NASA /CR-2003-212305 

Numerical Simulation of Ion Thruster Optics 

Cody C. Farnell, John D. Williams, and Paul J. Wilbur 

Colorado State University Fort Collins, Colorado 

July 2003 
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NASA /CR-2003-212305 

Numerical Simulation of Ion Thruster Optics 

Cody C. Farnell, John D. Williams, and Paul J. Wilbur 

Colorado State University, Fort Collins, Colorado 

Prepared for the 

28th International Electric Propulsion Conference 

cosponsored by the Centre National d'Etudes Spatlales (CNES), European Office of Aerospace 

Research and Development of the USAF, Snecma Moteurs, and European Space Agency (ESA) 

Toulouse, France, March 17-21,2003 

Prepared under Grant NAG3-1801 

National Aeronautics and 

Space Administration 

. Glenn Research Center 

July 2003 
Acknowledgments 

Financial support provided in part by the Ion Propulsion Program at the 

NASA Glenn Research Center is gratefully acknowledged. 

I 1 

This report contains preliminary 

findings, subject to revision as 

analysis proceeds. 

Available from 

NASA Center for Aerospace Information 

7121 Standard Drive 

Hanover, MD 21076 

National Technical Information Service 

5285 Port Royal Road 

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Available electronically at http: / /gltrs._erc.nasa.gov 
NUMERICAL SIMULATION OF ION THRUSTER OPTICS 

Cody C. Farnell*, John D. Williams, and Paul J. Wilbur 

Colorado State University 

Department of Mechanical Engineering 

Fort Collins, Colorado 80523 

ABSTRACT 

A three-dimensional simulation code (ffx) designed to analyze ion thruster optics is described. It is 

an extension of an earlier code and includes special features like the ability to model a wide range of grid 

geometries including cusp details and mis-aligned aperture pairs. The motivation for advancing the code 

was the need to accurately predict grid lifetimes and to study the physical evolution of the surfaces that are 

being eroded and surfaces where net material buildup occurs. The techniques used to model the sputter 

erosion processes and track the sputtered material are described. The ability of the ffx code to track material 

eroded from the grids that flows downstream into the ion thruster plume would be valuable to interpret data 

collected using advanced laser diagnostic systems. Ground-based life testing of ion thruster optics, essential 

to the understanding of the processes of grid erosion, can be both time-consuming and costly. Often these 

tests only provide limited data that is collected at the end of the test when it is too late to make any necessary 

changes to flight hardware. Even worse, data on only one test article are typically obtained, which are often 

viewed suspiciously due to test facility effects and the obvious lack of statistical significance. This situation 

is exacerbated for life tests of future ion thruster optics that are being designed for very ambitious 

applications requiring much larger propellant throughput. The goal of the ffx development effort is to 

provide a simulation tool to the ion propulsion community that can be used in combination with short-term 

tests of ion thrusters that are designed to alleviate the need to perform full-blown life tests. In this paper 

results of simulations are presented that describe wear profiles for several standard and non-standard 

aperture geometries, such as those grid sets with square- or slotted-hole layout patterns. The goal of this 

paper will be to introduce the methods employed in the ffi code and to briefly demonstrate their use. 

THE FFX CODE 

Many simulation codes have been developed to study various aspects of ion thruster optics. One 

such code is the igx code, developed by Nakayama and Wilbur for the high-speed, three-dimensional 

analysis of grid sets with axially aligned, hexagonal aperture layouts.' This code has been shown to agree 

well with experimental data in high specific impulse applications.* The ffx code analyzes a threedimensional, rectangular region with symmetry conditions applied on all sides. A uniformly spaced 

Cartesian mesh, efficient for cell inde~ing,~ is applied to the volume with each direction having a different 

mesh spacing. For the simulation of a typical aperture pair in a hexagonal aperture layout, a mesh of 

approximately 30 by 50 by 300 cells is applied to model two quarter-sized apertures. A cross-section of the 

typical geometry and applied potentials for a two-grid system is shown in Figure 1. The key simulation 

tasks in the ffx program are outlined in the flowchart in Figure 2. The following discussion will elaborate 

upon various aspects of these tasks. 

* Phone: (970) 491-8564; Fax: (970) 491-8671; ccf@engr.colostate.edu 

NASNCR-2003-212305 1 
Screen Grid ts lg ta Accel Grid \ I+&* f 

Upstream 

Boundary 

VPlaama VScreen VAccel \.I, % ~ I 

Figure 1. Typical two-grid thruster geometry. 

/ Set Operating Conditions and Geometry / 

stream 

idary . 

I 

Mesh Potentials 

Sheath /Neutralization Surfaces 

I Neutral Density 

I 

I 

Grid Sputtering 

n. 

-1- Electric Fields 

5m 4 Ion Trajectories 

E .- 

E w 

End Beamlet Loop ? Update Grid Geometry 

End Program 

Figure 2. Flowchart of the ffx simulation code. 

For the case of an electrostatic thruster, where magnetic fields can be neglected in comparison with 

the electric fields in the region near the accelerating grids, mesh potential values, denoted by I$, are described 

by the Poisson equation 

In this equation, p is space charge and c0 is the permittivity of free space. In the ffx code, both singly 

and doubly charged ions as well as electrons will contribute to the overall space charge. Thus the right-hand 

side of this equation will be 

(2) 

P -- P+ +P++ +P, -_ - 

EO EO 

where p+ and p,, are the space charge contributions from singly and doubly charged ions respectively, and 

pe is the space charge contribution from electrons. 

Poisson’s equation is solved for mesh point potentials at the start of each beamlet loop. Initially, no 

ion or electron space charge is applied upon the grid mesh. Consequently, the potential solution routine of 

the code first solves the Laplace equation, where the space charge is zero. 

NASNCR-2003-2 12305 2 
Electric fields in the region are related to potential by 

,!?=-V4 or E=- - (am, -i+-j+-k ; a4 A) 

ax az (3) 

The second order partial derivatives in the Poisson equation and the first order partial derivatives in the 

electric field equation are approximated by finite difference equations of the central difference type.4 For 

instance, in the x direction: 

Following other simulation codes,335 equations relating the electron space charge to ion space charge 

at each mesh point are solved within the potential solution routine of the code depending on the average 

upstream and downstream values of space charge and the current value of each mesh potential. In the region 

upstream of the grids within the discharge chamber plasma: 

In the region downstream of the grids within the beam plasma: 

In these equations, k is the value of the upstream plasma potential, $m is the value of the 

downstream plasma potential, $ is the current value of the mesh potential, Td is the upstream electron 

temperature, and T, is the downstream electron temperature. 

The Gauss-Seidel method is used to solve Poisson’s equation. In this method, Poisson’s equation is 

rearranged to obtain $ijs once the finite difference approximations for the partial derivatives have been 

substituted. Iterations are then performed through all mesh points that are not at a known potential until all 

of the mesh point potentials are changing by a value less than a preset limit. Relaxation is used to assist 

convergence of the solution by intuitively predicting each mesh point’s next potential value based on the 

current and last potential values. Relaxation is implemented using 

In this equation, a is the called the relaxation parameter. Without the electron density equations embedded 

within the potential solution, as in the first iteration, using a value for a between 1.0 and 2.0 will increase 

the rate of convergence, called overrelaxation. With the electron density equations added however, the 

electron density will change along with mesh potentials during the potential solution, making the solution of 

the equation non-linear. In this case, a value of a that is between 0.0 and 1 .O is used for convergence, called 

underrelaxation, because it helps to damp out oscillations that the electron equations create. 

#,- = a4,,m + (1 - where 0 < a < 2 . (7) 

Singly and doubly charged ions are represented by particles that are injected into the upstream 

boundary and terminate upon one of the grids or the downstream boundary. During the movement of these 

particles, they have the mass and charge of a single ion, whether they are singly or doubly charged. Each of 

these particles carries with it a much larger charge than a single ion in order to greatly reduce the number of 

particles required to simulate the ion beamlet. For a typical simulation, about 15,000 particle trajectories are 

tracked within each beamlet loop. Particles are injected into the upstream boundary of the analysis volume 

with axial velocities equal to the Bohm velocity: 

NASNCR-2003-212305 3 
where q is the particle’s charge, kTd is the upstream electron temperature, and m is the mass of the particle. 

Forces on charged particles are described by the Lorentz equation: 

F = q(G x ij+ E) (9) 

In this equation, q is particle charge, v is particle velocity, B is the magnetic field, and E is the electric field. 

In the electrostatic case, again neglecting magnetic field effects, the vector equations governing the motion 

of particles are given in equations 10a and lob. The resulting equations of motion in the x direction, for 

example, are given in equations 1 la through 1 IC. 

- -de F=qE=m- dt 

a!?- 

dt 

-_ -V 

qEx a, =- m 

1 

2 

x’ = x + v, At + -a, At2 

The time step At is constantly adjusted during particle motion to ensure that a particle’s movement is 

kept under a specified fraction of the smallest mesh spacing, h, in accordance with the Courant condition, i.e. 

1 

2 

Making sure particles do not travel beyond a certain distance in any given time step is required to ensure that 

details of the system are not lost. 

Av*At<-h (12) 

Throughout particle movement, a particle’s total energy remains constant and is given by the sum of 

its kinetic and potential energies: 

1 

2 

E =-mv2 +q$ 

Using the preceding equations of motion, however, the total energy of a particle is not guaranteed to remain 

constant. Thus at the conclusion of each time step, its total energy is forced to be constant by adjusting the 

length of its velocity vector according to the particle’s total initial energy and local value of potential. 

Specifically, the particle’s three velocity components are multiplied by the ratio of the required velocity 

magnitude for energy conservation to its current velocity magnitude in order to maintain the same direction 

of motion while keeping the total energy of the particle constant. 

As particle trajectories are followed through cells, the charge that each particle is carrying is applied 

to the eight surrounding mesh points by the method of volume weighting. In this method, the charge applied 

to each mesh point is proportional to the fraction of the total cell volume that the volume closest to the 

opposite corner occupies. For instance, the charge applied to Point 1 in Figure 3 is proportional to the 

fraction of the total volume that the volume near Point 8 occupies: 

During particle movement, the electric fields in each direction at the particle’s actual location are 

interpolated from the known electric fields at the eight corner mesh points of the cell using the same 

weighting scheme that is used to apply charges onto the mesh points in a reverse manner. The electric field 

at the particle in the x direction, for example, is the sum of the eight x direction electric fields at the corner 

points multiplied by the eight respective volume fractions. 

The neutral density in each cell, n,, is calculated by assuming a radially uniform, free molecular 

flow of neutral particles. Clawing factors, which are dependent on each grid’s thickness and aperture area, 

are used to adjust the neutral density through the grids. An upstream neutral density can be specified, or the 

beamlet current and propellant utilization efficiency can be used to determine the flow rate through the 

region. 

NASNCR-2003-2 12305 4 
Figure 3. Method of volume weighting. 

Once singly and doubly charged ion densities and the neutral density in each cell have been 

calculated, equations describing the volumetric charge exchange ion production rates in each cell are used to 

introduce charge exchange ions. Charge exchange ions are created when electrons transfer to fast moving 

ions from relatively slow moving neutral atoms, creating slowly moving ions and fast moving neutral atoms. 

Charge exchange ion production rates are described using equations of the form 

dn, = 71nn;v;o(v;) . dt 

Depending on the reaction in question, q can be the singly or doubly charged ion density, vi the velocity that 

a singly or doubly charged ion would be moving through the cell, and o(vJ the cross section of the specific 

reaction. 

Erosion of the grids can be due to both ions from the upstream plasma as well as charge exchange 

ions impacting the grids. The number of sputtered grid atoms is dependent on the energy and angle of 

incidence of the impacting particle. The angle of incidence is found using the local surface normal vector of 

the impacted cell. Because erosion will constantly change the grid surface, surface normal vectors are found 

at any time by calculating the location of the regional center of mass of the impacted cell relative to its 

geometrical center. The surface normal of the impacted cell is then defined as the line that extends from the 

center of mass in the region through the center of the impacted cell, as shown in Figure 4. 

Impinging Particle rJ Angle of 

Incidence 

Surface Normal 

Center of Cell 

Local Center 

of Mass 

Figure 4. Surface normal vectors and angle of incidence. 

Grid atoms are sputtered with a cosine distribution based on the surface normal vector of the impacted cell 

and are subsequently followed fiom the particle impact point until they deposit onto one of the grids or exit 

through the upstream or downstream boundaries. 

NASNCR-2003-2 12305 5 
Charge exchange ions originating from several centimeters downstream of the grids can flow back 

to the accel grid and cause erosion. To keep an analysis volume of reasonable size and a simulation of 

reasonable speed while still simulating charge exchange ions from far downstream, charge exchange ion 

production is adjusted near the downstream boundary to bring the current reaching the accel grid to the level 

measured during experimental tests conducted in vacuum facilities with very high pumping speeds. 

RESULTS 

To illustrate the use of the ffx code, simulations were run on four SUNSTAR (Scaled Up NSTAR) 

grid sets6 that have various aperture patterns. The geometries of these grids and the simulated operating 

conditions are shown in Figure 5. The parameter chosen to compare the grid sets is normalized perveance 

per unit grid area: 

In these equations, Jb is the beamlet current, ATG the aperture and web area, e, the effective grid spacing, and 

VT the total accelerating voltage. A comparison of the beamlet shapes at the perveance and crossover 

impingement limits is shown in Figure 6. At the perveance limit, where extraction currents are relatively 

high, ions impinge upon the accel grid on the same side of the aperture from which they started. At the 

crossover limit, where extraction currents are relatively low, ions can crossover the aperture centerline and 

impinge upon the opposite side of the accel grid. At the center of a thruster, aperture pairs would likely be 

operating closer to the perveance limit where ion densities are high. Conversely, at the edges of the thruster, 

aperture pairs would be operating closer to the crossover limit where ion densities are lower. 

Figure 7 compares the impingement limits found experimentally at CSU6 with those found using the 

ffx code for the four SUNSTAR aperture patterns. The ffx code generally predicted slightly greater 

perveance and crossover limits than those found experimentally. Overall, good agreement was seen between 

the relative limits among different aperture patterns. 

Figure 8 compares the sheath and neutralization surface shapes found near the perveance and 

crossover limits for each of the aperture patterns. The sheath and neutralization surfaces shown here are 

surfaces of constant potential: the sheath at the upstream plasma potential and the neutralization surface at 

the downstream plasma potential. 

Erosion tests were run on the four SUNSTAR grid sets at beamlet current values near the perveance 

limit for each grid set. In each case, the current reaching the accel grid was adjusted to 0.3% of the beamlet 

current. Figure 9 compares the erosion patterns produced on each of the grid sets at two values of propellant 

throughput, the second being twice as much as the first. Note that each of the grid sets would be run for 

different amounts of time to reach the same level of propellant throughput. Because the grids are operating 

near the perveance limit, high-energy charge exchange ions produced in the region between the grids quickly 

erode away the accel grid barrel from the upstream side of the accel grid in all of the aperture layouts. 

Following this period of immediate barrel erosion, charge exchange ions from the downstream region create 

erosion patterns in the downstream side of the accel grid with less accel grid barrel erosion. The familiar pit 

and groove pattern of erosion can be seen for the hexagonal aperture pattern. 

CONCLUSIONS 

A three-dimensional optics simulation code was developed that can be used to simulate ion optical 

behavior in complex grid geometries with reasonable efficiency. Results from the code were found to be 

consistent with experimental data in predicting impingement limits. Erosion patterns predicted by the code 

agree well with the known erosion patterns of grids with hexagonal aperture layouts, and the code yields 

erosion patterns for grids with square and slotted aperture layouts that are judged to be reasonable. 

NASNCR-2003-2 12305 6 
Hexagonal Pattern Square Pattern 

Half Slot Pattern 

d, or d, eCC 

eCC 

d, or d, 

Full Slot Pattern 

:---1 = ffx analysis region 

0 = region displayed in figures 

Figure 5. SUNSTAR grid aperture patterns. 

Plasma 

Sheath \ Charge Exchange Ions 

I 

/ 

Screen Grid Accel Grid 

Figure 6. Perveance and crossover impingement limits. 

Geometry / 

Operating Conditions 

I,, = 5000 s 

V, = 2.91 kV 

V, = 2.88 kV 

V, = -360 V 

v,=ov 

d, = 3.1 mm 

d, = 1.86 mm 

t, = 0.62 mm 

t, = 0.83 mm 

ecc = 3.6 ~zllll 

et,half = 4.8 mm 

ee.full = 12.0 mm 

Beamlet shape at the 

perveance limit 

Beamlet shape at the 

crossover limit 

NASNCR-2003-2 12305 7 
Experimental 

5 

Po 

h Crossover 

r Limits 

-2 ~ ~ 

k 5 10 - 

2 

g 8- 

0 

-2 

Full Slot Pattern 

~ ~___ ___ ~ 

02 0.3 04 05 06 

t 

0.1 

NORMALIZED PERVEANCE PER UNIT GRID AREA [PTG] 

ffx Code 

Hexagonal Pattern 

Square Pattern 

Half Slot Pattern 

Full Slot Pattern / 

I I Perveance 

Crossover Limits 1 ii f Limits 

0 0.1 0.2 0.3 0.4 0.5 0.6 

NORMALIZED PERVEANCE PER UNIT GRID AREA [PTG] 

Note: Charge exchange impingement current was not included in the ffx data. 

Figure 7. Comparison of experimental and ffx code predictions of crossover and perveance limits. 

NASNCR-2003-2 12305 8 
Hexagonal Pattern 

PTG = 0.13 

L J, = 0.25 mA 

Square Pattern 

PTG = 0.13 

J, = 0.29 mA 

Half Slot Pattern 

PTG = 0.20 

J, = 1.03 mA 

Full Slot Pattern 

PTG = 0.23 

J, = 2.19 mA 

Crossover Limit 

PTG = 0.3 1 

J, = 0.59 mA 

PTG = 0.26 

J, = 0.57 mA 

PTG = 0.32 

J,= 1.64 mA 

Perveance Limit 

c 

Accel Grid 

1 Neutralization 

Surface 

PTG = 0.40 

J, = 3.80 mA 

Figure 8. Sheath and neutralization surface shapes for the four SUNSTAR grid sets. 

NASNCR-2003-212305 9 
Propellant Usage 1 

(-5000 hours of operation 

for hexagonal grid) 

Propellant Usage 2 

(-10000 hours of operation 

for hexagonal grid) 

L 

Hexagonal Pattern 

I. 

PTG = 0.3 1 

J, = 0.59 mA 

Upstream 

Cells 

Square Pattern 

PTG = 0.26 

J, = 0.57 mA 

Half Slot Pattern 

PTG = 0.32 

J, = 1.64 mA 

Downstream 

Cells 

Full Slot Pattern 

PTG = 0.40 

J, = 3.80 mA 

Note: Colored cells are cells that have been eroded away. 

Figure 9. Accel grid cell erosion for the four SUNSTAR grid sets. 

NASAJCR-2003-2 12305 10 
REFERENCES 

Nakayama, Y. and Wilbur, P. J. “Numerical Simulation of High Specific Impulse Ion Thruster Optics.’’ 27” 1 

International Electric Propulsion Conference, IEPC-0 1-099, Pasadena, CA, October 200 1. 

.. Wilbur, P. J., Miller, J., Farnell, C., and Rawlin, V. K. “A Study of High Specific Impulse Ion Thruster Optics” 27” 2 

International Electric Propulsion Conference, IEPC-0 1-098, Pasadena, CA, October 2001. 

8 

Wang, J., Polk, J. “Three-Dimensional Particle Simulations of Ion Optics Plasma Flow and Grid Erosion.” 331d 3 

Plasmadynamics and Lasers Conference, AIAA 2002-2193, Maui, Hawaii, May 2002. 

Chapra, S. C., and Canale, R. P. Numerical methods for engineers, 3d ed., McGraw-Hill, New York, 1998. 

Brown, Ian. The physics and technology of ion sources, John Wiley & Sons, New York, 1989. 

Williams, J. D., Monthly Report to the Jet Propulsion Laboratory, December 18, 2002. See also in this conference 

J.D. Williams, D.M. Laufer, and P.J. Wilbur, “Experimental Performance Limits on High Specific Impulse Ion Optics,” 

28* International Electric Propulsion Conference, IEPC-03- 128, Toulouse, France, 17-2 1 March 2003. 

4 

5 

6 

NASNCR-2003-212305 11 
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4. TITLE AND SUBTITLE 

Numerical Simulation of Ion Thruster Optics 

5. FUNDING NUMBERS 

WBS-22-8W501 6. AUTHOR(S) 

NAG3-180 1 

Cody C. Farnell, John D. Williams, and Paul J. Wilbur 

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Colorado State University 

Fort Collins, Colorado 80523 

1 Colorado State 

3. SPONSORINWMONITORING AGENCY NAME(S) AND ADDRESS(ES) 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

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REPORT NUMBER 

E- 13882 

10. SPONSORINWMONITORING 

AGENCY REPORT NUMBER 

NASA CR-2003-212305 

Unclassified - Unlimited 

Subject Category: 20 

4. SUBJECT TERMS 

Ion thruster; Ion accelerator, Ion source; Space propulsion 

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20. LIMITATION OF ABSTRACT 

Available electronically at http://gltrs.wc.naa.gov 

This publication is available from the NASA Center for Aerospace Information, 3014214390. 

3. ABSTRACT (Maximum 200 words) 

A three-dimensional simulation code (ffx) designed to analyze ion thruster optics is described. It is an extension of an 

earlier code and includes special features like the ability to model a wide range of grid geometries, cusp details, and 

mis-aligned aperture pairs to name a few. However, the principle reason for advancing the code was in the study of ion 

optics erosion. Ground based testing of ion thruster optics, essential to the understanding of the processes of grid 

erosion, can be time consuming and costly. Simulation codes that can accurately predict grid lifetimes and the physical 

mechanisms of grid erosion can be of great utility in the development of future ion thruster optics designed for more 

ambitious applications. Results of simulations are presented that describe wear profiles for several standard and nonstandard aperture geometries, such as those grid sets with square- or slotted-hole layout patterns. The goal of this paper 

will be to introduce the methods employed in the ffx code and to briefly demonstrate their use. 

NSN 7540-01 -280-5500 Standard Form 298 (Rev. 2-89) 

Prescribed by ANSI Std. 239-18 

298-1 02