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The European Physical Journal D

Simple determination of Na

A. Crubellier, Olivier Dulieu, Françoise Masnou-Seeuws, M. Elbs, H. Knöckel, E. Tiemann

Published Jan 1, 1999• 52 citations
Publisher

Abstract

\n A simple model is developed to determine a scattering length from the\nexperimental values of the energy of the last, least bound, vibrational\nlevels of the ground state, either $^1\\Sigma _g^{+}$ or $^3\\Sigma _u^{+}$,\nof an alkali dimer. It is based on an extrapolation of the positions of the\nfew outermost nodes of the bound vibrational wave functions towards the\ndissociation threshold. It uses the asymptotic part of the molecular\npotential only. The method is applied to recently measured levels at the\nasymptote (3s+3s) of Na2. We obtain precise values for the two scattering\nlengths respectively associated to the dissociation limits $F_1=F_2=1,F=2$\n(55.1±1.6 a0) and $F_1=F_2=1,F=0$ (50.0±1.6 a0); the first\nvalue corresponds to the $a_{1,-1}$ scattering length usually considered in\ncold collisions. The extrapolation procedure is also applied to Li2,\nusing existing experimental data.\n\n

DOI
10.1007/s100530050303

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\n A simple model is developed to determine a scattering length from the\nexperimental values of the energy of the last, least bound, vibrational\nlevels of the ground state, either 1Sigmag+^1\\Sigma _g^{+}1Sigmag+​ or 3Sigmau+^3\\Sigma _u^{+}3Sigmau+​,\nof an alkali dimer. It is based on an extrapolation of the positions of the\nfew outermost nodes of the bound vibrational wave functions towards the\ndissociation threshold. It uses the asymptotic part of the molecular\npotential only. The method is applied to recently measured levels at the\nasymptote (3s+3s) of Na2. We obtain precise values for the two scattering\nlengths respectively associated to the dissociation limits F1=F2=1,F=2F_1=F_2=1,F=2F1​=F2​=1,F=2\n(55.1±1.6 a0) and F1=F2=1,F=0F_1=F_2=1,F=0F1​=F2​=1,F=0 (50.0±1.6 a0); the first\nvalue corresponds to the a1,−1a_{1,-1}a1,−1​ scattering length usually considered in\ncold collisions. The extrapolation procedure is also applied to Li2,\nusing existing experimental data.\n\n
The scattering lengths that characterize the low-energy elastic scattering of 23 Na atoms by 39 K, 40 K and 41 K atoms, when they interact via the X 1 Σ + and a 3 Σ + states of the NaK dimer, are calculated. The systematic variation of the short- and long-range parts of the interaction potentials reveals that the calculated scattering lengths are sensitive to uncertainties in the potentials. It is concluded that precise spectroscopic measurements of the least-bound ro-vibrational levels supported by the ground electronic state, which would reduce the uncertainties in both the molecular potentials, are needed for an accurate determination of the scattering lengths.
A multistep laser excitation scheme was used to access the last, least bound levels close to the lowest asymptote 3s+3s3s+3s3s+3s with the molecular states $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ and $a{}^{3}{\ensuremath{\Sigma}}_{u}^{+}$ of Na2{\mathrm{Na}}_{2}Na2​. All singlet levels within the last $10 {\mathrm{cm}}^{\ensuremath{-}1}$, and some triplet levels with sufficient singlet admixture, were observed. The level energies were determined with uncertainties of less than 22 MHz with respect to the dissociation limit. The hyperfine structure was resolved, and asymmetric line shapes due to predissociation by tunneling through the rotational barrier or electronic state coupling were recorded for levels above the dissociation limit.
A modified semi-empirical Tang-Toennies potential model is used to describe the a3Σu+ potentials of the alkali dimers. These potentials are currently of interest in connection with the laser manipulation of the ultracold alkali gases. The fully analytical model is based on three experimental parameters, the well depth De, well location Re, and the harmonic vibrational frequency ωe of which the latter is only slightly optimized within the range of the literature values. Comparison with the latest spectroscopic data shows good agreement for Na2, K2, Rb2, and Cs2, comparable to that found with published potential models with up to 55 parameters. The differences between the reduced potential of Li2 and the conformal reduced potentials of the heavier dimers are analyzed together with why the model describes Li2 less accurately. The new model potential provides a test of the principle of corresponding states and an excellent first order approximation for further optimization to improve the fits to the spectroscopic data and describe the scattering lengths and Feshbach resonances at ultra-low temperatures.
We report a preliminary value for the zero magnetic field Na 2 S(f = 1, m = Ϫ 1) + Na 2 S(f = 1, m = Ϫ 1) scattering length, a1,Ϫ1.This parameter describes the lowenergy elastic two-body processes in a dilute gas of composite bosons and determines, to a large extent, the macroscopic wavefunction of a Bose condensate in a trap.Our scattering length is obtained from photoassociative spectroscopy with samples of uncondensed atoms.The temperature of the atoms is sufficiently low that contributions from the three lowest partial waves dominate the spectrum.The observed lineshapes for the purely long-range 0g Ϫ molecular state enable us to establish key features of the ground state scattering wavefunction.The fortuitous occurrence of a p -wave node near the deepest point (Re = 72 a0) of the 0g Ϫ potential curve is instrumental in determining a1,Ϫ1 = (52 Ϯ 5) a0 and a2,2 = (85 Ϯ 3) a0, where the latter is for a collision of two Na 2 S(f = 2, m = 2) atoms.
The Morse/Long Range (MLR) potential has become one of the most reliable and highly \nused potential energy functions for diatomic molecules. It includes the theoretical long range \nbehaviour that diatomic molecules are known to exhibit as they approach the dissociation \nlimit. Heavy alkali metals with adjacent electronic states often exhibit strong coupling \nbetween the spin and orbital angular momentum. The ground state X¹Σg⁺ and the lowest \nlying triplet state aᶟΣᵤ⁺ of Cs₂ exhibit such coupling effects and as a result, modeling the \nhighest vibrational states of these states is a non-trivial problem. Utilizing scattering length \nvalues obtained from published analysis of 60 Feshbach resonances, the correct form of the \npotential energy function was determined. Moreover, the scattering length values were used \nto determine the correct leading dispersion coefficient that describes the true form of the \nlong-range potential energy functions. All previous attempts to determine global potential \nenergy functions for these states have considered only the optical spectroscopic data. This is \nthe first ever effort attempting to use scattering lengths determined from cold atom collision \nexperiments in a combined analysis with conventional spectroscopic data.
The experimental characterization of the electronic states correlated to the asymptote of ground state Na (3S) and Rb (5S) atoms was expanded by spectroscopic data on $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state levels using a high resolution Fourier transform spectroscopy technique. The hyperfine splitting of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state levels was partially resolved and analyzed for both Na85Rb\mathrm{Na}\phantom{\rule{0.2em}{0ex}}^{85}\mathrm{Rb}Na85Rb and Na87Rb\mathrm{Na}\phantom{\rule{0.2em}{0ex}}^{87}\mathrm{Rb}Na87Rb isotopomers. Transitions to high lying levels of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ and $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ states were recorded simultaneously which enables one to determine long range parameters of the molecular potentials. Coupled channels calculations based on the Fourier grid method were finally applied for deriving accurate potential energy curves of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ and $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ states capable of a reliable description of cold collisions between Na and Rb atoms in their ground states. Scattering lengths and Feshbach resonances were calculated for some quantum states.