Differential cross sections for rotationally inelastic ...houston.chem.cornell.edu/JCPPaperNOScattering.pdf · Differential cross sections for rotationally inelastic scattering of

JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 6 8 FEBRUARY 2001

Differential cross sections for rotationally inelastic scatteringof NO from He and D 2

M. S. WestleyDepartment of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853-1301

K. T. Lorenz and D. W. ChandlerCombustion Research Facility, Sandia National Laboratories, Livermore, California 94550

P. L. Houstona)

Department of Chemistry and Chemical Biology, Cornell University, Ithaca, New York 14853-1301

~Received 21 September 2000; accepted 14 November 2000!

State selective differential cross sections for rotationally inelastic scattering of NO (Ji50.5, 1.5,F1→Jf52.5– 12.5,F1 andJf51.5– 9.5,F2) from He and D2 measured by crossed molecular beamproduct imaging are reported. The differential cross sections were extracted from the data imagesusing a new basis image iterative fitting technique. The images typically exhibit a single broadrotational rainbow maximum that shifts from the forward to the backward scattering direction withincreasingDJ. The angle of the rainbow maximum was lower at a givenDJ for D2 than for He asa collision partner. At a collision energy of;500 cm21, primarily the repulsive part of the potentialsurface is probed, which can be modeled with a two-dimensional hard ellipse potential. This modelfor rotationally inelastic scattering is shown to qualitatively match the experimental differentialcross sections. A more advanced correlated electron pair approximation potential energy surface forNO1He does not give substantially better agreement with the experiment. The differences betweenscattering of He and D2 are partially attributed to their differing structure and partially to a smalldifference in collision energy used in the two experiments. ©2001 American Institute of Physics.@DOI: 10.1063/1.1338528#

n-seasArse

aianachegs

y

-hawic

rgsed

en

tions

calR

con-ers.ee-

ter-ularlar

h-NOh

peringcu-r,

forhatitudee

I. INTRODUCTION

Rotationally inelastic scattering of molecules is of fudamental importance in energy transfer of molecular gaIntegral and differential cross sections for rotationally ineltic scattering of NO from collisions partners such as He,NO, and N2 have been extensively studied. Along with theexperimental studies, potential energy surfaces~PES! forNO–Ar and NO–He have been generated and tested agexperimental results. The sheer volume of theoreticalexperimental work done on NO is attributable to several ftors. First, NO is experimentally convenient; it is one of tfew radical species that is stable enough to be stored in acylinder. Next, the spectroscopy of NO makes it very sentive to laser-based detection methods such as resonantlhanced multiphoton ionization~REMPI! or laser-inducedfluorescence~LIF!. Finally, NO is of theoretical interest, partially because of the extensive experimental work thatbeen done, but also because NO is a radical speciesnonzero orbital angular momentum in its ground state, whgives rise to complex potential interactions.

Some early measurements of collision-induced enetransfer in NO were carried out in a bulb using laser-baresonance techniques. For example, Subdo and Loy useIR UV double resonance~IRUVDR! technique to measurintegral cross sections for rotationally inelastic and spiorbit conserving and changing collisions of NO(v52) with

a!Electronic mail: [email protected]

2660021-9606/2001/114(6)/2669/12/$18.00

Downloaded 12 Mar 2001 to 128.253.219.94. Redistribution subject t

s.-,

nstd-

asi-en-

sithh

ydan

–

NO, He, Ar, N2 , CO and SF6.1,2 More recently, Islamet al.have performed studies of integral state–state cross secusing IRUVDR for NO(v52) scattering with He, Ar, andN2 and compared their results with the latest theoretipredictions.3 Other recent studies have used the IRUVDtechnique to look at rotational energy transfer of NO~collid-ing with He, Ar, and N2) at the exit of a Laval nozzle, aspecially shaped nozzle that produces a flow of gas at astant low temperature and pressure over several centimet4

The experimental data were found to be in excellent agrment with the latest theoretical calculations.

Measurements of integral cross sections for NO scating have also been performed using a crossed molecbeam technique. The rotational cooling of the molecubeam expansion produces rotationally cooled NO~typically.90% in J50.5, in F1 , the lower spin–orbit state, withequal L doublet populations! and laser-based probe tecniques can be used to examine specific final states of~including finalL doublet level, depending on which brancis used to ionize the NO!. Also, collisions which cause theNO to change from the ground spin–orbit state to the upspin–orbit state are routinely investigated. The pioneerwork by Joswig, Andresen, and Schinke used fixed molelar beams and LIF detection of NO after colliding with ANe, and He.5 Cross sections forF1→F1 scattering have beenmeasured or predicted to be 5–10 times larger than thoseF1→F2 scattering.6,7 These studies have also observed tthe integral cross sections, in general, decrease in magnwith increasingDJ.8 Additionally, several experiments hav

9 © 2001 American Institute of Physics

o AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html

foor

ros

tioec

hu

uls

u

te

ed

thgtra

o

ere

-uin

psetiontr

beoit

spita

rob

pl

n

rd

aith

tificlledif-

lartiven-byave-detheat-ark-f aewora-

ialies

almeeen

ofr-lara-ed.ly-

beda

to, aneri-ini-be-thegralrevi-the

ntstionilitytail-nell

2670 J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 Westley et al.

observedL doublet propensities in integral cross sectionsspin–orbit conserving and changing collisions fNO1Ar.9,10

Crossed molecular beam techniques combined withtatable detectors~or a fixed detector and rotatable beam!provide a method for measuring the differential cross secfor NO scattering. Using electron impact ionization to detHe scattering from NO, Keilet al. were able to accuratelymeasure diffraction oscillations of the scattered He and tmodel the attractive part of the NO–He potential.11 Thuiset al. used a hexapole selection technique and molecbeams to examine the angle dependence of scattering bycific spin–orbit states of aligned NO in the late 1970s.12,13

Gentry and co-workers used crossed, rotatable molecbeams and a laser ionization time-of-flight~TOF! detector torecord integral and differential cross sections for NO scating from Ar, NO, CO, and O2.7,10,14,15 Wodtke and co-workers made detailed integral cross-section measuremof rotationally inelastic scattering of vibrationally exciteNO using a crossed beam technique.16 They found that theintegral cross sections could be predicted by consideringvibrationally excited NO to have an extended bond lencorresponding to the equilibrium bond length in the vibtionally excited state. Meyer used counterpropagating mlecular beams, a TOF detector, and~211! REMPI throughthe E 2S1 state of NO to make measurements of the diffential cross sections, integral cross sections, and alignmof NO scattering from He.17 Meyer found differences in rotational rainbow structure for spin multiplet changing versconserving collisions which he attributed to differencesthe anisotropy for the two processes using a hard ellimodel. He also observed oscillations in integral cross stions superimposed on an overall decline in cross secwith DJ well modeled by an energy gap law, and alignmeof the scattered NO that could be explained by a geomeapse model.18–21

NO is an interesting candidate for theoretical studycause it has a2P ground state. A NO molecule has twnearly degenerateL doublet levels that represent, in the limof high J, the orientation of thep orbital of an electron withrespect to the rotation vector. The approach of anothercies to the NO, breaks the symmetry of these two orborientations. Thus, two potential energy surfaces, theA8 andA9 surfaces, are needed to adequately describe the intetions of NO with another species. The challenge of this prlem has made it appealing to theorists. Anab initio, electrongas model for NO1Ar was constructed by Nielsonet al. inthe late 1970s.22 This surface was used by Meyer~with achange in the reduced mass of the system to make it appriate for NO1He! to model his differential and integracross-section measurements.17 A more detailed theory for theinteractions between atoms and molecules in II electrostates was formulated by Alexander in 1984.23 Using thistheoretical framework and a CEPA~correlated electron paiapproximation! ab initio based method, Alexander producea PES for NO1Ar.6 A NO1He PES based on this NO1ArPES has also been produced by Yang and Alexanderresults from the NO1He surface are in good agreement wthe differential and integral cross-section data of Meyer.17,24


r

-

nt

s

arpe-

lar

r-

nts

eh--

-nt

s

ec-ntic

-

e-l

ac--

ro-

ic

nd

The latest PESs are available for use for the general sciencommunity through the use of a numerical package caHIBRIDON™, which was used in this report to generate dferential cross sections for NO1He. No PES for NO1D2 isknown to be presently available.

Imaging techniques, combined with crossed molecubeams and state selective ionization, provide an effectool for measuring angular distributions from rotationally ielastic scattering. NO scattering from Ar was first imagedHouston and co-workers, but since then improvements hbeen made on the technique.25,26 They observed two rotational rainbows for NO1Ar scattering, and no pronouncedifference in differential cross sections for spin multiplchanging versus spin multiplet conserving collisions. Tuse of velocity mapping and ion counting for imaging sctering reactions has recently been shown to produce remable results.27 The measurements reported here are part oseries of rotationally inelastic scattering studies using a ncrossed beam imaging apparatus at Sandia National Labtory ~the other systems studied include HCl1Ar andCO1Ne28,29!.

This work represents an effort to compare differentcross sections of NO scattering with two different spechaving approximately the same mass, He and D2. The state-selective detection of NO scattering from a light collisionpartner requires good angular resolution of slow lab-frascattering products. This resolution has only recently bpossible due to the advancements in velocity mappingproduct ions.30 The imaging technique for molecular scatteing lends itself to easy qualitative interpretation of angudistributions. However, for accurate quantitative informtion, substantial analysis of the images must be performThis article describes one method for performing this anasis, while other, very similar, methods have been descriin recent work.28,29 The focus of this work was to collectseries of differential cross sections as a function ofDJ forspin multiplet conserving (F1→F1) and spin multipletchanging (F1→F2) scattering events. We also wishedcompare the effect of having different scattering speciesatom versus a diatomic molecule, under very similar expmental conditions. Therefore, every effort was taken to mmize laser-dependent alignment effects. Furthermore,cause the strength of the imaging apparatus ismeasurement of angular distributions, and because intecross sections for NO scattering have been measured pously, integral cross sections are not reported. Instead,relative differential cross sections~which can be scaled bypreviously reported integral cross-section data! for rotation-ally inelastic scattering of NO (J50.5, F1)1X→NO(J52.5212.5F1 or J51.5– 9.5,F2)1X, where X is either Heor D2 , are reported in the following.

II. EXPERIMENT

The spin multiplet conserving scattering experimewere performed in a crossed molecular beam laser ionizaion imaging apparatus at the Combustion Research Facat Sandia National Laboratory, which was described in dein an earlier report.28 The spin multiplet changing experiments were performed in a very similar apparatus at Cor


ara

tio

siin

seams--

-C

rsiinatoa

ghopHe

gle

d

atu

ers

hetindi

f

p

nts

-

-xnd

nds

ef-ed

th

t-

g

-

f

toon.ima-hed tothensi-

uealhatfi-

iona

heosi-fordize

tingher a

u-idthredm-ract-ithd toels

2671J. Chem. Phys., Vol. 114, No. 6, 8 February 2001 Scattering of NO from He and D2

University.31 In each apparatus, two molecular beamsgenerated in differentially pumped source chambers, pthrough collimating apertures, and intersect in an interacchamber capable of maintaining a pressure of 731027 Torrwith both molecular beams pulsed at 30 Hz. The productthe scattering reaction are state-selectively ionized in theteraction region by a focused, pulsed probe laser. At thetersection region of the main chamber, a set of ion lenmounted perpendicular to the plane of the molecular beextract and focus30 the ions through a time-of-flight masspectrometer~TOFMS! and onto a position sensitive microchannel plate~MCP!/phosphor detector. The phosphor is imaged by a charge-coupled device~CCD! camera and the signal is integrated over a succession of shots either on the Cchip or in a computer.

For both sets of experiments, a 2% NO in argon mixtuwas used for the NO molecular beam, while a neat expanof D2 or He was used for the scattering beam. The backpressure behind each expansion was typically around 2The distribution of molecular or atomic velocities in eachthe beams could be measured by using the image of emolecular beam, the known flight time of the ions throuthe TOFMS, and the measured magnification of the iontics. For measurement of the velocity distribution of thescattering beam, a trace amount of NO was added to theto obtain the velocity distribution image. The velocity profiof the D2 beam was measured by ionizing the D2 directlyusing 211 REMPI with 204–205 nm light through theE,Fstate.32 The resulting velocity distributions are well modeleby Gaussian profiles with the following characteristics:vHe

51730 m/s, full width at half maximum~FWHM!-80 m/svD2

51834 m/s, FWHM-230 m/s; vAr/NO5574 m/s,FWHM562 m/s. These velocities are consistent withisentropic beam expansion through a nozzle at a temperaof approximately 310 K. In order to extract absolute numbfor the above-mentioned velocity distributions, it is necesary to know the magnification factor of the ion optics. Tmagnification factor was measured by least-squares fitthe diameters of the Newton spheres for scattering intoferent final rotational levels of NO.28 The results of this fit-ting give collision energy for the NO1He system of 491cm21 ~with a FWHM of 42 cm21! and a collision energy o545 cm21 ~with a FWHM of 125 cm21! for the NO1D2

system.The state distributions of the D2 and NO in the molecu-

lar beams were also measured. Due to the large energy sing of the rotational levels of D2 and theortho–para selec-tion rules for J changing collisions in D2 , the rotationaldegrees of freedom of D2 in the molecular beam expansioare not in equilibrium. Therefore, the ratio of specific heag, for the molecular beam expansion of D2 is between thelimits of 5/3 ~for no rotational degrees of freedom! and 7/5~for the two rotational degrees of freedom of a diatom!. Fur-thermore, the populations of D2 in the beam are not thermally distributed, and were directly measured by 211REMP133 to be 38%, 35%, 22%, and 5% inJ50 – 3, respec-tively, with the population inJ.3 less than 1%. The rotational distribution of NO in the molecular beam can be etracted from an ionization spectra of the NO and correspo


essn

ofn--ss

D

eongm.fch

-

as

nres-

gf-

ac-

,

-s

to a rotational temperature of about 2.5 K. This correspoto 92% of the NO inJ50.5, 8% of NO inJ51.5, and lessthan 1% inJ.1.5.

For the scattering measurements, NO ionization wasfected by lightly focusing the output of frequency doublYAG pumped dye laser~Sandia apparatus! or a XeCl exci-mer pumped dye laser~Cornell apparatus! to obtain approxi-mately 1 mJ/pulse of 225–227 nm light. The wavelengrange corresponds to 111 REMPI through theA 2S1 stateof NO. P andR transitions were used to ionize the NO scatered into higher rotational states of theF1 or F2 spin mul-tiplets by collisions with the scattering beam. ForF1→F1

scattering, product molecules forJf52.5– 5.5 were detectedby use of theP11 branch,Jf56.5– 9.5 were detected usinthe R21 branch, andJf.9.5 were detected using theR11

1Q21 branch. TheL doublets probed by this choice of transitions for detection is as follows:P11:P(A8) or e;R21:P(A9) or f; and R111Q21:P(A8) or e.34 For F1→F2

scattering, all of the transitions were probed using theR21

branch, which detectsP(A8) or e L doublets. The choice oa relatively high power probe laser, 111 ~as opposed to1118! REMPI, andP andR transitions of NO were made tominimize the possible preferential ionization of NO dueany correlation of rotational alignment and laser polarizatiAdditionally, because the measured product states are prrily low J states, we expect the rotational alignment of tproduct NO to be small. The data images were observebe independent of probe laser polarization direction forabove-mentioned conditions. The images were also insetive to the particular transition used to ionize the NO forF1 ,Jf54.5, 7.5, and 10.5, which implies that no differences dto final L doublet level were observed in the differenticross sections within experimental error. Note, however, tthe images presented in the following represent differentnal L doublet levels, depending on whichJf was probed.

The NO product ions were extracted from the interactregion by a set of ion optics which focuses the ions withgiven velocity projection in the molecular beam plane to tsame point on the MCP detector, regardless of starting ption of the ion. Due to the fact that the scattering sphereNO scattering from He or D2 is very small, the voltages useto extract the ions were set as low as possible to maximthe spatial extent of the scattering signal on the MCP. Setthe repeller voltage to 850 V extended the flight time of tNO ions so that the NO product ion scattering sphere fotypical scattering image was 50–60 pixels~5–6 mm! in di-ameter. The small spread in the lab frame velocity distribtion of the scattered NO also meant that the laser bandwwas sufficient to cover the entire Doppler spread of scatteproducts with relatively equal laser intensity. The data iages, collected by the CCD camera, are the result of subting an image with the scattering beam on from an image wthe scattering beam off. This subtraction procedure is usecorrect for the background population of any high J levpresent in the molecular beam expansion.



FIG. 1. Ion images forF1→F1 scattering of NO (Ji50.5,1.5) with D2 or He into a final rotational level as labeled above.

oaerys

hro

icnin

geta

it

p-ar-

o-is

ula-ble

touchbe-rgehate in

tec-tter-el

the

-

III. RESULTS

A. Image data

The background subtracted ion images forJi;0.5, F1

→Jf52.5– 12.5,F1 are shown in Fig. 1. The images forJi

;0.5, F1→Jf51.5– 9.5,F2 are shown in Fig. 2. Forwardscattering intensity appears in the upper left-hand portionthe scattering sphere, while backward scattering intensitypears in the lower right-hand section of the scattering sphIn order to orient the reader, a Newton diagram for the stem is overlaid on the image for NO (Jf55.5, F1) @Fig.3~a!#. Qualitative interpretation of the images leads to tconclusion that there is a single broad peak that moves fforward to backward scattering with increasingDJ as previ-ously observed for NO–He and NO–Ar inelastscattering.10,17,25,26Additionally, we observe that, for a giveJ, the position of the rainbow maximum is slightly morethe forward direction for D2 than for He.

B. Image characteristics

There are three major characteristics in the raw imathat must be understood and modeled if we are to obquantitative differential cross sections~DCSs! from the data:a region of low signal to noise in the net image due tolocalized peak in the background intensity, a lab veloc


fp-e.-

em

sin

ay

bias in detection efficiency, and a breakdown of the ion otics for ions at the edge of the detection region. These chacteristics are illustrated by Fig. 3~b!.

The first characteristic is that a region of low signal-tnoise ratio in the forward scattered portion of the imagesseen due to background NO in highJ levels present in themolecular beam expansion. Even though the beam poption of NO in J.1.5 is less than 1%, there is a measuraamount~compared to the intensity of scattered NO! of NO inthe molecular beam inJ52.5– 12.5. The ion signal from thisNO is localized in the image in a region that correspondsforward scattering. This region of the image thus has a msmaller signal-to-noise ratio than the rest of the imagecause the net signal is obtained by subtracting two lanumbers. This problem is analogous to the problem tcrossed beam, rotatable detector experiments experiencthe measurement of small angle scattering intensity.

The second characteristic is a velocity-dependent detion bias due to the laser-based method of detection. Scaing should exhibit cylindrical symmetry about the relativvelocity vector of the system~unless there is preferentiamolecular alignment before the collision!. However, as canbe seen from Fig. 3, the intensity of scattering towardzero lab velocity side of the image~upper right-hand portion!is higher than that on the high lab velocity side~lower left-hand portion!. Collisions of NO occur for tens of microsec


Tine

i-ioigecNetri

ricm

igpediothchinh.

necteri-Abel

er-

and


onds~the gas pulses are 100–200ms wide! before the laserpasses through the intersection of the molecular beams.population of scattered NO with different lab velocitiesthe laser ionization volume is directly related to the lab vlocity of the NO. NO with small lab velocities or lab velocties parallel to the laser direction leave the laser ionizatvolume at a much slower rate than scattered NO with hlab velocities or velocities perpendicular to the laser dirtion. Thus, there is a preferential detection of scatteredwith small lab velocities or lab velocities parallel to the lasdirection. Therefore, although the scattering is symmeabout the relative velocity vector in the center of mass framthe laser-based detection scheme destroys this cylindsymmetry about the relative velocity vector in the data iages.

The final characteristic in the data images is the ion snal outside of the Newton sphere that trails off to the upright-hand side. It was hypothesized that these ions wereto scattered NO ionized outside the effective focusing regof the ion optics. This hypothesis was confirmed throughuse of a Monte Carlo simulation of the experiment whiincorporated a finite difference model of the electric fieldsthe ion optics. Figure 4 shows a simulated image from tMonte Carlo program and the corresponding data image

FIG. 2. Ion images forF1→F2 scattering of NO (Ji50.5,1.5) with D2 orHe into a final rotational level as labeled above.


he

-

nh-Orce,al-

-ruene

is

C. Image simulation

The extraction of an angular distribution from an ioimage can be accomplished by various indirect and dirinversion procedures. Notably, for photodissociation expments and a few crossed beam experiments, an inverse

FIG. 3. ~a! Typical image, with beam and center of mass velocities supimposed.~b! The characteristics labeled in the image are labeled 1–3:~1!low signal/noise region due to molecular beam background,~2! intensityskew across cylindrical symmetry axis due to velocity detection bias,~3! nonvelocity mapped ions from edges of detection region.

FIG. 4. ~a! Monte Carlo simulation ofJf55.5, isotropic distribution, show-ing nonvelocity mapped ions trailing off toward top of image.~b! Data,Jf

55.5, He.


ioeth

omu

atelly

ionena-

nsete

e

ris

ait

nee-llyhitih

t ilgoat

ou

be

n-ity,ity.as-pen-d byllow

tae

en-

eerOF

thathegra-

graltheme

er-s is

thisly,s is

ofith

ethenedsde


transform has been used to extract an angular distributHowever, primarily due to the nonuniform blurring of thscattering intensity across its axis of symmetry, the use ofAbel transform to extract the differential cross sections frthe images in this experiment is not appropriate. A rigoroanalytical inversion for a crossed beam scattering apparis quite difficult and has not, to the authors’ knowledge, bedeveloped. Instead, iterative forward simulation is typicaused to extract the differential cross section.

The forward simulation process requires the evaluatof at least a nine-dimensional integral to convert the odimensional differential cross section into a two-dimensiosimulated image. Assuming~1! that there is no angular divergence in the molecular beams,~2! that the velocity map-ping is ideal, and~3! that there is no preferential ionizatioof NO based on alignment relative to the ionization lapolarization, it is necessary to evaluate the following ingral:

I ~xI !5 E Emolec beam

dv1 dv2Escat vol

dxs

3E dvpEts

tLdt NMB~v1 ,v2 ,xp ,t !I L~xp!

3g]s

]Vf t~vp ,xp ,xim!, ~1!

where xp5xs1(vp1vc.m.)(t2ts), where xI is the two-dimensional~2D! pixel position,v1 , v2 are the speeds of thradical ~NO! and scattering beam,xs is the position in theinteraction region at which the NO scatters,vp is the velocityof the product NO in the c.m. frame,t is time, ts is the timethat the scattering occurs, andtL is the time when the lasefires. The velocity of the product NO in the lab framegiven byvp1vc.m.. The quantities in the integrand are:NMB

the intensity dependence of the two molecular beams,I L , theintensity dependence of the laser,]s/]V, the differentialcross section, andf t , a function that relates the position ofdetected ion in the image plane to its position and velocafter being ionized. For ideal velocity mapping,f t can bewritten:

f t~v,x,xim!>d~v i2Mxim! f Jac~v:$vi5Mxim%!, ~2!

where vi5v2~v"ndet!ndet,

whereM is the magnification factor of ion lenses in m/s/mmndet is a unit normal vector perpendicular to the image plaand f Jac is the Jacobian for the overlap of the thredimensional scattering sphere onto the 2D detector. Initiaimages were simulated using a Monte Carlo technique wmanually iterating the input differential cross section unthe appearance of the data and simulated images matcHowever, utilizing some of the constraints of the system ipossible to construct a much faster image simulation arithm that reproduces the main features of the apparfunction using numerical integration.35

The development of the numerical integration methdepends on a few additional assumptions and a subseq


n.

e

susn

n-l

r-

y

,,

,leled.s-

us

dent

re-writing of the integral given by Eq.~1!. The first assump-tion is that the molecular beam intensity function canwritten as

NMB~v1 ,v2 ,xp ,t !5I MB,vel~v1 ,v2!I MB, t~ t !IMB,pos~xp!,~3!

whereI MB,vel is the velocity dependence of the beam intesity, I MB,time is the time dependence of the beam intensand I MB,pos is the spatial dependence of the beam intensThe time and velocity dependencies of the beams aresumed to be Gaussian. Furthermore, the beam spatial dedence is assumed to be constant within a cone determinethe molecular beam apertures. These assumptions aI MB,vel to be extracted from the integral overv1 andv2 . Theassumption thatf t can be written as a product of a delfunction andf Jaccombined with the energy constraints of thexperiment for a givenv1 and v2 ~they are fixed becausthey are in the outer integral, which fixes the collision eergy,Ecoll , andDE is fixed byDJ) allows the determinationof the projection ofvp onto the detector plane. This fixes thfull vector vp to one of two vectors: one with a positivcomponent along the TOF axis~perpendicular to the detectoplane! and one with a negative component along the Taxis. Thus, for a given pixel~the integral is written as afunction ofxI), the integral overvp can be replaced by a sumover these two possible vectors. The final assumption isonly NO present in the laser ionization volume when tlaser fires need to be accounted for by the numerical intetion, and that the sum overvp can be removed from theintegral over the scattering volume. This reduces the inteover the entire scattering volume to an integration overlaser ionization volume and an integration backwards in tito compute the amount of NO scattered withvp into the laserionization volume at the time when the laser fires. Furthmore, because the spatial density of the molecular beamassumed to be constant within their defining cones,backward integral in time can be performed analyticalgiven that the time dependence of the molecular beamassumed to be Gaussian. The limits of integration (t1 ,t2) forthe time integral can be found from the geometric overlapthe NO trajectory with the molecular beam cylinders. Wthese assumptions and simplifications, Eq.~1! can be ex-pressed as

I ~x1!5 E Emolec beam

dv1 dv2 NMB,vel~v1 ,v2!

3g(1

2]s

]Vf Jac~vp,i !E

laser voldxL I L~xL!E

t1

t2dt

3NMB,time,~ t ! ~4!

where vp,i ( i 51,2) are the two possible velocities of thproduct NO given a projection in the detector plane, andremainder of the quantities in this expression are as defipreviously. Using this algorithm for image simulation allowthe computation of a simulated image orders of magnitufaster than the Monte Carlo method.


fogeno

bth

vasac

tioau

s bntuisthmiosso

rinlt

taex-isse

tohas

larichffer-pa-

sedanan-ionhise aderted

ea-ri-

a-onto

sure-po-it-

esehen

pa-ityi-thetial

cate


D. Extraction of the differential cross section

Rather than running the image simulation programeach trial differential cross section in an iterative imasimulation process, it is possible to generate a set of nothogonal basis images from which a simulated image canconstructed. The differential cross section is input intosimulation program as an array of points~the DCS array!,which represent the differential cross section at discreteues ofu. Simulated images are a linear superposition of aof basis images that correspond to a unit value input for ediscreteu value~and zero for all otheru values! in the DCSarray. Figure 5 shows an example of the angular distribuand appearance of a basis image. By weighting these bimages with the trial DCS values and summing them, simlated images for the same set of experimental parameterdifferent trial DCS arrays can be generated almost instaneously. This makes the process of iterative simulation qfast. It is also possible to linearize the differential crossection extraction process by evaluating the overlap ofbasis images with the data images. This results in a systelinear equations that can be solved using matrix inversalgorithms. However, because the image inversion procenot unique~rapid oscillations added to a trial DCS array dnot change a simulated image due to the various blureffects! this linearization and inversion process are difficu

FIG. 5. ~a! Basis image for an angular spike at 85°.~b! Input angulardistribution for basis image.


r

r-e

e

l-eth

nsis-ut

a-te-eofnis

g.

Also, for this technique to work well, the simulated and daimages must have their relative pixel sizes and positionstremely well matched. An iterative extraction techniqueeffective at producing a simulated image that is a clomatch to the data image.

The process of iteratively varying the trial DCS arrayproduce a simulated image that matches that data imagebeen described in detail in a previous article.28 The processextracts a trial DCS array from the image from an annuintegral of the outer edge of the scattering sphere, whstrongly corresponds to the angular dependence of the diential cross section, but is skewed and blurred by the apratus function. Figure 6 shows a typical annular integral ufor the iterative fitting procedure. A simulated image withisotropic differential cross section is generated, and thenular integral of this image is used as a first-order correctto the annular angular function from the data image. Tfirst-order corrected DCS array is then used to producsimulated image, which is used to produce a higher orcorrection to the trial DCS array. This process is repeauntil the data and simulated images~their annular integrals!converge. The simulated images for several scattering msurements are shown in Fig. 6. In order to modify expemental parameters~such as the position of the laser ioniztion volume relative to the molecular beam intersectipoint!, it is necessary to run the image simulation programgenerate a new set of basis images. Independent meaments of all the experimental parameters, including lasersition and laser ionization volume, were performed. Theerative fitting procedure gives an internal check on thparameters, because it has been found to not converge wthere is a significant error in values of the experimentalrameters. For example, the relative skewing of ion intensaboutvrel is not correctly modeled unless the left/right postion of the laser is correct. Due to the sensitive nature ofgoodness of fit on experimental parameters, the differen

FIG. 6. Two examples of data and simulation fits. The dashed lines indithe annular regions used to compare the two images for fitting.


a

a

n-a

a-

on

esar

eitlens

prd

a

henelel

rastisw

si-t of

adr-

pul-viorWedell in

ofomnd

s


cross sections extracted using the above-given methodassumed to be accurate.

Plots of the extracted DCS arrays for the inelastic sctering of NO with He and D2 from Ji50.5, F1 to Jf

52.5– 12.5,F1 are contained in Fig. 7. The top panel cotains extracted DCSs for collisions of NO with He at a mecollision energy of 491 cm21. The middle panel displaysselected DCSs extracted from NO/D2 images with a meancollision energy of 545 cm21. The bottom panel showsdirect comparison of He and D2 scattering for selected product rotational states. The NO scattering from D2 is seen to begenerally more forward peaked, with a narrower distributithan the He scattered NO.

An error analysis of the images implies that the largerrors, approximately 15%, are associated with the forwscattered region. This is due to the interference of the NOthe molecular beam creating a ‘‘beam spot’’ on the imagnear the zero, or forward, deflection angle. The error limfor angles greater than 40° are generally believed to bethan 5%. The estimated error represents the propagatiocounting error~the number of ions for a given intensity waestimated from the raw data images! of the subtracted imagedata through the iterative simulation process discussedviously. Errors in the simulation parameters were assumebe minimal and were not treated in the error analysis.

Plots of the extracted DCS arrays for the inelastic sctering of NO with He and D2 from Ji50.5, F1 to Jf

FIG. 7. Extracted differential cross sections forF1→F1 scattering. Opensymbols are for He, while closed symbols are for D2 . In the top two panels,squares are forJf52.5, circles forJf55.5, triangles forJf57.5, diamondsfor Jf59.5, and inverted triangles forJf511.5. In the bottom panel, squareare forJf53.5, triangles forJf56.5, and diamonds forJf58.5.


re

t-

n

,

tdinssssof

e-to

t-

51.5– 9.5,F2 are shown in Fig. 8. The top panel displays textracted DCSs for NO scattering with He. The middle pashows the DCSs for NO/D2 scattering and the bottom panshows a direct comparison of He and D2 scattering for sev-eral selected rotational product states. Note that in contwith the spin–orbit conserving collisions, Fig. 7, thereseemingly little difference in the positions of the rainbomaxima between He and D2 scattering for a givenJf for thespin–orbit changing collisions. It is also seen that the potions of the rainbow peak are, to first order, independenthe spin–orbit state for a givenJf .

In general, the differential cross sections exhibit a brorainbow maximum that monotonically shifts from the foward to backward scattering direction with increasingDJ.This is consistent with scattering systems where the resive surface of the potential controls the scattering behaand can be well modeled by purely repulsive potentials.use one such potential, a 2D hard ellipse potential, to mothe differential cross section as will be discussed in detaithe following.

IV. DISCUSSION

We have obtained differential cross sections for mostthe available product rotational states for NO scattering frHe and D2, measured for both spin–orbit conserving a

FIG. 8. Extracted differential cross sections forF1→F2 scattering. Opensymbols are for He, while closed symbols are for D2 . In the top two panels,squares are forJf51.5, circles forJf53.5, triangles forJf54.5, diamondsfor Jf56.5, and inverted triangles forJf59.5. In the bottom panel, circlesare forJf52.5, diamonds forJf56.5, and inverted triangles forJf59.5.


-ES,

d in. Inultsrsedheel,e-l,nel,


FIG. 9. Comparison of spin–orbit conserving differential cross sections between data, Yang–Alexander Pand ellipse models for twoJf levels. The model datahave been used to simulate an image, then processethe same way as the data to produce the model plotsthe top panel, the solid squares are experimental resfor He andJ53.5, the solid line is the Yang–AlexandePES prediction, the open circles are a simulation baon the prediction of the Yang–Alexander surface, tdashed line is the prediction of a hard ellipse modand the3symbols are the simulation based on the prdiction of the hard ellipse model. In the bottom panethe symbols have the same meaning as in the top paexcept that the data are for He andJ57.5.

thcsin

gyap

ers

ese

is

s.he

d

tedthe

ticalen-ens

-O

n

ec-

. 9.

-hat

spin–orbit changing collisions. The general behavior oftwo systems is strikingly similar, indicating that the interation potentials are also similar since the reduced massethese two systems are identical. The more forward peakof the D2 system correlates with the higher collision enerfor this system, however as we will show in the following,portion of this effect may be attributed to a higher anisotroin the repulsive wall of the NO/D2 potential relative to thatof the NO/He potential.

In order to interpret the data, two models have beused. The CEPA PES for NO1He of Yang and Alexandehas been used to generate theoretical differential crosstions, which are then compared to experiment.24 This PES isfound to accurately reproduce the major features of theperimental data for NO1He with deviations in the positionof rainbow maxima. Due to the small differences betwethe differential cross sections for He and D2, the NO1HePES also does a reasonable job predicting the angular dbutions for scattering of NO1D2, suggesting that this PEScould be used to estimate NO1D2 scattering measurementAdditionally, due to the seemingly simple behavior of tdifferential cross section as a function ofDJ and scatteringspecies, a two-dimensional hard ellipse model was use


e-ofg

y

n

ec-

x-

n

tri-

to

predict the scattering behavior forF1→F1 scattering. In gen-eral, by adjusting the anisotropy of the ellipse, the predicrainbow maxima positions could be made to agree withrainbow maxima from the data.

The most recent theoretical PES for NO1He scatteringby Yang and Alexander has been used to produce theoredifferential cross sections using the numerical packagetitled HIBRIDON. HIBRIDON is a package of programs for thtime-independent quantum treatment of inelastic collisioand photodissociation written by Alexanderet al.36,37 TheNO was assumed to have an initial population of 0.92 inJ50.5, F1 and 0.08 inJ51.5, F1 with an equal distributionbetweenL doublet states. TheL doublet state of the scattered NO was fixed by the transition used to ionize the N~see Sec. II!. NO1He collisions were simulated at collisioenergies of 491 and 545 cm21. A comparison of the experi-mental differential cross sections with calculated cross stions from the Yang and Alexander~YA ! PES and a hardellipse model~details are provided in the following! for se-lected spin–orbit conserving collisions is presented in FigThe experimentally extracted NO/He DCSs forJf53.5 ~toppanel! andJf57.5 ~lower panel! are compared to the calculated YA DCSs and to DCSs extracted from images t


tri

thSYA

uCtai-isi

t-ta

onu

ipse

ohectab

eldCS.d

isto

wnat-ale

cedelythe

ofenc-tero-

e.

er-thebys

redddi-for-

or

singve

ber ate-nd

a,tes.cedoar-otthe

b

ofeg–


have been simulated using the calculated YA DCSs asinput cross sections. This simulation/extraction methodused in order to account for any distorting factors due toextraction technique itself. The simulation/extraction DCare shown as open circles in Fig. 9. Note that the inputPES DCS~solid line! and extracted DCS~open circles! arevery similar, a fact which highlights the goodness of oextraction technique. While the general shape of the Dcalculated from the YA PES is similar to the experimenDCS ~solid squares!, the calculated distributions are postioned significantly toward higher deflection angles. This dcrepancy could be accounted for by insufficient anisotropythe repulsive wall of the YA PES. With insufficient anisoropy, collisions lack the necessary torque to promote rotional excitation. Therefore, a more nearly head-on collisigenerating a larger deflection angle, is necessary to prodthe needed change in angular momentum. The hard ellDCSs shown in the dashed lines of Fig. 9 will be discusin the following.

Figure 10 shows experimental and calculated DCSstwo product states for spin–orbit changing collisions of tNO/He system. The theoretical DCSs have been extrafrom simulated images using input DCSs obtained from cculations using the YA PES. Consistent with the spin–or

FIG. 10. Comparison of spin–orbit changing differential cross sectionstween data and Yang–Alexander PES for twoJf levels. In the top panel, theopen circles are for He andJ53.5, whereas the solid line is the predictionthe Yang–Alexander PES. In the bottom panel, the open diamonds arHe and J56.5, whereas the solid line is the prediction of the YanAlexander PES.


alises

rSl

-n

-,cesed

f

edl-it

conserving collisions shown in Fig. 9, the bottom panshows that forJf56.5F2 , the theoretical DCS is peaketoward a higher scattering angle than the experimental DFor Jf53.5F2 ~top panel! both the experimental DCS anthat obtained from the YA PES peak atu50 ~forward scat-tering!. However, the width of the scattering distributionmarkedly broader in the experimental DCS. It is difficultattribute this difference to a specific part of the YA PES.

The two-dimensional hard ellipse model has been shoto accurately reproduce maxima in rotationally inelastic sctering angular distributions, known as rotationrainbows.38–40 The basic premise of the model is that thpotential energy surface for the interaction can be replaby a two-dimensional potential surface that has an infinithigh potential step on an elliptical contour. The shape ofellipse can be estimated from the classical turning pointsother potential surfaces. The rainbow maxima, for a givamount of rotational energy transfer, arise from the funtional form of the deflection angle versus impact parameand ellipse orientation angle for the collision. For a homnuclear, 2D ellipse model, withe5m/I!1, the rainbowmaxima can be approximated as

sin1

2uR5

D j

2~A2B!, ~5!

whereuR is the rainbow angle,D j 5 j /p0 , j is the final an-gular momentum of the rotor,p0 is the initial linear momen-tum, andA andB are the major and minor axes of the ellipsFor the systems studied here,e50.35 Å22, so the results ofthis formula will probably be somewhat inaccurate. Furthmore, for a heteronuclear molecule, the center of mass ofellipse is shifted from the geometric center of the ellipsean amountd. This shift gives rise to two rotational rainbowfor scattering from each end of the ellipse. For NO,d issmall, on the order of 0.05–0.10 Å, so NO can be considealmost a homonuclear molecule, and thus not much ational error is caused by use of the above-mentionedmula.

To simulate more rigorously the hard ellipse model fthe case of NO1He/D2 , F1→F1 , it is possible to simulatethe differential cross sections for the scattering process uthe formula for scattering angle as a function of effectiimpact parameter,D j ande from Bosanac and Buck:39

sinu5D j

2bn2 F4bn

22~D j !2~11ebn2!2

12e~D j !2 G1/2

, ~6!

where bn is the effective impact parameter which cancalculated from the appropriate geometric equations forange of impact parameters and ellipse orientations. By ingrating over impact parameter and ellipse orientation, astoring the results as a function ofD j and scattering angle,u,model hard ellipse DCS and corresponding rainbow maximuR , can be generated for different rotational product staFor heteronuclear molecules the center of mass is displafrom the geometrical center of the ellipse, giving rise to twunique rainbow maxima. Due to the near homonuclear chacter of NO and the experimental resolution limits, it is npossible to resolve these two rainbow maxima—even in

e-

for


fo

enfahens

ofips

euon

th-

lyng

exoinera

-c

pom

-s

d

flt

la

nd

nte

sg

lesiefo

eor

ec-

al

ic-tedret-ges

oftheod-ore,theof

theanicstter-t a

ur-g

the

t-

,

llipse

d


model calculation. Figure 9 shows 2D hard ellipse DCSsNO/He scattering forJf53.5 and 7.5~dashed lines!. Notethat the DCSs rise from zero in the classically forbiddregion to a maximum at the classical rainbow, and thenoff toward higher scattering angles. The position of tmaximum is determined solely by the collision energy athe relative sizes of the semimajor and semiminor ellipaxesA andB, respectively. Also included are extractionsthe DCSs from simulated images using these hard ellDCSs as inputs~line connecting3symbols!. The extractedDCSs are broadened due to the limited resolution in theperiment, due primarily to the velocity spread in the moleclar beams. This effect is especially pronounced in the fward scattering region, and is compounded by the additioa background ‘‘beam spot.’’

By adjustment of the ellipse parameters, especiallyanisotropy (A–B), the rainbow maxima positions of the extracted DCSs for the rotational states for each of the NO1Heand NO1D2 collision systems can be fit to a relativesimple hard ellipse model. Figure 11 shows a comparisorainbow maxima,uR , extracted from calculated DCSs usinthe YA PES, and those taken from the experimentallytracted DCSs, as well as a fit to the experimental rainbpositions. Data forJ59.5– 10.5 have not been includedthe fit because~1! the data do not conclusively show wheththere is a peak in the angular distribution or whether the dare simply backscattered, and~2! there is an intrinsic uncertainty in the extraction procedure that makes the nearly bascattered peak positions very sensitive to the assumedtion of the laser beam. Note that the rainbow maximuangles calculated usingHIBRIDON and the YA PES are consistently higher than the measured rainbow maxima. Thitrue for both the He~triangles! and the D2 ~crosses! scatter-ing. For NO1He (F1→F1), the results of fitting the ob-served rainbow maxima positions to a 2D hard ellipse mo~solid line fit to closed circles! give results of (A–B)50.62 Å andd50.04 Å ~usingA'2.7– 3.1 Å). It is not pos-sible to fit a value ofd with much certainty due to the lack oresolution of the two rotational rainbow peaks. The resufor NO1D2 (F1→F1) are similar~dashed line fit to opensquares!, but with noticeably higher anisotropy: (A–B)50.76 Å andd50.14 Å ~usingA'2.7– 3.1 Å). These modeellipse anisotropies can be compared to estimates of theisotropy in the repulsive wall of the CEPA PES of Yang aAlexander. The YA PES contours near 500 cm21 ~compa-rable to the collision energy of our experiment! give valuesfor A andB of ;2.8 and;2.4 Å, respectively. This results ina value of (A–B)50.4 Å, which is considerably less thathe anisotropy given by the 2D hard ellipse models extracfrom either the NO/He or NO/D2 data presented in thistudy. A lower repulsive anisotropy will result in scatterindistributions that are positioned at higher deflection angThis trend is shown in Fig. 11. Note that the rainbow potions, derived from the YA DCSs, are positioned to highdeflection angles relative to the 2D hard ellipse rainbowsall J values. In addition, the value of the YA theoreticaluR

rises more rapidly withJf than either of the hard ellipsmodels, to the point that there may not be sufficient anisropy for reaching the collisional energetic limit. This obse


r

ll

de

e

x--r-of

e

of

-w

ta

k-si-

is

el

s

n-

d

s.-rr

t--

vation is born out by CS calculations of integral cross stions, which show that at a collision energy of 508 cm21 thebest value ofJmax is about 11.5, corresponding to a rotationenergy of 240 cm21.24

As a final comparison between the theoretical predtions for the differential cross section and the DCSs extracfrom the experimental data, it is possible to use the theoical differential cross sections to produce simulated imato compare directly with the raw data images. The resultsthese comparisons illustrate the limitations in measuringdifferential cross section for the most forward scattered pructs due to the interference with the beam spot. Furthermwhile the hard ellipse calculations are successful in fittingrainbow maxima of the data reasonably well, the PESYang and Alexander naturally predicts more accuratelybroadness of the rainbow peaks, because quantum mecheliminates the singularities encountered in classical scaing differential cross sections. We therefore conclude thafully quantum mechanical calculation using a repulsive sface with an anisotropy similar to that which we find usinthe hard ellipse fit should be in excellent agreement withdifferential cross sections presented here.

V. SUMMARY

Differential cross sections for rotationally inelastic scatering of NO with He and D2 at ;500 cm21 collision energy

FIG. 11. Plot of rainbow maxima vsJf for data, the Yang–Alexander PESand the hard ellipse models. Closed circles are data for D2 /NO; opensquares are data for He/NO; solid and dashed lines are fits to a hard-emodel for the D2 and He data, respectively~see the text!; triangles andcrosses areHIBRIDON predictions for He/NO at collision energies of 491 an545 cm21, respectively.


isob

isaanhaa

pr

po

eNOs

nerb

f Aghg02

.De

g,

, J

m.

, J.

J. N.

ys.

s.

ys.

m.

um.

m.

.

g,


have been measured using crossed molecular beam ionaging and extracted using a new forward simulation baimage technique. The positions of the rainbow peakstained from the extracted differential cross sections forJf

,9.5 are found to be well fit by a hard ellipse model. Thmodel underestimates the broadness of the rainbow peThe differential cross sections obtained using the YangAlexander PES are typically more backward scattered tthe data, indicating that the anisotropy of the repulsive wof their potential may be too low. The data forJf.9.5 showa larger change in the rainbow scattering angle than isdicted by extrapolation of the low-J hard ellipse model. Thehigh-Jf peak positions are in closer agreement with thesitions calculated by Yang and Alexander.

For spin–orbit conserving collisions, the NO scatterfrom D2 is seen to be more forward scattered than thescattered from He. The differences in the differential crosections between NO1He and NO1D2 are small and arepartially, but not totally, attributable to the higher collisioenergy of the NO1D2 system. Only small differences arobserved between the rainbow positions of the spin–ochanging and spin–orbit conserving collisions.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the assistance oDixit and P. J. Pisano in collecting the spin multiplet chaning images from the apparatus at Cornell University. Twork at Cornell was supported by the Department of EnerBasic Energy Sciences, under Grant No. DE-FG88ER13934. The work at Sandia was supported by the UDepartment of Energy, Office of Basic Energy Sciences,partment of Chemical Sciences.

1A. S. Sudbo and M. M. T. Loy, Chem. Phys. Lett.82, 135 ~1981!.2A. S. Sudbo and M. M. T. Loy, J. Chem. Phys.76, 3646~1982!.3M. Islam, I. W. M. Smith, and M. H. Alexander, Chem. Phys. Lett.305,311 ~1999!.

4P. L. James, I. R. Sims, I. W. M. Smith, M. H. Alexander, and M. YanJ. Chem. Phys.109, 3882~1998!.

5H. Joswig, P. Andresen, and R. Schinke, J. Chem. Phys.85, 1904~1986!.6M. H. Alexander, J. Chem. Phys.99, 7725~1993!.7S. D. Jons, J. E. Shirley, M. T. Vonk, C. F. Giese, and W. R. GentryChem. Phys.97, 7831~1992!.


m-is-

ks.dnll

e-

-

d

s

it

.-ey,-S.-

.

8J. Troe, J. Chem. Phys.66, 4745~1977!.9A. Lin, S. Antonova, A. P. Tsakotellis, and G. C. McBane, J. Phys. CheA 103, 1198~1999!.

10S. D. Jons, J. E. Shirley, M. T. Vonk, C. F. Giese, and W. R. GentryChem. Phys.105, 5397~1996!.

11M. Keil, J. T. Slankas, and A. Kuppermann, J. Chem. Phys.70, 541~1979!.

12H. Thuis, S. Stolte, and J. Reuss, Chem. Phys.43, 351 ~1979!.13H. H. W. Thuis, S. Stolte, J. Reuss, J. J. H. van den Biesen, and C.

van den Meijdenberg, Chem. Phys.52, 211 ~1980!.14J. A. Bacon, C. F. Giese, and W. R. Gentry, J. Chem. Phys.108, 3127

~1998!.15M. T. Vonk, J. A. Bacon, C. F. Giese, and W. R. Gentry, J. Chem. Ph

106, 1353~1997!.16M. Drabbels, A. M. Wodtke, M. Yang, and M. H. Alexander, J. Phy

Chem. A101, 6463~1997!.17H. Meyer, J. Chem. Phys.102, 3151~1995!.18T. Orlikowski and M. H. Alexander, J. Chem. Phys.80, 4133~1984!.19V. Khare, D. J. Kouri, and D. K. Hoffman, J. Chem. Phys.74, 2656

~1981!.20V. Khare, D. J. Kouri, and D. K. Hoffman, J. Chem. Phys.74, 2275

~1981!.21H. Meyer, Chem. Phys. Lett.230, 519 ~1994!.22G. C. Nielson, G. A. Parker, and R. T. Pack, J. Chem. Phys.66, 1396

~1977!.23M. H. Alexander and T. Orlikowski, J. Chem. Phys.80, 1506~1984!.24M. Yang and M. H. Alexander, J. Chem. Phys.103, 6973~1995!.25L. S. Bontuyan, A. G. Suits, P. L. Houston, and B. J. Whitaker, J. Ph

Chem.97, 6342~1993!.26A. G. Suits, L. S. Bontuyan, P. L. Houston, and B. J. Whitaker, J. Che

Phys.96, 8618~1992!.27N. Yonekura, C. Gebauer, H. Kohguchi, and T. Suzuki, Rev. Sci. Instr

70, 3265~1999!.28K. T. Lorenz, M. S. Westley, and D. W. Chandler, Phys. Chem. Che

Phys.2, 481 ~2000!.29K. T. Lorenz, D. W. Chandler, and G. C. McBane~unpublished!.30A. T. J. B. Eppink and D. H. Parker, Rev. Sci. Instrum.68, 3477~1997!.31M. S. Westley, Ph.D. thesis, Cornell University, Ithaca, NY, 2000.32E. E. Marinero, C. T. Rettner, and R. N. Zare, Phys. Rev. Lett.48, 1323

~1982!.33K. D. Rinnen, M. A. Buntine, D. A. V. Kliner, R. N. Zare, and W. M

Huo, J. Chem. Phys.95, 214 ~1991!.34M. H. Alexanderet al., J. Chem. Phys.89, 1749~1988!.35G. C. McBane~unpublished!.36M. H. Alexander, D. E. Manolopoulus, H. J. Werner, and B. Follme

HIBRIDON 4.1 ed.~1996!.37M. H. Alexander and D. E. Manolopoulos, J. Chem. Phys.86, 2044

~1987!.38S. Bosanac, Phys. Rev. A22, 2617~1980!.39S. Bosanac and U. Buck, Chem. Phys. Lett.81, 315 ~1981!.40W. Schepper, U. Ross, and D. Beck, Z. Phys. A290, 131 ~1979!.


Documents

Differential cross sections for rotationally inelastic ...houston.chem.cornell.edu/JCPPaperNOScattering.pdf · Differential cross sections for rotationally inelastic scattering of