George C. Soulas

Glenn Research Center, Cleveland, Ohio

Modeling Neutral Densities Downstream

of a Gridded Ion Thruster

NASA/TM—2010-216930

December 2010

AIAA–2010–216930
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George C. Soulas

Glenn Research Center, Cleveland, Ohio

Modeling Neutral Densities Downstream

of a Gridded Ion Thruster

NASA/TM—2010-216930

December 2010

AIAA–2010–216930

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2010-216930 1 

Modeling Neutral Densities Downstream of a Gridded Ion Thruster 

George C. Soulas 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract

The details of a model for determining the neutral density downstream of a gridded ion thruster are presented. An 

investigation of the possible sources of neutrals emanating from and surrounding a NEXT ion thruster determined that 

the most significant contributors to the downstream neutral density include discharge chamber neutrals escaping 

through the perforated grids, neutrals escaping from the neutralizer, and vacuum facility background neutrals. For the 

neutral flux through the grids, near- and far-field equations are presented for rigorously determining the neutral density 

downstream of a cylindrical aperture. These equations are integrated into a spherically-domed convex grid geometry 

with a hexagonal array of apertures for determining neutral densities downstream of the ion thruster grids. The neutrals 

escaping from an off-center neutralizer are also modeled assuming diffuse neutral emission from the neutralizer keeper 

orifice. Finally, the effect of the surrounding vacuum facility neutrals is included and assumed to be constant. The 

model is used to predict the neutral density downstream of a NEXT ion thruster with and without neutralizer flow and a 

vacuum facility background pressure. The impacts of past simplifying assumptions for predicting downstream neutral 

densities are also examined for a NEXT ion thruster. 

Nomenclature 

a y-intercept in the non-dimensional wall flux equation

Ao reservoir particle emission area 

Aw cylindrical wall particle emission area 

b slope in the non-dimensional wall flux equation 

c neutral particle mean thermal speed 

D(γ, β) normalized neutral density distribution upstream of the thruster grids 

e charge of an electron (i.e., 1.602×10–19 C), 

f(v) Maxwell-Boltzmann speed distribution function 

Foa accelerator grid open area fraction 

Fscrn neutral density correction factor to account for the screen grid (i.e., 0.93 for the NEXT grids) 

Jb thruster beam current 

J

++/J+

 discharge chamber total doubly-to-singly-charged ion current ratio 

k hexagon side number 

kB Boltzmann’s constant (i.e., 1.3804×10–23 J/K) 

Kc transmission probability of neutrals through an accelerator grid aperture 

L length between the surface normal of dAo to the downstream neutral density location 

lcc grid aperture center-to-center spacing 

Lgrid length between the grid neutral emission site to the downstream neutral density location 

Lneut length between the neutralizer neutral emission area to the downstream neutral density location 

m number of apertures along the side of a hexagonal array of holes 

m ̇ d total flow rate into the discharge chamber 

mn molecular mass of the neutral gas particle 

m ̇ neut neutralizer mass flow rate 

n downstream neutral density 

nfar far-field equation for the downstream neutral density from an aperture 

nfar-gridsum far-field equation for the downstream neutral density from a grid by summing individual aperture 

contributions 

nfar-gridint far-field equation for the downstream neutral density from a grid by integration of the grid surface 

ngrid downstream neutral density from discharge chamber neutrals escaping through the grids 

nnear equation for the downstream neutral density from an aperture in the near- and far-fields 
NASA/TM—2010-216930 2 

nnear-gridsum equation for the downstream neutral density from a grid by summing individual aperture contributions 

that is valid in the near- and far-fields 

nneut downstream neutral density from neutralizer neutrals 

no reservoir or discharge chamber neutral density 

no,neut neutralizer reservoir neutral density 

ntank vacuum facility background neutral density 

ntot total downstream neutral density, including contributions from ngrid, nneut, and ntank 

nup downstream neutral density from reservoir neutrals through an aperture 

nwall downstream neutral density reflected from an aperture cylindrical wall in the near- and far-fields 

p number of successive rows defining a hexagonal array of apertures 

pmax maximum number of rows defining a hexagonal array of apertures 

ptank vacuum facility background pressure 

NASA National Aeronautics and Space Administration 

NEXT NASA’s Evolutionary Xenon Thruster 

NSTAR NASA’s Solar Electric Propulsion Technology Applications Readiness program 

R accelerator grid aperture radius 

Rgrid spherically-domed grid radius of curvature 

r downstream neutral density radial location 

rnear radial aperture-to-grid coordinate system transformation equation 

rno neutralizer keeper orifice radial location 

Rno neutralizer keeper orifice radius 

rnoff radial offset of the neutralizer keeper emission area 

ro reservoir radial location 

rof final reservoir, or discharge chamber, radial location limit of integration 

roi initial reservoir, or discharge chamber, radial location limit of integration 

roII radial limit of integration for the reservoir’s, or discharge chamber’s, contribution to the downstream 

 neutral density 

roIII radial limit of integration for the reservoir’s, or discharge chamber’s, contribution to the downstream 

 neutral density 

t accelerator grid aperture thickness 

T far-field angular distribution function for particles downstream of a cylindrical aperture 

To reservoir or discharge chamber neutral gas temperature 

Ttank vacuum facility background particle temperature 

v neutral particle speed 

z downstream neutral density axial location 

znear axial aperture-to-grid coordinate system transformation equation 

znoff axial offset of the neutralizer emission area 

zw aperture wall axial location 

zwf final aperture wall axial location for integration 

zwi initial aperture wall axial location for integration 

zwII axial limit of integration for the wall’s contribution to the downstream neutral density 

α angle between the dAo surface normal and the downstream neutral density 

αgrid angle between the grid surface normal and the downstream neutral density 

αneut angle between neutralizer emission area surface normal and the downstream neutral density 

β downstream azimuthal angle 

γ angle along spherically-domed grid 

γmax angle along spherically-domed grid to the edge of the perforated grid 

Γo reservoir or discharge chamber flux 

Γw aperture wall flux 

θ neutralizer keeper orifice azimuthal location 

κ aperture wall flux variable 

φ cylindrical aperture azimuthal angle 

φf final cylindrical aperture azimuthal angle limit of integration 

φi initial cylindrical aperture azimuthal angle limit of integration 
NASA/TM—2010-216930 3 

φoI angular limit of integration for the reservoir’s, or discharge chamber’s, contribution to the 

 downstream neutral density 

φomax angular limit of integration for the reservoir’s, or discharge chamber’s, contribution to the 

 downstream neutral density 

φwI angular limit of integration for the wall’s contribution to the downstream neutral density 

φwt angular limit of integration for the wall’s contribution to the downstream neutral density 

ωo solid angle from dAo 

ωw solid angle from dAw

Subscripts and Superscripts 

 ̄ non-dimensionalized value (dimensions are divided by R, densities by no, and fluxes by Γo) 

o reservoir, or discharge chamber 

w aperture wall 

I. Introduction 

The successful demonstration of the NSTAR (NASA’s Solar Electric Propulsion Technology Applications 

Readiness program) ion thruster on the Deep Space 1 mission and its subsequent use on the Dawn spacecraft have 

demonstrated the utility and reliability of ion thrusters for NASA’s deep space missions (Refs. 1 to 5). As a direct 

result of this success, NASA continued the development of ion propulsion technology by investing in NASA’s 

Evolutionary Xenon Thruster, or NEXT, which represents a next-generation ion propulsion system (Ref. 6). The 

objective of the NEXT project is to advance this next generation ion propulsion technology to NASA Technology 

Readiness Level 5, with significant progress towards Level 6, to support NASA Science Mission Directorate 

missions (Ref. 7). The NEXT program is developing an advanced ion propulsion system, that includes, in part, a 

high performance, 7 kW ion thruster with a spherically-domed beam extraction grid system with a chord diameter of 

36 cm. 

Although the NEXT ion thruster exhibits high propellant utilization efficiencies throughout its throttling range, 

some un-ionized propellant, or neutrals, will inevitably escape the ion thruster’s discharge chamber and neutralizer. 

For example, some neutrals will escape from the thruster discharge chamber through the grid apertures and expand 

into the vacuum of space. Some of these neutrals can undergo charge-exchange with the beam ion plume to create 

slow-moving charge-exchange ions. These charge-exchange ions are subsequently directed by potentials within the 

ion beam and charge-exchange plasmas away from the thrust axis and possibly towards spacecraft surfaces. 

Although the energies of these ions are low, modeling of the charge-exchange plasma is typically conducted to 

understand where these charge-exchange ions are directed and to assess their possible impact on the spacecraft. 

Properly modeling charge-exchange production requires an accurate understanding of the neutral density 

distribution downstream of and surrounding an ion thruster. 

In addition to charge-exchange plasma effects on the spacecraft, the neutrals escaping from the thruster can also 

affect plasma diagnostics’ measurements. For example, measurements of the doubly-charged beam ion current 

relative to the singly-charge current are affected by differences in charge-exchange cross-sections (Ref. 8). 

Specifically, the charge-exchange cross-section for singly-charged ions is much larger than that of doubly-charged 

ions (Ref. 9). This can lead to exaggerated doubly-to-singly-charged ion current ratios when measured far 

downstream of the ion thruster and at high background pressures (Ref. 8). Properly modeling this charge-exchange 

effect again requires an accurate understanding of the neutral density distribution downstream of the ion thruster. 

Many past studies have modeled the downstream neutral density emanating from an ion thruster. Unfortunately, 

most studies made simplifying assumptions that limited the utility of their models. These simplifying assumptions 

ignored the effect of the grid shape and the neutral efflux “beaming” effect of individual apertures of finite 

thickness, where thicker aperture walls direct more escaping neutrals along the aperture centerline (Refs. 10 and 11). 

Very early studies either treated the thruster as a point source for far-field analyses or merely assumed that the 

thruster was a flat surface with neutral particle emission throughout the surface, allowing for a simpler analytical 

expression to be used (Refs. 12 and 13). A large number of studies modeled the downstream neutral density using an 

analytical expression that merely assumed a neutrals point source upstream of the thruster grids (Refs. 14 to 21). 

Another study used an analytical expression that assumed that neutrals were emitted from a flat disk diffusely 

(Ref. 22). And yet other analyses utilized a Monte Carlo Particle-in-Cell technique to model the neutral efflux from 

ion thruster (Refs. 23 to 26). Such a technique has the benefit of including the impacts of charge exchange on the 

velocity distribution of the neutral efflux, for example, but computational difficulties with this technique led to 

 
NASA/TM—2010-216930 4 

simplifications that included computational boundaries that assumed a flat grid without individual apertures. The 

failure of all aforementioned studies to capture the effect of the grid shape and the “beaming” effect of individual 

apertures reduced the accuracy of the modeled results and limited the utility of the models to the far-field. 

Three studies did include the effect of the grid shape and two studies included neutral efflux from individual 

apertures (Refs. 27 to 29). Unfortunately, two studies did not accurately model the neutral efflux “beaming” effect 

of individual apertures (Refs. 27 and 28). The other study correctly captured this neutral “beaming” effect (Ref. 29). 

However, all three studies did not present the equations necessary to recreate the models. 

In addition to these simplifying assumptions, almost all of the aforementioned studies assumed that neutrals 

were only emitted through the thruster grids (Refs. 12 to 27 and 29). So, most of the studies neglected the 

contribution of the neutral efflux from the thruster’s neutralizer and the possible contribution of downstream 

reflections from other thruster surfaces. 

This paper will present the details of a model for determining the neutral density downstream of a gridded ion 

thruster. The model will include the effects of the grid’s spherical shape and the neutral efflux “beaming” through 

cylindrical apertures in both the near- and far-fields. The first section will examine all sources for neutrals 

emanating from an ion thruster. The following section will present near- and far-field equations for modeling neutral 

densities emanating from a discharge chamber through cylindrical apertures based on equations derived in other 

studies (Refs. 10, 11, and 30). These equations will then be integrated into the complex perforated grid geometry 

and this, along with the neutral efflux from the neutralizer, will be used to model neutral densities downstream of an 

ion thruster. The model will then be used to predict the neutral density downstream of a NEXT ion thruster with and 

without neutralizer flow and a vacuum facility background pressure. Finally, the impact of past simplifying 

assumptions for predicting downstream neutral densities will be assessed. 

II. Sources of Neutrals From and Surrounding an Ion Thruster 

Figure 1 shows sketches of a typical ion thruster with the various sources of neutrals that can emanate from and 

surround the thruster. As shown in the sketch, the sources of neutrals downstream of an ion thruster originate from 

the following thruster regions: 

1. un-ionized propellant escaping from the discharge chamber through: 

a. the apertures of the perforated grids; 

b. the thruster’s plasma screen; 

c. the gap between the front mask and accelerator grid; and 

2. un-ionized propellant escaping from the neutralizer; 

3. vacuum facility background neutrals. 

Through 

Grids

Through 

Neutralizer

Through Gap Between 

Front Mask & Grid 

Through Plasma 

Screen 

Neutralizer 

Reflection

Front Mask 

Reflection

Facility 

Neutrals

Chargeexchange Ion 

Reflection

Discharge 

Chamber 

Neutralizer Neutralizer 

Discharge 

Chamber 

Figure 1.—Unionize propellant escape paths from the thruster (right sketch) and downstream reflections of 

un-ionized propellant escaping from through the grids (left sketch). 
NASA/TM—2010-216930 5 

Also important are the reflections that un-ionized neutrals make after they exit the thruster, as shown in 

Figure 1. It will be shown that the discharge neutrals escaping through the grids are the most significant source of 

neutrals from the thruster. The downstream reflections of these neutrals include: 

1. neutrals reflected from the front mask; 

2. neutrals reflected from the neutralizer; and 

3. ions that impact the accelerator grid downstream surface and aperture walls and leave as neutrals. 

As the lists above and Figure 1 demonstrate, there are a numerous regions from an ion thruster where neutral 

particles can be ejected. 

Fortunately, there are two considerations that can reduce the complexity of determining downstream neutral 

densities. The first is related to the neutral densities within and surrounding the thruster. Ion thrusters typically 

operate at very low neutral densities. For a NEXT ion thruster operating at full power, for example, the neutral 

density within the discharge chamber is estimated to be about 2.5×1018 m–3. At this low particle density, the mean 

free path for neutral-to-neutral collisions is about 6 cm, which is 1/6 of the NEXT discharge chamber chord diameter 

and about 50x larger than an accelerator aperture diameter. Downstream of the thruster and within a vacuum facility 

at a 6.7×10–3 Pa (5×10–6 torr) background pressure, the mean free path for neutral-to-neutral collisions is about 

93 cm. Such large mean free paths define a rarified gas flow regime referred to as free molecular flow. In this flow 

regime, particle-to-particle collisions are so rare that they can be neglected. The benefit of modeling this regime is 

that the resulting downstream neutral density from the various neutral particle sources (shown in Fig. 1) can be 

modeled separately, and the resulting fluxes and densities can be added together to determine the total downstream 

neutral density. 

Only within the neutralizer, where pressures can reach as high as about 67 torr at full power for the NEXT 

neutralizer (Ref. 31), are the mean free paths for neutral-to-neutral collisions small enough for these collisions to be 

significant (i.e., mean free paths of about 3×10–6 cm at full power within the cathode insert region). However, these 

neutral-to-neutral mean free paths increase rapidly as the gas is ionized (and hence the neutral population decreases) 

and the neutral gas plume expands into vacuum. Because the neutral density downstream of and surrounding the 

thruster is desired, collisional effects within the neutralizer and immediately downstream of the neutralizer cathode 

orifice can also be ignored and the downstream free molecular flow conditions can be considered. 

The second consideration that can reduce the complexity of determining downstream neutral densities is the 

relative contribution of each neutral source and escape path. Although neutrals can escape from many regions of an 

ion thruster, some of the aforementioned emission sites can be neglected without any significant loss of accuracy. 

Although there are three escape paths for neutrals to leave the discharge chamber, the vast majority of particles will 

exit through the grid apertures. This is due to two reasons 1) the open area between the grids and discharge chamber 

is only about 11 percent that of the accelerator grid open area for a NEXT thruster; and 2) the transmission 

probability through the grid apertures is much higher. Regarding the latter, the escape paths from the discharge 

chamber to the plasma screen and front mask are typically significantly tortuous, involving multiple wall collisions 

before escaping, and exhibiting no direct line-of-sight. However, the escape path through the accelerator grid 

apertures is more direct and, as a result, will have a much higher transmission probability. The transmission 

probabilities through the front mask and plasma screen escape paths were crudely estimated for a NEXT ion 

thruster. This was done by treating the transmission between components within the thruster independently and 

assuming that the entrance conditions of the neutrals for each component was random (Ref. 32). In addition, other 

simplifying assumptions were made to handle bends and elbows, as well as the shapes through which neutrals were 

flowing. The total amount of propellant leaking through the front mask and plasma screen was estimated to be about 

1 percent of that escaping through the accelerator grid apertures. Of this 1 percent neutral loss, most escaped through 

the plasma screen. Because this represents such a small amount of the total leakage, this study will neglect neutrals 

escaping through the front mask gap and plasma screen. 

For a typical NEXT ion thruster, the accelerator current at full power is about 14 mA while the neutral loss rate 

from the discharge chamber is estimated to be 0.51 equivalent Amperes (Ref. 31). So, the total flux rate of 

neutralized accelerator current is only 2.7 percent that of the discharge chamber neutral loss rate. In addition to this, 

the accelerator current collected by the accelerator grid is a combination of charge-exchange ions collected on the 

downstream grid and within the accelerator aperture walls. While those ions collected on the downstream 

accelerator surface would be expected to contribute directly to the downstream neutral density, those collected 

 
NASA/TM—2010-216930 6 

within the aperture barrel would not necessarily do so. This is because a portion of those ions would be reflected 

upstream and could, therefore, re-enter the discharge chamber. This further reduces the contribution of reflected 

charge-exchange ions on the downstream neutral density. As a result, the total flux rate of neutralized accelerator 

current that contributes to the downstream neutral density is less than 2.7 percent that of the discharge chamber 

neutral loss rate. Because of this small contribution, ions that impact the accelerator grid downstream surface and 

aperture walls and leave as neutrals will also be neglected in this study. 

To determine if neutralizer and front mask reflections are significant, the flux of neutrals that escape from the 

grid apertures and impinge on the neutralizer and front mask was conservatively estimated. It was determined that 

the neutralizer and front mask intercept less than 0.2 percent and less than 0.9 percent, respectively, of the total 

neutral loss from the discharge chamber. Although the reflected neutrals can have a noticeable effect on the neutral 

density very close to the neutralizer and front mask surfaces, the effect further from these surfaces will be negligible. 

The effect of reflected neutrals from the neutralizer and front mask will, therefore, be neglected in this study. 

Vacuum facility background neutrals cannot typically be ignored. Within a vacuum facility at a 6.7×10–3 Pa 

(5×10–6 torr) background pressure, the neutral density would be about 1.6×1017 m–3, which is 6 percent that of the 

NEXT discharge chamber density operating at full power. More importantly, however, the background neutral 

density would be expected to be constant far downstream of an ion thruster, where the density of neutrals emanating 

from a thruster would be expected to be considerably lower than discharge chamber neutral densities. The effect of 

vacuum facility background neutrals will, therefore, be included in this study. 

Neutralizer flow rates are typically large in comparison to the discharge chamber neutral loss rate. For a NEXT 

thruster operating at full power, the neutralizer flow rate is 4.01 sccm. Although some of this neutralizer flow will 

exit the neutralizer as ionized gas, the majority is expected to exit as un-ionized propellant. Mikellides modeled a 

NEXT neutralizer during thruster operation at full power (Ref. 33). He found that the neutralizer neutral loss rate 

was 2.8 sccm during full power operation (Ref. 34). This represents 40 percent of the discharge chamber neutral loss 

rate. Because of the significance of this neutral loss contribution, the neutral loss from the neutralizer will be 

modeled in this study. 

One other source of neutrals, not shown in Figure 1 and neglected in this study, is the fast-moving neutrals 

created by charge-exchange between downstream neutrals and beam ions. Unfortunately, this requires knowledge of 

the spatial distribution of beam ion current density, which is beyond the scope of this study. Because of operation in 

the free molecular flow regime, however, the distribution of these neutrals can be determined by later modeling and 

the result added to that of this study, assuming the change in total neutral density is negligibly small. 

Table 1 summarizes the various neutral escape and reflection sites, their relative contributions, and whether they 

are modeled in this study. As the table shows, the significant contributors to the downstream neutral density include: 

1) the neutral flow from the discharge chamber through the grid apertures; 2) neutrals from the neutralizer; and 

3) background vacuum facility neutrals. The next section will develop the equations for modeling the downstream 

neutral density from these neutral sources. 

TABLE 1.—SUMMARY OF NEUTRAL PARTICLE SOURCES, THEIR RELATIVE CONTRIBUTIONS, 

AND WHETHER THEY ARE MODELED IN THIS STUDY 

Neutral particle source Relative contributiona Modeled in this study? 

Discharge chamber—Through grid 

apertures 

99 percent of discharge chamber neutral loss rate Yes 

Discharge chamber—Through gap 

between front mask and grid 

~0.2 percent of discharge chamber neutral loss rate No 

Discharge chamber—Through plasma 

screen 

~0.8 percent of discharge chamber neutral loss rate No 

Neutralizer 40 percent of discharge chamber neutral loss rate Yes 

Facility background neutrals 6 percent of discharge chamber neutral density Yes 

Reflected accelerator current 2.7 percent of discharge chamber neutral loss rate No 

Reflected from neutralizer <0.2 percent of discharge chamber neutral loss 

rate reflected 

No 

Reflected from front mask <0.9 percent of discharge chamber neutral loss 

rate reflected 

No 

From charge-exchange with beam ions Beyond scope of this study No a

Assumes a NEXT ion thruster operating at full power with a background pressure of 5×10–6 torr. 
NASA/TM—2010-216930 7 

III. Model Development 

The following sections will develop the equations necessary to determine the neutral density downstream of an 

ion thruster. The first section lists the assumptions and fundamental equations used to determine the downstream 

neutral density. Far- and near-field equations for determining the neutral density downstream of a cylindrical 

aperture are presented thereafter. The subsequent sections develop the base equations for determining downstream 

neutral density from a spherically-dome convex grid with a hexagonal array of apertures, an off-center neutralizer, 

and the surrounding vacuum facility neutrals. 

A. Assumptions and Fundamental Equations 

The equations used in the following sections are based on the fundamental equations presented within this 

section. The assumptions used to develop these equations include the following: 

1. the neutral density is sufficiently low that free molecular flow conditions exist throughout the modeled 

domain; 

2. neutrals from a reservoir (e.g., the discharge chamber) have a Maxwell-Boltzmann speed distribution; 

3. equilibrium conditions exist, so that the rate of neutrals leaving the reservoir is equal to the rate of neutrals 

fed into the reservoir; 

4. all wall reflections are diffuse; 

5. neutral adsorption onto the walls and surface diffusion are negligible; and 

6. wall temperatures are equal to the neutral particle temperature. 

The geometric relation between a differential element from a neutral particle emission site (i.e., a wall or the 

upstream reservoir, such as the discharge chamber) and the downstream neutral density location is shown in 

Figure 2. The variable α is the angle between the neutral emission site surface normal from the differential emitting 

area, dAo, to the location of the downstream neutral density, n. The variable L is the distance between the 

differential emitting area and downstream density. 

 The downstream neutral density from a reservoir of neutrals can be solved by starting with the MaxwellBoltzmann phase space speed distribution function (Ref. 35). Because equilibrium conditions are assumed, the speed 

distribution function, f(v), is independent of position (i.e., neutral velocities are isotropic). So, the neutral density 

downstream of an orifice to a reservoir is: 

       



 

 

            

o o

o o

0 0

o o 3 o n n f v d v f v n d dv f v dv n d (1) 

 

α

dAo, no 

Neutral Emission Site 

Downstream Neutral 

Density, n

Surface 

Normal L

Figure 2.—Geometric relation between a neutral 

emission site and the downstream neutral density. 
NASA/TM—2010-216930 8 

Here, no is the neutral density within the reservoir and the differential solid angle, dωo, is given by: 

 

2

o o L

dA cos d  

  (2) 

The first integral in Equation (1) can be shown to be 1/(4·π), so substituting this and Equation (2) into Equation (1) 

yields: 

         

Ao

o 2 o dA

L

cos

n

4

1

n (3) 

This equation is in a form that is useful for solving for the downstream neutral density when the emission site is the 

reservoir. When the emission site is a wall, it is more useful to re-express Equation (3) using a particle flux. This is 

done by noting that the flux in any one direction from a reservoir or diffusely-emitting surface is given by: 

4

n w c o,

o,w

   (4) 

where c is the mean thermal speed of the neutrals and the “w” subscript denotes a wall while the “o” subscript 

denotes a reservoir. Solving Equation (4) for no and substituting this into Equation (1) yields: 

    







         

w

w w

0

w 3 d

c

4 f v d v f v dv

c

4

n (5) 

Note that Γw appears within the second integral because the wall flux will be shown to be a function of the location 

on the wall. Substituting Equation (2) into Equation (5) yields: 

           

Aw

w w 2 dAw L

cos A

c

1

n (6) 

which is in a form that is useful for solving the downstream neutral density when a reflecting wall is the emission 

site. 

B. Equations for the Neutral Density Downstream of a Single Cylindrical Aperture 

The following sections will develop the equations for modeling the neutral density surrounding a grid cylindrical 

aperture. Two sets of equations will be presented. The first is an expression that was initially derived by Clausing 

and is only valid in the far-field (i.e., far enough from an aperture to consider it a point source) (Refs. 10 and 11). 

The second is a series of equations that can be used to rigorously determine neutral density within and downstream 

of a cylindrical aperture both in the near- and far-fields. 

The NEXT ion thruster utilizes a two grid system for accelerating beam ions. The upstream, or screen, grid is a 

thin grid with nearly cylindrical apertures and a high open area fraction. The thin screen grid thickness enhances 

thruster perveance while the small surface area reduces the likelihood that a discharge chamber ion will strike it and 

recombine with an electron, thus improving discharge chamber electrical efficiency. On the other hand, the 

downstream, or accelerator, grid is a thick grid with nearly cylindrical apertures and a much smaller open area 

fraction. The thicker accelerator grid enhances service life capability by providing more material for erosion and the 

smaller open area fraction and increased thickness reduce the neutral transmission probability, thus reducing the 

discharge chamber neutral loss rate. 

The two cylindrical apertures that a neutral particle has to pass through to escape the discharge chamber present 

a difficult problem to solve. Fortunately, though, Kuharski et al., modeled a two cylindrical aperture set that had a 

geometry that was very similar to that of the NEXT thruster (Ref. 29). They found that the angular distribution of 

particles exiting the accelerator grid was well modeled by assuming a single accelerator grid cylindrical aperture and 

 
NASA/TM—2010-216930 9 

assuming that the upstream neutral density was merely decreased by 0.93x due to the screen grid. This simplification 

was possible because of the large screen aperture diameter and low thickness relative to that of the accelerator grid. 

This same simplifying assumption will be exploited in this study. 

1. Far-field Equation 

Clausing was the first to develop a far-field expression for the angular distribution of particles downstream of a 

cylindrical aperture (Ref. 10). His equation assumes that the aperture is a point source and that the flux of particles 

from the walls of the aperture is linear as a function of axial thickness, which is a very accurate assumption over a 

range of aperture thickness-to-radius ratios (i.e., t/R) that span from 0 to about 8 (Ref. 30). The angular distribution 

of particles downstream of a cylindrical aperture, T(α), in the far-field is given by (Refs. 10 and 11):

 

         

 

 

 











































 



  

    





 



  































   

     























































     













     



t

2 if α arctan

2

t tan α

1 2 κ

3 π

4 κ

and

t

2 if α arctan

2

t tan α

2

t tan α 1 1

1 2 κ 3 π

4

2

t tan α 1

2

t tan α

2

t tan α 1 κ arcsin π2 1

T α

2

3

2

2

(7)

where 

t 4

4 2

t 4 t

2

2





    (8) 

Here, t is the accelerator aperture thickness and the bar symbol denotes non-dimensionalizing by dividing the 

thickness with the accelerator aperture radius, R. 

Figure 3 shows the geometry for a single aperture that is treated as a point source. It can easily be shown that: 

2 2 L  r  z (9) 

and 

  2 2 r z

z cos



  (10) 

Equations (9) and (10), along with T(α), can be substituted into Equation (3) to determine the downstream neutral 

density from a cylindrical aperture in the far-field. Note that all variables are independent of dAo because a point 

source is assumed, so that integral reduces to π·R2

. The resulting equation for the far-field neutral density, nfar, is: 

 

 2

3

2 2

2 2

2 o far scrn

r z

z

r z

z T arccos

4

n R

n r, z F







































    (11) 
NASA/TM—2010-216930 10 

Here, Fscrn represents the correction factor to account for the upstream screen grid, which is 0.93 for the NEXT grids 

as discussed earlier. This correction factor can also be set equal to 1 if no is the density upstream of accelerator grid. 

The far-field equation for a cylindrical aperture can be re-expressed in non-dimensional form by: 

 

 2

3

2 2

2 2 far scrn

r z

z

r z

z T arccos

4

1

n r, z F







































   (12) 

The bar symbol above each variable denotes a non-dimensional variable. Neutral density is non-dimensionalized 

with the upstream neutral density, no, and r and z are non-dimensionalized with the accelerator aperture radius, R. 

2. Near- and Far-field Equations 

Soulas developed equations for rigorously determining neutral density distribution surrounding a cylindrical 

aperture in the free molecular flow regime in the near- and far-fields (Ref. 30). His equations assume that the flux of 

particles from the walls of the aperture is linear as a function of axial thickness. Although he developed equations 

for the upstream, interior, and downstream regions of a cylindrical aperture, only the equations for the interior and 

downstream regions will be presented here. This section will only briefly review the development of these 

equations, although the full equation development can be found in Reference 30. 

The neutral density downstream of a cylindrical aperture is made up of two components: particles that have a 

direct line-of-sight to a downstream location from the reservoir and from the aperture cylindrical walls. In nondimensional form, as indicated by the bar over the variable, the downstream neutral density, nnear, is given by: 

 

   

     

      









  

      

     











  

      

    

 

 









wf

wi

f

i

of

oi

f

i

z

z

w

2

3

2 2 w

w w

r

r

o

2

3

o 2o 2 2

scrn o near

d dz

z z r 1 2 r cos

z 1 r cos

d dr

z r r 2 r r cos

z r

4

F n r, z

 (13) 

Upstream Reservoir, 

no 

Downstream Region

r

z L

Cylindrical 

Aperture 

α

n

Figure 3.—Geometry for determining downstream 

neutral density in the far-field. 
NASA/TM—2010-216930 11 

where all dimensions (r, z, t, ro, and zw) are non-dimensionalized with the aperture radius, R, and the wall flux, Γw, is 

non-dimensionalized with the reservoir flux, Γo. The wall flux was assumed to be linear and its non-dimensionalized 

equation is given by (Ref. 30): 

w z  a  b  z (14) 

where the above constants a and b are given by: 

a 1  (15) 

and 

t

2 1 b     (16) 

The first integral on the right hand side of Equation (13) represents the contribution of the reservoir to the 

downstream neutral density while the second integral represents the cylindrical wall’s contribution. The limits of 

integration are a function of the downstream location (i.e., r and z). The contribution of the reservoir and cylindrical 

wall areas depends on which portion of each area has a direct line-of-sight to the downstream location. This is 

illustrated in Figures 4 and 5. As shown in the figures, all reservoir and cylindrical wall areas have a direct line-ofsight to downstream locations within region I. But only a portion of the reservoir and cylindrical wall areas have a 

direct line-of-sight to downstream locations within regions II and III. And no reservoir particles have a direct-line of 

sight to downstream locations in region IV. Regions V and IV of Figures 4 and 5, respectively, lie within the walls 

of the cylinder. The reason for the separation of regions II and III of both areas and the further division of 

regions IIIa and IIIb for the reservoir was that it was mathematically necessary to correctly integrate these regions 

(Ref. 30). 

 Though complicated, the division of the downstream area into the regions illustrated in Figures 4 and 5 were 

used to define a set of equations with different limits of integration that can rigorously determine the neutral density 

downstream of a cylindrical aperture. The base equation for the downstream density, nnear, in non-dimensional form 

is given by: 

n r, z F n r,z nwallr,z near  scrn  up  (17) 

where the equations for the upstream (nup) and wall (nwall) contributions are listed in Tables 2 and 3, respectively, 

with the appropriate regional border conditions to determine which region, and therefore equation, to use. As before, 

Fscrn represents the correction factor to account for the upstream screen grid, which is 0.93 for the NEXT thruster. 

Because of the linear wall flux assumption, the double integrals of Equation (13) could be reduced to single integrals 

in Tables 2 and 3. Note that these equations are applicable to the aperture interior and downstream regions, and that 

the origin for these equations is the upstream entrance to the cylindrical aperture, as shown in Figures 4 and 5, 

unlike the far-field equation. The equations for the variable limits of integration used in Tables 2 and 3 are listed in 

Table 4. 
NASA/TM—2010-216930 12 

Upstream 

Reservoir, no 

Region I 

t

Region II 

Region III 

Downstream 

Region Shaded Wall 

Portions Have 

Direct Line-of-sight 

to Downstream 

Region 

Region IV 

R

r

z

Cylindrical 

Aperture Crosssection 

Figure 4.—Definition of downstream regions for the development of the variable limits of 

integration for the neutral particle flux from an upstream reservoir.

Figure 5.—Definition of downstream regions for the development of the variable limits of 

integration for the neutral particle flux from the cylindrical aperture walls.

Upstream 

Reservoir, no 

Region I 

R

t

Region II 

Region IIIa 

Region IIIb 

Region IV 

Cylindrical 

Aperture Crosssection 

Downstream 

Region Shaded Reservoir 

Portions Have 

Direct Line-of-sight 

to Downstream 

Region 

Region V 

r

z
TABLE 2.—EQUATIONS FOR THE CONTRIBUTION OF THE UPSTREAM RESERVOIR TO THE 

DOWNSTREAM NEUTRAL DENSITY FOR THE DIFFERENT REGIONS IN FIGURE 4 

Region Equation Conditionals 

I            

φd z φ sinr rz rz

φ cosr21z

φ sinr

rz

φ cosr

π 2

z

z,rn

π0

2 2 2

22

22 2 2 2

22

up 



 







     1 r 

II 

                  

 

         

φ

d z φ sinr rz rz

φ cosr21z

φ sinr

rz

φ cosr

π 2

z

φ

d z φ sinr rz rz

φ cosr

φ,z,rr2

φ,z,rrz

φ sinr

rz

φ cos

φ,z,rrr

π 2

z

z,rn

π

z,r φ

2 2 2

22

22 2 2 2

22

z,r φ

0

2 2 2

22

22

oII 2 oII 2 2 2

22

oII

up

oI

oI





 







   





 





 

    



z

t

r1  and tz 

IIIa 

 

  

         

  

           

 

 

     

  

           

φd

rz

φ cosr

φ,z,rr2

φ,z,rrz

φ sinr

rz

φ cos

φ,z,rrr

rz

φ cosr21z

φ sinr

rz

φ cosr

π 2

z

φd

rz

φ cosr

φ,z,rr2

φ,z,rrz

φ sinr

rz

φ cos

φ,z,rrr

rz

φ cosr

φ,z,rr2

φ,z,rrz

φ sinr

rz

φ cos

φ,z,rrr

π 2

z

z,rn

π

z,r φ

22

oIII 2 oIII 2 2 2

22

oIII

22 2 2 2

22

z,r φ

z,r φ

22

oIII 2 oIII 2 2 2

22

oIII

22

oII 2 oII 2 2 2

22

oII

up

oI

oI

maxo





   

  





  





   

  

  

 

  



 





 





 1 t

z 1 z

t r z

t

2

and tz 

IIIb  

 

     

  

           

φd

rz

φ cosr

φ,z,rr2

φ,z,rrz

φ sinr

rz

φ cos

φ,z,rrr

rz

φ cosr21z

φ sinr

rz

φ cosr

π 2

z

z,rn

π

z,r φ

22

oIII 2 oIII 2 2 2

22

oIII

22 2 2 2

22

up

oI





   

  





   



 





 





 1 z

t

2r1 t

z 1 z

t

2

and tz 

IV   0z,rnup





  1 z

t

2r

and tz 

V   0z,rnup

 1 r  and tz 

NASA/TM—2010-216930 13
TABLE 3.—EQUATIONS FOR THE CONTRIBUTION OF THE CYLINDRICAL WALLS TO THE 

DOWNSTREAM NEUTRAL DENSITY FOR THE DIFFERENT REGIONS IN FIGURE 5 

Region Equation Conditionals 

I      

       

   

     

 

φd

φ cosr21rz

φ cosr21rbzzba

φ cosr21rtz

φ cosr21rbtzzba

φ cosr21r

1 φ cosr

π 2

1

z,rn

π0

22

2

2 2

2

2 wall 







 













     1 r 

II 

     

       

   

         

     

  

 

 

       

   

     

 

 

φd

φ cosr21rz

φ cosr21rbzzba

φ cosr21rtz

φ cosr21rbtzzba

φ cosr21r

1 φ cosr

π 2

1

φd

φ cosr21r

φ,z,rzz

φ cosr21rb

φ,z,rzzzba

φ cosr21rtz

φ cosr21rbtzzba

φ cosr21r

1 φ cosr

π 2

1

z,rn

π

z,r φ

22

2

2 2

2

2

z,r φz,r φ

2 2

wII

2

2

wII

2

2

2 wall

wI

wt

wI









 













   









 



 











    





  1 z

t

2r1 And tz 

III      

       

   

         

     

 

φd

φ cosr21r

φ,z,rzz

φ cosr21rb

φ,z,rzzzba

φ cosr21rtz

φ cosr21rbtzzba

φ cosr21r

1 φ cosr

π 2

1

z,rn

π

z,r φ

2 2

wII

2

2

wII

2

2

2 wall

wt









 



 











     



  1 z

t

2r and tz 

IV   0z,rnup

 1 r  and tz 

NASA/TM—2010-216930 14
NASA/TM—2010-216930 15 

TABLE 4.—VARIABLE LIMITS OF INTEGRATION USED IN TABLES 2 AND 3 

Table use Type Equation 

Table 2 

Radial 

 

   

z

t 1

sin φ z

t r cos φ 1 zt r

r r, z, φ

2

2

oII



 





 





      



 

   

z

t 1

sin φ z

t r cos φ 1 zt r

r r, z,φ

2

2

oIII



 





 





      



Angular 

 



































 







   







 





   





 





  



z

t 1 zt r 2

z

t r

z

t 1 1

φ r, z arccos

2 2

oI

  



 





   t r

z φomax r, z π arcsin

Table 3 

Axial      

r 1

r t t 2 z 2 r t z cos φ z r, z,φ 2

2

wII 

         

Angular 

 











































   

  



z

t 2 r 1

t

z r 1 2

φ r, z arccos

2

wI

  



 



  r

1

φ wt r, z arccos

 
NASA/TM—2010-216930 16 

C. Convex Grid Geometric Equations and Downstream Neutral Density Equations From a Grid 

The following section will develop the geometric equations for determining downstream neutral density from a 

spherically-domed convex grid with a hexagonal array of apertures. The first part develops the geometric relation 

between the downstream neutral density and the grid surface. The second part develops aperture locations on the 

grid surface for a hexagonal array of holes. A cylindrical coordinate system will be used for the downstream neutral 

density location. 

Figure 6 shows the geometric relation between the downstream neutral density located at (r, z) in a cylindrical 

coordinate system and a neutral emission site located on a spherically-domed convex grid. The angle between the 

normal of the neutral emission surface and the desired downstream neutral density location is given by αgrid and is 

separated by distance Lgrid. The axial distance, z, is defined from the downstream center of the domed grids. The 

angle β is defined as the angle from the downstream neutral density to the emission site. The variable γ is the angle 

from the spherical dome center and γmax is the angle to the edge of the perforated grid. The radius of curvature for 

the grid’s spherical dome is given by Rgrid. 

To solve for the downstream neutral density, it is necessary to define αgrid and Lgrid as functions of γ, β, Rgrid, r, 

and z. Using the law of cosines, it can be shown that αgrid in non-dimensional form is given by: 

 

       





















































  

 

             















  



   

grid grid

grid

grid

2 2 2

grid

2

grid

2

grid

grid

grid

R L

cos( )

z R

2

R L z R tan r 2 z R tan r cos( ) cos( )

z R

r, z, , arccos (18) 

where all dimensions are non-dimensionalized with the accelerator aperture radius. Using the law of cosines again, 

the length Lgrid can be shown to be given in non-dimensional form by: 

    

2

grid grid

2 2

grid

2

grid L r, z, ,  r  R sin()  2 r R sin() cos()  z  R  1 cos() (19) 

r 

β

Lgrid

αgrid

Rgrid

γ

z 

Neutral Emission 

Site 

γmax

n Downstream Location

Domed Grid 

Figure 6.—The geometric relation between the downstream 

neutral density located at (r, z) and the neutral emission 

site located on spherically-domed convex grid.
NASA/TM—2010-216930 17 

For both of the aforementioned equations, all dimensions were non-dimensionalized with the accelerator aperture 

radius, R. 

A typical hexagonal pattern of apertures is shown in Figure 7. It will be assumed that a center aperture exists 

within a spherical aperture pattern whose boundary is defined with a maximum arc angle γmax (see Fig. 6). All other 

aperture locations can be determined as a function of γ and β in spherical coordinates relative to this center aperture 

by adding apertures along the sides of successively larger hexagons centered about this center aperture. Such an 

approach simplifies calculations because of symmetry because the hexagonal pattern can be divided into 6 sides. 

For this study, γ and β were defined as functions of three integer values and the aperture center-to-center 

spacing, lcc, explained below and illustrated in Figure 7. Note that for a spherically-domed grid, lcc is an arc length. 

Along a given hexagon’s side, k, and p·lcc distance along a major radius (e.g., along β = 0), where p is the number of 

apertures, aperture centerlines are located m·lcc away from the major radius. It can be shown that γ and β as 

functions of k, m, and p are given by: 

























































 



  







 



   

 

6 p cos

m

2

p

cos arctan

6

cos

R

p l

grid

cc

m,p (20) 

and 

  3

k 1

6 p cos

m

2

p

arctan

6 k,m,p



  





























 



  



    (21) 

Figure 7.—Typical hexagonal aperture pattern for a sphericallydomed convex grid. The center aperture is depicted in gray. 

γp,m

βp,m,k

p·lcc

m·lcc

k = 1

k = 2

k = 3 k = 4 k = 5

k = 6
NASA/TM—2010-216930 18 

To determine the γ and β locations of every aperture along a hexagon for a given p from the center aperture, γ and β

are calculated from m = 0 to p – 1, and these calculations for β are repeated from k = 1 to 6 to account for all 6 sides 

of the hexagon (because of symmetry, γ is independent of k). This process of determining aperture locations in γ and 

β coordinates as functions of k, m, and p by building successively larger hexagonal patterns of apertures is continued 

until p·lcc/Rgrid ≤ γmax. Beyond this, aperture radial locations must be checked against the aperture pattern boundary 

to ensure that apertures at γm,p > γmax are excluded because they lie beyond this boundary. This process is continued 

until p reaches a maximum value, pmax, which is defined as: 

































 



  

 

6

cos

R

l p

grid

cc

max max (22) 

where the brackets about the right hand side of the equation denote the ceiling function. 

With the locations of each individual aperture, αgrid, and Lgrid defined in the above equations, the downstream 

neutral density from a convex spherical dome can now be determined. The far-field neutral density can be solved as 

was done for Equation (12). It can be shown that with the far-field angular distribution function, the neutral density 

downstream of a convex grid, nfar-gridsum, is given in non-dimensional form by: 

        

 

       

 

  





























  

  

 

       



   

            

pmax

p 1

6

k 1

p 1

m 0

m,p max

m,p max

m,p k,m,p

2

grid

scrn grid m,p k,m,p grid m,p k,m,p

2

grid

scrn grid grid

far gridsum

0 if

and

 if

L r, z, ,

cos r, z, ,

4

F T r, z, ,

L r, z, 0, 0

cos r, z, 0, 0

4

F T r, z, 0, 0 n r, z

 (23) 

The first term on the right hand side of the equation above is the center aperture’s contribution to the downstream 

neutral density while the second term includes all other apertures. The second term on the right hand side also 

includes conditional statements to ensure that apertures located beyond the aperture pattern boundary are not 

included in the calculation. The variables αgrid, Lgrid, γ, β, and pmax are defined in Equations (18) to (22), respectively. 

This equation can be generalized further by accounting for variations in the upstream neutral density within the 

discharge chamber as a function of γ and β (i.e., across the grids). It is assumed that upstream neutral density 

distribution can be described by: 

n γ,β n Dγ,β o o   (24) 

where D is a distribution function and no is now an average upstream neutral density. So, Equation (23) can be 

generalized with Equation (24) to yield: 
NASA/TM—2010-216930 19 

         

 

         

 



 





























  

  

 

          



   

                 

pmax

p 1

6

k 1

p 1

m 0

m,p max

m,p max

m,p k,m,p

2

grid

scrn grid m,p k,m,p m,p k,m,p grid m,p k,m,p

2

grid

scrn grid grid

far gridsum

0 if

and

 if

L r, z, ,

cos r, z, ,

4

F T r, z, , D ,

L r, z, 0, 0

cos r, z, 0, 0

4

F T r, z, 0, 0 D 0, 0 n r, z

 (25) 

Equations (23) and (25) are non-dimensionalized with the average discharge chamber upstream neutral density, no. 

Further away from the grid, it will be shown that it is reasonable to neglect individual apertures and treat the 

entire grid surface as a neutral-emitting surface. The generalized solution can be solved by starting with 

Equation (3) and noting that: 

dA R sin(γ) dγ dβ 2 o grid     (26) 

Equations (24) and (26) can be substituted into Equation (3) to yield the neutral density downstream of a convex 

grid, nfar-gridint, in non-dimensional form: 

      

   



       

  

2 π

0

γ

0

2

grid

grid

grid

oa scrn 2grid

far gridint

max

sin(γ) dγ dβ

L r, z, γ,β

cos α r, z, γ,β D(γ,β) T α r, z, γ,β 4 πR F F n (r, z)

 (27) 

Here, the equation is non-dimensionalized with the average discharge chamber upstream neutral density (no), αgrid, 

Lgrid, and T are given by Equations (18), (19), and (7), respectively, and Foa is the accelerator grid open area fraction 

that is given by: 

2

cc

2

oa 3 l

2 π R F   

 (28) 

Equation (28) is only valid as long as lcc is very small in comparison to Rgrid. 

Equation (17), which is valid in the near- and far-fields, can also be applied to the grids. This is done by 

transforming r and z of the aperture coordinate system in Equation (17) to the grid coordinate system (i.e., by 

defining r and z in the aperture coordinate system as functions of γ, β, Rgrid, r, and z in the grid coordinate system of 

Figure 6). The contribution of all apertures can then be summed in a manner similar to Equation (25). It can be 

shown that the result in non-dimensional form is: 

         

       

  

























  

         



             

max p

p 1

6

k 1

p 1

m 0

m,p max

near near m,p k,m,p near m,p k,m,p m,p k,m,p m,p max

near gridsum near near near

0 if

and

n r r, z, , , z r, z, , D , if

n r, z n r r, z, 0, 0 , z r, z, 0, 0 D 0, 0

 (29) 

 
NASA/TM—2010-216930 20 

where nnear, γ, β, and pmax, are given by Equations (17), (20), (21), and (22), respectively, D is given by 

Equation (24), and 

r  L r, z, ,sin r, z, , near grid grid (30) 

and 

z L r, z, ,  cos r, z, ,  t near  grid    grid    (31) 

The variables αgrid and Lgrid are given by Equations (18) and (19), respectively. Equations (29) to (31) are 

non-dimensionalized with the average discharge chamber neutral density, no. 

D. Downstream Neutral Density Equation for a Neutralizer 

For the neutralizer, the gas upstream of the cathode orifice is collisional, so that the aforementioned equations for 

the “beaming” effects of cylindrical apertures in the free molecular flow regime do not apply. Although one option 

is to treat the gas exiting the cathode as a simple choked flow through an orifice, the discharge between the 

neutralizer cathode and its keeper and the ion beam will undoubtedly affect the angular distribution of neutral gas in 

the downstream region. Unfortunately, accurately modeling such effects is beyond the scope of this study. However, 

because the neutralizer discharge would likely randomize the spatial velocity distribution of the neutral gas particles, 

this study will assume that the neutrals exit the neutralizer keeper orifice diffusely. 

Figure 8 shows the geometric relation between the downstream neutral density located at (r, z) in a cylindrical 

coordinate system and the neutralizer keeper orifice. The angle between the normal of the neutralizer keeper orifice 

emission area and the downstream neutral density is given by αneut and is separated by distance Lneut. The axial and 

radial distances, z and r, are referenced to the downstream center of the domed grids, as was done for the grids in 

Figure 6. The angle β is defined as being 0° to the neutralizer centerline. The variables znoff and rnoff are the axial and 

radial offsets, respectively, of the neutralizer keeper orifice emission area center from the downstream center of the 

domed grid. The variables rno and θ are the radius and azimuthal angle for the neutralizer keeper orifice area, 

respectively. 

Figure 8.—The geometric relation between the downstream neutral density 

located at (r, z) and the neutralizer keeper orifice emission area located on 

the downstream end of the neutralizer. 

r 

β

Lneut

αneut

Rgrid

z 

Neutralizer Keeper 

Orifice 

(Relative Size 

Exaggerated for Clarity) 

γmax

n 

Downstream Location

Domed Grid 

znoff

rnoff

z - znoff 

θ

rno
NASA/TM—2010-216930 21 

To solve for the downstream neutral density, it is necessary to define αneut and Lneut as functions of r, z, β, rno, 

and θ. It can be shown that αneut is given in non-dimensional form by 

   











 

     L r, z, ,r ,

z z

r, z, ,r , arccos

neut no

noff

neut no (32) 

The length Lneut can be shown to be given in non-dimensional form by: 

 

 2

noff no noff

2 2

noff

2

noff no 2 2noff

neut no

 r r 2 r r cos( ) r 2 r r 2 r r cos( ) r cos( ) z z

L r, z, ,r ,

                   

  

 (33) 

For both of the aforementioned equations, all dimensions were non-dimensionalized with the accelerator aperture 

radius, R. 

With αneut and Lneut defined in the above equations, the downstream neutral density from the neutralizer can now 

be determined. The neutral density within the keeper orifice can be solved as was done for Equation (11) and noting 

that: 

2 n no

o,neut

neut m R

4

n c m        (34) 

where m ̇ neut is the neutralizer mass flow rate, no,neut is a reservoir neutral density, Rno is the neutralizer keeper orifice 

radius, and mn is the molecular mass of the neutral gas. It can be shown that the neutral density downstream of a 

neutralizer, nneut, is given in non-dimensional form by: 

 

   

 

























    

 

        

 

 



noff

noff

2

0

R

0

no no

no 2neut

neut no

o 2no 2 n

neut

neut

0 if z z

and

r dr d if z z L r, z, ,r ,

cos r, z, ,r ,

m c R n

m

n r, z,

no 

 (35) 

The variables αneut and Lneut are given by Equations (32) and (33), respectively. The neutralizer neutral density was 

non-dimensionalized with average discharge chamber upstream neutral density, no. The conditionals ensure that 

there is no contribution from the neutralizer for axial locations less than znoff because neutralizer neutrals have no 

direct line-of-sight to those axial locations. Note that the neutralizer downstream density is a function of β, unlike 

the equations for the downstream neutral density from the grids, which are nearly axisymmetric so that β can be 

ignored. 

E. Background Vacuum Facility Neutrals 

The background neutrals within the vacuum facility are those particles emitted from the thruster (i.e., neutrals, 

beam ions, charge-exchange ions, and neutralizer ions) that have been reflected from a facility surface. It is assumed 

in this study that free molecular flow conditions exist within the facility and that facility neutral particle velocities 

are isotropic. And although the vacuum facility particles will typically have a lower temperature than those particles 

emanating from the thruster (see, for example, Ref. 36 for NEXT ion thruster temperatures), it will be assumed that 

vacuum facility neutrals reflected from thruster surfaces will still have vacuum facility wall temperatures and are 

reflected diffusely. Finally, it is assumed that the neutral density throughout the vacuum facility is uniform, which is 

a reasonable assumption for thruster operation in large vacuum facilities. 

 
NASA/TM—2010-216930 22 

Because of the aforementioned assumptions, the inclusion of vacuum facility neutrals is straightforward. The 

background facility neutral density need only be added to the neutral densities from the neutrals escaping from the 

grids and neutralizer. The background facility neutral density, ntank, as a function of pressure and wall temperature is 

given in non-dimensional form by: 

B tank o

tank

o

tank

tank k T n

p

n

n n     (36) 

Here, ptank and Ttank are the vacuum facility background pressure and wall temperature and kB is the Boltzmann 

constant. The facility neutral density was non-dimensionalized with average discharge chamber upstream neutral 

density, no. 

F. Total Neutral Density Downstream of an Ion Thruster 

As discussed earlier, the downstream neutral density contributions of each neutral source can be added together 

because ion thrusters typically operate within the free molecular regime downstream of the thruster. So, the total 

neutral density downstream of an ion thruster, ntot, is given in non-dimensional form by: 

tot grid neut tan k n (r, z,)  n (r, z)  n (r, z,)  n (37) 

Here, ngrid is the discharge chamber contribution given by either Equations (25), (27), or (29); nneut is the neutralizer 

contribution given by Equation (35); and ntank is the vacuum facility contribution given by Equation (36). Note that 

only the neutralizer density is a function of β, which is 0° in the direction of the neutralizer. Although the hexagonal 

pattern of the apertures within the grid leads to a dependence on β for Equations (25) and (29), it is neglected in this 

equation because the pattern is nearly axisymmetric for the typically small accelerator grid apertures. 

All of the terms within Equation (37) are non-dimensionalized with the average discharge chamber neutral 

density, no. It can be shown that this neutral density is given by: 

    max oa c scrn 2

B o n grid

n d b

o 2 k T m R 1 cos F K F

J

J 1

J

J

2

1 1

e

m m J

n          



 

  

 









 (38) 

Here, m ̇ d is the total discharge flow rate, Jb is the beam current, e is the charge of an electron (i.e., 1.602×10–19 C), 

J

++/J+

 is the discharge chamber total doubly-to-singly-charged ion current ratio, and To is the discharge chamber 

neutral gas temperature. The variable Kc is the transmission probability of neutrals through an accelerator grid 

aperture and is given by (Ref. 11): 

             



K 2 T cos sin d

2

0

c (39) 

IV. Results 

The following sections will present the results of the neutral density model developed in this study. Neutral 

densities from a single aperture and a NEXT grid will be presented, and near- and far-field models will be 

compared. The neutral density downstream of a NEXT ion thruster will be presented with and without neutralizer 

flow and a vacuum facility background pressure. Finally, the impact of past simplifying assumptions for predicting 

downstream neutral densities will be assessed. 
NASA/TM—2010-216930 23 

A. Downstream Neutral Density From a Single Aperture 

The neutral density surrounding a single aperture is shown in Figure 9 for a NEXT accelerator grid cylindrical 

aperture. The figure was modeled with Equation (17) and the equations listed in Tables 2 to 4 for the aperture 

interior and downstream regions. Figure 9 also includes additional equations derived in Reference 30 for the 

upstream region. The figure shows a depleted density zone immediately upstream of the aperture, and that the 

neutral density drops rapidly from the upstream region to the downstream region, through the aperture. Figure 10 

shows just the neutral density downstream of the aperture. The neutral density drops to less than 10 percent of the 

upstream value within 1 aperture radius of the downstream surface. 

Figure 9.—Neutral density surrounding a single NEXT 

accelerator aperture. 

Figure 10.—Downstream neutral density from a single 

NEXT accelerator gird. 

Aperture Wall 

Aperture Wall 

r

z

nnear

Flow 

Direction 

r

z

nnear

Aperture Exit 

Flow Plane 

Direction 
NASA/TM—2010-216930 24 

The far-field non-dimensional neutral density equation (i.e., Eq. (12)) is compared to that calculated with 

Equation (17) in Figure 11. Here, the neutral density was determined at fixed radial distances from the downstream 

center of the aperture from 2 to 60 radii while the angle was varied (i.e., a spherical coordinate system was used) to 

determine angular distributions. The figure illustrates that at increasing distances from the aperture, the far-field 

angular distribution becomes equal to that of the Equation (17) and that the worst-case error in the far-field equation 

occurs along the axial centerline of the aperture. Also plotted in Figure 11 is the angular distribution of a simple 

diffuse assumption (i.e., T(α) = 1) corrected for the reduced transmission probability due to the aperture walls at 

60 radii from the aperture. The diffuse assumption produces a distribution that is more heavily weighted to larger 

angles and results that range from –36 to 107 percent that of the more rigorous calculation. 

Because Equation (17) represents a rigorous solution for downstream neutral density, the far-field equation of 

Equation (12) was compared to it in Figure 12 to determine the error as a function of non-dimensional radial 

distance from the downstream aperture center at various angles for a NEXT accelerator grid aperture. The figure 

shows that maximum far-field equation errors reduce to about 5 percent at about 25 aperture radii downstream for 

the 0° case, but within 9 aperture radii for the remaining angles. This figure demonstrates that the far-field equation 

will produce accurate results within 9 aperture radii downstream of a NEXT accelerator grid aperture. 

Figure 11.—Non-dimensional neutral density angular distributions for a 

variety of radial distances in a spherical coordinate system that is 

centered at the downstream aperture center for a NEXT accelerator 

aperture. Solid lines were determined from Equation (17) and dashed 

lines from Equation (12). 

Figure 12.—Far-field error as a function of radial distance at various 

angles in a spherical coordinate system that is centered at the 

downstream aperture center for a NEXT accelerator aperture. 

1.00E‐06

1.00E‐05

1.00E‐04

1.00E‐03

1.00E‐02

1.00E‐01

0 10 20 30 40 50 60 70 80 90

Non‐dimensionalDownstreamNeutral Density

Angle, degrees

2 2 ‐ Far‐field

10 10 ‐ Far‐field

60 60 ‐ Far‐field

60 ‐ Diffuse

Non‐dimensional Radial Distance:

0.01

0.1

1

10

100

0 20 40 60 80 100 120 140 160 180 200

Magnitude of Error in Far‐fieldNeutralDensity, %

Non‐dimensionalized Distance

0 degrees

20 degrees

60 degrees

80 degrees

40 degrees

Angle fromDownstreamAperture Center:
NASA/TM—2010-216930 25 

B. Downstream Neutral Density From the Ion Thruster Grids Only 

For this section, only the discharge chamber neutral loss through the grids is modeled. The contributions of the 

neutralizer and background vacuum facility neutrals are neglected so that near- and far-field comparisons can be 

made. In addition, the neutral density within the discharge chamber was assumed to be constant, so that D(γ, β) = 1 

in Equation (24), and the effect of the upstream screen grid was included (i.e., Fscrn = 0.93). 

The non-dimensional neutral density downstream of a NEXT ion thruster from neutrals escaping through the 

grid apertures is shown in Figure 13. This was determined using Equation (29), which includes the rigorous 

calculation of Equation (17) for the downstream neutral density from a single aperture. As the figure illustrates, the 

neutral density drops rapidly from the ion thruster grids into the downstream region due to the rapid expansion of the 

neutrals as they exit the accelerator apertures. 

Close to the thruster grids, the contribution of adjacent apertures to the downstream neutral density of any given 

aperture can be examined. The neutral density downstream of the center aperture of a NEXT accelerator grid (i.e., 

r = 0) from Equation (29) and the adjacent webbing (i.e., r = lcc/2) is compared in Figure 14 to that from a single 

aperture from Equation (17). The number densities at r = 0 and r = lcc/2 converge within 3 aperture radii for the 

multiple aperture case and within 6 radii for the single aperture case. More important, however, are the downstream 

magnitudes. The multiple apertures of the grids increase the downstream neutral density of the center aperture by 

10x compared to that of the single aperture at only about 6.5 aperture radii from the accelerator grid. This 

demonstrates the significant contribution of adjacent apertures to the downstream neutral density of any given 

aperture for the NEXT ion thruster grids. 

Figure 15 also demonstrates another simplifying assumption that can be employed for the NEXT grids. 

Equation (27) neglects the contribution of individual apertures by treating the entire domed grid surface as a neutralemitting surface whose angular distribution is corrected for the “beaming” effects of the cylindrical apertures. 

Because differences between the densities of Equations (27) and (29) decrease to 1 percent within about 5 

accelerator aperture radii from the accelerator grid, such a simplifying assumption also produces accurate neutral 

densities within about 3 mm of the NEXT thruster’s accelerator grid. Because of the ease of using this equation and 

the accuracy of the results, Equation (27) will be used for the remainder of this study. 

Figure 13.—Non-dimensional neutral density downstream of a NEXT 

ion thruster. The neutral density within the discharge chamber is 

assumed constant. 

r, cm 

z, cm 

Grid Location Flow 

Direction 

100 x n
NASA/TM—2010-216930 26 

Figure 14.—Non-dimensional downstream neutral density from a NEXT 

ion thruster along the grid centerline and at r = lcc/2 comparing the 

effect of the multiple apertures of a grid to that of a single aperture. 

Figure 15.—Non-dimensional downstream neutral density from a NEXT 

ion thruster along the grid centerline and at r = lcc/2 using 

Equations (25), (27), and (29). 

Although Equation (17) represents a rigorous solution for downstream neutral density of a single aperture, its use 

for determining the downstream densities from a perforated grid in Equation (29) can be cumbersome and 

computationally costly. However, the far-field assumptions used for solving the downstream neutral density of a 

perforated grid in Equations (25) and (27) are significantly less cumbersome to use and can be computed more quickly. 

So, although Equations (25) and (27) are accurate in the far-field, it would be of value to determine how accurate they 

are close to a NEXT ion thruster. Figure 15 compares the results of Equations (25) and (27) to Equation (29) along the 

centerline of the grids and at a radius of lcc/2 (i.e., between two apertures). As the figure illustrates, the far-field results 

converge rapidly onto the more rigorous results of Equation (29). In fact, differences reduce to 1 percent within about 5 

accelerator aperture radii from the accelerator grid. Because the NEXT accelerator aperture diameter is so small in 

comparison to the overall grid size, the far-field assumptions used in Equations (25) and (27) produce accurate neutral 

densities within about 3 mm of the NEXT thruster’s accelerator grid. 

0

0.05

0.1

0.15

0.2

0.25

0.3

012345678

Non‐dimensionalNeutral Density

Non‐dimensional Axial Location

Center Aperture of Grid,r = 0

OneAperture,r = 0

Center Aperture of Grid,r = lcc/2

OneAperture,r = lcc/2

0

0.05

0.1

0.15

0.2

0.25

0.3

012345678

Non‐dimensionalNeutral Density

Non‐dimensionalAxial Location

r = 0 Far‐field Summation Eq. (25)

r = 0 Rigorous Summation Eq. (29)

r = 0 & r = lcc/2 Far‐field Integration Eq. (27)

r = lcc/2 Rigorous Summation Eq. (29)

r = lcc/2 Far‐field Summation Eq. (25)
NASA/TM—2010-216930 27 

C. Total Downstream Neutral Density 

The contributions of the neutralizer and background vacuum facility neutrals are included in this section. For all 

of the following calculations, a NEXT ion thruster geometry was assumed. In addition, the neutral density within the 

discharge chamber was assumed to be constant so that D(γ, β) = 1 in Equation (24), and the effect of the upstream 

screen grid was included (i.e., Fscrn = 0.93). 

The effect of neutrals from the neutralizer on the downstream neutral density for an NEXT ion thruster operating 

at full power is shown in Figure 16, which excludes the effect of background vacuum facility neutrals and therefore 

represents operation in space. For a NEXT thruster operating at full power, the neutral loss rates are estimated to be 

7.05 sccm for the discharge chamber and 2.8 sccm for the neutralizer. The discharge chamber neutral temperature 

was assumed to be 300 °C while the neutralizer neutral temperature was assumed to be 1100 °C (Ref. 36). The 

figure illustrates the impact of the neutralizer neutrals on the downstream neutral density. The high neutral density 

surrounding the neutralizer was due to neutrals being distributed across such a small keeper orifice area compared to 

that of the grid area. Figure 17 plots the non-dimensional neutral density as a function of axial location with and 

without neutralizer neutrals at radial locations of 18 cm near and opposite the neutralizer (i.e., β = 0° and 180°, 

respectively). The 18 cm radial location was selected because it is the edge of the perforate grid radius. As expected, 

the neutralizer does not contribute to the density upstream of the neutralizer tip because there is no direct line-ofsight there. However, beyond that, the density increases by as much as 68 percent close to the neutralizer. Further 

downstream, the neutralizer’s off-center effect is minimized, but the density is still about 22 percent larger because 

of the neutralizer’s contribution. 

The effect of background neutrals from the vacuum facility on the downstream neutral density for an NEXT ion 

thruster operating at full power is shown in Figure 18. Here, the background pressure was set to 6.7×10–3 Pa 

(5×10–6 torr) with a background neutral temperature of 20 °C. The effect of neutralizer neutrals was included in this 

figure using the same assumptions as for Figure 16. The figure illustrates the significant impact of the facility 

neutrals on the downstream neutral density. Figure 17 includes the effect of vacuum facility neutrals on the nondimensional neutral density as a function of axial location at radial locations of 18 cm near and opposite the 

neutralizer. Although the contributions of the grid and neutralizer are significant close to them, the vacuum facility 

contribution represents the vast majority of the neutral density further away. 

r, cm 

z, cm 

100 x n

Neutralizer Tip 

Grid Location 

Flow 

Direction 

Figure 16.—Non-dimensional neutral density downstream of a NEXT ion thruster 

operating at full power in space. The neutral density within the discharge 

chamber was assumed constant and the neutral loss rate from the neutralizer 

was 2.8 sccm. 
NASA/TM—2010-216930 28 

Figure 17.—Non-dimensional neutral density downstream of a NEXT 

ion thruster operating at full power with and without neutralizer and 

vacuum facility contributions. The two radial locations shown include 

r = 18 cm in the direction of the neutralizer (i.e., β = 0°) and r = 18 cm 

opposite the neutralizer (i.e., β = 180°). The neutralizer tip is located 

at about z = 6 cm. 

Figure 18.—Non-dimensional neutral density downstream of a 

NEXT ion thruster operating at full power with vacuum facility 

neutrals. The neutral density within the discharge chamber 

was assumed constant and the neutral loss rate from the 

neutralizer was 2.8 sccm. Vacuum facility background 

pressure was set to 6.7×10–3 Pa (5×10–6 torr). 

0.001

0.01

0.1

0 20 40 60 80 100 120

Non‐dimensionalNeutral Density

Axial Location, cm

Near Neutralizer ‐ Grid, Neutralizer,& Facility

OppositeNeutralizer ‐ Neutralizer, & Facility

Near Neutralizer ‐ Grid &Neutralizer

OppositeNeutralizer ‐ Grid& Neutralizer

Either Side of Neutralizer ‐ Grid Only

Neutralizer

Tip
NASA/TM—2010-216930 29 

D. Assessment of Past Assumptions 

This section will assess some past simplifying assumptions that were made to more easily determine the 

downstream neutral density. These simplifying assumptions include neglecting the effect of neutralizer neutrals, 

neglecting the “beaming” effect of the cylindrical aperture walls, and assuming a flat grid. All of these assumptions 

will be examined with a NEXT ion thruster operating at full power. 

The impact of the neutralizer was already assessed in the prior section (see Fig. 17). It was shown that 

neglecting the neutralizer’s contribution to the downstream neutral density would lead to a neutral density that is 

22 percent lower than actual in the far-field for a NEXT thruster operating at full power. The impact of neglecting 

the “beaming” effect of the aperture walls is illustrated in Figure 19 along the grid centerline. In this figure, the 

effects of neutralizer and vacuum facility background neutrals were ignored to facilitate direct comparisons. And 

although neutrals were assumed to exit an aperture diffusely, the upstream neutral density was set equal to that 

where the “beaming” effect is included (i.e., the baseline case). The figure shows that the neutral density begins 

higher than the baseline case close to the grids because the diffuse angular distribution contributes more neutrals to 

the grid centerline close to the grid. But further away, this same effect leads to lower neutral densities as the neutrals 

expand more rapidly. So, ignoring the “beaming” effect of the aperture walls leads to an overestimation of 

14 percent close to the grid and an under-estimation that approaches 30 percent in the far-field. 

The impact of assuming a flat grid is also illustrated in Figure 19 along the grid centerline. The upstream neutral 

density was set equal to that of the domed grid case, and the “beaming” effect of the aperture walls is included. The 

figure shows that the neutral density is always higher than the baseline case because the flat grid collimates the 

outgoing neutrals. The flat grid assumption led to neutral density overestimations along the grid centerline of as 

large as 29 percent in the near-field and 14 percent in the far-field. 

Figure 19.—Non-dimensional downstream neutral density and errors 

as a function of axial location along the grid centerline for: a domed 

NEXT grid that includes the “beaming” effect of the aperture walls 

(Clausing, or the baseline case); a domed NEXT grid that excludes 

the “beaming” effect of the aperture walls (diffuse assumption); and 

a flat grid that includes the “beaming” effect of the aperture walls. 

‐40

‐30

‐20

‐10

0

10

20

30

40

0.001

0.01

0.1

0 20 40 60 80 100 120

Error, %

Non‐dimensionalNeutral Density

Axial Location, cm

FlatGrid, Clausing

Domed Grid, Clausing

Domed Grid, Diffuse

FlatGrid, Clausing ‐ Error

Domed Grid, Diffuse ‐ Error
NASA/TM—2010-216930 30 

Conclusions 

The details of a model for determining the neutral density downstream of a gridded ion thruster were presented. 

A rigorous examination of the possible sources of neutrals emanating from and surrounding a NEXT ion thruster 

concluded that the most significant contributors to the downstream neutral density included discharge chamber 

neutrals escaping through the perforated grids, neutrals escaping from the neutralizer, and vacuum facility 

background neutrals. Because ion thrusters were shown to operate in the free molecular flow regime, each neutral 

density source was modeled separately and the results added together. 

For the neutral flux through the grids, near- and far-field equations were presented for rigorously determining the 

neutral density downstream of a cylindrical aperture. These equations included the “beaming” effect of the aperture 

walls. These equations were subsequently integrated into a spherically-domed convex grid geometry with a 

hexagonal array of apertures for determining neutral densities downstream of the ion thruster grids. The neutrals 

escaping from an off-center neutralizer were also modeled assuming diffuse neutral emission from the neutralizer 

keeper orifice. Finally, the effect of the surrounding vacuum facility neutrals was included and assumed to be 

constant. 

The model was then used to predict the neutral density downstream of a NEXT ion thruster with and without 

neutralizer flow and a vacuum facility background pressure. Individual aperture results showed that far-field 

equation errors reduced to about 5 percent within 9 aperture radii, indicating that the far-field equation will produce 

accurate results close to a NEXT accelerator grid aperture. The neutral density from the NEXT thruster grids as a 

function of axial distance was found to decrease rapidly due to the rapid expansion of the neutrals as they exit the 

accelerator apertures. Because the NEXT accelerator aperture diameter is so small in comparison to the overall grid 

size, the far-field equations were shown to produce accurate neutral densities within about 3 mm of the NEXT 

thruster’s accelerator grid. One of these far-field equations neglected the contribution of individual apertures by 

treating the entire domed grid surface as a neutral-emitting surface whose angular distribution is corrected for the 

“beaming” effects of the cylindrical apertures. Because this equation was least cumbersome and computationally 

efficient, it was used for the remainder of the study. 

The impacts of the neutralizer neutral loss rate and the surrounding vacuum facility neutrals were separately 

assessed for a NEXT ion thruster operating at full power. Beyond the neutralizer, the density increased by as much 

as 68 percent at a radial location of 18 cm close to the neutralizer. Further downstream, the neutralizer’s off-center 

effect was minimized, but the density was still about 22 percent larger because of the neutralizer’s contribution. 

Proper modeling of charge-exchange production in the ion beam requires incorporating the effects of neutralizer 

neutral efflux. Vacuum facility effects were assessed at a background pressure set to 6.7×10–3 Pa (5×10–6 torr). 

Although the contributions of the grid and neutralizer were still significant close to them, the vacuum facility 

contribution represented the vast majority of the neutral density further away. 

The impact of past simplifying assumptions for predicting downstream neutral densities were also assessed for a 

NEXT ion thruster. Neglecting the “beaming” effect of the aperture walls along the grid centerline led to an 

overestimation of neutral density by 14 percent close to the grid and an under-estimation that approached 30 percent 

in the far-field. Assuming a flat grid led to neutral density overestimations along the grid centerline of as large as 

29 percent in the near-field and 14 percent in the far-field. 

References 

1. Rawlin, V.K., et al., “An Ion Propulsion System for NASA’s Deep Space Missions,” AIAA Paper 99-4612, 

September 1999. 

2. Polk, J.E., et al., “Demonstration of the NSTAR Ion Propulsion System on the Deep Space One Mission,” IEPC 

Paper 2001-075, October 2001. 

3. Brophy, J.R., et al., “The DS1 Hyper-extended Mission,” AIAA Paper 2002-3673, July 2002. 

4. Brophy, J.R., et al., “Implementation of the Dawn Ion Propulsion System,” AIAA Paper 2005-4071, July 2005. 

5. Brophy, J.R., et al., “Dawn Ion Propulsion System: Initial Checkout After Launch,” Journal of Propulsion and 

Power, vol. 25, no. 6, November-December 2009, pp. 1189–1202. 

6. Patterson, M.J. and Benson, S.W., “NEXT Ion Propulsion System Development Status and Performance,” 

AIAA Paper 2007-5199, July 2007. 

7. Benson, S.W. and Patterson, M.J., “NEXT Ion Propulsion System Progress Towards Technology Readiness,” 

AIAA Paper 2008-5285, July 2008. 
NASA/TM—2010-216930 31 

8. Pollard, J.E., et al., “Spatially-Resolved Beam Current and Charge-State Distributions for the NEXT Ion 

Engine,” To be presented at the 46th AIAA/ASME/SAE/ASEE Joint Propulsion Conference in July 2010. 

9. Miller, J.S., et al., “Xenon Charge Exchange Cross Sections for Electrostatic Thruster Models,” Journal of 

Applied Physics, vol. 91, no. 3, 2002, pp. 984–991. 

10. Clausing, P., “The Formation of Beams in Molecular Streaming,” Z. Physik, vol. 66, 1930, pp. 471–476. 

11. Dayton, B.B., “Gas Flow Patterns at Entrance and Exit of Cylindrical Tubes,” National Symposium on Vacuum 

Technology Transactions, Pergamon Press, New York, 1956, pp. 5–11. 

12. Reynolds, T.W. and Richley, E.A., “Distribution of Neutral Propellant from Electric Thrusters onto Spacecraft 

Components,” NASA Technical Note D-5576, December 1969. 

13. Hall, D.F., Newnam, B.E., and Womack, J.R., “Electrostatic Rocket Exhaust Effects on Solar-Electric 

Spacecraft Subsystems,” AIAA Paper 1969-271, March 1969. 

14. Samanta Roy, R.I., et al., “A Hybrid Particle–in-Cell/Fluid Model of Ion Thruster Plumes,” AIAA Paper 1994-

0421, January 1994. 

15. Samanta Roy, R.I., Hastings, D.E., and Gatsonis, N.A., “Ion Thruster Plume Modeling for Backflow 

Contamination,” Journal of Spacecraft and Rockets, vol. 33, no. 4, July-August 1996, pp. 525–534. 

16. Samanta Roy, R.I. and Hastings, D.E., “Three-Dimensional Modeling of Dual Ion-Thruster Plumes for 

Spacecraft Contamination,” Journal of Spacecraft and Rockets, vol. 33, no. 4, July-August 1996, pp. 519–524. 

17. Samanta Roy, R.I., Hastings, D.E., and Gatsonis, N.A., “Numerical Study of Spacecraft Contamination and 

Interactions by Ion-Thruster Effluents,” Journal of Spacecraft and Rockets, vol. 33, no. 4, July-August 1996, 

pp. 535–542. 

18. Wang, J., et al., “Deep Space One Investigations of Ion Propulsion Plasma Interactions: Overview and Initial 

Results,” AIAA Paper 1999–2971, June 1999. 

19. Wang, J., Brinza, D., and Young, M., “Modeling of Ion Propulsion Plasma Environment for Deep Space 1,” 

AIAA Paper 2000-3528, July 2000. 

20. Wang, J., Cao, Y., and Kafafy, R., “Electric Propulsion Plume Modeling Using Parallel Supercomputers,” 

AIAA Paper 2006-3559, June 2006. 

21. Cho, M., et al., “Interaction Between High Voltage Solar Array and Ion Thruster Plasma,” IEPC Paper 2003-

053, March 2003. 

22. Davis, V.A., et al., “Ion Engine Generated Charge Exchange Environment: Comparison Between NSTAR 

Flight Data and Numerical Simulations,” AIAA Paper 2000-3539, July 2000. 

23. Boyd, I.D., et al., “Monte Carlo Simulation of Neutral Xenon Flows in Electric Propulsion Devices,” Journal of 

Propulsion and Power, vol. 14, no. 6, November-December 1998, pp. 1009–1015. 

24. VanGilder, D.B., Font, G.I., and Boyd, I.D., “Hybrid Monte Carlo Particle–in-Cell Simulation of an Ion 

Thruster Plume,” Journal of Propulsion and Power, vol. 15, no. 4, August 1999, pp. 530–538. 

25. Nishida, M., et al., “DCMC-PIC Analysis of a Plume from a Small Ion Engine,” IEPC Paper 2001-110, 

October 2001. 

26. Fergason, S.D., “Ion Thruster Plume Simulation Using Clustered PC Workstations,” AIAA Paper 2005-966, 

January 2005. 

27. Kameyama, I., Monheiser, J.M., and Wilbur, P.J., “A Numerical Model of Thruster-Exhaust Plumes,” IEPC 

Paper 1997-180, August 1997. 

28. Mandell, M.J., et al., “Modeling the NEXT Multithruster Array Test With Nascap-2k,” IEEE Transactions on 

Plasma Science, vol. 36, no. 5, October 2005, pp. 2309–2318. 

29. Kuharski, R.A., et al., “Ion Engine Neutralizer Erosion in Lab and Space,” AIAA Paper 2005-3880, July 2005. 

30. Soulas, G.C., “Neutral Flux and Density Surrounding a Cylindrical Aperture in the Free Molecular Flow 

Regime,” to be published as a NASA Technical Memorandum. 

31. Soulas, G.C., et al., “NEXT Single String Integration Test Results,” AIAA Paper 2009-4816, August 2009. 

32. O’Hanlon, J.F., A User’s Guide to Vacuum Technology, John Wiley & Sons, Inc., New York, 1980, pp. 25–36. 

33. Mikellidis, I.G., et al., “Neutralizer Hollow Cathode Simulations and Validations with Experiments,” 

AIAA Paper 2009-5196, August 2009. 

34. Mikellidis, I.G., Private Communication, June 11, 2010. 

35. Gombosi, T.I., Gas Kinetic Theory, Cambridge University Press, Cambridge, 1994, pp. 18–35. 

36. Van Noord, J.L., “NEXT Ion Thruster Thermal Model,” AIAA Paper 2007–5218, July 2007. 
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Modeling Neutral Densities Downstream of a Gridded Ion Thruster 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

The details of a model for determining the neutral density downstream of a gridded ion thruster are presented. An investigation of the 

possible sources of neutrals emanating from and surrounding a NEXT ion thruster determined that the most significant contributors to the 

downstream neutral density include discharge chamber neutrals escaping through the perforated grids, neutrals escaping from the

neutralizer, and vacuum facility background neutrals. For the neutral flux through the grids, near- and far-field equations are presented for 

rigorously determining the neutral density downstream of a cylindrical aperture. These equations are integrated into a spherically-domed 

convex grid geometry with a hexagonal array of apertures for determining neutral densities downstream of the ion thruster grids. The 

neutrals escaping from an off-center neutralizer are also modeled assuming diffuse neutral emission from the neutralizer keeper orifice. 

Finally, the effect of the surrounding vacuum facility neutrals is included and assumed to be constant. The model is used to predict the 

neutral density downstream of a NEXT ion thruster with and without neutralizer flow and a vacuum facility background pressure. The 

impacts of past simplifying assumptions for predicting downstream neutral densities are also examined for a NEXT ion thruster.

15. SUBJECT TERMS

Ion thruster; Ion engine; Free molecular flow; Neutral density 

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