## 5 Translational Temperature Determination in HAI-Trap
### Spatial Distribution from Photodetachment Tomography  
The difference between the two distributions, shown in Section 5.1, can be further characterized by mapping out the spatial ion distribution along the cylindrical axis of the ion trap via a far-threshold photodetachment tomography. This technique determines the density distribution without relying on a closed cycling transition and precise imaging systems. If the photon energy of the laser is above the electron affinity of the anion, then the process of laser-induced photodetachment, \(\mathrm{OH^{-}+h\
u\to OH+e^{-}}\), causes neutralization of the anion--as a result, it is lost from the trap.  
For a narrow laser beam, this photodetachment loss rate is proportional to the photon flux and the ions' single particle column density at the position of the laser beam. The single particle column density \(n_{\mathrm{col}}\) can then be extracted by measuring the exponential decay rate \(\kappa\) (extracted from the ions' loss behavior following \(N(t)=N_{0}\exp{[-\kappa t]}\)) and the background loss rate \(\Gamma\) in the trap as  
\[n_{\mathrm{col}}(x_{L},y_{L})=\frac{\kappa(x_{L},y_{L})+\Gamma}{\sigma_{pd}P_{ L}/h\
u}, \tag{4}\]  
where \(\sigma_{pd}\) is the absolute photodetachment cross section and \(P_{L}\) the laser power. For \(\mathrm{OH^{-}}\), a free-running laser diode at a wavelength of 660 nm, far above the detachment threshold of 678 nm, was used. The laser light is focused to a beam diameter of 180 um and can be moved parallel to the trap's axis. A measurement along the trap axis is shown in Figure 6. The axial confinement created by the endcaps and the axial shielding plates can be well described by a harmonic potential. That implies a direct proportionality between the variance in position along that axis and the translational temperature of the ions.  
Note that for more accurate modeling of our trap, surface charges were introduced on both PCBs to account for the shift in mean position and width of the ion cloud. More details on this procedure can be found in Section 5.3. In Figure 6, the spatial distribution of the ions'  
Figure 5: Simulated energy distribution of ions thermalized with a 293 K helium buffer gas in the HAI-trap potential landscape. The energy of the ions is characterized via a Tsallis with \(T=313(2)\) K and \(q=1.070(2)\) as well as a Maxwell distribution with \(T=359(3)\) K. It can be seen that the Maxwell fit overestimates the relative amount of low-energy to mid-energy and underestimates the high-energy tails which are then better represented via a Tsallis distribution (as shown in graph inset).  
distribution described previously (in Figure 5) is shown. The simulated ion distribution \(n_{\text{ion}}\) is related to the potential as  
\[n_{\text{ion}}\sim\exp(-\frac{V_{\text{tot}}}{k_{\text{B}}T}) \tag{5}\]  
for a Maxwell distribution and as  
\[n_{\text{ion}}\sim\left(1-\left(1-q\right)\left(\frac{V_{\text{tot}}}{k_{\text{ B}}T}\right)^{\frac{1}{1-q}}\right) \tag{6}\]  
for a Tsallis distribution [38; 39], where \(q\) is the shaping parameter and \(V_{\text{tot}}\) is the sum of the static potentials as well as the effective potential calculated via Equation (1). A significant difference between the Tsallis and the Maxwell distribution can be seen near the peak of the distribution, with the Tsallis model exhibiting a narrower ion distribution. Although higher energy tails are predicted by the Tsallis distribution, the wings show little to no deviations.  
In conclusion, it can be said that in the case of collisions of ions with a buffer gas at room temperature, the deviations from the Maxwell distribution in an octupole trap can be neglected. However, changes in the ions' phase space density arising due to sympathetic cooling methods with different neutral-ion mass ratio might result in distributions with a higher \(q\)-parameter [40; 41; 42]. In this case, it becomes imperative to include a detailed characterization of the Tsallis energy distribution.