《航天推进 Space Propulsion》（英文版）Lecture 13-14: Electrostatic Thrusters

1 Introduction Electrostatic thrusters (“ion engines”) are the best developed type of electric propulsion device, dating in conception to the ‘50’s,(1) and having been demonstrated in space in 1964 on a suborbital flight of the SERT I spacecraft(2). The early history

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Lecture 13-14 Electrostatic Thrusters 1 Introduction Electrostatic thrusters c ion engines")are the best developed type of electric propulsion device, dating in conception to the 50's, )and having been demonstrated in space in 1964 on a suborbital flight of the SERT I spacecraft(2). The early history and concepts are well documented(),(3), and evolved through progressive refinements of various types of ion beam sources used in Physics laboratories, the and long life for these sources to be used in space. Of the various configurations improvements being essentially dictated by the needs for high efficiency, low mas discussed for example in Ref. 3(ca. 1973), only the electron bombardment noble gas type, plus(in Europe) the radio-frequency ionized thruster 4and(in Japan)th Electron Cyclotron Resonance thruster, have survived. Other interesting concepts such as Cesium Contact thrusters and duo-plasmatron sources have been largely abandoned, and one new special device, the field Emission Electrostatic(5)thruster has been added to the roster the electron bombardment thruster itself has evolved in the same time interval from relatively deep cylindrical shapes with uniform magnetic fields produced by external coils and with simple thermoionic cathodes, to by permanent magnets, and with hollow cathode plasma bridges used as cathode ed shallow geometrics using sharply nonuniform magnetic field configurations, prod and neutralizer. Where a typical ion production cost was quoted in Ref. (3)as 400 600 ev for Hg at 80% mass utilization fraction, recent work with ring-cusp thrusters has yielded for example a cost of 116 ev in Xenon at the same utilization o. Such reductions make it now possible to design for efficient operation(above 80% with environmentally acceptable noble gases at specific impulses below 3000 sec, a goal that seemed elusive a few years back. The major uncertain issues in this field seem now reduced to lifetime(measured in years of operation in orbit)and integration problems, rather than questions of cost and physical principle or major technological hurdles. Extensions to higher power(tens of kw)and higher specific impulse(to 7,000-8,000 s)are now being pursued by NASa for planetary missions requiring high△V 2 Principles of Operation Electrostatic thrusters accelerate heavy charged atoms(ions) by means of a purely electrostatic field Magnetic fields are used only for auxiliary purposes in the ionization chamber. It is well known that electrostatic forces per unit area(or energies per unit volume)are of the order of =c E, where e is the strength of the field(volts/m)and E, the permittivity of vacuum E,=8.85x10-12 Farad ypical maximum fields, as limited by vacuum breakdown or shorting due to imperfections are of the order of 10 V/m, yielding maximum force densities of roughly 5N/m2=5x105 atm This low force density is one of the major drawbacks of electrostatic engines and can be compared to force densities of the order of 10 N/m in self-magnetic devices such as MPD thrusters, or to the typical gas pressures of 10-10'N/m in chemical rockets. Simplicity and efficiency must therefore compensate for this disadvantage. 16.522, Space Propulsion Lecture 13-14 Page 2 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 2 of 25 Lecture 13-14 Electrostatic Thrusters 1 Introduction Electrostatic thrusters (“ion engines”) are the best developed type of electric propulsion device, dating in conception to the ‘50’s,(1) and having been demonstrated in space in 1964 on a suborbital flight of the SERT I spacecraft(2). The early history and concepts are well documented(1),(3), and evolved through progressive refinements of various types of ion beam sources used in Physics laboratories, the improvements being essentially dictated by the needs for high efficiency, low mass and long life for these sources to be used in space. Of the various configurations discussed for example in Ref. 3 (ca. 1973), only the electron bombardment noble gas type, plus (in Europe) the radio-frequency ionized thruster(4) and (in Japan) the Electron Cyclotron Resonance thruster, have survived. Other interesting concepts, such as Cesium Contact thrusters and duo-plasmatron sources have been largely abandoned, and one new special device, the Field Emission Electrostatic(5) thruster has been added to the roster. The electron bombardment thruster itself has evolved in the same time interval from relatively deep cylindrical shapes with uniform magnetic fields produced by external coils and with simple thermoionic cathodes, to shallow geometrics using sharply nonuniform magnetic field configurations, produced by permanent magnets, and with hollow cathode plasma bridges used as cathode and neutralizer. Where a typical ion production cost was quoted in Ref. (3) as 400- 600 eV for Hg at 80% mass utilization fraction, recent work with ring-cusp thrusters has yielded for example a cost of 116 eV in Xenon at the same utilization(6). Such reductions make it now possible to design for efficient operation (above 80%) with environmentally acceptable noble gases at specific impulses below 3000 sec, a goal that seemed elusive a few years back. The major uncertain issues in this field seem now reduced to lifetime (measured in years of operation in orbit) and integration problems, rather than questions of cost and physical principle or major technological hurdles. Extensions to higher power (tens of kW) and higher specific impulse (to 7,000 – 8,000 s) are now being pursued by NASA for planetary missions requiring high ∆V . 2 Principles of Operation Electrostatic thrusters accelerate heavy charged atoms (ions) by means of a purely electrostatic field. Magnetic fields are used only for auxiliary purposes in the ionization chamber. It is well known that electrostatic forces per unit area (or energies per unit volume) are of the order of 1 2 E 2 0 ε , where E is the strength of the field (volts/m) and 0 ε the permittivity of vacuum 12 Farad 8.85 10 m − 0 ⎛ ⎞ ε= × ⎜ ⎟ ⎝ ⎠. Typical maximum fields, as limited by vacuum breakdown or shorting due to imperfections, are of the order of 106 V/m, yielding maximum force densities of roughly 2 -5 5 N m = 5×10 atm. This low force density is one of the major drawbacks of electrostatic engines, and can be compared to force densities of the order of 104 N/m2 in self-magnetic devices such as MPD thrusters, or to the typical gas pressures of 106 -107 N/m2 in chemical rockets. Simplicity and efficiency must therefore compensate for this disadvantage

The main elements of an electrostatic thruster are summarized in Fig. 1. Neutral propellant is injected into an ionization chamber, which may operate on a variety of principles(electron bombardment, contact ionization, radiofrequency ionization.) The gas contained in the chamber may only be weakly ionized in the steady state, but ions are extracted preferentially to neutrals, and so, to a first approximation, we may assume that only ions and electrons leave this chamber. The ions are accelerated by a strong potential difference va applied between perforated plates (grids) and this same potential keeps electrons from also leaving through these grids. The electrons from the ionization chamber are collected by an anode and in order to prevent very rapid negative charging of the spacecraft(which has very limited electrical capacity), they must be ejected to join the ions downstream of the accelerating grid. To this end the electrons must be forced to the large tive potential of the accelerator (which also prevails in the beam), and they must then be injected into the beam by some electron-emitting device(hot filament, plasma The net effect is to generate a jet of randomly mixed (but not recombined) ions and electrons, which is electrically neutral on average, and is therefore a plasma beam The reaction to the momentum flux of this beam constitutes the thrust of the device Notice in Fig. 1 that, when properly operating, the accelerator grid should collect no ions or electrons, and hence its power supply should consume no power only apply a static voltage. On the other hand, the power supply connected to the neutralizer must pass an electron current equal in magnitude to the ion beam current and must also have the full accelerating voltage across its terminals; it is therefore this power supply that consumes (ideally)all of the electrical power in the device. In summary, the main functional elements in an ion engine are the ionization chamber the accelerating grids, the neutralizer, and the various power required Most of the efforts towards design refinement have concentrated on the ionization chamber, which controls the losses, hence the efficiency of the device, and on the power supplies, which dominate the mass and parts count. the grids are, of course, an essential element too and much effort has been spent to reduce their erosion by stray ions and improve its collimation and extraction capabilities. The neutralizer was at one time thought to be a critical item but experience has shown that with good design, no problems arise from it. Following a traditional approach(1) 3), we will first discuss the ion extraction system then turn to the chamber and other elements 3 Ion Extraction and Acceleration The geometry of the region around an aligned pair of screen and accelerator holes shown schematically in Fig. 2(from Ref. 7). The electrostatic field imposed by the strongly negative accelerator grid is seen to penetrate somewhat into the plasma through the screen grid holes. This is fortunate in that the concavity of the plasma surface provides a focusing effect which helps reduce ion impingement on the accelerator. The result is an array of hundreds to thousands of individual ion beamlets which are neutralized a short distance downstream as indicated the potential diagram in Fig 2 shows that the screen grid is at somewhat lower potential than the plasma in the chamber. Typically the plasma potential is near that of the anode in the chamber, while the screen is at cathode potential(some 30-60 volts lower, as we will see). This ensures that ions which wander randomly to the vicinity 16.522, Space Propulsion Lecture 13-14 Page 3 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 3 of 25 The main elements of an electrostatic thruster are summarized in Fig. 1. Neutral propellant is injected into an ionization chamber, which may operate on a variety of principles (electron bombardment, contact ionization, radiofrequency ionization…). The gas contained in the chamber may only be weakly ionized in the steady state, but ions are extracted preferentially to neutrals, and so, to a first approximation, we may assume that only ions and electrons leave this chamber. The ions are accelerated by a strong potential difference Va applied between perforated plates (grids) and this same potential keeps electrons from also leaving through these grids. The electrons from the ionization chamber are collected by an anode, and in order to prevent very rapid negative charging of the spacecraft (which has very limited electrical capacity), they must be ejected to join the ions downstream of the accelerating grid. To this end, the electrons must be forced to the large negative potential of the accelerator (which also prevails in the beam), and they must then be injected into the beam by some electron-emitting device (hot filament, plasma bridge…). The net effect is to generate a jet of randomly mixed (but not recombined) ions and electrons, which is electrically neutral on average, and is therefore a plasma beam. The reaction to the momentum flux of this beam constitutes the thrust of the device. Notice in Fig. 1 that, when properly operating, the accelerator grid should collect no ions or electrons, and hence its power supply should consume no power, only apply a static voltage. On the other hand, the power supply connected to the neutralizer must pass an electron current equal in magnitude to the ion beam current, and must also have the full accelerating voltage across its terminals; it is therefore this power supply that consumes (ideally) all of the electrical power in the device. In summary, the main functional elements in an ion engine are the ionization chamber, the accelerating grids, the neutralizer, and the various power supplies required. Most of the efforts towards design refinement have concentrated on the ionization chamber, which controls the losses, hence the efficiency of the device, and on the power supplies, which dominate the mass and parts count. The grids are, of course, an essential element too, and much effort has been spent to reduce their erosion by stray ions and improve its collimation and extraction capabilities. The neutralizer was at one time thought to be a critical item, but experience has shown that, with good design, no problems arise from it. Following a traditional approach(1),(3), we will first discuss the ion extraction system, then turn to the chamber and other elements. 3 Ion Extraction and Acceleration The geometry of the region around an aligned pair of screen and accelerator holes is shown schematically in Fig. 2 (from Ref. 7). The electrostatic field imposed by the strongly negative accelerator grid is seen to penetrate somewhat into the plasma through the screen grid holes. This is fortunate, in that the concavity of the plasma surface provides a focusing effect which helps reduce ion impingement on the accelerator. The result is an array of hundreds to thousands of individual ion beamlets, which are neutralized a short distance downstream, as indicated. The potential diagram in Fig. 2 shows that the screen grid is at somewhat lower potential than the plasma in the chamber. Typically the plasma potential is near that of the anode in the chamber, while the screen is at cathode potential (some 30-60 volts lower, as we will see). This ensures that ions which wander randomly to the vicinity

of the extracting grid will fall through its accelerating potential, while electrons(even those with the full energy of the cathode-anode voltage)are kept inside. The potential far downstream is essentially that of the neutralizer, if its electron-emission grid, in order to prevent backflow of electrons from the neutralizer through the or capacity is adequate this potential is seen to be set above that of the accelera accelerating system. In addition, by making the total voltage", VI, larger than the Net voltage",VN, the ion extraction capacity of the system is increased with no on lerated a third grid ( decelerator grid") is added to more closely define and control VN, and the neutralizer is set at approximately the same potential as this third grid It is difficult to analyze the three-dimensional potential and flow structures just described. It is however, easy and instructive to idealize the multiplicity of beamlets s a single effective one-dimensional beam the result is the classical Child-Langmuir space charge limited current equation The elements of the derivation are outlined below a)Poissons equation in the gap: dφ b) Ion continuity en v,=]= constant (2) c) Electrostatic ion free-fall: 2e(- Combining these equations, we obtain a 2n order, nonlinear differential equation for 9(x). The boundary conditions are 中(0)=0,中(X=d)= (4) In addition, we also impose that the field must be zero at screen grid This is because(provided the ion source produces ions at a sufficient rate) negative screen field would extract more ions which would increase the"in transit positive space charge in the gap. This would then reduce the assumed negative screen field and the process would stop only when this field is driven to near zero (positive fields would choke off the ion flux). At this point the grids are automaticall extracting the highest current density possible and are said to be space charge limited 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 4 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 4 of 25 of the extracting grid will fall through its accelerating potential, while electrons (even those with the full energy of the cathode-anode voltage) are kept inside. The potential far downstream is essentially that of the neutralizer, if its electron-emission capacity is adequate. This potential is seen to be set above that of the accelerator grid, in order to prevent backflow of electrons from the neutralizer through the accelerating system. In addition, by making the “total voltage”, VT, larger than the “Net voltage”, VN, the ion extraction capacity of the system is increased with no change (if VN is fixed) on the final velocity of the accelerated ions. In some designs, a third grid (“decelerator grid”) is added to more closely define and control VN, and the neutralizer is set at approximately the same potential as this third grid. It is difficult to analyze the three-dimensional potential and flow structures just described. It is however, easy and instructive to idealize the multiplicity of beamlets as a single effective one-dimensional beam. The result is the classical Child-Langmuir space charge limited current equation. The elements of the derivation are outlined below: a) Poisson’s equation in the gap: 2 i 2 0 d en = - dx φ ε (1) b) Ion continuity en v j = constant i i = (2) c) Electrostatic ion free-fall: ( ) i i 2e - v = m φ (3) Combining these equations, we obtain a 2nd order, nonlinear differential equation for φ (x ). The boundary conditions are () ( ) 0 = 0, x = d = -Va φ φ (4) In addition, we also impose that the field must be zero at screen grid: x=0 d = 0 dx ⎛ ⎞ φ ⎜ ⎟ ⎝ ⎠ (5) This is because (provided the ion source produces ions at a sufficient rate), a negative screen field would extract more ions, which would increase the “in transit” positive space charge in the gap. This would then reduce the assumed negative screen field, and the process would stop only when this field is driven to near zero (positive fields would choke off the ion flux). At this point, the grids are automatically extracting the highest current density possible, and are said to be “space charge limited

and, from (9),increasing the thrust density requires a reduction of the gap distance d. As noted before, this route is limited by eventual arcing or even by mechanica shorting due to grid warping or imperfections. For thruster diameters of, say, 10-50 cm,gap distances have been kept above 0. 5-1 mm The only other control, at this level of analysis, is offered by increasing the ion can be kept small, higher thrust(Equation 9). In addition to increasing thrust ided d molecular mass, m This allows increased voltages Va(Equation 10), and, prov density, higher molecular mass also reduces the importance of a given ion production cost Ao(See lecture 3), and hence increases the thruster efficiency The effect of ion deceleration past the accelerator grid (either through the use of a decel"grid or by relative elevation of the neutral incorporated in this 1-D model. For the usual geometries, the screen-accelerator gap still controls the ion current(Equation 6 with Va replaced by Vi, and d by da). This is because the mean ion velocity is high(and hence the mean ion density is low ) in the second gap, between the accelerator and the real or virtual decelerator so that no electrostatic choking occurs there. This is schematically indicated in Figure 4 by a break in the slope of the potential at the decelerator. More specifically it can be shown that Equation(6) still controls the current provided that (1R)(4+2R1);R= (for equal gaps, this is satisfied for all r between 0 and 0. 75, for instance at higher R, the second gap limits current). Accepting, then, Equation(6), the thrust is again given by f= mc, where m has not changed, but c is proportional to v,12.Hence we obtain instead of (9) F 8 18。(M a 9 Eo (12) The last form shows that for a given specific impulse(hence given VN), reducing R=VN/Vr increases thrust. It does so by extracting a higher ion current through the flux-limiting first gap Returning to Equation(6), if we imagine a beam with diameter D, we would predict a total beam current of 丌4 e 13) d where P is the so-called "perveance"of the extraction system. Equation(13)shows that this perveance should scale as the dimensionless ratio D, so that for example he same current can be extracted through two systems, one of which is twice the grid sp Itio 16.522, Space P rtinez- sanchez Lecture 13-14 Prof. Manuel martinez Page 6 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 6 of 25 and, from (9), increasing the thrust density requires a reduction of the gap distance d. As noted before, this route is limited by eventual arcing or even by mechanical shorting due to grid warping or imperfections. For thruster diameters of, say, 10-50 cm., gap distances have been kept above 0.5-1 mm. The only other control, at this level of analysis, is offered by increasing the ion molecular mass, mi. This allows increased voltages Va (Equation 10), and, provided d can be kept small, higher thrust (Equation 9). In addition to increasing thrust density, higher molecular mass also reduces the importance of a given ion production cost ∆φ (See lecture 3), and hence increases the thruster efficiency. The effect of ion deceleration past the accelerator grid (either through the use of a “decel” grid, or by relative elevation of the neutralizer potential) can be easily incorporated in this 1-D model. For the usual geometries, the screen-accelerator gap still controls the ion current (Equation 6 with Va replaced by VT, and d by da). This is because the mean ion velocity is high (and hence the mean ion density is low) in the second gap, between the accelerator and the real or virtual decelerator, so that no electrostatic choking occurs there. This is schematically indicated in Figure 4 by a break in the slope of the potential at the decelerator. More specifically, it can be shown that Equation (6) still controls the current provided that ( ) ( ) 1 2 d 12 12 N a T d V > 1 - R 1+ 2R ; R = d V (11) (for equal gaps, this is satisfied for all R between 0 and 0.75, for instance; at higher R, the second gap limits current). Accepting, then, Equation (6), the thrust is again given by F m c A A = i , where m A i has not changed, but c is proportional to 1 2 VN . Hence we obtain instead of (9) 2 2 32 12 TN T N 1 2 -3 2 2 a a a F8 8 8 VV V V = RR A9 9 d 9 d d 00 0 ⎛⎞ ⎛⎞ = = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ εε ε (12) The last form shows that for a given specific impulse (hence given VN), reducing R=VN/VT increases thrust. It does so by extracting a higher ion current through the flux-limiting first gap. Returning to Equation (6), if we imagine a beam with diameter D, we would predict a total beam current of 1 2 2 32 32 0 TT i 42 e D Ι V = PV 49 m d = ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ π ε (13) where P is the so-called “perveance” of the extraction system. Equation (13) shows that this perveance should scale as the dimensionless ratio 2 2 D d , so that, for example the same current can be extracted through two systems, one of which is twice the size of the other, provided diameter and grid spacing are kept in the same ratio

While the one-dimensional model is important in identifying many of the governing effects and parameters, its quantitative predictive value is limited Three-dimensional effects, such as those of the ratio of extractor to accelerator diameter the finite grid thicknesses, the potential variation across the beam etc(see Fig. 2)are all left out of account. So are also the effects of varying the properties of the upstream plasma such as its sheath thickness, which will vary depending on the intensity of the ionization discharge, for example. Also, for small values of R=VN/Vt, the beam potential(averaged in its cross-section) cannot be expected to approach the deep negative value of the accelerator, particularly for the very flattened hole geometry prevalent when d/D is also small. Thus, the perveance per hole can be expected to be of the functional form dD。ta (14) where the subscripts(s)and (a) identify the screen and accelerator respectively, t is a grid thickness, and vo is the discharge voltage, which in a bombardment ionizer controls the state of the plasma. These dependencies were examined for a 2-grid extractor in an Argon-fueled bombardment thruster in Ref. 7. Some of the salient conclusions of that study will be summarized here (1)Varying the screen hole diameter Ds while keeping constant all the ratios d/Ds, Da/Ds, etc. )has only a minor effect, down to D 0.5mm if the alignment can be maintained. This confirms the dependence upon the ratio (2) The screen thicknesses are also relatively unimportant in the range studied (t/D≈0.2-0.4) (3)Reducing R=VN/V always reduces the perveance, although the effect tends to disappear at large ratios of spacing to diameter(d/Ds), where the effect of the negative accelerator grid has a better chance to be felt by the ions. The value of d/Ds at which r becomes insensitive is greater for the smaller R values (4) For design purposes, when VN and not Vr is prescribed, a modified perveance va (called the"current parameter"in Ref. 7)is more useful. As Equation (13)shows, one would expect this parameter to scale as r-3/2, favoring low values of R(strong accel-decel design). This trend is observed at low R, but due to the other effects mentioned, it reverses for R near unity as shown in Fig. 5. This is especially noticeable at small gap/ diameter ratios when a point of maximum extraction develops at R07-0.8, which can give currents as high as those with R.0. 2. However, as Fig. 5 also shows the low -R portion of the operating curves will give currents which are independent of the gap/diameter ratio(this is in clear opposition to the 1-D prediction of Equation 13). Thus, the current, in this region, is independent of both d and Ds. This opens up a convenient design avenue using low R values: Fix the smallest distance d compatible with good dimensional control, then reduce the diameter ds to the smallest practicable size(perhaps 0.5 mm). This will 16.522, Space Propulsion Lecture 13-14 Prof. Manuel martinez-Sanchez Page 7 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 7 of 25 While the one-dimensional model is important in identifying many of the governing effects and parameters, its quantitative predictive value is limited. Three-dimensional effects, such as those of the ratio of extractor to accelerator diameter, the finite grid thicknesses, the potential variation across the beam etc. (see Fig. 2) are all left out of account. So are also the effects of varying the properties of the upstream plasma, such as its sheath thickness, which will vary depending on the intensity of the ionization discharge, for example. Also, for small values of R=VN/VT, the beam potential (averaged in its cross-section) cannot be expected to approach the deep negative value of the accelerator, particularly for the very flattened hole geometry prevalent when d/D is also small. Thus, the perveance per hole can be expected to be of the functional form aa s D ssss T d Dt t V P = p , , , ,R, DDDD V ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (14) where the subscripts (s) and (a) identify the screen and accelerator respectively, t is a grid thickness, and VD is the discharge voltage, which in a bombardment ionizer controls the state of the plasma. These dependencies were examined for a 2-grid extractor in an Argon-fueled bombardment thruster in Ref. 7. Some of the salient conclusions of that study will be summarized here: (1) Varying the screen hole diameter Ds while keeping constant all the ratios (d/Ds, Da/Ds, etc.) has only a minor effect, down to D 0.5 s ≈ mm if the alignment can be maintained. This confirms the dependence upon the ratio d/Ds. (2) The screen thicknesses are also relatively unimportant in the range studied ( s t/D 0.2 - 0.4 ≈ ). (3) Reducing R=VN/VT always reduces the perveance, although the effect tends to disappear at large ratios of spacing to diameter (d/ Ds), where the effect of the negative accelerator grid has a better chance to be felt by the ions. The value of d/ Ds at which R becomes insensitive is greater for the smaller R values. (4) For design purposes, when VN and not VT is prescribed, a modified perveance 3 2 N I V ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (called the “current parameter” in Ref. 7) is more useful. As Equation (13) shows, one would expect this parameter to scale as R-3/2, favoring low values of R (strong accel-decel design). This trend is observed at low R, but, due to the other effects mentioned, it reverses for R near unity, as shown in Fig. 5. This is especially noticeable at small gap/diameter ratios, when a point of maximum extraction develops at R~0.7-0.8, which can give currents as high as those with R~0.2. However, as Fig. 5 also shows, the low – R portion of the operating curves will give currents which are independent of the gap/diameter ratio (this is in clear opposition to the 1-D prediction of Equation 13). Thus, the current, in this region, is independent of both d and Ds. This opens up a convenient design avenue using low R values: Fix the smallest distance d compatible with good dimensional control, then reduce the diameter Ds to the smallest practicable size (perhaps 0.5 mm). This will

allow more holes per unit area(if the hole spacing varies in proportion to their size), hence more current per unit grid area. Due to this circumstance Ref. 7 recommends low R designs. (5) The perveance generally increases as Da/Ds increases, with the exception of cases with R near unity when an intermediate D,/D=0.8 is optimum (6)Increasing Vo/VT, which increases the plasma density appears to flatten th contour of the hole sheath( 8), which reduces the focusing of the beam. this results in direct impingement on the screen, and, in turn forces a reduction of the beam current Some appreciation for the degree to which Child-Langmuir's law departs from the observed current extraction capacity of real devices can be obtained from the data for the 30 cm. J-series thruster, as reported for example in Ref.( 9). In this case,we have d=0.5 mm ta=ts=0. 38 mm, Ds=1.9 mm, Da=1. 14 mm, and a total of 14860 holes. We will refer to data in Xe, for VNET/V=0.7 and Vo=31.2 Volts. VBeam=1200 V. Table Ill of Ref (9)then gives a beam current ]8=4.06 A. The correlation given in the same reference for various propellants is aM+25%6 (15) here a is a double-ion correction factor, given as 0.934 for this case, and M is the molecular mass in a. m u. The power of 2.2 instead of 1.5 for the effect of extractio oltage is to be noticed. This correlation yields for our case Ib=5.4 A, on the outer boundary of the error band For these data, if we apply the Child-Langmuir law(Equation 13)to each hole (diameter Ds), and use directly the spacing d=0.5 mm, we obtain a hole current of 3.83 mA, or, in total IB=57.1 A, i.e. 14 times too high. an approximate 3-D correction(Ref. s 10a, b)is to replace d by (d+t 2+D2/4 in Child-Langmuir's equation this gives now IB=8.4 A, still twice the experimental value. It is of interest to see how well the data of Rovang and Wilbur(ref. 7) can be extrapolated to the J thruster. We first use the data in Fig. 6a of Ref (7), which are for Ds=2mm ,R=0. 8 Da/Ds=0.66(lowest value measured), and VD/V=0. 1. Corrections for the actual Da/Ds=0.6 and Vo/VT=0.018 can be approximated from Fig. ' s 5 and (6a) of the same reference. The effects of R=0.7 instead of 0.8, as well as of the slightly different Ds should be small, according to Ref. 7. We obtain in this manner IB=5.2 A which is indeed as accurate as the correlation of Equation(15) Additional data on grid perveance are shown and assessed Ref.(10c)in the context of ion engine scaling To complete this discussion two limiting conditions should be mentioned he (a)Direct ion impingement on screen: At low beam current, the screen collects a very small stray current, which is due to charge-exchange ion-neutral collisions in the accelerating gap: after one such collision, the newly formed low speed ion is easily accelerated into the screen. The screen current takes however, a strong upwards swing when the beam current increases beyond some well defined limit. This is due to interception of the beam edges and 16.522, Space Propulsion Lecture 13-14 Prof. Manuel martinez-Sanchez Page 8 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 8 of 25 allow more holes per unit area (if the hole spacing varies in proportion to their size), hence more current per unit grid area. Due to this circumstance, Ref. 7 recommends low R designs. (5) The perveance generally increases as Da/Ds increases, with the exception of cases with R near unity, when an intermediate D /D 0.8 a s ≈ is optimum. (6) Increasing VD/VT, which increases the plasma density, appears to flatten the contour of the hole sheath(8), which reduces the focusing of the beam. This results in direct impingement on the screen, and, in turn, forces a reduction of the beam current. Some appreciation for the degree to which Child-Langmuir’s law departs from the observed current extraction capacity of real devices can be obtained from the data for the 30 cm. J-series thruster, as reported for example in Ref. (9). In this case, we have d=0.5 mm, ta=ts=0.38 mm, Ds=1.9 mm, Da=1.14 mm, and a total of 14860 holes. We will refer to data in Xe, for VNET/VT=0.7 and VD=31.2 Volts. VBeam=1200 v. Table III of Ref. (9) then gives a beam current JB=4.06 A. The correlation given in the same reference for various propellants is ( )2.2 T B 17.5 V 1000 J = + -25% α M (15) where α is a double-ion correction factor, given as 0.934 for this case, and M is the molecular mass in a.m.u.. The power of 2.2 instead of 1.5 for the effect of extraction voltage is to be noticed. This correlation yields for our case IB=5.4 A, on the outer boundary of the error band. For these data, if we apply the Child-Langmuir law (Equation 13) to each hole (diameter Ds), and use directly the spacing d=0.5 mm, we obtain a hole current of 3.83 mA, or, in total IB=57.1 A, i.e., 14 times too high. An approximate 3-D correction (Ref.’s 10a, b) is to replace d2 by 2 2 s s (d + t ) + D /4 in Child-Langmuir’s equation. This gives now IB=8.4 A, still twice the experimental value. It is of interest to see how well the data of Rovang and Wilbur (Ref. 7) can be extrapolated to the Jthruster. We first use the data in Fig. 6a of Ref. (7), which are for Ds=2mm., R=0.8 Da/Ds=0.66 (lowest value measured), and VD/VT=0.1. Corrections for the actual Da/Ds=0.6 and VD/VT=0.018 can be approximated from Fig.’s 5 and (6a) of the same reference. The effects of R=0.7 instead of 0.8, as well as of the slightly different Ds, should be small, according to Ref. 7. We obtain in this manner IB=5.2 A., which is indeed as accurate as the correlation of Equation (15). Additional data on grid perveance are shown and assessed Ref. (10c) in the context of ion engine scaling. To complete this discussion, two limiting conditions should be mentioned here: (a) Direct ion impingement on screen: At low beam current, the screen collects a very small stray current, which is due to charge-exchange ion-neutral collisions in the accelerating gap: after one such collision, the newly formed low speed ion is easily accelerated into the screen. The screen current takes, however, a strong upwards swing when the beam current increases beyond some well defined limit. This is due to interception of the beam edges, and

since the high energy ions are very effective sputtering agents, results in a very destructive mode of operation. All the perveance values reported in Ref. (7), for instance, are impingement-limited, i. e. correspond to the highest current prior to onset of direct impingement (b)Electron back-streaming: For R values near unity the barrier offered by the accelerator negative potential to the neutralizer electrons becomes weak, and beyond some threshold value of R, electrons return up the accelerator potential to the chamber. This results in screen damage, space charg distortion, and shorting of the neutralizer supply Kaufman (oa) gave the theoretical estimate 0.2 (16) D which was confirmed experimentally in Ref. 7, except that it was found to be a somewhat conservative estimate 4 Ion Production 4.1 Physical Process in Electron Bombardment Ionization Chambers In an electron bombardment ionizer, the neutral gas is partially ionized by an auxiliary dC discharge between conveniently located electrodes. Of these, the anode is the same anode which receives the electrons from the ionization process(see Fig 1). The primary electrons responsible for the ionization of the neutral gas are generated at a separate cathode which can be a simple heated tungsten filament, or for longer endurance, a hollow cathode. The cathode-anode potential difference vo is selected in the vicinity of the peak in the ionization cross-section of the propellant gas, which occurs roughly between three and four times the ionization energy i e. around 30-50 Volts for most gases). The structure of the potential distribution in the sheath near the cathode, and the body of the plasma is nearly equipotential, at a the electron saturation level, and so an electron-retarding voltage drop develops Ionization is due both to the nearly mono-energetic primary electrons(with energies of the order of evD)and to the thermalized secondary electrons themselves. These have typically temperatures(Tm)of a few ev, so that only the high energy tail of the Maxwellian energy distribution is above the ionization energy and can contribute to the process but their number density greatly exceeds that of the primaries and both contributions are, in fact, of the same order. It is therefore desirable to maximize the residence time of both types of electrons in the chamber before they are eventually evacuated by the anode this is achieved by means of a suitable distribution of confining magnetic fields. Fig 's 6(a),(b )and (c)show three types of These will be discussed in more detail later, but we note here that magnetic e nce magnetic configurations, of which only the last two are today of practical importa strengths can vary from about 10 to 1000 Guass, depending on type and location 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 9 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 9 of 25 since the high energy ions are very effective sputtering agents, results in a very destructive mode of operation. All the perveance values reported in Ref. (7), for instance, are impingement-limited, i.e., correspond to the highest current prior to onset of direct impingement. (b) Electron back-streaming: For R values near unity, the barrier offered by the accelerator negative potential to the neutralizer electrons becomes weak, and beyond some threshold value of R, electrons return up the accelerator potential to the chamber. This results in screen damage, space charge distortion, and shorting of the neutralizer supply. Kaufman(10a) gave the theoretical estimate max a a a 0.2 R =1- le t exp D D ⎛⎞ ⎛⎞ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ (16) which was confirmed experimentally in Ref. 7, except that it was found to be a somewhat conservative estimate. 4 Ion Production 4.1 Physical Process in Electron Bombardment Ionization Chambers In an electron bombardment ionizer, the neutral gas is partially ionized by an auxiliary DC discharge between conveniently located electrodes. Of these, the anode is the same anode which receives the electrons from the ionization process (see Fig. 1). The primary electrons responsible for the ionization of the neutral gas are generated at a separate cathode, which can be a simple heated tungsten filament, or for longer endurance, a hollow cathode. The cathode-anode potential difference VD is selected in the vicinity of the peak in the ionization cross-section of the propellant gas, which occurs roughly between three and four times the ionization energy (i.e., around 30-50 Volts for most gases). The structure of the potential distribution in the discharge is very unsymmetrical: most of the potential difference VD occurs in a thin sheath near the cathode, and the body of the plasma is nearly equipotential, at a level slightly above that of the anode (typically the anode current density is below the electron saturation level, and so an electron-retarding voltage drop develops). Ionization is due both to the nearly mono-energetic primary electrons (with energies of the order of eVD) and to the thermalized secondary electrons themselves. These have typically temperatures (Tm) of a few eV, so that only the high energy tail of the Maxwellian energy distribution is above the ionization energy and can contribute to the process, but their number density greatly exceeds that of the primaries, and both contributions are, in fact, of the same order. It is therefore desirable to maximize the residence time of both types of electrons in the chamber before they are eventually evacuated by the anode. This is achieved by means of a suitable distribution of confining magnetic fields. Fig.’s 6 (a), (b) and (c) show three types of magnetic configurations, of which only the last two are today of practical importance. These will be discussed in more detail later, but we note here that magnetic field strengths can vary from about 10 to 1000 Guass, depending on type and location

The ions generated in the active part of the discharge chamber are only weakly affected by the magnetic field, and so they wander at random, colliding rarely with neutral atoms before reaching any of the wall surfaces. Since these walls (or the cathode itself)are all negative with respect to the plasma, the ions penetrate the negative sheaths at a velocity of the order of the so-called Bohm velocity, or isothermal ambipolar speed of sound and are then further accelerated in the sheath proper. Those that happen to arrive at one of the extractor hole sheaths become thus the ion beam but those arriving at solid walls collide with them at an energy corresponding to that of the sheath, which often leads to sputtering, and are neutralized. They then return as neutrals to the plasma, where they are again subject to ionization or excitation processes 4.2 Nature of the losses Since electron-ion recombination, even if it did happen in the beam would contribute nothing to the engine thrust, the ionization energy per beam ion is the minimum energy expenditure required This would amount to 10.5 ev in Hg 15.8 in Argon or 400 or more eV. the sources of the additional losses can be identified from thelo 12. 1 ev in Xenon. In reality the energy loss per beam ion ranges from about 10 description of processes in the previous section (a)Some primary electrons reach the anode and surrender their high energy (b)The thermal electrons arrive at the anode with energies of a few ev. (c) Ions that fall to cathode-potential surfaces lose their kinetic energy to them. In addition they also lose the energy spent in their ionization (d) Metastable excited atoms surrender the excitation energy upon wall collision (e)short-lived excited atoms emit radiation, which is mostly lost directly of a different nature are the energy losses required to heat the cathode emitters or, in the case of Hg the vaporizers and chamber walls. Finally, not all the injected gas leaves in the form of ions(only a fraction nu, called the utilization factor"does ).At the best conditions, nu ranges from 75 to 95%. It is of interest to examine the relationship between nu and the degree of ionization a, in the chamber plasma. If ne is the electron(and ion) density the flux of ions being extracted is approximately T,=neva xe(ions/m?/sec) in o here vB is as in Equation(17)and os is the open area fraction of the screen grid The flux of neutrals through the same overall area is In=nnφ 16.522, Space P pessan Lecture 13-14 Prof. Manuel martinez Page 10 of 25

16.522, Space Propulsion Lecture 13-14 Prof. Manuel Martinez-Sanchez Page 10 of 25 The ions generated in the active part of the discharge chamber are only weakly affected by the magnetic field, and so they wander at random, colliding rarely with neutral atoms before reaching any of the wall surfaces. Since these walls (or the cathode itself) are all negative with respect to the plasma, the ions penetrate the negative sheaths at a velocity of the order of the so-called Bohm velocity, or isothermal ambipolar speed of sound, e B i KT V = m (17) and are then further accelerated in the sheath proper. Those that happen to arrive at one of the extractor hole sheaths become thus the ion beam, but those arriving at solid walls collide with them at an energy corresponding to that of the sheath, which often leads to sputtering, and are neutralized. They then return as neutrals to the plasma, where they are again subject to ionization or excitation processes. 4.2 Nature of the Losses Since electron-ion recombination, even if it did happen in the beam, would contribute nothing to the engine thrust, the ionization energy per beam ion is the minimum energy expenditure required. This would amount to 10.5 eV in Hg, 15.8 in Argon or 12.1 eV in Xenon. In reality the energy loss per beam ion ranges from about 100 to 400 or more eV. The sources of the additional losses can be identified from the description of processes in the previous section: (a) Some primary electrons reach the anode and surrender their high energy. (b) The thermal electrons arrive at the anode with energies of a few eV. (c) Ions that fall to cathode-potential surfaces lose their kinetic energy to them. In addition, they also lose the energy spent in their ionization. (d) Metastable excited atoms surrender the excitation energy upon wall collision. (e) Short-lived excited atoms emit radiation, which is mostly lost directly. Of a different nature are the energy losses required to heat the cathode emitters or, in the case of Hg, the vaporizers and chamber walls. Finally, not all the injected gas leaves in the form of ions (only a fraction ηu, called the “utilization factor” does). At the best conditions, ηu ranges from 75 to 95%. It is of interest to examine the relationship between ηu and the degree of ionization, α , in the chamber plasma. If ne is the electron (and ion) density, the flux of ions being extracted is approximately ( ) -1 2 2 Γ φ i e Bs = n v × e ions/m /sec (18) -1 2 s include e in φ where vB is as in Equation (17) and φs is the open area fraction of the screen grid. The flux of neutrals through the same overall area is n n n c = n 4 Γ φ (19)

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