Cold Atom Scattering

This problem is important for Bose-Eisnstein condensation of cold atomic gases.

Neutral atoms interact with each other by means of a so-called van der Waals potential. At distances much greater than the size of the atom this potential is attractive, and decreases as the inverse 6th power of the distance. The strength of this attraction is determined by the magnitude of the van der Waals constant. At smaller distances comparable to the atomic radius the potential becomes repulsive, due to an increasing overlap of the electron orbitals. Such potential typically has a minimum, and supports several bound states of the corresponding diatomic molecule. If the van der Waals constant is large and the atoms are heavy, the number of bound states can be about 100 or more, which means that the interactomic potential is in fact quite strong.

This situation creates a real problem for calculation of the scattering length (SL). If the energy of the upper bound s diatomic state is very small, the SL can be extremely large positive. If the potential is just a bit shallower, the energy of this state becomes zero, and the SL turns into infinity. When the depth of the potential is reduced a bit further the SL becomes very large negative. Thus, the actual magnitude of the SL is very sensitive to the shape of the potential, and one has to know it with very high accuracy to make unambiguous predictions of the sign and magnitude of the SL.

In our work [1] we have derived an analytical formula, which expresses the SL in term of an average SL, which is determined only by the van der Waals constant and the reduced mass of the atomic pair, times the factor, which depends on the actual shape of the potential at smaller distances. The potential influences the SL only through the semiclassical phase calculated at zero energy from the classical turning point to infinity. The average SL is positive, and can be determined with high accuracy. The true SL oscillates around the average value, so that if the uncertainty in the potential and the corresponding semiclassical phase is large, we can still assert that the SL is more likely to be positive (75% probability), than negative (25%). In other words, if one considers various atomic pairs, the SL will turn out to be positive in about 75% cases.

We expect that our theory can be extended to atomic scattering with non-zero angular momenta and non-zero energies. It can probably be generalized to account for the fact that when experiments with cold atoms are preformed in traps, the atoms are usually kept in a certain hyper-fine structure state with a given projection of the total angular momentum.


  1. G. F. Gribakin and V. V. Flambaum. Calculation of the scattering length in atomic collisions using the semiclassical approximation. Phys. Rev. A 48, 546 (1993).
  2. V. V. Flambaum, G. F. Gribakin, and C. Harabati. Analytical calculation of cold atom scattering. Phys. Rev. A 59, 1998-2005 (1999).

Back to Gleb Gribakin's home page