THE UNIVERSITY OF MICHIGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Scientific Report No. GS-3 THEORY OF CURRENT COLLECTION OF MOVING CYLINDRICAL PROBES Prepared on behalf of the project by: Madhoo Kanal ORA Projects 03484 and 04304 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION CONTRACT NO. NASr-15 WASHINGTON, D.C. and GEOPHYSICS.RESEARCH DIRECTORATE AIR FORCE CAMBRIDGE RESEARCH LABORATORIES OFFICE: OF AEROSPACE RESEARCH UNITED STATES AIR FORCE CONTRACT NO. AF 19(604)-6124 BEDFORD, MASSACHUSETTS administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR November, 1962

TABLE OF CONTENTS Page LIST OF FIGURES v LIST OF SYMBOLS vii ABSTRACT ix 1. INTRODUCTION 1 2. THEORY OF CURRENT COLLECTION OF MOVING CYLINDRICAL PROBES 3 2.1 General Considerations 3 2.1.1 Ion Current 4 2.1.2 Electron Current 5 2.1.3 Sign Convention 5 2.2 Trajectory of a Charged Particle in the Sheath 6 2,3 Superimposed Maxwellian Distribution 7 2.4 General-Ion-Current Function for an Accelerating Potential (-V) 9 2.4.1 Orbital-Motion-Limited Ion Current Function 13 2.4,2 Sheath-Area-Limited Ion Current Function 14 2.5 General-Electron-Current Function for Retarding Potential 15 2.5.1 Random-Electron-Current Function 16 35 DISCUSSION 19 3.1 Orbital-Motion-Limited Current Characteristics 19 3.2 Sheath-Area-Limited Current Characteristics 28 APPENDIX A. POLAR TRANSFORMATION OF THE GENERAL-ION-CURRENT FUNCTION FOR ACCELERATING POTENTIAL (-V) 31 APPENDIX B. STEPS INVOLVED IN ARRIVING AT THE SOLUTION OF THE GENERAL-ION-CURRENT FUNCTION 35 APPENDIX C. SOLUTION OF THE ORBITAL-MOTION-LIMITED ION CURRENT INTEGRAL 39 ACKNOWLEDGMENTS 45 REFERENCES 47 iii

LIST OF FIGURES Figure Page 1. Trajectory of a charged particle in the sheath. 6 2. Velocity space coordinates at the sheath surface. 8 3. Normalized orbital-motion-limited ion current Ino vs. V, X = 1, G = 0, 45~, 90~. 21 4. Normalized orbital-motion-limited ion current Ino vs. V, x 2, = 0O 450, 90. 22 5. Normalized orbital-motion-limited ion current Ino vs., x 5, = O450, 4 900~ 23 6. A predicted volt-ampere characteristic of a stationary thin cylindrical Langmuir probe, under typical F1 region conditions, showing primarily the ion saturation region of the current characteristic. 24 7. A predicted volt-ampere characteristic of a cylindrical probe illustrating the effect of orientation upon the ion current characteristic at a fixed velocity ratio, X = 1. 25 8. A predicted volt-ampere characteristic of a cylindrical probe illustrating the effect of orientation upon the ion current characteristic at a fixed velocity ratio, X = 2. 26 9. A predicted volt-ampere characteristic of a cylindrical probe showing primarily the electron current region of the curve from which the electron temperature may be derived. 27 10. Random current function drawn versus the velocity ratio illustrating the orientation effect. 29 11. Hyperbola generated in velocity-space coordinate system by p, as a function of ux. 31 v

LIST OF SYMBOLS W drift velocity of the probe Cm most probable velocity of a particle (= 42kT/m ) X = W/Cm 9 angle between the drift velocity W and axis of the cylinder K = \ sin G ux.,utIut components of a particle velocity along x', y', z' axes, respectively, fixed in space UxUy,Uz relative components of a particle velocity along x, y, z axes, respectively, when the probe is moving T temperature (~K) N number of particles per cubic meters e unit charge, 1.602 x 10-19 coulombs m mass of a particle in kgm k Boltzmann's constant 1.3803 x 10-23 joule/~K V potential of the collector with respect to the plasma normalized potential = eV/kT I current to the collector In normalized current = I/kiT/2mj NeAc Ac area of the collector a radius of the sheath r radius of the collector 7o symbol for /r2/(a2-r2) vii

LIST OF SYMBOLS (Concluded) L length of the collector 6V voltage applied between the cylindrical collector and the reference p closest point of approach of particle to collector viii

ABSTRACT The theory of current collection of a moving cylindrical probe is investigated. Volt-ampere relations are derived for two distinct cases: (i) The general-ion current for accelerating collector potential and its special cases, including general-ion current to the stationary probe, orbital-motionlimited current to the moving and the stationary probes, and sheath-arealimited current to the moving and the stationary probes; and (ii) The generalelectron current for retarding collector potential and its special cases, including general-electron current to the stationary probe and random-electron current to the moving and the stationary probe. Orientation of the cylinder with respect to the drift velocity vector is taken into account. Volt-ampere characteristics are included for illustrating the functional behavior of the current relations. ix

1. INTRODUJCTION As the study of the ionosphere has progressed through the years, refinements made in vertical sounding equipment and analysis techniques have exposed the more complex nature of the ionosphere and thus demonstrated the necessity for more direct measurements. Such measurements became possible after World War II with the advent of sounding rockets. The University of Michigan investigators suggested the use of Langmuir probes for such measurements, and in 1946 and 1947 three successful V-2 flights carried such probes in their payload. However, the design and location of the probes were dictated by the other instruments used, and consequently there was much uncertainty in the data. This uncertainty was attributed to: (a) failure of the probes to approximate any ideal geometry; (b) overlapping of the sheath of the electrode with that of the rocket; (c) perturbations in the density distribution of the particles caused by the high velocity of the rockets; and (d) contamination of the region around the rocket by rocket gases. In view of the uncertain results obtained in these earlier attempts to use rocket-mounted probes, complete ejection of'the probe from the rocket was considered necessary to reduce the ambiguities substantially. Two different probe configurations answering these requirements have been developed: a 1 2 Dumbbell-shaped bipolar probe,l 2 which in more recent flights was combined with cylindrical Langmuir probes; and a combination spherical ion trap and cylindrical Langmuir probe.3 In recent years, with the advent of multi1

experimental satellites, single cylindrical Langmuir probe experiments are also gaining popularity. In all these experiments, as is normal for rocketborne probes, the rocket velocity exceeds the characteristic velocity of the ions for much of the flight. Thus the data obtained indicate strong effects of the probe velocity on the ion current collected by the device, thereby drawing attention to the need for a theoretical development which would permit the volt-ampere relation to be predicted as a function of the probe velocity, and would thus aid in the reduction of the data from the flights. Mott-Smith and Langmuir4 published their classic paper on probe theory in 1926, in which they derived the volt-ampere characteristics for spherical, cylindrical, and planar probe geometries. In their treatment they assumed a stationary plasma having a Maxwellian distribution, and did not consider the effect of drift velocities except for the case of electron current collection by a thin cylindrical probe whose axis was at a right angle to the drift velocity. It is, therefore, necessary to investigate the general theory of current collection, particularly in regard to high probe velocities and collectors of various sizes. In this report the aim is fulfilled for cylindrical collectors only. The problem of moving spherical collectors was treated by the author in an earlier report.5 2

2. THEORY OF CURRENT COLLECTION OF MOVING CYLINDRICAL PROBES 2.1 GENERAL CONSIDERATIONS When a cylindrical electrode is immersed in a plasma consisting of positive ions and electrons having Maxwellian velocity distribution, the resulting collisions of the charged particles with the probe cause it to assume an equilibrium potential with respect to the plasma such that the net current to the collector is zero. If the medium is in thermal equilibrium, which means that the mean energies of the ions and electrons are equal, then the magnitude of the equilibrium potential is mainly determined by the square root of the ion-to-electron mass ratio and the resulting polarity of the probe is negative. The collector potential causes a region of positive charge to build up about the probe in which electrons are repelled and positive ions are attracted. Such a region is commonly called a positive-ion-sheath. The boundary of the sheath is defined as that distance beyond which the charged particles experience negligible force due to the probe potential. Since the primary purpose of this study is to evaluate the effects of the sheath upon the current collection when the probe is moving, some assumption in regard to the sheath configuration is necessary. Although the cylindrical shape of the sheath will not be maintained at high probe velocities, no sufficiently precise model is available which will justify empirically or theoretically any other shape. Therefore, as a first-order approximation a cylindrical sheath is assumed. 5

2.1.1 Ion Current In evaluating the effects of the sheath on the ion current collection, three distinct cases are encountered throughout the range of possible values of T/Nr2, where T is the mean temperature, N is the particle number density, and r is the radius of the collector. (a) When T/Nr2 < 5 x 10-6 the ion current collected by the electrode is termed "sheath-area-limited." In other words, all the ions that enter the sheath from the ambient plasma reach the collectors Mathematically this condition of collection is achieved by letting a/r + 1, where a is the sheath radius. (b) When T/Nr2 > 10-3, the ion current collected is termed "orbitalmotion-limited." Ion collection under this condition is dependent only upon the net voltage across the sheath and is practically independent of the sheath radius. Mathematically this may be expressed by letting a/r - oo. (c) When 5 x 10-6 < T/Nr2 < 10-3, the ion current collected is termed "intermediate." In this case the probability of collecting an ion is dictated by both the sheath radius and the net voltage across the sheath. As seen from the foregoing discussion the sheath-area-limited and the orbital-motion-limited conditions of ion collection constitute the asymptotic extremes of the general case. It should also be noted that the ranges of T/Nr2, where each class of ion collection is implied, are subject to some; change when the probe is moving with high drift velocity. However, for a

stationary probe these figures are generally accurate.1 2.1.2 Electron Current In the case of electron current collection which is independent of the sheath radius, essentially two distinct cases are encountered throughout the range of possible negative collector potential. (a) When the collector potential, V, is near zero with respect to the plasma, the electron current collected under this state is termed "random." (b) When a finite negative collector potential exists with respect to the plasma, the probability of an electron's reaching the collector is dictated by V, and the current collected by the electrode is termed "general." Since in most ionosphere experiments the probe velocities never even come close to the mean velocity of the electrons, the probe motion has little effect on the electron current collection and can be neglected. However, the probe motion must be taken into account in the case of negative ion collection because of the heavier mass and hence the low mean velocity. For this reason the theory developed here takes account of probe velocity for both the ion and electron collection cases, but in the discussion of volt-ampere characteristics (Section 3) the velocity effect on electron current is neglected. 2.1.3 Sign Convention In deriving the current functions for ions and electrons when the collector potential is either accelerating or retarding for any of the particles, 5

it is expedient to refer to the actual polarities of the collector and of the particles. Therefore in considering the trajectories of the charged particles in the sheath the polarities of both the particles and the collector are retained and a proper selection of the signs is made in setting up the current equations. 2.2 TRAJECTORY OF A CHARGED PARTICLE IN THE SHEATH Let "a" be the radius of the sheath concentric with the cylindrical collector of radius "r" and length "L" (L > r). Consider the moving probe such that its axis is perpendicular to the plane of the paper, as shown in Fig. 1. Let p and ux be the relative tangential and normal velocity components respectively of a charged particle (+e), at the sheath edge in a plane normal to the axis of the probe. Let Pc and uc be the corresponding quantities near the collector surface. Then from the laws of conservation of energy and angular momentum we have: p2 2+ U2 = p2 + U2 + 2(~e)(~V) p2+u~ pg~u~+ m (1) ap = Pc (2) og oe w-e ^ \.p Fig. 1. Trajectory of a charged particle in the sheath. 6

where e is the unit charge, V is the collector potential with respect to the plasma, and p is the closest point of approach of the particle to the collector. In Eq. (1) when we consider a positive ion (+e) encountering a negative collector potential (-V), the product (+e)(-V)-which represents gain in energy in the sheath —becomes negative, and in accordance with the physical situation that product must be subtracted from the right-,hand side of Eq. (1). In this way we can consider other cases of particle-field interaction and select the proper signs when required. Further consideration of Eqs. (1) and (2) suggests that only those particles will reach the collector for which Ux > 0, u2 > O. If we substitute Eq. (2) in Eq. (1) for pc, put uc = 0, P = Pi, P = r, and rearrange the terms we obtain P2 = a -r2 2 2(~e)(+V)j (3) Equation (3) thus describes the condition necessary for the collection of a charged particle for a given collector potential and dimension. Next we will consider the Maxwellian velocity distribution of the particles with respect to a coordinate system which is fixed in the plasma and then view the same distribution function from a coordinate system which is fixed on the moving probe. The latter we will term "the superimposed Maxwellian." 2.3 SUPERIMPOSED MAXWELLIAN DISTRIBUTION Consider an oblique view of the cylindrical sheath as shown in Fig. 2. Let x', y', z' be a set of three orthogonal axes fixed in space representing in direction and magnitude the three components uj, uy, uz of the particle 7

0 0 W I W sin 0 cos o X- x Z I IX X Fig. 2. Velocity space coordinates at the sheath surface. velocity, respectively. Let x, y, z be another set of three orthogonal axes, parallel to the first one, fixed on the moving probe, and representing in direction and magnitude the three relative components ux, uy, uz of the particle velocity, respectively. Choose y-axis along the axis of the cylinder as shown. If W is the probe velocity and 9 the angle between W vector and yaxis, then the components of W superimposed on ul, uy, ul are W sin Q cos A, W cos 9, W sin G sin p, respectively. In other words, U = u' + W sin G cos 3 Uy = u + W cos (4) uz = u' + W sin Q sin where P is the azimuth angle of W with respect to x-axis. Let N be the number density of one kind of particle in he number density of one kind of particle in the plasma, T the mean temperature, and m its mass; then the Maxwellian velocity distribution 8

of the particles with respect to the stationary coordinate system is N 2..1 2,271 f(uxuuz)duxduyduz = N32 exp - ( x2 + u2 + u duxdudu (5) where Cm is the most probable velocity of the particles defined as Cm = 2kT/m-, k being the Boltzmann's constant. With respect to the moving system the distribution function (5) is modified in a way determined by the linear transformations of the velocity components given in Eqs. (4). Thus for the moving system the new distribution function is F(uxUyuzP)duxduyduz = N -exp -(u W sin 9 cos P) + (uy - W cos 9 ((uz - W sin 9 sin ) duxuduyduz (6) which represents the superimposed Maxwellian velocity distribution. 2.4 GENERAL-ION-CURRENT FUNCTION FOR AN ACCELERATING POTENTIAL (-V) In Fig. 2 consider an infinitesimal strip of area (L a dp) on the sheath surface. The number of ions with velocity ranges between ux and ux+dux, uy and uy+duy, and uz and uz+duz that are expected to cross the infinitesimal area per unit time is given by LauxF(ux, Uy, Uz, P) duxduyduzdP (7) On multiplying Eq, (7) with the ionic charge and integrating between the proper limits we obtain the following equation for the ion current collected by the moving probe. Ii = Lae ux F(ux,uy,Uz,I) duxduyduzdP (8) =O ux=O uy=-oo Uz=-pi 9

In Eq. (8) the limits of Uy range from -co to oo because we have assumed that L > ro The limits of uz are from -pl to p1 because if in a plane normal to the cylinder ux is the radial component at the sheath surface, then uz represents the tangential component for which the trajectory (3) determines the values of ~plo It is trivial to integrate Eq. (8) for p and uy. Integration with respect to uz is carried out in Appendix A. The final result, in terms of the sum of two single integrals, is given in Eq. (9). i= k- NeAc e s(s2+V e-s2 Io(2Ks)ds +02''pYoJ 2 t+ 7-~ s2 eS Io(2Ks)d (9) 2 where K= sin G = /cW/C = W//S2kT/mi = _1r2/(a2r2) = eV/kT Ac = 2ctrL (area of the collector) and s is the new variable. For obtaining numerical values it is expedient to perform numerical integration of (9). However, it is desirable to present the analytical solution for the sake of completeness. Appendix B contains the elementary steps involved in arriving at the solution of Eq. (9) given in Eq. (10). 10

^ Y'2m(tkT eA' Lo [n + 2o + a X n $t (n +n 2 WO (10) Mir O C (n!)2 2' n=0 where (vx) and (v,x) are the incomplete gamma functions bearing the following relationship 00 rF ( x) =j et t-' dt = (v) - (v,x) x and Jn(x) is the Bessel function of order n. In order to visualize the effects of the sheath upon the current collection, let us define a normalized current, In, which is simply the collected current, I, divided by the current which would have resulted without the sheath. Mathematically In = -- - (11)!kT NeA, 2mir Thus, in the normalized form, the general-ion-current equation (10) becomes n=O 00 1 2n +^-^ 2.D3 ~ ^T^^-(12) + n!7 o (n2) —---- (n + 72V, (12) n=0 J When the probe velocity is small compared with the most probable velocity of the ions-in other words, when A is small or when the probe is moving with its axis pointing in the direction of the velocity vector, G = O-then K = 0. 11

Under these conditions it can be shown that Eq. (12) reduces to 0 -V Inil+O = ~ erf(7o) + e erfc [V y) (15) which, therefore, represents the general-ion-current function when K = 0. In Eq. (13) error functions are defined as usual: X erf(x) = - e dt 00 2 / 42 erfc(x) = - e-t dt Equation (13), which was previously derived by Langmuir4 in 1926, has been used in various studies of plasmas involving space charge tubes as well as low speed probes in the ionosphere measurements of temperature and density. With modern high speed rockets, which during most of their trajectories exceed the most probable velocity of the ions, it has become necessary to take into account the probe motion. In such a case Eq. (12) must be used. Because the cylindrical Langmuir probe is usually operated in the orbitalmotion-limited mode, this particular case is examined in detail. As a special case of the general-ion-current function given in Eq. (12) we can deduce the orbital-motion-limited ion current function by letting 7 0, since Y7 = -r(a2-r2). However, it is much simpler to let yo 0 in Eq. (9), which is the general-ion-current function still in the integral form, than to take the limit of Eq. (10) as such. Similarily, in deducing the sheatharea-limited ion current function it is easier to let yo + oo in Eq. (9) than in Eq. (12), 12

2.4.1 Orbital-Motion-Limited Ion Current Function By letting a/r + o or, correspondingly, by letting yo + 0 in Eqo (9), we obtain the ion current function which is independent of the sheath radius. It can be easily proven that when yo + 0, Eqo (9) reduces to ___ po00 In Ii 0 NeAc 4 2 s(s2+7)1/2e s2Io(2s)ds In~=io+o = o 2 m — ir T it Vomii e 0 (14) Solution of the above integral is given in Appendix C. The final result in the normalized form is In a more sophisticated form Eq. (15a) may be written as 00 Io = e 2n(2 K XF) n (n + 2 )Jn(2 T 2) (15b) n=O where Jn(x) and. Jn+3/2 (x) represent the Bessel functions of index n and n + 5/2, respectively, and r(n + 2, V) is the incomplete gamnma function. For a stationary probe or when 9. = 0, the current function (14) assumes a simpler formn:'nolK^ = - a + e2 erfc (JT) (16) For values of V h> 5, Eq. (16) can be approximated by 2I-o -.- e K oI | - 2 f(17) 13

Equations (16) and (17) are well known and have been used extensively in laboratory plasma studies.l14 As mentioned above, however, in the study of the ionosphere by means of probes carried by sounding rockets or satellites, the probe motion makes it imperative to use Eqo (15a) or (15b). 2.4.2 Sheath-Area-Limited Ion Current Function The sheath-area-limited condition is attained when all the ions that enter the sheath reach the collector. Mathematically, the functional representation of the current can be obtained by letting a/r - 1 or yo > oo in Eq. (9). The limit of Eq. (9) when y7 + o yields 00 Is = k NeAc (a/r) e s2 e2 I(2s)ds (18) b V 2m — 7 C " After solving the above integral, the normalized sheath-area-limited current is given by Tns = -e(a +2 [(1+K2)Io(2/2) + K211(2/2)] (19) where Io(x) and Il(x) are the modified Bessel functions of the order of zero and one, respectivelyo For stationary probe or when 9 - 0, Eq. (19) reduces to a insI[ Kl = r (20) Of course, the current is obtained by multiplying Eq. (20) with the normalization constant defined in Eqo (11). 14

2,5 GENERAL-ELECTRON-CURRENT FUNCTION FOR RETARDING POTENTIAL This section contains the derivation of the general-electron-current function for the case in which the electrons in the sheath encounter a retarding potential. In Eqo (8) the lower limit of the radial velocity component ux of the ion was zero for the accelerating collector potential. When an electron enters the sheath and experiences a retarding potential, then the least radial velocity component, u-, necessary for its collection is \J2eV/m. Hence, in integrating the right-hand side of Eq. (8) the lower limit of ux = ul =;2em must be used. Also, in the trajectory relation (3), care must be exercised in selecting the proper sign of eV; in this case the sign is clearly positive since both e and V are negative. The integral expression for the general-electron current, Ie, is, therefore, given by 2if oo 00 p1 Lae / ux F(ux,uy,uz,p)duxduyduz d (21) o Ui'-00 -P I where the symbols have their usual meaning. The solution of Eq. (21) is discussed in Appendix C. The final result in the normalized form is given below, 00 Ine = exp[- (7T+2) ] (2n+l) (/ Jn In(2 2) n=O (n!)222k where In(x) is the modified Bessel function of the first kind and nth order. The series converges very rapidly for small values of K; for instance, when K = 035, the first three terms of the series gives the result correct to 15

within four parts in ten thousand. Equation (22) was also derived by MottSmith and Langmuir,4 except that their result was for the case of orientation angle G = 90~ Since Eq. (22) is derived for the case in which the collector potential, V, is negative, the probe velocity, which never even begins to approach the most probable velocity of the electrons, can always be ignored. Consequently, for K = 0, Eq. (22) reduces to Ine = e (23) When the ion current to the collector which is at positive potential with respect to the plasma is considered, for x > 1, Eq. (22) must be used. Equation (23) for the retarded ion current holds only when \ = 0 or ~ = 0. 2.5.1 Random-Electron-Current Function When T is negligibly small, in other words, when the collection of electrons is random, it can be shown that Eqs. (21) and (22) reduce to Inr = e2 [(1+)2)Io(12/2) + K2I1( 2/2)] (24) where Inr represents the normalized random electron current. For XK % 0, which is always true in the case of electrons, Eq. (24) becomes Inrl = 1 (25) Comparison of Eq. (24) with the sheath-area-limited ion current function (19) shows that the two cases differ functionally from each other only by the 16

factor (a/r). In Section 3 we will use the current functions in predicting the voltampere characteristics of the cylindrical probe. 17

3. DISCUSSION Since in practice Langmuir probe characteristics are usually interpreted in the voltage range which is negative with respect to the plasma, the current characteristics will be presented for this range only. In other words we need consider only the accelerated ion and the retarded electron currents. A close examination of Eq. (12) for the accelerated general-ion current shows that the equation contains two parameters, yo and 7, which are not independent of each other. Since by definition yo = r2/(a2-r2) and V = eV/kT, an independent relation between (a/r) and V to solve for one of these two parameters is needed in conjunction with Eq. (12) to obtain the actual ion current characteristics. 3.1 ORBITAL-MOTION-LIMITED CURRENT CHARACTERISTICS Since the orbital-motion-limited current, Eq. (15a) or Eq. (15b), is independent of (a/r), we do not need an independent relation, as we need for the general case, to solve for a/r as a function of V. In order to obtain the ion current characteristics for a given collector radius and ion density, we first multiply Eq. (15b) with the normalization constant given in Eq. (11). Thus I = 2i NeAc Ino (26) where Io represents the orbital-motion-limited current and Ino is given by Eq. (15a) or (15b). Since the right-hand side of Eq. (26) is a function of 19

the probe velocity X and of the orientation angle 9, we fix X and parameterize G. Figure 3 illustrates the normalized ion current drawn versus T-for \ = 1 and g = 0, 45~, 90~. The curve for 9 = 0~ also describes the stationary probe characteristic, i.e., X = 0. This, of course, follows from the fact that (K = \ sin G) = 0 when either X = 0 or G = 0. Similarly in Figs. 4 and 5, which are drawn for X = 2 and X = 3 respectively, both 9 = 0 curves correspond to the stationary probe case. As one would expect, and as is demonstrated in Figs. 4 and 5, the orientation effect is relatively pronounced at higher probe velocities. Figure 6 illustrates the stationary probe characteristic, X = 0, in which the net current is drawn versus the applied difference of potential 6V between the collector and the reference, for T = 1600~K and N = 105 particles/ cc. Of course for A = 0, no orientation effect is involved. Figure 7 is drawn for X = 1 and the range of 9 as shown. Here, as one would expect, the orientation effect is visible mostly in the positive ion current region. Figure 8 demonstrates the similar behavior at A = 2, where the orientation effect is even more pronounced. Thus, in reducing the ion density from experimental data, it is evident from Figs. 7 and 8 that the effects of orientation must be considered. For instance, we must know the measured current, probe velocity, 9, V, and assumed values of T and ion mass to determine the ion density from Eq. (26). Figure 9 illustrates the electron current characteristic for X = 2 and for all 9. Since the net ion current component for all 9 is small compared with the electron current component, the orientation angle has a negligible 20

A =1 K 0 It 0 5 10 15 20 V Fig. 3. Normalized orbital-motion-limited ion current Ino vs. V, = 1, G = 0, 45~, 90~

A 62 7.0 \ Il 6.0 4/ 0 = 90~ 0 = 45~ 4.0 - 0Z 0 3.0 2.0 1. 0 0 5 10 1I 20 Fig. 4. Normalized orbital-motion-limited ion current Ino vs. V, X = 2, = = 0, 45~, 90.

7 I 0 5 10 15 20 x = 5, = 0, h45, 90~.

-4W 0a X. = "m E -3Cm o All 8 lo N = 10 Ions/cc x T 1600~K mp=16,(0+ Ions) Z -2 nQ I -1.0 -0.4 0 0.4 1.0 APPLIED VOLTAGE 8V (Volts) 2 3 -4 8 W 5 All _ Ion Saturation Region i6 Fig. 6. A predicted volt-ampere characteristic of a stationary thin cylindrical Langmuir probe, under typical F1 region conditions, showing primarily the ion saturation region of the current characteristic. 24

-4 W E Cm e 10 9 0=,450,90~ x N =10 Particles/cc z-2 T=1600~K 2l mp= 16 1 -1 w- I I I -1.0 -0.4 0 0.4 1.0 APPLIED 8V(Volts) I, 2 3 4 8 6= 45 8= 45 8= 90~'6 Fig. 7. A predicted volt-ampere characteristic of a cylindrical probe illustrating the effect of orientation upon the ion current characteristic at a fixed velocity ratio, A = 1. 25

E - 00~~~0 X 2 ^ 3 0= Qo,45o, 90 D N= 105 Porticles/cc - _ T= 1600 OK z w 1.0 -0.4 0.4 1.0 APPLIED 8V (volts) 8=45~~"4 0=90~ -6 Fig. 8. A predicted volt-ampere characteristic of a cylindrical probe illustrating the effect of orientation upon the ion current characteristic at a fixed velocity ratio, x = 2. 26

-2.0 X= 2 All 8 N =105 Particles/cc T= 1600 ~K (0 I -1.5 10 x Z I WI - I- Io - - 1.0 I / 0 I~E -— 0.5 Inet _07 -0.5 -0. -0.3 -0.1 I i i I''- 1 1 1 1 * 1 ~ll O 0.1 0.3 0.5 0.7 All80 @APPLIED 8V (Volts) Fig. 9. A predicted volt-ampere characteristic of a cylindrical probe showing primarily the electron current region of the curve from which the electron temperature may be derived. 27

effect on determination of the electron temperature. Thus with the help of Eq. (23) we can determine the electron temperature by plotting the natural logarithm of the electron current versus the applied voltage. Rearrangement of Eq. (23) for the electron temperature yields Eq. (27): Te = -e/k d- (loge Iel ) (27) where subscript e refers to the electron parameters. 3.2 SHEATH-AREA-LIMITED CURRENT CHARACTERISTICS For the extreme case in which all the ions that enter the sheath reach the collector, the current function given by Eq. (19) is dependent on (a/r) and K only. For this reason we do not need an independent relation to solve for the current, as we do in the case of the general-ion-current function (12). In Fig. 10, which illustrates the functional behavior of Eq. (19), f(~) is drawn versus X with ~ parameterized. Since f(K) alone represents the random-current function, Fig. 10 is also representative of the random-electroncurrent function, which is inherently independent of "a." For electrons, however, X is nearly zero even for the satellite velocities; hence the curve for G = 0 in Fig. 10 represents the relevant electron-current characteristic. Functionally the behavior of the random-electron-current function is exactly the same as that of the sheath-area-limited ion current function. 28

6 i i i I i I f(K)VS. V 2 0o0V S *^ V> eT[^w^( 2, j y22 d A i f(K) -e K2I )+KzIK( -5 where K' sin e l f( K) 30a = 0 0 1 2 3 4 5 6 7 8 9 10 Fig. 10. Random current function drawn versus the velocity ratio illustrating the orientation effect.

APPENDIX A POLAR TRANSFORMATION OF THE GENERAL-ION-CURRENT FUNCTION FOR ACCELERATING POTENTIAL (-V) It can be shown by plotting ux, P1 as rectangular coordinates of a point that Eq. (3), when V is negative and e is positive, reads: r2 2eV~ 1 a2_r2 This equation is a hyperbola whose semi-axes are 2eV 1/2/ 2 2eV V/2 (2_) / (ia - 1), i(m-) on the pi and ux axes respectively. Since integral (8) is an even function of uz, it is sufficient to show only one branch of the hyperbola. This is illustrated in Fig. 11 where o = Jr2/(a2-r2). After integrating Eq. (8) for P and Uy, we have 4LNae -2 2/ 7 s r l i~s2 1 12UU2 I = X e ux exp ~ (Ux + u)| Ioe2 (ux+u) duxduz cm'-b' LCm J_ LcC -J (A-l) LR1 ux Fig. 11. Hyperbola generated in velocity-space coordinate system by p1 as a function of ux. 51

and the domain of integration is the one enclosed by the hyperbola shown shaded in Fig. 11. Let U/Cm = s cos uz/Cm = P1/Cm = s sin' (A-2) duxduz = C s ds dr Now divide both sides of Eq. (3) with C2, obtaining p = 2 (72 + 7) (A-3) where V = 2eV/mCg = eV/kT, the normalized voltage (not to be confused with the normalized current). Substitute Eq. (A-2) in Eq. (A-3) to obtain 1 = sin-1 /( s2+V)/s2(l+yg) (A-4) This means that if we divide the hyperbolic domain shown in Fig. 11 into two regions, R1 and R2, then Eq. (A-4) yields the' variance in region R2 with respect to the hyperbola. Thus the limits of integration of Eq. (A-l) in region R2 are yO IF< s < oo (A-5) and in region R1 the limits are o < << =t/2 (A-6) o < s < 70 IT 32

Insert the polar transformation of the coordinates given in Eqs. (A-2) and the limits of integration given by Eqs. (A-5) and (A-6) in Eq. (A-l) to obtain I = 4LNae Cm e - s2cos $ e- Io(2Ks)ds dr + s2cos I e-S Io(2s) ds dj R2 R1 which, on integrating with respect to 4 and rearranging the terms, becomes 00 I N~ e- s(s e Io(2s)ds + _ + f / 2 e-S2 Io(2s)ds2 (A-7) This corresponds to Eq. (9) given in Section 2.4. 33

APPENDIX B STEPS INVOLVED IN ARRIVING AT THE SOLUTION OF THE GENERAL-ION-CURRENT FUNCTION There are two integrals given in Eq. (9) which we desire to solve. In the first integral I1 = s(s2+V)1/2 e-S2 Io(2Ks)ds (B-l) YoF Put s2 + = t dt sds = - 2 to obtain I = 1 e t /2 et Io(2N/t- )dt (B-2) V(l+y2) and expand the Bessel function. Thus 00 v1 V K~ 2k / 00 = 1 eV J J t1/2(tVyk e-t dt (B-5) k=O v(l+y2) Binomial expansion of the term (t-Vk in the integrand of(B-3) yields 0o k 21 7T~ CC -t (K t - I1= L e t e dt (B-4) k=O n=O k'(k-n)'n Now according to the definitions of the incomplete gamma functions7 (vx) = e^ t dt (B-5) 00 (vx) ='e't t- dt (B-6) 55

the right-hand side of Eq. (B-4) may be written as oo k e- Y Z K2k( ) k-n I1 = e......... Y 3 ( 2) 2 n=O k (k-n):n [n + V(l+ ] (B-7) k=O n=O In Eq. (B-7) use the following lemma:8 oo k oo C / A(k,n) = A(k+n,n) (B-8) k=O n=O k,n=O Thus we obtain 00 -e K l2k+2n( -7)k 1 - L2 e (k+n)k!n [n + 2 l+y) ] (B-9) k,n=O Since the Bessel function of order n is defined as 00 J, ) C (.)(_,kz 2k+n Jn(z) = (-1) (z/2) k=O k!(n+k); Then Eq. (B-9) after summing with respect to k may be written as 00 e = 2e L /n: [n +, V(l+) ]Jn(2 (B-10) n=O Hence Eq. (B-10) is the solution of integral (B-l), the first integral of Eq. (9). Similarly, the second integral of Eq. (9) may be solved as follows: o 2 I2 = s2 e Io(2Ks)ds 0 00 2n 0 2n+ -s2 = s2n+2 s e ds n=O o 1 K2n YOT n+1/2 n=0 Using the definition (B-5) of the incomplete gamma function we obtain

12 = l (n +2' 72 (B-ll) n=O Thus Eq. (B-ll) provides the solution for the second integral of Eq. (9). 37

APPENDIX C SOLUTION OF THE ORBITAL-MOTION-LIMITED ION CURRENT INTEGRAL The required integral to be solved is Eq. (14), which is given by 00 1/2 -s2 I = s(s2+V)l2 e S Io(2s)ds (C-l) o where we have omitted the factor exp (-K2)4/. There are various ways of solving Eq. (C-l). We will choose two methods which provide the solutions in power series; one of the two can be computed easily. But before proceeding further, it would be expedient to define what is called a "confluent hypergeometric function." Any solution of Kummer's confluent hypergeometric differential equation, given as Eq. (C-2), is called a "confluent hypergeometric function."9 d2 dy x d + (b-x) d - ay = (C-2) The simplest solution of Eq. (C-2) is Kummer's hypergeometric function: a a(a+l) x2 a(a+l)(a+2) x3 1F1(a;b;x) = 1 + E x + - + x.+... (C-3) b(b+l) 2: b(b+l)(b+2) 3! There are several notations in use for series (C-3). In this work it will be denoted by 00 1Fl(a;b;x) = () (c-4) where (a)n and (b)n are the "factorial functions" defined as 39

(a) - l(a+n) r(a) n a= (a+j-1) j=l a(a+l)(a+2) *.. (a+n-l) (C-5) It is clear from the definition (C-5) that the factorial function is the generalized form of the gamma function. The integral representation of the confluent hypergeometric function (C-4) is xt a-i b-a-1,Fl(a;b;x) = e t (l-t) dt (c-6) p(a) r(b-a) o in which Re(b) > Re(a) > 0. In the process of integrating Eq. (C-l) we will also use the following lemma 8 oo n 00oo oo00 j j B(k,n) = B(k,n+k) (C-7) n=O k=O n=0 k=O and Lengendre's duplication formula8 f r(2z) = 2 (z) r(z + ) (C-8) With this background we can proceed toward the solution of Eq. (C-l), in which we substitute s2+T = Wy to obtain 00 I / e y/2 eVy Io(2K1 / I1 ) dy Now expand the modified Bessel function and break the integral into two parts. 40

Thus 00.- - I =e r yl/2( ly) ne dy - y l/2(y) e 2 2 n=O (n')2 o (c-9) Take each integral in (C-9) separately: n 00 k 00 Syl/2(l y) n. dy (-1) a k+1/2 VeYd o k=o (nk)!k! n -) kn P(k+. /2 L (-k) n2 k+32 (C-9a) (n-k)!k: v k=0o From the integral representation of the confluent hypergeometric function as given by Eq. (C-6) we obtain the solution for the second integral in Eq. (C-9), in which a = 3/2, b = n+5/2, and z = -T. Thus yl/2(1y) e VYdy = (3/ r( ) 1F1(3/2;n+5/2;-7) (C-9b) o r (n+5/2) Substitute (C-9a) and (C-9b) in Eq. (C-9) to obtain r(3/2)Pr(n+l) 1Fl(5/2;n+5/2;-T) oo n V~3/2 2- 00n+k 2n V3/2 e Z -) (-1) (kn -1(k+3/2)() 2 Lno k=O n!(n-k) k k3/2 0n=0n)2 k=O 00 r(3/2) (-l)n( 1T2 lFl(3/2;n+5/2);-V) n=O n (n+5/2) (-) 41 ~~~~-1 (/2 (,,_pn, r.....

In Eq. (C-10) use Lemma (C-7) in the first part and expand the hypergeometric function in the second part according to the definition (C-4), obtaining ~3/2 [ 7/2 nT - 0 (-l)n( 2n r(k+/2) 2 = - -nkO n!(n+k) k!()l k+3 O00.k=O (l)n(- K -)2n+k+3/2 (-) k (k+3/2) (dT) n,k=O n' r(n+k+5/2) k! k+3/2V3/4 (C-ll) Now according to the definitions of the cylindrical and the spherical Bessel functions, we have 00 Jk(2Z) = ( )n( z2n+k (C-lla) n!(n+k): n=O 00 Jk+3/2(2z) = (l)n( )2n+k+/ 2 ( b) n!r(n+k+3/2) n=O Using (C-lla) and (C-llb) in Eq. (C-ll) we obtain i =1 e r (.k+3/2) (/IV) Jk 2 ) kk k=O 00 k=O From Legendre's duplication formula, (C-8), we obtain r (k+3/2) = 2k+) 2k+l k 2 k+1(k+l) 22k+l k' 42

Hence 00 L 2k! 2 [j/I)k Jk(2k)7 k=O (_)k (/) 3/ Jk+/2 (2K )i (C-13) Multiplication of Eq. (C-13) with the factor exp(-K2)4/Nf yields the right-hand side of Eq. (15a). The steps involved in obtaining the second solution of Eq. (14), given by Eq. (15b), are shown in Appendix B. 43

ACKNOWLEDGMENTS The author wishes to express his deep gratitude to the following individuals: Andrew F. Nagy for making many useful suggestions in the preparation of this report; particularly Walter Hoegy for reading and helping to organize the report; Hugo DiGiulio for programming the current equations; Salma Khammash for plotting the graphs; Rosann Burke, secretary of the project, for typing the manuscript; and George Carignan and L. H. Brace, present and past directors of the project, for many useful discussions. 45

REFERENCES 1. Boggess, R. L., Electrostatic Probe Measurements of the Ionosphere, Univ. of Mich. ORA Report No. 2521, 2816-1, 03484-1-S, Ann Arbor, Nov. 1959. 2. Hoegy, W. R., and Brace, L. H., The Dumbbell Electrostatic Ionosphere Probe: Theoretical Aspects, Univ. of Mich. ORA Report No. 03599-5-S, Ann Arbor, Sept. 1961. 3. Kanal, M., Brace, L. H., and Caldwell, J. R., An Ejectable Ion TrapLangmuir Probe Experiment for Ionosphere Direct Measurements, Univ. of Mich. ORA Report No.03484-3-S, Ann Arbor, July 1962. 4. Mott-Smith, H. M., and Langmuir, I., "Theory of Collectors in Gaseous Discharges," Phys. Rev. 28 (Oct. 1926). 5. Kanal, M., Theory of Current Collection of Moving Spherical Probes, Univ. of Mich. ORA Report No. 03599-9-S, Ann Arbor, April 1962. 6. Langmuir, I., "Electrical Discharges in Gases," Part II, Rev. Mod. Phys. 3 (April 1931). 7. Bateman Manuscript Project, Higher Transcendental Functions, Vol. 2, McGraw-Hill Book Co,, 1953. 8. Rainville, E. D. (Univ. of Mich.), Special Functions, Macmillan, 1960. 9. Slater, L. J., Confluent Hypergeometric Functions, Cambridge Univ. Press, 1960o See also: Magnus, W., and Oberhettinger, F., Functions of Mathematical Physics, Chelsea, 1954. 47