## 5 Translational Temperature Determination in HAI-Trap
### Correlation between Ions' TOF Distribution and Their Translational Temperature  
To model the extraction of the ions from the trap onto the detector, the ions are initially placed in the extraction potential of the trap at randomized initial positions and velocities. The velocity vectors are randomly sampled from a Maxwell-Boltzmann distribution at a given temperature. The ion's position, though randomly sampled, is weighted with the simulated trapping potential, and expected ion distribution is calculated as \(\rho_{\text{ion}}\sim\exp(-\frac{V_{\text{He}}}{k_{\text{B}}T})\). It is imperative to account for the simulated potential so as to accurately describe the appearance of pockets, i.e., regions of global maxima in the ion density distribution. Figure 8a shows a cut of the radial ion distribution at the axial local potential minimum (trap center) at 373 K, while Figure 8b shows the same at 30 K. At higher temperatures, the ion distribution is very homogeneous, but the effect of local potential minima becomes particularly pronounced at lower temperatures. The origin of these pockets lies in the symmetry-breaking of the potential, as reported in [32]. Here, the axial and vertical shielding electrodes have a strong influence in the trap center, thus leading to symmetry-breaking in the trap. This results in the two pockets, as the potential landscape is strongly dominated by the two pairs of shielding electrodes. The depth of the pockets is calculated from the energy of ions \(E\) via the relation \(T=\frac{2E}{k_{\text{B}}}\) to \(\sim\)60 K. As previously mentioned, the initial positions and velocities of the ions are randomly sampled, prior to  
Figure 7: Characterization of surface charge accumulation on PCBs. The surface charge distribution is defined such that the tomography measurements for 80 V endcap voltage matches the simulated ion distribution at a temperature of 373 K. Once the charges are fixed, the modified potential is used to obtain the ion distribution for a configuration with 40 V endcap voltage, which matches the measured ion density distribution for the chosen endcap voltage. the extraction from the trap. The arrival time of the ions is then binned and subsequently represented by a Gaussian fit. The expectation for the relationship between the variance of the ions' TOF distribution and their temperature is a linear function. This assumption is based on the following considerations: Firstly, the TOF spread is primarily dominated by the velocity vectors along the main axis, pointing towards or opposite to the detector. The TOF spectrometer was adjusted to create a symmetric ion distribution via the axial shielding plates, as described in Section 3. The axial velocity spread is described by a Gaussian distribution with \(\sigma_{v}^{2}\propto k_{\mathrm{B}}T\). Secondly, the axial spatial distribution is governed by the harmonic potential defined via a Gaussian distribution, where \(\sigma_{s}^{2}\propto k_{\mathrm{B}}T\) holds valid again. Lastly, the extraction potential is well-approximated via sequential regions of linear acceleration. Thus, the axial velocity distribution of the ions can be linearly mapped onto the ions' time of flight to the detector. In agreement with this expected behavior, a linear relationship between the variance, i.e., \(\sigma^{2}\), of ions' TOF distribution and their translational temperature is seen in Figure 9.  
It is important to note that for temperatures lower than 60 K, a deviation from the initial linear slope is seen. This can be attributed to the aforementioned occurrence of pockets in this temperature regime. The ions get trapped locally in these pockets which exhibit a more quadrupole-like character. During the extraction, this causes the radial velocity component of the ions pointing radially outwards to rapidly turn around, thus decreasing the time spent in the trap and hence, the TOF spread.