In the case of a molybdenum surface placed 60 cm next to a 30 cm diameter grating, almost 90% of the sputtering is due to the 5% of ions with the highest collision exit energies. To facilitate faster execution of the calculations, simplifications based on the predicted behavior of the CEX ions are implemented.
Nomenclature
F(˜x) Total flux of CEX ions across a spherical surface at the target point S(˜x); non-directed CEX ion flux. F(˜x,n)ˆ˜ Total ion flux CEX through projected area δA˜ at target point S(˜x); Directional CEX ion flow.
Introduction
- The Ion Engine
- Modeling the CEX Process
- Limitations to Ion Engine Grid Design
- Thesis Overview
In this chapter we give a brief description of the operation of an ion engine (Section 1.1). Once ionized as a result of the charge exchange process, the CEX ions are subjected to the electric field downstream of the engine.
Mathematical Formulation of the General Problem
Derivation of the Objective Function: Sputtering Rate
- Energy- and Angle-Dependent Rate
- Angle-Independent Sputtering Rate
Jeans [19] and Chapman and Cowling [20] provide thorough discussions of the velocity distribution function, and Guggenheim [21] provides an excellent explanation of the differential cross section. By the definition of the velocity distribution function [20], fp(u), the number of primary ions classified within a volume δV is located at P(x).
Description of an Optimization Problem
This is similar to the statement that any small step in any permissible direction away from the solution point results in an increase in the value of the objective function. The parameters ξ chosen to determine the value of the sputtered flow Υ will determine how sophisticated the model will need to be.
Neutral Density Distribution Determination
- Rarefied Flow through Holes of Varying Depth
- DSMC Calculation of Neutral Distribution for an NSTAR Two-Hole ApertureTwo-Hole Aperture
- Single Aperture in an Infinite Grid
- Single Aperture with a Pseudo-Periodic Hole Pattern
- NSTAR Aperture Density Function and Clausing Factor
- Neutral Distribution Due to Multiple Holes
- Pseudo-Code: Calculating the Neutral Density
- Neutral Density Downstream of an NSTAR Grid
The density for different radii, as a function of angleχmeasured from the axis of rotation, is shown in Figure 3.4 in non-dimensional units. The procedure for calculating the density at any point downstream of the electricity grid is detailed in section 4.4.1.
Computation of CEX-Ion Flux and Sputtering Rate
Obtaining the Electrostatic Potential
The contribution to the density and energy distribution atS from ions scattering at θ∗ will be different from the contribution from those scattering at θ. First, the number of ions scattered in a solid angle δΩ will be different for each angle due to the different values of the differential cross section dσ/dΩ.
The Charge Density: Computing the Electric Potential
- Main Beam Ions
- CEX Ions
- Pseudo-Code: Calculating the Electric Potential
Due to the logarithmic dependence of the potential on the ion density (Equation 4.4), the small number of CEX ions (compared to the number of primary ions) makes only a small correction to the potential within the main beam. According to equation 4.4, the electric field is proportional to the gradient of the logarithm of the ion density. Because the majority of CEX ions move slowly, it is reasonable to assume that large gradients in CEX ion density (especially compared to the gradients formed at the edge of the main beam) do not have a chance to form.
The potential is then calculated from the ion density using the barometric potential law.
Computing Ion Trajectories Through the Electric Field
- The Velocity-Verlet Method
- Pseudo-Code: Time-Adaptive Velocity-Verlet Algorithm
In the form introduced for the original verlet method [54], the position and velocity of the particle at a specific time are found using a central difference scheme;. In addition to the limited velocity accuracy in this algorithm, our knowledge of the velocity always lags behind the positions by one time step. Each time step requires only one evaluation of the function, f(x(t+h)) and storage of the 9N positions, velocities and accelerations, f(x(t)), of the N particles at time.
However, as the particles are accelerated from the main beam, the large time step appropriate for the initially small velocity of these ions may be too large to satisfy the Courant condition for the subsequent accelerated velocity.
Output x, v, and h
- The Differential Cross-Section
- The Sputter Yield
- Ion Behavior: Scattering Angle Solutions and Stream- tube Divergencetube Divergence
- A 2D Example
- Extension to a 3D Example
- Pseudo-Code: Scattering Angle Partial Gauss-Newton Search Al- gorithmgorithm
- Computing the Ion Flux and Sputtering Rate
- The Beamlet Shell
- Example: Computing Quantities for One NSTAR Beamlet Shell
- Example: Sputtering Rate Contribution from One NSTAR Beamlet
- Chapter Summary
The target point S(˜x) is located on the surface of the sphere and defines the radius,Rs=|˜x−x|. The energies of the scattered ions, at the target point, are shown in the fourth figures from the top. Simpson integration of the lines at constant E of Figure 4.27, over shell radius, gives the contribution to the CEX ion flow energy distribution from the beam of our example.
For several reasons, the difference in the two calculated values of the CEX ion density at the chosen target point is not unexpected. The curve can be obtained from the product of the sputtering yield with the energy distribution (see. More important, however, is the sensitivity of the sputtering to the energy of the impinging ions.
Approximation Technique
LOS Approximation Method
- Description of the Approximation
- Comparison of Results with the Full Trajectory Analysis
- Pseudo-Code: Adjusting the Scattering Angle
Only mesh points that contribute to the CEX ion distribution at the target point are shown (Dmin= 0). The contributing area of the beamlet shell to the ion distribution at the point of impact is smaller for the LOS method than for the full trajectory analysis method. Top: The contribution to the energy distribution of CEX ions, as a function of energy and beam radius, calculated using the full trajectory analysis.
Top: The contribution to the energy distribution of CEX ions, as a function of the energy and the beam radius of the jets, calculated using the full trajectory analysis.
In the Case of Grid-Shape Asymmetry
The estimated contributions from the 2D approximation are shown for both the full-orbit analysis assuming a 2D potential, as well as for the LOS energy- and angle-corrected approximation. For the second beamlet, the 2D full-path approximation overestimates the sputter velocity contribution by approx. 11%, and the 2D energy- and angle-corrected approximation underestimates the contribution by approx. 9%. The 2D approximation was insufficient to estimate the contribution of the third beamlet shown in Figure 5.11.
Although the 2D approach produces inaccurate values for the third bundle, the error contribution is small.
Perforated Shell Structure Analysis
Previous Results
Effective moduli predicted from theory and measured from experiments for fractions with varying open area of plates in the thin (t/R1) and thick (t/R1) boundaries. Clearly the theory predicts a large difference in the effective properties depending on whether the plate is considered thin or thick. Note that for thick slabs under bending, the effective Poisson's ratio, ν∗, is expected to be greater than the undrilled bulk value, ν, for all open-surface fractions.
The effective properties found by theory, finite element analysis and experiment are shown in Figure 6.2.
Shallow Shells vs. Flat Plates
A finite element analysis tool specifically adapted for the analysis of thin shells developed at Caltech was used to perform the structural analysis.
The Loading
Not only is the magnitude of the load an unknown quantity, the direction in which that load is expected to be applied to the network is also unclear. The effective moduli were calculated by interpolating the data in Figure 6.2 and summarized in Table 6.1. In order to get an idea of the magnitudes of expected displacements due to different loads, the NSTAR grids were modeled in a similar way to the flat plates above.
It was found that as the thickness of the simulated grid decreased, the time required to complete the computational analysis increased, and it was less likely to converge to a solution.
Optimization Procedure and Results
Control Mesh and Limit Surface Construction
Instead of using the mesh that defines the location of the 15,000 holes in the grid as input to the finite-element code, we need a mesh with many fewer points and elements. The positions of these new mesh points are calculated from the positions of the original control mesh points. There is no restriction on the height of the labeled points (1-5) imposed by these requirements.
The same procedure was used for each mesh so that the evolution of the control mesh to the finite element mesh and finally the boundary surface was consistent.
Setting Up the Problem
- The Optimization Routine
A matrix expressing the equations of the linear inequality constraints: NULL b vector of the values of the inequality constraints: NULL (A x≤b) An eq matrix expressing the equations of the linear equality constraints: NULL b eq vector of the values of the equality constraints: NULL (Aeqx=beq) param min vector of minimum values the optimization parameters can have param max vector of maximum values the optimization parameters can have a nonlincon fun function that evaluates the nonlinear parameter constraints.
The Load and Constraints
The finite element code did not output the resulting stresses in the shell due to loading, so we have limited the constraints of our optimization to only the maximum allowable displacement of the lattice. Under these loads, the maximum deflection of the NSTAR accel grating at two and four times the actual thickness is expected to be (Figure 6.3)δ2NSTAR= 0.133. It was decided to assume that the maximum deflection any mesh could undergo was approximately twice that of the NSTAR mesh under the same load.
In addition to the deflection constraint, two constraints were imposed on the grid shape such that the grid's maximum and minimum heights could not exceed specified values.
Optimization Results
- Sensitivity: param init
- Sensitivity: Evaluation Points
- Sensitivity: Options
- Sensitivity: Neutral-Density Effects
- Sensitivity: Sputter-Yield Effects
- Thickness Effects
The thickness, t, of the mesh in question is shown in the second column, and the height of the target point, ˜z, is given in the third column. The sixth column tabulates the maximum grid displacement calculated at the solution point. The thickness t of the mesh in question is shown in the second column, and the height of the target point ˜z is given in the third column.
While none of these sputtering yields are derived from measurements on actual materials, they are constructed to emphasize one energy range or the other: “low” or “high.” The first artificial sputtering yield increases the effect of ions with energies below 60 eV, compared to the Mo-Xe+ yield, but is the same as xenon on molybdenum for energies higher than 60 eV.
Discussion of Results and Conclusions
Summary of Work Done and Discussion of Results
Such interpolation allowed a highly resolved estimate, by numerical integration, of the contribution to the CEX ion energy distribution at the target point from each shell. Most of the CEX ions that reach the target originate from this region. It was found that the minimization solution depended on the values of the initial optimization parameters.
The location of the target point was found to have little influence on the shape of the optimally shaped grating.
Future Work
All results indicated that changing the lattice shape can result in reduced sputtering of certain surfaces and the reduction is expected to be on the order of 20-30% compared to the expected sputtering using the NSTAR lattice. Regarding the practical implementation of the model in an optimization problem, increasing the number of control network points could provide more flexibility in terms of the feasible shapes of the optimal grid. Of course, increasing the number of control points also brings with it the associated problems of possible convergence instability and the feasibility of the electricity grid.
Any effort invested in continuing this work will need to be considered in terms of the balance between the extra computational time or resources required and the expected improvement in results.
Ion engine-generated charge-exchange environment: Comparison between NSTAR flight data and numerical simulations. In 37th AIAA/ASME/SAE/ASEE Joint Propulsion Conference and Exhibit, Salt Lake City, Utah, 2001. Three-Dimensional Particle Simulations of Ion Propulsion Plasma Environment for Deep Space 1.Journal Of Spacecraft And Rockets.
