebarrier, then the angle turned through is very large.

Figure 3 shows the distribution of angles for the3Pg

dissociation. The curve shown there is used as a weightingfunction to calculate the rotational correction described be-low. Figure 4 defines the directions and angles of interest.The N1 analyzer and detector is placed on thez axis. TheAuger electron analyzer and detector is placed in thez–yplane at angleu with respect to thez axis. If the axial recoilapproximation were strictly true, the detection of an N1 ionwould signal that the dissociating molecular ion was orientedwith its internuclear axis parallel to thez axis. The coinci-dent detection of an N1 ion and an Auger electron wouldmean that the Auger electron was emitted at an angleu withrespect to the internuclear axis of the parent molecule.

If the target molecule is rotating at the time of the initialionizing collision and subsequent dissociation, then an N1

ion detected on thez axis was not oriented along thez axis atthe time of Auger electron emission. The unit vectorA inFig. 4 indicates one possible orientation of the molecularaxis which is drawn for the case that the molecule will rotatethrough angled during the dissociation process and will as-ymptotically follow thez axis and be detected. The vectorAdoes not necessarily fall in thez–y plane. The family ofpossible vectorsA, which rotate through angled, describe acone of half-angled centered on thez axis.

The unit vectorB in Fig. 4 points in the direction of theelectron detector. The anglea betweenA andB is the angleof interest to the experimentalist measuring the angular dis-tribution of Auger electrons emitted by the molecular target.Unfortunately, only the angleu can be measured directly. If

one computes the inner product ofA and B, one finds arelationship betweena, u, d, andf, wheref is the conven-tionally defined azimuthal angle

cosa5sin d sin f sin u1cosd cosu. ~2!

The circle followed by the tip of unit vectorA asA isallowed to rotate through all possible azimuthial anglesf isthe source of all the N1 ions which rotate through angledand reach the detector located on thez axis. If one considersthe population of parent molecules which rotate throughangles betweend andd1dd, the area element at the tip ofAis

da5sin d dd dw. ~3!

Of course, not all of the molecules oriented along vectorA inFig. 4 will rotate so that the N1 ion resulting from dissocia-tion moves outward along thez axis. Molecular ions whichstart with axes alongA and do not rotate through angledtowards thez axis will not be detected. Furthermore, molecu-lar ions which start with axes alongA and rotate throughangled will be uniformly distributed on a circular ring cen-tered onA. Figure 5 illustrates the geometry. Only a smallportion of the yield produced by the molecules originallyoriented alongA will reach the analyzer and detector.

The fraction of the N1 ions in the ring centered onAwhich are detected depends on the size of the analyzer aper-ture. If that aperture has widthw, then the fraction detectedis equal to the ratio ofw to the circumference of the ring.That is, the fraction is given by

f5w

2p sin d. ~4!

The yield per unit area emanating from the elementda isdetermined by the cross section for the emission of an Auger

FIG. 4. The directions and angles of interest in accounting for the rotationaleffect in the angular distribution of Auger electron. The N1 detector is seton the negativez axis. The unit vectorB in the y–z plane points in thedirection of the electron detector which makes an angleu relative to thepositivez direction. The unit vectorA makes an angled with the positivez direction and indicates one possible orientation of the molecular axis. Theangle betweenA andB is denoted asa.

FIG. 5. The scattering geometry of Fig. 4 redrawn to illustrate the threedimensional nature of the correction process. Only a small portion of themolecules oriented along the unit vectorA will rotate through the angledand end up aligned with thez axis. The ring centered onA indicates all thepossible orientations ofA after the rotation.

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electron at anglea with respect to the internuclear axis. If theyield per unit area is represented byI ~a!, then the yielddetected is

dy5 f I ~a!da. ~5!

Combining Eqs.~3!, ~4!, and~5!, one finds

dy5w

2pI ~a!dd dw. ~6!

The quantityI ~a! is the significant quantity as far as theoverall experiment is concerned. It is a function which de-scribes the angular distribution of the Auger electrons emit-ted in a particular reaction channel. Reference 2 describeshow I ~a! is calculated using a two-center effective potentialexpressed in prolate spheroidal coordinates. If the axial re-coil approximation were strictly true, the angular distributionmeasurements described in Ref. 1 would be compared di-rectly to the calculatedI ~a!. The rotational correction resultsin a function I ~u!, which represents the predicted angulardistribution which should be compared to the measured data.

Returning to the discussion of Eq.~6!, the net yield de-tected with the Auger electron detector at an angle ofu with

respect to thez axis is obtained by integrating overd andf.The integrations are not simple becausea depends onu, d,and f as shown by Eq.~2!. In this work the calculationswere carried out by a computer code which calculated thecontribution to the total yield for a large number off valuesfor each of a large number ofd values. Each of thed valuesis weighted according to the distribution shown in Fig. 3.

III. TRANSLATIONAL CORRECTION

There is a second correction needed because of the non-zero temperature of the target gas. It stems from the thermaldistribution of translational velocities of the N2 target mol-ecules. The effect can be explained in terms of Fig. 4. Sup-pose one of the dissociation fragments is launched along thedirection ofA and acquires a velocityv0 with respect to thecenter of mass of the molecule. The thermal velocityvt ofthe molecule must be added tov0 in order to obtain theresultant velocityvf of the N

1 ion. If vf lies along thez axisthe N1 will be detected. The angled illustrated in Fig. 4 canbe calculated by consideringvt' the component ofvt perpen-dicular to the vectorA and using the equation

tan d5vt' /v0 . ~7!

The component ofvt parallel toA affects the net energy ofthe ion, but the direction the ion takes will be toward thedetector if Eq.~7! is satisfied. The net effect is to allow someN1 fragments which were not originally directed toward theN1 analyzer to reach the analyzer and be detected. The de-tailed analysis of the translational correction is very similarto that for the rotational correction. In particular, the geom-etry illustrated in Figs. 4 and 5 is the same for both correc-tions. The Maxwell–Boltzmann velocity distribution for N2at 300 K is used to calculate a probability distribution for theangled in the calculations.

IV. INSTRUMENTAL CORRECTION

A third correction is needed because of the finite size ofthe slits in front of the positive ion and electron analyzers.The finite sizes permit a spread in the angle of orientation ofthe internuclear axis and the angle of emission of the Auger

FIG. 6. The effects of applying the rotational correction for the N221 3Pg

dissociation process to a step function at 45°. The rise, which describes thesignificance of the correction, is described.

FIG. 7. The effects of applying the translational correction to a step functionat 45°.

FIG. 8. The effects of applying the instrumental correction to a step functionat 45°.

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electron. Figure 1 shows that the nominal angle of ejection ofthe electron relative to the internuclear axis isunom5180°2uA2uB . This is true if the axial recoil approxi-mation is correct and we have both a point source and pointanalyzers. In fact, the finite analyzer slits permit a certainlength of the target along the electron beam to be observedby both analyzers. The length of the target visible to theanalyzers depends on the angles at which the analyzers areplaced. Furthermore the angleu depends on the pathsthrough the slit systems followed by the N1 ion and theassociated Auger electron.

A computer code was written to sample the differentpoints of origin, and the different paths followed by the ionand the electron. For each nominal angleunom, a distributionfunction describing the possible anglesu was calculated.This procedure is carried out for each settinguA of the elec-tron detector.

Each of the analyzers used in this work was preceded byan identical pair of slits. The first slit was 1.016 mm wide,3.68 mm high, and 19.05 mm distant from the center of thetarget. The second slit was 2.54 mm wide, 2.54 mm high,and 44.45 mm distant from the center of the target.

V. COMPARISONS

The relative importance of the three corrections was de-termined by applying each of them individually to a rectan-gular step function arbitrarily placed at 45°. Each correctionsmoothed the sharp step. In order to make quantitative com-parisons, the angular range over which the corrected distri-

bution changed from 10% to 90% of the step size was cal-culated. In the figures which follow the angular range isreferred to as the rise. Figures 6 through 8 present the results.The rotational correction is the largest of the three with a riseof 16.9°. The translational and instrumental corrections areabout the same size, having rises of 8.0° and 7.8°, respec-tively.

It is important to note that the translational and instru-mental corrections are independent of the dissociation path-way, while the rotational correction is strongly dependent onthe specific potential-energy curve governing the dissocia-tion. Figure 6 represents the rotational correction for the N2

21

3Pg dissociation process. The rotational corrections for theother dissociation pathways discussed in this work are 11.5°for the 1Sg

1 state, and 9.7° for the1Dg state.As stated above, the measurement and analysis of the

angular distribution of the Auger electrons emitted in theprocesses involving the three states of N2

21 are discussed inRefs. 1 and 2. The results of the model calculations of theangular distributions and the fully corrected angular distribu-tions are presented there.

1Q. Zheng, A. K. Edwards, R. M. Wood, and M. A. Mangan, Phys. Rev. A52, 3940~1995!.

2Q. Zheng, A. K. Edwards, R. M. Wood, and M. A. Mangan, Phys. Rev. A52, 3945~1995!.

3R. Wood and A. K. Edwards,Accelerator-Based Atomic Physics Tech-niques and Applications, edited by S. M. Shafroth and J. C. Austin~AIP,New York, 1997!.

4R. N. Zare, J. Chem. Phys.47, 204 ~1967!.5S. V. O’Neil ~private communication!.

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