Collisional properties of trapped cold chromium atoms Zoran Pavlovi´c1, Bj¨orn O. Roos2, Robin Cˆot´e1, and H. R. Sadeghpour3 1Physics Department, University of Connecticut, 2152 Hillside Rd., Storrs, CT 06269-3046 2Department of Theoretical Chemistry, Chemical Center, P.O.B. 124 S-221 00 Lund, Sweden and 3ITAMP, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138. (Dated: February 12, 2018) We report on calculations of the elastic cross section and thermalization rate for collision between two maximally spin-polarized chromium atoms in the cold and ultracold regimes, relevant to buffer- gas and magneto-optical cooling of chromium atoms. We calculate ab initio potential energy curves for Cr2 and the van der Waals coefficient C6, and construct interaction potentials between two colliding Cr atoms. We explore the effect of shape resonances on elastic cross section, and find that they dramatically affect the thermalization rate. Our calculated value for the s-wave scattering length is compared in magnitude with a recent measurement at ultracold temperatures. Collisions of atoms at ultracold temperatures have re- ergy curves dissociating to two ground Cr atoms were ceived considerable attention because of their importance obtained. The van der Waals interaction coefficient, in cooling and trapping of atoms [1] and their role in high C6, was obtained semi-empirically from available bound precision spectroscopy [2] and Bose-Einstein condensa- atomic transition matrix elements and photoionization tion [3]. Processes occurring in the cold regime are sen- cross sections to be C6 = 745 ± 55 a.u., where 1 a.u. = sitive to the details of the interaction potentials between 9.57 × 10−80 Jm6. The elastic cross section and colli- the colliding systems over an extended range of inter- sion rate coefficient were calculated using the newly con- nuclear separations. Recent experiments with chromium structed potential curves and compared against two re- [4, 5, 6, 7] emphasize the need for theoretical studies of cent measurements of elastic rate coefficients [5, 7]. We Cr scattering properties. The interest in cooling Cr stems also investigated the bound and resonance structure in 7 13 + from its particular properties; in its ground state S3, it the Σg potential energy curve and determined the res- possesses a very large magnetic moment, 6 µB (µB: Bohr onance positions and widths as a function of rotational magneton), making it an ideal atom for buffer-cooling in angular momenta. a purely magnetic trap [7], as well as for magneto-optical The potential curves for Cr2, shown in Fig. 1, were trapping [4]. In addition, anisotropic long-range interac- constructed from three regions joined smoothly together. tions, such as chromium’s magnetic dipole-dipole inter- We first computed ab initio potential curves using the actions, may lead to novel phenomena in BECs [9, 10]. CASSCF/CASPT2 method [11, 12]. The CASSCF wave The existence of a stable fermionic isotope, 53Cr, opens function is formed by distributing 12 electrons in the the possibility of obtaining fermionic degenerate gas us- 3d and 4s derived active orbitals while keeping the in- ing sympathetic cooling. Chromium was also used in a active 1s, 2s, 2p, 3s, and 3p derived orbitals occupied. new cooling scheme [5], where ultracold chromium atoms The remaining dynamical electron correlation energy were loaded from a MOT into an Ioffe-Pritchard mag- is obtained through second order perturbation theory netic trap, and cooled below 100µK. However, a not (CASPT2). The basis set used in the calculations is of so-desirable byproduct of large-spin collision is inelastic the atomic natural orbital (ANO-RCC) type contracted “bad” scattering rates that deplete the trap. to 9s8p7d5f3g. This basis set is relativistic and includes On the theoretical front, the electronic spectrum of functions for correlating the 3s and 3p electrons [13]. The the Cr2 dimer poses a considerable numerical challenge. Douglas-Kroll Hamiltonian was used with Fock-type cor- Chromium is the first atom in the periodic table with a rection 0.5 ∗ g1, see [14]. The full counterpoise method half-filled d shell (the ground electronic configuration is was used to correct energies for the basis set superpo- 5 7 −10 Cr(3d 4s, S)) and the Cr2 dimer is one of the most ex- sition error (BSSE). Convergence to 10 , in hartrees, treme cases of multiple metal-metal bonding. To date, was achieved, and numerical accuracy in computed bind- − the best attempt to calculate its interaction potential ing energies was about 10 8. For separations R ≤ Rλ, where Rλ is the smallest sepa- arXiv:physics/0309076v1 [physics.atom-ph] 16 Sep 2003 curves is a multiconfiguration second-order perturbation 1 1 theory with complete active space self-consistent field ration of the ab initio data for the potential energy Vλ(R), (CASSCF/CASPT2) [11, 12]. While some information each curve was joined smoothly to the exponential form 1 + on the spectroscopy of the ground electronic state ( Σg Vλ(R) = cλ exp(−bλR), with the coefficients cλ and bλ molecular symmetry) exists, there is no data available for determined by matching both the potential curve and its λ the interaction of two maximally spin-stretched Cr atoms first derivative continuously at R1 . At large values of R, 13 + ab initio in the Σg molecular symmetry, i.e. total spin, S = 6. the data were matched to the asymptotic form In this communication, we explore the collisional prop- C6 ν −βR erties of Cr atoms in the cold and ultracold tempera- Vλ(R)= − + AλR e , R6 ture regimes by revisiting the electronic structure of the dimer. More accurate Born-Oppenheimer potential en- where C6 is the van der Waals coefficient [15], and the 2 where α is the fine-structure constant, a0 is the Bohr 13Σ+ 0 g 7Σ+ radius, and σph is the photonization cross section. The u 11Σ+ dimensionless df/dǫ is obtained by multiplying Eq.(3) by u 5Σ+ |E | (also in a.u.). g 0 9Σ+ -0.02 g We assess the quality of the semi-empirical computa- tion of the van der Waals coefficient by satisfying two 3Σ+ u sum rules, namely, the zero-th and the inverse second -0.04 0 V (a.u.) moments, 1 + Σ -0.001 13Σ+ g g -0.002 S(0) = f0i = N -0.06 i -0.003 X 5 10 15 20 2 2 e ¯h f0i S(−2) = α0 = 2 2 + G(1) , (4) m|E0| v -0.08 ( i i ) 2 3 4 5 6 7 8 9 10 X R (a.u.) where N is the number of electrons (24 for Cr), and α0 is the static polarizability. We obtained i f0i = 22.3 and FIG. 1: Potential curves for the ground state manifold of Cr2 α0 = 85.00 a.u., in agreement with a recommended value computed with the CASSCF/CASPT2 method. The max- P 13 + 82±20% a.u. [21]). The resulting dispersion coefficient is imally spin-streched electronic state, Σg , is shown in red expected to have an accuracy of about 7%, C = 745±55 and in detail in the inset. This state supports 30 vibrational 6 levels. a.u. Using these potential curves, we computed the elastic λ σel. cross section for the maximally spin-aligned molecu- 13 + parameters of the exchange energy are determined ac- lar state, i. e. the Σg state. [Results for other molec- 7 cording to Smirnov and Chibisov [16]: ν = 2I − 1 and ular states and also inelastic and spin relaxation cross β = 2I, where I is the ionization energy of the atom sections will be provided in a future publication.] The 52 (I =0.248664314 a.u. for Cr). The parameters Aλ were elastic cross section for the collision of two Cr atoms, found by fitting the ab initio curves at separations where composite bosons, expanded over the rotational quantum the exchange energy was still considerable (e.g. R be- number, l, is [22] 13 + tween 10 and 14 a.u. for Σ ). The C6 coefficient was g 8π calculated using σλ (E)= (2l + 1) sin2 δλ , (5) el. k2 l l even 2 2 2 X 3 e ¯h 1 f0if0j C6 = 3 2 2 2 m |E | vivj (vi + vj ) where E =¯h k /2µ is the kinetic energy of relative mo- 0 i,j λ X tion, µ is the reduced mass, and δl (k) is the l-th scatter- ∞ f G(1 + v ) (df/dǫ)G(2 + ǫ)dǫ ing phase shift in electronic state λ. In the low-energy +2 0i i + , (1) v (1 + ǫ) limit, the elastic cross section behaves as i i Z0 ) X 2 λ E→0 8πaλ where contributions of bound-bound, bound-free, and σel.(E) −→ , (6) 2 2 1 2 2 free-free transitions are given by the first, second, and k aλ + 1 − 2 rλaλk third term, respectively. This expression is derived from where scattering length aλ is determined at the zero- the London’s [17] formula, assuming vi =1 − Ei/E0 and 1 λ th energy s-wave limit, aλ = − limk→0 tan δ (k), while ǫ = −E/E0, where E0, Ei, and E are the ground, the i k 0 the effective range rλ can be found by fitting the cross excited state and continuum energies, respectively, f0i are the oscillator strengths pertaining to transitions to section, from an integral expression [22], or using quan- tum defect theory[23]. the ground states, and df0E accounts for the continuous The cross section as a function of the collision energy spectrum. The auxiliary function G(z) is given by 13 + ∞ is shown in Fig. 2 for three different Σg interaction po- (df/dǫ)dǫ tentials; each constructed by matching at large distances G(z)= .