Neutral Density Distribution Determination
3.4 Neutral Distribution Due to Multiple Holes
The DSMC simulations described in Section 3.2 only simulated the neutral density distribution resulting from atoms effusing downstream through an individual aperture. Earlier, the mean free path within the discharge chamber was found to be approximately 20 cm. The density downstream of an aperture is limited by the upstream density, so the mean free path downstream is, at a minimum, the same as in the discharge chamber. Since the mean free path is on the order of the size of the engine grids (30 cm), we expect very few collisions between atoms, and so it is reasonable to assume that the total density downstream of the engine grids can be found by superposition of the density contributions from all grid holes. The neutral density downstream of an NSTAR-shaped grid was computed using superposition of both the cosine distribution and the distribution resulting from a single hole in an infinite plane. The pseudo-code for the method is detailed in Section 3.4.1, and the res ults are discussed in Section 3.4.2.
3.4.1 Pseudo-Code: Calculating the Neutral Density
The following pseudo-code outlines the procedure used to determine the neutral atom density at any point. The density is computed at the desired point by superposition of the contributions from all grid holes. The code was implemented using Matlab.
1. Letxbe the location of the point at which to evaluate the density,N be the total number of holes, andRbe the accel grid hole radius.
2. Letρ0 be the neutral density inside the engine discharge chamber.
3. k= 0.
4. k=k+ 1.
5. Letyk be the coordinates of the center of holek, and ˆnk the normal vector to holek.
6. Compute the vector between holekand the evaluation point,k =x−yk. 7. Computek =k .
8. Computeχk = cos−1
−1k k·nˆk
.
9. If 0≤χk ≤90◦, go to step 10; otherwisen0,k= 0, go to step 11.
10. Calculate the contribution to the density from holek,
n0,k= ρ0f(χk)
2 1− 1
1 + (R/k)2
. (3.12)
11. Ifk < N, go to step 4; otherwise go to step 12.
12. Sum all contributions,n0(x) =
kn0,k. 13. Outputn0(x).
3.4.2 Neutral Density Downstream of an NSTAR Grid
The total neutral density distribution downstream of an NSTAR-shaped grid was modeled using superposition of the contributions from all grid holes. In order to prevent near-infinite contributions at positions very close to the surface of the grid, the distribution contribution from each hole was calculated using the following (see Equations 3.5 and 3.6):
n0(, χ) =ρ0f(χ)
2 1− 1
1 + (R/)2
. (3.13)
At large distances,/R1, Equation 3.13 approaches that of Equation 3.6; at a distance of 0.5 cm (∼10R), the density contributions computed from each relation differ by less than 1%.
The total density was computed using both the distributions resulting from a single aperture in an infinite plane, and from an infinitely thin hole (cosine distribution). The total flux from each hole was normalized by reducing the total flux through the infinitely thin hole, i.e., f(χ) = Ws cosχ.
The procedure for computing the density at any point downstream of the grid is detailed in Section 4.4.1. The resulting density distributions, using both the single aperture and infinitely thin hole, are shown in Figure 3.9.
Comparing the two resulting distributions yields the following conclusions. In the region near the grid axis and up to approximately 13 cm downstream, the expected neutral density is higher using the thin hole than using the single aperture, especially within the first couple of centimeters from the grid surface. Beyond approximately 13 cm in the region near the grid axis, the density drops off more rapidly for the cosine distribution than for the single aperture distribution. Also, at high angles with respect to the grid axis, the density drops off less rapidly using the thin hole than using the single aperture. As we develop the model in Chapter 4, we will find that the high energy CEX ions that reach points inclined at large angles, with respect to the grid axis (greater than∼80◦), predominantly originate from the region near the edge of the main beam. As a result, the differences in the neutral density in this region, due to the different neutral distribution models, could have a significant impact on the predicted sputtering rates.
00
0.15 0.45
0.75 1.05
1.35 1.65 2.25 1.95 10
10 20
20 30
30 Radial Distance (cm)
AxialDistance(cm)
(a) Density distribution resulting from modeling each hole as a single aperture in an infinite plane.
00
0.15
0.45 0.75 1.05 1.35 1.95 1.65 2.55 2.25 10
10 20
20 30
30 Radial Distance (cm)
AxialDistance(cm)
(b) Density distribution resulting from modeling each hole using the cosine distribution.
Figure 3.9: NSTAR neutral density distribution. The density at any individual point is obtained from superposition of the contributions from all grid holes. Each grid hole is modeled to have the density distribution of (a) a single aperture in an infinite plane, and (b) an infinitely thin hole (cosine distribution). In the case of the infinitely thin hole, the total flux through each hole was normalized to be the same as for the single aperture, i.e.,f(χ) =Wscosχ. Contour line quantities are in terms of percentage of the upstream density,ρ0.
In the derivation of the model equations in Chapter 2, we assumed that the CEX ions do not undergo further collisions once they are created from a charge-exchange collision. Now that we have estimates to the neutral density distribution around an NSTAR grid, we are in a position to test whether this assumption is reasonable or not. According to the Beer-Lambert Law, the attenuated flux,F, of a beam passing through some medium (due to scattering collisions) is
F =F0exp
− d
0
n0σ0ds
, (3.14)
whereF0 is the flux of the beam at the starting point,s= 0,dis the path length traversed through the medium,n0 is the density of scattering centers, andσ0 is the total scattering cross-section [23].
In Chapter 4, we will find that the total charge-exchange cross-section for xenon ion/atom collisions is approximately 55 ˚A2for ion energies of 300 eV. Using this cross-section, integrating down the grid axis of both neutral distributions (Figure 3.9) yields an attenuation of less than 1% for a beam of ions travelling from the grid surface to a position 30 cm downstream. Though there are collisions between the CEX ions and the neutral atoms, in light of this small attenuation rate, we feel the assumption made in deriving the model equations is justified. The assumption of no collisions between CEX ions and neutral atoms may not be appropriate for describing the plume of an engine operating in a vacuum chamber, since the background pressure may significantly raise the neutral density and increase the number of scattering centers [37].
Now, having established a method for determining the neutral density at any point downstream of the engine grid, we proceed to develop a method for modeling the remaining quantities in the model equations of Chapter 2.