Cold ion-atom chemistry driven by spontaneous radiative relaxation: a case study for the formation of the YbCa^{+} molecular ion
CCold ion-atom chemistry driven by spontaneousradiative relaxation: a case study for the formationof the
YbCa + molecular ion B. Zygelman , Zelimir Lucic and Eric R. Hudson Department of Physics and Astronomy, University of Nevada, Las Vegas, Las VegasNV 89154, USA Department of Physics and Astronomy, University of California, Los Angeles,California 90095, USAE-mail: [email protected]
Abstract.
Using both quantum and semi-classical methods, we calculate the ratesfor radiative association and charge transfer in cold collisions of Yb + with Ca. Wedemonstrate the fidelity of the local optical potential method in predictions for thetotal radiative relaxation rates. We find a large variation in the isotope dependenceof the cross sections at ultra-cold gas temperatures. However, at cold temperatures,1mK < T < . × − cm s − . It is is about five orders ofmagnitude smaller than the chemical reaction rate measured in [Rellergert et al., PRL
07, 243201 (2011)].PACS numbers: 34.70.+e,34.50.Ez,34.50.Lf,33.15.-e a r X i v : . [ phy s i c s . a t m - c l u s ] A ug old chemsitry
1. Introduction
Advances in the cooling, trapping, and manipulation of ultra-cold atoms have openednew vistas in our understanding of quantum degenerate matter. In recent years,laboratory techniques have advanced so that it is now feasible to cool ions andexplore their interactions with neutral matter in the sub- to milli-Kelvin temperaturerange. The development of hybrid, ion-atom traps (Schmid et al. 2010, Zipkeset al. 2010, Ratschbacher et al. 2012, Rellergert et al. 2011, Hall et al. 2011, Hallet al. 2013) has allowed researchers to explore competing pathways for reaction betweencold atoms and ions, including non-radiative and radiative charge transfer as wellas radiative association, in which ions and atoms combine to form a molecule atcold temperatures. Among possible applications for cold ion-atom chemistry arequantum-limited control of chemical reactions and buffer gas cooling of single ionclocks (Ratschbacher et al. 2012). The aforementioned reactions are also important inastrophysical applications (Rellergert et al. 2011, Stancil & Zygelman 1996, Zygelmanet al. 1998). Laboratory efforts in measuring accurate rate coefficients of the latterenhances the atomic data base employed in astrophysical models.In a recent laboratory study (Rellergert et al. 2011), Rellergert et al. used ahybrid trap to investigate the interactions between cold Yb + ions with Ca atoms.They observed a large, on the order of 2 × − cm s − , chemical reaction ratecoefficient and, based on a preliminary theoretical estimate, suggested that radiativecharge transfer was the dominant process behind this rate. However, their theoreticalestimate also predicted that nearly half of the chemical reactions should produceYbCa + molecules through radiative association, which disagreed with the experimentalobservation that the fraction of reactions leading to molecule formation was ≤ . + + Ca → Yb + Ca + + ¯h ω Yb + + Ca → YbCa + + ¯h ω. (1)Our results are several orders of magnitude smaller than that given in (Rellergertet al. 2011) and raises doubts on the suggestion, given in that paper, concerning the roleof radiative relaxation. Our calculations employed the local optical potential method,essentially a semi-classical theory, whose utility at ultra-low collision energies has beenlargely(Zhou et al. 2011) un-tested. In addition, we used molecular data that wasgleaned from the illustrations given in (Rellergert et al. 2011). At these energies small old chemsitry
2. Theory
The cross section for the radiative association processYb + + Ca → YbCa + + ¯h ω (2)where ¯ hω is the energy of the emitted photon is given by the expression(Zygelmanet al. 1998) σ RA = (cid:88) J (cid:88) n π ω nJ c k (cid:104) ( J + 1) M J +1 ,J ( k, n ) + M J − ,J ( k, n ) (cid:105) M J,J (cid:48) ( k, n ) = (cid:90) ∞ dR f J ( kR ) D ( R ) φ J (cid:48) n ( R ) (3)where D ( R ) is the transition dipole moment between the X Σ + and A Σ + states ofthe YbCa + molecular ion. φ nJ is a rho-vibrational eigenstate of the X Σ + groundstate, with energy eigenvalue (cid:15) nJ , and is characterized by the angular and vibrationalmomentum quantum numbers J , n respectively. f J ( kR ) is the wavefunction thatsatisfies the radial Schrodinger equation f (cid:48)(cid:48) J ( kR ) − J ( J + 1) R f J ( kR ) + 2 µV A ( R ) f J ( kR ) + k f J ( kR ) = 0 (4)where V A ( R ) is the Born-Oppenheimer (BO) energy of the excited A Σ + state, µ is thereduced mass of the collision system and k is the wavenumber for the incident collision old chemsitry f J ( kR ) → (cid:115) µπk sin( kR − J π δ J ) , (5)where δ J is a phaseshift, as R → ∞ . The energy of the emitted photon is given by¯ hω nJ = ¯ hk µ + V A ( ∞ ) − (cid:15) nJ − V X ( ∞ ) . (6) (cid:209) Ω A (cid:83) (cid:43) X (cid:83) (cid:43) (cid:45) (cid:45) R (cid:72) a (cid:76) V (cid:72) a . u . (cid:76) Figure 1. (Color online) Illustration of the BO molecular potential curves (solid thicklines) participating in the radiative association process. In the A Σ + entrance channelthe wave function is shown by the light undulating line. The oscillations are due tothe strong polarization force in the entrance channel leading to a potential minimumat R ≈ . Association is precipitated by the emission of a photon of energy ¯ hω near the classical turning point. The final bound rho-vibrational state, in the X Σ + channel, is shown by the thin line. The cross section for the radiative charge transfer process,
Y b + + Ca → Y b + Ca + + ¯ hω (7)is given by(Zygelman et al. 1989) σ CT = (cid:90) ω max dω dσdω ,dσdω = (cid:88) J π ω c k (cid:104) J M J,J − ( k, k (cid:48) ) + ( J + 1) M J,J +1 ( k, k (cid:48) ) (cid:105) (8)where M J,J (cid:48) ( k, k (cid:48) ) = (cid:90) ∞ dR f J ( kR ) D ( R ) f J (cid:48) ( k (cid:48) R ) . (9) old chemsitry f J ( kR ) is a solution to (4) and f J (cid:48) ( k (cid:48) R ) obeys the corresponding equation for the, X Σ + , exit channel with wavenumber and partial wave k (cid:48) , J (cid:48) respectively. The radialwavefunctions are normalized as in (5) and¯ hω = ¯ hk µ − ¯ hk (cid:48) µ + ∆ E ∆ E ≡ V A ( ∞ ) − V X ( ∞ ) . (10)According to (10) the maximum angular frequency ω max is given by¯ hω max = ¯ hk µ + ∆ E. (11)The sums given by (8) can be evaluated as in (Stancil & Zygelman 1996), but here weuse a simplified expression, derived in the Appendix, in which σ CT is replaced by itsupper bound, i.e. σ CT < ˜ σ CT = 83 π ω max c k (cid:88) J (2 J + 1) (cid:90) ∞ dRf J ( kR ) D ( R ) . (12) An alternative approach for the calculation of the total radiative loss cross section isgiven by the local optical potential method(Zygelman & Dalgarno 1988). In it, thecollision system in the incoming A Σ + state experiences, in addition to the BO energy V A ( R ), a complex absorptive potential that has the form V opt = iA ( r )2 A ( R ) ≡ c D ( R )( V A ( R ) − V X ( R )) (13)where A ( R ) is an R -dependent Einstein-A coefficient that is illustrated in figure 2. Thecross section for radiative quenching is given by σ = πk (cid:88) J (2 J + 1) (cid:16) − exp( − η J ) (cid:17) (14)where η J is the imaginary part of the J’th partial wave phase shift δ J for the radialwave f J ( kR ) that satisfies f (cid:48)(cid:48) J ( kR ) − J ( J + 1) R f J ( kR ) + 2 µ ( V A ( R ) + V opt ( R )) f J ( kR ) + k f J ( kR ) = 0 . (15)
3. Ultra-cold limit
In the limit of ultra-cold temperatures in which only the s-wave of the entrance channelparticipates, the total radiative association cross section takes the form(Zygelmanet al. 2001) σ = (cid:88) n µ π ω n c k (cid:12)(cid:12)(cid:12) (cid:90) R dR φ ( R ) D ( R ) φ nJ =1 ( R ) (cid:12)(cid:12)(cid:12) (16) old chemsitry R (cid:72) a (cid:76) A (cid:72) un it s o f s (cid:45) (cid:76) Figure 2. (Color online) Einstein A coefficient as a function of internuclear distance where φ ( R ) is the s-wave solution to (4) subject to the boundary condition(Zygelmanet al. 2001) d Φ( R ) /dR | R = 1at some, sufficiently large radius R and φ nJ =1 ( R ) are J=1 rho-vibrational states of the X Σ + potential. Because the overlap integral in (16) is independent of the incomingwavenumber k , (16) predicts that the association cross section, in the ultra-cold regime,scales as the inverse of the collision velocity and, therefore, the rate tends to a constant.In calculating φ ( R ) one typically matches the numerical solution for f J ( kR ) with theasymptotic form given by expression (5). Because of the polarization potential C /R ,one must typically integrate far into the asymptotic region to achieve convergence.Exact solutions for the C /R potential are given by radial Mathieu functions andbetter convergence can be achieved by employing the latter in the evaluations for thephaseshifts e.g (Spector 1964, Holzwarth 1973).
4. Results
Figure (1) illustrates the mechanism for radiative association for the Yb + ion and Caatom that approach in the A Σ + electronic BO state. The BO energies where takenfrom the data of the ab-initio calculations reported in (Rellergert et al. 2011). At largeinternuclear distances this potential has the form V A ( R ) → − C A R C A = 78 . . (17)In the incident A Σ + channel the system can relax via the emission of a photon, andin the case of association, the final state is a bound rho-vibrational level of the X Σ + channel. In radiative charge transfer the collision partners can exit in that channel, asa re-arranged Yb - Ca + pair. In the exit channel, V X ( R ) → − C X R + ∆ E C X = 71 . E = − . old chemsitry R → ∞ . In calculating the radiative association cross sections given by (3) we needto itemize all bound states supported by the X Σ + channel. The total number of boundstates can be approximated using the JWKB expression n = (cid:88) L Floor (cid:104)(cid:90) ∞ R c dR (cid:115) − µ ( V X ( R ) − ∆ E ) + ( L + 1 / R − / (cid:105) ≈ ,
803 (19)where R C is a classical turning point and Floor[ x ] is the integer lower bound of x .Because of the large reduced mass µ , the number of bound states contributing is muchlarger than that for association of lighter species in which typically several hundred rho-vibrational are supported e.g. see (Zygelman et al. 1998). However, at cold temperaturesthe centrifugal repulsion in the entrance channel limits the number of partial waves thatparticipate and so limits, because of the J ± J up to the value ≈
15 contribute to the association rate. In figure (2)we present the results of our calculations for a collision temperature of 1 mK. In thatfigure the circles represent the partial wave association cross sections obtained usingthe FGR expression (8), the symbol X in that figure represents the upper limit fortotal radiative relaxation, which is obtained by adding the association cross sections(8) with those given by expression (12). The square icons represent the cross sectionspredicted by expression (14). It is evident, from this figure, that for J <
10 the opticalpotential method provides an excellent approximation for the total cross sections, andfor
J > J max (Zygelman & Hunt 2012), J max = (cid:113) µk C A = (cid:113) µ k B T C A ≈
12 (20)the optical potential method is somewhat less reliable, though still gives reasonableorder of magnitude estimates. J max is the critical angular momentum for which thecollision system, approaching in the incident channel at a given energy, has sufficientcollision energy (here given by 3 / k B T , where k B is the Boltzmann constant and T isthe temperature in Kelvin) to overcome the centrifugal potential barrier(Zygelman &Hunt 2012). For larger J tunneling resonances can access the inner region where thetransition dipole moment is non-negligible and induce a radiative transition. In (1) wetabulate the various cross sections at several representative collision temperatures. Inthe second column we itemize the association cross section obtained using the FGRmethod described above. For the radiative charge transfer cross sections, itemized inthe third column, we use expression (12) . Thus the upper bound for the total radiativerelaxation cross sections are given in column 4. The last column gives the resultsobtained using the local optical potential method. The table shows that, over thetemperature range considered, the local optical potential method predicts cross sectionsthat are less than the upper bound itemized in column 4. Secondly, the differencesbetween the predictions of the two theories are small. The optical potential crosssection differs by less than 4 % from the upper limit values over the entire temperaturerange, including the ultra-cold region. We also note that the optical potential methodpredicts cross sections that are larger than the radiative association cross sections which old chemsitry X X X X X X X X X X X X X X X J Σ J (cid:72) a . u . (cid:76) Figure 3. (Color online) Plot of various cross sections as a function of incomingpartial wave J . Circles represent data for the radiative association cross sections, X ’s represent the upper bound for total (association + radiative) charge transfer, andsquares represent the data for the total radiative relaxation obtained using the opticalpotential method. Table 1.
Radiative relaxation (RR) cross sections, in units of a , as a function ofgas temperature.T(K) Association RCT Total RR Optical Potential1 pK 2 . × . × . × . × . × . × . × µ K 29.23 2.049 31.28 31.1110 µ K 6.294 0.441 6.735 6.698100 µ K 1.909 0.134 2.043 2.03151 mK 0.626 0.0439 0.670 0.66610 mK 0.201 0.0132 0.214 0.214 underscores an observation cited made in ((Zygelman et al. 1989)), that the opticalpotential method provides a reliable upper bound for the total (RR) cross section. Infigure (4) we plot the total radiative relaxation cross section, obtained using the opticalpotential method, for the gas temperature range 1mK < T < K . Though the opticalpotential method provides a good approximation for the total radiative relaxation rate,calculation of the photon emission spectrum requires the use of the FGR method. Infigure (5) we illustrate the association cross sections σ nJ , at T = 1mK, for the individualrho-vibrational levels as function of the frequency of the emitted photon. The structurein the emission pattern, which shows regions of suppressed and enhanced emission is aresult of the oscillations in the incoming wave illustrated in figure (1).In the limit as T → a ≡ − k tan δ ( k ) = 1 . × − i . k → . (21) old chemsitry Energy (cid:72) units of Kelvin (cid:76) C r o sss ec ti on (cid:72) a . u . (cid:76) Figure 4. (Color online) Total radiative relaxation cross section as a function ofcollision energy expressed as E = k B T, where T is the temperature in Kelvin.
Therefore, the total RR cross section, according to the optical potential method, hasthe limiting value σ = 4 πk | Im [ a ] | = 4 πk . . (22)Defining the rate coefficient k RR ≡ (cid:104) vσ (cid:105) T → k RR ≈ . × − cm s − and is about three orders of magnitude largerthan the corresponding rate in the temperature range 1mK < T <
50 100 150 200 2500.51.01.52.0
Frequency (cid:72) units of THz (cid:76) Σ n J (cid:180) (cid:72) a . u . (cid:76) Figure 5. (Color online) Emission spectrum for the radiative association process ata gas temperature of 1mK. plot the effective rates k ≡ (cid:104) v (cid:105) σ for different isotopes of the Y b + ion. At temperaturesT > µ K the three rates, corresponding to the isotopes labeled in that figure, merge toa common value of about 1 . × − cm s − . This feature can be attributed to Langevinbehavior (Vogt & Wannier 1954) which predicts that at low, but high enough that manypartial waves contribute, temperatures ion-atom cross sections scale as the inverse ofthe collision velocity and therefore the rate coefficient tends to a constant. The value old chemsitry /v behaviorin the cross sections is also operative, e.g. see (21), but for a different reason. Whereasin the Langevin regime the cross sections are governed largely by the C coefficient, theultra-cold s-wave phaseshift is also sensitive, as required by Wigner-threshold theory, toshort-range parameters. So the presence of a real, or virtual, bound state near thresholdcan strongly influence that cross section. As a consequence, radiative quenching rateswhich are nearly constant in both the Langevin and ultra-cold regions, can suffer rapidvariations in the temperature range that adjoins the two territories. This behavior isillustrated in figure (6) by the rate for the Y b isotope. For this isotope, a boundstate near the threshold leads to significant enhancement in the s-wave cross sectionat ultra-cold temperatures, the corresponding rate differs significantly from that in theLangevin region. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) T (cid:72) K (cid:76) Yb Yb Yb k (cid:72) un it s o f c m s (cid:45) (cid:76) Figure 6.
Plot of the effective rate k ≡ (cid:104) v (cid:105) σ , where σ is the total radiative chargetransfer cross section, for various isotopes of Ytterbium ions in collisions with cold andultracold Ca atoms. The horizontal axis denotes the effective temperature T definedso that the center of mass relative velocity, v = √ B T where k B is the Boltzmannconstant.
5. Summary and discussion
We have presented a computational study of the collision induced radiative processes (1)at gas temperatures that range from the cold to ultra-cold regimes. We found that atcold temperatures the total effective rates for theses radiative processes is no larger thanabout 10 − cm s − . We evaluated the fidelity of the local optical potential method(Westet al. 1982, Zygelman & Dalgarno 1988) in its ability to predict radiative quenchingrates, and found that it provides very accurate estimates for the latter even in the ultra-cold regime. We validated Langevin behavior (which predicts nearly constant ratesas a function of temperature) at higher temperatures but found dramatic departures,and a strong isotope dependence, in the transition from the cold to ultra-cold regimes. old chemsitry + + Ca → YbCa + (A Σ + ) + ¯h ω. (24)where association proceeds into the weakly bound ( A Σ + ) molecular state in whichthe collision partners initially approach. Because process (24) is driven by a dipolemoment that, at large R , is proportional to the internuclear distance, one mightanticipate a significant rate for it. However, we estimate that this rate is negligibleat the temperatures operative in this experiment. Therefore, we conclude that bothadditional experimental and theoretical studies are necessary in order to understandand reveal cold chemistry of the Ca + Yb system. On the experimental side, revisitingthis reaction with new methods recently developed to better probe both the productsof the reactions (Schowalter et al. 2012) and the role, if any, of excited electronicstates (Sullivan et al. 2012) may help elucidate the relevant pathways. While, on thetheoretical front, improved ab initio molecular potentials might help better understandthe potential role of non-adiabatic effects in this reaction (Zygelman et al. 1989).
Acknowledgments
We thank Prof. Svetlana Kotochigova for use of the ab-initio molecular data.
Appendix A. Bound on RCT cross sections
According to (8) - (10) the frequency of the emitted photon, during an RCT transition,is¯ hω = ¯ hk µ − ¯ hk (cid:48) µ + V A ( ∞ ) − V X ( ∞ ) = ¯ hk µ − E (cid:48) + V A ( ∞ ) − V X ( ∞ ) . (A.1)Now d (¯ hω ) = − dE (cid:48) and ¯ hω max = ¯ hk µ − E (cid:48) + V A ( ∞ ) − V X ( ∞ ), which corresponds to E (cid:48) = 0, and ¯ hω = 0 for E (cid:48) max = ¯ hk µ + V A ( ∞ ) − V X ( ∞ ). Therefore (8) can be written as σ = 83 π c k (cid:90) E (cid:48) max dE (cid:48) ω ( E (cid:48) ) (cid:104) J M J,J − ( k, E (cid:48) ) + ( J + 1) M J,J +1 ( k, E (cid:48) ) (cid:105) . (A.2)We have the inequality σ < π ω ( E (cid:48) ) max c k (cid:90) ∞ dE (cid:48) (cid:104) J M J,J − ( k, E (cid:48) ) + ( J + 1) M J,J +1 ( k, E (cid:48) ) (cid:105) . (A.3)Consider the integral (cid:90) ∞ dE (cid:48) J M J,J − ( k, E (cid:48) ) = (cid:90) ∞ dE (cid:48) (cid:90) ∞ dR f J ( kR ) D ( R ) f J − ( k (cid:48) R ) (cid:90) ∞ dR (cid:48) f J ( kR (cid:48) ) D ( R (cid:48) ) f J − ( k (cid:48) R (cid:48) ) . (A.4) old chemsitry (cid:90) ∞ dE (cid:48) J M J,J − ( k, E (cid:48) ) < (cid:88) E (cid:48) J M J,J − ( k, E (cid:48) ) = J (cid:90) ∞ dRf J ( kR ) D ( R ) . (A.5)where the sum (cid:80) E (cid:48) includes all, bound and continuum states of the exit channel, andthe second inequality follows from closure properties for the final states for a given valueof J . Therefore we obtain the inequality σ < π ω ( E (cid:48) ) max c k (cid:88) J (2 J + 1) (cid:90) ∞ dRf J ( kR ) D ( R ) . (A.6) References
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