Nuclear Instruments and Methods 189 (1981) 55--70 North-Holland Publishing Company
55
ION EXTRACTION A N D OPTICS ARITHMETIC * J.H. WHEALTON Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830, U.S.A.
Arithmetic algorithms for ion extraction from a plasma are examined, instabilities are delineated, and ancillary results are presented.
Tile mathematical formulation for extraction flora a collisionless plasma is typically taken to be the following Poisson-Vlasov equation:
V% =ffdu- e-s , u" V r f -
}
Z>Zo,
e VrO" Vt).f= 0 ,
(1) (2)
m
where q~ and f a r e the electric potential and ion distribution functions, respectively Boundaries arranged so that either
~=(~0
f=fo
(Dirichlet),
(3)
Vo±f = 0
(Neumann) ,
(4)
or Vr±(~ = O ,
are considered to render the problem well posed. The Dirichlet boundary conditions [eq. (3)] are straightforward to specify, except for the source plasma boundary, where the traditional ploy is to connect with the 1-D collisionless sheath problem [ 1 ],
pedigree formulation of ion extraction for which solution is attempted (we will call this formulation I); this is because no ansatz on the shape of the sheath need be advanced. Higher pedigree solutions are formulated [2,3]. The difficulty in the solution is of course that the extremely nonlinear eq. (1) must be solved without benefit of boundary stabilization. Nevertheless, eqs. ( 1 ) - ( 5 ) are fraught with inconsistency. To begin with, the use of a Vlasov equation instead of a Boltzmann equation [ 4 - 8 ] and the use of the square root term of eq. (5) are the hallmarks of co]lisionless plasma calculations. If this is the case, then why is the Boltzmann distribution electron density term on the RHS of eq. (1) an appropriate electron source term? Would not a better formulation be:
v2¢=f:i dui--fie e
Ue'V~L+ Vr¢'%L =o, m
m" v ~ j l Ot2 d2~(2 ')
2
dz z
#
g ( y ) dy
- J x/[¢~z)--~Cv)] - eo(z)' z < Z o , (5) o
where eqs. ( 1 ) - ( 4 ) hold for some z > Z o and eq. (5) holds for z ~< Zo (a is the inverse ionization mean free path on the scale of a Debye length and g is a source function for ionization). If Zo is chosen sufficiently small so that the problem is one-dimensional, i.e., X Vr~b = 0 ,
(6)
then eqs, (1) -(5) are thought to exhibit the highest * Research sponsored by the Office of Fusion Energy, U.S. Department of Energy, under contract W-7405-eng-26 with the Union Carbide Corporation.
0029-554X/81/0000-0000/$02.50
© 1981 North-Holland
e m
v,~. vu~] = o ,
(7)
z > zo,
(8)
(9)
where fe is the electron distribution function? In fact, such a formulation has been solved for a magnetic field direct-recovery device [9]. This inconsistency can be shown to be moot by demonstrating that the ion optics and the optimum perveance are virtually independent of the shape of the electron density distribution so long as the Debye length is short compared with electrode dimensions (see fig. 1). The second manifestation of the collisionless inconsistency is that eqs. (1) and (2) neglect ionization processes, whereas eq. (5), to which they couple, depends on them for the very existence of a solution (assuming typical boundary data) [10]. It is not enough that the ionization mean free path be large compared with a II. ION SOURCES AND BEAM OPTICS
J.H. Whealton / Ion extraction and optics arithmetic
56
z o
.
1.6
.
.
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.
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"
1 'e' !~
.
-
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•
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"o
E o.8
m
1.
0 t
2
.
t0
5
.'..i
20
50
_1 ....
tOO 200
',
,.
500 tOOO
kTe/e(eV) .
100
50
2O
.
i
J
I0
5
ELECTRODE WIDTH (k o)
Fig. i. Effect of electron temperature on ion optics showing that as long as the Debye length is m u c h sm',dler than the electrode dimensions, t h e optics is unaffected b y the electron distribution. Note no wall or plasma potentials were changed with electron temperature. This is customarily done with such studies to maintain zero net flux walls, b u t here we are studying only the effect o f the electron distribution on ion optics.
Debye length since the criterion is actually that the ionization mean free path be large compared with the conditions for unidimensionality [eq. (6)], and it may take .hundreds of Debye lengths to obtain this for large diameter apertures [11]. Ironically, the more exact one wants to become on the condition of emitter unidimensionality, the more glaring is the collisionless inconsistency. This could be cured in principle by including generation of ions in the volume, but that would have to be done a little more heavyhandedly than is the ~b@) term in eq. (5), since the path length is not uniquely determined from the difference in potential for the multidimensional problem [12t. The least inferior alternative formulation takes the form of eqs. (1)-(5) but does not require eq. (6)for the emitter surface; rather, it selects a surface at Z >> Zo in which the normal field V.L~ matches Oc~/Oz of the solution to eq. (5) over an equipotential (formulation II). The problem with this scheme is that the field in the actual 2-D problem is not even constant over such an equipotential in a multidimensional problem, much less equal to the 1-D result. Even with this scheme (formulation !1), solution is fraught with computational peril since the nonlinear
electron Boltzmann distribution in eq. (1)raises havoc when the slightest excursion from a totally paraxial situation is contemplated. By far the most popular formulation [13-17], since the computational effort is trivial by comparison, is that of eqs. (1)-(4) modified by neglecting the Boltzmann electron term in eq. (i) and instead demanding an emitting surface with zero electric field [ 18,19] (formulation IIl); this is usually accompanied by some noises about space-charge-limited flow being bandied about as rationalization [11,20,21]. Unfortunately, solutions considering electrons show that the electric field is in fact a maximum at the very place where this formulation assumes it to be zero. The emitting surface is adjusted such that the ion current density is constant across it; however, the ion current density is not constant across an equipotential corresponding to that of the first electrode as assumed, as numerious examples herein illustrate. Fortunately, formulations II and III (optimized for uniform j) can yield approximately correct results in many cases [22-29], but these cases are similar in that the optics is paraxial virtually everywhere and thus susceptible to analytic reduction [30-41]; the computational effort of the paraxial solution is trivial in comparison to even the relatively simple algorithms of procedure III. A counter example to the above observation is an application of the procedure III code to tetrodes to examine the effect of field ratio on ion optics [29]. This is a subject fraught with peril since it is well known that the high field ratio enables us to obtain an improvement h! optics by deliberately running overdense in the first gap; this cancels out aberrations due either to the applied fields, in the case of highly aberrated systems [36,42,43] or the radial sheath fields, in the case of an attempt at a pseudo-Pierce design. Since such a mechanism is a strain for procedure II calculations, it is no wonder that the results of these calculations show a diminished effect of the field ratio on the ion optics as compared to the procedure II [42] and procedure I [36] calculations. The computation for procedure III [eq. (2) and eq. (1) without the e -~ term] [44-47] involved the first solutin of the Poisson equation [eq. (1)] for with a fixed ion density p: p =f/'d,,.
(10)
The Vlasov equation [eq. (2)] was solved by repeated computation of ion trajectories from reasonable initial data through the potential ¢ computed by the
J.H. Whealton / Ion extraction and optics arithmetic
57
previous solution of the Poisson equation. Thereafter, the. charge was computed by integration, as in eq. (10). Generally, this was done by finding the centroid of charge and depositing the nodal charge accordingly. The iteration procedure was continued by resolving the Poisson equation for 4)with the new source term p just computed and then using more trajectories from the new q~to solve for a newer P and so on. This iteration procedure is not guaranteed to converge; if it does converge, it is not guaranteed to converge to a unique answer. Before this paper, in fact, no serious mention of the problem has been made. We will, however, show that the procedure does not in general converge and that it will sometimes oscillate between two different results, neither of which is the correct one (this is known as the bunching instability). In any event, inclusion of the electron source term e - e in eq. (1), even with a Dirichlet boundary nearby (procedure II) for constraint-imposed stability, causes more insurmountable computational problems. Direct electron under-relaxation such as shown in fig. 2 of ref. 48 frequently does not converge at all; rather it blows up (this is the runaway instability). This occurs at high currents or when the potential stagnation region (area of impingement on inside of first grid hole) is large on the scale of a Debye length. Parametric cures include making the emitting surface very close to the electrode, making the electron temperature large, or making the mesh very coarse. From the algorithm developed for slot geometry [49,50] or cylinder geometry [51 ], a typical solution with such an algorithm starts only a few Debye lengths back from the first electrode. An example of the latter algorithm is shown in fig. 2. The electron underrelaxation coefficient in these codes is on the order of a few percent so that the computational time was very large. The slot code [49,50] deposits charge on a triangular mesh while the cylindrical code [5]]
deposits on three corners of a rectangular mesh; this produces a chunky charge distribution that hides the Vlasov iteration bunching instability and induces the runaway instability. For the cylindrical case [51], the algorithm attempted to ameliorate the problem by axial mesh stretching and charge smoothing by underrelaxation. The slot code ameliorated the problem by dealing only with cases in which the stagnation region was negligibly small, i.e., Pierce-like geometries [44-47,49,52]. This luxury could not be afforded by the cylindrical opticians since the transparency of the electrodes would be inordinately small. The slot (procedure II) algorithm [49,50] has been utilized for openers for Tokamak Fusion Test Reactor (TFTR) designs [53] and for accelerators for double charge exchange negative ion source [54] arid steering calculations [40]. The cylinder (procedure IT) algorithm [51] has been utilized in tetrode and triode designs for TFTR and Princeton Large Torus (PLT) [55]. In addition, it has been utilized to do steering calculations [56], such as those shown in fig. 3 utilizing a paraxial suppositior~ to do this intrinsically 3-D problem. While minor improvements in procedure II were made (see fig. 3 of ref. 48) with a comparison of expediency made in fig. 4 (compare curves A and B), clearly something more was needed if convergence was desirable in a nontrivial case. That "something" did not turn out to be very profound (see fig. 4 of ref. 48). By accelerating the electron underrelaxation to 100% so that nothing transpired on subsequent iterations (thus ensuring convergence to something or other) and simultaneously preventing the Poisson iteration from running to completion (which is easy for the SOR Poisson iteration scheme) [44] and thus not allowing the runaway instability to develop, convergence without blowup was ensured [48]. The interesting feature about this scheme without status is that wherever testable, it
L
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:
s
,
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Fig. 2. Typical solution o f Poisson--Vlasov equation for procedure II showing the emitting surface close to the f'trst electrode. This triode configuration o f July 1975 is not unlike the proposed PLT accelerator or the O R M A K accelerator. 11. ION SOURCES AND BEAM OPTICS
58
J.IL Whealton l i o n extraction and optics arithmetic
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Fig. 3. Cylinder steering calculation done in 1975 utilizing the paraxial superposition of two cylindrical solutions and a procedure II algorithm.
~O6
-l
1
--
-
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A
/
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-
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agreed with the straight electron under-relaxation scheme; and for extension of the parameters into heretofore unreachable domains (fig. 4, curve C), the results were not unreasonable. Furthermore, the arithmetic was cheap (see fig. 4) by comparison and applicable to any code, since it is explicit. This code has found nmch application [ 15,43,57]. This modest degree o f success from the crassest of heavy-handed computational techniques conceivable inspired my colleagues to consider the matter with the attention it deserved, knowing that a method with status must be just around the corner. Accordingly, the Newton method was applied [58] with much more improvement in computational time and accuracy [59,60] (see fig. 5). This implicit scheme has limitations on applicability to some existing algorithms, but that was of no concern to us. Implicit schemes are far more convergent generally than explicit schemes. Uniqueness and global convergence are proved for this nonlinear iteration scheme, so that the runaway instability is gone away forever. A copy of the algorithm is available [60]. A 3-D code being developed at our institution [ 6 1 - 6 4 ] uses an explicit Taylor series expansion of the electron density about the perturbed potential with remarkable success (see fig. 6) and universality of application.
D = 4.O2 405 ()A
O.3
P/PcL, R E L A T I V E
'1.O PERVEANCE
50
Fig. 4. Comparison of computational time for the three schemes considered as a function of perveance on the scale o f the C h i l d - L a n g m u i r . p e r v e a n c e . The arrow on the abscissa in the perveance where the ion beam is most convergent.
J.H. Whealton l i o n extraction and optics arithmetic
1°5 I- ....... i
. METHOD
I
........ METHOD TI SINGLE PRECISION i 10 4
METHOD Tr DOUBLE PRECISION --a (MESH SIZE)
/
~
-
g o
1
59
An interesting variant was examined by Marder [65], who also did a Taylor series expansion of the ion charge density using an assumed trial function potential variation. (He was also the first to do the electron charge Taylor series, but his work in this regard' went unnoticed.) Marder claims that computation of the entire sheath region is generally possible without the divergence, provided an astute trial function
"~ 10 3
S
H i = F/tO e 0
10 2 I0 0
tO-5
10-lo
10-15
RESIDUAL
Fig. 5. Computational work against the absolute residual for the two methods; the abscissa for Method I is the total number of point successive overrelaxation (SOR) passes; for Method II the abscissa is the number of SOR passes plus the number of Newton iterations divided by the number of mesh points; the ordinate in each case is the maximum absolute residual. The curve for Method I was calculated in IBM 360 single precision (32-bit work len~h); curves for Method II were calculated in both single- and double-precision arithmetic. The number (n × n) labeling each curve is the mesh size for that part of the curve.
Fig. 6. 3-D cylindrically symmetrical sheath region calculation.
is utilized. Again his work goes unnoticed, since there are still algorithms that are occasionally divergent without benefit of this technique [49,50,66]. With new potential precision, as shown in fig. 5, attention focused on the heretofore unmentioned blotch instability induced by inaccuracies in the Vlasov solver (compare fig. 7 with the "cure" of fig. 8). The blotch instability was deemed to be due to inaccuracies in charge deposition (3 nodes out of 4) and possibly, although less likely~ orbit computation. Since the Vlasov computation was not unchanged during the aforementioned evaluation o f the Poisson solver demonstrated in fig. 5, the Vlasov computation emerged as the dominant computation, instead o f being a relatively insignificant fraction of resource utilization. This cylindrical arithmetic uses a deferred limit integrator to explicitly integrate the equations o f motion [67]. The slot arithmetic [49,50] used a constant field in a cell with a local parabolic orbit. Using parabolas with selective refinement [68] obtained from a generally nonconstant intercell field computation, the Vlasov computational time is reduced by a factor of 15 and the total computational time is improved b y a factor o f 10 (see fig. 9) over that in rcfs. 51, 59, 60. In addition, the aforementioned threecornered charge deposition was determined by fitting cubic~ of various refinement trajectory computations as previously described [51,60]. The variation o f computational time with charge deposition refinement is shown in fig. 10, while the need for refinement for reduction of the blotch instability is indicated in fig. 1 I. Nevertheless, a more satisfying solution was sought that would deposit on all four corners o f the rectangular cells. An examination shows that four-node deposition is ambiguous in terms o f the information supplied and that an auxiliary condition must be applied [69], but the unnoticed study of ref. 65 seems to have found a suitable algorithm. Nevertheless, a better solution is to I1. ION SOURCES AND BEAM OPTICS
J.H. Whealton ~1on extraction and optics arithmetic
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Fig. 7. Character plot showing potential (top) and charge deposition (bottom) utilizing the 3-node charge deposition and orbit computation (deferred limit integrater) as in ref. 52. The symbols refer to the fraction of the extreme values for the potential or charge. The potential has its minimum value on the LIIS of fig. 7a and a maximum value elsewhere; the charge has its maximum value on the LHS. The symbols and the positions they denote are as fo[lows: - Potential more extreme than minimum values; O Quantity = 0; F 0 < Quantity < 10-s; E 10-s < Quantity < 10-4; D 10-4 < Quantity < 3 x 1 0 4 ; .3×10-4< Quantity < 10-3; C 10 -3 < Quantity < l 0-2; B 10 .2 < Quantity < 0.1; 1 0.1 < Quantity < 0.3; 3 0.3 < Quantity < 0.5;5 0.5 < Quantity < 0.7; 7 0.7 < Quantity < 0.9; 9 0.9 < Quantity < 0.95; + 0.95 < Quantity < 0.98; * 0.98 < Quantity.
deposit the charge t w o nodes at a time, not three or four, by only worrying about the ion velocity at cellcrossing time and depositing only on the t w o adjacent nodes. This is both more accurate than the three- or four-node depositions and cheaper, since it is a oneline arithmetic statement per deposition as opposed
to fig. I0. All the cubics, centroid determinations, etc., are eliminated. The result is fig. 8, which was referred to previously as "the cure". The effective time saved for the totality o f arithmetic for ahnost comparable accuracy and stability is about a factor of 60, the cheaper one being still slightly more stable
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extraction and optics arithmetic
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(compare fig. 8A with fig. 11E). Now with the blotch instability and the runaway instability gone forever, the bunch instability and the axial artifact rear their ugly heads. The axial artifact has appeared unmentioned on numerous high ion temperature cases [20,21,36,52,43,70-73] and to a lesser extent for low ion temperature [1 I], but only for cylindrical geometry. The problem is that the singular point r = 0 in the cylindrical Vlasov equation requires special treatment, only some of which it is receiving. With the exclusion of the centripetal term in the Vlasov equation, not made in refs. 51 and 65 but subsequently made in ref. 60, it becomes inordinately improbable for a trajectm3~ to be near the axis for finite ion temperature. This is because the azimuthal initial ion velocity distribution function, for openers, has to be continuous, whereas it has routinely been ~vcn a discrete number of speeds. Even
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II. ION SOURCES AND BEAM OPTICS
62
J.H. Whealton ~Ion extraction and optics arithmet&
for a continuous initial radial ion distribution function such a cavalier azimuthal distribution would cause an artifact. We have taken a policy of benign neglect on this matter without ever testing explicitly whether sufficient statistics would cure the problem. The first method of dealing with the axial artifact under this program is to deal with only low ion temperature [ 1 1 , 7 4 - 7 8 ] . This makes the centripetal force term very small and nullifies any bulbous perturbance. The second possibility is to consider high ion temperature in slot geometry where the centripetal force is zero (see fig. 5B of ref. 79). If, in view of these two procedures, we desire more satisfaction in the status of results for high ion temperature cylinders, we can throw away the centripetal force term altogether and insert a Neumann boundary condition [eq. (4)] on the axis (ref. 79, fig. 2 or 5A; also refs. 51, 65, 80). This incredibly flagrant disregard for physical principle has one redeeming feature: for small nonuniformity of acceleration in the region where the centripetal force is important, the emerging angle from the centripetal force region is approximately the same as for a specular reflection, which is what we will obtain by imposition of eq. (4). No claim is made on the radial position error due to this aberrant replacement of physical law, but for neutral beam injectors, unlike neutron generators and the like, only the divergence angle is of importance. The above rationalization raises the question of how much error is made for those axial regions in which the axial acceleration is significant. Consistent with the rigor with which we have pursued this artifact, we will defer this question to a future investigation and leave the subject by suggesting that if the axial region of significant centripetal force is amenable to the paraxial approximation, there will be no significant angular difference between reality and Ncumann boundary imposition. The bunch instability is a manifestation of the Poisson-Vlasov iteration scheme. It occurs in situations where the stagnation region is large. This may happen for a fixed-boundary case by increasing the number of transverse mesh nodes or in a fixed-nodenumber case by decreasing all the radial dimensions. In the former scenario, the number of radial nodes has been increased from 16 to 32, with disastrous results, as shown in fig. 12. Any instability caused by the Poisson-Vlasov iteration scheme may be altered by ion underrelaxation [44,51]. For one-half-ion underrelaxation, the situation comparable to fig. 12 is shown in fig. 13 and for 9/10 under-relaxation, the
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comparison is shown in fig. 14. As can be seen, the instability is mitigated for the half under-relaxation and virtually eliminated for the large under-relaxation. However, for the large under-relaxation, the convergence is slow and, indeed, has not yet been ob-
63
J.H. Whealton ~Ion extraction and optics arithmetic ORNL-DWG 8 0 - 2 8 8 7
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tained for the 48 Vlasov iterations shown here. Fig. l 5 shows a superimposed emittance diagram with each Vlasov iteration at intermediate radial cell number (24) for the three values of ion underrelaxation considered above. The nature o f the oscillatory
instability as a function of Vlasov iterations is shown in fig. 16, which has the same conditions as fig. 12 except for one less Vlasov iteration (47 as opposed to 48). A phenomenon qualitatively similar to the case shown here of increasing the number of radial nodes occurs for decreasing the radial dimensions. Presumably, the bunch instability does not occur for the ion charge Taylor expansion scheme o f ref. 65. In any event," experimental evidence confirms the results of procedure I calculations if some care is taken in their interpretation [81 ]. For the first time, this 2-D code can be applied to steering phenomena for large stagnant potential configurations. It has always been true, and it is substantiated here, that the steering due to displaced apertures is substantially greater [32] than that provided by the most trivial analysis available [34]. Recently, 11. ION SOURCES AND BEAM OPTICS
J.H. lChealton ~Ion extraction and optics arithmetic
64
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J.H. Whealton /Ion extraction and optics arithmetic (a)
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a partial explanation of this cause has been delineated [40], but a total explanation awaits justification or replacement of the space-charge-limited flow field correction [40]. The change o f steering with perveance is shown in fig. 17; for perveance such that no electrode interception occurs (see fig. 18c and d), the steering is independent o f perveance. This further adds to the dimensional analysis argument typically used [11 ] to show that the space-charge-limited flow field correction has inadequate foundation. Variation of ion density in the plasma might be a source of beam steering; however, a variation o f 3% in the ion current density over the width o f a slot causes only a 0.2 ° divergence (see fig. 19). Large a c c e l - d e c e l systems are o f interest for high current, low potential beams and have found use in the most extensively implemented negative ion program until recent years [40,54]. What is not generally Fig. 19. Effect on beam steering due to a +-3% change in ion density in the source .plasma; the bottom view shows a blowup of the sheath region of the top view. II. ION SOURCES AND BEAM OPTICS
J.H. lChealton ~Ion extraction and optics arithmetic
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touted about these systems, and what we will delineate here, is that while the current does get higher with increased decel potential (see fig. 20; for the system considered, see fig. 21) the beam optics gets markedly worse (see fig. 20). The optics is, as usu,,d [I 1,20,21, 36,70-76, 79] aberration dominated as shown by the emittance diagram (fig. 22 for 1560 transmitted orbits shown). Although this kind of information should be derivable from the results of other algorithms used to examine the situation [49.50] in the context of the negative ion program and the steering study done by Conrad [40], no mention of this has
Fig, 22. Emittance d i a ~ a m for the case in fig. 21 showing 431 X 2 orbits.
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J.tl. Whealton /Ion extraction and optics arithmetic
been published to my knowledge. It was noted in an earlier version [82] of ref. 40 but was suppressed in revision. At the very least, the observer was not alerted. A study of electrode radius effects on ion optics was done for cylindrical geometry for a typical triode
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triode similar to that of fig. 23. II. ION SOURCES AND BEAM OPTICS
68
J.EL Whealton /Ion extraction and optics arithmetic
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Fig. 29. Triode used for sample slot and effect calculation.
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phenomena that serve to reduce electrode aberrations. One obtains at large radii; the other, as can be seen from fig. 24, is that the sheath at optimum perveance is located very near the gap as opposed to the large radius case (fig. 24D). Thus, the aberration fields are mitigated by the source plasma itself. This is a common mode of aberration reduction for high field ratio tetrodes as well. At large radii, however, the electrode thickness becomes negligible (compare fig. 23D to fig. 23A). Therefore, in this case of large radii, most of the aberrations are not directly caused by the electrode but by the sheath itself and its nonperfect foundation. Where neither of these effects is dominant, we have the maximum shown at intermediate values of r. This effect could not be delineated [11 ] with the algorithm explained in ref. 60 with its three-node change deposition; the noise of the former algorithm is exhibited in the emittance diagram of fig. 27, which has a temperature of 0.4 eV versus that for the latter algorithm in fig. 28, which has a 4 eV ion temperature (the optics is still aberration dominated).
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Fig. 30. Parallel slot optics including end effects for the triode shown in fig. 29.
J.H. lChealton /Ion extraction and optics arithmetic Another effect that is under examination is the slot-end effect. This is an intrinsically 3-D problem and so has heretofore been devoid of consideration. But due to the 3-D arithmetic being developed at one institution [ 6 1 - 6 4 ] , the problem is amenable to treatment. The triode considered is shown in fig. 29, and the parallel optics is shown in fig. 30 for a 5 : 1 slot. In this case, the perpendicular r.m.s, divergence is about 4.6 ° while the parallel divergence is 3 ° , totally dominated by aberrations. I am indebted to R.W. McGaffy, J.W. Wooten, J.C. Whitson, E.F. Jaeger, L J . Drooks, and J. Smith for their indispensable help. Also, I am indebted to my colleagues with a vested interest in efficient neutral beam generators, whose insight was necessary in the elucidation of important parameters; amongst these parties are L.D. Stewart, J. Kim, W.L. Stifling, L.R. Grisham, C.C. Tsai, W.L. Gardner, M.M. Menon, P.M. Ryan, G.G. Kelley, O.B. Morgan, G. Schilling, W.S. Cooper (LBL), D.E. Schechter, and H.H. Haselton.
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