Non-Sticking of Helium Buffer Gas to Hydrocarbons
NNon-Sticking of Helium Buffer Gas to Hydrocarbons
James F. E. Croft and John L. Bohn
JILA, NIST, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440, USA (Dated: October 6, 2018)Lifetimes of complexes formed during helium-hydrocarbon collisions at low temperature are es-timated for symmetric top hydrocarbons. The lifetimes are obtained using a density-of-states ap-proach. In general the lifetimes are less than 10-100 ns, and are found to decrease with increasinghydrocarbon size. This suggests that clustering will not limit precision spectroscopy in helium buffergas experiments. Lifetimes are computed for noble-gas benzene collisions and are found to be inreasonable agreement with lifetimes obtained from classical trajectories as reported by Cui et al [1].
PACS numbers: 37.10.Mn, 36.40.-c
I. INTRODUCTION
In the gas phase, no two molecules can truly stick uponcolliding, unless there exists some mechanism for releas-ing their binding energy. Most often, this mechanisminvolves a third molecule, hence molecule clustering pro-ceeds more slowly as the density of the gas is reduced.Thus in molecular beam expansions, the occurrence ofseed gas atoms (typically noble gasses) adhering to en-trained molecular species can be controlled by varyingthe pressure of the expansion. Likewise, reduced three-body recombination rates in extremely rarefied ultracoldgases are what allow phenomena such as Bose-Einsteincondensation to be studied at all.More precisely, three-body recombination, scaling as n (where n is the number density), dwindles in compar-ison to the two-body collision rate which scales as n . Acomplication occurs when the two-body collisions resultin the formation of a collision complex with sufficientlylong lifetime τ that collision with another molecule canoccur during the interval τ , resulting in a bound stateof two particles and a release of energy, a process knownas the Lindemann mechanism [2]. The importance ofthis mechanism relies crucially on the scale of τ for agiven low-density gas. Recently it has been suggested– though not yet empirically verified – that τ can growquite long for extremely low-temperature gases, even oforder 100 ms for alkali-metal dimer molecules collidingin microKelvin gases [3]. If true, this process would leadto the decay of such a gas.Between the regimes of supersonic jet expansions andultracold molecules lies a host of experiments on coldmolecules, notably, those cooled to the temperature of anambient helium buffer gas in a cold cell at temperatureson the order of 1-10 K. The buffer-gas cell has proven tobe a reliable source of cold molecules, at temperaturessufficiently low to extend the reach of precision spec-troscopy [4–7]. The success of these experiments requiresthe helium buffer gas not to stick to the molecules understudy, as the spectral lines would thereby be shifted. Un-der typical experimental conditions, the collision rate ofthe buffer gas with the molecule is tens of µ sec [8]; spec-troscopy is safe if the collision complex lives for shortertimes than this. Thus far, no empirical evidence has emerged suggest-ing that the sticking occurs in the buffer gas environ-ment [4, 8–11], a conclusion that is supported by de-tailed classical trajectory calculations [1, 12, 13]. Thisappears to be true even for a relatively floppy moleculesuch as trans-stilbene, where comparatively low energyvibrational modes might have been expected to promotesticking [8]. The existing evidence suggests, therefore,that the transient lifetime of a hydrocarbon-helium com-plex in the buffer gas cell remains comfortably less than ∼ µ sec.In this article we argue that such short lifetimes arenatural and perhaps even generic under these circum-stances. The argument is based on considerations drawnfrom the theory of unimolecular dissociation, in which acomplex molecule with sufficient energy to dissociate nev-ertheless experiences a time delay before actually doingso. In this theory, the dwell time of the complex standsat the balance between excitation of degrees of freedomthat cannot dissociate while conserving energy (thus con-tributing to longer dwell times), and degrees of freedomthat can (thus contributing to shorter dwell times). Forcomplexes consisting of a hydrocarbon molecule with atransiently attached helium atom, both these densities ofstates may increase with increasing molecule size, so thatthe dwell time τ depends weakly on the specific hydrocar-bon. Based on simple ideas, we give order-of-magnitudeestimates for typical lifetimes in such a gas. II. BUFFER GAS ENVIRONMENT
Here we contemplate a buffer gas cell, in thermal equi-librium at temperature T , containing helium gas withnumber density n a and hydrocarbon molecules with den-sity n m (cid:28) n a , in which case the majority of collisionsthe molecules suffer will be with atoms. Upon intro-ducing molecules into the gas, atom-molecule collisionsoccur at a rate K am n a n m , defined by a rate constant K am . In the Lindemann model , these collisions produceshort-lived complexes that are characterized by numberdensity n c , and that decay at a mean rate γ = 1 /τ . Un-der these circumstances the atomic density is not signif-icantly depleted, and the collisions are described by the a r X i v : . [ phy s i c s . a t o m - ph ] D ec rate equations ˙ n m = − K am n a n m + γn c ˙ n c = K am n a n m − γn c . (1)After an equilibration time ∼ ( K am n a ) − , the fraction ofmolecules temporarily absorbed in complexes is n eqc n eqm ≈ K am n a τ, (2)a fraction that is negligible unless the dwell time τ is atleast comparable to the inverse collision rate.To place approximate numbers to this constraint, con-sider a typical helium number density of n a = 2 × cm − [8], and a collision cross section approximated bythe Langevin capture cross section [2], σ L = π (cid:18) (cid:19) / (cid:18) C k B T (cid:19) / ≈ × − cm , (3)assuming a van der Walls coefficient of C = 100 atomicunits (see below). The atom-molecule rate constant isthen K am ¯ vσ L ≈ × − cm /s. whereby the fractionof complexes is approximately n eqc n eqm ≈ τ µ s . (4)Thus for dwell times significantly less than 10 µ s, the com-plexes should be rare. In what follows, we estimate thelifetime, finding it to be at most 10-100 ns. Therefore, inthe buffer gas we expect fewer (probably far fewer) thanone molecule in a hundred to be involved in a collisioncomplex at any given time. III. RATES AND LIFETIMES
We are interested here in identifying an upper boundfor the sticking lifetime of helium atoms on hydrocarbonmolecules. The sticking process is denoted schematicallyasHe + M(X) → (He + M) ∗ (JM J ) τ (J , M J ) −−−−−→ He + M(X (cid:48) ) (5)where X are a set of quantum numbers, including molec-ular rotation N , which completely describe the state ofthe molecule. For the duration of the collision, the atomand molecule are assumed to reside in a complex withtotal angular momentum J . This angular momentum isregarded as the vector sum, in the quantum mechanicalsense, of the molecule’s rotation N and the partial waveof the atom-molecule relative motion, L . τ ( J, M J ) is thelifetime of a complex for total angular momentum J andprojection M J .At a given collision energy E c , collisions can occur inany of a set of incident channels, whose number is thenumber of energetically open channels, N ( J, M J ), for a given total angular momentum. The relevant mean stick-ing lifetime in the experiment is therefore the lifetime ofeach collision complex averaged over all J and M J combi-nations, and weighted by the number of incident channelsleading to that combination:¯ τ = (cid:80) J,M J τ ( J, M J ) N o ( J, M J ) (cid:80) J,M J N o ( J, M J ) (6)Within the Rice-Ramsperger-Kassel-Marcus (RRKM)model [2, 14, 15] the dwell time of a complex is approxi-mated as τ ( J, M J ) = 2 π ¯ h ρ ( J, M J ) N o ( J, M J ) , (7)where ρ ( J, M J ) is the density of available ro-vibrationalstates (DOS) of the complex for the given total angularmomentum. Thus the mean sticking lifetime is given by¯ τ = 2 π ¯ h (cid:80) J,M J ρ ( J, M J ) (cid:80) J,M J N o ( J, M J ) . (8)It must be emphasized that this approximation is an up-per limit to the lifetime, as it assumes that all the pos-sible states contributing to ρ are in fact accessible tobe populated in a collision. This assumption disregards,for example, barriers in the potential energy surface thatforbid a given entrance channel from probing a certainregion of phase space. This circumstance would reducethe effective density of states, hence also the lifetime. IV. DENSITY OF STATES
Following Mayle et al [3, 16] we estimate the densityof states ρ ( J, M J ) by a counting procedure. This be-gins by somewhat artificially separating the degrees offreedom of the He-molecule complex into those coordi-nates { X } necessary to describe internal motions of themolecule, and a coordinate R giving the relative motionof the atom and molecule. The enumeration of molecu-lar states follows from the spectrum of the molecule. Theatom-molecule states are approximated by postulating aschematic atom-molecule potential V ( R ).For a given molecular state with energy E ( X ), the po-tential V X,L ( R ) = V ( R ) + ¯ h L ( L + 1) / µR + E ( X ) isconstructed, so far as L and the molecular rotation areconsistent with the total angular momentum J underconsideration. The number of bound states N am ( X, L ) of V X,L , lying within an energy range ∆ E , centered aroundthe collision energy, is found. The density of these statesis then given by the sum ρ ( J, M J ) = 1∆ E (cid:48) (cid:88) X,L N am ( X, L ) . (9)The prime on the summation sign is a reminder thatthe sum is taken over those quantum numbers for whichenergy and angular momentum conservation are satisfied.In the model, the potential V ( R ) is assumed to beof Lennard-Jones form. This potential has a realisticvan der Waals coefficient for the He-hydrocarbon inter-action, as well as a reasonable minimum. A key point inthe lifetime analysis is that the parameters of this poten-tial are weakly dependent on the particular hydrocarboninvolved. Table I shows the equilibrium distance, vander Waals coefficient, and energy minimum for a vari-ety of Helium-Hydrocarbon systems. While this tablecomprises a variety of hydrocarbons of different shapesand sizes, the equilibrium distance and energy minimumvary little between them. This is because the heliumatom only interacts with the nearby atoms in the hydro-carbon. Further, the reduced mass for the collision of ahydrocarbon with helium is to a very good approxima-tion simply the helium mass.Thus the potentials V X,L ( R ), and the numbers ofstates N am ( X, L ) that they hold, vary little between dif-ferent helium-hydrocarbon systems. We therefore makethe approximation ρ ( J, M J ) ≤ ρ am (cid:48) (cid:88) X,L ≡ ρ am N m ( J, M J ) , (10)where N m ( J, M J ) is the number of states of the moleculeconsistent with angular momentum and energy conserva-tion. The quantity ρ am is a kind of representative atom-molecule density of states. It is conveniently approxi-mated by the inverse of the lowest vibrational excitation, ρ am = 1 E v =1 ,L =0 − E v =0 ,L =0 (11)Thus the complex lifetime is approximately boundedabove by ¯ τ ≈ π ¯ hρ am × (cid:80) J,M J N m ( J, M J ) (cid:80) J,M J N o ( J, M J ) . (12) System R e (˚A) V min (K) C (au) ρ am (K − )Helium+Methane 3.2 52 16 0.05Helium+Ethane 3.3 77 27 0.03Helium+Propane 3.7 82 62 0.03Helium+Butane 3.5 108 59 0.02Helium+Pentane 3.2 126 41 0.02Helium+Hexane 3.4 131 55 0.02Helium+Benzene 3.0 130 30 0.02Helium+Naphthalene 3.2 159 45 0.01Helium+Propandiol 3.4 115 56 0.02TABLE I: Equilibrium distance and energy minimum forthe helium-hydrocarbon interaction for a variety of differentsystems. Equilibrium distance and potential were obtainedin GROMACS [17] with the OPLS-AA force field [18, 19]. Ineach case, these data can be used to construct a schematicLenard-Jones potential V ( R ), leading to the atom-moleculedensity of states factor ρ am , defined in (11). Here the first factor has a generic approximate value forany He-hydrocarbon interaction (see Table I), while thesecond factor elaborates on the distinction between dif-ferent molecules. Upon increasing the density of statesof the molecule, both the numerator and the denomina-tor of this factor could increase. The counting exercisemust be done to find its ultimate effect on the molecularlifetime.
V. EFFECT OF ROTATIONAL STATES
Rotational splittings in large hydrocarbons tend to besmall compared to the collision energy in a buffer gas∆ E rot (cid:28) E c ≈
10 K while in general the vibrationalsplitting is larger ∆ E vib > E c . The dominant contribu-tion to N m and N o , in equation (12), arises therefore fromthe rotational levels of the molecule. Figure 1 shows thelowest rotational energy levels for both hexane and ben-zene. The rotational constants were obtained from theComputational Chemistry Comparison and BenchmarkDatabase (CCCBD) [20] and the energy levels computedwith PGOPHER [21]. It is seen that these two systemshave very different rotational energy levels.Shown in Figure 1 are two dashed lines. The lower oneis 10 K, the approximate collision energy at buffer gastemperatures. The upper one represents the the collisionenergy plus the depth of the schematic potential V ( R )between the atom and the molecule, a quantity denoted E max = E c + | V L =0 ( R min ) | . Ignoring for the momentconsiderations of angular momentum conservation, thetotal number of states belonging to any potential V X,L and lying in energy below E max denote potentially res-onant states that contribute to increasing the lifetime;states higher in energy than this do not satisfy energyconservation.In more detail, the number of states N m must becounted in a way consistent with the conservation of an-gular momentum. Thus for a given fixed total J thepossible rotation N and partial wave L quantum num-bers are considered, and the potentials V N,L = V ( R ) +¯ h L ( L + 1) / µR + E ( N ) constructed. If the minimumof this potential lies below the collision energy, then thisstate is energetically allowed and is counted as part of N m ; otherwise not (see Figure 2). Likewise, if the cen-trifugal barrier of the potential V N,L lies below the colli-sion energy, then the state is counted toward the numberof open channels N o . Otherwise, the collision is assumednot to tunnel through this barrier and does not countas an entrance or exit channel. This requirement is es-sentially the same as assumed in the Langevin capturemodel of collisions.Computing the sum in this way, we find lifetimes ofhexane and benzene to be approximately 36 ps and 44ps, respectively. Gratifyingly, the lifetime for benzene isconsistent with the far more detailed classical trajectorycalculations of Cui, Li, and Krems [1].Within such a model, we can consider the lifetimes for E n e r g y ( K ) E n e r g y ( K ) FIG. 1: (Color online) Rotational energy levels for hex-ane (upper panel) and benzene (lower panel). The dashedlines show the collision energy E c = 10 K and the highestpossible threshold energy that can contribute to the DOS, | V L =0 ( R min ) | + E c . While hexane has more levels above E c that can contribute to sticking, is also possesses more levelsbelow E c that can lead to dissociation of the atom-moleculescomplex. many hypothetical molecules, characterizing their rota-tional spectra by the symmetric top energy levels E ( N, K ) = BN ( N + 1) + ( A − B ) K . (13)These lifetimes assume, as above, that ρ am is approx-imately the same for all such molecules. To make thecalculation concrete, we assume the same value of ρ am and the schematic potentials V L,N as for hexane.Figure 3 shows the lifetime of symmetric top moleculeswithin this model, as a function of rotational constants A and B , at two different collision energies, E c = 1 and10 K. The longest lifetimes, for both collision energies,occur when B ≈ E c . The lifetime is only weakly depen-dent on A . As B decreases below E c the lifetime rapidlydecreases, because in this circumstance rotational levels Distance E n e r g y FIG. 2: (Color online) Schematic showing contributions to N m and N o . The dashed horizontal lines represent zero energyand the collision energy. The horizontal solid black lines onthe right represent rotational levels of the molecule E ( N ).Combinations of N and L consistent with angular momentumconservation lead to potentials V N,L , only a few of which areshown for clarity. Potentials V N,L whose minimum lies belowthe collision energy contribute to N m ; these potentials arecolored green. Likewise potentials V N,L with a threshold andcentrifugal barrier below the collision energy contribute to N o and are colored red. Potentials which cannot contributeto either are colored blue. lying below E c contribute to N o in addition to N m . Inaddition, as B increases above E c the lifetime slowly de-creases, as rotational levels are pushed higher and fewercontribute to N o . We therefore conclude that maximumlifetimes occur when B > ∼ E C . Finally, for a given molec-ular spectrum, the lifetimes are larger for lower collisionenergy E c , since relatively more of the molecular statescontribute to N m than to N o .For buffer gas experiments at 10 K this means the max-imum lifetime actually occurs for light species such asmethane, where B = 7 . µ s required for clustering to occur.This result is quite promising for the prospect of coolinglarge hydrocarbons. It is also worth remembering thatthe RRKM lifetime is an upper-bound on the actual life-time as it assumes ergodicity, an assumption that appearsjustified for helium-benzene collisions [1].These remarks are derived for symmetric topmolecules, but little should change for asymmetric topmolecules. The lifetime of a symmetric rotor at a givenbuffer gas temperature is primarily determined by itsprincipal rotational constant B . The rotational energylevels of an asymmetric top are intermediate betweenprolate and oblate limits. As such it is expected that,as with symmetric tops, the lifetime of asymmetric topswill also be primarily determined by B .Thus far we have considered lifetimes of collisioncomplexes of helium with large hydrocarbons, but onemay also contemplate buffer gas cooling large biological -2 -1 B (K)10 -2 -1 A ( K ) hexane benzene 100100010000 li f e t i m e ( p s ) -2 -1 B (K)10 -2 -1 A ( K ) hexane benzene 101001000 li f e t i m e ( p s ) FIG. 3: (Color online) Lifetime of helium symmetric-topmolecule clusters as a function of rotational constants A and B at E c = 1 K (upper panel) and E c = 10 K (lower panel).Using ρ am and V L,N as for hexane. Labeled are the lifetimefor the prolate top hexane and oblate top benzene, the pointlabeled is at the bottom left of the word. molecules such as nile-red [8]. In general such moleculeshave a hydrocarbon backbone with functional groupscontaining elements such as oxygen, nitrogen etc. Whilethe interaction of helium with such elements can bestronger (compare propandiol and propane in table I).this should be a minor effect. Indeed, that the estimatedlifetimes for propane and propandiol are 54 and 39 psrespectively.
VI. INFLUENCE OF VIBRATIONAL STATES
Including vibrational energy levels of the molecule willpresumably increase the lifetime of the complex, by in-creasing the density of states ρ without significantly in-creasing the number of open channels N o (this latter factfollows because the vibrational constant is likely to belarger than 10 K). The longest increase in lifetime will occur when a vibrational level exists just above the col-lision energy ∆ E vib > ∼ E c ≈
10 K, so that it contributesto N m but not to N o . Even in this case, perhaps tenvibrational levels would occur in the energy range upto E max , meaning that the lifetimes could increase fromthe estimates in the previous section by perhaps an or-der of magnitude, up to tens to hundreds of nanosecondsat E c = 10K. This short lifetime is consistent with thelack of clustering observed in trans-stillbene and nile redwhere low energy vibrational modes might have been ex-pected to promote sticking [8]. VII. ALTERNATIVE NOBLE GASES
Other noble gas (NG) atoms are potential candidatesfor buffer-gas cooling and supersonic-expansion experi-ments [22]. Cui et al [1] have reported noble-gas benzenecomplex lifetimes, for temperatures in the range 5-10 K,from classical trajectory simulations. Table II comparesthe DOS lifetimes with those of Cui et al at 10 K. Wecompute lifetimes separately for each of the cross-sectionsof the NG-benzene potential reported in [1]. As for theclassical trajectory lifetimes the DOS lifetime increaseswith NG mass, as deeper potentials lead to higher N and L quantum numbers contributing to N m .The DOS lifetimes always overestimate the classicaltrajectory lifetime and by an amount increasing with themass of the NG atom. The DOS lifetime assumes er-godicity, that is the full density of rotational states isactually populated in a collision. If this is not the casethen the lifetime of the cluster is reduced. We interpretthe increasing overestimation of the lifetime, with NGmass, as evidence that high rotational states available inthe collision are not necessarily populated. Intuitivelythis can be understood as the the shape of the benzenehindering the rotation of the NG atom around it. Never-theless, the estimates for experimentally relevant heliumbuffer gas remain fairly accurate, in cases where the com-parison can be made. τ (ps)System Out-of-plane Vertex-in-plane Side-in-plane Cui et al Helium 40 40 30 ∼ ∼ ∼ ∼ ∼ VIII. CONCLUSIONS
In the present work we have developed a method for es-timating helium-hydrocarbon complex lifetimes, using adensity-of-states approach, at low collision energies. Thismodel distinguishes between degrees of freedom that donot have energy to dissociate (contributing to longer life-times), and degrees of freedom that do (contributing toshorter lifetimes). The lifetime of a complex is deter-mined by the balance between these. We obtain lifetimesfor generic symmetric-top hydrocarbons finding that thelifetime decreases with increasing hydrocarbon size. Thisresult is extremely encouraging for using helium as abuffer gas for cooling large biological molecules, whichrelies on helium buffer gas not to stick to the molecules.This result is in agreement with all empirical evidence[4, 8–11] and other theoretical calculations [1, 12, 13]based on classical trajectories. Our approach comple-ments these, enabling a rough survey of molecular species and their behavior in the buffer gas environment.Finally, we note that in some case the lifetimes arenot always many orders of magnitude below 10 µ sec, butin some cases may be as high as tens to hundreds ofnanoseconds. Moreover, lifetimes increase at lower col-lision energies, while collision rates increase at higherbuffer-gas densities. Thus sticking may be an observableeffect, in slightly colder, denser helium cells, for well-chosen molecules. Acknowledgments
This work was supported by the Air Force Office ofScientific Research under the Multidisciplinary Univer-sity Research Initiative Grant No. FA9550-1-0588. Weacknowledge useful conversations with J. Piskorski, D.Patterson, and J. Doyle. [1] J. Cui, Z. Li, and R. V. Krems, J. Chem. Phys. ,164315 (2014).[2] R. D. Levine,
Molecular reaction dynamics (CambridgeUniversity Press, 2005).[3] M. Mayle, G. Qu´em´ener, B. P. Ruzic, and J. L. Bohn,Phys. Rev. A , 012709 (2013).[4] D. Patterson, M. Schnell, and J. M. Doyle, Nature ,475 (2013).[5] N. Tariq, N. A. Taisan, V. Singh, and J. D. Weinstein,Phys. Rev. Lett. , 153201 (2013).[6] V. A. Shubert, D. Schmitz, D. Patterson, J. M. Doyle,and M. Schnell, Angewandte Chemie International Edi-tion , 1152 (2014).[7] J. Baron, W. C. Campbell, D. DeMille, J. M. Doyle,G. Gabrielse, Y. V. Gurevich, P. W. Hess, N. R. Hutzler,E. Kirilov, I. Kozyryev, et al., Science , 269 (2014).[8] J. Piskorski, D. Patterson, S. Eibenberger, and J. M.Doyle, ChemPhysChem pp. 3800–3804 (2014).[9] D. Patterson, E. Tsikata, and J. M. Doyle, Phys. Chem.Chem. Phys. , 9736 (2010).[10] D. Patterson and J. M. Doyle, Molecular Physics ,1757 (2012).[11] J. H. Piskorski, Ph.D. thesis, Harvard University (2014). [12] Z. Li and E. J. Heller, J. Chem. Phys. , 054306 (2012).[13] Z. Li, R. V. Krems, and E. J. Heller, J. Chem. Phys. ,104317 (2014).[14] R. A. Marcus, J. Chem. Phys. , 352 (1952).[15] R. A. Marcus, J. Chem. Phys. , 355 (1952).[16] M. Mayle, B. P. Ruzic, and J. L. Bohn, Phys. Rev. A ,062712 (2012).[17] B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl,J. Chem. Theory Comput. , 435 (2008).[18] W. L. Jorgensen, D. S. Maxwell, and J. Tirado-Rives, J.Am. Chem. Soc. , 11225 (1996).[19] G. A. Kaminski, R. A. Friesner, T. J. Rives, and W. L.Jorgensen, J. Phys. Chem. B , 6474 (2001).[20] R. D. J. III, NIST computational chemistry comparisonand benchmark database,nist standard reference databasenumber 101 release 16a , http://cccbdb.nist.gov/ (2013).[21] C. M. Western,
PGOPHER, a program for simulatingrotational structure , http://pgopher.chm.bris.ac.uk, uni-versity of Bristol.[22] D. Patterson, J. Rasmussen, and J. M. Doyle, New J.Phys.11