Quantum reactive scattering of ultracold NH($X\,^3Σ^-$) radicals in a magnetic trap

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by  Liesbeth M. C. Janssen

Quantum reactivescattering ofultracoldNH(X3Σ−)radicalsina magnetictrap
Liesbeth M. C. Janssen,∗ Ad van der Avoird, and Gerrit C. Groenenboom†
Radboud University Nijmegen, Institute for Molecules and Materials,
Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
(Dated:January10,2013)
WeinvestigatetheultracoldreactiondynamicsofmagneticallytrappedNH(X3Σ−)radicalsusingrigorous
quantum scattering calculations involving three coupled potential energy surfaces. We find that the reactive
NH+NHcrosssectionisdrivenbyashort-rangedcollisional mechanism, anditsmagnitude isonlyweakly
dependentonmagneticfieldstrength. Unlikemostultracoldreactionsobservedsofar,theNH+NHscattering
3
dynamics is non-universal. Our results indicate that chemical reactions can cause more trap loss than spin-
1
inelasticNH+NHcollisions,makingmolecularevaporativecoolingmoredifficultthanpreviouslyanticipated.
0
2

n The ability to produce and trap molecules at sub-kelvin eration[23,24]andbuffergascoolingtechniques[15,25,26],
a
temperaturesoffers numerousexcitingpossibilities in chem- andearliertheoreticalstudies–basedonnon-reactivescatter-
J
istry. Recent experiments have demonstrated that ultracold ingcalculations–haveindicatedthatNHisapromisingcan-
9
chemical reactions can be efficiently manipulated using an didateforsecond-stagecoolingexperiments[12–17,19,27].

] external electromagnetic field [1–3], providing new tools to Within a magnetic trap, NH is polarized in the low-field-
h
controlreactionpathwaysandrateconstants.Thepronounced seekingstate|S = 1,M = 1i,withS denotingthe
p NH SNH NH
quantumbehaviorof ultracoldmattermayalsolead tonovel total electronic spin and M its projection onto the mag-
- SNH
m phenomenasuch as“superchemistry”,a processin whichan netic field axis. A collision complex of two such molecules
e atomic and molecular Bose-Einstein condensate (BEC) are is in the high-spin quintet |S = 2,MS = 2i state, with S
h coherentlycoupledtostimulatechemicalreactivity[4]. Upto andM referringtothedimerspinquantumnumbers. Spin-
S
c thepresent,however,studiesonultracoldchemicalreactions inelasticNH+NHcollisionscanchangeeithertheM quan-
S
.
s have focused only on (bi-)alkali-metalsystems with a rather tumnumberofthequintetstateorthetotalspinS toproduce
c
limited chemistry. Moreover,mostof the observedultracold singlet(S =0)ortriplet(S =1)complexes.SincetheS =0
i
s reactiveprocessesexhibituniversalbehavior,i.e.,thedynam- and 1 states are chemicallyreactive [21], S-changingtransi-
y
icsarecompletelydeterminedbylong-rangeinteractions[5]. tionsmayalsoinitiatechemicalreaction.
h
p Coldreactivecollisionsinthenon-universalregime,aswellas In order to study the reaction dynamics of cold NH radi-
[ coldreactionsinvolvingnon-alkaliandopen-shellmolecules, cals, we have employeda single-arrangementquantumreac-

arestilllargelyunexplored. tive scattering method that allows for the calculation of to-
1
v Coldchemistryisalsorelevantinthecontextofevaporative tal NH + NH reaction probabilities. The cross sections are
1 and sympathetic cooling. These second-stage cooling meth- obtained from full quantum scattering calculations on three
3 ods, which rely on strong elastic collisions between trapped (S = 0,1,2) coupled potential energysurfaces. Our results
9 particles, represent the final step towards full quantum de- indicatethatchemicalreactionscancausemoretraplossthan
1
generacy and Bose-Einstein condensation [6]. As a rule of inelastic collisions, implying that reactive channels must be
.
1 thumb,theratiobetweenelasticandnon-elastic(inelasticand takenintoaccountwhenassessing thefeasibilityofevapora-
0
reactive)crosssectionsshouldbeatleasttwoordersofmag- tive and sympathetic cooling. We also find that the total re-
3
nitudeinorderforsecond-stagecoolingtosucceed.Forpolar action probabilityis stronglydependenton the detailsof the
1
: molecules,whosepropertiesareexpectedtofindwideappli- (short-range)interactionpotentials,providingoneofthefirst
v
cationsinultracoldphysics[7–11],coolingintothequantum- examplesofnon-universalcoldchemistry.
i
X degenerateregimeisyettobeachievedexperimentally. Var- We focus on collisions between two magnetically trapped
r ious theoretical studies suggest that molecular second-stage bosonic15NH(X3Σ−)moleculesandtreatthemonomersas
a
coolingisfeasible[12–19],buttheseinvestigationsarebased rigidrotors.Thecoordinatesystemconsistsoftheintermolec-
on the assumption that chemical reactions are strongly sup- ularvectorR,withlengthR,thepolaranglesθ andθ ofthe
A B
pressed in a (magnetically trapped) spin-polarized gas [20] monomers,andthedihedralangleφ. ThescatteringHamilto-
and that the dynamics evolves on a single non-reactive po- nianisgivenby
tential. To our knowledge, this assumption has not yet been
¯h2 ∂2 Lˆ2
validatedbyexplicitreactivescatteringcalculations. Hˆ = − R+
2µR∂R2 2µR2
InthisLetter,wepresentrigorouscoupled-channelscalcu-
lationsforthereactiveNH(X3Σ−)+NH(X3Σ−)systemin + X |S,MSiVS(R,θA,θB,φ)hS,MS|
the presence of a magnetic field. Ultracold reactive NH + S,MS
NH collisions can yield as many as eight different product +V (R,Sˆ ,Sˆ )+Hˆ +Hˆ , (1)
magn.dip A B A B
arrangements[21, 22], making it a versatile system for cold
chemistrystudies. NHhasalreadybeencooledtomillikelvin where µ is the reduced mass of the complex, Lˆ2 is the
temperaturesandstoredinamagnetictrapusingStarkdecel- angular momentum operator associated with rotation of R,
2
10-11
Elastic
Elastic, S=2 only
20000 10-12 Inelastic
Inelastic, S=2 only
Reactive
10000 10-13
-1V (cm) 0 2σ (cm)10-14
10-15
-10000
0 45 10 10-16
90 8
135 6
θ (degrees) 180 4 R (a)
0
10-6 10-4 10-2 100
E (K)
FIG.1:Cutsthroughthequintet(cyan),triplet(magenta),andsinglet
(darkblue)potentialenergysurfacesofNH(X3Σ−)–NH(X3Σ−)
forφ=180◦andθ =θ =θ. FIG.2: Crosssectionsformagneticallytrapped15NH+15NHasa
A B
functionofcollisionenergy,calculatedforamagneticfieldstrength
of1G.Thesolidlineswereobtainedbyincludingallthreepotentials
inthescatteringcalculations, whilethedashedlineswere obtained
V (R,θ ,θ ,φ) is the potential energy surface for total
S A B usingonlythenon-reactivequintetpotential.
spinS,V (R,Sˆ ,Sˆ )istheintermolecularmagnetic
magn.dip A B
dipoleinteractionbetweenthetwomonomertripletspins,and
Hˆ andHˆ aretheHamiltoniansoftheindividualmonomers. the relevantcollision cross sections. Our algorithmamounts
A B
The latter account for the monomer rotation, intramolecular tosolvingthecoupled-channelsequationsfortwosetsofsolu-
spin-spin coupling, spin-rotation coupling, and Zeeman in- tions,andsubsequentlyapplyingreactivescatteringboundary
teraction [17]. The terms that couple the S = 0,1, and 2 conditionstoobtaintheS-matrix.Detailscanbefoundinthe
potentials and that ultimately drive spin-changing processes SupplementaryMaterial. Weusedasymmetry-adaptedchan-
aretheintermolecularmagneticdipoleinteractionandthein- nel basis set with even permutationsymmetry and even par-
tramolecularspin-spinandspin-rotationcouplings. ity, and a space-fixed total angular momentum projection of
TheNH–NHpotentialenergysurfaceshavebeenobtained M= 2(seealsoSupplementaryMaterial). Thisbasisallows
fromhigh-levelabinitiocalculations,asdescribedpreviously fors-wavecollisionsbetweenidenticalbosonicmoleculesin
[28]. Figure1showsacutthroughthethreeNH–NHpoten- the same initial quantum state. The radial grid ranged from
tialsforφ = 180◦ andθ = θ = θ. Itcanbeseenthatthe 4.5to1500a ,withaminimumof10gridpointspersmallest
A B 0
singlet and triplet potentials are strongly attractive at small (local) de Broglie wavelength. The cross sections were cal-
intermoleculardistances,whichisduetotheirchemicallyre- culatedforcollisionenergiesE of10−6 to1Kandmagnetic
active nature. In view of these deep potential energy wells fieldstrengthsB of10−1to104G.
we may assume that, once a reactive singlet or triplet NH– Figure2showstheelastic,spin-inelastic,andreactivecross
NHcomplexisformed,thesystemreadilyundergoesexoergic sections for two magnetically trapped NH molecules as a
chemical rearrangement. For instance, NH + NH may react function of the collision energy for B = 1 G. We find that
intoN H (providedthatathirdbodycandissipatetheexcess the elastic cross section at small E is constant as a function
2 2
kineticenergy)orintoabinaryproductconfigurationsuchas ofenergy,consistentwithWigner’sthresholdlawfor s-wave
N2+H2[22]. Inordertocalculatethetotalreactionprobabil- elastic scattering [30], σelastic ∝ ELin+Lout. Here Lin and
ity,weconsideronlytheNH+NHreactantarrangementand L denotethepartialwavesfortheincomingandoutgoing
out
apply“capture”boundaryconditionsat shortrange. Thatis, channels, respectively. The inelastic and reactive cross sec-
atasufficientlysmallvalueoftheradialcoordinateR,weal- tionsexhibitE−1/2 thresholdbehavior,asexpectedfromthe
lowfluxtodisappearintoreactivechannels.Suchanapproach ELin−1/2 lawforexoergics-wave(L = 0)collisions[30].
in
iscommonlyusedin(reactive)scatteringproblemsinvolving Inordertocompareourresultswiththenon-reactivecase,we
deep potential energy wells [29]. We note that not all colli- have also plotted the cross sections obtained from scattering
sions on the singlet and triplet potentials are reactive, since calculationson the non-reactivequintetpotential. As can be
thesesurfacesarerepulsiveforcertaingeometries. Hence,S- seeninFig.2,theinclusionofchemicallyreactive(S =0and
changing collisions will not necessarily lead to chemical re- 1)potentialshasanalmostnegligibleeffectontheelasticand
action.Thisisalsocontainedinourboundaryconditions(see spin-inelastic cross sections, confirming our earlier expecta-
SupplementaryMaterial). Collisionsoccurringonthequintet tionsreportedinRef.[27]. Wethusconcludethatmostofthe
potentialareentirelynon-reactive. non-reactive,inelastictraplossoccursonthequintetsurface,
We have developed a novel reactive scattering algorithm and thatcollisions on the S = 0 and 1 potentialsare almost
based on the renormalized Numerov propagatorto calculate 100%reactive.
3
10-10
B = 0
B = 0.1 G
10-12 Elastic B = 10 G
B = 1000 G
Universal limit
10-12
2σ (cm) 2σ (cm)10-14 Reactive
10-14
Elastic
Inelastic
Inelastic, Vmagn.dip=0 10-16 Inelastic
Reactive
Reactive, Vintra-SS=0
10-16
10-1 100 101 102 103 104 10-6 10-4 10-2 100
B (gauss) E (K)
FIG.3: Crosssectionsformagneticallytrapped 15NH+15NHasa FIG. 4: Cross sections for magnetically trapped 15NH + 15NH as
functionofmagneticfieldstrength,calculatedatE = 10−6 K.The afunctionof collisionenergy, calculatedforseveral magneticfield
spin-inelastic cross sections obtained with theintermolecular mag- strengths. The elastic cross sections are the same for all magnetic
netic dipole term switched off, and the reactive cross sections ob- fields considered, and the reactive cross sections are the same for
tained withtheintramolecular spin-spin coupling switched off, are B = 0,0.1, and10G.Theresultsoftheuniversalquantum-defect
alsoplotted. modelarealsoshown.
Figure 2 also indicates that (for B = 1 G) the elastic-to- is essentially a consequence of the ELin−1/2 law for exoer-
reactivecrosssectionratioismuchsmallerthantheelastic-to- gic collisions: the threshold behavior is determined only by
inelasticratio,thedifferencebeing2to3ordersofmagnitude. thecentrifugalbarrierintheentrancechannel,andtheenergy
Thisleadsustoreconsidertheprospectsformolecularevap- of the outgoing(reactive) channelhas no effecton the cross
orativecooling.Whilepreviousstudiesbasedonnon-reactive section. Notethatadifferentthresholdlawappliesforshort-
scatteringcalculationshavefoundthatevaporativecoolingof range-induced inelastic processes, for which the centrifugal
NH is feasible [17, 19, 27], our present results indicate that barrierandenergyoftheoutgoingchanneldoplayarole[34].
chemical reactions can pose a major constraint on the effi- Figure4comparesthescatteringresultsfordifferentmag-
ciencyofthecoolingprocess. Thus,second-stagecoolingof netic field strengths, including B = 0, as a function of col-
NHmightbemoredifficultthanpreviouslyexpected.Wewill lisionenergy. We findthatthereactivecrosssectionsarein-
returntothistopiclater,whendiscussingtheeffectofuncer- dependent of magnetic field for B < 100 G at virtually all
taintiesinthepotentials. energiesconsidered,andthe{B0,ELin−1/2}regimeextends
InFig.3,wepresentthecollisioncrosssectionsasafunc- down to B = 0. The inelastic cross sections, however, are
tion of magnetic field for a collision energy of 10−6 K. We differentforallBvalues,andtheybecomeconstantasafunc-
have already established in previous work [17, 31] that the tionofenergyforB =0[36]. Thus,thereactiveandinelastic
intermolecularmagneticdipolartermis the dominantsource cross sections show fundamentally different behavior in the
of inelastic trap loss at ultralow energies. Indeed, when dis- ultracoldregime.
carding the intermolecular magnetic dipole-dipole coupling Let us now consider ultracold NH + NH scattering in the
(V ) in our reactive scattering calculations, the spin- universal limit. Knowledge of the degree of universality is
magn.dip
inelastic cross section decreases by several ordersof magni- useful to understand and interpret the (generally complex)
tude. For the reactive cross sections, however, we find that full-dimensional reaction dynamics in terms of simple few-
the main contribution comes from the intramolecular spin- parametermodels. Inordertoestablishwhethertheultracold
spin coupling(denotedas V in Fig. 3). Theeffectof chemistryofNH +NHisuniversal,wecompareourscatter-
intra−SS
theintermoleculartermV onthechemicalreactivityis ing results with the single-channelquantum-defectmodel of
magn.dip
almostnegligible.Thiscanalsobeunderstoodbyconsidering IdziaszekandJulienne[5]. Detailsofthismodelaregivenin
that the intermolecular spin-spin interaction is long-ranged the Supplementary Material. In the universal limit, all scat-
[31], while a chemical reaction can only proceed when the tering flux that reaches the short range disappears into reac-
reactantsapproacheachothertoaveryshortdistance. Hence, tivechannels. Figure4 comparesthe results ofthe universal
theintramolecularspin-spincoupling,whichactsthroughthe quantum-defectmodel with those of the numerical coupled-
potential anisotropy at short range [32, 33], plays the most channelscalculations. In thes-wave regime(E < 10−3 K),
importantroleinenablingthereactiontooccur. theelasticcrosssectionsareunderestimatedbytheuniversal
Itmayalsobe seenthatthereactivecrosssection atsmall model, while the reactive cross sections are clearly overesti-
fields (B < 100 G) is constant as a function of B. This mated. The ratio between the universal elastic and reactive
4
10-9 approachalsoprovidesameanstosampletheeffectsofuncer-
tainties in the interaction potentials. Althoughthe potentials
10-10 Elastic BB == 11 0G0 G have been obtained from state-of-the-art methods, there is a
Universal limit smallremaininginaccuracyin the ab initio data whichgives
10-11 rise to an uncertainty in the calculated cross sections. This
alsocarriesimplicationsfortheprospectsformolecularevap-
2cm)10-12 Reactive orative cooling. More specifically, one should evaluate the
σ ( elastic-to-reactivecrosssectionratioforallrelevant λvalues
toobtaina realistic estimate ofthecoolingefficiency. In the
10-13
case ofλ = 1, we find thatreactiveNH +NH collisionsare
moreprobablethanelasticones(cf.Figs.3and4),whilefor
10-14 Inelastic λ≈0.95andλ≈1.03theelasticcrosssectionsareaboutone
orderofmagnitudelargerthanthereactiveones.Theseresults
suggestthatevaporativecoolingofmagneticallytrappedNH
0.9 0.95 1 1.05 1.1
λ
mightstillbefeasible,buttheprobabilityofsuccessissignif-
icantlysmallerthanestimatedearlierfromnon-reactivescat-
FIG. 5: Cross sections for magnetically trapped 15NH + 15NH as
teringcalculations. We notethatthesefindingsarestillvalid
afunctionofthescalingfactorλ,calculatedfortwomagneticfield
whenthesizeofthechannelbasissetisincreased,asdetailed
strengths (B = 1 and 100 G) at E = 10−6 K. The resultsof the
intheSupplementaryMaterial.
universalquantum-defectmodelarealsoshown.
ItcanalsobeseeninFig.5thattheinelasticcrosssections
change ratherdramatically from B = 1 to 100 G, while the
cross sections differs by more than one order of magnitude reactivecrosssectionsshowonlyaweakdependenceonmag-
(a factor of ≈ 26) from the numerical data. These differ- neticfield.Nevertheless,forcertainvaluesofλ,thereactivity
ences suggest that the NH + NH reaction dynamics is non- can increase by almost one order of magnitude as the mag-
universal, because a significant fraction of the incident flux netic field strength is changed. Thus, it may be possible to
is reflected at short range. It should be noted, however, that control the NH + NH reaction rate by means of an external
thequantum-defectresultsapplytosingle-channelscattering field. Formostλvalues,however,thereactivecrosssections
onasingleisotropicpotential,whilethenumericaldatahave showB0 behaviorandmagneticfieldcontrolwillnotbepos-
been obtainedfrom multi-channelcalculationson three cou- sible. We note that the final product-statedistribution might
pled anisotropic potentialenergysurfaces. Nevertheless, the be more sensitive to the magnetic field strength than the to-
collision dynamics in the ultracold regime is s-wave domi- tal reaction probability, but our scattering method does not
nated,andtheeffectiveNH–NHlong-rangepotential–which allowthecalculationofproduct-state-resolvedreaction cross
is the same for all three spin states – is governedmainly by sections.
the isotropic interaction. Thus, the single-channelquantum- In summary, we have presented the first rigorous quan-
defectmodelshouldbeapplicabletoNH–NHintheuniversal tum scattering study of ultracold reactive molecular colli-
regime. sions in the presence of a magnetic field. Our results illus-
An alternative way to establish the degree of universality trate the importance of chemical reactions in a magnetically
istotesttheeffectofsmallmodificationsinthe(short-range) trappedmoleculargas,andcalltoreconsidertheprospectsfor
potentials. Ifthescatteringisuniversal,thecrosssectionsare molecular evaporative cooling and magnetically-controlled
completely determined by the long-range features of the in- coldchemistry.Thisworkmayserveasabenchmarkforother
teraction potentials. For instance, a scaling of the potentials ultracold paramagnetic and dipolar molecules, the dynamics
by a factor of λ (or, equivalently, a scaling of the reduced ofwhicharestillvirtuallyunexplored.
mass by λ) should change the universalelastic cross section LMCJandGCG thankthe CouncilforChemicalSciences
byλ1/2. Figure5showstheuniversalquantum-defectresults oftheNetherlandsOrganizationforScientificResearch(CW-
asafunctionofthescalingparameterλ(0.9 ≤ λ ≤ 1.1)for NWO)forfinancialsupport. AvdAthankstheAlexandervon
E = 10−6 K. The corresponding numerical cross sections, HumboldtFoundationforaHumboldtResearchAward.
obtainedbyreduced-massscaling,arealsoshownforB = 1
and100G. Itis evidentthatthe numericalresultsare highly
sensitive to the details of the potentials, and that the univer-
sal modelis inaccurate forall valuesof λ. In fact, the reso-
nance featuresin the numericalcross sections are signatures ∗ Electronicmail:[email protected]
† Electronicmail:[email protected]
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