# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Relative Doppler and Density Profiles across the Grid Face
Rough estimates for source and background densities are given in Table 2. Accel grid current due to CEX ions created from background neutrals was estimated from the relation  
\[I_{a}(bkgnd)=n_{bd}\sigma_{cxx}I_{d}I_{b}, \tag{3}\]  
where \(\sigma_{cxx}\) is the CEX cross section for Xe+-Xe [10], \(I_{d}\) is the effective distance over which charge exchange ions are collected, \(I_{b}\) is thruster beam current, and \(n_{bd}\) is the density of background xenon neutrals in the region where CEX ions are created and collected. This region can be considered to extend from a downstream surface outside the extraction grid system up to the plasma sheath near the screen grid. However, we ignore the intergrid region, as its contribution is relatively minor and stops at the downstream accel face. Each  
Figure 12: Relative molybdenum density profiles for high and low utilization DTL28. of the parameters is well known except for \(l_{d}\) and possibly \(n_{bd}\), which might be enhanced by a pileup of backstreaming neutrals at the downstream face of the accel [17, 18, 19]. A similar equation is written for source neutrals:  
\[f(x)=a_{1}\exp\left(-\left(\frac{x-a_{2}}{a_{4}}\right)^{2}\right)+a_{3}, \tag{1}\]0  
where \(n_{su}\) is the intergrid (upstream) source density, and \(n_{sd}\) is source density just downstream from the accel grid. The contribution of intergrid Xe+-Xe scattering to Ia has been neglected. For source flow, the accel current associated with downstream neutrals is expected to be larger than the upstream contribution, but not necessarily dominate the other. The \(l_{d}\) parameter is an unknown for source flow, as it is for background gas, and will vary with thruster operating parameters, such as beam and accel voltage [20]. Identical capture probabilities are assumed for downstream CEX ions, whether derived from source or background flow. The same 100% capture probability is assumed for intergrid ions, which can be a significant overestimate. Equating the ingoing transmitted flux for an accel aperture to the flux just downstream, we write  
\[f(x)=a_{1}\exp\left(-\left(\frac{x-a_{2}}{a_{4}}\right)^{2}\right)+a_{3}, \tag{1}\]1  
where \(v_{su}\) is the intergrid thermal velocity, \(v_{sd-ax}\) is the downstream average axial velocity, and K is the Clausen factor for the aperture. Estimating K as 0.6 and equating the two velocities based on Monte Carlo simulation results [21], we obtain \(n_{su}\cong 7n_{sd}\). Assuming \(l_{d}=15l_{u}\), because the collection length for downstream ions is much longer, we rewrite Equation (4) as  
\[f(x)=a_{1}\exp\left(-\left(\frac{x-a_{2}}{a_{4}}\right)^{2}\right)+a_{3}, \tag{1}\]2  
Summing Equations (3) and (6) and allowing for an additional current source, we obtain  
\[f(x)=a_{1}\exp\left(-\left(\frac{x-a_{2}}{a_{4}}\right)^{2}\right)+a_{3}, \tag{1}\]3  
Here, C accounts for current that remains when \(\eta=100\%\) and the facility has infinite pumping speed. Fixing the value of \(1.5n_{sd}\) at \(1\times 10^{11}\) cm\({}^{-3}\) for DTL28-H-H (compare with density figures in Table 2), scaling \(n_{sd}\) according to \((1-\eta)/\eta\), and applying Equation (7), the data plotted in Figure 8a can be modeled easily. For the model result shown, \(n_{bd}\) is 3.1 \(\times\) 10\({}^{11}\) cm\({}^{-3}\), \(l_{d}\) is 1.70 cm, and C is 1 mA. Increasing C to 2 mA drives \(n_{bd}\) down to 2.6 \(\times\) 10\({}^{11}\) cm\({}^{-3}\), with \(l_{d}\) equal to 1.68. Conversely, reducing C to zero pushes \(n_{bd}\) up to 3.6 \(\times\) 10\({}^{11}\) cm\({}^{-3}\), with \(l_{d}\) equal to 1.68. In all cases the \(\frac{n_{bd}}{n_{sd}}\) ratio is substantially above the estimate of 1.4 obtained from a simple calculation of average densities (see Table 2). It is not unreasonable to expect the background density at the downstream grid face to be elevated above the ion gauge measurement at the wall. A portion of the neutral flux impinging on the accel downstream face may backflow from the beamstop with directional bias and elevated axial velocity [18, 19].