Cold atom-dimer reaction rates with 4 He, 6,7 Li and 23 Na
Mahdi A. Shalchi, Marcelo T. Yamashita, Tobias Frederico, Lauro Tomio
CCold-atom-dimer reaction rates with He, , Li and Na M. A. Shalchi , M. T. Yamashita , T. Frederico and Lauro Tomio Instituto de F´ısica Te´orica, Universidade Estadual Paulista, 01405-900 S˜ao Paulo, Brasil. Instituto Tecnol´ogico de Aeron´autica, DCTA, 12228-900 S˜ao Jos´e dos Campos, Brasil. (Dated: December 18, 2020)Atom-dimer exchange and dissociation reaction rates are predicted for different combinationsof two He atoms and one of the alkaline species among Li, Li and Na, by using three-bodyscattering formalism with short-range two-body interactions. Our study was concerned with low-energy reaction rates in which the s − , p − and d − wave contributions are the relevant ones. The He is chosen as one of the atoms in the binary mixture, in view of previous available investigationsand laboratory accessibilities. Focusing on possible experimental cold-atom realizations with two-atomic mixtures, in which information on atom-dimer reaction rates can be extracted, we predict theoccurrence of a dip in the elastic reaction rate for colliding energies smaller than 20 mK, when thedimer is the He Na molecule. We are also anticipating a zero in the elastic p − wave contributionfor the He + He Li and He + He Na reaction processes. With weakly-bound molecules reactingwith atoms at very low colliding energies, we interpret our results on the light of Efimov physicswhich supports model independence and robustness of our predictions. Specific sensitivities on theeffective range were evidenced, highlighted by the particular inversion role of the p − and d − wavesin the atom exchange and dissociation processes. I. INTRODUCTION
The fast development of methods to control atomand molecules in ultra-cold experiments, which have fol-lowed after the realization that Feshbach resonance tech-niques [1] could be used to manipulate the two-body in-teraction [2–4], made possible to emerge ultra-cold chem-istry as a new field of interest with an intense researchactivity in recent years. On this regard, some previousreviews can be found on collisions with ultra-cold atomsand related discussions on possible experimental inves-tigations [5–8], which were followed by Ref. [9], wherethe basis are settled for investigations on collisions andreactions with ultracold molecules. This work [9], inwhich it was concluded that the magnitude of elasticcollision cross sections depends more on the mass andsymmetry than on the interaction, came just after theexperimental realization of Bose-Einstein condensationof Cooper-paired molecules of Li [10] and K [11].Inelastic molecule-molecule and molecule-atom collisionswere also characterized within the study of dissociationand decay of ultra-cold sodium molecules driven by theFeshbach mechanism [12]. The investigations on ultra-cold molecules via Feshbach mechanism can be exem-plified by recent production of Na Rb molecules re-ported in Ref. [13] and the more recent investigation onultra-cold Na K ground-state molecules [14], whichcan provide conditions for fully controlled studies withultra-cold molecular collisions. Furthermore, it was re-cently reported progress with extreme mass imbalancedFermionic mixtures where for the first time was observedmagnetic Feshbach resonances in Li and
Yb ultra-cold atoms [15].With the complex structure of molecules, new oppor-tunities can also be opened for research, such as the re-alization of quantum fluids of bosons with anisotropicinteractions, as well as quantum coherent chemistry by considering atom-molecule conversions.Some pioneer experimental investigations with trappedultracold atom-molecule collisions have been reported in2006, in Refs. [16, 17], by considering cesium (Cs) atomsin Cs molecules. These works are followed by experi-ments with molecular collisions with tunable halo dimers,exploring four-body processes with identical bosons [18],inelastic collision rates of p − wave Li molecules [19],RbCs molecule collisions with Rb and Cs atoms [20]. Ap-plications in cold controlled chemistry, with the possibili-ties to manipulate molecule collisions by electromagneticfields, were discussed in Refs. [21–23], which togetherwith references therein are covering both theoretical andexperimental aspects in the first stages of investigationsalong these lines. As shown in these works, collisions ofmolecules with temperatures below 1 Kelvin can be ma-nipulated by external electromagnetic fields. In Ref. [23],by considering potassium-rubidium molecules, it was alsopointed out that, even when the cooled molecules haveno energy to collide, exothermic atom exchange reac-tivity processes can occur through quantum mechanicaltunneling. Nowadays, the reported experimental setupsare showing an increasing level of control on atomic andmolecular states, as one can trace from ongoing experi-mental and theoretical investigations, which can be ex-emplified by Refs. [24–33]. For more recent experimen-tal activities reported in the last five years, on ultracoldatom-dimer collisions, by using tunable Feshbach reso-nances, we should also mention the Refs. [34–41].Our study is also motivated by the interest on ver-ifying manifestations of Efimov physics [42] in atomicreactions, following Ref. [43], which could support pre-vious theoretical studies with three-body weakly-boundsystems [44–52]. In particular, the existence of aweakly-bound excited state in helium was stablished inRef. [48]. This matter was further investigated in theEfimov physics context in Refs. [53–64], motivated by a r X i v : . [ phy s i c s . a t m - c l u s ] D ec FIG. 1.
Illustration of the atom-dimer reaction (for each panel, the entrance channel is in the left side, with theexit channel at the right side). The He is represented by the identical particles, with the dimers representedby He β ( β = Li, Li or Na) and He bound states. Assuming the He β is more bound than He , theatomic exchange reaction is endothermic in the upper-right panel; and exothermic in the lower-right one. the remarkable small binding energy of the He dimer: B He = 1.31 mK, which was first reported independentlyin Refs. [65, 66], being more recently confirmed experi-mentally [67, 68]. The experimental success in verifyingsuch long-time theoretical prediction, together with theresults of previous experimental investigations of Efimovphysics in cold atom laboratories [69–71], which are ex-tended to mixed atomic-molecular combinations [72–75],became highly motivating for more deeper studies withsingle or mixed atomic species [76–81]. Quite remark-able are the advances in the laboratory techniques, suchthat one can even consider the possibility to alter thetwo-body interaction by using Feshbach resonance mech-anisms (originally proposed in the nuclear physics con-text) [3]. The possibility of tuning the two-body scat-tering length in ultracold atomic experiments can alterin an essential way the balance between the non-linearfirst few terms of the mean-field description, as it wasexplored when modeling the atomic Bose-Einstein con-densation [82].The above mentioned experimental possibilities in coldchemistry laboratories and the interest to find furthermanifestations of the universal Efimov physics, moti-vate our study focusing on dissociation and atom ex-change reactions with weakly-bound atom-molecular sys-tems. Our aim is to explore the associated universalproperties, which emerge in the studies of the reactionmechanisms of cold atoms with weakly-bound molecules.By following a previous investigation on s − wave scatter-ing properties with the atomic species He, Li, Li and Na [43], with our present study we are providing a de-tailed analysis on atom-dimer dissociation and exchangereactions, in which we discuss and quantity the s , p and d lower partial-wave contributions to the associated reac-tion rates. The choice of these four atomic samples relieson the fact that the dimer and trimer binding energiesare enough known, either from available realistic poten-tial model calculations or from experiments as in the caseof He , which provide support to our theoretical predic-tions. In this introduction we are including a pictorialillustration of the reactions that we are considering (seeFig. 1), by assuming the common case in which the He β molecule ( β = Li, Li or Na) is more bound than the He dimer. For the entrance channel, in the upper rowof this illustration we have the collision of a single Heatom with the He β molecule; in the lower row, the single β atom collides with the He dimer.In the present work, the Faddeev formalism wasadopted [83] with renormalized zero-range (ZR) [84] andfinite-range Yamaguchi [85] separable (FR) two-body in-teractions, using as inputs the available results reportedin Ref. [86] for the dimer and trimer binding energiesobtained from realistic interactions. These realistic in-teraction binding energies reported in Ref. [86] are se-lected from different potential models discussed in theliterature, being guided by the corresponding availableexperimental results.In the next Sect. II, we present the basic three-bodyformalism for two-atom mixtures, in which our main fo-cus is on the case of atom-dimer collision, for separabletwo-body interactions. Details on reaction rates and ex-istent models are presented in Sect. III. In the next threesections (IV, V and VI) we have separated the corre-sponding results for the mixtures of He with Li, Li and Na, respectively. Our final remarks and conclusions arepresented in Sect. VII. In addition, three appendices areadded to detail the three-body framework.
II. THREE-BODY FRAMEWORK
The standard Faddeev formalism for three-body sys-tems [83, 87, 88] with two atomic species interactingthrough one-term separable potentials including the zero-range one is presented in this section, by considering theelastic and rearrangement scattering amplitudes, fromthe collision of one of the atoms with a dimer formedby the remaining ones. (The detailed derivation of thescattering equations are presented in Appendix A.) Asexplained in our motivation for this study (with three-body system having two distinct atoms), we choose theidentical two particles, labelled α , as the He atom, withthe third atom, labelled β , being Li, Li, or Na.This particular choice is related to the available dataobtained from experimental and well-known potentialmodels for the three-body and two-body binding ener-gies, when assuming different combinations of the corre-sponding three-particle atom-dimer systems. Our moti-vation also relies on the actual interest in the produc-tion of ultracold molecular systems [13, 14], and in therecent investigations on binary molecular collisions [34–36, 38, 40], together with experimental studies with theseatomic samples [33, 41].In this study, we assume that the selected three-bodysystem ( ααβ ) is always bound, as well as the possiblesubsystems ( αβ ) and ( αα ), with the corresponding avail-able data being used as inputs coming from different re-alistic potential models (given by previous available stud-ies), together with existent experimental data. So, we as-sume as fixed the He binding energy and correspondingscattering length, such that E αα = − B αα = − . a αα =100˚A. For the other input binding energies,we select two specific models which have been discussedin Ref. [43], that we found enough representative to ex-plore the sensibility of our results on dimer binding en-ergies variation. As it will be discussed, by using differ-ent potential models, we expect our results on reactionrates, together with possible future experimental data,be useful in selecting the appropriate two-body potentialmodel. A. Bound-state ααβ three-body system
We start by recovering some details from Ref. [43], inorder to fix the notation of the formalism, summariz-ing the s − wave three-body bound-state coupled equa-tion when considering separable potentials with all sub- systems being bound. We assume units such that (cid:126) = 1(with energies given in mK), with m ≡ m α = m He and a mass ratio defined by A ≡ m β /m α . So, forthe reduced masses we have µ αα = m/ µ αβ = Am/ ( A + 1) for the αα and αβ subsystems, respectively.The corresponding three-body reduced masses are givenby µ α ( αβ ) = m ( A + 1) / ( A + 2) for the α − ( αβ ); and µ β ( αα ) = m (2 A ) / ( A + 2) for the β − ( αα ). The bound-state energies for the two- and three-body systems aregiven by E αα ≡ − B αα , E αβ ≡ − B αβ and E = − B ,respectively; with the energies of the s − wave elastic col-liding particle given by E k α ≡ k α / [2 µ α ( αβ ) ] ≡ E − E αβ and E k β ≡ k β / [2 µ β ( αα ) ] ≡ E − E αα . The extensionto higher partial waves will be done along the follow-ing subsection. In this case, the Faddeev coupled equa-tions are reduced to integral equations for the momen-tum space spectator functions of the particles α and β ,which are given by χ α ( q ; E ) and χ β ( q ; E ). Consider-ing the s − wave case, by redefining these functions, as χ j = α,β ( q ; E ) ≡ χ j ( q ; E ) / ( q + | k j | ) , the correspondingcoupled formalism is given by χ α ( q ; E ) = τ α ( q ; E ) (cid:90) ∞ dkk (cid:20) K ( q, k ; E ) χ α ( k ; E )( k + | k α | )+ K ( q, k ; E ) χ β ( k ; E )( k + | k β | ) (cid:35) , (1) χ β ( q ; E ) = τ β ( q ; E ) (cid:90) ∞ dkk K ( k, q ; E ) χ α ( k ; E )( k + | k α | ) , where K ( q, k ; E ) and K ( q, k ; E ) are the appropriatemomentum space kernels, which will be explicitly givenaccording to the kind of form-factors one considers forthe two-body interaction. The respective two-body t-matrix elements for the αβ and αα bound subsystemsare τ α ( q ; E ) and τ β ( q ; E ), in which τ j and the s − wavekernels K , will be appropriately extended to (cid:96) partialwaves in the Appendix B, when considering the specificpotential models. B. Atom-dimer collision
By considering atom-dimer collision with two differentatomic species, we have three separate channels in thecontinuum: elastic scattering, rearrangement (exchangereaction), and breakup (dissociation reaction). For thescattering of α by the αβ bound subsystem, by using thesymmetry properties applied to identical bosons, the (cid:96) -partial-wave scattering amplitudes, h (cid:96)α and h (cid:96)β , with one-term separable potentials, can be written as follows (seeAppendices A and B): h (cid:96)α ( q ; E ) = τ α ( q ; E ) (cid:40) π K (cid:96) ( q, k α ; E ) + (cid:90) ∞ dkk (2) × (cid:34) K (cid:96) ( q, k ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) + K (cid:96) ( q, k ; E ) h (cid:96)β ( k ; E ) q − k β − i (cid:15) (cid:35)(cid:41) , h (cid:96)β ( q ; E ) = τ β ( q ; E ) (cid:40) π K (cid:96) ( k α , q ; E ) (3)+ (cid:90) ∞ dkk K (cid:96) ( k, q ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) (cid:41) , where k α and k β are the on-shell momenta (already de-fined), with h (cid:96)α ( k α ; E ) representing the on-shell elas-tic amplitude and h (cid:96)β ( k α ; E ) the corresponding on-shellatom-exchange amplitude. K (cid:96) , which have been intro-duced (for (cid:96) = 0) in Eq. (1) are momentum space kernelsin (cid:96) -wave. From the scattering amplitudes, we can obtainthe cross sections with the corresponding reaction rates.For the elastic scattering, we obtain (see Appendix C) σ el ( E k α ) = 4 π (cid:88) (cid:96) (2 (cid:96) + 1) | h (cid:96)α ( k α ; E ) | , (4)with the exchange reaction cross section given by σ ex ( E k α ) = 2 πf α f β (cid:115) µ α ( αβ ) µ β ( αα ) (cid:18) − E αα − E αβ E k α (cid:19) × (cid:88) (cid:96) (2 (cid:96) + 1) | h (cid:96)β ( k α ; E ) | , (5) where the on-shell h (cid:96)α and h (cid:96)β are obtained from Eqs. (2)and (3).For the scattering of particle β by the αα bound sub-system, the scattering amplitudes can be calculated bythe following coupled equation (see Appendices A andB): h (cid:96)α ( q ; E ) = τ α ( q ; E ) (cid:40) πK (cid:96) ( q, k α ; E ) + (cid:90) ∞ dkk (6) × (cid:34) K (cid:96) ( q, k ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) + K (cid:96) ( q, k ; E ) h (cid:96)β ( k ; E ) q − k β − i (cid:15) (cid:35)(cid:41) ,h (cid:96)β ( q ; E ) = τ β ( q ; E ) (cid:90) ∞ dkk K (cid:96) ( k, q ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) . (7) Here h (cid:96)α represents the exchange reaction and h (cid:96)β repre-sents the elastic scattering. In this case elastic and ex-change reaction cross sections can be calculated by (seeAppendix C): σ el ( E k β ) = π (cid:88) (cid:96) (2 (cid:96) + 1) | h (cid:96)β ( k β ; E ) | , (8)and σ ex ( E k β ) = 2 πf β f α (cid:115) µ β ( αα ) µ α ( αβ ) (cid:115) − E αβ − E αα E k β × (cid:88) (cid:96) (2 (cid:96) + 1) | h (cid:96)α ( k β ; E ) | , (9)where the on-shell h (cid:96)α ( k β ) and h (cid:96)β ( k β ) are calculated fromEqs. (6) and (7). The f α and f β factors, together withdetailed derivation of the scattering amplitudes and crosssections for each potential, are described in the Ap-pendix C. The scattering amplitude for breakup or dissociationreaction can also be calculated by the summation overhalf-off-shell results of h α and h β using the Eqs. (2) and(3) (for α − αβ scattering) and by using Eqs. (6) and (7)(for β − αα scattering). The breakup amplitude is notdetailed in the presented work. However, the dissociationrate is obtained through the inelasticity parameter of theelastic amplitude and rearrangement rate, as explainedin the following section. III. REACTION RATES AND MODELS
The reaction rates are defined in terms of the productof the corresponding cross sections (which are detailed inthe Appendix C) and the respective velocity of the collid-ing particle. By assuming that the atom-dimer reactionhas initially the particle α or β colliding with the dimerformed by the particles αβ or αα , respectively, with en-ergy E k ≡ k / (2 µ ), with µ being the atom-dimer reducedmass, the total elastic case is such that α + αβ → α + αβ or β + αα → β + αα . Therefore, the elastic reaction rateis given by corresponding product of the elastic cross sec-tion with the velocity (cid:126) k/µ , such that K elas ( E k ) = (cid:126) kµ σ el ( E k ) . (10)Analogously, the atom-exchange reaction rate can be de-fined, corresponding to the processes α + αβ → β + αα (or β + αα → α + αβ ), which is given by K ex ( E k ) = (cid:126) kµ σ ex ( E k ) . (11)The third possible process refers to dissociation processes(possible for energies above the three-body continuum),when for example α + αβ → α + α + β . It is given by K diss ( E k ) = (cid:126) kµ σ diss ( E k ) . (12)Given the above, in terms of the corresponding cross-sections, we define the loss-rate coefficient by K loss ( E k ) = K ex ( E k ) + K diss ( E k ) . (13)which refers to the exchange and dissociation, orbreakup, cross-sections, respectively.As in our approach we have two different initial con-figurations for two-identical particles α with a particle β ,the above defined quantities (momentum k , energy E k and reduced mass µ ) should be understood such that:(i) If α is the colliding particle, k ≡ k α , E k ≡ E k α and µ ≡ µ α ( αβ ) = m α ( m α + m β ) / (2 m α + m β ); (ii)if β is the colliding particle, k ≡ k β , E k ≡ E k β and µ ≡ µ β ( αα ) = (2 m β m α ) / (2 m α + m β ).The corresponding partial wave (cid:96) decomposition of theabove reaction rates [Eqs. (10) - (13)] can be expressed interms of the non-diagonal S-matrix, for the elastic and forthe atom-exchange channel. Given that K x ( E k ) ≡ K x ,for x = (elas, ex, diss, loss), we have the following: K elas = π (cid:126) µk ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) | − S (cid:96) el ( E k ) | , (14) K ex = π (cid:126) µk ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) | S (cid:96) ex ( E k ) | , (15) K diss = π (cid:126) µk ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1)(1 − | S (cid:96) el ( E k ) | − | S (cid:96) ex | ) , (16) K loss = π (cid:126) µk ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1)(1 − | S (cid:96) el ( E k ) | ) . (17)Our aim is to study the reaction rates for the atom-dimercollisions, in which α ≡ He, with β ≡ Li, Li, and Na,by considering all possible combinations for the three ααβ systems. The relevant information about these systems areprovided in Tables I and II, with the dimer and trimer bindingenergies being obtained from Refs. [86] (a1) and [49] (a2).The respective configuration-space behaviors of the two-bodypotentials V ( r ), which were used to obtain the dimer energiesprovided in Refs. [49] and [86], are represented in Fig. 2.The corresponding two- and three-body ground-state bind-ing energies (absolute values, given in mK) are presented inTable I, in which we also include the given excited three-body binding energies, whenever known. More specifically,the results for (a1) and (a2) were obtained by consideringinteractions from Refs. [89] and [90], respectively, as shownin Fig. 2. Other model calculations exist for the systems weare studying, beyond the selected (a1) and (a2) models, asdescribed in Ref. [86]. They are from Ref. [60], with αα and αβ potentials given by Refs. [89, 91, 92]; from Ref. [59, 60],with potentials given by Ref. [93, 94]; from Ref. [62], with αα potential given by Ref. [93], and αβ by Ref. [89]. However, weselect only (a1) and (a2), among the ones which are discussedin Refs. [43, 86], as they are providing values for all the caseswe are considering, having binding energies enough distinctfor a significant comparative study.One should notice that the binding-energy difference be-tween the dimers He − Li and He − Li, shown in Table I,refers to the corresponding isotopic mass difference. Morerelevant to be noticed in the table, for such weakly-bound sys-tems, is the binding energy sensibility on the potential depthdifferences, as verified between the (a1) and (a2) models forthe He-Li system (see the inset of Fig. 2), which changes thekind of the atom-exchange reaction (endothermic or exother-mic).The Table II refers to the kind of each reaction channelconsidered in the two potential models we have used (ZR andFR) with both different inputs. In all the cases under ouranalysis, in which we have two He atoms, identified by α and a third atom identified by β =( Li, Li, Na), a weaklybound molecule exists, such that six entrance channels arepossible for the atom-dimer reactions, which are the following:(1) He + He Li , (2) Li + He , (3) He + He Li , (4) Li + He , (5) He + He Na , (6) Na + He . (18)For each one of these atom-dimer initial reaction channels, we can obtain the elastic, atom exchange and dissociation re-action rates. In order to compute these quantities, amongthe available potential models which were investigated inRefs. [49, 86], we select two of them, as we already men-tioned, which provide less-similar values for the dimer andtrimer binding energies that we are considering. These po-tential models will be used to provide the necessary bindingenergies for the inputs to adjust the parameters of our zero-range and finite-range s − wave separable interactions, withina Faddeev three-body formalism for the atom-dimer collision(for details, see sect. II B). r (a.u.) -10-505 V (r) ( K ) He - HeHe - NaHe - Li
He-Li (a2)(a1)
FIG. 2. Dimer potentials for the He-He (dotted-line withtriangles), He-Na (dashed-line with squares) and He-Li (solidline) systems, used in the three-body calculations of [86] (a1)and [49] (a2), are shown in the main frame. The He-Li (a1)and (a2) potentials, respectively derived in Refs. [89] and [90],differ by tenth’s of mK, being indistinguishable and repre-sented by a single line in the main frame. They are shown inthe inset by the solid (a1) and dashed (a2) lines.TABLE I. This table provides the available dimer and trimerbinding energies (in mK), combining two He with Li, Liand Na atoms. The given results for (a1) are from Ref. [86]and for (a2) from [49]. With (*) we have the available excitedbinding energies, from Ref. [86].Molecule (a1) (a2) He He Li 1.515 0.12 He Li 5.622 2.16 He Na 28.98 28.98 He − Li 57.23 31.4( He − Li) ∗ He − Li 79.36 45.7( He − Li) ∗ He − Na 150.9 103.1In order to compute the reaction rates, as mentioned before,for each set of binding energies (a1) and (a2) provided inTable I, we have adjusted our zero-range and finite-range two-body interactions, for which the corresponding kernels of the
TABLE II. Classification of reaction channels, identifiedin (18), as endothermic or exothermic according to dimerbinding energies of models (a1) and (a2), given in Ta-ble I, for elastic and exchange processes. For dissociation( → He+ He+ Li), all channels are endothermic.Channel Model Elastic Exchange(1) He + He Li He + Li(a1) endothermic(a2) exothermic(2) Li + He He + He Li(a1) exothermic(a2) endothermic(3) He + He Li He + Li(a1) endothermic(a2) endothermic(4) Li + He He + He Li(a1) exothermic(a2) exothermic(5) He + He Na He + Na(a1) endothermic(a2) endothermic(6) Na + He He + He Na(a1) exothermic(a2) exothermicscattering equations are detailed in Appendix B. In the zerorange model, we have the two-body amplitude parametrizedby the diatomic binding energies, together with a regularizingmomentum parameter fixed by the triatomic molecule. Forthe finite-range interaction, we assume a rank-one separableYamaguchi potential, given by V ij ( p, p (cid:48) ) = λ ij p + γ ij p (cid:48) + γ ij , (19)where ij = αα or αβ , respectively, for the αα or αβ two-bodysubsystems. λ ij and γ ij refer to the strengths and rangesof the respective two-body interactions. As in the presentapproach we consider only bound (negative) two-body sub-systems, E ij = − B ij , the corresponding relations for thestrengths and ranges are given by λ − ij = − πµ ij γ ij ( γ ij + κ ij ) , r ij = 1 γ ij + 2 γ ij ( γ ij + κ ij ) , (20)where r ij are the effective ranges, with κ αα ≡ (cid:112) − µ αα E αα , κ αβ ≡ (cid:112) − µ αβ E αβ . (21)For the case of FR potential given in Eq. (19), the param-eters with corresponding ranges and scattering lengths, areshown in Table III, given in three blocks for the cases with( He Li), ( He Li) and ( He Na). We observe that, in all thecases, for the dimer He binding energy, the accepted value B αα =1.31 mK is being considered, with the correspondingparameters given in this table.The results for the reaction rates are organized according tothe Table II and are calculated with the ZR and FR potentialmodels, in order to exhibit the model independent features of TABLE III. Parameters used in the FR s − wave separableinteraction, with the corresponding ranges and scatteringlengths, in order to reproduce the respective binding energiesgiven in Table I for the model potentials (a1) and (a2).Dimer Model γ αβ (˚A − ) r αβ (˚A) a αβ (˚A) He Li (a1) 0.17 15.85 90.38(a2) 0.14 20.04 300.37 He Li (a1) 0.17 14.77 50.08(a2) 0.14 19.02 77.43 He Na (a1) 0.16 12.44 25.34(a2) 0.09 19.0 34.24 He (a1)&(a2) 0.39 7.34 100the present results, for up to the d − wave contribution. Thechoice of the potential, ZR or FR, affects mainly the s − wavecontribution, which is more sensitive to the effective range.Reminding that the models fit the same diatomic and tri-atomic binding energies, we found that the bulk results arealmost model independent; particularly, for the higher par-tial waves the results are to a large extend universal. Suchfeatures gives robust outcomes of our model calculations, aswe will present in the following. However, the reaction ratesare bounded by the binding energies provided in Refs. [86]and [49], while our predictions allow to discriminate betweenthe very different values given for the model (a1) and (a2).We perform calculations up to kinetic energies of 0.1 K. Notealso that, we have carefully checked that almost no deviationfrom the unitarity appears in our numerical solutions of thescattering equations, for all the colliding energies being con-sidered. In particular, we verified that the coupled elastic andatom-exchange channels S-matrix is unitary below the disso-ciation threshold, as obtained numerically for both zero- andfinite-range interaction models.TABLE IV. For the molecules identified in the first column,in the 2nd, 3rd and 4th columns we have the correspondingtwo-body energies (mK), scattering lengths (˚A), and trimerenergies (mK) (obtained from Ref. [86]), respectively. In the5th to 7th columns we have the first excited bound-state en-ergies (in mK). Results obtained from Refs. [86] and [43], asindicated in the last row. In addition, for the excited states,we show results obtained by using zero-range (ZR) and finite-range (FR) one-term separable interactions. The parametersfor the He- He subsystem are given in Tables I and III. molecule He-Li a HeLi He Li (He Li) ∗ (He Li) ∗ (He Li) ∗ He - Li 1.515 100 57.23 1.937 1.901 1.977 He - Li 5.622 48.84 79.36 5.642 - 5.672Ref. [86] [43] [86] [86] (ZR) [43] (FR) [43]
Before closing this section, it is necessary to point out thatthe molecules He Li and He Li present excited Efimovstates close to the lowest scattering thresholds as verified inRef. [86] and for the ZR and FR potentials in [43]. Theseresults corresponding to model (a1) are provided in Table IV,where in [43] slightly different scattering lengths were usedwith respect to Table III, which is not relevant. It is impor-tant to note that weakly bound triatomic states close to the threshold affect the reaction rates at low energies.In what follows we will present our results for the atom-dimer reaction rates by considering the possible three-atomsystems ααβ with the particle α being He and β being oneof the species Li, Li or Na. We split the presentation inthe next three sections: In Sect. IV we consider β ≡ Li; inSect. V, β ≡ Li; and in Sect. VI, β ≡ Na. In each of thesethree sections, we have two subsections for the reaction-rateresults, such that α + αβ reactions are presented in (A), withthe β + αα reactions presented in (B), with all the possibilitiesbeing presented in Table II. IV. THREE BODY REACTIONS WITHHELIUM-4 AND LITHIUM-6A. Reaction rates for He + He Li The calculations for the reaction rates, i.e., elastic, ex-change and loss, for the He + He Li collision are shownin the following. The parameters of the zero-range and rank-1 separable s − wave potential (see Table II) models are fittedto reproduce the binding energies obtained from the choicesof potentials models given in Table I. The results are pre-sented in Fig. 3 for model (a1) and in 4 for model (a2), and ineach figure the outcome of the zero-range and separable finiterange potential are shown.The exchange and loss rates for the He + He Li withmodel (a1), shown in Fig. 3, correspond to endothermic re-actions: the threshold energy in order to open the exchangechannel is 0.2 mK, with the three-body dissociation at 1.5mK. The s − wave elastic rate is dominant below opening thethreshold attaining values as larger as 10 − cm /s, for boththe zero-range and finite-range potentials. Once the atom ex-change channel opens, i.e. He + He Li → He + Li, the p − wave becomes noticeable and right away gives the majorcontribution to both the atom exchange and loss rate, evenabove the dissociation channel.The p − wave dominance comes from the well-known one-atom exchange diagram, which due to the involved smallbinding energy has a pole close to the scattering threshold.Physically in the exchange process, the incoming He picksthe other one from the He Li molecule, and the remaining Li is moving backwards with respect to the incoming He,enhancing the p − wave contribution to the reaction process.The s − wave projection of the one-atom exchange diagramis also associated with the appearance of the Efimov effect, assuch diagram comes also as the kernel of the integral equa-tions for the scattering and bound state, both for the zero-range and separable potential models. One can observe thatboth models predict very similar reaction rates, as these low-energy process are quite universal, and determined by thetwo-atoms low energy observables (in this case the bindingenergies) and one three-body input, as the binding energiesof the triatomic molecules. The separable interaction modelhas indeed scattering lengths almost three times the effectiveranges (see Table II), characterizing well a short range poten-tial with three-body low energy observables weakly dependenton the range. It is important to emphasize that the s − waveobservables are more sensitive to the potential, as they re-quire for the limit of zero-range interaction the informationof the triatomic binding energy, while the p , d , .... waves -5 -4 -3 -2 -1 E k (K) -13 -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -5 -4 -3 -2 -1 E k (K) -13 -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 3. Reaction rates for He + He Li, using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model.are essentially sensitive to the on-shell low energy two-atomamplitude.The reaction rates for He + ( He Li) obtained with model(a2) are shown in Fig. 4. Now, the atom exchange reac-tion is exothermic (see Table III), with the three-atom con-tinuum opening at 0.12mK. The s − wave is dominant up tothis threshold with rates above 10 − cm /s. The s − wave elas-tic rate has a minimum at the inelastic threshold, with the p − wave emerging as the dominant one above it. The d − waveis in general less important, but can be comparable with the s − wave for higher energies. The manifestation of the min-imum in the s − wave can be clearly seen in the elastic rate.The d − wave phase shift has also a zero around 1mK, reflectedas a minimum of the elastic amplitude, due to the absorption.This feature is model independent, as one can observe by com-paring the two potential models. The atom exchange rate isdominated by the p − wave above the dissociation threshold,with the s − and d − waves representing less than 20% of therate. The calculated loss rate for the reaction He + He Li is -8 -7 -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -12 -11 -10 -9 -8 K e x ( c m / s ) -12 -11 -10 -9 -8 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -8 -7 -6 -5 -4 -3 -2 -1 E k (K) -13 -12 -11 -10 -9 -8 K e l a s ( c m / s ) -12 -11 -10 -9 -8 K e x ( c m / s ) -12 -11 -10 -9 -8 K l o ss ( c m / s ) S_waveP-waveD-wavetotal(a)(b)(c)
FIG. 4. Reaction rates for He + He Li, using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.also shown in Fig. 4 for the model (a2). A noticeable dip ap-pears at 0.1 mK, which comes from the fast increase of the p − and d − waves contribution to the dissociation channel, whilethe s − wave becomes less relevant. The dissociation channelis more efficiently populated by the higher partial waves, asthe relevant inputs are only the binding energies of the Hedimer and the He Li, which in this case are pretty small.As already mentioned, beyond s − wave the three-body scat-tering amplitudes are dominated by the on-shell two-atomlow-energy T-matrix, which sizes the kernel of the three-bodyscattering equation as can be clearly verified, for example, inthe zero-range model. In this way, the bound state pole of thevery weakly bound He Li molecule enhances even more thekernel and together the contribution of the p − and d − waves tothe loss rate. The elastic, atom exchange and loss rates clearlydistinguish the models (a1) and (a2), providing a mean to in-directly access the on-shell quantities. The difference in thebinding energy of the He Li molecule in models (a1) and (a2) is relevant for the s − wave, while it is barely perceivedby the p − and d − waves, that show sensitivity to the differ-ent values of the He Li energy even above the dissociationthreshold.
B. Reaction rates for Li + He The results for the reaction rates in the Li + He colli-sion are shown in Figs. 5 and 6, with the Li + He → He+ He Li reaction being exothermic for model (a1) and en-dothermic for model (a2) (see Table III). For the potentialmodel (a1), the dissociation channel opens at 1.3 mK, domi-nating the losses only above 10 mK, as one can observe fromthe corresponding panels of Fig. 5. The elastic rate is domi-nated by the s − wave up to 10 mK, when the p − wave takesover. The atom exchange rate, already discussed, has a majorcontribution from the p − wave, which raises above the s − wavearound 0.1 mK, while the d − wave up to 0.1 K is not relevant.The dissociation component of the loss rate has an over-whelming contribution from the p − wave up to about 10 mK,when the d − wave becomes competitive due to its contribu-tion, essentially, to the dissociation process. We have to re-mind that the p − and d − waves are determined by the bind-ings energies of the diatomic molecules He and the veryweakly bound He Li, which gives a long range tail for theattractive Efimov like potential [42], that of course dampsthe centrifugal barrier enhancing the importance of the higherpartial waves, even at low energies. The relative relevance ofthe d − wave contribution can be clearly seen in the loss rate.Comparing the calculations of the zero- and finite-rangepotential models, for the same inputs from the set (a1), it isclear the independence of the bulk results on the detail of thepotential beyond the dimer and trimer binding energies. Bothmodels exhibit a strong enhancement of the atom exchangeand loss rates between 2-3 mK, increasing up to 10 − cm /s,with the he p − wave playing a major role.For the model (a2), the Li + He → He + He Li re-action is endothermic, as seen from the Table III. The corre-sponding results are given in Fig. 6, where the atom exchangechannel opens at 1.2mK and the dissociation at 1.3mK, offer-ing an interesting interplay between the competing losses inthese two channels. For the elastic reaction rate, the s − wavecontribution is relevant below 1mK, with the p − wave beingdominant above 10mK, while the d − wave is marginal. In theatom exchange rate the s − and d − waves are present, althoughthe p − wave contributes to a large extend.The loss rate corresponds essentially to the atom exchangerate up to 5mK, while above 10mK it receives contributionfrom the dissociation process in both p − and d − waves, whichattains values on the bold part of 10 − cm /s. The relevantcontributions from the p − and d − waves amounts to the small-est of the binding energy of the He- Li molecule, which ex-tends the Efimov long range potential, increasing the rele-vance of p − and d − waves to the reaction process. -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 5. Reaction rates for Li + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model. V. THREE BODY REACTIONS WITHHELIUM-4 AND LITHIUM-7A. Reaction rates for He + He Li In this section, we report our results on the elastic, ex-change and loss rates for the system containing the Li iso-tope, namely, when considering the He + He Li reaction,which has the diatomic and triatomic molecules more boundthan the previous case where the Lithium-6 isotope was in-teracting with the other two He atoms. For the inputs of thezero and finite-range models, the corresponding parametersfor the potentials (a1) and (a2) are given in Table I. For bothsets (a1) and (a2) the reaction He + He Li → He + Li isendothermic, with the corresponding binding energies beingquite close. As verified, the results for the rates are very much -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 K e x ( c m / s ) -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 6. Reaction rates for Li + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.similar, as it will be detailed in our discussion of Figs. 7 and 8.In this case, the separable potential has the scattering lengthsmuch larger than the effective range, namely, about one orderof magnitude and above, as seen in Table II, making the out-come of the zero-range model closer to the results obtainedwith the finite range potential, as the corrections due to theeffective range are marginal.In Fig. 7, the reaction rates for the parameter set (a1)are shown, where the atom exchange channel opens around4.3mK, and close to this energy the p − wave elastic phaseshift has a zero, clearly seen for the zero-range and finiterange models. The s − wave amplitude gives the elastic rateup to the atom exchange threshold, when the p − wave be-comes dominant, this is noticed by a depression in the rate,independent of the model. The atom exchange rate is essen-tially defined by the p − wave, as we have discussed before.The loss rate in the d − wave becomes relevant at energies ofabout 100 mK.The calculations of the reaction rates with the parameter -4 -3 -2 -1 E k (K) -12 -10 -8 -6 K e l a s ( c m / s ) -11 -10 K e x ( c m / s ) -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 K e x ( c m / s ) -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 7. Reaction rates for He + He Li, using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model.set (a2) are shown in Fig. 8 for the He + He Li collision.In this case the atom exchange reaction opens close to 1mK.Interesting enough is the zero of the elastic p − wave phase shiftclose to the threshold, as seen by the minimum of the elasticrate close to the atom exchange threshold. This minimumseems a universal property of the p − wave also appearing inthe elastic process with model (a1). The difference betweenthe two parameter sets is the binding energy of the He Limolecule, as the triatomic binding energy is not relevant forthe p − wave, which is also the protagonist on both the atomexchange and loss rate. The s − wave has a minor contributionon these two rates, and the d − wave appears in the loss rateabove 100mK. The bulk of the results for the reaction ratesare unaffected by the change of the potential range. -4 -3 -2 -1 E k (K) -13 -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 -8 K e x ( c m / s ) -11 -10 -9 -8 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 -8 K e x ( c m / s ) -11 -10 -9 -8 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 8. Reaction rates for He + ( He Li), using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.
B. Reaction rates for Li + He Our results on the reaction rates for Li + He , presentedin the Figs. 9 and 10, are discussed in this subsection. Asrepresented in Table II, the atom exchange channel is exother-mic for both set of potential models (a1) and (a2), with thedissociation threshold appearing at 1.3mK for both two cases.The results for the zero and finite-range models with pa-rameters obtained from (a1), given in Table I, are shown inFig. 9. The s − wave dominates the elastic rate up to the dis-sociation threshold, when the p − wave takes over and gives thetotal value for the rate. The exothermic atom exchange reac-tion, i.e., Li + He → He + He Li, has the major contribu-tion from the s − wave up to about 0.1mK, when the p − wavedominates. The loss rate shows the relevance of the p − waveup to 100mK, when the d − wave starts to compete. Bothzero-range and separable potential models provide very simi-lar model independent results. So, for the results obtained in -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 K e l a s ( c m / s ) -12 -11 -10 -9 K e x ( c m / s ) -12 -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 9. Reaction rates for Li + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model.these calculations, it seems not relevant the interaction range.The next set of calculations were performed for the zero andfinite-range models with the parameters (a2), given in TableI, and presented in Fig. 10. Comparing with the bindingenergies from the set (a1), the corresponding values for thediatomic and triatomic molecules are more weakly bound, ofabout half of the corresponding values. This is reflected in the s − wave elastic rates that are about three times larger below0.1mK, while the maximum is about the same and somewhatbelow 10mK, with value of 10 − cm /s. This is a consequenceof the large difference in the binding energies of He − Limolecule, which for set (a2) is considerably smaller than set(a1). The p − wave outcome is only sensitive to the diatomicbinding energies, with He and He Li changing from 5.6 to2.16 mK, that explains the results being quite close, as wellas for the d − wave.The exothermic atom exchange reaction rate shown in Fig.10 has the dominance of the s − wave up to about 0.1mK, -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -6 -5 -4 -3 -2 -1 E k (K) -12 -11 -10 -9 -8 K e l a s ( c m / s ) -11 -10 -9 K e x ( c m / s ) -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 10. Reaction rates for Li + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.when the p − wave rises and gives the bulk part of the rate.The p − wave also dominates the loss rate up to 10mK, whenthe d − wave starts to compete. The comparison between thezero-range and separable potential model results shows againthe model independence, with the finite range playing a minorrole to build the bulk values of the calculated rates. VI. THREE BODY REACTIONS WITHHELIUM AND SODIUMA. Reactions rates for He + He Na The collision He + He Na process has some distinctivefeatures compared to the previous cases composed by heliumand lithium atoms. Two salient differences will be apparent inthe rate results. For this case, the parametrization of the sep- arable potential gives an effective range, which is only abouthalf of the scattering length, as shown in Table II. The otherfeature is the presence of Efimov zeros in the elastic s − wavephase-shift [43]. -2 -1 E k (K) -13 -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -2 -1 E k (K) -13 -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 11. Reaction rates for He + He Na, using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model.The Efimov zeros are a sequence of zeros/dips in the elasticscattering amplitude, that are controlled by both the scat-tering lengths and the actual value of the triatomic bindingenergy [43]. The He + He Na s − wave phase-shift presentssuch a zero/dip, which is sensitive both to the two-atom lowenergy parameters and to the short range physics summa-rized in the binding energy of He − Na molecule. The zeroturns into a dip, if it appears above the threshold of the atom-exchange or dissociation channels, due to the probability fluxfrom the elastic channel to those ones.The atom-exchange reaction is now endothermic for He+ He Na → He + Na and this channel opens around28mK, while the dissociation reactions happens above 29mK,for both parameter sets (a1) and (a2) (see Table III). Ourresults for the reaction rates obtained with the zero-range -2 -1 E k (K) -14 -13 -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -2 -1 E k (K) -13 -12 -11 -10 -9 K e l a s ( c m / s ) -12 -11 -10 -9 K e x ( c m / s ) -12 -11 -10 -9 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 12. Reaction rates for He + He Na, using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.and separable potential models with parameter sets (a1) and(a2) are shown in Figs. 11 and 12, respectively.In Figs. 11 and 12, the calculations of the reaction ratesfor the helium collision with ( He Na) molecule are shownfor both zero and finite range separable potentials. It is inter-esting to observe that the Efimov zero of the elastic s − wavephase-shift and corresponding reaction rate appears for thezero-range and one-term Yamaguchi separable potentials re-spectively at 20mK and 15mK for the set (a1). In the case ofthe parameter set (a2), the zero of the elastic rate for the po-tential comes around 2.5mK, while for the zero-range modelthe zero is not present in the scale of the figure. The effectiverange moves the zero to somewhat larger values of the energy.These zeros are quite sensitive to the difference in the on-shellparameters as one can appreciate in Tables I and II.The two potential models have identical diatomic and He − Na binding energies for the parameter set (a1), butthey differ in the effective range and scattering lengths, which combined produce the change in the position of the s − wavezero, while keeping the same qualitative features in the elasticrate. The zero of the s − wave elastic reaction rate gives to-gether with the dominant p − wave contribution a pronounceddip, which for (a1) comes around 12-15mK, and the elasticrate from this energy up to 50 mK, increases two orders ofmagnitude to about 10 − cm /s. The same qualitative fea-ture is found for the parameter set (a2), with the dip around2.5mK for the separable potential, while for the zero-rangemodel it is not present in the scale of the figure. The positionof the minimum is sensitive to the effective range, as alreadyobserved for the parameter set (a1). For both parameter sets(a1) and (a2) the p − wave turns to be relevant between 10and 30mK, while for the one-term separable potential withset (a2) it presents a zero.The atom exchange and loss rates are dominated by thehigher waves, with particular preponderance of the p − wave,as found for the triatomic systems with the lithium isotopes.When considering the sets (a1) and (a2) parametrized by Ya-maguchi potentials, the influence of the effective range is ver-ified in both cases by the importance of the d − wave withrespect to the p − wave contribution. Naively, one can under-stand the effect of the effective range on the imbalance of the p − and d − waves, as it cuts the long-range Efimov potential,which is relatively more important for the p − wave than forthe d − wave. In the zero range model, without the weakeningof the long range potential, the p − wave is dominant on theexchange and loss rates. B. Reaction rates for Na + He The reaction Na + He has an exothermic channel withthe atom exchange to form the He − Na molecule (see Ta-ble III), and the dissociation threshold is quite low at 1.31mK,for both set of parameters (a1) and (a2). The results areshown in Figs. 13 and 14, for both zero-range and one-termseparable Yamaguchi potentials with parameter sets (a1) and(a2), respectively. The magnitude of the reaction rates arefound to grow up to 10 − cm /s. These values are one orderbelow the reaction rates obtained with the isotopes of lithiumwith mass numbers 6 and 7, as we have throughly discussedin sections IV B and V B.The elastic reaction rate is dominated by the s − wave upto about 10mK, independent of the parameter set (a1) or(a2) and potential range, as shown in Figs. 13 and 14. The p − wave importance above such energies increases over the s − and d − waves. The atom exchange process is open forthis entrance reaction channel, and the p − wave contributesto the bulk of this rate above 1mK, while the d − wave becomesmore relevant with respect to the p − wave for the finite rangepotential. This happens due to the fact that the finite rangeof the potential has the effect to deplete the p − wave rate, aswe have discussed before. This behavior is independent of theparameter set (a1) or (a2).Finally the dissociation rate component of K loss is essen-tially determined by the p − and d − waves, with the s − wavecontributing marginally. The inversion of the importance ofthe p − and d − waves to the dissociation process comes as aconsequence of the consideration of the effective range. By an-alyzing separately the elastic and loss rates, one can disentan-gle the two and three-atomic low energy input informationscontained s − , p − and d − waves. Specifically, the s − wave re- action rates depends on two and three-atomic low energy in-put informations, while the higher partial waves mainly carrythe diatomic low energy parameters and being insensible tothe triatomic molecular energy. By taking into account ex-perimental differential angular reaction rates the individualpartial waves could be extracted. Following this procedure,real possibilities exist to separate the effects of the triatomicmolecule binding energy from the diatomic low energy infor-mations, namely, binding energy, scattering length and effec-tive range, which are contained in the elastic atom exchangeand total loss rate from eventual experimental data below100mK. -5 -4 -3 -2 -1 E k (K) -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -5 -4 -3 -2 -1 E k (K) -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 13. Reaction rates for Na + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a1) model. VII. CONCLUSION
In the present work, we provide predictions for severalatom-dimer reaction rates, by considering particular two-atomic systems for different configurations having two He -5 -4 -3 -2 -1 E k (K) -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c) -5 -4 -3 -2 -1 E k (K) -12 -11 -10 K e l a s ( c m / s ) -12 -11 -10 K e x ( c m / s ) -12 -11 -10 K l o ss ( c m / s ) S-waveP-waveD-wavetotal(a)(b)(c)
FIG. 14. Reaction rates for Na + He , using zero-range(upper-set of panels) and finite-range (lower-set of panels)interactions, considering the (a2) model.atoms with one of the species, among Li, Li and Na,taking the advantage that dimer and trimer binding ener-gies are available for these atom-dimer systems from experi-mental data or predicted by realistic potential model calcu-lations. Within the Faddeev formalism approach, these real-istic binding energies are directly applied, in our study withzero-range and finite-range two-body interactions. We pre-dict reaction rates for cold He elastic collisions with He Li, He Li and He Na molecules, as well as by considering theatomic species Li, Li and Na reacting with helium dimersfor center-of-mass kinetic energies up to 100 mK.For our study we use the diatomic and triatomic parame-ters from two atomic models given in [86]. The elastic, atomexchange and loss rates are computed with a zero-range anda s − wave one-term separable potential in order to access therelevance of the effective range, besides the diatomic bindingenergies from [86]. This work follows a previous explorationof the cold s − wave cross-sections for the He on the diatomicmolecules formed by the helium atom with the isotopes of lithium and sodium [43] using the zero-range and one-termseparable potentials.Specifically, the elastic, atom-exchange and dissociationchannels are investigated for the following six reactions: • He + He Li and Li + He , in Section IV; • He + He Li and Li + He , in Section V; • He + He Na and Na + He , in Section VI.The main characteristic found in these studies is the p − wavedominance of the atom-exchange and dissociation reaction,allowing the separation of the effects of the on-shell low en-ergy diatomic properties from the triatomic binding energy,which is determinant for the elastic reaction rate below thedissociation threshold. In particular, for the , Li and Nareactions with the He dimer, the atom exchange channel isalways open, which allows to separate the s − wave contribu-tion below the dissociation threshold from the higher partialwave rates.Of possible experimental relevance, is the presence of a min-imum in the s − wave elastic reaction rate for the He → ( He- Na) scattering, following the previous findings of Ref. [43].This minimum is kept once the other partial waves are in-cluded in the elastic rate, and thus could be of interest fromthe point of view of an actual experiment. Such minimum isreminiscent of the Efimov effect that translates in the scatter-ing region as a log periodicity of zeros in the elastic s − wavephase shift in the collision of the atom with a weakly bounddimer. Another interesting property is the presence of azero/dip for some of the elastic rate in the p − wave, whichwas particularly seen in the case of He + ( He Li) and He+ ( He Na) between 1 and 10mK.The atom exchange and dissociation rates for all exampleswe have studied is dominated by the p − wave, as it is under-stood from the importance of the one atom exchange mech-anism, which gives more weight to the process at backwardangles. Pictorially, one of the atoms in the initial moleculeis picked-up by the incoming atom, and the newly formedmolecule is propagating backwards. We remind that alreadythe kernel of the integral equations and inhomogeneous termfor the zero-range and s − wave one term separable potentialcorresponds to an atom exchange amplitude.The range correction to the reaction rates computed withthe one-term separable potential, is particularly noticeableas an inversion of the relative importance between the p − and d − waves, with respect to the results obtained with thezero-range model. This effect comes from the softening ofthe long-range Efimov type potential at the short distancesdue to the finite range, still perceptible for the p − wave, whilethe d − wave is less sensitive to such modification due to thestronger centrifugal barrier. Therefore, such an effect empha-sizes even more the possibility to disentangle the on-shell lowenergy two-atom informations and the triatomic binding en-ergy, with the last one is determinant for the s − wave rates,but irrelevant for the higher wave contributions. The p − and d − wave dominate the atom exchange and dissociation rates,while presenting different sensitivities to the low energy di-atomic parameters.Although the results for the reaction rates present somesensitivity to the potential range when the fitted potentialpresents effective ranges comparable to the scattering length,which is found particularly for the He- He Na, still the bulkvalues obtained with the zero range and the finite range poten-tial are similar, supporting the robustness of our predictions. Closing our summary, we should point to some future di-rections of our investigation. The methods employed here canbe extended to ultracold atoms in atomic traps to study con-trolled chemical reactions, by means of tuning the scatteringlengths with Feshbach resonances, and also moving the tri-atomic state by induced few-body forces [95, 96]. In addition,more complex reactions considering the collision of diatomicmolecules widens the scope of our investigation. Another per-spective is the manipulation of the aspect ratio of the trap,changing the effective dimension in which the reaction takesplace in a continuous way from three to two dimensions [97–100], therefore we hope that in the future not only the inter-action can be tuned but also the effective dimension.
ACKNOWLEDGMENTS
This work was partially supported by Funda¸c˜ao de Am-paro `a Pesquisa do Estado de S˜ao Paulo [2017/05660-0 (T.F.and L.T.), 2019/00153-8 (M.T.Y.)], Coordena¸c˜ao de Aper-fei¸coamento de Pessoal de N´ıvel Superior (M.A.S.), Con-selho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico[303579/2019-6 (M.T.Y.), 308486/2015-3 (T.F.), 304469-2019-0(L.T.), and Project INCT-FNA 464898/2014-5].
Appendix A: Atom-dimer transition operators
As considering the particles i and j (= 1 , , i is theprojectile, can be written in operator form as [87, 88] U ji = δ ji G − + (cid:88) k δ kj t k G U ki (A1)where U ji s are transition amplitudes, t k s are two body T-matrices and G is three-body free propagator. If particle 1is the projectile U is related to elastic scattering amplitudeand U and U are related to rearrangement amplitudes. Bymultiplying the both sides of Eq. (A1) from right and left to G : G U G = G t G U G + G t G U G G U G = G + G t G U G + G t G U G G U G = G + G t G U G + G t G U G (A2)If we use separable potential for two-body interactions, wewill have the following two-body T-matrices. t i = | g i (cid:105) τ i (cid:104) g i |(cid:104) p q | t i | p (cid:48) q (cid:48) (cid:105) = δ ( p − p (cid:48) ) δ ( q − q (cid:48) ) τ ( E − q µ iq ) , (A3)where | g i (cid:105) is called the form factor. By introducing the fol-lowing operators: X ij = (cid:104) g i | G U ij G | g j (cid:105) ,Y ij = (cid:104) g i | G | g j (cid:105) (A4)we will have: X = Y τ X + Y τ X ,X = Y + Y τ X + Y τ X ,X = Y + Y τ X + Y τ X . (A5) We consider X ij ( q , q (cid:48) ) = (cid:104) q | X ij | q (cid:48) (cid:105) and Y ij ( q , q (cid:48) ) = (cid:104) q | Y ij | q (cid:48) (cid:105) . For the scattering of a particle α by the αβ bound subsystem, we have 1 → α , 2 → α and 3 → β . Us-ing symmetry property for the identical bosons, we consider X ( q , q (cid:48) ) + X ( q , q (cid:48) ) = X ( q , q (cid:48) ) for elastic scattering and X ( q , q (cid:48) ) = X (cid:48) ( q , q (cid:48) ) for rearrangement. We can also calcu-late: Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = K ( q , q (cid:48) )2 ,Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = K ( q , q (cid:48) )2 , (A6)and we can write the partial wave decomposition of the cou-pled equations (A5), as follows: X (cid:96) ( q, q (cid:48) ) = K (cid:96) ( q, q (cid:48) )2+ 4 π (cid:90) k dk K (cid:96) ( q, k )2 τ αβ ( k ; E ) X (cid:96) ( k, q (cid:48) )+ 4 π (cid:90) k dkK (cid:96) ( q, k ) τ αα ( k ; E ) X (cid:48) (cid:96) ( k, q (cid:48) ) ,X (cid:48) (cid:96) ( q, q (cid:48) ) = K (cid:96) ( q (cid:48) , q )2+4 π (cid:90) k dk K (cid:96) ( k, q )2 τ αα ( k ; E ) X (cid:96) ( k, q (cid:48) ) , (A7)where τ ( E − k µ α ( αβ ) ) = τ ( E − k µ α ( αβ ) ) = τ αβ ( k ; E ) and τ ( E − k µ β ( αα ) ) = τ αα ( k ; E ). By removing the singularityof two-body T-matrices: τ αβ ( q ; E ) = τ αβ ( q ; E ) q − k α − i (cid:15) ,τ αα ( q ; E ) = τ αα ( q ; E ) q − k β − i (cid:15) . (A8)We introduce the scattering and rearrangement reducedamplitudes as follow: h (cid:96)α ( q, q (cid:48) ) = 2 π τ αβ ( q ; E ) X (cid:96) ( q, q (cid:48) ) ,h (cid:96)β ( q, q (cid:48) ) = 4 π τ αα ( q ; E ) X (cid:48) (cid:96) ( q, q (cid:48) ) , (A9)and using Eq. (A9) in Eq. (A7) we find the following equa-tions for the half-on-shell scattering and rearrangement re-duced amplitudes: h (cid:96)α ( q ; E ) = τ α ( q ; E ) (cid:40) π K (cid:96) ( q, k α ; E ) + (cid:90) ∞ dkk ×× (cid:34) K (cid:96) ( q, k ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) + K (cid:96) ( q, k ; E ) h (cid:96)β ( k ; E ) q − k β − i (cid:15) (cid:35)(cid:41) ,h (cid:96)β ( q ; E ) = τ β ( q ; E ) (cid:40) π K (cid:96) ( k α , q ; E )+ (cid:90) ∞ dkk K (cid:96) ( k, q ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) (cid:41) , (A10)where q (cid:48) = k α = (cid:112) µ α ( αβ ) ( E − E αβ ) and τ α ( q ; E ) = 2 πτ αβ ( q ; E ) ,τ β ( q ; E ) = 4 πτ αα ( q ; E ) . (A11) For the scattering of a particle β by the αα bound sub-system, we have 1 → β , 2 → α and 3 → α . Usingsymmetry property for the identical bosons, we consider X (cid:48) ( q , q (cid:48) ) ≡ X ( q , q (cid:48) ) for elastic scattering and X ( q , q (cid:48) ) ≡ X ( q , q (cid:48) ) + X ( q , q (cid:48) ) for rearrangement. We can also cal-culate: Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = K ( q (cid:48) , q )2 ,Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = Y ( q , q (cid:48) ) = Y ( q (cid:48) , q ) = K ( q , q (cid:48) )2 , (A12)and we can write Eq. (A5) as follows: X (cid:96) ( q, q (cid:48) ) = K (cid:96) ( q, q (cid:48) )+ 4 π (cid:90) k dk K (cid:96) ( q, k )2 τ αβ ( k ; E ) X (cid:96) ( k, q (cid:48) )+ 4 π (cid:90) k dkK (cid:96) ( q, k ) τ αα ( k ; E ) X (cid:48) (cid:96) ( k, q (cid:48) ) X (cid:48) (cid:96) ( q, q (cid:48) ) = 4 π (cid:90) k dk K (cid:96) ( k, q )2 τ αα ( k ; E ) X (cid:96) ( k, q (cid:48) )(A13)Using Eq. (A9) in (A13) we have the following equations forthe half-on-shell scattering and rearrangement reduced ampli-tudes: h (cid:96)α ( q ; E ) = τ α ( q ; E ) (cid:40) πK (cid:96) ( q, k α ; E ) + (cid:90) ∞ dkk ×× (cid:34) K (cid:96) ( q, k ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) + K (cid:96) ( q, k ; E ) h (cid:96)β ( k ; E ) q − k β − i (cid:15) (cid:35)(cid:41) ,h (cid:96)β ( q ; E ) = τ β ( q ; E ) (cid:90) ∞ dkk K (cid:96) ( k, q ; E ) h (cid:96)α ( k ; E )( k − k α − i (cid:15) ) , (A14)here q (cid:48) = k β = (cid:112) µ β ( αα ) ( E − E αα ), where h α representsthe rearrangement and h β represents the elastic scatteringamplitudes. Appendix B: Scattering equation kernels
When using zero-range interactions, a momentum cut-off isrequired to regularize the integral equations for the s − wavestate, within a renormalization procedure, where the bindingenergy of the triatomic molecule is kept fixed. For that, in thekernels a subtraction is performed with a regularizing momen-tum parameter µ (see e.g. [95]), such that the kernels K , , τ j and f α,β used in the bound-state and scattering equationsare given by K (cid:96)i =1 , ( q, k ; E ) ≡ G (cid:96)i ( q, k ; E ) − G (cid:96)i ( q, k, − µ ) δ l ,G (cid:96) ( q, k ; E ) = (cid:90) − dx P (cid:96) ( x ) E + i (cid:15) − q m − k µ αβ − kqxm G (cid:96) ( q, k ; E ) = (cid:90) − dx P (cid:96) ( x ) E + i (cid:15) − q + k µ αβ − kqxAm , (B1) τ α ( q ; E ) ≡ µ α ( αβ ) πµ αβ [ κ αβ + κ ,αβ ( E )] , (B2) τ β ( q ; E ) ≡ µ β ( αα ) πµ αα [ κ αα + κ ,αα ( E )] , (B3) where κ αα ≡ (cid:112) − µ αα E αα , κ αβ ≡ (cid:112) − µ αβ E αβ κ αα ( E ) ≡ (cid:115) − µ αα (cid:20) E − q µ β ( αα ) (cid:21) ,κ αβ ( E ) ≡ (cid:115) − µ αβ (cid:20) E − q µ α ( αβ ) (cid:21) . (B4) f α = µ αβ / (cid:112) k αβ f β = µ αα / √ k αα (B5)In the case of the Yamaguchi separable potential fromEq. (19), K , and τ j are given by the following: K (cid:96) ( q, k ; E ) = (cid:90) − dx (cid:20) q + k qkx + γ αα (cid:21) − (B6) × (cid:20) k + q A ( A + 1) + 2 qkAx ( A + 1) + γ αβ (cid:21) − × (cid:20) E + i (cid:15) − q m − k µ αβ − qkxm (cid:21) − P (cid:96) ( x ) ,K (cid:96) ( q, k ; E ) = (cid:90) − dx (cid:20) k + q ( A + 1) + 2 qkx ( A + 1) + γ αβ (cid:21) − × (cid:20) q + k ( A + 1) + 2 qkx ( A + 1) + γ αβ (cid:21) − (B7) × (cid:20) E + i (cid:15) − ( q + k )2 µ αβ − qkxAm (cid:21) − P (cid:96) ( x ) ,τ α ( q ; E ) ≡ µ α ( αβ ) πµ αβ (cid:34) γ αβ ( γ αβ + κ αβ ) γ αβ + κ αβ ( E ) + κ αβ (B8) × [ γ αβ + κ αβ ( E )] [ κ αβ + κ αβ ( E )] (cid:35) ,τ β ( q ; E ) ≡ µ β ( αα ) πµ αα (cid:34) γ αα ( γ αα + κ αα ) γ αα + κ αα ( E ) + κ αα (B9) × [ γ αα + κ αα ( E )] [ κ αα + κ αα ( E )] (cid:35) .f α = 2 πµ αβ / (cid:112) k αβ γ αβ ( k αβ + γ αβ ) f β = 2 πµ αα / (cid:112) k αα γ αα ( k αα + γ αα ) (B10)In our approach, the parameters of the separable interac-tions are fixed by the corresponding bound-state energies, aswell as by the effective ranges (when considering finite-rangeinteractions). Appendix C: Cross-sections
The scattering observables are obtained from the scatteringamplitudes, which for elastic reaction is U el = U + U andfor exchange reaction U ex = √ U . For the elastic scattering,we write that: dσ el d Ω = (2 π ) µ α ( αβ ) |(cid:104) q f φ α | U el | q i φ α (cid:105)| , (C1) where φ α is the bound state of αβ subsystem and q i = q f = (cid:112) − µ α ( αβ ) ( E − E αβ ) = (cid:112) − µ α ( αβ ) E k .We will use the following relations between two-body boundstate and the form factor when we are using the one-termseparable potential: V | φ α (cid:105) = λ | g α (cid:105)(cid:104) g α | φ α (cid:105) = f α | g α (cid:105) ; (C2)consequently, in subsystem αβ for on-shell momentum, wehave: τ αβ ( k ; E ) = 2 µ α ( αβ ) f α ( k α − k ) , ( k → k α ) (C3)and finally we can relate the transition amplitudes to ourscattering function that we have calculate in Eq. (A10) h α ( k ; E ) = 2 π τ αβ ( k ; E ) X ( k , k (cid:48) )= − π µ α ( αβ ) f α X ( k , k (cid:48) )= − π µ α ( αβ ) f α (cid:104) k | (cid:104) g α | G U el G | g α (cid:105) | k (cid:48) (cid:105) = − π µ α ( αβ ) f α (cid:104) k φ α | V G U el G V | k (cid:48) φ α (cid:105) /f α = − π µ α ( αβ ) (cid:104) k φ α | U el | k (cid:48) φ α (cid:105) (C4)considering all the above equations the elastic cross sectioncan be written as follow: dσ el d Ω = | h α ( k ; E ) | , (C5)For the exchange reaction cross section we have: dσ ex d Ω = (2 π ) µ β ( αα ) (cid:114) µ α ( αβ ) µ β ( αα ) (cid:114) − E αα − E αβ E k × |(cid:104) q f φ β | U ex | q i φ α (cid:105)| (C6) where q f = (cid:112) µ β ( αα ) ( E αβ − E αα + E k ), with the samemethod we can show that h β ( k , E ) = − π µ β ( αα ) f β √ f α (cid:104) q f φ β | U ex | q i φ α (cid:105) , (C7)resulting in: dσ ex d Ω = f α f β (cid:114) µ α ( αβ ) µ β ( αα ) (cid:114) − E αα − E αβ E k | h β ( k ; E ) | . (C8)In the case of scattering of β from αα we need scatteringamplitudes for elastic, U el = U , and exchange channels, U ex = ( U + U ), and the cross-sections are written as: dσ el d Ω = 14 | h β ( k ; E ) | , (C9)and dσ ex d Ω = f β f α (cid:114) µ β ( αα ) µ α ( αβ ) (cid:114) − E αβ − E αα E k | h α ( k ; E ) | , (C10)where h α and h β are calculated from (A14).[1] H. Feshbach(1958), Unified theory of nuclear reactions,Annals of Physics , 357 (1958).[2] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner,D.M. Stamper-Kurn, W. Ketterle, Observation of Fesh-bach resonances in a Bose-Einstein condensate, Nature , 151 (1998).[3] E. Timmermans, P. Tommasini, M. Hussein, and A.Kerman, Feshbach resonances in atomic Bose-Einsteincondensates, Phys. Rep. , 199 (1999).[4] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Fes-hbach resonances in ultracold gases, Rev. Mod. Phys. , 1225 (2010).[5] P. D. Lett, P. S. Julienne, and W. D. Phillips, Photoas-sociative spectroscopy of laser cooled atoms, Ann. Rev.Phys. Chem. , 423 (1995).[6] D. J. Heinzen, “Collisions of ultracold atoms in opticalfields”, in Atomic Physics (D. J. Wineland, C. E.Wieman, and S. J. Smith, Eds.), 369-388, AIP Press,New York, 1995.[7] J. Weiner, V. S. Bagnato, S. C. Zilio, and P. S. Julienne,Experiments and theory in cold and ultracold collisions,Rev. Mod. Phys. , 1 (1999).[8] W. C. Stwalley and H. Wang, Photoassociation of ul-tracold atoms: A new spectroscopic technique, Journalof Molecular Spectroscopy , 194 (1999).[9] W. C. Stwalley, Collisions and reactions of ultracold molecules, Can. J. Chem. , 709 (2004).[10] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S.Reidl, C. Chin, J. Hecker Denschlag, and R. Grimm,Bose-Einstein condensation of molecules, Science ,2101 (2003).[11] M. Greiner, C.A. Regal, and D.S. Jin, Emergence of amolecular Bose-Einstein condensate from a Fermi gas,Nature , 537 (2003).[12] T. Mukaiyama, J. R. Abo-Shaeer, K. Xu, J. K. Chin,and W. Ketterle, Dissociation and decay of ultracoldsodium molecules, Phys. Rev. Lett. , 180402 (2004).[13] M. Guo, B. Zhu, B. Lu, X. Ye, F. Wang, R. Vexiau,N. Bouloufa-Maafa, G. Qu´em´ener, O. Dulieu, and D.Wang, Creation of an ultracold gas of ground-state dipo-lar Na Rb molecules, Phys. Rev. Lett. , 205303(2016).[14] K. K. Voges, P. Gersema, T. Hartmann, T. A. Schulze,A. Zenesini and S. Ospelkaus, A pathway to ultracoldbosonic Na K ground state molecules, Phys. Rev.Lett. , 083401 (2020).[15] A. Green, H. Li, J. H. S. Toh, X. Tang, K. C. Mc-Cormick, M. Li, E. Tiesinga, S. Kotochigova, and S.Gupta, Feshbach resonances in p − wave three-body re-combination within Fermi-Fermi mixtures of open-shell Li and closed-shell
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