## The experimental results for the
u _{mathrm{HF}}^{--} and
u _{mathrm{HF}}^{-+} in comparison with three-body QED calculations, where
u _{
m {mathrm{HF}}} denotes the SSHF transition frequencies, δ_{exp} is the relative error of the measured frequencies and Γ the resonance linewidth

**Table 2.** The experimental results for the \nu _{\mathrm{HF}}^{--} and \nu _{\mathrm{HF}}^{-+} in comparison with three-body QED calculations, where \nu _{\rm {\mathrm{HF}}} denotes the SSHF transition frequencies, δ_{exp} is the relative error of the measured frequencies and Γ the resonance linewidth. The relative deviation of experiment and theory is defined as δ_{th–exp} = (ν_{exp} − ν_{th})/ν_{exp}. The quoted theoretical precision is ~5 **×** 10^{−5} from the limitation of the Breit–Pauli approximation that neglects terms of relative order α^{2}. This does not include numerical errors from the different variational methods used. For [11], \Delta \nu _{\mathrm{HF}}^{\pm } was calculated from the difference of the tabulated antiproton spin-flip transitions J^{--+}\longrightarrow J^{---} and J^{+-+}\longrightarrow J^{+--}, resulting in a relative error of 3 **×** 10^{−4}.

**Abstract**

In this work, we describe the latest results for the measurements of the hyperfine structure of antiprotonic ^{3}He. Two out of four measurable super–super-hyperfine (SSHF) transition lines of the (*n*, *L*) = (36, 34) state of antiprotonic ^{3}He were observed. The measured frequencies of the individual transitions are 11.125 48(08) GHz and 11.157 93(13) GHz, with the increased precisions of about 43% and 25%, respectively, compared to our first measurements with antiprotonic ^{3}He (Friedreich *et al* 2011 *Phys. Lett.* B **700** 1–6). They are less than 0.5 MHz higher with respect to the most recent theoretical values, still within their estimated errors. Although the experimental uncertainty for the difference of 0.032 45(15) GHz between these frequencies is large as compared to that of theory, its measured value also agrees with theoretical calculations. The rates for collisions between antiprotonic helium and helium atoms have been assessed through comparison with simulations, resulting in an elastic collision rate of γ_{e} = 3.41 ± 0.62 MHz and an inelastic collision rate of γ_{i} = 0.51 ± 0.07 MHz.