Ultracold collisions in the system of three helium atoms
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tracold Collisions in the System of Three Helium Atoms E. A. Kolganovaa, b, A. K. Motovilova, and W. Sandhasb a
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia b Physikalisches Institut, Universität Bonn, Bonn, Germany
Abstract—The Faddeev differential equations for a system of three particles with a hard-core interaction are described. Numerical results on the binding energies of the 4He3 and 3He4He2 trimers and on ultracold collisions of 3, 4He atoms with 4He2 dimers obtained with the help of those differential equations are reviewed. The results obtained for the hard-core model using the Faddeev equations are compared with analogous results obtained by alternative methods. PACS numbers: 21.45.-v, 31.15.-p, 34.50.-s DOI: 10.1134/S106377960902004X
1. INTRODUCTION Investigations of the systems of three Helium atoms are of great interest for various branches of physical chemistry and molecular physics. For instance, investigations of 4He dimers and trimers is an important step towards understanding the properties of liquid helium droplets and the superfluidity in 4He films, see, e.g., the papers [1, 2]. Great interest in the properties of clusters containing several 4He atoms is born by the tremendous activity in the research on the Bose-Einstein condensation in ultracold gases, see, e.g., the papers [3, 4]. Recall that Helium, as a chemical element, has only two stable isotopes 4He and 3He. The 4He atoms are bosons, while the 3He atoms are fermions. Because of that, ultracold systems of many 4He and 3He particles exhibit entirely different physical properties. For ultra low temperatures, which can be obtained by the modern technique of laser cooling, Helium 3He behaves as an ordinary liquid. At the same time, liquid Helium 4He transits into a macroscopically ordered superfluid state, see the paper [5]. Because of a small mass for the Helium atoms, and a relative weakness of the attractive component in the interaction of these atoms, under normal pressure Helium remains in a liquid state even at the temperature equal to absolute zero, see [6]. It is well known that a weak van der Waals interaction between neutral atoms and molecules arises due to the interaction between constant and/or induced electric moments (dipole, quadrupole, and so on). This was pointed out in the beginning of the 1920s, several years before the creation of quantum mechanics, in pioneering works by Debye [7] and Keesom [8, 9]. In 1924 Jones [10] established that in order for the second virial coefficient to be finite, it is necessary for the interaction
forces between the molecules in a rarefied gas to decay at large distances at least as r–4 or faster. As the simplest model for such interactions he proposed the following potential λ λ V ( r ) = – -----mm- + ----n-n , r r
λ m, λ n ≥ 0,
(1.1)
where m and n are integer numbers greater than or equal to 3. Since then, the model (1.1) found numerous applications and became widely known under the name Lennard-Jones potential.1 The formats “(m –
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