Formation and dynamics of van der Waals molecules in buffer-gas trapsw

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1 PCCP Dynami Artile Links View Online Cite this: DOI: /1p21317b PAPER Formation and dynamis of van der Waals moleules in buffer-gas trapsw Nathan Brahms,* a Timur V. Tsherbul, b Peng Zhang, Jaek K$os, d Robert C. Forrey, e Yat Shan Au, bf H. R. Sadeghpour, A. Dalgarno, b John M. Doyle bf and Thad G. Walker g Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b Reeived 26th April 2011, Aepted 27th June 2011 DOI: /1p21317b We show that weakly bound He-ontaining van der Waals moleules an be produed and magnetially trapped in buffer-gas ooling experiments, and provide a general model for the formation and dynamis of these moleules. Our analysis shows that, at typial experimental parameters, thermodynamis favors the formation of van der Waals omplexes omposed of a helium atom bound to most open-shell atoms and moleules, and that omplex formation ours quikly enough to ensure hemial equilibrium. For moleular pairs omposed of a He atom and an S-state atom, the moleular spin is stable during formation, dissoiation, and ollisions, and thus these moleules an be magnetially trapped. Collisional spin relaxation is too slow to affet trap lifetimes. However, 3 He-ontaining omplexes an hange spin due to adiabati rossings between trapped and untrapped Zeeman states, mediated by the anisotropi hyperfine interation, ausing trap loss. We provide a detailed model for Ag 3 He moleules, using ab initio alulation of Ag He interation potentials and spin interations, quantum sattering theory, and diret Monte Carlo simulations to desribe formation and spin relaxation in this system. The alulated rate of spin-hange agrees quantitatively with experimental observations, providing indiret evidene for moleular formation in buffer-gas-ooled magneti traps. Finally, we disuss the possibilities for spetrosopi detetion of these omplexes, inluding a alulation of expeted spetra for Ag 3 He, and report on our spetrosopi searh for Ag 3 He, whih produed a null result. The ability to ool and trap moleules holds great promise for new disoveries in hemistry and physis. 1 4 Cooled and trapped moleules yield long interation times, allowing for preision measurements of moleular struture and interations, tests of fundamental physis, 5,6 and appliations in quantum information siene. 7 Chemistry shows fundamentally different behavior for old moleules, 8,9 and an be highly ontrolled based on the kineti energy, external fields, 10 and quantum states 11 of the reatants. a Department of Physis, University of California, Berkeley, California, USA. nbrahms@berkeley.edu b Harvard-MIT Center for Ultraold Atoms, Cambridge, Massahusetts, USA ITAMP, Harvard-Smithsonian Center for Astrophysis, Cambridge, Massahusetts, USA d Department of Chemistry and Biohemistry, University of Maryland, College Park, Maryland, USA e Department of Physis, Pennsylvania State University, Berks Campus, Reading, Pennsylvania, USA f Department of Physis, Harvard University, Cambridge, Massahusetts, USA g Department of Physis, University of Wisonsin, Madison, Wisonsin, USA w Eletroni supplementary information (ESI) available. See DOI: /1p21317b This journal is the Owner Soieties 2011 Experiments with old moleules have thus far involved two lasses of moleules: Feshbah moleules and deeply bound ground-state moleules. Feshbah moleules are highly vibrationally exited moleules, bound near dissoiation, whih interat only weakly at long range, due to small dipole moments. They are reated by binding pairs of ultraold atoms using laser light (photoassoiation), a magneti fields (rf assoiation), or slowly varying d magneti fields (magnetoassoiation). By ontrast, ground-state heteronulear moleules often have substantial dipole moments and are immune to spontaneous deay and vibrational relaxation, making these moleules more promising for appliations in quantum information proessing 7 and quantum simulation. 12 These moleules are either ooled from high temperatures using tehniques suh as buffer-gas ooling or Stark deeleration, or are reated via oherent deexitation of Feshbah moleules. This artile introdues a third family to the hierarhy of trappable moleules van der Waals (vdw) omplexes. vdw moleules are bound solely by long-range dispersion interations, leading to the weakest binding energies of any ground-state moleules, on the order of a wavenumber (Bone kelvin). This is three orders of magnitude smaller than the binding energy of a typial ionially bound moleule.

2 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b The nulei are bound at long range, and ground state eletroni wavefuntions often differ from the onstituent atomi wavefuntions only by small perturbations. These unique harateristis have made vdw omplexes an attrative platform for the study of hemial reation dynamis and surfae interations. 18 They also play a key role in nonlinear optial phenomena and deoherene of dense atomi and moleular gases. 19 One formed, the vdw moleules an deay via ollision-indued dissoiation, 20 hemial exhange 14,16,17 and eletroni, vibrational, rotational, and Zeeman predissoiation. 18,21 In vdw omplexes with small binding energies (suh as He O), the Zeeman predissoiation an be ontrolled with an external magneti field. 21 A partiularly important lass of vdw moleules, formed by an S-state metal atom (M) binding with a rare gas atom (Rg), has been the subjet of extensive researh for deades. 18,22 These studies have used a variety of experimental tehniques to produe and ool vdw moleules, inluding (1) supersoni expansions, (2) immersion in dense rare gas, (3) immersion in bulk liquid 4 He, and (4) doping of superfluid 4 He nanodroplets. We present a brief overview of this earlier work below, with a partiular emphasis on vdw omplexes of alkali- and noble-metal atoms with Rg of relevane to the present work. (1) In a typial moleular beam experiment, vdw lusters are produed in a supersoni expansion of M + Rg mixtures and probed via laser spetrosopy Examples inlude studies of NaNe and KAr via mirowave spetrosopy, 23,24 CuAr, 27 AgAr, and Ag 2 Ar via two-photon ionization spetrosopy 25,28 and laser-indued fluoresene exitation spetrosopy, 26 and AuRg (Rg = Ne, Kr, Xe) via multiphoton ionization spetrosopy These studies provide valuable spetrosopi information on fine and hyperfine interations in vdw moleules in their ground 23,24 and exited 29,30 eletroni states. (2) vdw moleules also play an important role in the dynamis of speies interating with a dense rare gas vapor. Spontaneous formation of vdw moleules plays a key role in the deoherene of alkali vapors in buffer-gas ells. They were first observed in the spin relaxation of Rb due to the formation of RbAr and RbKr moleules. 35 Suh moleules mediate effiient spin-exhange between alkali atoms and rare-gas nulei 34 and have been shown to limit the preision of vapor-ell based atomi loks. 19 Dense Rg vapors have also been used to spetrosopially observe MRg eximers. 35 (3) For vdw moleules omposed of a helium (He) atom dispersively bound to a moiety, binding energies are typially on the order of a wavenumber, orresponding to temperatures below one kelvin. Due to their weak binding energies and vulnerability to ollisions, He-ontaining vdw moleules are typially not observed in supersoni expansions, but an be produed by a number of alternative tehniques suh as immersing atoms in liquid 4 He or dense 4 He gas. 35,36 These studies have enabled the observation of vdw lusters formed from ions 37,38 or exited-state neutral atoms. 39 (4) Yet another powerful experimental tehnique for studying vdw moleules is based on He nanodroplet spetrosopy. Researh has inluded spetrosopi investigations of the formation of RbHe exiplexes on the surfae of a superfluid He droplet, 40,41 and of the effets of spin orbit interations on the formation of NaHe and KHe eximers upon optial exitation of alkali-doped nanodroplets. 42,43 He nanodroplet spetrosopy an also give information on the harater of M Rg fores, showing, for example, that Ag atoms reside at the enter of He nanodroplets In this artile, we explore the formation and dynamis of He-ontaining van der Waals omplexes in buffer-gas ooling experiments. 47 This method is similar to the dense gas prodution method (2) above. In buffer-gas ooling experiments, however, the prodution of van der Waals moleules is favored by ombining moderate Rg vapor densities with temperatures muh older than the vdw moleules binding energy. In ontrast to the methods above, He-ontaining vdw moleules in suh a system have relatively long hemial lifetimes (on the order of 10 to 100 milliseonds), and are favorably produed in their absolute rovibrational ground state. These harateristis offer a unique laboratory for studying the dynamis of these moleules, inluding their formation proesses, photohemistry, and sattering properties. In addition, buffer-gas ooling experiments are readily integrated with external fields, espeially magneti traps. A large number of He-ontaining vdw moleules are paramagneti, raising the possibility for vdw moleule trapping with lifetimes up to hundreds of seonds, and perhaps even ooling into the ultraold regime. We begin with a general model for the formation of He vdw moleules, desribing the thermodynamis and hemial kinetis of the system. We predit that a wide variety of reatants, inluding metal and nonmetal atoms, as well as moleular speies, will readily bind in these systems. We desribe the ollisional properties of these vdw moleules within magneti traps. We provide a detailed model for the ase of Ag 3 He. Using ab initio models of the moleular struture and quantum ollision alulations, we desribe the formation and ollision dynamis of Ag 3 He. In a previous letter, 48 we argued that the observed dynamis 49 of Ag in a buffer-gas trap with 3 He provided indiret evidene for the formation of Ag 3 He moleules; here we use our theoretial model to present this argument in detail. Finally, we disuss the possibility for spetrosopi observation of vdws formed in buffer-gas ooling experiments, introduing a new tehnique enabled by the sensitivity of buffer-gas trapping to the ollisional dynamis of vdw moleules. 1 General model Buffer gas ooling experiments 47 operate by introduing a hot vapor (either via ablation or via a thermal beam) of a target speies X (atom, moleule, or ion) into a moderately dense, ryogenially ooled vapor, typially He. After approximately 100 ollisions, X is ooled to the temperature of the oolant vapor, whih an be as old as 300 mk for 3 He, or 700 mk for 4 He. The old X then diffuse through the He until they reah the walls of the ryogeni ell. Without a magneti trap, this method typially provides densities of m 3 for atomi speies, or m 3 for moleular speies. The timesale for diffusive loss of the ooled speies lies between a few and a few hundred milliseonds, This journal is the Owner Soieties 2011

3 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b proportional to the density of the buffer gas, whih is ontrolled between a few and m 3. Due to these low temperatures, the formation of van der Waals omplexes an be thermodynamially favored. We onsider formation of vdws due to inelasti three-body proesses between a X and two He atoms. Suh ollisions an lead to pair formation via the proess: X þ He þ HeÐ K XHe þ He: ð1þ D Here K and D are the rate onstants for three-body reombination and ollision-indued dissoiation. We speifially ignore ollisions involving multiple X partners, as the density of He is typially 4 to 6 orders of magnitude larger than the X density. The pair formation and dissoiation kinetis for this proess are : n XHe = n : X = n X /t f n XHe /t d, (2) where n i denotes the density of speies i, and the formation time t f and dissoiation time t d obey 1/t f = Kn 2 He and 1/t d = Dn He. If the timesale for pair formation and dissoiation is fast ompared to the lifetime of X within the experiment, the density of pairs will ome into thermal equilibrium with the : free X and He densities. In thermal equilibrium, n XHe =0 implies n XHe = k(t)nxn He, where k(t) =K/D is the hemial equilibrium oeffiient, derived from statistial mehanis: 50 k ¼ n XHe n X n He ¼ l 3 db X g i e e i=k B T : Here k B is Boltzmann s onstant, T is the temperature, and e i is the energy of moleular state i, having degeneray g i, where e i is zero at dissoiation. Note that bound states have negative e i. l db is the thermal de Broglie wavelength p of the omplex, of redued mass m, given by l db ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 =2pmk B T. Eqn (3) shows that n XHe inreases exponentially as the temperature is lowered, or as the binding energy of the moleule is made more negative. A large number of vdw omplexes have binding energies that are omparable to or greater than 1 K, and thus formation is thermodynamially favored in buffer gas ooling experiments. (Alkali-metal He pairs 51 are a notable exeption, having binding energies below 0.03 m 1.) Table 1 gives a sampling of andidate X speies, showing binding energies and predited population ratios n XHe n X ¼ kn He at standard temperatures and buffer gas densities. Eqn (2) implies that the timesale to reah thermal equilibrium is t 1 eq = t 1 f + t 1 d = Dn He (1 + kn He ). (4) An exat alulation of D requires knowledge of the speifi XHe He interation potential. However, an estimate an be obtained by assuming that dissoiation ours when the energy of the XHe He ollision omplex exeeds the moleular binding energy. The fration of ollisions with energy greater than the binding energy is, in the low-temperature limit, Be e0/kbt. The formation rate is therefore estimated by letting D = sv m e ei/kbt, where s is the elasti ross setion and v m is the average relative ollision veloity. This gives This journal is the Owner Soieties 2011 i ð3þ t 1 eq E sn He v m (e ei/kbt + g i l 3 dbn He ). (5) Table 1 Predited ground-state energies e 0 and pair population ratios n XHe n X ¼ kn He of some speies ompatible with buffer gas ooling, for whih XHe internulear potentials are available Moleules are assumed to be in their absolute rovibrational ground state, with interation potentials taken from ref. 58 (NH He), 59 (CaH He), 60 (YbF He), and 61 (MnH He) X State X 3 He e 0 a n XHe n X b Typial buffer gas ooling parameters are n He =10 16 m 3, with T = 300 mk when using 3 He, or T = 700 mk when using 4 He. Using m E 3 or 4 amu and a worst-ase elasti ross setion of s =10 15 m 2 gives an equilibrium time t eq r 4ms for pairs with e 0 o 1m 1 and t eq r 600 ms for all values of e 0. This timesale an be ompared to the typial lifetime of a buffer gas trapped speies, around 100 ms without a magneti trap, and \1 s with a magneti trap. We therefore expet, in general, that the vdw pair density will reah thermal equilibrium in buffer gas ooling experiments. Until now, we have negleted the formation of larger vdw omplexes. These will form by proesses similar to the pair formation proess, via XHe m þ He þ HeÐ Km X 4 He e 0 a n XHe n X b Si 3 P Ge 3 P N 4 S 3/ P 4 S 3/ As 4 S 3/ Bi 4 S 3/ O 3 P S 3 P Se 3 P F 2 P 3/ Cl 2 P 3/ Br 2 P 3/ I 2 P 3/ Li 2 S 1/2 d Na 2 S 1/2 d Cu 2 S 1/ Ag 2 S 1/ Au 2 S 1/ NH 3 S CaH 2 S YbF 2 S MnH 7 S a Energies in m 1,1m 1 E 1.4 K. b Pair population ratios for n He = m 3, at 300 mk for X 3 He and at 700 mk for X 4 He moleules, for the level with energy e 0. At these parameters, pairs are subjet to runaway lustering. The equilibrium density an be hosen by raising T or lowering n He. d No bound states are predited for Li 3 He or Na 3 He. D m XHe mþ1 þ He: ð6þ In thermal equilibrium, the density of lusters ontaining m + 1 He atoms is related to the density of those having m He atoms by the hemial equilibrium oeffiient k m for this proess. If we assume that all the k m are approximately equal (reasonable for small m, where the He He interations are negligible), then the density of lusters with m + 1 He atoms is Bn X (kn He ) m. Higher-order lusters should therefore be favored one kn He \ 1. See Table 1 for examples of atoms where lustering should our. For e 0 on the order of a few m 1, lustering an be ontrolled by adjusting the ell

4 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b temperature, with pairs favored for k B T \ e 0 /8. We posit that runaway lustering might be responsible for the heretofore unexplained rapid atom loss observed in experiments using 300 mk Au and Bi. 52, vdw omplexes in magneti traps For paramagneti speies, both the density and lifetime of buffer-gas-ooled speies an be signifiantly inreased by the addition of a magneti trapping field. Suh a trap is usually omposed of a spherial quadrupole field, with a magnetifield zero at the trap enter, and a magneti-field norm rising linearly to a few Tesla at the trap edge. A fration (approximately half) of X (the weak-field seekers ) will have magnetimoment orientations suh that their energy is minimized at the enter of the trap. Suh a trap typially results in a two order of magnitude inrease in density, and lifetimes up to tens of seonds. However, in order to stay trapped, the magneti orientation of the speies with respet to the loal field must be stable. Spin-hanging ollisions with the buffer gas, in partiular, will lead to relaxation of the magneti orientation, and subsequently the relaxed partiles will be lost from the trap. We now show that, for the speial ase where X is an S-state atom, the XHe moleule will be spin-stable, and remain trapped, even through formation and ollision proesses. Beause the X + He + He ollision omplex laks any strong diret oupling between the spin orientation of X and the degrees of freedom of the He atoms, we expet the spin orientation to be proteted during both vdw assoiation and dissoiation. A quantitative estimate for spin hange during assoiation and dissoiation proesses an be obtained by assuming this rate to be similar to the spin-hange rate in two-body X + He proesses. For speies ompatible with buffer-gas trapping, 47 this rate is typially negligible, on the order of t10 6 per formation or dissoiation event. Van der Waals pairs that have formed from a trapped X and a He atom are, therefore, also trapped. These pairs may, however, suffer spin-hange at a rate faster than the unbound X. Spin-hange of vdw pairs has previously been studied in optial pumping experiments, 33 in whih a hot (4300 K) spinpolarized alkali vapor diffused in a Kr, Xe, or N 2 buffer gas. In these experiments, vdw moleules were formed between the alkali metal and the buffer gas, and suffered spin-hanging ollisions with additional buffer gas. In this proess, the eletron spin preesses internally due to either the spinrotation or the hyperfine interations; this preession an deohere during a ollision. We onsider these interations using the moleular Hamiltonian ^H mol ¼ e N þ A X I X S þ B ð2m B S þ m X I X þ m He I He Þ rffiffiffiffiffi 8p X 2 þ gn SþA He I He S þ Y 2;q 15 ð^rþ½i He SŠ ð2þ q : q¼ 2 ð7þ Here e N is the rovibrational-eletroni level energy and B is the magneti field. N is the rotational angular momentum, S is the eletron spin, g is the spin-rotation onstant, I X is the nulear spin of X with moment m X, A X is the atomi hyperfine onstant, and m B is the Bohr magneton. The last two terms in eqn (7) desribe the isotropi and anisotropi hyperfine interation of S with a 3 He nulear spin I He. We neglet both the small nulear-spin orbit interation and the weak anisotropi part of the I X S interation. The interation parameters g, A He, and an be estimated using the approximate methods ontained in ref. 34. Now onsider an interation of the form asj, where J represents, e.g., N or I He. This interation mixes the spinpolarized state with less-strongly trapped states. Collisions an ause deoherene of this mixing, e.g. via angular momentum transfer, moleular dissoiation, or nulear exhange. We overestimate the spin-hange rate by making the gross approximation that all ollisions ause deoherene with 100% probability. That is, we assume that ollisions serve as projetive measurements of S z. The probability of spin-hange in a ollision is now simply the overlap between the moleular eigenstate and more weakly trapped spin states. For relatively large magneti fields (2m B B \ a, whih is the ase for all but the entral mm 3 of the trap), this overlap an be found using perturbation theory. To first order, the interation only auses an overlap with the m S = s 1 state: ji js; m jiþ ffiffiffi a p 2s 4m B B C j;mjs 1; m j þ 1i : ð8þ 1 þ a2 s C 2 8m 2 B B2 j;m p Here C j;m ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðj m j Þðj þ m j þ 1Þ. The probability p SC that the spin relaxes is taken from an average over possible values of m j, giving p SC a2 sjðj þ 1Þ 12m 2 : ð9þ B B2 Finally, we average this probability over the magneti-field distribution of an anti-helmholtz quadrupole trap: p SC 1 2m 3 Z B B 2 e 2m BB=k B T p SC ðbþ db; 2 k B T ð10þ a2 sðj 2 þ jþ 6k 2 : B T2 For typial internulear separations of B10 a 0, the hyperfine onstants are Bh 1 MHz. For the ontat hyperfine with T E 0.3 K, this beomes an average per-ollision spin-hange probability of a few With mean ollision times on the order of a few miroseonds, we find that this type of spinhanging ollision will be too rare to signifiantly impat the trap lifetime. A similar analysis as above an be applied to tensor interations suh as anisotropi hyperfine, with Dm = 2 transitions also allowed at first order. However, the results are of a similarly small magnitude. We note that for ground-state moleules N = 0. Spin-rotation interations, therefore, an only our as a virtual oupling within ollisions. We expliitly onsider this mehanism in our treatment of Ag 3 He 3 He ollisions below. We find that the spin is similarly proteted by the trap magneti field, with a spinhange probability too small to play a role in trap loss. This journal is the Owner Soieties 2011

5 Finally, the anisotropi hyperfine interation an ouple trapped ground states to untrapped exited rotational levels. At trap magneti fields where these levels ross, this oupling auses an avoided rossing, and moleules an adiabatially transfer between trapped and untrapped states. In ertain ases this an be the dominant loss proess. We detail this proess in our treatment of Ag 3 He in Setion 2.5 below. Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b 2 Ag 3 He moleules We now apply our analysis to the reently reported experimental work 49 that studied silver (Ag) trapped using buffer-gas ooling with 3 He. In this experiment, B10 13 Ag atoms were ooled to temperatures between 300 and 700 mk using buffer-gas densities between and m 3. For all experimental parameters, exponential loss of the trapped atoms was observed. By fitting the loss rate as a funtion of buffer-gas density at eah temperature, the ratio of the rate of Ag spin-hange to the elasti Ag 3 He ollision rate was extrated. This ratio displayed a strong empirial T 6 temperature dependene. In this setion we apply quantum ollision theory analysis to show that the observed spinhange rate ould not result from a standard Ag 3 He inelasti ollisional proess; we show that the formation and subsequent spin-hange of Ag 3 He moleules quantitatively explains the observed spin-hange rate. 2.1 Moleular struture We begin our analysis by onstruting an ab initio internulear potential energy urve for Ag He. Potentials for this system have previously been onstruted 55,62,63 in studying the AgHe* exiplex. To onstrut our potential, shown in Fig. 1, we employed the partially spin-restrited oupled luster method with single, double and perturbative triple exitations (RCCSD(T)) 64 as implemented in the MOLPRO suite of programs. 65 The referene wave funtions for the eletroni ground Ag( 2 S) He and exited Ag( 2 P) He and Ag( 2 D) He omplexes have been obtained from restrited Hartree Fok alulations (RHF). We employed the augmented, orrelation-onsistent basis set (aug--pvqz) for the He atom. 66 For the Ag atom, we used an effetive ore potential from the Stuttgart/Cologne group, 67 whih desribes the first 28 eletrons of the Ag atom (ECP28MCDHF), oupled with a pseudo-potential based on the aug--pvqz-pp basis set of Peterson and Puzzarini 68 to desribe the remaining 19 eletrons. This basis was additionally enhaned by using bond funtions omposed of 3s3p2d2f1g1h funtions, with their origin half-way between the Ag and He atoms. The bond funtions had the following exponents: sp, 0.9, 0.3, 0.1, df, 0.6, 0.2 and gh, 0.3. The bond funtions were added to assist the inomplete atomi-entered basis set used in the desription of the van der Waals interation. The interation energy is orreted for the basis-set superposition error by employing the ounterpoise proedure of Boys and Bernardi. 69 We monitored the T 1 -diagnosti to ensure that the referene wave funtions are mostly desribed by a single determinant. During oupled-luster alulations the T 1 diagnosti was around for Ag( 2 S) He, for Ag( 2 P)-He and Fig. 1 Ab initio interation potentials for the X 2 S + (lower panel) and O = 1/2, 3/2, and 5/2 (upper panel) states of the AgHe omplex as funtions of the internulear separation r. The energy of the N =0 ground state is shown. The exited-state potentials are alulated by diagonalizing the Hamiltonian matrix using the non-relativisti AgHe potentials of S, P, and D symmetry omputed in the present work (see text). The inset shows the region of (avoided) rossing between the potential energy urves orrelating with the 2 P 3/2 and 2 D 5/2 eletroni states of Ag. for Ag( 2 D) He, so we ould apply a single-referene RHF/ RCCSD(T) approah for all omplexes. The positions and well depths of the potentials are haraterized in Table 2. The potentials are quite shallow, exept that of the A 2 P state (the depth originates from the 2 P term of the Ag atom). Our potential may be ompared to the results of previous works (Table 2), whih use varying Table 2 Equilibrium distanes r e and well depths D e for nonrelativisti AgHe omplexes. The minima are reported with respet to the asymptote of eah eletroni term of the Ag atom. Spin orbit interations are not inluded. The labeling sheme is that of Jakubek and Takami 62 Complex State ( S L) r e /a 0 D e /m 1 Ref. Ag( 2 S) He X 2 S Present Ag( 2 P) He A 2 P Present A 02 S Present Ag( 2 D) He B 2 S Present B 02 P B 002 D This journal is the Owner Soieties 2011

6 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b basis sets, bond funtions, and levels of theory. Our results are losest to those of Tong et al., 70 who used a similar basis set (d-aug--pvqz) and the ab initio method with CCSD(T). Those of Cargnoni et al., 55 whih used the CCSDT level of theory and the d-aug--pvqz set, and of Gardner et al., 71 whih used RCCSD(T) and the d-awcvnz basis set, show a slightly shorter minimum radius and B10% deeper depth. The older potential of Jakubek and Takami 62 used the CIS/MP2 level of theory, and shows a muh deeper depth. The A 02 S + state orrelating to the 2 P term is pratially repulsive; we find a very shallow minimum approximately 1m 1 deep loated very far from the Ag nuleus, at around 15 a 0. The A 2 P potential of the AgHe* exiplex exhibits the deepest minimum relative to its asymptoti limit. Due to the effet of the bond funtions, our minimum is deeper than the one of Jakubek and Takami 62 and of Cargnoni et al. 55 Cargnoni and Mella 63 have reently published new alulations of the Ag( 2 P)-He potentials, whih used CISDT alulations, obtaining a shallower well depth for the A 2 P state. The potentials are shown, inluding spin orbit effets, in Fig. 1. Notably, the spin orbit oupling is seen to add an effetive barrier to the A 2 P 1/2 state, ausing its bound levels to have lassial turning points at radii o6 a 0. The binding energies of vdw moleules ontaining S-state Ag were alulated by solving the one-dimensional Shro dinger equation using the DVR method, 72 yielding the rovibrational energy levels e vn and wavefuntions vn (r). For vdw moleules formed by P-state Ag, both V S and V P potentials were inluded in bound-state alulations and the variation of the spin orbit oupling oeffiient with r was negleted. 73 The binding energies of atom moleule vdw omplexes were alulated using the variational method of ref. 74 assuming the validity of the rigid-rotor approximation for rotational energy levels of the monomer. The AgHe moleular potential supports one vibrational bound state, with rotational quantum numbers N = 0, 1, 2 at energies e N = 1.40, 1.04, 0.37 m 1, shown in Fig. 1. We also note the existene of a quasibound N = 3 state at e 3 = 0.48 m 1, with a alulated lifetime of B1 ns. The predited hemial equilibrium oeffiient for Ag 3 He populations is shown in Fig. 2, alongside the predition using the potentials of ref. 55. We observe that, due to its Boltzmann fator, k(t) is very sensitive to fine details of the ab initio interation potentials. At T = 0.5 K, a 10% hange in the binding energy (from 1.40 to 1.54 m 1 ) leads to a 50% inrease in k. Measurements of the moleular population should therefore serve as preise tests of intermoleular interations, whereas even aurate theoretial alulations of interation potentials will provide only estimates of moleular populations. The ab initio permanent eletri dipole moment for the ground eletroni state of AgHe is shown in Fig. 3. We observe that the moleule is slightly polar, having a dipole moment expetation value of h 00 (r) d(r) 00 (r)i = 0.004ea 0 in its ground state. The Zeeman spetrum of Ag 3 He, onsisting of idential hyperfine manifolds for eah rotational level, is shown in Fig. 4. To alulate the r-dependent isotropi and anisotropi hyperfine interation onstants A He (r) and (r), we employed Fig. 2 Chemial equilibrium onstants for AgHe alulated as funtions of temperature using the RCCSD(T) interation potential omputed in this work (full line) and the RCCSDT interation potential (dashed line) from ref. 55. Fig. 3 Ab initio dipole moment of the AgHe( 2 S 1/2 ) omplex as a funtion of r (irles). The ground-state rovibrational wavefuntion of AgHe is superimposed on the plot (dashed line, arbitrary units). quasirelativisti density funtional theory (DFT) using the perturbatively orreted double hybrid funtional B2PLYP, 75 whih ombines the virtues of DFT and seond-order perturbation theory to improve the desription of the eletron orrelation. Relativisti effets have been taken into aount by the zeroth order regular approximation (ZORA) 76,77 of the Dira equation, whih has shown good performane for hyperfine onstants of heavy elements. 78,79 A fully unontrated and modified WTBS 80 basis (31s21p19d7f4g) was used for the Ag atom, where the f and g funtions were taken from the d funtions, and a set of spdf diffuse funtions were added by the even-tempered manner, and two extra tight S funtions were augmented by multiplying the largest S exponent of the parent basis by a fator of 3. For the He atom, a fully unontrated (12s7p4d3f2g) basis 81 was adopted. The lose agreement between the experimental hyperfine onstant of the This journal is the Owner Soieties 2011

7 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b Fig. 4 Zeeman energy levels of the AgHe moleule as alulated by diagonalizing the Hamiltonian of eqn (7). The avoided rossing between the N = 0 and N = 2 rotational levels enirled in panel (a) is sequentially magnified in panels (b) and (). The magnitudes of the splittings are given in khz in panel (). Ag atom, A Ag /h = 1713 MHz, 82 and the omputed asymptoti value of 1694 MHz validates the urrent approah. For the spin-rotation parameter g(r), we use the perturbative result from ref. 83: gðrþ ¼ 2h6 a 2 D SO 3m 2 f 2 e mr2 S ðrþf2 P ðrþ; D 3 SP ð11þ where D SO /h = m 1 is the spin orbit splitting of the lowest exited 2 P term of Ag, D SP /h = m 1 is the splitting between the 2 S and 2 P terms, a = a 0 is the S-wave sattering length for eletron He ollisions, 84 and m e is the eletron mass. The radial wavefuntions, normalized as R f(r) 2 r 2 dr = 1, are taken from the Hartree Fok alulation of ref. 85 for the 5s state and alulated using the quantum defet method 52 for the 5p state. These interations, as funtions of nulear distane, are shown in Fig. 5. Their values, averaged over the N = 0 nulear wavefuntion, are A He = h 0.9 MHz, = h 1.04 MHz, and g = h 180 Hz. 2.2 Ag 3 He ollisions We now alulate the spin-hange rate g SC of buffer-gas trapped Ag due to two-body Ag 3 He ollisions. We first alulate the Ag 3 He elasti and diffusion ross setions s d using the ab initio AgHe potential alulated in this work (Setion 2.1) by numerially integrating the Shro dinger equation for ollisional angular momentum up to l = 5 to produe the ross setions as a funtion of ollision energy, shown in Fig. 6(a). The experimentally measured rate of atomi spinhange an be extrated from the measured elasti inelasti ratio x using g SC = xs d v m. The experimental spin-hange rates are shown in Fig. 7. To show that two-body atomi ollisions annot aount for the measured thermal behavior of the spin-hange rate, we performed quantum ollision alulations of spin exhange and spin relaxation rates in Ag( 2 S) 3 He ollisions using the same quantum sattering approah as developed earlier for the Fig. 5 Spin-oupled interations in the AgHe omplex as a funtion of the internulear separation: isotropi hyperfine interation A He (red dotted line), anisotropi hyperfine interation (blak dashed line), and unsaled (f s = 1) spin-rotation interation g (blue solid line). The zero-energy lassial inner turning point of the AgHe potential is shown by the green vertial line. Note the rapid falloff of the spinrotation interation with r. Fig. 6 Elasti (upper panel) and spin relaxation (lower panel) ross setions for Ag 3 He ollisions as funtions of ollision energy. The solid line in the lower panel shows the ontribution due to hyperfine interations, while the dashed line shows the ontribution due to an inflated (by a fator of 9000) spin-rotation interation. This journal is the Owner Soieties 2011

8 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b Fig. 7 Spin-hange rate of buffer-gas trapped Ag for n He = m 3. The experimental data (green squares) disagree with the alulated maximum ontributions from Ag 3 He (blak dotted line) and Ag 3 He 3 He ollisional spin relaxation (red dashed line). The data are well explained by Monte Carlo simulations of loss due to adiabati following at the anisotropi-hyperfine-indued avoided rossings (blue line, shaded area indiates 68% onfidene interval). alkali-metal atoms. 86,87 The Hamiltonian of the Ag( 2 S) 3 He ollision omplex in an external magneti field B is similar in form to that given by eqn (7) H _ ol ¼ h2 2mr 2 r þ 2 2mr 2 þ A AgI Ag S þ B ð2m B S þ m Ag I Ag þ m He I He ÞþgðrÞ S rffiffiffiffiffi 8p X 2 þ A He ðrþi He S þ ðrþ Y 2;q 15 ð^rþ½i He SŠ ð2þ q : q¼ 2 ð12þ where r is the interatomi separation, m is the redued mass of 107 Ag 3 He and l is the orbital angular momentum by the ollision (replaing the rotational angular momentum N in eqn (7)). Unlike the moleular Hamiltonian of eqn (7), the Hamiltonian given by eqn (12) depends expliitly on r. The three last terms in eqn (12) desribe, respetively, the spinrotation, isotropi hyperfine, and anisotropi hyperfine interations. We expliitly ignore the small effet of the anisotropi modifiation of the Ag hyperfine interation due to the interation with the He atom. Having parametrized the Hamiltonian of eqn (12), we solve the sattering problem by expanding the wavefuntion of the AgHe omplex in the fully unoupled basis: Sm S i I X m IX i I He m IHe i lm l i, (13) where m S, m IX, and m IHe are the projetions of S, I Ag, and I He on the magneti field axis. The system of lose-oupled Shro dinger equations for the radial wavefuntions is solved for fixed values of the ollision energy and magneti field. The sattering matrix is evaluated in the basis 87 whih diagonalizes the asymptoti Hamiltonian given by the third and the fourth terms on the right-hand side of eqn (12), whih yields the probabilities for ollision-indued transitions between different hyperfine states of Ag. We onsider ollisions of Ag atoms initially in the low-field-seeking hyperfine state F =0,m F =0i. Fig. 6(a) shows the elasti ross setion for Ag 3 He ollisions plotted as a funtion of the ollision energy. The ross setion displays a pronouned peak near 0.5 m 1, whih is due to an l = 3 shape resonane in the inident ollision hannel. The alulated spin-relaxation ross setion is shown in Fig. 6(b), and is, not surprisingly, dominated by ontributions from the hyperfine interations. By averaging the ross setions shown in Fig. 6(b) with a Maxwell Boltzmann distribution of ollision energies, we obtain the inelasti Ag 3 He ollision rates as funtions of temperature, shown in Fig. 7. The spin-relaxation rate, alulated for the average trap field of B = 0.4 T,z inreases only slowly with temperature. The rate remains small in absolute magnitude ompared to the moleular spin relaxation rates onsidered in the next setion. We also perform the alulation with a grossly exaggerated spin-rotation parameter (by a fator of 9000), inflated to math the experimental measurement at 700 mk. The disagreement of this alulation with measurement at low temperatures indiates that, even if the magnitude of the perturbative result is inorret, Ag 3 He ollisions are not responsible for the observed spin-hange rate. Furthermore, the observed loss was exponential in time, rather than following the 1/(a + t) profile that would result if Ag Ag ollisions were responsible for the Ag spin relaxation. We therefore onlude that single atomi ollisions annot be responsible for the experimentally observed behavior. In ontrast, we see a marked similarity between the thermal behavior of the measured relaxation rate and the temperature dependene of the AgHe hemial equilibrium oeffiient (see Fig. 2). This agreement strongly suggests that moleules form within the trap, and that it is these moleules whih suffer spin relaxation, thereby ausing the observed Ag trap loss. We begin our treatment of the moleular dynamis by alulating the moleular formation kinetis of Ag 3 He. 2.3 Moleular formation We onsider two mehanisms for the formation of Ag 3 He moleules. The first is formation via the l = 3 shape resonane shown in Fig. 6. In this mehanism, ground state moleules form in a sequene of two-body ollisions. In the first ollision, an Ag and a He atom form a quasibound pair. During the lifetime of this pair, another He atom ollides with the omplex, ausing rotational relaxation of the pair into a bound rotational level. Additional ollisions ause rotational relaxation into the rotational ground state: Ag + 3 He " Ag 3 He*(N = 3), (14) Ag 3 HeðNÞþ 3 g N;N 0 HeÐ Ag 3 HeðN 0 Þþ 3 He: ð15þ g N 0 ;N To alulate the formation rate of ground-state Ag 3 He pairs, we apply the resonant three-body reombination model z The Ag 3 He rate varies only slightly with field. This journal is the Owner Soieties 2011

9 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b developed by Roberts, Bernstein, and Curtiss (RBC) 88 (also known as the Lindemann reombination mehanism). Under this model, the rate oeffiient for bound pair formation is simply the produt of the equilibrium oeffiient for quasibound pairs times the rotational relaxation rate oeffiient: K r ¼ kðe 3 Þ X g 3;N ¼ 7l X 3 db e e 3=k B T g 3;N ; ð16þ N N where the fator of 7 arises from the degeneray of the N =3 state. The rotational relaxation rate oeffiients are alulated using the atom moleule ollision theory desribed in the next setion. For the state-to-state rotational relaxation rates from the N = 3 quasibound state we find g 3,2 = , g 3,1 = , and g 3,0 = m 3 s 1 at T = 0.5 K. The alulated formation rates are shown vs. temperature in Fig. 8, and are dominated by resonant ombination into the N =2 level. The formation rate is between 0.8 and m 6 s 1 for all temperatures in the experiment. After formation in the N = 2 level, rotational thermalization proeeds via additional rotational relaxation. The rate onstants for rotational relaxation from the N = 2 and N = 1 rotational levels are similar to those from the N = 3 level, so the timesale for rotational thermalization is fast (by a fator Bk(e 3 )n He ) ompared to the moleular reombination rate, and K r therefore sets the timesale for resonant ground-state moleule formation. Fig. 8 (a) The equilibrium onstant for the formation of metastable AgHe*(N = 3) omplex as a funtion of temperature. (b) Rate onstants for three-body reombination Ag + He + He - AgHe(N 0 ) + He alulated using the RBC model as funtions of temperature. Eah urve is labeled by the final rotational state N 0 of AgHe. Beause K r n 2 He r 100 ms is muh less than the trap lifetime t trap Z 400 ms for all values of n He and T used in the experiment, the moleular density, and hene the moleular spin-hange dynamis, an be alulated assuming thermal equilibrium. The seond formation mehanism is diret formation via the three-body proess Ag þ 3 He þ 3 HeÐ K d D d Ag 3 He þ 3 He: ð17þ An exat alulation of the formation rate via this mehanism lies outside the sope of this artile. However, the rate may be approximated by extending the sequential RBC orbiting resonane theory to inlude ontributions from the nonresonant two-body ontinuum. Beause the time delays of the non-resonant states are negligible, this approah gives pure third-order kinetis for all densities. We use the index i to denote an unbound or quasibound initial state of AgHe and ompute the reombination rate oeffiient using X K r ¼ l 3 db k if g i e E i=k B T ; ð18þ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 8 1 k if ¼ s if ðe T Þe E T=k B T E T de T : pmk 3 B T3 0 if ð19þ The energy E T = E E i is the translational energy in the ith hannel, and s if (E T ) is the ollision ross setion for transition to a bound final state. The energy E i is a positive energy eigenvalue of the diatomi Shro dinger equation in a Sturmian basis set representation, whih may orrespond to a resonane or to a disretized non-resonant ontribution in a numerial quadrature of the ontinuum. 89 We use a Sturmian basis set representation onsisting of 100 Laguerre polynomial L (2l+2) n funtions of the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a s n! f l;n ðrþ ¼ ða s rþ lþ1 e asr=2 L ð2lþ2þ n ða s rþ; ð20þ ðn þ 2l þ 2Þ! with a sale fator a s = 10. To assess the quality of this representation, the positive energy eigenstates were used to ompute the Ag + He elasti sattering ross setion for l = 3. The results are shown in Fig. 9 along with the exat results obtained from numerial integration. The lowest energy v = 0 eigenstate is learly assoiated with the resonane, whereas the v 4 0 eigenstates may be assoiated with the non-resonant bakground. The ollision ross setions s if are omputed by solving a set of oupled hannel (CC) equations using the general inelasti sattering program MOLSCAT. 90 The reombining He atom is assumed to be distinguishable from the olliding He atom and to be nearest to the Ag atom. As disussed below, the interation PES beomes more anisotropi with inreasing r. We investigated the sensitivity of the CC alulations to this anisotropy by setting V(R, r, y) =V(R, r max, y) for r Z r max and onsidering r max = R/n with n = 1 4. The results are shown in Fig. 10. Also shown are results of the l = 3 resonant ontribution using the rigid rotor approximation with r =3a 0. The CC alulations that inlude vibrational oupling were This journal is the Owner Soieties 2011

10 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b Fig. 9 Elasti sattering ross setion for Ag + 3 He with l = 3. The solid blak urve was omputed using numerial integration, and the red irles were omputed using the Sturmian basis set representation. The v = 0 eigenstate is assoiated with the resonane, and the v 4 0 eigenstates with the non-resonant bakground. Fig. 10 Three-body reombination rates for the formation of AgHe moleules in different rotational states (N) as funtions of temperature: N = 2 (upper panel), N = 1 (middle panel), and N = 0 (lower panel). Solid lines results alulated using the ontinuum disretization method (see text), symbols results obtained using the RBC model. found to onverge using v max = 6 and j max = 6 for a total of 49 basis funtions. Convergene of the Legendre expansion VðR; r; yþ ¼ Xlmax V l ðr; rþp l ðos yþ: l¼0 ð21þ was found for n =3usingl max = 6. For onsisteny, these parameters were used for eah of the CC alulations along with amathingdistaner max =100a 0, a maximum total angular momentum quantum number J max = 10, and a 20 point numerial integration over y. The figure shows that the CC results with n = 4 are similar to the RBC results using the rigid rotor approximation. The diret three-body reombination mehanism is essentially negligible in this ase. The vibrational oupling to the non-resonant bakground is more substantial as n dereases ausing the reombination rate for eah N to inrease. It is diffiult to pinpoint preisely how muh inrease may be expeted for the exat potential, however, the n =1 results provide a reasonable estimate. The onvergene of both the Legendre expansion and the basis set representation begins to break down as n is redued further, whih suggests that a hemial exhange mehanism may be signifiant for this system. This possibility will be onsidered in a future study. 2.4 Ag 3 He 3 He ollisions One formed, the Ag 3 He moleules an undergo spin relaxation in ollisions with 3 He atoms, whih onvert low-fieldseeking states to high-field-seeking states, leading to trap loss. In this setion, we estimate the rate for spin-flipping Ag 3 He 3 He ollisions, and show that it is too small to aount for the experimentally measured 49 trap loss rates. Low-temperature ollisions involving vdw moleules may lead not only to inelasti spin relaxation, but also hemial exhange and three-body breakup. A proper theoretial desription of these proesses requires the use of hyperspherial oordinates, 20 and is beyond the sope of this work. In order to estimate spin relaxation probabilities in Ag 3 He 3 He ollisions, we instead assume that: 1. The energy spetrum of Ag 3 He is desribed by the rigidrotor Hamiltonian of eqn (7). This approximation is justified for the first three rotational levels (N = 0 2). The N = 3 level orresponds to the long-lived shape resonane shown in Fig. 6, and an thus be treated in the same manner as the (truly bound) lower rotational states. 2. The ontributions to spin relaxation ollisions due to hemial exhange (AgHe + He 0 - AgHe 0 + He) and ollision-indued dissoiation (AgHe + He - Ag + He + He) an be negleted. Beause of the short-range nature of these proesses, they might result in a more effiient spin relaxation than inelasti ollisions alone. Therefore, our estimates for spin relaxation are best thought of as lower bounds to true moleular spin relaxation rates. 3. The ontributions of the hyperfine interations to the loss rate an be desribed by the perturbative result in eqn (9) (in fat, this equation should overestimate their ontribution). We will therefore only perform a quantum ollision alulation of the ontribution from the spin-rotation interation. In order to plae an upper limit on this ontribution, we will use the saled value of Setion The interation potential for AgHe He is the sum of pairwise interation potentials for Ag He and He He evaluated at a fixed AgHe distane of 3.0 a 0. We hoose this value in order to ensure the onvergene of the Legendre expansion of the AgHe 2 interation potential (see below). This journal is the Owner Soieties 2011

11 Downloaded by University of Wisonsin - Madison on 02 August 2011 Published on 01 August 2011 on doi: /c1cp21317b Under these assumptions, the Hamiltonian of the Ag 3 He 3 He omplex an be written as 2 ^H ¼ h2 2 R þ L2 2mR 2 þ VðR; r; yþþ ^H mol ; ð22þ where R stands for the atom moleule separation, r is the internulear distane in AgHe, y is the angle between the unit vetors rˆ = r/r and Rˆ = R/R, L is the orbital angular momentum for the ollision, m is the AgHe He redued mass, V(R, r, y) is the interation potential, and Hˆ mol is given by eqn (7). The eigenstates of Hˆ mol are the Zeeman energy levels of AgHe shown in Fig. 4. We hoose the following low-fieldseeking states of AgHe as the initial states for sattering alulations: N, m N, m I, m S i = 0, 0, 1/2, 1/2i and 1, 0, 1/2, 1/2i. The Ag 3 He 2 interation potential is represented as the sum of the pairwise Ag He and He He potentials. We express the potential in the Jaobi oordinates illustrated in Fig. 11. The He He potential is taken from ref. 91. The number of terms in the Legendre expansion of the interation potential (21) inreases with inreasing r, as the interation potential beomes more anisotropi. At r 4 r, where r is some ritial value, the topology of the interation potential hanges dramatially and the expansion in eqn (21) beomes inadequate, as illustrated in the lower panel of Fig. 11. The hanges in topology inlude the appearane of short-range minima orresponding to the insertion of the He atom into the strethed AgHe bond. Furthermore, the pairwise additive approximation is expeted to fail at short range, whih may lead to unphysial effets in the three-body exhange region. In order to avoid these problems, we hoose to fix r at 3.0 a 0 rather than keeping the real AgHe equilibrium distane (r = 8.5 a 0 ). This proedure is onsistent with the assumption of negligible diret three-body proesses (see Setion 2.3). The wave funtion of the Ag 3 He 2 omplex is expanded in the fully unoupled basis 60 Nm N i Sm S i Im I i Lm L i, (23) where Lm L i are the partial waves desribing the orbital motion of the ollision partners. The asymptoti behavior of the expansion oeffiients defines the sattering matrix and the probabilities for ollision-indued transitions between the different Zeeman states of AgHe. As in the ase of Ag 3 He ollisions desribed above, the sattering boundary onditions are applied in the basis 60 whih diagonalizes the asymptoti Hamiltonian Hˆ mol. The asymptoti transformation mixes different M N and M S, but is diagonal in L and M L. We integrate the oupled-hannel equations for the radial oeffiients in eqn (23) numerially in a yle over the total angular momentum projetion M = m N + m S + m I + m L from R =3a 0 to 50 a 0 with a step size of 0.04 a 0. The alulations are onverged to better than 50% with respet to the maximum number of rotational states (N r 7) and partial waves (L r 7) inluded. The total moleular spin-hange rate %g SC is determined by the thermal and trap average of g SC (E, B). We perform the trap average using the averaging distribution in eqn (10) and thermally averaged Ag 3 He 3 He inelasti ollision rates alulated on a log-spaed grid of 41 points in the range B = 10 4 to 5 T as desribed above (Fig. 12). The ontribution of moleular spin relaxation to the overall spin-hange rate of Ag Fig. 11 Top: Pairwise AgHe 2 interation PES, evaluated at r = 8.75 a 0, orresponding to the bottom of the AgHe potential well. Color sale is logarithmi, with green to red indiating energies from 1 to 10 5 m 1, and yan to blue indiating energies from 1 to 15 m 1. Bottom: Lowest-order expansion oeffiients V l (R) of the AgHe 2 PES as funtions of R, evaluated with r = 8.6 a 0. Note the divergene in the viinity of R = 8.5 a 0. Fig. 12 Rate oeffiients for Ag 3 He 3 He spin hange as a funtion of magneti field, for ollision energy E ol = 0.5 m 1. Cirles are results of the oupled-hannel alulations for saled spin rotation. The lines are the asymptoti forms of eqn (10), evaluated for the isotropi (red dashed line) and anisotropi (green dotted line) hyperfine interations. This journal is the Owner Soieties 2011