Weakly bound LiHe 2 molecules in the framework of three-dimensional Faddeev equations
aa r X i v : . [ phy s i c s . a t m - c l u s ] F e b Few-Body Systems manuscript No. (will be inserted by the editor)
E. A. Kolganova · V. Roudnev
Weakly bound LiHe molecules in the framework ofthree-dimensional Faddeev equations the date of receipt and acceptance should be inserted later Abstract
A method of direct solution of the Faddeev equations for the bound-state problem with zero totalangular momentum is used to calculate the binding energies. The results for binding energies of He Li andHe Li systems and helium atom - HeLi dimer scattering length are presented. The results show that modernpotential models support two bound states in both trimers. In both cases the energy of the excited state is veryclose to the energy of the lowest two-body threshold.
Keywords
Efimov effect · triatomic systems · Faddeev approach · helium-alkali trimers The Efimov effect is a remarkable phenomenon, which is an excellent illustration of the variety of possibilitiesarising when we transit from the two-body to the three-body problem. In 1970 V.Efimov [1,2] proposed thatthree-body systems with short range interaction can have an infinite number of bound states when none of thetwo-particle subsystems has bound states but at least two of them have infinite scattering lengths. In such acase the scattering length a is much lager than the range of the interaction r . The simplest situation describedby Efimov [2] corresponds to three identical neutral bosons interacting via short-range resonant interactionstreated in the zero-range theory framework. In this theory it is assumed that the short-range region details ofthe interaction can be neglected and the wave function in the asymptotically free region can be parametrizedby the scattering length. In order to reproduce correctly the two-body wave function in the region outside ofthe range of interaction r one can use the Bethe-Peierls boundary conditions [3] for the three-body wavefunction Ψ when the two particles separated by r come in contact − r Ψ ∂ r Ψ∂ r −−→ r → a . (1)The simplification which is used in the zero-range theory is to keep the same form of the wave function downto r =
0, although this is unphysical at distances r < r . To describe the three-body system Efimov used thefree Schr¨odinger equation written in hyperspherical coordinates [4] with Bethe-Peierls boundary conditions(1) which lead to the equation for a radial function (cid:18) − d dR − R ddR + s n R (cid:19) F n ( R ) = EF n ( R ) , (2) E. A. KolganovaBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research and Dubna State University, 141980 Dubna,Russia E-mail: [email protected]. RoudnevDepartment of Computational Physics, St Petersburg State University, 7/9 Universitetskaya nab., St Petersburg, 199034, Russia where R is the hyperradius and s n is a solution of the following transcendental equation [2,5] − s n cos (cid:16) s n π (cid:17) + √ (cid:16) s n π (cid:17) = . (3)All the solutions of equation (3) are real, except one s = . i which is purely imaginary, which resultsin an attractive effective potential in equation (2) for n =
0. This attraction is the origin of the Efimov effect. Inorder to prevent the Thomas collapse [6] an additional three-body boundary condition can be used to fix someof the three-body observables (the ground state energy or the particle-dimer scattering length). This boundarycondition breaks the scale invariance under arbitrary scale transformations but still keeps the scale invarianceunder some discrete set of scale transformation with scaling factors being powers of λ = exp ( π / | s | ) . Thus, inthe limit a → ∞ there is an infinite number of bound states, forming a geometric series of energies accumulatedat the threshold. The following relationship holds for three identical bosonslim n → ∞ E n + E n = exp ( − π / | s | ) ≡ λ − ≈ . ≈ . . (4)One of the best theoretically predicted three-body system with an excited state of the Efimov type is anaturally existing molecule of the helium trimer He (see, [7,8] and refs. therein). The interaction betweentwo helium atoms is quite weak and supports only one bound state with the energy about 1mK and a ratherlarge scattering length about 100 ˚A. Only recently the long predicted weakly-bound excited state of the heliumtrimer was observed for the first time using a combination of Coulomb explosion imaging and cluster massselection by matter wave diffraction [9].The first experimental evidence of the Efimov resonance was observed in an ulracold gas of Cs atomsin 2006 [10]. Experimentally, they observed a giant three-body recombination loss when the strength of two-body interaction was varied. More recently, the second Efimov resonance has been observed and the scalingfactor for the Efimov period has been found to be 21 . λ = . λ (it depends only on the masses of the particles) gets smaller as the mass imbalance in-creases. Experimentally, heteronuclear Efimov states have been searched for in K Rb [16], K Rb [17], K Rb [18], Li Rb [19], Li Cs [20,21] systems. Recent reviews on Efimov effect could be foundin [22,23].There is a growing interest in the investigation of He - alkali-atom van-der-Waals systems, that are ex-pected to be of Efimov nature. In addition to the Helium dimer, the He - alkali-atom interactions are evenshallower and also support weakly bound states. In triatomic He -alkali-atom systems presence of Efimovlevels can be expected. Three-body recombination and atom-molecular collision in Helium-Helium-alkali-metal systems at ultracold temperatures have been studied using adiabatic hyperspherical representation inRef. [24,25,26]. Here we use the Faddeev equations in total angular momentum representation to calculatethe He Li binding energies and a scattering length, which has not been studied before.
The configuration space of three particles after elimination of the center of mass can be described in terms ofthree sets of Jacobi coordinates x i = (cid:18) m j m k m j + m k (cid:19) / ( r j − r k ) y i = (cid:18) m i ( m j + m k ) m i + m j + m k (cid:19) / (cid:18) r i − m j r j + m k r k m j + m k (cid:19) (5)The set of coordinates i describes a partitioning of the three particles into a pair ( jk ) and a separate particle i .The Jacoby vectors with different indexes are related by an orthogonal transform x j = c ji x i + s ji y i y j = − s ji x i + c ji y i . (6) The coefficients c ji and s ji are expressed through the masses of the particles c ji = − (cid:18) m i m j ( m i + m k )( m j + m k ) (cid:19) / , s ji = ( − ) j − i sign ( j − i ) (cid:0) − c ji (cid:1) / and satisfy c ji + s ji = H = H + ∑ i V i ( x i ) , (7)where H stands for the kinetic energy of the three particles and V i ( x i ) is the interaction potential acting in thepair i . Faddeev decomposition represents the wave function Ψ in terms of the Faddeev components Φ i Ψ = ∑ i Φ i ( x i , y i ) , (8)which satisfy the following set of equations [27] ( − ∆ x i − ∆ y i + V i ( x i ) − E ) Φ i ( x i , y i ) = V i ( x i ) ∑ k = i Φ k ( x k , y k ) , (9)where E is the total energy of the system. In case of zero total angular momentum the angular degrees offreedom corresponding to collective rotation of the three-body system can be separated [28] and the kineticenergy operator reduces to H = − ∂ ∂ x i − ∂ ∂ y i − ( x i + y i ) ∂∂ z i ( − z i ) ∂∂ z i , (10)where x i , y i and z i are intrinsic coordinates x i = | x i | , y i = | y i | , z i = ( x i , y i ) x i y i , x i , y i ∈ [ , ∞ ) , z i ∈ [ − , ] . (11)Due to (5) and (6) these coordinates are related by x j = q c ji x i + c ji s ji x i y i z i + s ji y i , y j = q s ji x i − c ji s ji x i y i z i + c ji y i , x j y j z j = ( c ji − s ji ) x i y i z i − c ji s ji ( x i − y i ) . As a result we have a set of three-dimensional differential Faddeev equations ( H + V i ( x i ) − E ) φ i ( x i , y i , z i ) = − V i ( x i ) ∑ k = i φ k ( x k , y k , z k ) . (12)When two particles of a system are identical, the Faddeev equations can be simplified. For example, for theHe Li atomic systems particles 1 and 2 corresponding to He atoms are identical and the Faddeev components φ ( x , y , z ) and φ ( x , y , z ) transform into each other under an appropriate rotation of the coordinate space.Therefore, it is sufficient to consider only two independent Faddeev components. In the case of three identicalbosons all the Faddeev components take identical functional form, which makes it possible to reduce thesystem of three equations (12) to one equation.Using the fact that the both dimers – He and HeLi – have a unique bound state, the asymptotic boundarycondition for a bound state as ρ = p x + y → ∞ and/or y → ∞ reads as follows (see [27,29]) φ ( x , y , z ) = ψ d ( x ) exp ( i √ E − ε d y ) a ( z ) + exp ( i √ E ρ ) √ ρ A ( y / x , z ) , (13)where ε d stands for the corresponding dimer energy while ψ d ( x ) denotes the dimer wave function. The coef-ficients a and A ( y / x , z ) describe contributions into φ ( x , y , z ) from (2+1) and (1+1+1) channels respectively.The last term can be neglected for the states below the three-body threshold. For bound state calculationsDirichlet or Neumann boundary conditions can also be employed. The Li-He interaction is described by the KTTY potential [30], theoretically derived by Kleinekath¨ofer, Tang,Toennies and Yiu with more accurate coefficients taken from [31,32]. Calculated values of the binding energyfor He Li is 1.512 mK and for He Li is 5.622 mK. Such small values of binding energy give indication onpossible existence of Efimov states in the corresponding He Li triatomic systems.In our calculations we use two different model potentials for He-He interaction - TTY [33] and Przy-bytek [34]. The purely theoretical TTY potential derived by Tang, Toennies and Yiu [33] is based on theperturbation theory and is described by a relatively simple analytical expression. The recent Przybytek [34]potential includes relativistic and quantum electrodynamics contributions as well as some accuracy improve-ments. Each of these potentials supports a single weakly bound state of the dimer. Calculated values of thebinding energy ε d for the corresponding dimers, the inverse wavenumber κ − = ( ¯ h / µε d ) / ( µ is a reducedmass) and the atom-atom scattering length a are presented in Table 1. Atomic masses for different isotopesare taken from [35]. Table 1
Absolute values of dimer energies | ε d | (in mK), the inverse wavenumber κ − ( in ˚A) and the scattering length a (in ˚A)for He-He and He − alkali-atoms calculated for the potentials used.Dimer | ε d | (mK) κ − ( ˚A) a ( ˚A) Dimer | ε d | (mK) κ − ( ˚A) a ( ˚A) He He a He Na 28 .
97 15.7 23.37 He He b He K 11 .
20 24.4 33.32 He Li 1.515 81.63 89.42 He Rb 10 .
27 24.9 34.02 He Li 5.622 41.14 48.84 He Cs 4 .
945 35.5 45.32 a Using the He-He potential from Ref. [33] b Using the He-He potential from Ref. [34]
From Table 1 we can see that the inverse wave number is a good approximation for the scattering length,which indicates that the zero-range potential model (1) is applicable. The calculated binding energy of thehelium dimer with Przybytek potential [34] is very close to its recent experimental value of − . ± . ≈ − . ± .
15 mK [36] while the energy obtained with TTY potential [33] is closer to the previousexperimental estimation of − . + . − . mK ≈ − + − neV [37]. The choice of the He-He potential is espe-cially important for He Li, as switching between the two model potentials swaps the order of the two-bodythresholds in the system. In the case of TTY potential the lowest two-body threshold corresponds to the LiHesystem, while for the Przybytek potential it is the He dimer which is bound stronger.All He-alkali-atom dimers are weakly bound, but the binding energies of HeLi and HeCs systems are of thesame order as the binding energy of the Helium dimer. It suggests that in the corresponding He -alkali-atomtriatomic systems Efimov states might exist. Indeed, in calculations [24,25,26,29] an excited state locatedvery close to the HeLi threshold has been found.To calculate the binding energy of He Li and He Li trimers, we employed the equations (12), andthe bound-state asymptotic boundary condition (13). The details of the numerical procedure are describedin [38,39,40]. The three-body interaction is expected to be small as in the case of helium trimer [41] andwe do not take it into account. Convergence tables for bound states of the He Li trimer calculated with thePrzybytek potential [34] are shown in Table 2. The bound state energies (with respect to the three-body break-up threshold) are presented for different numbers of grid points. The number of grid points in coordinates x and y are set equal, and the number of grid points in angular coordinate z is varied independently. As it is seenfrom Table 2, the excited state is much less sensitive to the angular grid. Similar behavior had been observedfor the helium trimer [38,39].Our results for He Li and He Li trimers binding energies as well as the results obtained by otherauthors are summarized in Table 3. The results show that the both potential models support two bound statesin the both trimers. The energy of the excited state is very close to the energy of the lowest two-body threshold.Different He-He potentials give 0 . ∼ . Li trimers, although the energy of excitedstate of He Li is practically unchanged (difference is ∼ . Lisystem the lowest threshold is different for different potentials: for TTY it corresponds to the energy of HeLi
Table 2
Convergence for the He Li bound states energies with respect to the number of grid points for the Przybytek poten-tial [34]. Ground state energy (mK) N x = N y N z = N z = N z = N z = N z =
520 -78.4124 -80.0524 -80.4523 -80.6422 -80.662230 -78.3750 -79.9701 -80.3877 -80.5630 -80.593740 -78.3849 -79.9695 -80.3853 -80.5690 -80.597750 -78.3877 -79.9727 -80.3869 -80.5712 -80.5990Excited state energy (mK)20 -5.6457 -5.6522 -5.6538 -5.6545 -5.654630 -5.6447 -5.6510 -5.6527 -5.6534 -5.653540 -5.6448 -5.6511 -5.6527 -5.6534 -5.653650 -5.6448 -5.6511 -5.6527 -5.6535 -5.6536
Table 3
Bound-state energies for the He Li systems and helium atom - HeLi dimer scattering length a . The energies aregiven in mK and are relative to the three-body dissociation threshold. The scattering length is given in ˚A. The present results arecompared to results given in references.E (mK)/ a ( ˚A) present present [25] [26] [43] [43] [44] [45] [46] [47]He-He potential TTY Przybytek LM2M2 LM2M2 Jeziorska Jeziorska LM2M2 LM2M2 TTY HFDBHe-Li potental KTTY KTTY KTTY KTTY KTTY KTTY Cvetko Cvetko KTTY KTTY | E Li He | | E ∗ Li He | a
683 553 | E Li He | | E ∗ Li He | a
191 144 ∗∗ Li atom - He dimer scattering length bound state, while for Przybytek it is the bound state energy of the He dimer. So, for different potentials theabsolute value of the excited state energy changes slightly, but the relative energy with respect to the two-bodythreshold remains practically the same.Results of other authors in Table 3 are based on solving the Shr¨odinger equation. The adiabatic hyper-spherical approach has been employed in [24,25,26,43,44] and variational calculations has been performedin [45,46,47]. The fourth column of Table 3 contains the results obtained by Suno in [25] using the adiabatichyperspherical method. For the He-Li interaction he has used KTTY potential [30] as in our calculations,but for the He-He interaction the LM2M2 potential [48] has been used. However, LM2M2 and TTY poten-tials support He with the energies ε d = − .
31 mK and ε d = − .
32 mK, correspondingly [39]. So goodagreement between our results and results from [25] are not surprising (see columns 1 and 3 in Table 3).The fifth column of Table 3 contains the results obtained by Wu et al. [26] using the mapping methodwithin the adiabatic hyperspherical framework [42]. The next two columns are the results of calculationsby H. Suno, E. Hiyama and M. Kamimura [43] using the Gaussian expansion method and the adiabatic hy-perspherical representation respectively, although with different He-He potentals. They employed the He-Hepotential suggested by Jeziorska et al. [49], which gives the helium dimer biding energy − .
74 mK whichis lower than for Przybytek potential. The two methods give different results, but authors in [43] mentionedthat the adiabatic hyperspherical representation was less accurate. The next column is the results of calcu-lations by Suno and Esry [24,44] by the adiabatic hyperspherical method. They also employed the He-Hepotential from [49], but for Li-He interaction Cvetko potential from [50] has been used. The potential pro-posed by Cvetko et al. [50] gives the HeLi smaller binding energy than the KTTY potential, namely − . He Li and − .
31 mK in case of He Li dimer. The ninth column contains the results obtained byBaccarelli et al. [45] with the same potential as in [44], but using a different computation method - variationalcalculations in terms of distributed Gaussian functions. The last two columns contain the results of MonteCarlo calculations by Di Paola et al. [46] and Stipanovi´c et al. [47] using TTY [33] and HFDB [51] as He-Heinteractions.
We should also mention the first results obtained by Yuan and Lin [52] using the adiabatic hypersphericalmethod which gives an upper bound to the ground state − . He Li and − . He Li andthe prediction of the bound state energies made by Delfino et al. [53] using the scaling ideas and zero-rangemodel calculations. The preliminary Faddeev calculation using bipolar partial-wave expansion for searchingEfimov states in He Li system have been performed in [29]. In these papers, however, the contribution ofhigher partial waves was underestimated because of the computational restrictions.As it has been demonstrated in [29], the excited state of He Li has a Efimov-type behavior similar tohelium trimer system [54]. To check for the Efimov-like state the original Li-He potential has been multipliedby a factor λ . An increase of the coupling constant λ makes the potential more attractive and Efimov levelsbecome weaker and disappear with further increase of λ . Indeed this situation is observed for the excited stateenergy of He Li in contrast to the ground state energy whose absolute value increases continuously withincreasing attraction.The results for the He-atom – HeLi-dimer scattering length are presented in the last line of the Table 3for each Li isotopes. The helium atom – helium-alkali-atom collisions at ultralow energies are studied inRef. [55] by Suno and Esry using the adiabatic hyperspherical representation. In particular, they calculatedthe total cross section also for HeLi + He → HeLi + He elastic scattering. Our estimation of the crosssection σ = π a at the threshold is 5 . × − cm using TTY potential [33] and 3 . × − cm usingPrzbytek potential [34]. These values agree with the value ≈ × − cm obtained in Ref. [55] using SAPTpotential [49] at the energy 10 − mK above the threshold. We have used direct solution of the Faddeev equations for the bound-state and scattering problems with zerototal angular momentum. The numerical algorithm is based on spline expansion of the Faddeev componentscombined with the tensor trick preconditioning and the Arnoldi algorithm for eigenanalysis. Calculationsof the He Li and He Li ground and excited states show that the method is very efficient and allows oneto obtain stable convergent results. Apparently, it performs better than the previously exploited method ofthe bipolar partial-wave expansion. Our results for He Li and He Li trimers binding energies show thatdifferent potential models support two bound states in both trimers. The energy of the excited state is veryclose to the energy of the lowest two-body threshold. In case of the He Li system the lowest thresholdis different for different potentials but the relative energy with respect to the lowest two-body threshold ispractically the same.
Acknowledgements
One of the author (EAK) would like to thank T. Frederico, A. Kievsky and P. Stipanovi´c for stimulatingdiscussions and also W. Sandhas and A.K. Motovilov for their constant interest to this work.
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