Final-State Spectrum of $^{3}$He after β − Decay of Tritium Anions T − Alexander Stark and Alejandro Saenz ∗ AG Moderne Optik, Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin, Newtonstr. 15, D – 12 489 Berlin, Germany (Dated: November 14, 2018) The final-state spectrum of β decaying tritium anions T − was calculated. The wavefunctions de- scribing the initial T − ground state and the final $^{3}$He states were obtained by the full configuration- interaction method. The transition probability was calculated within the sudden approximation. The transition probability into the electronic continuum is extracted from the complex-scaled resol- vent and is shown to converge for very high-energies to an approximate analytical model probability distribution. PACS numbers: 31.15.-p, 23.40.Bw, 14.60.Pq spectrum was further adapted to specific needs (isotope distribution and temperature) of the KATRIN experi- ment [15]. The neutrino rest mass is a very important parameter for cosmology, astrophysics, and the standard model of elementary particles. The existence of neutrinos, already postulated by Pauli and put into a mathematical frame- work of β decay by Fermi [1] long time ago, was verified by Reines and Cowan in 1956 [2]. However, despite the high solar neutrinos flux of about billions per m − $^{2}$s − 1 on earth the answer to the question about their rest mass is one of the big unknowns in physics. Since neutrino-flavor oscillations have been observed in the late nineties at the Super-Kamiokande experiment [3] a non-vanishing neu- trino rest mass has to be expected. Unfortunately, this type of experiments reveals only mass differences between neutrino flavors. Although a high purity of the molecular tritium source is expected for KATRIN, the produced β electrons can in- teract with other gas molecules and thus produce tritium species different from T$_{2}$. One of the expected processes is the dissociative attachment where T − formation occurs. Despite the relative small cross section compared to, e. g., the one for vibrational excitation of the T$_{2}$ molecule, this process is very impor- tant. The reason is the higher endpoint energy of the β spectrum for the decay of T − compared to the one of T$_{2}$. Due to this fact the occurrence of T − ions leads to a systematical error and hence to a possible limitation of the sensitivity of KATRIN, if it is not properly accounted for [16]. The presently constructed KATRIN (Karlsruhe tri- tium neutrino-mass) experiment with an expected sensi- tivity of about 0 . 2 eV (90% C.L.) should have the ability to determine the absolute value for one of the flavors or at least a new upper limit to it [4]. This so called next- generation tritium β -decay experiment is only based on kinematic relations and energy and momentum conser- vation. Thus KATRIN provides a model-independent direct measurement of the antineutrino rest mass m$_{¯}$ ν$_{e}$ (more accurately the mass of the antineutrino in a given mass-flavor mixture mostly attributed to the electronic neutrino). In more detail, m 2 ¯ ν$_{e}$ will be extracted in a fit procedure from the shape of the β spectrum. Besides the precise measurement of the β -electron energy spectrum it is crucial for the mass extraction to know how the β spectrum is modified by the final-state spectrum of the decay product. As in the previous most recent tritium neutrino-mass experiments in Mainz and Troitsk the T$_{2}$ molecule is chosen as tritium source. T$_{2}$ comprises a compromise between experimental accessibility and the- oretical treatability. The final-state spectrum of its decay product $^{3}$HeT + was therefore subject of a number of very detailed calculations [5–13], finally accumulating in the one covering the whole energy regime [14]. Recently, the To the authors’ knowledge, there exist so far only two theoretical predictions for the final-state spectrum fol- lowing β decay of T $^{−}$. However, in [17] only transition probabilities to 4 final states are reported. Furthermore, the results in [17] disagree substantially from the ones given in an earlier work [18] that was, however, also lim- ited to 10 final states. The aim of this work was thus to provide a complete final-state spectrum for the decay process and to shed some light on the disagreeing earlier results. The calculation of the non-relativistic eigenstates of the atomic systems is performed within the approxi- mation of an infinitely heavy mass of the nuclei, i. e. T ≈ $^{∞}$H and $^{3}$He ≈ $^{∞}$He . This is justifiable due to the large mass difference of the nucleus and the elec- trons. The calculation of the final-state spectrum can be performed analytically for neutral T atoms and reveals a II. METHOD AND COMPUTATIONAL DETAILS I. INTRODUCTION arXiv:1001.0120v1 [physics.atom-ph] 31 Dec 2009 ∗ Electronic address: alejandro.saenz@physik.hu-berlin.de T − → $^{3}$He + e − + ν$_{e}$ . (2) e − + T$_{2}$ → T − + T (1)