We have calculated the s-wave scattering length of all the even isotopes of
calcium (Ca) and barium (Ba), in order to investigate the prospect of
Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular
potential based on detailed spectroscopic data. Our calculations show that Ca
does not provide other isotopes alternative to the recently Bose condensed 40Ca
that suffers strong losses because of a very large scattering length. For Ba we
show by using a model potential that the even isotopes cover a broad range of
scattering lengths, opening the possibility of BEC for at least one of the
isotopes.Comment: 4 page

Dammalapati, U.

Knoop, S.

Willmann, L.

English

arXiv

We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) in order to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular potential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopes alternative to the recently Bose condensed (40)Ca that suffers strong losses because of a very large scattering length. For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths, opening the possibility of BEC for at least one of the isotopes.</p

University of Groningen

University of GroningenScattering lengths of calcium and barium isotopesDammalapati, U.; Willmann, L.; Knoop, S.Published in:Physical Review ADOI:10.1103/PhysRevA.84.054703IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2011Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dammalapati, U., Willmann, L., & Knoop, S. (2011). Scattering lengths of calcium and barium isotopes.Physical Review A, 84(5), Article 054703. https://doi.org/10.1103/PhysRevA.84.054703CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 25-12-2025PHYSICAL REVIEW A 84, 054703 (2011)Scattering lengths of calcium and barium isotopesU. Dammalapati and L. Willmann*Kernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The NetherlandsS. Knoop†LaserLaB Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands(Received 25 October 2011; published 21 November 2011)We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) inorder to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecularpotential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopesalternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length.For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths,opening the possibility of BEC for at least one of the isotopes.DOI: 10.1103/PhysRevA.84.054703 PACS number(s): 34.20.−b, 34.50.Cx, 67.85.−dI. INTRODUCTIONKnowledge of the s-wave scattering length plays a crucialrole in the achievement of Bose-Einstein condensation (BEC)in ultracold atomic gases [1]. A positive sign of the scatteringlength indicates that a Bose-Einstein condensate is stable,whereas a negative sign indicates an unstable or collapsingcondensate. Also, its magnitude is crucial in the formation ofBEC, as it determines the elastic and inelastic collision rates.For efficient evaporative cooling, a large scattering length isrequired; however, for a too-large scattering length, three-body recombination loss limits the formation and collisionalstability of a BEC. This gives rise to a range of about 50–200 a0 (a0 = 0.5292 nm) that can be considered favorable.Alkaline-earth-metal elements and the alike ytterbium (Yb)system have been the center of attraction in ultracold atomresearch because of their unique atomic structure and theavailability of different stable isotopes with zero nuclear spin(bosons) and with large nuclear spin (fermions). They arebeing used in frequency metrology [2–4] and are proposed forquantum information processing [5]. Recently, BEC has beenachieved in 40Ca [6], 84Sr [7,8], 86Sr [9], and 88Sr [10]. Also,BEC of different Yb isotopes has been reported: 174Yb [11],170Yb [12], 176Yb [13], and 168Yb [14], of which the last onehas an abundance of only 0.13%, highlighting the possibilityof using rare isotopes to achieve BEC.In general, isotopes of an element have different scatteringlengths. Therefore, the ability of BEC formation dependscrucially on the chosen isotope. For instance, the scatteringlength of 84Sr is 123 a0 [15], which made it an ideal candidateto achieve BEC despite its low abundance of only 0.6% [16],as compared to the more abundant 86Sr (10%, 800 a0 [15])and 88Sr (83%, −2 a0 [15]), either suffering strong losses [9]or requiring sympathetic cooling with another isotope [10],respectively. The 40Ca isotope has a large scattering length ofabout 440 a0 [6], limiting the size and stability of BEC.Motivated by the success of BEC in Sr and Yb even isotopesand 40Ca, we carried out calculations to obtain the scattering*l.willmann@rug.nl†s.knoop@vu.nllengths for all the even (bosonic) isotopes of Ca and Ba,for which mass and abundance are summarized in Table I.Laser cooling and trapping of all stable Ca isotopes havebeen reported [17,18]. The demonstration of magneto-opticaltrapping of Ba [19] has opened its use for ultracold collisionstudies, photoassociation spectroscopy, and Bose-Einsteincondensation. Recently, an optical clock based on ultracoldBa has been proposed [20].II. THEORYThe scattering properties of ground-state atoms are ob-tained from the underlying two-body ground-state potentials.For alkaline-earth-metal atoms, there is only one molecularground-state potential, namely the singlet X1+g potential.Furthermore, the lack of nuclear spin and, therefore, hyperfinestructure for the even (bosonic) isotopes highly reduces thenumber of collision channels compared with the alkali metalsystems.To calculate the s-wave scattering length, we solve the 1Dradial Schrödinger equation with zero angular momentum andvanishing kinetic energy,ψ ′′(R) + 2μh̄2[E − V (R)]ψ(R) = 0, (1)where μ is the reduced mass, R the internuclear distance, V (R)the Born-Oppenheimer potential (here the X1+g potential),and E the kinetic energy (below μK). The asymptotic formof the wavefunction is ψ(R) ∝ sin[k(R − a)], where k =√2μE/h̄2, and a is the scattering length. We fit the asymptoticform to the solution of Eq. (1) for large R in order to obtain a.Within the Born-Oppenheimer approximation, the potentialsare identical for all the isotopes of a particular atomic system.Therefore, for a given potential V (R), one only needs to adjustμ to obtain a for all the isotopes.To obtain reliable scattering lengths, accurate potentialsare needed. With the exception of systems with only a fewelectrons, like metastable helium [22], ab initio potentials arein general not accurate enough and constraints from experi-mental data are required. For Ca2, an analytic representation ofthe X1+g potential based on an extensive set of experimentaldata from the Tiemann group is available [23]. For Ba2,054703-11050-2947/2011/84(5)/054703(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)TABLE I. Mass and abundance of all stable even isotopes of Caand Ba [21].CalciumIsotope 40 42 44 46 48Mass (amu) 39.963 41.959 43.955 45.954 47.953Abundance (%) 96.941 0.647 2.086 0.004 0.187BariumIsotope 130 132 134 136 138Mass (amu) 129.906 131.905 133.904 135.905 137.905Abundance (%) 0.11 0.10 2.42 7.85 71.70such an accurate potential is not available. Here we rely onthe analytical representation according to the Tang-Toenniespotential model [24], which has been shown to be able toreproduce accurately the X1+g potentials of Ca2 [25] andSr2 [24]. The used Ca2 and Ba2 potentials are shown inFig. 1.III. RESULTSA. CalciumInformation on the Ca2 X1+g potential has been gatheredby several spectroscopic methods, including photoassociationspectroscopy [26–28], Fourier-transform spectroscopy [29],and filtered laser excitation technique [23]. Based on thesedetailed spectroscopic data, an interval of the scattering lengthfor 40Ca is determined to be 340–700 a0 [28]. In the 40Ca BECexperiment, a scattering length of a ≈ 440 a0 was estimatedfrom a measurement of the chemical potential [6], withoutquoting an uncertainty. Therefore, we take the scattering lengthrange of 340–700 a0 as the starting point of our calculations.We have taken the analytical representation and parametervalues of the Ca2 X1+g potential from Ref. [23] (see Fig. 1).We allow the C6 long-range coefficient to vary within afactor 0.991 to 1.003 from its reported value [23], in order toreproduce the scattering length interval for 40Ca. We then takethe mass as a variable parameter and calculate the scattering5 10 15 20 2540302010010R (unitsof a )0VhTHzCa2 Ba21S+1SFIG. 1. (Color online) The X1+g potentials of Ca2 and Ba2 as afunction of internuclear distance. The solid blue curve is the potentialof Ca2 taken from Ref. [23]. The dashed red curve is the Tang-Toennies potential of Ba2 taken from Ref. [24].40 42 44 46 4840020002004006002Μ amuaunitsofa 0FIG. 2. (Color online) The mass dependence of the scatteringlength for Ca, using the X1+g potential of Ref. [23], showingcalculations with C6 coefficients that give rise to 340 a0 (red closedcircles) and 700 a0 (blue closed squares) for 40Ca. The plotted massis twice the reduced mass and the dashed vertical lines indicate themass of the even isotopes of Ca.length for the full mass interval spanned by the stable evenisotopes. In this way we transfer the knowledge on 40Ca to theother isotopes.The results for Ca are shown in Fig. 2. The plotted massis twice the reduced mass, which for homonuclear collisionssimply is the atomic mass. The dashed vertical lines indicatethe masses of the even isotopes, from which the scatteringlengths for homonuclear collisions are directly read off. Twocalculations are shown, obtained from potentials with C6 coef-ficients that give rise to 340 a0 (red closed circles) and 700 a0(blue closed squares) for 40Ca. The scattering length showsthe expected behavior as a function of mass, with a regularpattern of scattering resonances. They can be understood fromthe mass dependence of the vibrational splitting and, therefore,the number of bound states. A scattering resonance appears atthose mass values at which a new vibrational state becomesbound, i.e., where the least bound vibrational state has zerobinding energy. We find that accidentally all even isotopes arelocated close to such a scattering resonance, giving rise to largepositive or negative scattering lengths.The scattering length intervals for the different isotopesare given in Table II. It is clear that Ca does not providean isotope with a favorable scattering length. At most onecan state that 42Ca has a slightly smaller scattering lengththan 40Ca. In addition, the interisotope scattering lengthsare given for collisions between 40Ca and the other isotopesTABLE II. Scattering lengths of even Ca isotopes (left), as wellas interisotopic scattering lengths for all combinations with 40Ca and42Ca+44Ca (right).a (units of a0) a (units of a0)40 + 40 +340 . . . +700 [28] 40 + 42 +45 . . . +5042 + 42 +320 . . . +640 40 + 44 ±∞44 + 44 +460 . . . +1800 40 + 46 +62 . . . +6846 + 46 ±∞ 40 + 48 −66 . . . −3948 + 48 −230 . . .−120 42 + 44 +47 . . . +52054703-2BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)and 42Ca + 44Ca. Here we find favorable scattering lengthsfor 40Ca + 42Ca, 40Ca + 46Ca, and 42Ca + 44Ca. Taking intoconsideration the large scattering length of 44Ca and 46Ca,the only interesting mixture in view of sympathetic coolingis 40Ca + 42Ca.B. BariumKnowledge of the Ba2 X1+g potential is sparse because of alimited amount of experimental data [30,31], which cover onlya small part of the vibrational spectrum. Similarly, theoreticaldata is scarce. The binding energies of Ba2 were calculatedby Jones [32]. First wave function based quantum chemicalcalculation was reported by Ref. [33] and most recently byRef. [34]. The most accurate calculations of the long-rangecoefficients C6, C8, and C10 are reported by Porsev andDerevianko [35,36].We have used the Tang-Toennies potential [37] of Ref. [24]to model the Ba2 X1+g potential asV (R) = Ae−bR −5∑n=3[1 − e−bR2n∑k=0(bR)kk!]C2nR2n, (2)where the short-range parameters (in a.u.) A = 105.4 andb = 0.9657 are obtained from rescaling an accurate Sr2 X1+gpotential [38], using the measured fundamental vibrationfrequency [31], and taking the long-range coefficients fromRefs. [35,36] (see Fig. 1).Although we do not expect the potential to be accurateenough to predict the scattering length, we can use it toinvestigate its mass dependence. The result is shown in Fig. 3(black closed circles). We observe that the spacing betweenthe resonances is larger than that between the even isotopes,giving rise to a variety of scattering lengths among the differentisotopes. Therefore, it is very probable that at least one130 132 134 136 13840020002004006002Μ amuaunitsofa 0FIG. 3. (Color online) The mass dependence of the scatteringlength for Ba, using the Tang-Toennies model to represent the X1+gpotential with the parameter values from Ref. [24] (black closedcircles). The arrow indicates the effect of the uncertainty in thetheoretical C6 coefficient, showing its lower (blue closed triangles)and upper (red closed squares) limit. The plotted mass is twice thereduced mass and the dashed vertical lines indicate the mass of theeven isotopes of Ba.of the isotopes has a favorable scattering length for BECformation.In addition, we have investigated the effect of the theoreticaluncertainty in the reported C6 coefficient, 5160 ± 74 a.u.[35,36]. We have repeated the above calculation with the upper(red closed squares) and lower (blue closed triangles) limits(see Fig. 3). Note that these adjustments in C6 also requiresmall changes in A and b [24]. The uncertainty in C6 givesrise to a large spread in the scattering length, which can alsobe seen as a large shift along the mass axis, as indicated bythe arrow. This means that even if an accurate short-rangepotential would be available, still no precise prediction ofthe scattering length is possible. On the other hand, whereasthe scattering length itself is extremely sensitive to detailsof the potential, the spacing between the scattering resonancesis not. Therefore, once scattering length information of oneof the isotopes becomes available, the simple Tang-Toenniespotential is sufficient to predict the scattering lengths of allthe other isotopes,1 at least at an accuracy that allows one tochoose the best isotope to achieve BEC.IV. CONCLUSION AND OUTLOOKIn summary, we have investigated the s-wave scatteringlengths of the Ca and Ba even (bosonic) isotopes. Anexperimentally accurate ground-state potential of Ca2 enabledus to calculate the scattering lengths for all the isotopes,taking knowledge of 40Ca scattering length as a reference.We have observed that accidentally scattering resonancesappear near all even isotopes, leading to large positiveand negative scattering lengths for all isotopes. Only 42Cashows a small improvement in terms of scattering lengthcompared to 40Ca. Therefore, alternative methods to thatof evaporative cooling are interesting for Ca, such as di-rect laser cooling into quantum degeneracy using a narrowtransition [39].In addition, scattering lengths for all the even Ba isotopeshave been investigated using a Tang-Toennies model potential.In contrast to the Ca case, here the scattering lengths varystrongly over the isotopes, resulting in a high probabilitythat at least one isotope has a favorable scattering length toproduce a BEC. More accurate knowledge of the ground-stateBa2 potential, including the long-range coefficients, based onexperimental data that covers the full vibrational spectrum isneeded for an improved determination of the scattering lengths.However, once experimental knowledge on the scatteringlength of one isotope becomes available, the scattering lengthsof all the other isotopes can be predicted with the presentmodel potential.ACKNOWLEDGMENTSWe thank Erling Riis and Wim Vassen for careful readingof the manuscript and useful comments. This work has been1We have applied the Tang-Toennies potential model for Ca2 [25]and Sr2 [24], from which we have obtained similar results as inSec. III A and Ref. [15], respectively, when fine-tuning the C6coefficient to reproduce the scattering length of one particular isotope.054703-3BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)performed as part of the research program of the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM) throughprogramme 114 (TRIμP), which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO). S.K. acknowledges financial support from NWO(VIDI grant).[1] K. Burnett, P. S. Julienne, P. D. Lett, E. Tiesinga, and C. J.Williams, Nature (London) 416, 225 (2002).[2] A. D. Ludlow et al., Science 319, 1805 (2008).[3] N. D. Lemke et al., Phys. Rev. Lett. 103, 063001 (2009).[4] A. Derevianko and H. Katori, Rev. Mod. Phys. 83, 331(2011).[5] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett.101, 170504 (2008).[6] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev.Lett. 103, 130401 (2009).[7] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck,Phys. Rev. Lett. 103, 200401 (2009).[8] Y. N. Martinez de Escobar et al., Phys. Rev. Lett. 103, 200402(2009).[9] S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A82, 041602(R) (2010).[10] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan,B. J. DeSalvo, and T. C. Killian, Phys. Rev. A 81, 051601(R)(2010).[11] Y. Takasu et al., Phys. Rev. Lett. 91, 040404 (2003).[12] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76,051604(R) (2007).[13] T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Phys. Rev.A 79, 021601(R) (2009).[14] S. Sugawa, R. Yamazaki, S. Taie, and Y. Takahashi, Phys. Rev.A 84, 011610(R) (2011).[15] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171(2010).[16] T. Zelevinsky, Physics 2, 94 (2009).[17] S. Hoekstra, A. K. Mollema, R. Morgenstern, H. W. Wilschut,and R. Hoekstra, Phys. Rev. A 71, 023409 (2005).[18] U. Dammalapati, I. Norris, C. Burrows, A. S. Arnold, andE. Riis, Phys. Rev. A 81, 023424 (2010).[19] S. De, U. Dammalapati, K. Jungmann, and L. Willmann, Phys.Rev. A 79, 041402(R) (2009).[20] Y. Geng-Hua, Z. Jia-Qi, L. Rung-Bing, W. Jin, andZ. Ming-Sheng, Chin. Phys. Lett. 28, 073201 (2011).[21] J. E. Sansonetti and W. C. Martin, J. Phys. Chem. Ref. Data 34,1559 (2005).[22] M. Przybytek and B. Jeziorski, J. Chem. Phys. 123, 134315(2005).[23] O. Allard, C. Samuelis, A. Pashov, H. Knöckel, and E. Tiemann,Eur. Phys. J. D 26, 155 (2003).[24] P. Li, J. Ren, N. Niu, and K. T. Tang, J. Phys. Chem. A 115,6927 (2011).[25] D. D. Yang, P. Li, and K. T. Tang, J. Phys. Chem. 131, 154301(2009).[26] G. Zinner, T. Binnewies, F. Riehle, and E. Tiemann, Phys. Rev.Lett. 85, 2292 (2000).[27] C. Degenhardt et al., Phys. Rev. A 67, 043408 (2003).[28] F. Vogt et al., Eur. Phys. J. D 44, 73 (2007).[29] O. Allard, A. Pashov, H. Knöckel, and E. Tiemann, Phys. Rev.A 66, 042503 (2002).[30] R. M. Clements and R. F. Barrow, J. Chem. Soc., Faraday Trans.2 81, 625 (1985).[31] M. A. Lebeault, J. Viallon, V. Boutou, and J. Chevaleyre, J. Mol.Spectrosc. 192, 179 (1998).[32] R. O. Jones, J. Chem. Phys. 71, 1300 (1979).[33] A. Allouche, M. Aubert-Frecon, G. Nicolas, and F. Spiegelmann,Chem. Phys. 200, 63 (1995).[34] A. V. Mitin, Russ. J. Phys. Chem. 83, 1160 (2009).[35] S. G. Porsev and A. Derevianko, Phys. Rev. A 65, 020701(R)(2002).[36] S. G. Porsev and A. Derevianko, J. Exp. Theor. Phys. 102, 195(2006).[37] K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 (1984).[38] A. Stein, H. Knöckel, and E. Tiemann, Phys. Rev. A 78, 042508(2008).[39] C. S. Adams, S. G. Cox, E. Riis, and A. S. Arnold, J. Phys. B36, 1933 (2003).054703-4

Scattering lengths of calcium and barium isotopes

We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) in order to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular potential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopes alternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length. For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths, opening the possibility of BEC for at least one of the isotopes. © 2011 American Physical Society

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ARTS repository - University of Groningen

University of GroningenScattering lengths of calcium and barium isotopesDammalapati, U.; Willmann, L.; Knoop, S.Published in:Physical Review ADOI:10.1103/PhysRevA.84.054703IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2011Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dammalapati, U., Willmann, L., & Knoop, S. (2011). Scattering lengths of calcium and barium isotopes.Physical Review A, 84(5), Article 054703. https://doi.org/10.1103/PhysRevA.84.054703CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 24-12-2025PHYSICAL REVIEW A 84, 054703 (2011)Scattering lengths of calcium and barium isotopesU. Dammalapati and L. Willmann*Kernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The NetherlandsS. Knoop†LaserLaB Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands(Received 25 October 2011; published 21 November 2011)We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) inorder to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecularpotential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopesalternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length.For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths,opening the possibility of BEC for at least one of the isotopes.DOI: 10.1103/PhysRevA.84.054703 PACS number(s): 34.20.−b, 34.50.Cx, 67.85.−dI. INTRODUCTIONKnowledge of the s-wave scattering length plays a crucialrole in the achievement of Bose-Einstein condensation (BEC)in ultracold atomic gases [1]. A positive sign of the scatteringlength indicates that a Bose-Einstein condensate is stable,whereas a negative sign indicates an unstable or collapsingcondensate. Also, its magnitude is crucial in the formation ofBEC, as it determines the elastic and inelastic collision rates.For efficient evaporative cooling, a large scattering length isrequired; however, for a too-large scattering length, three-body recombination loss limits the formation and collisionalstability of a BEC. This gives rise to a range of about 50–200 a0 (a0 = 0.5292 nm) that can be considered favorable.Alkaline-earth-metal elements and the alike ytterbium (Yb)system have been the center of attraction in ultracold atomresearch because of their unique atomic structure and theavailability of different stable isotopes with zero nuclear spin(bosons) and with large nuclear spin (fermions). They arebeing used in frequency metrology [2–4] and are proposed forquantum information processing [5]. Recently, BEC has beenachieved in 40Ca [6], 84Sr [7,8], 86Sr [9], and 88Sr [10]. Also,BEC of different Yb isotopes has been reported: 174Yb [11],170Yb [12], 176Yb [13], and 168Yb [14], of which the last onehas an abundance of only 0.13%, highlighting the possibilityof using rare isotopes to achieve BEC.In general, isotopes of an element have different scatteringlengths. Therefore, the ability of BEC formation dependscrucially on the chosen isotope. For instance, the scatteringlength of 84Sr is 123 a0 [15], which made it an ideal candidateto achieve BEC despite its low abundance of only 0.6% [16],as compared to the more abundant 86Sr (10%, 800 a0 [15])and 88Sr (83%, −2 a0 [15]), either suffering strong losses [9]or requiring sympathetic cooling with another isotope [10],respectively. The 40Ca isotope has a large scattering length ofabout 440 a0 [6], limiting the size and stability of BEC.Motivated by the success of BEC in Sr and Yb even isotopesand 40Ca, we carried out calculations to obtain the scattering*l.willmann@rug.nl†s.knoop@vu.nllengths for all the even (bosonic) isotopes of Ca and Ba,for which mass and abundance are summarized in Table I.Laser cooling and trapping of all stable Ca isotopes havebeen reported [17,18]. The demonstration of magneto-opticaltrapping of Ba [19] has opened its use for ultracold collisionstudies, photoassociation spectroscopy, and Bose-Einsteincondensation. Recently, an optical clock based on ultracoldBa has been proposed [20].II. THEORYThe scattering properties of ground-state atoms are ob-tained from the underlying two-body ground-state potentials.For alkaline-earth-metal atoms, there is only one molecularground-state potential, namely the singlet X1+g potential.Furthermore, the lack of nuclear spin and, therefore, hyperfinestructure for the even (bosonic) isotopes highly reduces thenumber of collision channels compared with the alkali metalsystems.To calculate the s-wave scattering length, we solve the 1Dradial Schrödinger equation with zero angular momentum andvanishing kinetic energy,ψ ′′(R) + 2μh̄2[E − V (R)]ψ(R) = 0, (1)where μ is the reduced mass, R the internuclear distance, V (R)the Born-Oppenheimer potential (here the X1+g potential),and E the kinetic energy (below μK). The asymptotic formof the wavefunction is ψ(R) ∝ sin[k(R − a)], where k =√2μE/h̄2, and a is the scattering length. We fit the asymptoticform to the solution of Eq. (1) for large R in order to obtain a.Within the Born-Oppenheimer approximation, the potentialsare identical for all the isotopes of a particular atomic system.Therefore, for a given potential V (R), one only needs to adjustμ to obtain a for all the isotopes.To obtain reliable scattering lengths, accurate potentialsare needed. With the exception of systems with only a fewelectrons, like metastable helium [22], ab initio potentials arein general not accurate enough and constraints from experi-mental data are required. For Ca2, an analytic representation ofthe X1+g potential based on an extensive set of experimentaldata from the Tiemann group is available [23]. For Ba2,054703-11050-2947/2011/84(5)/054703(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)TABLE I. Mass and abundance of all stable even isotopes of Caand Ba [21].CalciumIsotope 40 42 44 46 48Mass (amu) 39.963 41.959 43.955 45.954 47.953Abundance (%) 96.941 0.647 2.086 0.004 0.187BariumIsotope 130 132 134 136 138Mass (amu) 129.906 131.905 133.904 135.905 137.905Abundance (%) 0.11 0.10 2.42 7.85 71.70such an accurate potential is not available. Here we rely onthe analytical representation according to the Tang-Toenniespotential model [24], which has been shown to be able toreproduce accurately the X1+g potentials of Ca2 [25] andSr2 [24]. The used Ca2 and Ba2 potentials are shown inFig. 1.III. RESULTSA. CalciumInformation on the Ca2 X1+g potential has been gatheredby several spectroscopic methods, including photoassociationspectroscopy [26–28], Fourier-transform spectroscopy [29],and filtered laser excitation technique [23]. Based on thesedetailed spectroscopic data, an interval of the scattering lengthfor 40Ca is determined to be 340–700 a0 [28]. In the 40Ca BECexperiment, a scattering length of a ≈ 440 a0 was estimatedfrom a measurement of the chemical potential [6], withoutquoting an uncertainty. Therefore, we take the scattering lengthrange of 340–700 a0 as the starting point of our calculations.We have taken the analytical representation and parametervalues of the Ca2 X1+g potential from Ref. [23] (see Fig. 1).We allow the C6 long-range coefficient to vary within afactor 0.991 to 1.003 from its reported value [23], in order toreproduce the scattering length interval for 40Ca. We then takethe mass as a variable parameter and calculate the scattering5 10 15 20 2540302010010R (unitsof a )0VhTHzCa2 Ba21S+1SFIG. 1. (Color online) The X1+g potentials of Ca2 and Ba2 as afunction of internuclear distance. The solid blue curve is the potentialof Ca2 taken from Ref. [23]. The dashed red curve is the Tang-Toennies potential of Ba2 taken from Ref. [24].40 42 44 46 4840020002004006002Μ amuaunitsofa 0FIG. 2. (Color online) The mass dependence of the scatteringlength for Ca, using the X1+g potential of Ref. [23], showingcalculations with C6 coefficients that give rise to 340 a0 (red closedcircles) and 700 a0 (blue closed squares) for 40Ca. The plotted massis twice the reduced mass and the dashed vertical lines indicate themass of the even isotopes of Ca.length for the full mass interval spanned by the stable evenisotopes. In this way we transfer the knowledge on 40Ca to theother isotopes.The results for Ca are shown in Fig. 2. The plotted massis twice the reduced mass, which for homonuclear collisionssimply is the atomic mass. The dashed vertical lines indicatethe masses of the even isotopes, from which the scatteringlengths for homonuclear collisions are directly read off. Twocalculations are shown, obtained from potentials with C6 coef-ficients that give rise to 340 a0 (red closed circles) and 700 a0(blue closed squares) for 40Ca. The scattering length showsthe expected behavior as a function of mass, with a regularpattern of scattering resonances. They can be understood fromthe mass dependence of the vibrational splitting and, therefore,the number of bound states. A scattering resonance appears atthose mass values at which a new vibrational state becomesbound, i.e., where the least bound vibrational state has zerobinding energy. We find that accidentally all even isotopes arelocated close to such a scattering resonance, giving rise to largepositive or negative scattering lengths.The scattering length intervals for the different isotopesare given in Table II. It is clear that Ca does not providean isotope with a favorable scattering length. At most onecan state that 42Ca has a slightly smaller scattering lengththan 40Ca. In addition, the interisotope scattering lengthsare given for collisions between 40Ca and the other isotopesTABLE II. Scattering lengths of even Ca isotopes (left), as wellas interisotopic scattering lengths for all combinations with 40Ca and42Ca+44Ca (right).a (units of a0) a (units of a0)40 + 40 +340 . . . +700 [28] 40 + 42 +45 . . . +5042 + 42 +320 . . . +640 40 + 44 ±∞44 + 44 +460 . . . +1800 40 + 46 +62 . . . +6846 + 46 ±∞ 40 + 48 −66 . . . −3948 + 48 −230 . . .−120 42 + 44 +47 . . . +52054703-2BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)and 42Ca + 44Ca. Here we find favorable scattering lengthsfor 40Ca + 42Ca, 40Ca + 46Ca, and 42Ca + 44Ca. Taking intoconsideration the large scattering length of 44Ca and 46Ca,the only interesting mixture in view of sympathetic coolingis 40Ca + 42Ca.B. BariumKnowledge of the Ba2 X1+g potential is sparse because of alimited amount of experimental data [30,31], which cover onlya small part of the vibrational spectrum. Similarly, theoreticaldata is scarce. The binding energies of Ba2 were calculatedby Jones [32]. First wave function based quantum chemicalcalculation was reported by Ref. [33] and most recently byRef. [34]. The most accurate calculations of the long-rangecoefficients C6, C8, and C10 are reported by Porsev andDerevianko [35,36].We have used the Tang-Toennies potential [37] of Ref. [24]to model the Ba2 X1+g potential asV (R) = Ae−bR −5∑n=3[1 − e−bR2n∑k=0(bR)kk!]C2nR2n, (2)where the short-range parameters (in a.u.) A = 105.4 andb = 0.9657 are obtained from rescaling an accurate Sr2 X1+gpotential [38], using the measured fundamental vibrationfrequency [31], and taking the long-range coefficients fromRefs. [35,36] (see Fig. 1).Although we do not expect the potential to be accurateenough to predict the scattering length, we can use it toinvestigate its mass dependence. The result is shown in Fig. 3(black closed circles). We observe that the spacing betweenthe resonances is larger than that between the even isotopes,giving rise to a variety of scattering lengths among the differentisotopes. Therefore, it is very probable that at least one130 132 134 136 13840020002004006002Μ amuaunitsofa 0FIG. 3. (Color online) The mass dependence of the scatteringlength for Ba, using the Tang-Toennies model to represent the X1+gpotential with the parameter values from Ref. [24] (black closedcircles). The arrow indicates the effect of the uncertainty in thetheoretical C6 coefficient, showing its lower (blue closed triangles)and upper (red closed squares) limit. The plotted mass is twice thereduced mass and the dashed vertical lines indicate the mass of theeven isotopes of Ba.of the isotopes has a favorable scattering length for BECformation.In addition, we have investigated the effect of the theoreticaluncertainty in the reported C6 coefficient, 5160 ± 74 a.u.[35,36]. We have repeated the above calculation with the upper(red closed squares) and lower (blue closed triangles) limits(see Fig. 3). Note that these adjustments in C6 also requiresmall changes in A and b [24]. The uncertainty in C6 givesrise to a large spread in the scattering length, which can alsobe seen as a large shift along the mass axis, as indicated bythe arrow. This means that even if an accurate short-rangepotential would be available, still no precise prediction ofthe scattering length is possible. On the other hand, whereasthe scattering length itself is extremely sensitive to detailsof the potential, the spacing between the scattering resonancesis not. Therefore, once scattering length information of oneof the isotopes becomes available, the simple Tang-Toenniespotential is sufficient to predict the scattering lengths of allthe other isotopes,1 at least at an accuracy that allows one tochoose the best isotope to achieve BEC.IV. CONCLUSION AND OUTLOOKIn summary, we have investigated the s-wave scatteringlengths of the Ca and Ba even (bosonic) isotopes. Anexperimentally accurate ground-state potential of Ca2 enabledus to calculate the scattering lengths for all the isotopes,taking knowledge of 40Ca scattering length as a reference.We have observed that accidentally scattering resonancesappear near all even isotopes, leading to large positiveand negative scattering lengths for all isotopes. Only 42Cashows a small improvement in terms of scattering lengthcompared to 40Ca. Therefore, alternative methods to thatof evaporative cooling are interesting for Ca, such as di-rect laser cooling into quantum degeneracy using a narrowtransition [39].In addition, scattering lengths for all the even Ba isotopeshave been investigated using a Tang-Toennies model potential.In contrast to the Ca case, here the scattering lengths varystrongly over the isotopes, resulting in a high probabilitythat at least one isotope has a favorable scattering length toproduce a BEC. More accurate knowledge of the ground-stateBa2 potential, including the long-range coefficients, based onexperimental data that covers the full vibrational spectrum isneeded for an improved determination of the scattering lengths.However, once experimental knowledge on the scatteringlength of one isotope becomes available, the scattering lengthsof all the other isotopes can be predicted with the presentmodel potential.ACKNOWLEDGMENTSWe thank Erling Riis and Wim Vassen for careful readingof the manuscript and useful comments. This work has been1We have applied the Tang-Toennies potential model for Ca2 [25]and Sr2 [24], from which we have obtained similar results as inSec. III A and Ref. [15], respectively, when fine-tuning the C6coefficient to reproduce the scattering length of one particular isotope.054703-3BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)performed as part of the research program of the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM) throughprogramme 114 (TRIμP), which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO). S.K. acknowledges financial support from NWO(VIDI grant).[1] K. Burnett, P. S. Julienne, P. D. Lett, E. Tiesinga, and C. J.Williams, Nature (London) 416, 225 (2002).[2] A. D. Ludlow et al., Science 319, 1805 (2008).[3] N. D. Lemke et al., Phys. Rev. Lett. 103, 063001 (2009).[4] A. Derevianko and H. Katori, Rev. Mod. Phys. 83, 331(2011).[5] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett.101, 170504 (2008).[6] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev.Lett. 103, 130401 (2009).[7] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck,Phys. Rev. Lett. 103, 200401 (2009).[8] Y. N. Martinez de Escobar et al., Phys. Rev. Lett. 103, 200402(2009).[9] S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A82, 041602(R) (2010).[10] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan,B. J. DeSalvo, and T. C. Killian, Phys. Rev. A 81, 051601(R)(2010).[11] Y. Takasu et al., Phys. Rev. Lett. 91, 040404 (2003).[12] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76,051604(R) (2007).[13] T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Phys. Rev.A 79, 021601(R) (2009).[14] S. Sugawa, R. Yamazaki, S. Taie, and Y. Takahashi, Phys. Rev.A 84, 011610(R) (2011).[15] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171(2010).[16] T. Zelevinsky, Physics 2, 94 (2009).[17] S. Hoekstra, A. K. Mollema, R. Morgenstern, H. W. Wilschut,and R. Hoekstra, Phys. Rev. A 71, 023409 (2005).[18] U. Dammalapati, I. Norris, C. Burrows, A. S. Arnold, andE. Riis, Phys. Rev. A 81, 023424 (2010).[19] S. De, U. Dammalapati, K. Jungmann, and L. Willmann, Phys.Rev. A 79, 041402(R) (2009).[20] Y. Geng-Hua, Z. Jia-Qi, L. Rung-Bing, W. Jin, andZ. Ming-Sheng, Chin. Phys. Lett. 28, 073201 (2011).[21] J. E. Sansonetti and W. C. Martin, J. Phys. Chem. Ref. Data 34,1559 (2005).[22] M. Przybytek and B. Jeziorski, J. Chem. Phys. 123, 134315(2005).[23] O. Allard, C. Samuelis, A. Pashov, H. Knöckel, and E. Tiemann,Eur. Phys. J. D 26, 155 (2003).[24] P. Li, J. Ren, N. Niu, and K. T. Tang, J. Phys. Chem. A 115,6927 (2011).[25] D. D. Yang, P. Li, and K. T. Tang, J. Phys. Chem. 131, 154301(2009).[26] G. Zinner, T. Binnewies, F. Riehle, and E. Tiemann, Phys. Rev.Lett. 85, 2292 (2000).[27] C. Degenhardt et al., Phys. Rev. A 67, 043408 (2003).[28] F. Vogt et al., Eur. Phys. J. D 44, 73 (2007).[29] O. Allard, A. Pashov, H. Knöckel, and E. Tiemann, Phys. Rev.A 66, 042503 (2002).[30] R. M. Clements and R. F. Barrow, J. Chem. Soc., Faraday Trans.2 81, 625 (1985).[31] M. A. Lebeault, J. Viallon, V. Boutou, and J. Chevaleyre, J. Mol.Spectrosc. 192, 179 (1998).[32] R. O. Jones, J. Chem. Phys. 71, 1300 (1979).[33] A. Allouche, M. Aubert-Frecon, G. Nicolas, and F. Spiegelmann,Chem. Phys. 200, 63 (1995).[34] A. V. Mitin, Russ. J. Phys. Chem. 83, 1160 (2009).[35] S. G. Porsev and A. Derevianko, Phys. Rev. A 65, 020701(R)(2002).[36] S. G. Porsev and A. Derevianko, J. Exp. Theor. Phys. 102, 195(2006).[37] K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 (1984).[38] A. Stein, H. Knöckel, and E. Tiemann, Phys. Rev. A 78, 042508(2008).[39] C. S. Adams, S. G. Cox, E. Riis, and A. S. Arnold, J. Phys. B36, 1933 (2003).054703-4

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VU Research PortalScattering lengths of calcium and barium isotopesDammalapati, U.; Willmann, L.; Knoop, S.published inPhysical Review A. Atomic, Molecular and Optical Physics2011DOI (link to publisher)10.1103/PhysRevA.84.054703document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Dammalapati, U., Willmann, L., & Knoop, S. (2011). Scattering lengths of calcium and barium isotopes. PhysicalReview A. Atomic, Molecular and Optical Physics, 84, 054703. 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May. 2021PHYSICAL REVIEW A 84, 054703 (2011)Scattering lengths of calcium and barium isotopesU. Dammalapati and L. Willmann*Kernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The NetherlandsS. Knoop†LaserLaB Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands(Received 25 October 2011; published 21 November 2011)We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) inorder to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecularpotential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopesalternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length.For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths,opening the possibility of BEC for at least one of the isotopes.DOI: 10.1103/PhysRevA.84.054703 PACS number(s): 34.20.−b, 34.50.Cx, 67.85.−dI. INTRODUCTIONKnowledge of the s-wave scattering length plays a crucialrole in the achievement of Bose-Einstein condensation (BEC)in ultracold atomic gases [1]. A positive sign of the scatteringlength indicates that a Bose-Einstein condensate is stable,whereas a negative sign indicates an unstable or collapsingcondensate. Also, its magnitude is crucial in the formation ofBEC, as it determines the elastic and inelastic collision rates.For efficient evaporative cooling, a large scattering length isrequired; however, for a too-large scattering length, three-body recombination loss limits the formation and collisionalstability of a BEC. This gives rise to a range of about 50–200 a0 (a0 = 0.5292 nm) that can be considered favorable.Alkaline-earth-metal elements and the alike ytterbium (Yb)system have been the center of attraction in ultracold atomresearch because of their unique atomic structure and theavailability of different stable isotopes with zero nuclear spin(bosons) and with large nuclear spin (fermions). They arebeing used in frequency metrology [2–4] and are proposed forquantum information processing [5]. Recently, BEC has beenachieved in 40Ca [6], 84Sr [7,8], 86Sr [9], and 88Sr [10]. Also,BEC of different Yb isotopes has been reported: 174Yb [11],170Yb [12], 176Yb [13], and 168Yb [14], of which the last onehas an abundance of only 0.13%, highlighting the possibilityof using rare isotopes to achieve BEC.In general, isotopes of an element have different scatteringlengths. Therefore, the ability of BEC formation dependscrucially on the chosen isotope. For instance, the scatteringlength of 84Sr is 123 a0 [15], which made it an ideal candidateto achieve BEC despite its low abundance of only 0.6% [16],as compared to the more abundant 86Sr (10%, 800 a0 [15])and 88Sr (83%, −2 a0 [15]), either suffering strong losses [9]or requiring sympathetic cooling with another isotope [10],respectively. The 40Ca isotope has a large scattering length ofabout 440 a0 [6], limiting the size and stability of BEC.Motivated by the success of BEC in Sr and Yb even isotopesand 40Ca, we carried out calculations to obtain the scattering*l.willmann@rug.nl†s.knoop@vu.nllengths for all the even (bosonic) isotopes of Ca and Ba,for which mass and abundance are summarized in Table I.Laser cooling and trapping of all stable Ca isotopes havebeen reported [17,18]. The demonstration of magneto-opticaltrapping of Ba [19] has opened its use for ultracold collisionstudies, photoassociation spectroscopy, and Bose-Einsteincondensation. Recently, an optical clock based on ultracoldBa has been proposed [20].II. THEORYThe scattering properties of ground-state atoms are ob-tained from the underlying two-body ground-state potentials.For alkaline-earth-metal atoms, there is only one molecularground-state potential, namely the singlet X1+g potential.Furthermore, the lack of nuclear spin and, therefore, hyperfinestructure for the even (bosonic) isotopes highly reduces thenumber of collision channels compared with the alkali metalsystems.To calculate the s-wave scattering length, we solve the 1Dradial Schrödinger equation with zero angular momentum andvanishing kinetic energy,ψ ′′(R) + 2μh̄2[E − V (R)]ψ(R) = 0, (1)where μ is the reduced mass, R the internuclear distance, V (R)the Born-Oppenheimer potential (here the X1+g potential),and E the kinetic energy (below μK). The asymptotic formof the wavefunction is ψ(R) ∝ sin[k(R − a)], where k =√2μE/h̄2, and a is the scattering length. We fit the asymptoticform to the solution of Eq. (1) for large R in order to obtain a.Within the Born-Oppenheimer approximation, the potentialsare identical for all the isotopes of a particular atomic system.Therefore, for a given potential V (R), one only needs to adjustμ to obtain a for all the isotopes.To obtain reliable scattering lengths, accurate potentialsare needed. With the exception of systems with only a fewelectrons, like metastable helium [22], ab initio potentials arein general not accurate enough and constraints from experi-mental data are required. For Ca2, an analytic representation ofthe X1+g potential based on an extensive set of experimentaldata from the Tiemann group is available [23]. For Ba2,054703-11050-2947/2011/84(5)/054703(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)TABLE I. Mass and abundance of all stable even isotopes of Caand Ba [21].CalciumIsotope 40 42 44 46 48Mass (amu) 39.963 41.959 43.955 45.954 47.953Abundance (%) 96.941 0.647 2.086 0.004 0.187BariumIsotope 130 132 134 136 138Mass (amu) 129.906 131.905 133.904 135.905 137.905Abundance (%) 0.11 0.10 2.42 7.85 71.70such an accurate potential is not available. Here we rely onthe analytical representation according to the Tang-Toenniespotential model [24], which has been shown to be able toreproduce accurately the X1+g potentials of Ca2 [25] andSr2 [24]. The used Ca2 and Ba2 potentials are shown inFig. 1.III. RESULTSA. CalciumInformation on the Ca2 X1+g potential has been gatheredby several spectroscopic methods, including photoassociationspectroscopy [26–28], Fourier-transform spectroscopy [29],and filtered laser excitation technique [23]. Based on thesedetailed spectroscopic data, an interval of the scattering lengthfor 40Ca is determined to be 340–700 a0 [28]. In the 40Ca BECexperiment, a scattering length of a ≈ 440 a0 was estimatedfrom a measurement of the chemical potential [6], withoutquoting an uncertainty. Therefore, we take the scattering lengthrange of 340–700 a0 as the starting point of our calculations.We have taken the analytical representation and parametervalues of the Ca2 X1+g potential from Ref. [23] (see Fig. 1).We allow the C6 long-range coefficient to vary within afactor 0.991 to 1.003 from its reported value [23], in order toreproduce the scattering length interval for 40Ca. We then takethe mass as a variable parameter and calculate the scattering5 10 15 20 2540302010010R (unitsof a )0VhTHzCa2 Ba21S+1SFIG. 1. (Color online) The X1+g potentials of Ca2 and Ba2 as afunction of internuclear distance. The solid blue curve is the potentialof Ca2 taken from Ref. [23]. The dashed red curve is the Tang-Toennies potential of Ba2 taken from Ref. [24].40 42 44 46 4840020002004006002Μ amuaunitsofa 0FIG. 2. (Color online) The mass dependence of the scatteringlength for Ca, using the X1+g potential of Ref. [23], showingcalculations with C6 coefficients that give rise to 340 a0 (red closedcircles) and 700 a0 (blue closed squares) for 40Ca. The plotted massis twice the reduced mass and the dashed vertical lines indicate themass of the even isotopes of Ca.length for the full mass interval spanned by the stable evenisotopes. In this way we transfer the knowledge on 40Ca to theother isotopes.The results for Ca are shown in Fig. 2. The plotted massis twice the reduced mass, which for homonuclear collisionssimply is the atomic mass. The dashed vertical lines indicatethe masses of the even isotopes, from which the scatteringlengths for homonuclear collisions are directly read off. Twocalculations are shown, obtained from potentials with C6 coef-ficients that give rise to 340 a0 (red closed circles) and 700 a0(blue closed squares) for 40Ca. The scattering length showsthe expected behavior as a function of mass, with a regularpattern of scattering resonances. They can be understood fromthe mass dependence of the vibrational splitting and, therefore,the number of bound states. A scattering resonance appears atthose mass values at which a new vibrational state becomesbound, i.e., where the least bound vibrational state has zerobinding energy. We find that accidentally all even isotopes arelocated close to such a scattering resonance, giving rise to largepositive or negative scattering lengths.The scattering length intervals for the different isotopesare given in Table II. It is clear that Ca does not providean isotope with a favorable scattering length. At most onecan state that 42Ca has a slightly smaller scattering lengththan 40Ca. In addition, the interisotope scattering lengthsare given for collisions between 40Ca and the other isotopesTABLE II. Scattering lengths of even Ca isotopes (left), as wellas interisotopic scattering lengths for all combinations with 40Ca and42Ca+44Ca (right).a (units of a0) a (units of a0)40 + 40 +340 . . . +700 [28] 40 + 42 +45 . . . +5042 + 42 +320 . . . +640 40 + 44 ±∞44 + 44 +460 . . . +1800 40 + 46 +62 . . . +6846 + 46 ±∞ 40 + 48 −66 . . . −3948 + 48 −230 . . .−120 42 + 44 +47 . . . +52054703-2BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)and 42Ca + 44Ca. Here we find favorable scattering lengthsfor 40Ca + 42Ca, 40Ca + 46Ca, and 42Ca + 44Ca. Taking intoconsideration the large scattering length of 44Ca and 46Ca,the only interesting mixture in view of sympathetic coolingis 40Ca + 42Ca.B. BariumKnowledge of the Ba2 X1+g potential is sparse because of alimited amount of experimental data [30,31], which cover onlya small part of the vibrational spectrum. Similarly, theoreticaldata is scarce. The binding energies of Ba2 were calculatedby Jones [32]. First wave function based quantum chemicalcalculation was reported by Ref. [33] and most recently byRef. [34]. The most accurate calculations of the long-rangecoefficients C6, C8, and C10 are reported by Porsev andDerevianko [35,36].We have used the Tang-Toennies potential [37] of Ref. [24]to model the Ba2 X1+g potential asV (R) = Ae−bR −5∑n=3[1 − e−bR2n∑k=0(bR)kk!]C2nR2n, (2)where the short-range parameters (in a.u.) A = 105.4 andb = 0.9657 are obtained from rescaling an accurate Sr2 X1+gpotential [38], using the measured fundamental vibrationfrequency [31], and taking the long-range coefficients fromRefs. [35,36] (see Fig. 1).Although we do not expect the potential to be accurateenough to predict the scattering length, we can use it toinvestigate its mass dependence. The result is shown in Fig. 3(black closed circles). We observe that the spacing betweenthe resonances is larger than that between the even isotopes,giving rise to a variety of scattering lengths among the differentisotopes. Therefore, it is very probable that at least one130 132 134 136 13840020002004006002Μ amuaunitsofa 0FIG. 3. (Color online) The mass dependence of the scatteringlength for Ba, using the Tang-Toennies model to represent the X1+gpotential with the parameter values from Ref. [24] (black closedcircles). The arrow indicates the effect of the uncertainty in thetheoretical C6 coefficient, showing its lower (blue closed triangles)and upper (red closed squares) limit. The plotted mass is twice thereduced mass and the dashed vertical lines indicate the mass of theeven isotopes of Ba.of the isotopes has a favorable scattering length for BECformation.In addition, we have investigated the effect of the theoreticaluncertainty in the reported C6 coefficient, 5160 ± 74 a.u.[35,36]. We have repeated the above calculation with the upper(red closed squares) and lower (blue closed triangles) limits(see Fig. 3). Note that these adjustments in C6 also requiresmall changes in A and b [24]. The uncertainty in C6 givesrise to a large spread in the scattering length, which can alsobe seen as a large shift along the mass axis, as indicated bythe arrow. This means that even if an accurate short-rangepotential would be available, still no precise prediction ofthe scattering length is possible. On the other hand, whereasthe scattering length itself is extremely sensitive to detailsof the potential, the spacing between the scattering resonancesis not. Therefore, once scattering length information of oneof the isotopes becomes available, the simple Tang-Toenniespotential is sufficient to predict the scattering lengths of allthe other isotopes,1 at least at an accuracy that allows one tochoose the best isotope to achieve BEC.IV. CONCLUSION AND OUTLOOKIn summary, we have investigated the s-wave scatteringlengths of the Ca and Ba even (bosonic) isotopes. Anexperimentally accurate ground-state potential of Ca2 enabledus to calculate the scattering lengths for all the isotopes,taking knowledge of 40Ca scattering length as a reference.We have observed that accidentally scattering resonancesappear near all even isotopes, leading to large positiveand negative scattering lengths for all isotopes. Only 42Cashows a small improvement in terms of scattering lengthcompared to 40Ca. Therefore, alternative methods to thatof evaporative cooling are interesting for Ca, such as di-rect laser cooling into quantum degeneracy using a narrowtransition [39].In addition, scattering lengths for all the even Ba isotopeshave been investigated using a Tang-Toennies model potential.In contrast to the Ca case, here the scattering lengths varystrongly over the isotopes, resulting in a high probabilitythat at least one isotope has a favorable scattering length toproduce a BEC. More accurate knowledge of the ground-stateBa2 potential, including the long-range coefficients, based onexperimental data that covers the full vibrational spectrum isneeded for an improved determination of the scattering lengths.However, once experimental knowledge on the scatteringlength of one isotope becomes available, the scattering lengthsof all the other isotopes can be predicted with the presentmodel potential.ACKNOWLEDGMENTSWe thank Erling Riis and Wim Vassen for careful readingof the manuscript and useful comments. This work has been1We have applied the Tang-Toennies potential model for Ca2 [25]and Sr2 [24], from which we have obtained similar results as inSec. III A and Ref. [15], respectively, when fine-tuning the C6coefficient to reproduce the scattering length of one particular isotope.054703-3BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)performed as part of the research program of the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM) throughprogramme 114 (TRIμP), which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO). S.K. acknowledges financial support from NWO(VIDI grant).[1] K. Burnett, P. S. Julienne, P. D. Lett, E. Tiesinga, and C. J.Williams, Nature (London) 416, 225 (2002).[2] A. D. Ludlow et al., Science 319, 1805 (2008).[3] N. D. Lemke et al., Phys. Rev. Lett. 103, 063001 (2009).[4] A. Derevianko and H. Katori, Rev. Mod. Phys. 83, 331(2011).[5] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett.101, 170504 (2008).[6] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev.Lett. 103, 130401 (2009).[7] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck,Phys. Rev. Lett. 103, 200401 (2009).[8] Y. N. Martinez de Escobar et al., Phys. Rev. Lett. 103, 200402(2009).[9] S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A82, 041602(R) (2010).[10] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan,B. J. DeSalvo, and T. C. Killian, Phys. Rev. A 81, 051601(R)(2010).[11] Y. Takasu et al., Phys. Rev. Lett. 91, 040404 (2003).[12] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76,051604(R) (2007).[13] T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Phys. Rev.A 79, 021601(R) (2009).[14] S. Sugawa, R. Yamazaki, S. Taie, and Y. Takahashi, Phys. Rev.A 84, 011610(R) (2011).[15] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171(2010).[16] T. Zelevinsky, Physics 2, 94 (2009).[17] S. Hoekstra, A. K. Mollema, R. Morgenstern, H. W. Wilschut,and R. Hoekstra, Phys. Rev. A 71, 023409 (2005).[18] U. Dammalapati, I. Norris, C. Burrows, A. S. Arnold, andE. Riis, Phys. Rev. A 81, 023424 (2010).[19] S. De, U. Dammalapati, K. Jungmann, and L. Willmann, Phys.Rev. A 79, 041402(R) (2009).[20] Y. Geng-Hua, Z. Jia-Qi, L. Rung-Bing, W. Jin, andZ. Ming-Sheng, Chin. Phys. Lett. 28, 073201 (2011).[21] J. E. Sansonetti and W. C. Martin, J. Phys. Chem. Ref. Data 34,1559 (2005).[22] M. Przybytek and B. Jeziorski, J. Chem. Phys. 123, 134315(2005).[23] O. Allard, C. Samuelis, A. Pashov, H. Knöckel, and E. Tiemann,Eur. Phys. J. D 26, 155 (2003).[24] P. Li, J. Ren, N. Niu, and K. T. Tang, J. Phys. Chem. A 115,6927 (2011).[25] D. D. Yang, P. Li, and K. T. Tang, J. Phys. Chem. 131, 154301(2009).[26] G. Zinner, T. Binnewies, F. Riehle, and E. Tiemann, Phys. Rev.Lett. 85, 2292 (2000).[27] C. Degenhardt et al., Phys. Rev. A 67, 043408 (2003).[28] F. Vogt et al., Eur. Phys. J. D 44, 73 (2007).[29] O. Allard, A. Pashov, H. Knöckel, and E. Tiemann, Phys. Rev.A 66, 042503 (2002).[30] R. M. Clements and R. F. Barrow, J. Chem. Soc., Faraday Trans.2 81, 625 (1985).[31] M. A. Lebeault, J. Viallon, V. Boutou, and J. Chevaleyre, J. Mol.Spectrosc. 192, 179 (1998).[32] R. O. Jones, J. Chem. Phys. 71, 1300 (1979).[33] A. Allouche, M. Aubert-Frecon, G. Nicolas, and F. Spiegelmann,Chem. Phys. 200, 63 (1995).[34] A. V. Mitin, Russ. J. Phys. Chem. 83, 1160 (2009).[35] S. G. Porsev and A. Derevianko, Phys. Rev. A 65, 020701(R)(2002).[36] S. G. Porsev and A. Derevianko, J. Exp. Theor. Phys. 102, 195(2006).[37] K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 (1984).[38] A. Stein, H. Knöckel, and E. Tiemann, Phys. Rev. A 78, 042508(2008).[39] C. S. Adams, S. G. Cox, E. Riis, and A. S. Arnold, J. Phys. B36, 1933 (2003).054703-4

University of Groningen Research Database

University of GroningenScattering lengths of calcium and barium isotopesDammalapati, U.; Willmann, L.; Knoop, S.Published in:Physical Review ADOI:10.1103/PhysRevA.84.054703IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2011Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dammalapati, U., Willmann, L., & Knoop, S. (2011). Scattering lengths of calcium and barium isotopes.Physical Review A, 84(5), Article 054703. https://doi.org/10.1103/PhysRevA.84.054703CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 27-12-2025PHYSICAL REVIEW A 84, 054703 (2011)Scattering lengths of calcium and barium isotopesU. Dammalapati and L. Willmann*Kernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The NetherlandsS. Knoop†LaserLaB Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands(Received 25 October 2011; published 21 November 2011)We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) inorder to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecularpotential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopesalternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length.For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths,opening the possibility of BEC for at least one of the isotopes.DOI: 10.1103/PhysRevA.84.054703 PACS number(s): 34.20.−b, 34.50.Cx, 67.85.−dI. INTRODUCTIONKnowledge of the s-wave scattering length plays a crucialrole in the achievement of Bose-Einstein condensation (BEC)in ultracold atomic gases [1]. A positive sign of the scatteringlength indicates that a Bose-Einstein condensate is stable,whereas a negative sign indicates an unstable or collapsingcondensate. Also, its magnitude is crucial in the formation ofBEC, as it determines the elastic and inelastic collision rates.For efficient evaporative cooling, a large scattering length isrequired; however, for a too-large scattering length, three-body recombination loss limits the formation and collisionalstability of a BEC. This gives rise to a range of about 50–200 a0 (a0 = 0.5292 nm) that can be considered favorable.Alkaline-earth-metal elements and the alike ytterbium (Yb)system have been the center of attraction in ultracold atomresearch because of their unique atomic structure and theavailability of different stable isotopes with zero nuclear spin(bosons) and with large nuclear spin (fermions). They arebeing used in frequency metrology [2–4] and are proposed forquantum information processing [5]. Recently, BEC has beenachieved in 40Ca [6], 84Sr [7,8], 86Sr [9], and 88Sr [10]. Also,BEC of different Yb isotopes has been reported: 174Yb [11],170Yb [12], 176Yb [13], and 168Yb [14], of which the last onehas an abundance of only 0.13%, highlighting the possibilityof using rare isotopes to achieve BEC.In general, isotopes of an element have different scatteringlengths. Therefore, the ability of BEC formation dependscrucially on the chosen isotope. For instance, the scatteringlength of 84Sr is 123 a0 [15], which made it an ideal candidateto achieve BEC despite its low abundance of only 0.6% [16],as compared to the more abundant 86Sr (10%, 800 a0 [15])and 88Sr (83%, −2 a0 [15]), either suffering strong losses [9]or requiring sympathetic cooling with another isotope [10],respectively. The 40Ca isotope has a large scattering length ofabout 440 a0 [6], limiting the size and stability of BEC.Motivated by the success of BEC in Sr and Yb even isotopesand 40Ca, we carried out calculations to obtain the scattering*l.willmann@rug.nl†s.knoop@vu.nllengths for all the even (bosonic) isotopes of Ca and Ba,for which mass and abundance are summarized in Table I.Laser cooling and trapping of all stable Ca isotopes havebeen reported [17,18]. The demonstration of magneto-opticaltrapping of Ba [19] has opened its use for ultracold collisionstudies, photoassociation spectroscopy, and Bose-Einsteincondensation. Recently, an optical clock based on ultracoldBa has been proposed [20].II. THEORYThe scattering properties of ground-state atoms are ob-tained from the underlying two-body ground-state potentials.For alkaline-earth-metal atoms, there is only one molecularground-state potential, namely the singlet X1+g potential.Furthermore, the lack of nuclear spin and, therefore, hyperfinestructure for the even (bosonic) isotopes highly reduces thenumber of collision channels compared with the alkali metalsystems.To calculate the s-wave scattering length, we solve the 1Dradial Schrödinger equation with zero angular momentum andvanishing kinetic energy,ψ ′′(R) + 2μh̄2[E − V (R)]ψ(R) = 0, (1)where μ is the reduced mass, R the internuclear distance, V (R)the Born-Oppenheimer potential (here the X1+g potential),and E the kinetic energy (below μK). The asymptotic formof the wavefunction is ψ(R) ∝ sin[k(R − a)], where k =√2μE/h̄2, and a is the scattering length. We fit the asymptoticform to the solution of Eq. (1) for large R in order to obtain a.Within the Born-Oppenheimer approximation, the potentialsare identical for all the isotopes of a particular atomic system.Therefore, for a given potential V (R), one only needs to adjustμ to obtain a for all the isotopes.To obtain reliable scattering lengths, accurate potentialsare needed. With the exception of systems with only a fewelectrons, like metastable helium [22], ab initio potentials arein general not accurate enough and constraints from experi-mental data are required. For Ca2, an analytic representation ofthe X1+g potential based on an extensive set of experimentaldata from the Tiemann group is available [23]. For Ba2,054703-11050-2947/2011/84(5)/054703(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)TABLE I. Mass and abundance of all stable even isotopes of Caand Ba [21].CalciumIsotope 40 42 44 46 48Mass (amu) 39.963 41.959 43.955 45.954 47.953Abundance (%) 96.941 0.647 2.086 0.004 0.187BariumIsotope 130 132 134 136 138Mass (amu) 129.906 131.905 133.904 135.905 137.905Abundance (%) 0.11 0.10 2.42 7.85 71.70such an accurate potential is not available. Here we rely onthe analytical representation according to the Tang-Toenniespotential model [24], which has been shown to be able toreproduce accurately the X1+g potentials of Ca2 [25] andSr2 [24]. The used Ca2 and Ba2 potentials are shown inFig. 1.III. RESULTSA. CalciumInformation on the Ca2 X1+g potential has been gatheredby several spectroscopic methods, including photoassociationspectroscopy [26–28], Fourier-transform spectroscopy [29],and filtered laser excitation technique [23]. Based on thesedetailed spectroscopic data, an interval of the scattering lengthfor 40Ca is determined to be 340–700 a0 [28]. In the 40Ca BECexperiment, a scattering length of a ≈ 440 a0 was estimatedfrom a measurement of the chemical potential [6], withoutquoting an uncertainty. Therefore, we take the scattering lengthrange of 340–700 a0 as the starting point of our calculations.We have taken the analytical representation and parametervalues of the Ca2 X1+g potential from Ref. [23] (see Fig. 1).We allow the C6 long-range coefficient to vary within afactor 0.991 to 1.003 from its reported value [23], in order toreproduce the scattering length interval for 40Ca. We then takethe mass as a variable parameter and calculate the scattering5 10 15 20 2540302010010R (unitsof a )0VhTHzCa2 Ba21S+1SFIG. 1. (Color online) The X1+g potentials of Ca2 and Ba2 as afunction of internuclear distance. The solid blue curve is the potentialof Ca2 taken from Ref. [23]. The dashed red curve is the Tang-Toennies potential of Ba2 taken from Ref. [24].40 42 44 46 4840020002004006002Μ amuaunitsofa 0FIG. 2. (Color online) The mass dependence of the scatteringlength for Ca, using the X1+g potential of Ref. [23], showingcalculations with C6 coefficients that give rise to 340 a0 (red closedcircles) and 700 a0 (blue closed squares) for 40Ca. The plotted massis twice the reduced mass and the dashed vertical lines indicate themass of the even isotopes of Ca.length for the full mass interval spanned by the stable evenisotopes. In this way we transfer the knowledge on 40Ca to theother isotopes.The results for Ca are shown in Fig. 2. The plotted massis twice the reduced mass, which for homonuclear collisionssimply is the atomic mass. The dashed vertical lines indicatethe masses of the even isotopes, from which the scatteringlengths for homonuclear collisions are directly read off. Twocalculations are shown, obtained from potentials with C6 coef-ficients that give rise to 340 a0 (red closed circles) and 700 a0(blue closed squares) for 40Ca. The scattering length showsthe expected behavior as a function of mass, with a regularpattern of scattering resonances. They can be understood fromthe mass dependence of the vibrational splitting and, therefore,the number of bound states. A scattering resonance appears atthose mass values at which a new vibrational state becomesbound, i.e., where the least bound vibrational state has zerobinding energy. We find that accidentally all even isotopes arelocated close to such a scattering resonance, giving rise to largepositive or negative scattering lengths.The scattering length intervals for the different isotopesare given in Table II. It is clear that Ca does not providean isotope with a favorable scattering length. At most onecan state that 42Ca has a slightly smaller scattering lengththan 40Ca. In addition, the interisotope scattering lengthsare given for collisions between 40Ca and the other isotopesTABLE II. Scattering lengths of even Ca isotopes (left), as wellas interisotopic scattering lengths for all combinations with 40Ca and42Ca+44Ca (right).a (units of a0) a (units of a0)40 + 40 +340 . . . +700 [28] 40 + 42 +45 . . . +5042 + 42 +320 . . . +640 40 + 44 ±∞44 + 44 +460 . . . +1800 40 + 46 +62 . . . +6846 + 46 ±∞ 40 + 48 −66 . . . −3948 + 48 −230 . . .−120 42 + 44 +47 . . . +52054703-2BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)and 42Ca + 44Ca. Here we find favorable scattering lengthsfor 40Ca + 42Ca, 40Ca + 46Ca, and 42Ca + 44Ca. Taking intoconsideration the large scattering length of 44Ca and 46Ca,the only interesting mixture in view of sympathetic coolingis 40Ca + 42Ca.B. BariumKnowledge of the Ba2 X1+g potential is sparse because of alimited amount of experimental data [30,31], which cover onlya small part of the vibrational spectrum. Similarly, theoreticaldata is scarce. The binding energies of Ba2 were calculatedby Jones [32]. First wave function based quantum chemicalcalculation was reported by Ref. [33] and most recently byRef. [34]. The most accurate calculations of the long-rangecoefficients C6, C8, and C10 are reported by Porsev andDerevianko [35,36].We have used the Tang-Toennies potential [37] of Ref. [24]to model the Ba2 X1+g potential asV (R) = Ae−bR −5∑n=3[1 − e−bR2n∑k=0(bR)kk!]C2nR2n, (2)where the short-range parameters (in a.u.) A = 105.4 andb = 0.9657 are obtained from rescaling an accurate Sr2 X1+gpotential [38], using the measured fundamental vibrationfrequency [31], and taking the long-range coefficients fromRefs. [35,36] (see Fig. 1).Although we do not expect the potential to be accurateenough to predict the scattering length, we can use it toinvestigate its mass dependence. The result is shown in Fig. 3(black closed circles). We observe that the spacing betweenthe resonances is larger than that between the even isotopes,giving rise to a variety of scattering lengths among the differentisotopes. Therefore, it is very probable that at least one130 132 134 136 13840020002004006002Μ amuaunitsofa 0FIG. 3. (Color online) The mass dependence of the scatteringlength for Ba, using the Tang-Toennies model to represent the X1+gpotential with the parameter values from Ref. [24] (black closedcircles). The arrow indicates the effect of the uncertainty in thetheoretical C6 coefficient, showing its lower (blue closed triangles)and upper (red closed squares) limit. The plotted mass is twice thereduced mass and the dashed vertical lines indicate the mass of theeven isotopes of Ba.of the isotopes has a favorable scattering length for BECformation.In addition, we have investigated the effect of the theoreticaluncertainty in the reported C6 coefficient, 5160 ± 74 a.u.[35,36]. We have repeated the above calculation with the upper(red closed squares) and lower (blue closed triangles) limits(see Fig. 3). Note that these adjustments in C6 also requiresmall changes in A and b [24]. The uncertainty in C6 givesrise to a large spread in the scattering length, which can alsobe seen as a large shift along the mass axis, as indicated bythe arrow. This means that even if an accurate short-rangepotential would be available, still no precise prediction ofthe scattering length is possible. On the other hand, whereasthe scattering length itself is extremely sensitive to detailsof the potential, the spacing between the scattering resonancesis not. Therefore, once scattering length information of oneof the isotopes becomes available, the simple Tang-Toenniespotential is sufficient to predict the scattering lengths of allthe other isotopes,1 at least at an accuracy that allows one tochoose the best isotope to achieve BEC.IV. CONCLUSION AND OUTLOOKIn summary, we have investigated the s-wave scatteringlengths of the Ca and Ba even (bosonic) isotopes. Anexperimentally accurate ground-state potential of Ca2 enabledus to calculate the scattering lengths for all the isotopes,taking knowledge of 40Ca scattering length as a reference.We have observed that accidentally scattering resonancesappear near all even isotopes, leading to large positiveand negative scattering lengths for all isotopes. Only 42Cashows a small improvement in terms of scattering lengthcompared to 40Ca. Therefore, alternative methods to thatof evaporative cooling are interesting for Ca, such as di-rect laser cooling into quantum degeneracy using a narrowtransition [39].In addition, scattering lengths for all the even Ba isotopeshave been investigated using a Tang-Toennies model potential.In contrast to the Ca case, here the scattering lengths varystrongly over the isotopes, resulting in a high probabilitythat at least one isotope has a favorable scattering length toproduce a BEC. More accurate knowledge of the ground-stateBa2 potential, including the long-range coefficients, based onexperimental data that covers the full vibrational spectrum isneeded for an improved determination of the scattering lengths.However, once experimental knowledge on the scatteringlength of one isotope becomes available, the scattering lengthsof all the other isotopes can be predicted with the presentmodel potential.ACKNOWLEDGMENTSWe thank Erling Riis and Wim Vassen for careful readingof the manuscript and useful comments. This work has been1We have applied the Tang-Toennies potential model for Ca2 [25]and Sr2 [24], from which we have obtained similar results as inSec. III A and Ref. [15], respectively, when fine-tuning the C6coefficient to reproduce the scattering length of one particular isotope.054703-3BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)performed as part of the research program of the Stichtingvoor Fundamenteel Onderzoek der Materie (FOM) throughprogramme 114 (TRIμP), which is financially supported by theNederlandse Organisatie voor Wetenschappelijk Onderzoek(NWO). S.K. acknowledges financial support from NWO(VIDI grant).[1] K. Burnett, P. S. Julienne, P. D. Lett, E. Tiesinga, and C. J.Williams, Nature (London) 416, 225 (2002).[2] A. D. Ludlow et al., Science 319, 1805 (2008).[3] N. D. Lemke et al., Phys. Rev. Lett. 103, 063001 (2009).[4] A. Derevianko and H. Katori, Rev. Mod. Phys. 83, 331(2011).[5] A. J. Daley, M. M. Boyd, J. Ye, and P. Zoller, Phys. Rev. Lett.101, 170504 (2008).[6] S. Kraft, F. Vogt, O. Appel, F. Riehle, and U. Sterr, Phys. Rev.Lett. 103, 130401 (2009).[7] S. Stellmer, M. K. Tey, B. Huang, R. Grimm, and F. Schreck,Phys. Rev. Lett. 103, 200401 (2009).[8] Y. N. Martinez de Escobar et al., Phys. Rev. Lett. 103, 200402(2009).[9] S. Stellmer, M. K. Tey, R. Grimm, and F. Schreck, Phys. Rev. A82, 041602(R) (2010).[10] P. G. Mickelson, Y. N. Martinez de Escobar, M. Yan,B. J. DeSalvo, and T. C. Killian, Phys. Rev. A 81, 051601(R)(2010).[11] Y. Takasu et al., Phys. Rev. Lett. 91, 040404 (2003).[12] T. Fukuhara, S. Sugawa, and Y. Takahashi, Phys. Rev. A 76,051604(R) (2007).[13] T. Fukuhara, S. Sugawa, Y. Takasu, and Y. Takahashi, Phys. Rev.A 79, 021601(R) (2009).[14] S. Sugawa, R. Yamazaki, S. Taie, and Y. Takahashi, Phys. Rev.A 84, 011610(R) (2011).[15] A. Stein, H. Knöckel, and E. Tiemann, Eur. Phys. J. D 57, 171(2010).[16] T. Zelevinsky, Physics 2, 94 (2009).[17] S. Hoekstra, A. K. Mollema, R. Morgenstern, H. W. Wilschut,and R. Hoekstra, Phys. Rev. A 71, 023409 (2005).[18] U. Dammalapati, I. Norris, C. Burrows, A. S. Arnold, andE. Riis, Phys. Rev. A 81, 023424 (2010).[19] S. De, U. Dammalapati, K. Jungmann, and L. Willmann, Phys.Rev. A 79, 041402(R) (2009).[20] Y. Geng-Hua, Z. Jia-Qi, L. Rung-Bing, W. Jin, andZ. Ming-Sheng, Chin. Phys. Lett. 28, 073201 (2011).[21] J. E. Sansonetti and W. C. Martin, J. Phys. Chem. Ref. Data 34,1559 (2005).[22] M. Przybytek and B. Jeziorski, J. Chem. Phys. 123, 134315(2005).[23] O. Allard, C. Samuelis, A. Pashov, H. Knöckel, and E. Tiemann,Eur. Phys. J. D 26, 155 (2003).[24] P. Li, J. Ren, N. Niu, and K. T. Tang, J. Phys. Chem. A 115,6927 (2011).[25] D. D. Yang, P. Li, and K. T. Tang, J. Phys. Chem. 131, 154301(2009).[26] G. Zinner, T. Binnewies, F. Riehle, and E. Tiemann, Phys. Rev.Lett. 85, 2292 (2000).[27] C. Degenhardt et al., Phys. Rev. A 67, 043408 (2003).[28] F. Vogt et al., Eur. Phys. J. D 44, 73 (2007).[29] O. Allard, A. Pashov, H. Knöckel, and E. Tiemann, Phys. Rev.A 66, 042503 (2002).[30] R. M. Clements and R. F. Barrow, J. Chem. Soc., Faraday Trans.2 81, 625 (1985).[31] M. A. Lebeault, J. Viallon, V. Boutou, and J. Chevaleyre, J. Mol.Spectrosc. 192, 179 (1998).[32] R. O. Jones, J. Chem. Phys. 71, 1300 (1979).[33] A. Allouche, M. Aubert-Frecon, G. Nicolas, and F. Spiegelmann,Chem. Phys. 200, 63 (1995).[34] A. V. Mitin, Russ. J. Phys. Chem. 83, 1160 (2009).[35] S. G. Porsev and A. Derevianko, Phys. Rev. A 65, 020701(R)(2002).[36] S. G. Porsev and A. Derevianko, J. Exp. Theor. Phys. 102, 195(2006).[37] K. T. Tang and J. P. Toennies, J. Chem. Phys. 80, 3726 (1984).[38] A. Stein, H. Knöckel, and E. Tiemann, Phys. Rev. A 78, 042508(2008).[39] C. S. Adams, S. G. Cox, E. Riis, and A. S. Arnold, J. Phys. B36, 1933 (2003).054703-4

University of GroningenScattering lengths of calcium and barium isotopesDammalapati, U.; Willmann, L.; Knoop, S.Published in:Physical Review ADOI:10.1103/PhysRevA.84.054703IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2011Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Dammalapati, U., Willmann, L., & Knoop, S. (2011). Scattering lengths of calcium and barium isotopes.Physical Review A, 84(5), Article 054703. https://doi.org/10.1103/PhysRevA.84.054703CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 30-12-2025PHYSICAL REVIEW A 84, 054703 (2011)Scattering lengths of calcium and barium isotopesU. Dammalapati and L. Willmann*Kernfysisch Versneller Instituut (KVI), University of Groningen, Zernikelaan 25, 9747 AA Groningen, The NetherlandsS. Knoop†LaserLaB Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands(Received 25 October 2011; published 21 November 2011)We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) inorder to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecularpotential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopesalternative to the recently Bose condensed 40Ca that suffers strong losses because of a very large scattering length.For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths,opening the possibility of BEC for at least one of the isotopes.DOI: 10.1103/PhysRevA.84.054703 PACS number(s): 34.20.−b, 34.50.Cx, 67.85.−dI. INTRODUCTIONKnowledge of the s-wave scattering length plays a crucialrole in the achievement of Bose-Einstein condensation (BEC)in ultracold atomic gases [1]. A positive sign of the scatteringlength indicates that a Bose-Einstein condensate is stable,whereas a negative sign indicates an unstable or collapsingcondensate. Also, its magnitude is crucial in the formation ofBEC, as it determines the elastic and inelastic collision rates.For efficient evaporative cooling, a large scattering length isrequired; however, for a too-large scattering length, three-body recombination loss limits the formation and collisionalstability of a BEC. This gives rise to a range of about 50–200 a0 (a0 = 0.5292 nm) that can be considered favorable.Alkaline-earth-metal elements and the alike ytterbium (Yb)system have been the center of attraction in ultracold atomresearch because of their unique atomic structure and theavailability of different stable isotopes with zero nuclear spin(bosons) and with large nuclear spin (fermions). They arebeing used in frequency metrology [2–4] and are proposed forquantum information processing [5]. Recently, BEC has beenachieved in 40Ca [6], 84Sr [7,8], 86Sr [9], and 88Sr [10]. Also,BEC of different Yb isotopes has been reported: 174Yb [11],170Yb [12], 176Yb [13], and 168Yb [14], of which the last onehas an abundance of only 0.13%, highlighting the possibilityof using rare isotopes to achieve BEC.In general, isotopes of an element have different scatteringlengths. Therefore, the ability of BEC formation dependscrucially on the chosen isotope. For instance, the scatteringlength of 84Sr is 123 a0 [15], which made it an ideal candidateto achieve BEC despite its low abundance of only 0.6% [16],as compared to the more abundant 86Sr (10%, 800 a0 [15])and 88Sr (83%, −2 a0 [15]), either suffering strong losses [9]or requiring sympathetic cooling with another isotope [10],respectively. The 40Ca isotope has a large scattering length ofabout 440 a0 [6], limiting the size and stability of BEC.Motivated by the success of BEC in Sr and Yb even isotopesand 40Ca, we carried out calculations to obtain the scattering*l.willmann@rug.nl†s.knoop@vu.nllengths for all the even (bosonic) isotopes of Ca and Ba,for which mass and abundance are summarized in Table I.Laser cooling and trapping of all stable Ca isotopes havebeen reported [17,18]. The demonstration of magneto-opticaltrapping of Ba [19] has opened its use for ultracold collisionstudies, photoassociation spectroscopy, and Bose-Einsteincondensation. Recently, an optical clock based on ultracoldBa has been proposed [20].II. THEORYThe scattering properties of ground-state atoms are ob-tained from the underlying two-body ground-state potentials.For alkaline-earth-metal atoms, there is only one molecularground-state potential, namely the singlet X1+g potential.Furthermore, the lack of nuclear spin and, therefore, hyperfinestructure for the even (bosonic) isotopes highly reduces thenumber of collision channels compared with the alkali metalsystems.To calculate the s-wave scattering length, we solve the 1Dradial Schrödinger equation with zero angular momentum andvanishing kinetic energy,ψ ′′(R) + 2μh̄2[E − V (R)]ψ(R) = 0, (1)where μ is the reduced mass, R the internuclear distance, V (R)the Born-Oppenheimer potential (here the X1+g potential),and E the kinetic energy (below μK). The asymptotic formof the wavefunction is ψ(R) ∝ sin[k(R − a)], where k =√2μE/h̄2, and a is the scattering length. We fit the asymptoticform to the solution of Eq. (1) for large R in order to obtain a.Within the Born-Oppenheimer approximation, the potentialsare identical for all the isotopes of a particular atomic system.Therefore, for a given potential V (R), one only needs to adjustμ to obtain a for all the isotopes.To obtain reliable scattering lengths, accurate potentialsare needed. With the exception of systems with only a fewelectrons, like metastable helium [22], ab initio potentials arein general not accurate enough and constraints from experi-mental data are required. For Ca2, an analytic representation ofthe X1+g potential based on an extensive set of experimentaldata from the Tiemann group is available [23]. For Ba2,054703-11050-2947/2011/84(5)/054703(4) ©2011 American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)TABLE I. Mass and abundance of all stable even isotopes of Caand Ba [21].CalciumIsotope 40 42 44 46 48Mass (amu) 39.963 41.959 43.955 45.954 47.953Abundance (%) 96.941 0.647 2.086 0.004 0.187BariumIsotope 130 132 134 136 138Mass (amu) 129.906 131.905 133.904 135.905 137.905Abundance (%) 0.11 0.10 2.42 7.85 71.70such an accurate potential is not available. Here we rely onthe analytical representation according to the Tang-Toenniespotential model [24], which has been shown to be able toreproduce accurately the X1+g potentials of Ca2 [25] andSr2 [24]. The used Ca2 and Ba2 potentials are shown inFig. 1.III. RESULTSA. CalciumInformation on the Ca2 X1+g potential has been gatheredby several spectroscopic methods, including photoassociationspectroscopy [26–28], Fourier-transform spectroscopy [29],and filtered laser excitation technique [23]. Based on thesedetailed spectroscopic data, an interval of the scattering lengthfor 40Ca is determined to be 340–700 a0 [28]. In the 40Ca BECexperiment, a scattering length of a ≈ 440 a0 was estimatedfrom a measurement of the chemical potential [6], withoutquoting an uncertainty. Therefore, we take the scattering lengthrange of 340–700 a0 as the starting point of our calculations.We have taken the analytical representation and parametervalues of the Ca2 X1+g potential from Ref. [23] (see Fig. 1).We allow the C6 long-range coefficient to vary within afactor 0.991 to 1.003 from its reported value [23], in order toreproduce the scattering length interval for 40Ca. We then takethe mass as a variable parameter and calculate the scattering5 10 15 20 2540302010010R (unitsof a )0VhTHzCa2 Ba21S+1SFIG. 1. (Color online) The X1+g potentials of Ca2 and Ba2 as afunction of internuclear distance. The solid blue curve is the potentialof Ca2 taken from Ref. [23]. The dashed red curve is the Tang-Toennies potential of Ba2 taken from Ref. [24].40 42 44 46 4840020002004006002Μ amuaunitsofa 0FIG. 2. (Color online) The mass dependence of the scatteringlength for Ca, using the X1+g potential of Ref. [23], showingcalculations with C6 coefficients that give rise to 340 a0 (red closedcircles) and 700 a0 (blue closed squares) for 40Ca. The plotted massis twice the reduced mass and the dashed vertical lines indicate themass of the even isotopes of Ca.length for the full mass interval spanned by the stable evenisotopes. In this way we transfer the knowledge on 40Ca to theother isotopes.The results for Ca are shown in Fig. 2. The plotted massis twice the reduced mass, which for homonuclear collisionssimply is the atomic mass. The dashed vertical lines indicatethe masses of the even isotopes, from which the scatteringlengths for homonuclear collisions are directly read off. Twocalculations are shown, obtained from potentials with C6 coef-ficients that give rise to 340 a0 (red closed circles) and 700 a0(blue closed squares) for 40Ca. The scattering length showsthe expected behavior as a function of mass, with a regularpattern of scattering resonances. They can be understood fromthe mass dependence of the vibrational splitting and, therefore,the number of bound states. A scattering resonance appears atthose mass values at which a new vibrational state becomesbound, i.e., where the least bound vibrational state has zerobinding energy. We find that accidentally all even isotopes arelocated close to such a scattering resonance, giving rise to largepositive or negative scattering lengths.The scattering length intervals for the different isotopesare given in Table II. It is clear that Ca does not providean isotope with a favorable scattering length. At most onecan state that 42Ca has a slightly smaller scattering lengththan 40Ca. In addition, the interisotope scattering lengthsare given for collisions between 40Ca and the other isotopesTABLE II. Scattering lengths of even Ca isotopes (left), as wellas interisotopic scattering lengths for all combinations with 40Ca and42Ca+44Ca (right).a (units of a0) a (units of a0)40 + 40 +340 . . . +700 [28] 40 + 42 +45 . . . +5042 + 42 +320 . . . +640 40 + 44 ±∞44 + 44 +460 . . . +1800 40 + 46 +62 . . . +6846 + 46 ±∞ 40 + 48 −66 . . . −3948 + 48 −230 . . .−120 42 + 44 +47 . . . +52054703-2BRIEF REPORTS PHYSICAL REVIEW A 84, 054703 (2011)and 42Ca + 44Ca. Here we find favorable scattering lengthsfor 40Ca + 42Ca, 40Ca + 46Ca, and 42Ca + 44Ca. Taking intoconsideration the large scattering length of 44Ca and 46Ca,the only interesting mixture in view of sympathetic coolingis 40Ca + 42Ca.B. BariumKnowledge of the Ba2 X1+g potential is sparse because of alimited amount of experimental data [30,31], which cover onlya small part of the vibrational spectrum. Similarly, theoreticaldata is scarce. The binding energies of Ba2 were calculatedby Jones [32]. First wave function based quantum chemicalcalculation was reported by Ref. [33] and most recently byRef. [34]. The most accurate calculations of the long-rangecoefficients C6, C8, and C10 are reported by Porsev andDerevianko [35,36].We have used the Tang-Toennies potential [37] of Ref. [24]to model the Ba2 X1+g potential asV (R) = Ae−bR −5∑n=3[1 − e−bR2n∑k=0(bR)kk!]C2nR2n, (2)where the short-range parameters (in a.u.) A = 105.4 andb = 0.9657 are obtained from rescaling an accurate Sr2 X1+gpotential [38], using the measured fundamental vibrationfrequency [31], and taking the long-range coefficients fromRefs. [35,36] (see Fig. 1).Although we do not expect the potential to be accurateenough to predict the scattering length, we can use it toinvestigate its mass dependence. The result is shown in Fig. 3(black closed circles). We observe that the spacing betweenthe resonances is larger than that between the even isotopes,giving rise to a variety of scattering lengths among the differentisotopes. Therefore, it is very probable that at least one130 132 134 136 13840020002004006002Μ amuaunitsofa 0FIG. 3. (Color online) The mass dependence of the scatteringlength for Ba, using the Tang-Toennies model to represent the X1+gpotential with the parameter values from Ref. [24] (black closedcircles). The arrow indicates the effect of the uncertainty in thetheoretical C6 coefficient, showing its lower (blue closed triangles)and upper (red closed squares) limit. The plotted mass is twice thereduced mass and the dashed vertical lines indicate the mass of theeven isotopes of Ba.of the isotopes has a favorable scattering length for BECformation.In addition, we have investigated the effect of the theoreticaluncertainty in the reported C6 coefficient, 5160 ± 74 a.u.[35,36]. We have repeated the above calculation with the upper(red closed squares) and lower (blue closed triangles) limits(see Fig. 3). Note that these adjustments in C6 also requiresmall changes in A and b [24]. The uncertainty in C6 givesrise to a large spread in the scattering length, which can alsobe seen as a large shift along the mass axis, as indicated bythe arrow. This means that even if an accurate short-rangepotential would be available, still no precise prediction ofthe scattering length is possible. On the other hand, whereasthe scattering length itself is extremely sensitive to detailsof the potential, the spacing between the scattering resonancesis not. Therefore, once scattering length information of oneof the isotopes becomes available, the simple Tang-Toenniespotential is sufficient to predict the scattering lengths of allthe other isotopes,1 at least at an accuracy that allows one tochoose the best isotope to achieve BEC.IV. CONCLUSION AND OUTLOOKIn summary, we have investigated the s-wave scatteringlengths of the Ca and Ba even (bosonic) isotopes. Anexperimentally accurate ground-state potential of Ca2 enabledus to calculate the scattering lengths for all the isotopes,taking knowledge of 40Ca scattering length as a reference.We have observed that accidentally scattering resonancesappear near all even isotopes, leading to large positiveand negative scattering lengths for all isotopes. Only 42Cashows a small improvement in terms of scattering lengthcompared to 40Ca. Therefore, alternative methods to thatof evaporative cooling are interesting for Ca, such as di-rect laser cooling into quantum degeneracy using a narrowtransition [39].In addition, scattering lengths for all the even Ba isotopeshave been investigated using a Tang-Toennies model potential.In contrast to the Ca case, here the scattering lengths varystrongly over the isotopes, resulting in a high probabilitythat at least one isotope has a favorable scattering length toproduce a BEC. More accurate knowledge of the ground-stateBa2 potential, including the long-range coefficients, based onexperimental data that covers the full vibrational spectrum isneeded for an improved determination of the scattering lengths.However, once experimental knowledge on the scatteringlength of one isotope becomes available, the scattering lengthsof all the other isotopes can be predicted with the presentmodel potential.ACKNOWLEDGMENTSWe thank Erling Riis and Wim Vassen for careful readingof the manuscript and useful comments. This work has been1We have applied the Tang-Toennies potential model for Ca2 [25]and Sr2 [24], from which we have obtained similar results as inSec. III A and Ref. 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We have calculated the s-wave scattering length of all the even isotopes of calcium (Ca) and barium (Ba) in order to investigate the prospect of Bose-Einstein condensation (BEC). For Ca we have used an accurate molecular potential based on detailed spectroscopic data. Our calculations show that Ca does not provide other isotopes alternative to the recently Bose condensed (40)Ca that suffers strong losses because of a very large scattering length. For Ba we show by using a model potential that the even isotopes cover a broad range of scattering lengths, opening the possibility of BEC for at least one of the isotopes

https://doi.org/10.1103/PhysRevA.84.054703

Scattering lengths of calcium and barium isotopes

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