Theoretical problems involving antihydrogen

Antihydrogen () contains a positron and an antiproton and is the simplest atom that is made up of antimatter. Two large experimental groups, involving collaborators in several countries, have been working hard for several years on the construction at CERN of the experimental apparatus needed to trap and store at very low temperatures. The projects that they are working on are called ATHENA and ATRAP. For a review of the background to research on, see Charlton et al., Physics Reports 241, 65 (1994), Holzscheiter and Charlton, Rep. Prog. Phys. 62, 1 (1999) and Gabrielse, Adv. At. Mol. Phys. 45, 1 (2001).
In the summer of 2002, the experimentalists working on the ATHENA project were successful in preparing about 50,000 atoms at a temperature of about 15 K. See Nature 419, 456 (2002). was subsequently prepared by the ATRAP project. See Phys. Rev. Lett. 89. 213401,233401 (2002). This is a very exciting development. It should lead on to experiments to study the properties of . Of particular interest are tests of the CPT invariance of quantum field theory and also of Einstein's principle of equivalence, which is the cornerstone of the general theory of relativity. To be able to carry out these experiments, it will be necessary to trap and store at temperatures below 1 K. The first work we carried out on was the calculation of an upper bound to the critical distance below which a proton and an antiproton are unable to bind an electron and a positron and they separate off to form positronium, J. Phys. B 31, L679 (1998). This is of interest as the region in which the light particles are unbound plays an important role in rearrangement processes in which protonium, a hydrogen-like atom made up of a proton and an antiproton, and positronium are formed. This led on to scattering calculations. The Kohn variational method was used to carry out a detailed calculation of the scattering of by an H atom at very low temperatures, J. Phys. B 35, L489 (2002). Much work remains to be done on this scattering process. Yimin Liu is a PDRA supported by EPSRC. He is currently working on the scattering of by helium (He) at very low temperatures. This process is of interest for two reasons. Firstly, experimentalists on the ATHENA project are very keen to know if collisions with ultracold He could be used to cool to the very low temperatures required for trapping. Secondly, He is likely to be one of the main impurities in the ATHENA trap. Detailed information about the effect on of scattering by He would make it possible to choose the conditions in the trap so as to maximise the lifetime of the. The He- system contains three light particles, two electrons and a positron. To carry out very accurate calculations on this system we need to include Hylleraas-type functions in the basis sets in our variational calculations. These functions contain the distance between two of the light particles as a linear factor. To include these functions, it has been necessary to devise ways of calculating integrals involving up to three distances between light particles, two of them linear factors, the other to the power -1. Integrals involving one and two such factors were required for Kohn calculations carried out previously of elastic scattering of a positron by a hydrogen molecule (H₂), J. Phys. B 23, 3057 (1990). A method has been devised to calculate the additional integrals involving r_ijr_ik/r_jk, where r_ij is the distance between particles i and j, required in He- calculations. These are nine-dimensional integrals. In this method, integration is carried out analytically over seven dimensions, using prolate spheroidal coordinates, so that numerical integration is only required for the two remaining dimensions. This has made possible the evaluation of a Born-Oppenheimer potential for He- from which elastic scattering cross sections can be calculated. In positron scattering the positron may annihilate with a target electron to form two or three gamma rays. The annihilation is brought about by the electromagnetic interaction and in most scattering processes it has only a small effect. In scattering there is the additional possibility that the antiproton may annihilate when it comes in contact with a nucleus. This process is brought about by the strong nuclear.interaction and is much more probable than positron annihilation. Taking it into account is interesting. The probability of antiproton annihilation may be sufficiently high that He is unsuitable for cooling . The next stage will be to calculate cross sections for rearrangement processes that give rise to, for example, antiprotonic He + positronium or a hydrogen-like ion made up of a He nucleus and an antiproton + positronium minus, a bound state of two electrons and a positron. These will be calculated by evaluating the T-matrix elements corresponding to these processes, using the Born-Oppenheimer elastic scattering wave function as an approximation to the exact scattering wave function. It is hoped eventually to carry out these evaluations using the Kohn method. It seems that likely that H₂ would be more suitable for cooling , as the nuclei have charge +1 as opposed to + 2, as in the case of He, and are not located at the nuclear centre of mass. This makes it likely that the probability that the antiproton in will annihilate is lower in H₂ scattering than in He scattering. Also, H₂ is the likely to be the other main impurity in the trap, besides He. Thus, scattering by H₂ is a very interesting process. The H₂- system contains three light particles, as in the case of He-. However, H₂ contains two nuclei which is a serious complication. As a first step, we intend to use an adapted quantum chemistry code to calculate a potential energy surface for H₂- and use it to calculate cross sections for elastic scattering. We will also calculate a cross section for antiproton annihilation. This will give us some indication of the suitability of H₂ for cooling . We are keeping in contact with Professor Charlton and his group at Swansea, one of the main groups working on the ATHENA project. We are interested in any of the problems they experience for which a theoretical treatment would be helpful.