6 ready the direct term (first term beyond the sudden ap- proximation) is by a factor η 2 smaller than the sudden approximation. Close to the end point of tritium β decay one finds for the Sommerfeld parameter η ≈ − 0 . 0271 and for T$_{2}$ it was explicitly shown that the first-order correc- tion to the sudden approximation is itself of the order of η 2 and thus of the order of 0.01 % [11], in accordance with corresponding system-independent sum rules given in [21]. 100 1000 Energy [eV] 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Probability Density[eV -1 ] 50 100 0 0.004 FIG. 2: Final-state continuum probability density of He after β decay of T − (solid) and a model tail with Z e = 1 . 3074 (dashed). The inset shows the probability density on a linear scale. IV. CONCLUSION Acknowledgments In this work the complete final-state probability distri- bution of He following the nuclear β − decay of tritium anions has been calculated. For the small number of bound states considered previously in [17] the agreement is very good for the dominant ground and first excited states, but especially for the (very weakly populated) n = 3 state a deviation by about an order of magnitude was found. It is preliminarily attributed to a possible typographical error. The agreement with an earlier rel- ativistic multi-configuration Dirac-Fock calculation [18] is on the other hand very poor. Since such a large size of relativistic effects is not expected, especially not for the light nuclei involved, this deviation is attributed to a possibly too small basis set used in [18]. Nevertheless, the present study may stimulate further theoretical work to clarify the discrepancies to [17] and of both works to the relativistic ones in [18]. With the expectation value of 〈 r − $^{1}$〉 and the ground state energy E$_{0}$ (T $^{−}$) reported by Frolov in [17] the mean ex- citation energy calculated with (15) is E = 27 . 469 eV. Frolov also reported the expectation values for the case of a finite mass of the tritium anion nuclei. Again us- ing (15) for this case one obtains E ′ = 27 . 479 eV. This comparison confirms the quality of the final-state distri- bution obtained in this work and validates furthermore the use of the approximation of an infinitely heavy nu- cleus approximation. Frolov noted in [17] that his calculated bound-state probability appears to imply a continuum contribution of about 30 %. Since this value is about 10 time larger than the continuum probability known for neutral T atoms, he speculated that in fact a large number of decays (about 15 to 20 %) may end up in triplet states of helium, leaving a much smaller fraction in the singlet continuum. The reasoning in [17] is that the more diffuse ground state of T − is not sufficient to explain such a large continuum contribution, as follows from a comparison to results ob- tained for Rydberg states of neutral T atoms. On the other hand, triplet states may be populated by the (vir- tual) interaction with the β electron omitted in the sud- den approximation. This argumentation is, however, er- roneous. As has been discussed in detail in [21], the sum rule for the sudden approximation gives always unity, in- dependently on higher-order corrections to it. This is also (as discussed above) fulfilled by the present calcu- lation which indeed confirms the about 30 % continuum probability indirectly found but rejected in [17]. Finally, the exchange interaction is expected to be much smaller than the direct one (in [30] it was found for atomic tritium to be by a factor η 2 smaller), but al- The authors acknowledge financial support from the Stifterverband f¨ ur die Deutsche Wissenschaft and the Fonds der Chemischen Industrie . In order to further test the accuracy of the present cal- culation, the bound-state energies were compared to very accurate literature data and were found to agree very ac- curately with them. Furthermore, the mean excitation energy obtained from the complete final-state spectrum was compared to the value predicted on the basis of clo- sure. Again, very good agreement was found. There- fore, the results of this work should be reliable and of direct importance for the tritium neutrino-mass experi- ment KATRIN that is presently under construction. In order to allow the use in the experimental analysis and for predicting how much a possible T − admixture to the T$_{2}$ source spoils the extracted neutrino mass, the contin- uum transition probability is given in binned form, but also available numerically on request. Finally, it may be noted that a controlled admixture of T − to the tritium source may in fact be used for the analysis of the experi- mental sensitivity to the atomic and molecular final-state spectrum.