THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report VOLT-AMPERE CHARACTERISTICS OF CYLINDRICAL AND SPHERICAL LANGMUIR PROBES FOR VARIOUS POTENTIAL MODELS M. Kanal W.G. Dow E.G. Fontheim ORA Project 06106 supported by: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NOo NsG-525 WASHINGTON, DoC. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR May 1967

ACKNOWLEDGMENTS The authors wish to acknowledge the help of Mr. George R. Carignan, Director of the Space Physics Research Laboratory in making several suggestions in regard to the text and in coordinating the research effort. We also wish to express our thanks to the following individuals for eager help; Prof. Andrew F. Nagy in reading the manuscript and for several enthusiastic discussions, Mr. Stanley Woodson for writing the computer program, Mrs. Martha Beard for doing the computations, and Mrs. Kristin Blalock for typing the original manuscript. Last, but by no means least, Mrs. Sylvia A. Kana.l for proofreading the manuscript. 111

PROLOGUE In the early stages of the research presented here, the only intention of the authors was to findan analytic expression for the voltampere characteristics of spherical and cylindrical probes for sufficient.ly genera.l types of potential functions and to check our calculations for the Maxwell-Boltzmann distribution against the self-consistent numerical calculations of La.framboiseo In the course of this investigation the classic works of Mott-Smith and Langmuir were subjected to a. critical analysis, in terms of the boundary conditions encountered in ionospheric experiments. This analysis revealed serious defects in Langmuir' s finite sheath model and put certain limitations on their orbital-motion-limited theory. Furthermore, there seemed to be no agreement among the existing theories as to the mathematical definition of the t"sheath" which is the central part of any theory describing the plasma-probe interactions. It was therefore felt that it would be of general benefit to examine in some detail the basic physics of the interaction between the probe and the surrounding plasma. The first two chapters and the beginning of the third chapter of this report, thus, are in the nature of a textbook discussion and should be read in this spirit. iv

TABLE OF CONTENTS Page LIST OF TABLES vii LIST OF FIGURES viii I. INTRODUCTION 1 II. THE POTENTIAL INSIDE THE SHEATH AND ITS CONTINUITY AT THE PLASMA-SHEATH BOUNDARY 6 2.1 The Sheath Radius and "Sheath Edges" 6 2.2 General Problem of the Potential Distribution in a Finite Sheath, and Electric Field Continuity at the Plasma-Sheath Boundary 13 2.3 Need for, and Consequences of, Abandonment of the Concept of a Well-Defined Sheath Boundary by Making the Sheath Radius Formally Infinite 17 2.4 Potential Function Models and Charge Densities for the Infinite Sheath 19 III. ORBIT ANALYSIS 25 3.1 The "Effective Potential" for Accelerated Trajectories 25 3.2 Relationship of the Occurrence of Maxima and Minima of the Effective Potential be to the Potential Structure 27 3.3 Properties of the Effective Potential of Langmuir's Finite Sheath Model; the "Admissible Space" Diagram for Particles Reaching the Probe 30 3.4 Utility of the t-Function in Determining Applicability of the Langmuir Volt-Ampere Relations 37 3.5 Dependence of the Mott-Smith and Langmuir CurrentCollection Equations on the Presence of a Discontinuity in the Potential Gradient at the Sheath Edge 42 3.6 Admissible Space for the Orbital-Motion-Limited Condition in the Mott-Smith and Langmuir Theory 43 3.7 A Class of Potential Functions, Giving Rise to Both a Maximum and a Minimum in the Effective Potential 46 3.8 The Unique Angular Momentum Mk for which the Maximum Value of Effective Potential Equals that at the Collector Surface 47 v

TABLE OF CONTENTS (Concluded) Page 359 Proof that for rc > ri the Effective Potential Maximum Can Equal the Effective Potential at the Probe Surface Only by Occurring at the Probe Surface: i.e., rk = rc if ri < rc 50 3o10 Evaluation of Kv and v, for the class of Potential Functions Defined by the Relation be* = KVM2V 54 3.11 The Admissible Space for the Condition ri > rc 57 3.12 Abandonment of the Concept of a Well-Defined Sheath Outer Bound in Favor of an Infinite Sheath Radius 57 3.13 Orbital Behavior and Admissible Space for a.n InversePower-Law Potential Function 60 3.14 Discussion of Potential Functions Different From Any Inverse Power Law but with Separable Locus-of-Maxima Equations 68 3.15 Admissible Spaces for Arbitrary Positive Values of the Exponent in the Inverse-Power-Law Potential 69 IV. THE VOLT-AMPERE EQUATIONS 76 4.1 General Expressions for the Current 76 4.2 Integration of the General Expression for the Current 80 4.3 Collected Current Expressions for the Inverse-PowerLaw Potential 83 4.4 Current Collection for a Power-Law Approximation to the D'ebye Potential Distribution for a tLarge Sphere 84 4.5 Discussion of the Volt-Ampere Relations 86 4.6 Comparison with the Results of a Self-Consistent Analysis 94 Vo CONCLUSION 107 APPENDIX: DERIVATIONS OF THE MOTT-SMITH AND LANGMUIR EQUATIONS FOR CURRENT TO CYLINDRICAL OR SPHERICAL PROBES FOR FINITE SHEATH MODELS 109 REFERENCES 113 vi

LIST OF TABLES Table Page I, Attracting Cylindrical Collector 88 II. Attracting Spherical Collector 89 III. Cylindrical Probe 95 IV. Spherical Probe (i = 1) 96 Vo Cylindrical Probe ( = 0) iLC: VI. Spherical Probe (6 = 1) 101 vii

LIST OF FIGURES Figure Page 1. Illustrates the meaning of Eqs. (1) and (2) 8 2. Radial and tangential velocity components u and ut for an approaching charged particle at a distance r from the center of a cylindrical or spherical electrostatic probe. 25 3. Graph of the function r(r), Eq. (28), for a form of this function that increases monotonically up to the sheath radius rs and has the value 4s at rs. 29 4. Plots of the effective potential energy Oe(r), for various values of M2. 33 5a. be(r) for M2 < M2 and total energy E for which collection occurs. 36 s 5b. b (r) for M2 > M2 and total energy E for which collection occurs. 36 6. The shaded area is the "admissible space" in the (E,M2) plane for Langmuir's finite sheath model. 38 7o Graph of the singular *-function (43) corresponding to the Eq. (42) form of 0L, which is the limiting form for validity of the volt-ampere equations of the Langmuir finite sheath model. 41 80 Plots of the effective potential if rs + oA, giving the Mott-Smith and Langmuir orbital-motion-limited condition. 44 9. "Admissible space" diagram, for the orbital-motion-limited condition, occurring for the Mott-Smith and Langmuir model when rs -+ a 45 10. Graph of a t-function, displaying a maximum inside the sheath. 48 11. Graph of a *-function for the case of a potential for which ri < r. 51 12. Graphs of e.(r) showing in each of three cases the maximum (e at r = r*. 52 15. Admissible space diagram from Eqs. (67-1), (67-2), for a V-function of the form illustrated in Fig. 10 with r. > rc. 58 viii

LIST OF FIGURES (Concluded) Figure Page 14. Graph of the effective potentials, be(r) for an inversepower-law potential function 0=0c(rc/r)' with a > 2; also shown is the locus 0*(r*). 64 15. Graphs of t(r) for various values of a in the power-law-potential Eq. (68-1). 67 16.o Graph of v versus a as given by Eq. (75-1) in the case of the power-law-potential given by Eq. (68-1). 70 17. Graphical representation of 0e when a = 2 in the potential (68-1). 72 18. Admissible space diagram, from Eq. (83-3), for the infinitely thin sheath around an electron accelerating probe described by letting a o in the potential function (68-1). 75 19. Cylindrical probe (6=0). Plot of the dimensionless current density (Jco/Jo) vs. dimensionless voltage (-Ooc) for various values of o. 90 20. Cylindrical probe (6=0), 91 21. Spherical probe (6=1). Plot of the dimensionless accelerated current density (Jcl/Jo) vs. the dimensionless potential (-(oc) for various values of a. 92 22~ Spherical probe (S=1). 93 23. Cylindrical probe (6=0). 97 24~ Spherical probe (8=1). 98 25. Cylindrical probe. 102 26. Cylindrical probe (6=0). 103 27. Spherical probe (6=1). 104 28. Spherical probe (6=1)o 105 ix

I INTRODUCTION The collisionless theory of electrostatic probes immersed in a plasma involves the two following basic physical concepts, (a) The concept of a "lsheath" surrounding the probe, in which the potential change from that of the plasma to that of the probe occurs, as propounded by Langmuir, and (b) The existence of distinct velocity distributions of the electrons and positive ions within the region of interest, the sheath. The sheath is fully described when one knows both velocity distributions and the potential distribution in the sheath. These two properties of the sheath are, of course, intimately interrelated. In principle, therefore, the problem of deriving the voltampere relations of an electrostatic plasma probe becomes one of a simultaneous self-consistent solution of the Poisson and Vlasov equations and to give the potential and velocity distributions, subject to appropriate boundary conditions in the plasma and at the probe surface, The collected current is then obtained from the current density at the collector surface which can be calculated from the electron and ion velocity distributions at the probe radius. The self-consistent approach to the problem is very complicated, besides being usually obtained for a relatively narrow range of boundary conditionso Consequently, for a physically meaningful, tractable, and 1

reasonable widely useful solution, certain simplifying assumptions become necessaryo This paper will present certain such assumptions, and compare some 2 of the results with those obtained by Laframboise who used a numerical approacho But before going into the details of our assumptions, we will give a review of the relationship of our work to the most important existing literature on electrostatic probe theory. For the past several decades, a considerable amount of effort has been devoted to plasma, research by means of electrostatic probes. The most important contributions are those by Mott-Smith and Langmuir, 4 Langmuir and Compton,3 Bernstein and Rabinowitz, and Gurevitch5 The first unified treatment of electrostatic probe theory as a. whole, was that by Hok, in a report which has been a valuable resource in much of the work on rocket investigation of the ionosphere by The University of Michigan. Hok discusses the concept of a "potential well" or transition region, important to bipolar electrostatic probeso The classic works of Mott-Smith and Langmuir and of Langmuir and Compton, have been extensively used in laboratory plasma and ionosphere research, and for this reason part of this paper is devoted to giving a criticial review of certain aspects of their theory, particularly in regard to details of their assumed sheath model and its physical significanceo Our early discussion of the "Sheath and Sheath Edge" introduces an examination of the classical theory as to the self-consistency of the sheath model from the standpoint of potential theoryo This is then 2

extended into a. discussion of the continuity of the potential across the plasma-sheath boundary, and then to a study of some general properties of potential functions for the infinite sheath model. The "Orbital Analysis" portion of this study employs throughout the concept of the "effective potential." This is defined so that, by incorporating the effect of angular momentum, it becomes a function which produces a fictitious force field governing the radial motion of the charged particles. By systematically examining the mathematical properties of the effective potential, the treatment then derives the Langmuir criteria for collection of particles by the probe and in particular demonstrates that the discontinuity of the electric field at the sheath edge is a necessary condition for Langmuir's theory to hold. Finally the infinite sheath limit in Langmuir's theory is discussed. A two-variable, separable, mathematical form for the maxima, of the effective potential is then introduced which relates in conceptually useful ways to the "admissible space" (Langmuir's term). The admissible space is defined by the limits of integration employed in obtaining the probe volt-ampere characteristic. This separable expression is then used to investigate the volt-ampere cha.racteristics for a. general inversepower-law potential function. The underlying conceptual basis of the orbital analysis is similar to that used by Bernstein and Rabinowitz, and Gurevitch,5 in that it neglects collisions in the sheath, and provides a, framework for determi5

ning current collection by treating the whole region from the collector surface to the undisturbed plasma by means of probability distribution functions Following the orbital analysis, the probe volt-ampere relations are derived in analytic form, for cylindrical and spherical probe geometrieso Among the assumptions that the whole procedure employs in arriving at the volt ampere relationships, the following two are especia.lly basic: (a.) The probability distribution of the particles is Maxwellia.n in the undisturbed plasma. (b) It is reasonable to approximate a part or the whole of the potential distribution by a suitable inverse power law potentia.lo To check the feasibility and utility of our second assumption, we will compare our volt-ampere characteristics with those obtained by La.framboise. The treatment of Bernstein and Rabinowitz is restricted to monoenergetic distributions and to probes of large radius. Gurevitch5 and La.framrboise2 have extended the theory to Maxwell-Boltzmann distributions and to probes of arbitrary radius: but only Laframboise had carried out the calculations of the volt-ampere characteristics by solving the whole problem, including the effects due to space charge, in a numerically self-consistent wayo His results are, therefore, a logical source for the compa.risono To summarize the purposes of this study, we shall attempt to: (a) Give a critical review of the classical probe theory of MottSmith and Langmuir, examining their sheath model in detail, 4

especially in regard to discontinuities of the field at the plasma.-sheath boundaryo (b) Review the properties of approximations to the potential function. Then, by means of a suitable inverse-power-.law potential function, derive the probe volt-ampere relations in analytic form, and compare the results with those obtained numerically by La.framboisee The first phase of the analysis will deal with a, few classical definitions in electrostatic probe theory and their employment in dealing with certain rather general aspects of sheath potential distributions and of the merging of the sheath into the undisturbed plasma.o 5

II. THE POTENTIAL INSIDE THE SHEATH AND ITS CONTINUITY AT THE PLASMA-SHEATH BOUNDARY 2o1 THE SHEATH RADIUS AND "SHEATH EDGES " Mott-Smith and La.ngmuir define the sheath and the sheath edge in the following wayo Let r be the radius of the probe (cylindrical or spherical) and rs that of the sheath. Further let 0(r)/q be the electric potential of a particle of charge q at a. radius r in the field of an accelerating probe, D(rc)/q = 0,/q the electric potential at the probe and let D(rs)/q =,Ds/q = 0 be the plasma potential. We now quote Mott-Smith and Langmuir, who define the sheath edge in the following wa.y: "If we assume any distribution of potential between rs and rc, we can always find a cylinder (or sphere) of radius r' intermediate betweenrs and rc such that, for this cylinder (sphere) or any other of smaller radius, the condition 2 v2 2 r r -r c (1) -' 2 r2 r' -r r c is satisfiedo In other words, such a. surface can be taken to be the edge of the sheath if the distribution for the velocities of the ions crossing it is knowno As far as the equations of orbital motion determine it, the sheath edge is therefore, simply a. surface on which we know the velocities of the ions and within which the above condition is sa.tisfiedo" 6

Note that this does not compel the potential gradient to be zero at, or just beyond, a "sheath edge" where the velocities are known. Figure 1 illustrates the physical meaning of Eq. (1), and the relationship of a particular "sheath edge" r' = re to the sheath radius rs. To aid in constructing Fig. 1, Eq. (1) has been rearranged into the following form: 2 r - 2 r cl = felcL (2) e -1 rc2 When the sheath edge becomes the sheath radius, re = rs, the equation changes to r2 - s l > (D —- -cj = -1>1 z (5) r2 I rc The definitions of the functions fs and fe are contained in these equations. Note in particular that the slope of the factor fs goes to zero only as r- oo, and that, therefore, the slope of the curve is nonzero at r=rs. Thus, as illustrated in Figo 1, O(r) must also have a nonzero slope as it approaches r=rs if it is to remain below fsc, as Eq. (3) requires (for <c < 0, i.e., attractive potentials). 7

I A POSSIBLE "SHEATH EDGED ri' L II ~ ~ Z tC I I 0i6 EI Fig. 1. Illustrates the meaning of Eqs. (1) and (2). (rs2/r2)-l (re2/r2)-l (rs2/rc2)- (re2/rc2) - 8

To obtain Eq. (2) which applies when re < rs it is necessary to observe only that: fe < fs, for r < re, if re < rs, because rc < ro (4) This is illustrated in Figo 1 where the fe$c curve lies everywhere above the f sc curve. Here fe and fs are the bracketed factors multiplying I|cl in Eqso (2) and (3), respectivelyo One can note, immediately, the following serious weakness in the reasoning underlying Eqso (2) and (3)~ In the case where re = rs. Eqo (3), and therefore Eqo (1) with r' = r describes a model in which there must be a discontinuity in the potential gradient at r = rs in order to have = 0 beyond rs Such a discontinuity is physically nonrealizable, as discussed later in detailo This weakness does not appear when rT is given a value re that is significantly less than rs, because at such an re, the potential is still rising, and the gradient is not compelled by the boundaries of the model to have any particular value at and beyond the selected re~ Subsequent sections deal with various methods of probe model analysis chosen to avoid this weaknesso The above quotation from Mott-Smith and Langmuir, as illustrated by Figo 1 and discussions thereof, leads one to conclude that, in general, what Mott-Smith and Langmuir call "sheath edge" need not coincide with the sheath radius rs at which the potential becomes 9

equal to that of the undisturbed plasma. Furthermore, one concludes that Mott-Smith and Langmuir considered that it is only inside any "sheath edge" that the behavior is dominated by the dynamics, thus making the velocity distribution different from Maxwellian. Outside the sheath edge, presumably, the analysis would be that applying to a plasma region in which there is a gentle potential gradient and a moderate-to-small flux density of particles. It is therefore reasonable, for analytical purposes, to place a sheath edge as close to the probe as one may reasonably expect the velocity distribution to be Maxwellian, with relatively little reference to where this may lie with respect to the potential distribution. The right-hand side of the inequality (1), and its equivalent (2), changes sign when r' < r < rs, and the inequality is reversed when |O(r)l < Ifecl for r < rs (see Fig. 1). However, these two aspects are of little concern, since in the region beyond the sheath edge (i.e., r > r') one presumably expects to carry out the analysis not in terms of particle trajectories in a sheath, but rather in terms of particle behavior in a plasma with a gentle inward gradient insufficient to cause significant departure of the velocity distribution from that in the undisturbed plasma (although it may affect the particle density significantly). Still more generally, in view of the Mott-Smith and Langmuir distinction between sheath edge and sheath radius, one may well ask these two questions as to placement of an r' < rs: (a) Can in any realistic model such a sheath edge surface r exist and be usefully placed, within which the Mott-Smith and Langmuir 10

condition (1) [also Eq. (2)] is satisfied for all values of r < r'? It has already been pointed out earlier that potential distributions can exist that do not satisfy Eqs. (1) and (2) for all values of r', particularly if r' is close to or at rs. (b) If such a surface does exist, what would be the velocity distribution at r=r'? Presumably, with r=r' properly chosen, this would be a Maxwellian distribution having parameters governed by the plasma in which the probe is immersed but this merely changes the question of properly locating a sheath edge r' within the sheath model. In the strict sense, answers to these two questions cannot be given prior to solving the whole problem of the potential distribution irn-the selfconsistent way. Supposing that the self-consistent method gave rise to a potential function which did not satisfy the inequality (1) for the entire range rc < r < rs, one would then expect the corresponding volt-ampere characteristics not to coincide with those of Mott-Smith and Langmuir. It is worth while to note that the Mott-Smith and Langmuir "sheath edge" may well be interpreted as the boundary between the two regions which Hok calls, respectively, the sheath and the potential well. The latter is a transition region exhibiting some slow radial flux of particles and a gentle potential gradient toward the bipolar probe electrode system. He envisions this region as having properties similar to those 7 8,9 in laboratory plasmas exhibiting what Tonks and others have called a.mbipolar diffusion. In such plasma, regions the ion and electron densities are very nearly equal, and both are, among themeselves, in thermal equilibrium, but at different temperatures. Thus, at Hok's boundary bell

tween his sheath and the potential'well, he is able to postulate Maxwellian velocity distributions of the particles. Yet he need not postulate the potential to be that of the plasma, nor that there be a zero potential gradient, nor a. discontinuity of the gradient at this boundary. These are just the attributes of a, Mott-Smith and Langmuir sheath edge, distinct from their sheath radius, and well within the sheath radiuso All contributors, including Mott-Smith and Langmuir, Langmuir and Compton,3 and Hok,6 give essentially their whole attention to studying what happens inside what is really a "sheath edgeo" Yet it is frequently unclear just where this sheath edge is relative to the sheath radius, and still more often the sheath edge is identified with the sheath radius, with various illogical consequences. It should be clear that, in general, the defining of a. sheath edge is justified only if it serves to simplify the problem of determining the volt-ampere relations~ The remarkable simplicity of the classical Langmuir probe theory1'3 lies precisely in the fact that, by setting r' = rs one is able to bypass the problem of solving for the potential function. Thus, by La.ngmuir s sheath model, we mean the sheath region described by a. potentia.l function 0 such that 22 2 r -r r S c 0, for r > rs (5) This is consistent with the inequality (2), if the inequality (4) is satisfiedo 12

In general, Langmuir' s sheath model will be shown not to be adequate for most plasma probe experiments, Particularly in ionospheric measurements, a potential function describing a finite sheath region and satisfying Eqo (5) cannot exist, unless one assumes the presence of a charge shell at r = rs Thus, the curve of 0(r) in Fig. 1 exhibits a discontinuity in its derivative'at r = rs, which could only exist if there were a shell of surface charge to terminate the flux due to the potential gradient just within r o No such shell can realistically be assumed. 2.2 GENERAL PROBLEM OF THE POTENTIAL DISTRIBUTION IN A FINITE SHEATH, AND ELECTRIC FIELD CONTINUITY AT THE PLASMA-SHEATH BOUNDARY Let us define the electric potential function for a. finite sheath by: A(r) = (r)e (r -rc) e(rs-r) (6) where q is the electric charge per particle, taken to be positive in this discussiono 0(r) is, as heretofore, the potential energy and 8(x) is the unit step function: e(x) o1 if x > 0 (7) 0 ifx < 0 with ad(x) _= (x), -d (-x) = -6(x) (8) dx dx where 6(x) is Dira.c' s delta function. In the above we have considered 153

only the radial variable, because of the symmetry of the field for both cylindrical and spherical probes. O(r) is considered to be a function that is continuous from r = 0 to r = o. If Eq. (6) is inserted into Poisson's equation, V2A = - (9) 0 one obtains p 2 + -e = v2A = A q V 3(r-rc)s(rs-r) + (D'5(r-rc)((rs-r) (10) -t' 5(rs-r)e(r-rc) (10) where p is interpreted to include both space-charge density and surface charge density, p becoming infinite for the latter, The prime on 0 represents the derivative with respect to r. Now the surface charge density a on any arbitrary surface or r set of surfaces S enclosing a. region outside of which the electric potential gradient is zero is given by a = -o v X1 (11) r ~0 * V evaluated on S. Here X is the electric potential function in the region enclosed by S, and n is the unit normal vector pointing into the region where the electric field exists. VX is, of course, to be identified with O'/q in Eq. (10). At r = r, the unit vector points radially outward, so that f = +1, whereas at r = r it points radially inward, so that s A = -1. Thus Eq. (11) becomes 14

= - - aat r = r (12) q D (rs) r q E at r = r, (13) Hence, the expressions for the surface charge densities rc and ars at r = rc and r = rs respectively, are, EO -c (14-1) rc q ~~re q rs = (14-2) rs q where = (r) r r and I = (r)lr = r These algebraic signs c s are consistent with Fig. 1 as long as q is positive, for in that case the O(r) curve in Fig. 1 can equally well represent potential energy per particle and electric potential. If it represents electric potential, the surface charge at rc is clearly negative, and at rs positive, from the construction of the figure. Now let Pv denote the volume charge density. Then we may write 2 () 0 It is now instructive to obtain an overall expression for the charge density by inserting Eqs. (14-1), (14-2), and (15) into Eq. (10) to obtain P pv 8(r-rc) 3(rs-r)-,arc(r-rc) o(rs-r) -ars(rs-r)e(r-rr) (16) This is consistent with Eqs. (12) and (13). The delta functions identify the fact that the charge density becomes infinite for a surface charge. 15

Of course in an integration to obtain total volume charge and total surface charges, the delta-function terms ma.ke finite contributions. From Eq. (14-1) it is seen that ors = 0 only if s = 0; i.,e, at the sheath outer bound, the electric field must vanish in addition to the vanishing of the potential provided by Eq. (6). If only 0 vanishes at r but not (I, then there would exist a. surface charge layer of strength given by Eq. (14-2). In the interior of a plasma., remote from physical boundaries, the existence of such a shell is obviously unphysical, because it implies a. discontinuity in the electric fieldo We have already seen in the previous section that a. nonvanishing (' at r, approaching from the left in Fig. 1, is required by the Mott-Smith and Langmuir model in which r' = rs in Eq. (1). Therefore, as discussed in more detail later, this nonvanishing of ~ at rs is a necessary requirement in deriving their volt-ampere relations for a finite sheath in which their "sheath edge" coincides with the sheath outer bound. In other words, those current expressions will be strictly valid only under the unrealizable condition in which this sheath edge, placed at sheath radius, is replaced by a zero-potential conductor of the same type of geometry as the probe. Presumably Mott-Smith and Langmuir were aware of this, but felt their model, even though containing this element of unrealizability, was an adequate first approximation, and so it has been for very many years. 16

2.3 NEED FOR, AND CONSEQUENCES OF, ABANDONMENT OF THE CONCEPT OF A WELL-DEFINED SHEATH BOUNDARY BY MAKING THE SHEATH RADIUS FORMALLY INFINITE The finite sheath model has weaknesses that go considerably beyond that of the potential gradient discontinuity at the sheath outer bound, discussed in the previous section. That particular weakness could be formally overcome by employing for $(r) some simple function that monotonically rises from the probe to sheath radius, but whose slope becomes monotonically less steep, finally reaching zero at sheath radius. Or, one could employ one kind of potential function within a sheath edge, and another between sheath edge and sheath radius, with potential gradients forced to be equal at the sheath edge, and that at sheath radius forced to be zero. But in all such highly artificial potential models, the second and perhaps higher derivatives would have discontinuities at these bounda.ries. The second derivative is intimately related to space-charge density, and it is almost a.s unrealistic to presume an abrupt discontinuity in spa.ce-charge density at a sheath edge or sheath radius as it is to presume a discontinuity in the potential gradient. In any realistic model, not only the potential but also all of its radial derivatives must be presumed to be continuous through the sheath into the plasma. This type of continuity demands that the sheath potential approach the plasma potential asymptotically with increasing radius; thus, in reality, there can be no well defined sheath radius. There ca.n of course be described a. general range of values of the radius within which the sheath region with its steep gradients merges 17

into the plasma. with its zero gradient. Any plasma will have random potential variations, and the sheath may be considered to have merged into the plasma. within any range of values of the radius for which the difference between the sheath potential and that of the undisturbed p.la.sma. is of the order of the random variations in plasma potential. In terms of experimental systems, this may occur at relatively small distances from the probe surface. Mathematically, the merging of the sheath into the plasma. is provided for by letting the sheath radius become infinite (roo) and requiring that both the potential and the potential gradient be zero at an infinite radius. That is, <D(r)=O, and' (r) = 0 at r = rs = oo. Of course this makes the sheath include the two regions discussed earlier in Section 2.1, namely the steep gradient sheath region proper, and the gentle gradient plasma-like transition region. It also makes the sheath include what Hok6 ha.s called the potential well that exists a.round a. bipolar probe, which represents a reasonably good conceptual approach to realityo Thus when we later, for purposes of ana.lysis, extend the sheath ra.dius to infinity, we must recognize that we a.re including in the sheath two types of regions having wholly different properties. In the region close to the probe, the potential gradient is steep and particles of one or the other polarity dominate so that space charge ha.s a major effect on the potential distribution In the outer or transition region, the positive ion and electron densities are nearly, but not quite, equalo Particles 18

of each po.l.a.rity a.re in thermal equilibrium, but maybe at different temperatureso There may exist a close parallel to Tonks' a.mbipola.r diffusion region 7''9 The potential gradient and flux of particles to the probe are not zero. The flow of heavy particles (ions) to the probe may be governed by mobilities affected by collisions, or they might pursue orbits with negligibly few collisions. Any self-consistent analysis of the sheath should, to be complete, be equipped to treat both regions, a.s for example by using what has been called the "combined plasma-sheath equat ion " 2,4 POTENTIAL FUNCTION MODELS AND CHARGE DENSITIES FOR THE INFINITE SHEATH It is clear from earlier discussions, that the general problem of actually determining the true potential function for a physically realistic model is very complicated, for in Poisson's Eqo (15) the density function p will, in general, involve 0 implicitly. Instead of solving this selfconsistent problem we shall examine the properties of various potential models and their corresponding volt-ampere characteristics. Each potential model of course implies a. certain charge distribution in the sheatho Our investigation will be restricted to potential models having the common property that 0 and T' approa.ch zero a.s r->oo An illustrative choice of a class of potential models, to be discussed below) is the function ~ = Oc -c r, (17) where Oc is the potential energy per particle at the probe or collector, 19

and a is a positive number. For an accelerating potential, (- c) is numerically positive. The potential energy gradient is car - ( C) c6+1 (18) r Clearly, both the potential and its gradient vanish as r->oo. The exponent a has a very simple physical interpretation. Using Eq. (14-1) in Eq. (18) when r - rc we get rc (rc) (19) in which, for an accelerating potential, both ( ) and are in which, for an accelerating potential, both (-arcq) and (-$c) are positive quantities. That is, a is proportional to the ratio of the collector surface charge density to the collector potential. In any realistic model, to which Eq. (17) can of course only be an approximation, one can consider arc to be the sum of the charge density on the probe that would exist if it were at the potential Tc in a spacecharge-free environment, and the induced charge on it due to the space charge in the sheath region. Before criticizing this concept, let us state the charge density distribution called for by Eq. (17). By using it in Eq. (10), the result is ----. ~o(-c) a(a- )r0C P = ) c a + arc6(r-rc). (20) qC+2 rcS(re ) q r where ~ = 0 for the infinite cylinder and 5 = 1 for the sphere and a is given by Eq. (14-1). Equation (20) applies only if a > 1, as will be explained below. In that case no third term appears on the right20

hand side because, a.s r>oo, the surface charge density at the sheath boundary goes to zero fa.ster than the surface area of the sheath boundary increases For the spherical geometry, we use 5 = 1 in Eqo (20), This means that if 5 = 1 in spherical geometry, Eq. (17) describes a. space-chargefree "sheath" potential (there will generally be substantial space charge in the sheath of a. useful electrostatic probe in a. plasma, if the probe potential differs appreciably from the plasma. potential ) For the case where a = 1 in spherical geometry, there is a. well-defined crc on the probe surface, found by using Eqo (18) with O = 1 in Eqo (14-1). Since there is no volume spa.ce charge, the electric flux that originates at a must terminate at infinite radius Thus, in this case there would be a well-defined charge 4ntr2rc on the probe, and an equal and opposite charge at infinite radius where 0 = 0, with 0 = Tc on the probe. But since this finite cha.rge is distributed over a spherical shell of infinite radius, the charge density arc goes to zero in such a. way that the tota.l charge remains constant, as r->oo. This also corresponds to the result of Eqo (18) that with c. = 1, limp' = O0 Of course in the integration over all volume r-oo elements from r = rc to r=o, Eq. (20) must in principle integrate to zero; for O = 1 in spherical geometry, this can be accomplished only by adding another term lim arsb(rs - r) whose contribution to the charge r sco integral is equa.l in magnitude, and opposite in sign, to the total charge on the spherical. probe, If, for spherical geometry (l=1), we use a > 1 in Eqso (18) and 21

(.19) the situation is straightforward, in that the volume integral of charge is found to equal the area integral of arc; in this case Eq. (20) includes all electric charges. All electric flux lines then originate at the probe surface and terminate in the volume charge in the sheatho The result of the integration is: Total Volume 4' \ Cha.rge = (4q 0 crc(-(Dc) Coulombs (21) and arc has an equal but negative value, But if 0 < a < 1 for spherical geometry (t = 1), Eqo (20) predicts a volume charge of the same sign a.s the surface charge on the probe. Such a. model obviously is physically unrealistic and will, therefore, not be discussed in this treatment. Thus, in summary, for spherical geometry, only the range a > 1 in Eq. (17) has any real interest. But the above comments do suggest the possible utility of a model in which 0 is the sum of two terms like Eqo (17), in one of which i > 1 and in the other ( = 1o In fact such a. model probably closely describes the rea.l potential which ca.n be constructed a.s the sum of, (a.) a. contribution due to the space charge and its induced charge on the probe, and (b) that for a space-charge-free structureo This two-term potentia.l model will not be dealt with in the present papero For the infinite-cylinder geometry we use 0 = 0 in Eqo (20). This means that no expression of the form of Eq. (17) can describe the spacecha.rge-free potential structure which is, of course, logarithmic in formn Furthermore, that.logarithmic form is uninteresting because it becomes infi22

nite as r-mo (with, however, a zero-value O' at r = o)0 With the logarithmic potential used in Poisson' s equation, the space-charge density term in the new Eq. (20) vanishes and, to make the charge integral balance, we would have to add a term lim crs (r -r) which r->oo rs s would make a finite contribution to the charge integral. Of course, any cylindrical electrostatic probe used in the ionosphere is not an infinite cylinder. Even if completely isolated from any other conductor, the field of a charged cylinder of finite length would, at a sufficiently large radius, become essentially the same as that for a charged sphere. So there is always, in fact, a finite space-charge-free potential for such a body carrying a finite charge. But our infinite-cylinder analysis is not adequate for studying that aspect of the problem. If, for the infinite cylinder geometry (E = 0), we use a > 0 in Eqs. (18) and (19) the volume integral of the charge equals the area integral of arc and Eqo (20) identifies all the charges. The result of the integration of Eqo (20) is~ uotal volume charge, per _4 Eo ac unit length of cylinder q 2 e and orc has an equal, but negative, value. In summary, we can say that by postulating a relatively simple potential function (eogo, Eqo (17)), we have obviously no a priori assurance that the charge density implied by such a potential is physically realistice However, it will be shown later that in the range 25

rc/ADo 5 the volt-ampere relations obtained from the potential (17) for certain values of a agree very well with those obtained by the selfconsistent calculations of Laframboise 2 Here \D is the Debye length. We shall now proceed to the orbit analysis, which includes a detailed discussion of the behavior of the particles in the sheath region and the selection of those trajectories, which intersect the probe. 24

III. ORBIT ANALYSIS 5.1 THE "EFFECTIVE POTENTIAL" FOR ACCELERATED TRAJECTORIES The solution of the equations of motion of a particle approaching a cylindrical or spherical electrostatic probe (Fig. 2), leads to two Fig. 2. Radial and tangential velocity components ur and ut for an approaching charged particle at a distance r from the center of a cylindrical or spherical electrostatic probe. constants of motion; viz., E, the total energy, and M, the angular momentum. These are given by: E = m (u2 + ut2) + 0, (23) 2 r t M = mrut, (24) where m is the mass of the particle and ur and ut are the components of the particle velocity in the radial and tangential directions, respectively, at some point in space located a distance r from the center; D is the potential energy. In the cylindrical geometry, ur and ut are in a plane perpendicular to the axis of the cylinder (taken 25

along the z axis of a right-handed coordinate system) so that E is the total energy due to motion in that plane and M is the component of angular momentum in the z direction. Since the cylinder is considered to be infinite in length, the velocity component uz in the z direction remains constant and does not play a role in the classification of the orbits. In the spherical geometry, ur is the radial component and ut is the total tangential component of the particle velocityo Thus, in this case, E is the total energy and M is the total angular momentum. For both geometries, 0 is the potential energy at r, relative to a. zero value in the plasma. Since we will be dealing exclusively with the' situation in which the particles are accelerated toward the collector, we will have $ < 0 and V' > 0, where the prime on ( denotes differentiation with respect to ro The classification of the orbits will now be carried out without reference to whether the geometry is cylindrical or sphericalo Now let us introduce the concept of the so-called "effective potential energy" by substituting ut from Eqo (24) into Eqo (23) to get E = mu2 + 2 (25) where 0 +t -^ D. ^ (26) - 2 + ~ 2mr The effective potential energy e governs the radial motion of the particles. The points, where ur = 0 are the turning points of the orbitso At these points the effective potential energy be equals the total energy Eo 26

In principle, the mathematics describes two such points, the apogees and the perigees of quasi-elliptical orbits. However, the present physical model deals only with the perigees, as the apogees would be beyond the sheath region, or more generally beyond the region of interest for the present study. 35.2 RELATIONSHIP OF THE OCCURRENCE COF MAXIMA AND MINIMA OF THE EFFECTIVE POTENTIAL be TO THE POTENTIAL STRUCTURE In order to investigate the qualitative behavior of particle trajectories, we need to examine the behavior of the effective potential be as a function of r for various values of the angluar momentum M. In particular we must study the extrema of be. On equating to zero the 2 derivative of be with respect to r, keeping M constant, in Eqo (26), we obtain M = mr3, when e 0 (27-1) that is, *(r) = M2, when e = 0, (27-2) where, for convenience, we use a function 4 (r) defined as follows: 4(r) - mr35' (28) t(r) is a function of the field structure, that has the unique value M2 at values of the radius for which the effective potential < has a maximum or minimum. Of course, the solution of Eq. (27-2) for r is dependent 27

on the form of V(r). If r(r) is a. monotonically increasing function of r, as illustrated in Fig. 3, then for every given value of M2 there exists one and only one value of r for which Eq. (27-2) is satisfied: therefore, there exists only one value of r at which be has an extreme value, and this extreme value can only be a minimum. A sufficient condition for r(r) to increase monotonically would be for 0(r) to obey an inverse power law varying less steeply than 1/r2, for all values of r within the region of interest. Therefore l(r) would increase monotonically, as shown in Fig. 3, if D(r) were to vary as Cl/ro In contrast, if 0(r) were to vary as Cl/r5, then t(r) would be a monotonically decreasing function of r. In such a case there would be one and only one value of r at which 0e(r) would have an extreme value; but this would be a maximum, resulting in a. potential barrier inside the sheath. The steepness of the potential gradient ((r) is partly determined by the geometry of the probe (i.e., whether it is spherical or cylindrical). In addition the radial dependence of D(r) is governed by the extent to which space charge is present in the region of interest The potential D(r) is, in general, describable as the sum of a space-charge free term of positive gradient (for example varying as 1/r or as In r) and a. spacecharge dependent term. This relates to the discussion below Eqo (20), for the condition 1 = 1 and ca > 1 (for spherical geometry), and e = 0 and CO > 0 (for cylindrical). The usual physical model for accelerated particles is that of a negative potential probe drawing a current of positive 28

+ F_...___ —---—. —---— M2 t(r) / s r r Fig. 3. Graph of the function *(r), Eq. (28), for a form of this function that increases monotonically up to the sheath radius rs and has the value is at rs. An illustrative value of M2 is shown corresponding to a trajectory for which the angular momentum is +M (determined at entry into the sheath). 29

ions through the sheath. For this model, the space-charge density is positive throughout the sheath. For a potential model that incorporates a space-charge free potential plus a potential due to space-charge content the space-charge free potential term by itself gives rise to a monotonically increasing r(r), for either the spherical or the cylindrical geometry. Thus, departure from such a monotonicity comes only when the rate of change of the space charge term in r(r) is dominant in a region where this rate of change is negative. As discussed later in connection with Fig. 15, wherever the space-charge density locally declines as l/rn with n<4, it contributes to the monotonic increase of the function r(r). But if 4(r) will increase monotonically only because of the dominance of the space-charge free term over that due to space charge, in the range of radii of interest. Thus it appears that a sufficient, but not always necessary, condition for i(r) to increase monotonically is an adequate domination of the space-charge free effect on the potential distribution over the spacecharge effects. 5.5 PROPERTIES OF THE EFFECTIVE POTENTIAL OF LANGMUIR' S FINITE SHEATH MODEL; THE "ADMISSIBLE SPACE' DIAGRAM FOR PARTICLES REACHING THE PROBE We will first examine 0 in the light of Langmuir' s finite sheath model, that is fO(r) when r < rs; (r) = -s (29) 0 when r > rs; 30

with < satisfying the inequality (3) for rc < r < rs, where rc and rs are the radius of the probe and the radius of the sheath, respectively. An illustrative plot of 4(r) is shown in Fig. 3. We will show that Langmuir' s expression for the accelerated current to the probe, for either cylindrical or spherical geometry, can be derived only when it is assumed that Lim' (r) t 0, (30) r- rs that is, if it is assumed that the potential gradient does not vanish as r approaches rs from within the sheath. Let 4s denote the value of V (r) a.s r-*rs. Then, for M2>s, there exist no solutions of Eqs. (27-1) and (27-2) in the range r < rs. In the range 0 < M2< V_, one value of r is obtained for each given M2 from Eq. (27-2), and at this value of r, Oe has a minimum. At r = rs, the value of be is, from Eq. (26) given by 2 e(rs) = M (31) 2mrs because 0 is defined to be zero at and beyond r = rs. This expression tells us that, for nonzero M2, qe(rs) is always positive. The derivative of (e at rs is (rs) = (4-=2) (32) Mrs where's = V(r ). It will be recalled from Eq. (28) that the meaning of this is: 31

'e(rs) = (mr (r) - M (33) mrs From Eq. (32) it is clear that from Oe(rs) > 0 follows M2 < *s, (34) whereas for other values of M2, e(rs) < O. Plots of be(r) are shown in Fig. 4. Now, there exists a value of 2 = Ms2 such that Oe(rc) = Oe(rs). For any potential which satisfies the boundary condition O(rs) = O, Eq (26) can be used to show that 2mrs 2rc2 M -2 = -- (-(C). (35-1) r2 2 r-2 rc s C Note that for a very large sheath, when rs22 r 2 this reduces to: M2 = 2mrc2 (-Dc) (35-2) In reference to Fig, 4, it is seen from Eq. (26) that the M2 = 0 curve describes equally well: (a) The potential distribution in the sheath: (b) The energy of a particle that is wholly radially directed and has zero velocity at r = rs; such a particle falls freely from rest at r = r into the probe; it must begin this fall, because in the Langmuir model now being described,' (r) is positive at r = rS In accordance with the discussion following Eqs. (27) and (28) we now consider a potential that is qualitatively like the M2 = 0 curve in Fig. 32

Probe - --- Sheath a Plasma M2 > I'\ M2= M2 0| -c ---— / 0'=0 Fig. 4. Plots of the effective potential energy be(r), for various values of M2. Ms is defined as the value of M2 for which be is the same at the probe surface as at the sheath radius. 1s is the value of M2 for which the minimum in be occurs at sheath radius (see Fig. 3). 33

4 and has a monotonically increasing V function as shown in Fig. 3. Then each effective potential be(r) has one and only one minimum in the range r < r as shown in Fig. 4. If M is nonzero but very small, the minimum occurs for r < rc, and so has no physical significance for the present study. As M2 increases, the minimum moves across r = rc. We are interested in two particular values of M2. One of these is of course M2 = M2, for which a particle with zero initial radial velocity grazes the probe at perigee. If M2 < s < M2, a particle with zero initial velocity will not start inward, because be < 0. An illustrative example for such a 0 curve is shown in Fig. 4. e The second unique value of M occurs at M = s, for which, from Eq. (32), the minimum of be occurs just at r = rs. A particle with this value of M2 and zero initial radial velocity will not move inward from r = rs even though't(rs) is positive, because the gradient e'(rs) of the effective potential is zero at rS and negative for r< r. From the above comments it is clear that M 2< ts as long as the potential is such that x(r) increases monotonically as illustrated in Fig. 2. The relationship of particle trajectories to effective potentials of the type shown in Fig. 4 will be discussed with reference to Figs. 5a and 5b. Figure 5a shows, a typical effective potential be(r) for a. case in which 2 < M2 a.nd the total energy E is sufficiently large so that the trajectory will intersect the probe, i.e., a. particle along such a 34

2 2 trajectory will be collected, If M < Ms, the requirement for collection (see Fig. 5a) is that E > e(rs) since then also E > e(rc) be2 ca.use of the way in which Ms is defined. Figure 5b shows, a. typical Oe(r) for the case in which M2 > Ms2, with the total energy E larger than Oe(rc). Thus the trajectory for this E and this M will also intersect the probe, a.nd collection will occur. Note that in this case particles with energy less than %e(rc) could exist inside the sheath. But in such a case the perigee (E = Oe) would be at a. larger ra.dius than rc implying tha.t collection would not occur. In Fig. 5a (M2 < M2) the requirement that Imu 2 cannot be negative, s 2 r limits the possible values of E to the range E > Oe(rs) and since Oe(rs) > ( (r ) in Fig 5a., any energy that exceeds e (r ) will give rise to a e c e s trajectory reaching the probe. In contrast to this, if M2 > Ms (Fig. 5b), the particle reaches the probe only if -mu > 0 at r = r which 2r c limits the possible values of E to the range E > _e(rc)o Therefore, we can now define a, so-called "admissible space.t Each particle can be 2 2 represented by a point in the (E, M ) space, since both E and M are constants of the motion. Then the admissible space is that subspa.ce of (E, M2) space which contains the representative points of the collected particleso Based on the above discussion the admissible space is defined by the following relations: F > (r2 if 0 < MrM 2 (36-1) -~~ e2 S -_ 2 s<_ 55

\ l 1Total Energy E -- Probe P Sheath -- Plasma \ e (r) l S21/2 mu r I t I C e(r) 2e (r) e =M s <,q/s M2/2m S Total Energy E M 2 c 2m 2 rs E/ui2 / cleto ocr\ / mrs. o -s \y- o 1\./! rc r _r re Yr s M2<M [, -Probe ---- Sheath —- - PlasmaFig. 5a. Oe(r) for M2 < M2s and total energy Fig. 5. (r) for M2 > Ms2 and total energy E for which collection occurs E for which collection occurs.

E e(rs) = 2 + c if M2 > M2 (56-2) 2m.rc The shaded area. in Fig. 6 illustrates this admissible space, 3.4 UTILITY OF THE *-FUNCTION IN DETERMINING APPLICABILITY OF THE LANGMUIR VOLT-AMPERE RELATIONS The V-function (28) plays a very important role in deriving criteria for the validity of Langmuir's theory. We have seen that, if f(r) is a monotonically increasing function of r, then for each given M2, be(r) has a minimum. For a finite sheath each be has a minimum provided M2 < Vs, where = (rs). We wish to show in a. slightly different way from the treatment of Mott-Smith and La.ngmuir that there indeed exists a limiting potential function $L (r), with the property that any other potential function that varies less steeply than AL' for all values of r, will give rise to Langmuir's current expressions, while a.ny potential function varying more steeply than L will give rise to current expressions different from those of Langmuir. First, we note that if < has a maximum (for a given M2<Ms2) which lies to the left of rs (see Fig. 4), the condition for collecting the particle corresponding to that M2 will not be determined at ra, but at a smaller distance than rs. This is contrary to Langmuir' s requirement that all trajectories intersecting the probe must be determined by initial conditions at rs. Hence the class of potential functions which satisfy this requirement must be such that they produce no maxima in Oe in the region r < r < rs o As discussed earlier, e has minima when j(r) has a positive -c ~se slope and maxima. when it has a. negative slope. The necessary condition 57

ADMISSIBLE SPACE \\\\NS^X2) E M2 ^E= 2mrc2' " Fig. 6. The shaded area is the "admissible space" in the (E,M2) plane for Langmuir's finite sheath model. To the left of Ms2, Eq. (36-1) applies, conditions as illustrated in Fig. 5a. To the right of M52, Eq. (36-2) applies, conditions as in Fig. 5b. 38

for the applicability of Langmuir' s theory can, therefore, be restated by saying that the *-function cannot have a. negative slope. Therefore, 4 (r) can, at most, be a. constant in this range of r, in order for Langmuir's volt-ampere equations to be valid. Thus, by setting 4(r) = mCa, introducing the potential given in Eq. (6), and employing Eq. (28), we have dOL dL 9 (r-rC) ) (r -r) +'L 5 (r-rc) (rs-r) Cd -sL 6(r -r) ~ (r-r C) (37) r Upon integrating this, we get Ca OL(r) + Oc = Cb - 2 (38) 2r On employing the boundary conditions L(r) = s at r = r, $L(r) = 0 at r = r, (39) we obtain C C a.- 2 ( 4 o.-..) b 2 2 (402rc C C - a _, (40-2) 2rs and therefore 2(- ) r 2r 2 Ca =- - ( ) c, (41-1) 2 2 rs - rc 39

C C 2 2(- ). (41-2) b 2 C 2rc Now inserting these values of Ca and Cb into Eq. (38) we get r 2-r2 r 2 (L(r) - 2) (rs-r). (42) L c 2 2 2 r -r r Equation (42) defines the desired limiting potential function. It is seen to be of the same form a.s the potential defined by Eq. (5) which ha.d been employed in a.n earlier section to identify La.ngmuir' s sheath model. 0L is graphically represented by the fs.ac curve in Fig. 1. The corresponding V(r) is obtained by using the radial derivative of Eq. (42) for O' in Eq. (28), which yields 2mr2 (r) = (t) f- — ((r -r) (43)' rS2 (r-r) r2 i,. 0,-.1 rc As shown in Fig. 7, this is a. horizontal straight line for r < r where it has a step function drop to zero. The conclusion is that for a finite sheath radius and given boundary value' (rs) any monotonically increasing function 4(r) (corresponding to some potential function) which lies in the rectangle, illustrated in Figo 7, will give rise to Langmuir' s volt-ampere relations. The dashed curves in the figure represent two such possible 4(r) functions. For the infinite-sheath case (rS-oO) the right-hand side of Eqo (42) reduces to the inverse square law. Thus for an infinite sheath the limiting potential function (corresponding to the limit of the range for the orbitalmotion-limited mode of collection)is simply 40

((r) FOR (L / / i(r) | / 1 (r) FOR / / 0 rc - rs Fig. 7. Graph of the singular i-function (43) corresponding to the Eq. (42) form of ~L, which is the limiting form for validity of the volt-ampere equations of the Langmuir finite sheath model. <L is the fs.Oc curve in Fig. 1, for r < rs, and of course:L = 0 for r > rs. 41

ipl = Oc( (44) 3.5 DEPENDENCE OF THE MOTT-SMITH AND LANGMUIR CURRENT-COLLECTION EQUATIONS ON THE PRESENCE OF A DISCONTINUITY IN THE POTENTIAL GRADIENT AT THE SHEATH EDGE When one employs the domain of integration given by Eqs. (36-1) and (36-2) as illustrated by Fig. 6, the resulting expressions for the accelerated currents to the cylindrical and the spherical probes are the ones obtained by Mott-Smith and Langmuirl (see the Appendix for the derivations). It is important to note here that it was necessary to assume a discontinuity in <' across rs (indicated at r = rs by the M2 = 0 curve of Fig. 4) in order to obtain these current expressions. If we were now to assume that 0 has a continuous derivative at rs, and in fact that Lim'0 = 0 from the left and from the right this would amount to requiring that, in the neighborhood of rs the variation of D be as (r - rs)n, where n > 1. This contradicts the gross-aspect requirement, used by Mott-Smith and Langmuir, that the variation of 0 must approximate that of an inverse power less than 2. More specifically, if one were to require that O' = 0 at r = r, this would in turn require that rs=mrs3 (dO/dr) = 0. For a finite r, this is wholly inconsistent with the requirement illustrated by Fig. 7 that r(r) be a monotonically increasing function of r for rc < r < r s Therefore, as will be seen later, the current expressions obtained by using V' = 0 at r = rs are quite different from those of Mott-Smith and Langmuir. One may also note that in the finite sheath model, with Lim #' O, a slight error in the estimation of rs can contribute a sig42

nificant error in the current calculation. This is so because the slope of Line (1) in Fig. 6 is inversely proportional to r2. 3.6 ADMISSIBLE SPACE FOR THE ORBITAL-MOTION-LIMITED CONDITION IN THE MOTT-SMITH AND LANGMUIR THEORY The case, where the sheath radius is large as compared with the probe radius, is of special interest in many applications. Following La.ngmuir, this is referred to as the orbital motion limited case. As r increases, s the slope of the function E = M2/2mrs (see line 1 in Fig. 6) becomes small and in the limit of infinite sheath radius goes to zero. r(r) becomes then a monotonically increasing function from r = 0 to r = oo. Consequently, for every given M2, Eqs. (27) are satisfied for some value of r. In other words, he(rs) = (e(oo) = 0, and he(r) has a minimum for every M2; this is illustrated in Fig. 8. The corresponding admissible space is illustrated in Fig. 9, a.nd is generated by the following expressions obtained a.s modifications of Eqs. (36-1) and (36-2): O < E < oo, if O < M2 < M 2; (45-1) 2 E > 2 + 0c; if MS < 2 < M oo, (45-2) 2mrc where Ir -oo in Eq. (55-1) i S M 2 =2mrc2( - ). (45-3) Note that the admissible space illustrated in Figo 9 governs the MottSmith and Langmuir orbital-motion-limited condition. 45

\ Total Energy E be (r) \ _ __ /___ e ) \ \i M 1/2 mu2 r ~ C \\ rCr 1/2 m u2 r \ 2 any positive E will result in collection, as expressed by Eq. (45-2 to occur, as expressed by>M Eq. (45-2). See Fig. 9, for the corl l >/M2.<M Eq. (45-1) Fig. 8. Plots of the effective potential if rsa, giving the MottSmith and Langmuir orbital-motion-limited condition. If M2 < MS2 any positive E will result in collection, as expressed by Eq. (45-1), if M2 > Ms2 E must equal or exceed Oe(rc) for collection to occur, as expressed by Eq. (45-2). See Fig. 9, for the corresponding admissible space. 44

ADMISSIBLE I\\\Y SPACE EQ (45-1) E'1 (45-2) M2 \\\^\^ E = Zr? + E= Ms= 2Mrc2 (-r c) Fig. 9. "Admissible space" diagram, for the orbital-motion-limited condition, occurring for the Mott-Smith and Langmuir model when rs-wo. The two shaded regions correspond respectively to the Fig. 8, curves for M2 < Ms2, as in Eq. (5-1), and for M2 > M2 as in Eq. (45-2). 45

3.7 A CLASS OF POTENTIAL FUNCTIONS, GIVING RISE TO BOTH A MAXIMUM AND A MINIMUM IN THE EFFECTIVE POTENTIAL Next, we shall examine the effective potential ( (r) for a class of potential functions with the following properties: The associated *-function increases montonically to a certain value of r denoted by ri, where it reaches a maximum, and then decreases montonically, as shown in Fig. 10. At r = 0, t(r) = O. Consider for a time the condition r < ri, and define M. so that t(r) = Mi2 at its maximum point r = ri. Thus with this form of t(r) we have no solutions of Eqo (27-2) for M > Mi2, which means that for these larger values of M the function 0e(r) monotonically decreases with increasing r, for all r < o. Also, the curve for (De(r,Ml) ha.s an inflexion point at r = r For M <, Eq. (27-2) has two solutions corresponding to the values of r where eD has a maximum and a minimum. e Let e *,r*, and e* symbolize, respectively, the maximum value of b (r), the radius at which it occurs, and the potential at that radius, all for some given M. Evidently r* > ri. Thus we may e *2 + (e (46-1) 2mr Of course r* is the solution of Eq. (27-2) which identifies the radius at which this maximum occurs: r* is itself a function of M2. It is clear that in order to find the locus of the maxima of be one must first know the form of e. However, instead of restricting ourselves to a particular form of e, we shall consider a class of potential functions for which the loci of the maxima of $ (r) as a function of M2, are of the form 46

(De = Kv M2 r* > ri (46-2) Using the definition (26) of De we therefore obtain 0 * = M + + * = KVM2 (46-3) e 2mr De* is said to be separable for a. given potential D if it can be written in the form (46-2) such that v and Kv are independent of M2. These parameters depend only on the probe geometry and potential. In Eqs. (46-2) and (46-3) one must always take M2V a.s (M2) for all values of v, in order to retain the symmetry be* must have with respect to the angular momentum M. Note particularly that Eqs. (46-2) a.nd (46-3) describe only the beha.vior of be a.t the maxima, and do not restrict the location of the minima.. 358 THE UNIQUE ANGULAR MOMENTUM Mk FOR WHICH THE MAXIMUM VALUE OF EFFECTIVE POTENTIAL EQUALS THAT AT THE COLLECTOR SURFACE We shall now study the relation of the maximum value De* to the value De(rc) of the effective potential at-the probe surface for a, *function of the type illustrated in Figo 10o Such a. study leads to an evaluation of Kv a.nd -v for the particular class of potentials identified by Eqs. (46-2) and (46-3), but also of general interest as regards any potential function whose associated *-function varies a.s shown in Fig. 10. For any particular potential function having two extrema in the expression for De(r), the probe radius rc may in principle be less than, 47

S(r) ~/ j \ X~ ~ILLUSTRATIVE M2 o r Fig. 10. Graph of a i4-function, displaying a maximum inside the sheath. In this case the effective potential has: a minimum if r < ri and M2 < Mi2; a maximum if r > ri and M2 < Mi2; and inflection point if r = ri and M2 = Mi2; neither maximum nor minimum if M2 > Mi2. 48

equal to, or larger than ri. This location of rc provides an important basis for classification of the dynamic behavior. Of course, only the region r > rc is of physical interest, wherever rc may lie relative to ri For any relationship of r to r., that is, for rc ri, there can exist a value of M2, denoted by M2, such that Oe(rcMk) = e (rk, ( ioeo, when M2 = Mk, e at the maximum equals that at the probe surface. Use of Eqso (26) and (46-1) gives:. + = +. (48) 2mr2 c 2m2 k 2mr, 2mr This is also expressible a.s M 2 2mrr2 2 k - rc k c= -rk 2 2 k rc (49) rk - rc Using Eq. (27-1) M2 can also be expressed in the following way: M< = mrrk3 (50) where Ok =' (r ). A relation between rk,k and'. can be obtained. by eliminating Mk2 between Eqso (50) and (48). Useful forms of the result are r3 rk 2 k + 2' k (5 2rc and 49

rk r r $k 2 2 2 ___ 2= Xk c (52) These equations apply to values of rc over the whole range r > r. ran e c < 1 For a *-function of the type shown in Fig. 10 let k = i(rk), (55) of course Mk = M2. Thus, in Fig. 11, l k is some value of i(r) that is less than Mi2. For the moment let us consider the case where r < r.. It is clear c 1 from the concepts underlying Fig. 10 and Eq. (47), (without requiring that e* = KM2V) that if rc < ri the function e(rr 2) will exhibit the following properties in the range r < r < rk: (a) A maximum at r = rk > ri in accordance with the definition of Mk2, (b) a point of inflection at a radius between the two intersections of Mk2 with *(r), (c) a minimum at some radius r < ri for which i(r) = k = Mk2, (d) a negative slope at r = rc, (e) $ (rkDM2 ) = e((rcM2k). Figure 12b represents this behavior for a collector radius rc. The case where r > r. is of special interest. An important aspect of it will be dealt with in the next section. 50

KI - PROBE -SHEATH iL(r ~fa ) 2+_ —-----—.... —-- M M / I X mmmm mm m m ^ —mm- O F e/ I/ 0 r, r, --- Fig. 11. Graph of a i-function for the case of a potential for which ri < rc. 51

- ri M j2 ) (a) For M2 = Mi2, e(r) e (r) has a zero-slope inflexIeI Is ] ion point at r = ri Probe - Sheath II i 0 rc' r r rs \t | —_ _ e (rc, Mk2) (b) For M2 = k2, the e -(r) e \/,(r|, Mk2 ) | maximum *(rk,Mk2 occurs e e k' I at rk = rk. 2 |I M2M2 0 _ ^/ ~~| i: ^ ~ I| ~ ((c) The general case, [~ ~ ig~ -^~ ~ M2 < Mk2, with $e oco rc' rS she ath |iI /^ for M2 =0 | I/ I Fig. 12. Graphs of Me(r) showing in each of three cases the maximum 0* at r = r*. Two possible sheath radii are shown, rc > ri and r( T ri. 52 C ~ ~ ~ crigisdC h

3.9 PROOF THAT FOR rc > ri THE EFFECTIVE POTENTIAL MAXIMUM CAN EQUAL THE EFFECTIVE POTENTIAL AT THE PROBE SURFACE ONLY BY OCCURRING AT THE PROBE SURFACE: i.e., rk = rc if ri < rc The title of this section describes an important aspect of the behavior of Oe(r) for the conditions existing when rc > ri (see Fig. 11). It is intuitively evident that for ri < rc, rk must be equal to rc. From Eq. (47) rk is the radius at which the maximum Ce*(rk,Mk) equals the value ~e(rkM2) at the probe surface. If the maximum 0e*(rkMk) were to occur at a radius greater than rc, then'e(rcMk) could equal ~e*(rkM~) only if there were to exist a minimum of De(r,M2) between rc and rk. But according to the definition of ri, the function Oe(r,M2) has no minima in the range r > ri which includes the range r > rc since rc > ri. Thus clearly rk = rc if rc > ri; that is, the particular maximum oe*(rkM2) occurs at the surface of the probe, for the case ri < rc presently under discussion. This is indicated in Fig. 11 and its significance and consequences clarified by Fig. 12. For an analytical proof that rk = rc when rc > ri we examine the identity rk dO e(r,) - c ) = r dr (54) e rkM e c rc dr Let us assume that rc t rk. In that case dOe/dr is positive in the range rc < r < rk and zero at rk, since be has a maximum at rk and no minimum for r > rc (since all the minima are located in the range r < ri and we consider now the case r > ri) It follows that the integral on the right-hand side of Eq. (54) is positive definite. However, according to the definition of rk (Eq. (47)) the left-hand side of Eq. (54) vanishes. These two conclusions are contradictory. Therefore, the initial ass.umption that rc / rk must be wrong, and we conclude that 53

rk = r, if r. < r (55) k C 1 C Consider for a moment the applicability of Eqs. (48) and (52) to Eq. (55). Obviously, Ok = (Dc when rk = rc. Therefore, both Eqs. (48) and (52) are identically satisfied when rk = r. 3.10 EVALUATION OF Kv AND v, FOR THE CLASS OF POTENTIAL FUNCTIONS DEFINED BY THE RELATION be* = K.M2V 2 In this section the properties of e (r,Mk) will be used to evaluate K and v, first for arbitrary ri and then for the special case ri < rc. If ri is arbitrary, we can have rk> rc. By definition of M2 in Eqo (47), the unique maximum e*(rkM2) has the same value as $e(rc,Mk). If ri > rc, then rk > rc. This is illustrated in Fig. 12b for a collector surface with radius r'. c Using Eq. (46-3), and its derivative with respect to M2, both evaluated at M2 = M, we can solve for K and v for arbitrary ri. From Eqs. (46-3) and (47) we obtain, at r = rc,, M2 k + c K M2'v e*(rkk) = (r k) = - +c = KM (56) 2mrc Solving this for K. gives + M2k/2mr2 c k c Kv =- (57) M2 v k In order to obtain an expression for v, we take the derivative of both sides of Eq. (46-3) with respect to M2 and evaluate it at M = K. For the left-hand side of Eq. (46-3) we obtain

____ = ___- _ + I* I (58) (M2) M2 M2) M 2mr*2 ( M2 In carrying out the differentiation we must recall that r* is a function of M2, and that it becomes rk when M2 = M2. We then obtain (M22 2 k 2 8O*(M2)| S k - mr3k) 8(M2) Mk 2mrk mrk / (M2) In Eq. (59) we used the notation a-* = - D (60) 6r* = r k rk rk Because of Eq. (50) the parenthesis in the second term on the right-hand side of Eq. (59) vanishes, so that the slope of Oe*(M2) at Mk2 is oa*(M2) e 1= 1 (61) 2 2 2 6(M ( 2mrk Differentiating the right-hand side of Eq. (46-3) we obtain a(K M2v) -( — - j:= M vKkv v-2 (62)'' k On equating Eqs. (61) and (62) we get Mk2 V = (63) 2 2v 2mrk KvMk If Eq. (57) is substituted into Eq. (63) and 2 is replaced by k' the expression for v becomes 55

k/2mr2 V k/ rk (64) (c+D k/2mrc which is correct for arbitrary values of ri, i.e., r. It should be pointed out that v and Kv are by definition independent of M2. Therefore, the evaluation of Eqs. (61) and (62) for M2=M2 is merely a matter of convenience. The expressions for v and Kv are valid for all values of M for which > has a maximum. Equation (57) can be rewritten by setting e M = k'. This gives c c + /2mr2 K,' c (65) k where the exponent v in the denominator is given by Eq. (64). Equations (64) and (65) are the desired expressions for v and Kv for r r. If ri > rc, then the effective potential Oe(r,Mk) has both a maximum and a minimum in the sheath region. In that case the value k', and thus also the parameters v and K given by Eqs. (64) and (65), depend on the particular form of the potential function O(r). If, however, ri<r, then rk = r, and Eq. (50) can take the form ~k = mr3c'. Using this in Eqs. (64) and (65) one obtains for ri < r 20 + r $' K c c (66-1) 2(mrrcC) r. 1 v = c (66-2) 20 + r O' c c c It is clea.r that for the range of values ri < rc once the potential energy function e(r) is known, K and v can be found, provided e(r) beV

longs indeed to the class of functions defined by Eq. (46-2) in terms of the loci of the maxima of the effective potential. e 3.11 THE ADMISSIBLE SPACE FOR THE CONDITION r. > rc Next we examine the "admissible space," in the (E, M2) plane, for effective potentials satisfying the condition (46-2) and for the case ri > rc which implies both a maximum and a minimum in $e(r). (See Fig. 12 with probe radius r'.) From the definition of M2 Eqo (47) it follows that any inward moving particle of angular momentum Mk and energy E > De*(rk,M2) will be able to reach the probe. For M2 < M2, a. particle with energy E > be* (rM2) can reach the probe, because then Oe(rc,M2) will be lower than the maximum e*(r*,M2). On the other hand, if M2 > M, then e(r2,M2) >e *(r*,M2), as illustrated in Fig. 12o Therefore, in order for such a particle to be collected, it is not sufficient for its energy to be larger than the maximum Oe*(r*,M2). Instead the requirement in this case is E > D (r2 M2). Thus, when ri > rc, the admissible space is generated by the following relations, illustrated in Fig. 13 (see Fig. 12 with collector radius at r'): c E > * = K M2v, if 0 < M2 < M2 (67-1) e v k E > (r) = +, if M < M2 < o (67-2) - e c 2 c 2mr c 3.12 ABANDONMENT OF THE CONCEPT OF A WELL-DEFINED SHEATH OUTER BOUND IN FAVOR OF AN INFINITE SHEATH RADIUS As discussed in the Introduction, a.nd in Cha.pter III, there a.re 57

ADMISSIBLE SPACE 2) Q ( 67-1 EQ. (67E ~w 1 ~~ 2mr,2 Fig. 13. Admissible space diagram from Eqs. (67-1), (67-2), for a $-function of the form illustrated in Fig. 10 with ri > rc. 58

serious logical inconsistencies in presuming that there is a. well-defined outer bound to the sheath. Realistica.lly not only the potential energy function but also all of its radial derivatives must be presumed to be continuous through the sheath into the plasma. Such continuity demands that the sheath potential approach the p.la.sma potential asymptotically with increasing radius; thus there can be no well-defined sheath radius. There exists a range of values of the radius within which the merging of the sheath into the plasma. occurs. Any plasma will exhibit random potential variations, and the sheath may be considered to have merged into the plasma. when the difference between the sheath potential and tha.t of the undisturbed plasma is sma.ll relative to the random spatial variations in plasma potential. In terms of experimental systems, this may occur at a relatively small distance from the probe surface. Mathematically the merging of the sheath into the plasma is provided for by letting the sheath radius become infinite (rs->oo) and requiring that both the potential and the potential gradient be zero at infinite radius. That is, D = 0 and (' = 0 at r = rs = 0. Of course this makes the sheath include the region some authors have,,6 described as the "potential well1, a region that resembles the plasma in having very nearly equal electron and ion concentrations, but resembling the sheath in exhibiting a. significant ra.dial potential gradient and a. significant flow of charged particles of one sign or the other or both toward the probe. The "potential well" concept represents a. reasonably good approximation to reality. It corresponds in general to the concept 59

of an ambipolar diffusion envelope around the probe, paralleling in some on7 8,9 degree descriptions by Tonks and others 9 of ambipolar diffusion in laboratory plasmas. Thus when, for purposes of orbital analysis, we extend the sheath radius to infinity, we must recognize that we are including in the sheath two types of regions having wholly different properties. Any self-consistent ana.lysis of the sheath potential must be handled accordingly, as for example by using what ha.s been called the "combined,,7,8 plasma-sheath equation'. In the region close to the probe, particles of one or the other polarity dominate, so that space charge has a major effect on the potential distribution. In the outer or transition region the positive ion and electron charge densities a.re undoubtedly very nearly, equal; particles of each polarity are themselves in thermal equilibrium, but may be at different temperatures. The potential gradient and net flux of particles to the probe are not zero. The flow of the heavy particles (ions) to the probe may either be governed by a "mobility" affected by collisions, or they might fall essentially freely without collisions. 3.13 ORBITAL BEHAVIOR AND ADMISSIBLE SPACE FOR AN INVERSE-POWER-LAW POTENTIAL FUNCTION In this section the orbital behavior of particles moving in a potential of the form O X (68-1) 60

will be discussed. Here (c is some positive number and Tc is negative. The sheath radius for such a potential is infinite, i.e., -+0 and' -+0 asa r-o. The gradient is = — (-c) + 1 (68-2) r It is useful also to state the corresponding expression for the spacecharge density, obtained by employing Eq. (68-1) in Poisson's equation. The results are: For the infinite cylinder: 2 p c a-rc (68-3) r For the sphere ca(a -,]): P = (4() - l) (68-4) For the sphere the case = 1 corresponds to the space-charge-free situation; for the infinite cylinder the space-cha.rge-free potential ha.s a. logarithmic variation and so is not included in the family of potential functions defined by Eqo (68-1). This zero-space-charge potential for an infinite cylinder has a zero gradient at infinite radius like the potentials of the form (68-1), but unlike them it becomes infinite at infinite radius. We note the following aspects of the potential (68-1): 61

(a.) The use of an arbitrarily assumed potential function corresponds to assuming a specific radial distribution of spa.cecharge density, as in this case described by Eqs. (68-3) and (68-4). Note that for Ua > 2 the space-charge density must.a+2 vary a.t 1/r (b) If a > 2 in Eq. (68-1) then V(r) of Eq. (28) is a. monotonically decreasing function of r, because r(r) then has the form: r J((r) = m(-@c ) -~ (69) As discussed earlier, the locus of the extrema of the effective potential e*(r*) is obtained by solving the equation M2 =r(r*). If 41 (r*) > 0 the extremum is a. minimum, and if 4s (r*) < 0 it is a maximum. Hence, the effective potentials associated with a. potentia.l of the class (68-1) have only maxima if a > 2. Consequently, if a > 2, the logic leading up to Eq. (55) applies and requires that rk = rc; that is, the maximum of the function e(rk,,Mk2) must occur at the probe surface, r = rc. (c) For any non-negative value of Oa in the power-la.w potential (68-1) the separability condition (46-2) is always satisfied; this will shortly be shown for the ca.se al > 2. For the moment, the treatment will be restricted to the ca.se Ce > 2. Then rk = rc as noted in (b) above. This situation is illustrated in Fig. 14 for M2 < M 2, M2 = M 2, and M2 > M 2. We may then write Eq. (46-3) as k k k follows: 62

r*' * + * = Kv(mr*'*)V (70) where we have substituted mr *3' for M2 from Eq. (27-1). Now, Eqs. (68-1) and (68-2) become a.t the maximum e * (DI'*: (- o+) (71-1) r.(C + 1) * D = -(- o)() (71-2) Introducing these into Eq. (70) yields the following equation (- ) ( - 1) = )r r(2 - )v + a (72) As r* is a function of M2, while none of the other quantities depends on M2, this equation is satisfied only when the exponent of r* is zero. Therefore we find that v = I7 (73) a- 2 According to Eq. (46-3) this value of v, together with the corresponding value of Kv given below, determines the maximum De* and its locus r*, provided that the potential 0(r) is of the form (68-1). These maxima have meaning only if r* > rc. Direct evaluation of v from Eq. (66-2), confirms the value for v given by Eq. (73). The value of K corresponding to the v of Eq. (73) is, (-~o)(a - 2) Kv ~ rv^,/( -).

Ii c M2>Mk2 e (D -e (Mk2) / \ ^s M2:Mk / \M / Locus of / | <>err D* ( r* r e r V M2 =0, i.e. CD =<C rc atr —— c /c /ge / / Fig. 14. Graph of the effective potentials, be(r) for an inversepower-law potential function ~=~c(rc/r)a with a > 2; also shown is the locus e*(r*). 64

For convenience Eq. (73) and an alternative form of Eqo (74) will be repeated in Eq. (75). These expressions evidently apply to the case a > 2, and to some extent more broadly a.s discussed in a later section; they are valid for any M2 v = (75-1) - 2 c -2 K. a/(a-2./(-2) 75-2) 2(-c) 2 nrc) This la.st equation is precisely of the form that is directly obtainable from Eq. (66-1). Since Eq, (66-1) applies to the ca.se of a. monotonically decreasing 4(r), it is therefore valid in the range U > 2 (see Eq. (69)). The fact that Eqs. (75-1) and (75-2) are independent of M2 shows that the separability condition (46-2) is satisfied in the case of the inverse-power-law potential (68-1) with a > 2, a.s has been claimed above. Substituting Eq. (71-1) into Eq. (27-1) with r = r* = rk = rc we obtain the following expression for Mk2: M2k = Camr (-c). (76) This expression is valid for potentials of the form (68-1) with a > 2. It defines the angular momentum k for which the corresponding effective potential he ha.s its maximum at the probe radius. Figure 14 illustrates the locus, he*(r*), of the maxima, for the potential function ~ given by Eq. (68-1). The equation for the locus curve for the given model is obtained from Eq. (46-2) by using Eq. (27-1) 65

for M2, with 1' as in Eq. (68-2), and with Eqs. (75-1) and (75-2) used for v and KV. The locus equation so obtained is ( r*) -=(D^c ) - () (77) The value of r*/rc as a function of M2 for any locus is obtained by eliminating 0'* between Eqs. (27-1) and (68-2), evaluated at r*, and using Eq. (76). The result is */M 2\l/(2 i) r = - 2 (78) rc wVik Combination of the last two equations give. (e*(r*) as a function of M2; thus *(M2) = (_) - 2M2 /(a-2) Pe*( Ma2) = ( r )-22 (79) 2 Mk2 when Eq. (68-1) applies, and a > 2. Equation (78) can be rewritten in terms of the function *t(r) since at the maximum of <e by definition M2 = r(r*). Hence, one obtains, 4(r*) = Vk 1, 2 (80) (r./re)..- 2 (r*/rC) This is illustrated by Fig. 15. Figure 13, with Mk2 given by Eq. (76), represents the admissible space for the dynamical system based on the inverse power law (68-1) with a > 2, because this system satisfies Eq. (46-2), and Fig~ 13 is adequate for any system that does so. 66

t(r) _ Fig. 15. Graphs of ip(r) for various values of a in the poweroc= 2.5 law-potential Eq. (68-1 ). <67.5 Fig. 15, Graphs of *(r) for various values of a in the powerlaw-potential Eq. (68-1).

3.14 DISCUSSION OF POTENTIAL FUNCTIONS DIFFERENT FROM ANY INVERSE POWER LAW, BUT WITH SEPARABLE LOCUS-OF-MAXIMA EQUATIONS Illustrative potential functions different from Eq. (68-1) might be postulated that would satisfy Eq. (46-2) or some equally useful separability condition, and at the same time have one or more of the following properties not possessed by Eq. (68-1). Thus such functions might conceivably (a) be of forms giving the curve of t(r) both increasing and decreasing portions as illustrated in Fig. 10, thus permitting minima as well as maxima of,e' with the locus of the maxima obeying Eq. (46-2) or some equally separable function. (b) be of forms that permit the model to have a finite sheath radius, with both 0(rs) = 0 and D' (rs) = 0. The model outlined in (b) has the weakness that it permits the second, third, and higher derivatives of 0 to be discontinuous at the sheath edge. It is intuitively reasonable to expect the true model to exhibit continuity in all derivatives of the potential at all values of r. One of the merits of Eq. (68-1) is that, by abandoning the concept of a finite sheath radius, it does retain continuity of all derivatives of 0 for all values of the radius. However, it is certain that the actual potential variation is much more complex than Eq. (68-1), and that the loci of the maxima of the effective potential do not obey Eq. (46-2), nor any similarly separable function. Nature is not that kind to us. Yet one learns a great deal by studying the model (68-1) for (X > 2, which satisfies the separability 68

property (46-2) and pushes rs out to infinity. Undoubtedly one useful potential model would be the sum of two potentials, one corresponding to the space-charge-free potential variation, the other describing the potential due to the existing space-charge density, whatever it may be. For spherical geometry this would employ O = 1 in Eq. (68-1) to describe the space-charge-free behavior with some other function describing the effect of the space charge. This second term would be of opposite sign to the first, because the charge causing it would be of opposite sign to that on the surface of the probe. For the cylindrical geometry, the space-charge-free term would have the form < = Oc (ln r/rc + 1), which is obviously not of the form (68-1) 3.15 ADMISSIBLE SPACES FOR ARBITRARY POSITIVE VALUES OF THE EXPONENT IN THE INVERSE-POWER-LAW POTENTIAL (68-1) It is desirable to discuss the properties of a model that uses the potential function (68-1) with arbitrary positive a. Negative values of a are excluded because the purpose of introducing the potential function D = 0e( r/r)? was to set up a model having an infinite sheath radius, ioe., to provide an asymptotic approach to 0 = 0 a.s r oo. If we plot v vs. C from Eqo (75-1) a.s is done in Figo 16, we get a singularity a.t a = 2, and we find that v + 1 as a + co Only non-negative values of v are of interest because be* in Eqs. (46-2) and (46-3) is identified with the maxima of Oe(r). In any realistic model the maximum value of the effective potential must of necessity be an increasing function of the a.ngular momentum. This is a. direct consequence of the definition of the effective potential Eq, (26). We note that for an arbitrary potential. >(r) the effective potential ~e(r) increases with 69

I I -2 - I 6aoI 4 —cc — 1 2-2- -2t -4t Fig. 16. Graph of V versus, as given by Eq. (75-1) in the case f the powerlaw-potential given by Eq. (6870

M2 for fixed values of r. Therefore the be(r) curves for different values of the parameter M2 (eog., Fig. 4) cannot cross one another, and it follows that the maximum value Oe*(rl*,Ml2) must exceed the maximum value Oe*(r2*,M22) if M12 > M22, i.e., e*(r*,M2) is an increasing function of M2, and therefore negative values of v have to be ruled out. According to Eq. (73) a must be larger than 2 to assure positive values of v. From Eq. (69), furthermore it can be seen that 4 is a monotonically increasing function of r if a < 2, and, as has been explained earlier, in that case the effective potential 0e has no maxima, but only minima. Since the parameters v and K have been defined in connection with the maxima of cbe, it follows that the separability property (46-2) is satisfied by inverse power potentials of the form (68-1) only if a > 2. Attention is next given to the special case a = 2. In that case v is not defined by Eq. (73). In fact, the effective potential becomes then e(r) + c )/r2 (81) which has no extrema. Since this treatment is concerned with attractive potentials (i.e., ~c < 0), the effective potential vanishes for all values of r if M2 = -2mrc2(c. (82) This value of M2 is easily seen to be the value of Mk2 for a = 2 (see Eq. (76)). Figure 17 shows a set of curves of ~e(r) with = 2. It should a.lso be noted that the corresponding 4f-function is a constant, i.e., 71

I I rI I (r) M2 2 01 —---. 72 ~Mk I I Fig. 17. Graphical representation of ~e when o = 2 in the potential (68-1). These second-degree hyperbolas represent Eq. (81). 72

i = -2mr2c. (83) Hence, there exists only one unique value of M2 for which M2 =, the value given by Eqs. (82) and (83). Obviously this does not represent an extremum of the effective potential. The first derivative of be vanishes for this particular value of M2 for all values of r simply because 0e itself is constant. It ha.s been discussed in Section 3.4 that any potential with a nondecreasing *-function satisfies the conditions for applicability of Langmuir's equations. This also includes the ca.se of constant fo In particular it has been shown that the inverse square potential is the orbital-motionlimited case for the set of potentials having constant 4 functions. Therefore, the admissible space for potentials of the form (68-1) with c = 2 is given by Eqs. (45-1) and (45-2). Considering next the case 0 < c < 2 we note that the i-function is now monotonically increasing as illustrated in Fig. 15. The effective potentials corresponding to this range of a have minima, as shown in Figo 8. This leads again to the admissible space described by Eqs. (45-1) and (45-2). As has been explained in Section 2.4, the range a <.1 is physically meaningless in the case of a. spherical probe. We conclude therefore that potentials of the form (68-1) give rise to the orbital-motion-limited current-voltage characteristics derived by Langmuir, if 0 < O < 2 in the case of a cylindrical probe or 1 < a < 2 in the case of a. spherical probe. For very large values of O the sheath thickness becomes small and the collection is no longer determined by the orbital motions. In the 73

limit a -+ oo, the potential q(r) vanishes for all values r > rc and equals ic at the collector surface, r = rc. Hence, the potential gradient is zero for r > rc and becomes infinite at r = rc. The collected current should approach the random current in the limit a + oo. In order to verify this statement, we note first that the maximum be can be expressed by substituting Eq. (76) into Eq. (79) to obtain e(M2) = (-) ac - 2 cM2 (De*(X2) = -'DC ) 2 [2rc2( - (79-1) It is easy to show that lim e*(M2) = M2 for 0 < M2 < o. (79-2) a- oo 2mr c From Eq. (78) it follows furthermore that lim r* = r cQ oO Hence, since Oe has its maximum value at the collector surface, the requirement for collection is that E > Oe(rc) for all values of M2 which means in view of Eq. (79-2), E > --- M for 0 < M2 < oo. - 2mr The admissible space defined by this expression is illustrated in Fig. 18. By integation of Eq. (97) below it can be easily verified that this admissible space leads to the random current as has been claimed above. 74

ADMISSIBLE SPACE EQ. (83- 3) M2 / E E= //'c / V -$M 2 / 2mr ~/~~ Fig. 18. Admissible space diagram, from Eq. (83-3), for the infinitely thin sheath around an electron accelerating probe described by letting aO-l in the potential function (68-1). 75

IV. THE VOLT-AMPERE EQUATIONS 4.1 GENERAL EXPRESSIONS FOR THE CURRENT In this final chapter the expressions for the current collected by an accelerating probe will be set up in general form and then integrated between the limits of integration described by appropriate admissible spaces. Let f(U) be the velocity distribution of the particles, and U the velocity vector of a particle at any instant. The flux density Jr of electric A current carried toward the probe by the particles crossing a particular closed surface a.t which the velocity distribution is known is given by: Jr = qdU u. n f(u) (84) where q is the electric charge carried by ea.ch particle a.nd n is the inward-pointing normal unit vector. The integration is to be carried out only over inward-bound particles and, among them, only those destined to arrive at the probe. Thus "admissible space" criteria govern the integration limits. A t. n is taken positive when pointing in the direction of the center of the force field. The closed surface chosen for this integration ca.n be the outer radius rs of the sheath, or it might be a. "sheath edge" surface a.s introduced earlier in Section 2.1. Because the inward orbital motion involves loss of radial energy to a.ngular energy, the orbital behavior in the accelerating field causes marked departures of directions of motion from an initial random distribution. Therefore, the closed surface at which the integration is to take place should be far enough so that this effect of the probe 76

on the velocity distribution is as yet insignificant. If the known initial velocity distribution is the Maxwell-Boltzmann distribution function, then for a cylindrical probe Eq. (84) may be written as: uz + UJ Ur +(2>/m) r = Nq f f S duz dut dur ur exp uz +ut + ur 2 (2/m) ( o2)312 L2 (85) where, as earlier in this paper, Uz, ut, ur are the velocity components in the axial, tangential, and radial directions, respectively, c = Jk7T/m the most probable velocity, k is Boltzmann's constant, T the temperature, and ~(r) the potential energy of the particle at the radial distance r. For a. cylinder of infinite length, which is the model used here, the admissible values of uz, are from -oo to +oo, and uz is independent of ur and ut. Therefore Eq. (85) may immediately be integrated with respect to uz and rewritten for the cylinder as Jr -= 7 J dut durur exp (ur U+ u m m ) (86) Tr c c For a spherical probe we define ur, ug, u/ to be the components of i along the radial, polar, and azimuthal directions, respectively. Since the motion of a particle in a. centrally symmetric field is planar, the components ue and u/ may be reduced to one tangential component ut by introducing polar coordinates as follows: uG = Ut cos Y (87-1) 77

uo = ut sin 7 (87-2) where 7 is the angle between the plane of motion and the plane containing ug and the polar axis. With this change of variables Eq. (84) may be written for the sphere as Jr f f f dut dur dy ur ut exp - (U2 + (88) (nc2)52f Lc r t (88) where, once again, the integration is to be carried out over the inwardbound pa.rticles and with.limits established by admissible space criteria., Since all planes of motion are equivalent we integrate Eq. (88) with respect to 7 from 0 to 2it and write for the sphere 2N0q - - Jr / rt f dut ut du u exp - (u2 + u2 + 2 ) (89) 12Jr - /3UrUt -rr r m We may now transform the variables of integration of both Eq. (86) and Eq. (89) from the velocity space (ur, ut) to (M,E) spa.ce, as follows: E -= 2 (u 2+ u 2) + (90-1) 2 t r M = mrut. (90-2) The second of these equations can be used to eliminate ut2 from the first resulting in explicit equations for the velocities in terms of M and E, u 2 (E - ), (91-1) r m2 m Ut = -M. (91-2) mr The transformation of the (dut dur) area. element in(ur, ut) space to the 78

(dE dM) area element in (M, E) space takes the form ( ur, ut ) dur dut = -- -- dE dM. (92) a (E,M) By the use of Eqs. (91-1) and (91-2) the Jacobian10 is found to be 6(ur,ut) 1 ( (93) 6(E,M) m rur Therefore, du dut = - dE dM (94) m rur The variable r in Eqs.(90), (91), (93), and (94) is of course the radius of the surface at which the velocity distribution is known. But the current density is that at the probe surface, denoted by Jc. The two current densities are related by the equations J = J r (cylindrical geometry) (95-1) c r r-c 2 r= (spherical geometry) (95-2) Jc = J -2 rc Applying the transformation (91-1), (91-2) to the integrals in Eqs. (86) and (89) and using the relations (95-1) and (95-2), one obtains the following expressions for the current densities at the probe surface. For cylindrical geometry: scr rMi dEdM exp(-m (96-1) For spherical. geometry: 79

(/2N.q3>2 ), dE dM M exp(- E(962) The last two equations may jointly be written in the form Jc = Jo hs6 M fE dE dM M6 exp(-.) (97) where 6 = 0 for the cylinder, 6 = 1 for the sphere, and Jo = Noq \v2m, (98) 2 h = (99) kT (2mrckT) 1 ) /2 For 6 = 0 and 6 = 1, respectively, h6 reduces to: 1 ho= /7 - - (cylindrical geometry) (100-1) (kT) /2Tmr2 hl = T2 (spherical geometry) (100-2) mr2(kT) 4.2 INTEGRATION OF THE GENERAL EXPRESSION FOR THE CURRENT The limits of integration are determined by the assumed potential model. Various potential models have been discussed in earlier sections. For example, the admissible space defined by Eqs. (67-1) and (67-2) applies to effective potentials which have the property (46-2). This property is quite genera.l a.nd includes the case of effective potentials which ha.ve both a minimum and a maximum, as has been discussed in Section 3.11. In Eq. (97), we may set M2 = x, and write 80

h x=M2 E=oo/2 Jc Jo~ 5 L f dE dx x(S / exp(-E/kT) 2 = x=o E=Kvx + f X fJ dE dx )/2 exp(-E/kT) (101) x=M E= 2 r+ The integration with respect to E is straightforward, leading to hkT x=Mk dx x'(65)/2 exp KxV -oo exp x(i)/2 ( i + exp(- d) dx exp (102) kT 22flh17kT/j This can be rearranged to yield the following expression: hskT kT +I)./2 yM(K/kT) 1/2 )/2 J, = j0 26~ f dy x.)2 ep(-yV) 2 (=J2Kv / Y 2 ^v// y=o + exp ( (2mrkT)) y=M dy ()/2 exp(-) Y k c kT " - J y=M /2mr kT J (103-1) where the variables of integration have been changed by using y = (Kv/kT)1 x in the first integral, and y = x/2mrckT in the second integral. Now let yv = z in the first integral, to convert the expression into the following: hskT~kT kT +1)/2)v 1 Z KVMl /kT J = Jo k -2! dz Kz('l)/2v exp(-z) 81

+ exp (- )(2mrckT)(6 /2 2 2 dy -+(+) /2 exp( T / y= 2/2mr 2kT dy y elp)d1 J (103-2) This may be expressed as J =7,\3()+1)/2 2 I)*I, - r _ (2mrckT) K] ) ) + / e cpl gm f i ( 04 ) + exp t k? ^''2mrkT t=x (,x) dt t exp(-t) (105-1) t=o and we recognize that t =o ir(,x) f- dt t-1 exp(-t) =r(S) - y(,x) (105-2) t=x Equation (104) represents the accelerated-particle current to the probe, with 5 = 0 corresponding to the cylindrical model and 6 = 1 to the spherical model. The radius of the probe is arbitrary, so that Eq. (104) applies to a *-function of the general shape shown in Fig. 10, whether rc > rc, or ri < rc, but satisfying condition (46-2) on the maxima of b(r). However, the case of special interest to us is the one where ri < rc (see Fig. 11), where in the range r > r there exists either no extremum in e or only a maximum as illustrated in Fig. 14. In that case rk = rc according to Eq. (55) that is, the maximum value of e*( r*) equals its value ~e(rc) at the probe surface only if the maximum occurs at the probe 82

surfaceo For the class of potential functions, which have the property (46-2) and for which ri < rc, the coefficients K and v are given by Eqs. (66-1) and (66-2), i.e., 2c + rc (c 2mrc c r <, v c c (106-2) 20c + r c' This is the condition for which Mk = *k = c = mrc c (107) 4.3 COLLECTED CURRENT EXPRESSIONS FOR THE INVERSE-POWER-LAW POENTIAL For an inverse-power-law potential function, as given by Eq. (68-1) we obtained expressions (75-1) and (75-2) for v and K,,, which can be written in the following forms~ v = (108-1) (-c)(-) ( 2) Kv [ 22 ( )8a/-2) where M2k = c >= amr2(-~c)o Now by substituting these expressions for v, Kv, and Mk2 into Eq. (104), we obtain 85

o ( -/- +l (0-2 1- (6t1)(0a-2)/2a -0./2 (c<~ ) 2) exp ( j kT I, Q o (109) Equation (109) represents the accelerated current collected by a, probe for the inverse-power-law potential (68-1). The mode of orbital-limited current collection is obtained by setting a = 2. Denoting the orbital-motionlimited current density at the probe surface by Joml, Eq. (109) becomes for the case ( = 2; 1J J~;_ + exp r 2, -c (110) For 6 = 0 and 6 = 1, corresponding to the cylindrical and spherical geometries respectively, Eq. (110) takes the familiar forms given by Eq. (A-13) and Eq. (A-14) in the Appendix. For = oo, the right-hand side of Eq. (109) reduces to J0o Consequently, in this mode of collection the probe collects the random current, as was anticipated in comments in the last paragraph of Section 3515. 404 CURRENT COLLECTION FOR A POWER-LAW APPROXIMATION TO THE DEBYE POTENTIAL DISTRIBUTION FOR A LARGE SPHERE As a, final example we will consider the Debye potential distribution for a. large spherical probe (\D < ri < rc) and find a power law which approximates it. The new notation is as follows: 84

OD(r) = The Debye potential function ~D = The Debye length. Thus OD = ce,e T ~ ~D = c X- exprM (111) kD By using this potential in Eqo (106) we get r +\D = c (112) r -~ c D We are now looking for a particular potential of the form (68-1) which has the property that the exponent v, given by Eq. (108-1) is equal to the expression of v (112) for a Debye potential. The two values of v are equal if rc+%D rc+a (113) Hence, the desired potential is /r ( (rc+ )D)/ - ~oc ~r (114) It can be shown that for values of r close to rc the potential (114) is approximately equal to the Debye potential if rc/%D > 1 In order to show this we let r = rc + 6r with br/rc < 1. Then Eq. (114) can be rewritten in the following form, @ = ( 11rc/r 8c rc(ll) 85

rc/r ( rc = r c ( _r r "-c - exp - ) i if /D <<1. (117) c r yHence, the volt-ampere relation for a spherical probe of large radius for the case of a Debye potential can be obtained by using the expression (113) for aC and 5 = 1 in Eq. (109). The result is Jc=.o [.. i) \ kT 2 Y XX+1 kT 2 + exp, -2 + ekT (7 (118) where X = rc/D. 4.5 DISCUSSION OF THE VOLT-AMPERE RELATIONS The accelerated current density at the collector (cylindrical or spherical), for a power law potential, is given by Eq. (109). For the cylindrical geometry we set 5 = 0 and obtain, Jco = J oc0 (a-2)c22 a2 oc + exp(-%Doc). erfc(N-o) (109-1) where Ooc = (c/kT < 0 is the dimensionless potential energy at the collector surface. If we plot Jco/Jo against (-$oc) for fixed values of a we would an86

ticipate that for increasing G~ the ratio Jco/Jo would uniformly decrease i.e., if l1 > 62 then Jco(Cl) < Jco(C2) for all values of (- oc) In other words, the larger the values of cQ the smaller is the current density at the probe. The plot of ~o/Jo against (-%oc) for various values of C0 is shown in Fig. 19. The first curve for Ca = 2 is the oribital-motion-limited form of Eqo (109-1). The rest of the curves are for higher values of Oa as shown. One observes that for Qz > 2 the saturation region is attained progressively faster. The decreasing character of Jco/Jo for increasing 06 is illustrated in Fig, 20. These curves also show that in the range 0 < Ki < 2 the current is independent of Ca, which is not surprising since Langmuir s theory applies in this range and his volt-ampere characteristics are independent of the detailed structure of the potential. Table I lists the values of Jco/Jo for different ad and Toc calculated from Eq. (109-1). For the spherical probe (6 = 1) Eq. (109) reduces to, Jcl = <oci 2/cz /Q o-2c 0- 2,2 y, - o(c-2)) + exp(-oc(la) (109-2) where %oc again is the dimensionless potential energy (ec/kT) at the collector surface, and for attracting potential it is negative. Figure 21 illustrates the behavior of Jcl/Jo as a function of ( -oc) (Eq.(109-2)) for various values of a. For any fixed value of (-(oc) the decreasing character of Jcl/Jo for increasing a. is evident. Figure 22 illustrates that behavior for some representative values of (-Ioc)' These curves have been drawn for the range Oc > 1 since the case ca < 1 is physically meaningless as has been discussed in Section 2,4. Table II 87

TABLE I ATTRACTING CYLINDRICAL COLLECTOR POweR POTETIAL, 0 = (r/r) )a COMPUTED VALUES OF THE DIMENSIONLESS CURRENT AS A FUNCTION OF THE DIMENSIONLESS COLLECTOR POTENTIAL WITH ALPHA PARAMETERIZED. DELTA.000 ____________ ___________________________________________ PHI ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA ALPHA __________ 3.0 3.5 4.0 4.5 5.0 6.0 7.0 8.0 9.0 10.0.0- 1.OOCCE 00 1.0000E 00.l0000E 00 —.OOOOE 00C 1.0000E 00 1.OOOOE 00~1.OO E00 1-.OOOOE 00 —.OOOOE 00.OOOOE 00.5 1.3084E 00 1.2987E 00 1.2889E 00 1.2793E 0C 1.2702E 00 1.2532E 00 1.2382E 00 1.2248E 00 1.2128E 00 1.2021E 00 1.0 1.5215E 00 1.4965E0C 1.4719E 00 74486E 00-1.4270E 00 1.3889E 00-1.3567E 00-1.3294E 00 1.3061E 00-1.2859E 00 1.5 1.6943E 00 1.6519E CO 1.6113E 00 1.5741E 00 1.5405E 00 1.4829E 00 1.4362E 00 1.3979E 00 1.3659E 00 1.3390E 00 2.0 1.8417E 00 1.7812E 00 1.7249E 00 1.6744E 00 1.6296E 00 1.5551E 00 1.4962E 00 1.4490E 00 1.4103E 001.3780E 00 2.5 1.9713E 00 1.8926E 00 1.8211E 00 1.7582E 00 1.7033E 00 1.6138E 00 1.5447E 00 1.4898E 00 1.4.457E 00 1.4092E 00 OD.3.0 2.0874E 00 1.9908E 00 1.904TE 00 1.8303E 0 1.7664E 00 1.6636E 00 1.5853E 00 1.5243E 00 1.4753E 001.4351E 00 3.5 2.1929E 00 2.0788E OC 1.9790E 00 1.8939E 00 1.8216E 00 1.7067E 00 1.6207E 00 1.5539E 00 1.5007E 00 1.4574E 00 4.0-2.2898E 00-2.1587E 00 2.0458E 0 1. 9508E0 00 87C4E00-1.7452E 00 1.6519E 00 1.5801E 00 1.5232E 00-1.477OE 00 4.5 2.3796E 00 2.2321E OC 2.1069E 00 2.0019E 00 1.9152E 00 1.7798.E 00 1.6799E 00 1.6035E 00 1.5432E 00 1.4945E 00 5.0 2.4634E 00 2.3000E 00 2.1618E 00 2.0496E 00 1.9561E 00 1.8114E.00 1.7.054E 00 1.6248E 00 1.5614E 001.5103E 00 5.5 2.5421E 00 2.3634E 00 2.2144E 00 2.0936E 00 1.9938E 00 1.8404E 00 1.7288E 00 1.6443E 00 1.5780E 00 1.5248E 00 6.0 2.6164E-00-.208E -00 2.26 34E-00 2. 1346E CW0 2TC288Eh8 -1.8673E 00 1774E 00 1.6622E 00 1.5934E 00-1.5381E 00 6.5 2.6868E 00 2.4774E 00 2.3093E 00 2.1729E 00 2.0616E 00 1.8923E 00 1.7705E 00 1.6790E 00 1.6076E 00 1.5505E 00 "7.02 7. 5 3 - C'8E"- -0- -2* 3-5-26E 00 2 —200 E 2 2^4~ — 1. 00951.6946E 00-1.6209E 00-1.5620E 00 8.0 2.8756E 00 2.6297E 00 2.4325E 00 2.2756E 00 2.1490E 00 1.9590E 00 1.8239E 00 1.7231E 00 1.6451E 00 1.5830E 00 9.0-2.99 20E00 7T199 C2.5C52E 2.33$59E~0-2.2002E 00 1.9978E 00 1.8548E 00 1.7487E 00 1.6668E 00-1.6017E 00 10.0 3.0996E 00 2.8031E 00 2.5721E 00 2.3912E 00 2.2471E 00 2.0332E 00 1.8829E 00 1.7718E 00 1.6864E 00 1.6187E 00 11.0 8T Y20E 00 Y )EU0 27.634 1E 00 2. 44 2420ZZ9WETh -Z70658E 00 1.9087E 00 1.7931E 00 1.7044E 00-1.6342E 00 12.0 3.2944E 00 2.9530E 00 2.6921E 00 2.4901E 00 2.3305E 00 2.0959E 00 1.9326E 00 1.8127E 00 1.7209E 00 1.6485EO0 13.0-3.3836E JO0r 3.2i3E 00 " 2.7465E 00 2. 5348F S 273681E 00 2.1241E 00 1.9548E 00 1.8309E 00 1.7363E 00-1.6617E 00 14.0 3.4683E 00 3.0860E 00 2.7978E 00 2.5769E 00 2.4035E 00 2.1505E 00 1.9756E 00 1.8479E 00 1,7507E 00 1.6741E 00 15.0 -3.5490E 00-3.1 4E00 2.8465E 00 -2-167E 00 2.4369E 00 -2.1753E 00 1.9952E 00 -1.8640E 00 -1.7641E 00 —737F 0 16.0 3.6263E 00 3.2060E 00. 2.8928E 00 2.6545E 00 2.4685E 00 2.1989E 00 2.0137E 00 1.8791E 00 1.7768E 001.6966EO0 17.0 -3.7003E 00 3.2620ECE Z.3T0TrOE 00 29.o6 D55ESOZ-2.4987E 00 2.2212E 00 2.0312E 00 1.8933E 00 1.7888E 00-17069EO00 18.0 3.7715E 00 3.3157E OC 2.9793E 00 2.7249E QO 2.5274E 00 2.2425E 00 2.0479E 00 1.9069E 00 1.8002E 00 1.7167E 00 19.0 3.8401E 00 3.3674EC- 3 00..7578E 002.5549E 00 2.2628E 00 2.0637E 00 1.9199E 00 1.8111E 00-1.7260E 00 20.0 3.9063E 00 3.4171E OC 3.0588E 00 2.7895E 00 2.5812E 00 2.2822E 00 2.0789E 00 1.9322E 00 1.8214E 00 1.7349E 00

TABLE II ATTRACTING SPHERICAL COLLECTOR POWER POTNTIAL, I = e (rc/r)" CCMFUTED VALUES OF THE DIMENSIONLESS CURRENT AS A FUNCTION OF THE DIMENSIONLESS COLLECTOR POTENTIAL WiITH ALPHA PARAMETERIZED. DELTA =1.QOC0 PHI ALPHA ALPHA ALP;HA ALPHA ALPHA ALPHA ALPH A A LPHA ALPHA ALPHA t.o 3.5 4.0 4.5 5.0 6.C 7.0 8.0 9.0 10.0.0t 1.000CE 00 tC l1.i;tOOE 1.000 C 1.000,0 00t 10.i CE C,? 0 1., 0E C 1.On CO E C:. 1. no E'o 0E'.5 1.4851E;:0 1.4738E 00 1.4622E 00 1.45:58 00 1.4392E Cr 1.417 0 1.3975E 00 1.3791E CO 1.3622E 00 1.3466E 00 1.0 1.943CE 0 1.9210 C 1.8615 OC 13 8228E 00 1. 764E 0 1.72:9'2E 1 6645E C O 1.6158E; n 1.5737E 00 1.5369E o0 1.5 23 2.771E 2 2.233E C 2.2132c O0 2.1394E 0C 2. 723E C0 1.9570E 00 1.8629E 0C 1.7853E'C 1.7206E 00 1.6660E 0n 2.0 2.7904E Oi2 2.654C0E OC 2.5279E 003 2.4153E,0r 2.3158E,f.0 2.15I07E'"G 2.2213E 00 1.9180 E 00 1.8340E 00 1.7643E 0'0 2.5 3.1854E 2:0 2.9893E OC 2.8135E 00 2.o6'6E 20r 5 2.5287E iC 2.3159E 00 2.1541E 00 2.0274E 00 1.9271E n00 1.8451E 0 3.0 3.5641E 00 3.3f 34E C 3.0759E C: 2.32824E C0 2.7187E 00 2.4607E CO 2.2686t C0 2.1224E 00 2.0069E 00 1.9137E 00 3.5 3.9284E 30 3.5994E OC 3.3191E 00 3.2!856E,0 8 2.8911E 0C 2.5895E:, 2.3712E CO 2 2.2058E 0C 2.0769E 00 1.9736E?0 a)3 4.0 4.2797rE r 3.:600sEE rC 3.5466E OC 3.2737E 3,0 3.C47C E 00 2.7079E CO 2.4635E 00 2.2808E 0O 2.1394E 00 2.0270E.00 4.5 4.6195E O0 4.1472E iC 3.7609E O0 3.4457E "C 3.1954E 00 2.8166E C0 2.5479E 00 2.3489E 00 2.1952E 00 2.0754E 00 5.0 4.94689E:04 4.4, 28E ~0 3.9571E 30 3.6126E 00 3.3335E CO 2.9173E 00 2.6257E r0 2.4116E 00 2.2482E 00 2.1196E 00 5.5 5.2688E::0 4.64b2E C 4.15301 E 0 3.7698E 00 3.4634E 00 3.0115E CO 2.6982E 00 2.4698E 00 2.2963E 00 2.1603E 00 6.0 5.5801E iL 4.8733E rf0 4.3393E 00 3.9190 3.5 E 3.5862E 0 3.1,1 C 2.7661E 00 2.5241E 00 2.3411E O0 2.1983E 00 6.5 5.8836E -', 5.1i49tE rC 4.5175E 00 4.0612E 03 3.7(29E GO 3.1840E C: 2.8301E 00 2.5751E 00 2.3832E 00 2.2337E 0 7.0 6.1494E 0C: 5.3282E CC 4.6886E OC 4.1973E 0 3.8144E 00 3.2636E 00 2.8907E 00 2.6232E 0 2.4227E 00 2.2671E 00 8.0 6.7344E 00', 5 7535F OC 5.R 129E 0 4.4540E 00 4. 237E 00 3.4122E 00 3.0031E 00 2.7123E 00 2.4957E 00 2.3285E 00 9.0 7.2921E'05 6.1554ct C 5.3172E iC: 4.6934E ^0 4.21768E 0 3.5488E 00 3.1059E 00 2.7933E 00 2.5619E.".0 2.3840E 00 10.0 7.8272E:0 6.5381E bL 5.6 49E 90' 4.9185E C" 4.3993E nO 3.6756E (0 3.2008E 00 2.8679E C 2.6226E 30 2.4347E n0 11.0 8.3434E 00 6. r 43E-( 5.8785E 0j 5.1313E 0O 4.5703E;C 3.7943E 00 3.2892E 00 2.9371E 00 2.6787E 00 2.4816E 00 12.C 8.8434E?' 7.2564c'Cf 6.14C00E 00 5.3336E OC 4.7322E 00 3.9b60E 00 3.3720E 00 3.0016E 00 2.7310E nO 2.5251E 00 13.0 9.32'91E'0 7.5961 E OC 6 390 7E -3 5.5268E C 4.8861E ^0 4.0 116E 03 3.4500E 00 3.0623E 0O 2.7800E 00 2.5659 14.0 9.8C22E 0: 7.9247F OC 6.6319t 30 5.7118E 0'C 5.0331E 0- 4.1119E 00 3.5238E 00 3.1196E 00 2.8262E 00 2.6042E 00 t5.0 1.0264E 01 8.2434. CO 6.8647E 00 5.8897E 0! 5.1740E C. 4.2076E CO 3.5939E 00 3.1739E 00 2.869,9E 00 2.6404E "'0 16 0 1.0715E 01 8.5531F 00 7.0898E 30 6.0611E 0 5.3093E 00 4.2991E 00 3.6608E 00 3.2255E 00 2.9113E 30 2.6747E 00 17.0 1.1158E 01 8. d546E 00, 7.3':8CE'00 6.2266E 0t.' 5.4396E OC 4.3868E 00 3.7248E 00 3.2747E 0 0 2.9508E 00 2.7073E 00 18.0 1.1591E 0i 1 9.1486H 0 7.5199E 00 6.3868E 00 5.5654E 00 4.4712E CO 3.7861E 00 3.3219E 00 2.9885E 00 2.7384E 00 19.0 1.2016E 0.1 9.4356E OC 7.7259E r0 6.5421E 20 5.6871E n0 4.5525E 00 3.8451E 00 3.3671E 0 003.0246E 00 2.7682E 00 20.C 1.2434E 01 I9.7163 C'O 7.9267E C0 6.6930E 00 5.80500.E 00 4.6310E O0 3.9018E 00 3.4105E 00 3..0593E 00 2.7968E 00

I " ~~~~~~~~~~~~~~~~~~~~~a=3 2 1-a=4 0001 ~~~~~~~~~~~~~~~~a=5 6 6 a — a=10 0 1 2 31 4 5 6 ~ 8 9 1 0 1 1 1 2 1 3 j4 ~15 6 1 7 1 8 1 9 2 0 Pig. 19 Cylindrica l Probe (-o)P o th c 4rrent vs. dimensionless votg 0.Plot Of the -iesols vo I for -various values Of c l. oc f a u re t d n i y

DIMENSIONLESS CURRENT, I/I0 - (1r) I I o 0 0 ~- ~ ~ ~ ~ ~~~~N0 0 0 o p * (D _b 0 0 4(D (D Pi v

14 13 12 11-, 10 o 8 00 4 66 a = 2,,=,=a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Fig. 21. Spherical probe (61). Plot of the dimensionless accelerated current density (Jcl/Jo) vs. the dimensionless potential (-%oc) for various values of a.

20 19 18 17 16 -Cl= 15 15 14 13 12 11-Oc1 I 10 o 7 0 1 2 3 4 5 6 7 8 9 10 a -- Fig. 22. Spherical probe (6=l). Plot of the dimensionless accelerated current density (Jcl/Jo) vs. o for various values of (->oc). 93

lists the values of Jcl/Jo for different values of a and 0oc calculated from Eq. (109-2). In the volt-ampere characteristics of cylindrical and spherical probes (Figs. 19 and 21) the values of the parameter OG were considered independent of (oc and X, where X is the ratio of the probe radius rc to the Debye length \D. If one represents the potential distribution in the sheath by an inverse power law of the form (68-1)) the exponent Ca will of course depend on the probe potential 0oc and the parameter X. This is merely a. consequence of the fa.ct that the rate at which the potential disturbance decreases radially (which is controlled by Oc) depends on the probe potential and on the electron temperature and density. In a realistic sheath model, as pointed out earlier, the potential function must be determined in a self-consistent way together with the charge density. In the following section the relation between a, %ocn and X, will be obtained numerically by comparison of Eqs. (109-1) and (109-2) with the volt-ampere relations obtained by a self-consistent treatment of the problem. 4.6 COMPARISON WITH THE RESULTS OF A SELF-CONSISTENT ANALYSIS Using numerical methods Laframboise2 has recently carried out a selfconsistent field analysis of the potential and volt-ampere current relations for cylindrical and spherical probes. Tables III and IV contain his calculated values for the respective probes and Figs. 23 and 24 show the corresponding characteristic curves. For the determination of a as a. function of ~oc and X we took the values of the current from Tables III and IV, inserted them in Eqs. (109-1) and (109-2) respectively) and in94

TABLE III CYLINDRICAL PROBE Computed Values of Attracted-Species Current Density Ratio (J /J ) for T. /T 1 /_,, _. ~ _ co7 ion electron (Both Species Maxwellian) o v 0 o 1 1..5 2 3 4 5 10 20 30 40 0 100 0 1.0 1.0 1.0 1.0000 1.0 1.0 1.0 1.0 1. 1. 1. 1.0 0.1 1.0804 1.0804 1.0804 1.0804 1.0804 1.080 4 1.0084 1.84 1.0803 1.0803 0.3 1.2101 1.2101 1.2101 1.2101 1.2101 1.2101 1.2101 1.2100 1.208 1.205 1.198 1.194 0.6 1.3721 1.3721 1.3721 1.3721 1.3721 1.3721 1.3721 1.371 1.362 1.348 1.327 1.314 \o 1.0 1.5560 1.5560 1.5560 1.5560 1.5560 1.5560 1.554 1.549 1.523 1.486 1.439 1.409 1.5 1.7551 1.7551 1.7551 1.7551 1.7551 1.754 1.747 1.735 1.677 1.605 1.525 1.478 2.0 1.9320 1.9320 1.9320 1.9320 1.9320 1.928 1.913 1.893 1.798 1.689 1.576 1.518 3.0 2.2417 2.2417 2.2417 2.2417 2.237 2.226 2.192 2.151 1.98 1.801 1.68 1.561 5.0 2.7555 2.7555 2.7555 2.750 2.731 2.701 2.626 2.544 2.22 1.940 1.703 1.599 7.5 3.2846 3.2846 3.2846 3.266 3.227 3.174 3.050 2.920 2.442 2.060 1.756 1.628 10.0 3.7388 3.7388 3.735 3.703 3.645 3.567 3.402 3.231 2.622 2.157 1.798.6 15.0 4.5114 4.5114 4.493 4.439 4.342 4.235 5.990 3.749 2.919 2.319 2.082 1.868 1.686 20.0 5.1695 5.1695 5.141 5.o60 4.936 4.789 4.489 4.183 3.166 2.455 2.177 2.025 1.929 1.719 25.0 5.7526 5.7525 5.711 5.607 5.462 5.291 4.926 4.565 3.384 2.576 2.262 2.092 1.983 1.747

TABLE IV SPHERICAL PROBE (& = 1) Computed Values of Attracted-Species Current Density Ratio (Jcl/Jo) for Tion/Telectron = 1 (Both Species Maxwellian) X = rc/D -oc 0 0.2 0.3 0.5 1 2 3 5 7.5 10 15 20 50 100 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.1 1.1 1.1 1.1 1.1 1.0999 1.0999 1.0999 1.099 1.099 1.098 1.097 1.095 1.094 0.3 1.3 1.3 1.3 1.3 1.299 1.299 1.293 1.288 1.288 1.280 1.269 1.255 1.245 0.6 1.6 1.6 1.6 1.6 1.595 1.584 1.572 1. 2 1.552 1.518 1.481 1.433 1.402 Co 1.0 2.0 2.0 2.0 2.0 1.987 1.955 1.922 1.869 1.869 1.783 1.694 1.592 1.534 1.5 2.5 2.5 2.5 2.493 2.469 2.599 2.529 2.219 2.219 2.050 1.887 1.719 1.632 2.0 3.0 3.0 5.0 2.987 2.945 2.824 2.709 2.529 2.529 2.266 2.030 1.803 1.694 3.0 4.0 4.0 4.0 3.970 3.878 3.632 3.406 3.068 3.o68 2.609 2.235 1.910 1.762 5.0 6.0 6.0 6.0 5.917 5.687 5.126 4.640 3.957 3.957 3.119 2.516 2.037 1.833 7.5 8.5 8.5 8.5 8.324 7.871 6.847 6.007 4.887 4.094 3.620 2.779 2.148 1.891 10.0 11.0 11.0 11.0 10.704 9.990 8.460 7.258 5.710 4.658 4.050 3.002 2.241* 1.938* 15.0 16.0 16.0 16.0 15.405 14.085 11.482 9.542 7.167 5.645 4.796 3-383 2.597 2.022 20.0 21.0 21.0 21.0 20.031 18.041 14.314 11.636 8.473 6.518 5.455 4.318 3.716 2.532* 2.097* 25.0 26.0 25.765 25.462 24.607 21.895 17.018 13.603 9.676 7.318 6.053 4.719 4.018. 2.658 2.166 *Obtained by Graphical Interpolation.

7 5 \o 3 -J X~~~~~~~~~~~~~~~~~~~~~~~~~~- =5 20 x=10 X=20 — 4 ~0 -"o ~~~~~~~~- I-. —-- -~ —-'=-X =50 2 ~~~~~~~~~~~~~~~~~~~X:100 0 0 1 23 4 5 6 8 9 OC 1 12~ 13 2O Fig - 23. 1 15 6 (i / Cylindrical Poe0 71 2be oc) for uari ous ltO Plote crvees are from'values of a ra e dimens Lafamoj'? Tmpertr- rati current density Table 1I, Tion/7electron = i.

14 1312 - 10 9 8 7~~~~~~~~~~~~~~~~~~~~~~~~~~~C' 4 0 0 —— ~ ----- * — 0o -- - - 0 =20 3 2 =50 1=1 X a 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 -Fig. 24. Spherical probe (= —-). Fig. 24. Spherical probe (6==l).

verted these expressions numerically to obtain the corresponding values of a for given values of oca and X. Figure 25 illustrates O, for the cylindrical probe as a function of (- c) for various values of X and for the temperature ratio Tion/Telec = 1. Figure 26 shows a, for the cylindrical probe, as a function of X for various values of (-(oc). Table V contains these calculated values. Similarly, for the spherical probe, Fig. 27 illustrates cO as a function of (-(oc) for various values of X, and Fig. 28 shows C as a function of X for various values of (-Ooc). The temperature ratio for all cases is unity. Table VI contains these calculated values. In order for an inverse-power-law potential to be realistic, the exponent Ca should not depend on the probe potential 0oc' Figures 25. and 27 show that CO is fairly insensitive to changes in the probe potential oc for values of X less than about 5, except in the immediate neighborhood of the plasma. potential. In the range X < 5?, C is essentially constant for (Jooc) > 4. For a. plasma of 2000~K and a value of ( -oc) = 4 the probe potential is about 0.7 volt. Electron density data are generally obta.ined at higher voltages, where a does not change anymore with probe potential. From Fig. 25 it is seen that a = 2.8 if X = 5 and oc > 4. If one compares the X = 5 curve of Fig. 27 with the a = 2.8 curve obtained by interpolation from Fig. 19, one finds that the two curves are practically identical for all values of Ooc(even for| ocl< 4). For X < 5 the agreement improves of course. In the case of cylindrical probes used for ionospheric measurements 99

TABLE V CYLINDRICAL PROBE (6 = 0) Temperature ratio Tion/Telectron = 1. Values of 0a as a function of (-Doc), for various values of X = rc/\D, derived from Laframboise' s Table III of volt-ampere characteristics. - X;=5 X=10 X=20 X=50 X=100 OC 0.0 2.00 2.00 2.00 2.00 2.00 0.1 2.235 2.236 0.3 2.508 2.892 3.648 4.070. 6 2.195 2.719 3.304 4.141 4.679 1.0 2.365 2.696 3.710 4.717 5.450 1.5 2.483 3.203 4.082 5.281 6.093 2.0 2.564 3.359 4.349 5.693 6.596 3.0 2.680 3.559 4.720 6.299 7.367 5.0 2.774 3.780 5.10 7.026 8.382 7.5 2.793 3.860 5.295 7.488 9.071 10.0 2.809 3.881 5.375 7.738 9.507 15.0 2.795 35.866 5.401 7.964 10.001 20.0 2.774 3.834 5.371 8.025 10.203 25 0 2.754 3. 797 5. 323 8.028 10.337 100

TABLE VI SPHERICAL PROBE (6 = 1) Temperature ratio Tion/Telectron = 1. Values of a as a function of (-(oc), for various values of X = rc/kD, derived from Laframboise s Table IV of volt-ampere characteristicso x=5 X=10 X=20 x=50 X=100 a O a a a oc __ C6 ~a 0.0 2.00 2,00 2.00 2,00 2,00 01o 35397 4o336 5.242 7o054 7.956 0,3 35767 4.644 5.893 7.570 8.860 0.6 3.831 4.893 6.157 80053 9.503 1 0 3.909 5.053 6.451 8.555 10.092 1o5 35956 5o178 6.720 9o031 10o711 2 0 35993 5.278 6.915 9.422 11.226 350 4.018 5~397 7o210 10,044 12 177 5o0 4 006 5.459 7.481 10.773 13.398 7.5 35968 50468 7.568 11.189 14.209 10.0 35931 5.431 7.556 11.348 14.648 15o0 3.878 5~340 7.465 11.419 14o976 20,0 3.829 5.243 75331 11.356 15.023 25o0 35789 5.172 7.218 11219 14.961 10.1

14 13 - 12 10 r\) X=I 0 =10 X — 1 _0 X =50 thei ~1 3 I9 1 12 13 =4 5 16 89 B collector potentia. Potental > =. c With TTo ion electron -I.

14 13 12 10 r~~~~~~~~p,-~~~~~~~~D 25 Oc' ~~ —------— ~~~~~-(D =I — OC 6~~~~~~~~~~~~~~~~~ =3. w 0.000 -.05 60 70 80 9 100 Fig. 2 6. Cylin i2c ps of -—,Doc Temperatu rati Ti0/T I re ratio TionT VS' X for(Do ous 0.= 0e elec~~~~~~~~t~~~~ron 1.~~~~~~4O

14 13 - 12 - X =50 11 10 9 8 0t 7 ~~~~~~~~~~~~~~X:20 6 5 4 54 — X =5 - 3 2 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 oc -C)OC Fig. 27. Spherical probe. (6=1). Plot a vs. (-Ooc) with X = rc/XD parameterized. Temperature ratio Tion/Telectron =1.

14 1312 - ---- 11 - - ^r~~~~~~~~~~~~~~~~~~~~~0 10 - Om ~ OC 9 0 10 0 30 4 0 60 70 8 0 0 oc 8 7 0H~ X =AQ/\O -— ~~ =o \J11 5 4 3 2 0- " 1 0 10 20 30 40 50 60 70 80 90 100 X =&C/XD Fig. 28. Spherical probe (6=1). Plot of a vs. x for various values of (-Qoc). Temperature ratio Tion/Telectron = 1.

X is usually considerably less than unity so that a = 2, and Langmuir s volt-ampere characteristics correctly give the collected currento 106

V. CONCLUSION Experimental data of excellent quality a.re now being obtained from electrostatic probes both in the ionosphere and in laboratory plasmas. The detailed shapes of the measured volt-ampere curves obtained would give valuable information about plasma properties if the theory of these curves were better understood. This requires the use of realistic models of the potential structure in the sheath region. The need for such realistic, yet mathematically tractable, potential models provided the stimulation for undertaking this studyo This report illustrates the artificial nature of a.ny probe theory based on the use of potential functions that exhibit discontinuities of the potential gradient, or of higher radial derivatives of the potential at any specified sheath radiuso Discontinuity of the gradient presumes a surface charge suspended in space; discontinuity of the second derivative presumes a step function in space-charge density; neither of these corresponds to any physical reality. Thus a model might be considered in which the potential and its first and second derivatives are all matched a.t a "sheath edge"' surface interior to the outer bound of the sheatho. In this case the region outside the sheath edge would be a low-space-charge transition region67'8'9, between the plasma and the high-space-charge steep-gradient sheath region, One of the most interesting aspects of the material presented is the comparison between the results of the self-consistent field analysis by Laframboise2 and the results of our calculations based on the inverse107

power-law potential (68-1)o In particular our results show tha.t in the range of parameters of interest in ionospheric applications our probe chara.cteristics agree very well with those of Laframboiseo Of course the "infinite" sheath radius should not literally be taken to be infinite. It means merely that the sheath potential approaches the plasma potential asymptoticallyo At a. range of radius values in which the difference between the sheath potential and the plasma potential has become small relative to the inherent random variations in plasma. potential, one is outside the sheatho Thus the "r infinite radius" of the sheath can in fact be a very short distance~ One ought to think in terms of some mean value radius, as one thinks of a. time constant of an asymptotically decaying circuit transiento In summary, this study emphasizes the need for recognizing that, to be reasonably in accord with reality, one must indeed use a sheath model in which the sheath potential approaches the plasma potential as ympt ot i c a.lly It appears from the present work tha.t it is feasible to devise a simple potential function model that serves these purposes wello Of course other equally good or better models may be found, but the inversepower-law function proposed here seems to provide a good compromise between the need for representing reality and the need for mathematical. tractability 108

APPENDIX DERIVATIONS OF THE MOTT-SMITH AND LANGMUIR EQUATIONS FOR CURRENT TO CYLINDRICAL OR SPHERICAL PROBES FOR FINITE SHEATH MODELS This derivation is carried out by using the domain of integration shown shaded in Figo 6 of the text, described as follows in accordance with Eqso (36-1) and (36-2) of the text: E M2' if 0 < M2 < M_2 (A-l) 2mrs E > M2 +, if M 2 < < (A-2) 2 c s 2mrc where 2 2mrs2rc 2(c) Ms 2 2 (A-3) (rs - rc To obtain the current density J to a cylindrical or spherical probe for a sheath of finite radius rs we apply the above Eqso (A-l), (A-2) and (A-3) in Eqo (107) of the text, with symbolism as therein, and using M = x;; the result is h: S2 I x dE dx l2 exp(-P) ic o 27 f o 2Emr kT 2mrs x oo E oo + f F dE dx xx1)/2 exp( - x M + 2mrc ( (A-4) Integration with respect to E gives: 109

hkT x= M2 +_ 1 -2- x _J. h ^k dx x exp(c e 2) 2 Lx =.o 2mkTrs 6+1 1 + exp V ) J =M2 dx x exp - (A-5) \kTJ x = M 2mkrc2 s 2mkTrc Use of new variables of integration y = x/2mkTrs2 in the first integral, and y = x/2mkTrc2 in the second converts this to:,Jo, _ 2mk_,s21/'2 ^ s/2kTr= 22 c L b 2mkTrs2 2 / 2 2 dy y exp(-y) 26 ly =o (6+1)72 6 y = +00 ]+ _ + (2mkTr 2) exp(- ) f/ dy y exp(-) (A6) i R y = Ms2/2mkTr2 2 Now substitute for h, from Eq. (1.09) in the text, and for Ms2 from Eqo (A-3), and express the integrals in terms of incomplete gamma functions as in Eqso (115-1) and (115-2), to get r ~~/ 2 o s +1 c c \ Jc = \ 1 c Y kT r~T r2 + exp A^~r(+l c r 2 +exp) (2 9 k (A-7) kT, 2- rs 2 2 For cylindrical geometry = 0. The incomplete gamma functions then become / X2-< r 2 l~c r c2 c c y. ______ = f = - -- t-./2 r5t 2 = kT rs t dt t -1/2 exp(-t) r ^.t =o s r er - rc2 -=-~ erkT.2 - 2 ( (A-8) 110

and / c -r 2 \ -^ r 2 r TC 2 =-2 erfIc I (A-9) Thus the expression for the current collection to the inwa.rdly-a.ccelerating infinite-cylinder probe may be written a.s r r2 (c r2 2 rs' rIC IC Jo =- J erf - - exp ( - erfc C r c kT r 2 kT r2 -r c2 \ /. s C (A-10) For the spherical probe, we obtain by a. similar procedure after using 6 = 1 in Eqo (A-7): 2 2 2 2 s ( ICr IC c ic JoPTc exp (A-li) Jc = Jo 2 [ ( 2 rc (A-r 11 rs _ rv These Eqs. (A-10) and (A-ll) for the current were originally derived by Mott-Smith and Langmuir using the velocity space domain~ In the orbital-motion-limited mode of collection, r /rC oo and Eqo (A-7) becomes Joml = (l 2 L'+ a' exp (5kT) " (A-12) onl1 J, C+ 6+1 k( k For the infinite cylinder, 6 = O, we get Joml = Jo + exp erfc(c ) (A-13) For the sphere, 6 = 1 we get J 1= J (1- ) (A-14) oml. kT/ 111

Equations (A-13) and (A-14) are the well known orbital motion limited voltampere equations of Mott-Smith and Langmuir. 112

REFERENC ES 1. Mott-Smith, H. M. and Io Langmuir, "The Theory of Collectors in Gaseous Discharges." Phys. Rev. 28, 727 (1926). (In Eq. (28a) the fa.ctor exp. 7 should appear in the last term.) 2. Laframboise, J. G., "Theory of Spherical and Cylindrical Langmuir Probes in a. Collisionless, Maxwellian Plasma at Rest." UTIAS Report No. 100, University of Toronto, 1966. 3. Langmuir, I.,. and K. T. Compton, "Electrical Discharges in Gases, Part II, Fundamental Phenomena in Ga.seous Discharges," Rev. Modern Physics, 3, 191 (April 1931). 4. Bernstein, I. B., and I. N. Rabinowitz, "Theory of Electrostatic Probes in a. Low-Density Plasma.." Physics of Fluids, 2, 112 (1959). 5. Gurevitch, A. V., "The Distribution of Particles in a. Centrally Symmetric Field."' Geomagnetism and Aeronomy, 2, 151 (1963). 6. Hok, Gunna.r, N. W. Spencer, A. Reifman, and W. Go Dow,'"Dynamic Probe Measurements in the Ionosphere," Scientific Report No. FS-3, The University of Michigan Research Institute, Spa.ce Physics Research Laboratory, Electrical Engineering Department, November, 1958. Issued under Contract No. AF 19(604)-18435, being a reprint of the original issue in August 1951 under Contract No. AF 19(122)-55, both contracts with the Geophysics Division. 7. Tonks, Lewi and I Langmuir, "A General Theory of the Plasma of an Arc " Phys. ReVo, 34, 876 (September 1929)o 80 Cobine, J, D., pp. 321, 236, Gaseous Conductors:' McGraw-Hill Book Co., New York, 1941, reprinted in paperback circa 1955. 9. vo Engel, A., and M. Steenbeck, Elektrische Gasentla.dungen, ihre Physilc uo Technik, Vol. 2, pp- 83-92, Julius Springer, 1934, 10. Kaplan, Wilfred, Advanced Calculus, pp. 94, 200, 289, AddisonWesley Publishing Co., Reading, Ma.ss., 1952. 113