THE UNIVERSITY OF MI C H IGAN COLLEGE OF ENGINEERING Department of Electrical Engineering Space Physics Research Laboratory Annual Report No. 1 THE PHOTOELECTRON ENERGY DISTRIBUTION IN THE IONOSPHERE THEORETICAL STUDY OF THE MOVING LANGMUIR PROBE IN THE PRESENCE OF A MAGNETIC FIELD THEORETICAL STUDY OF THE RESPONSE OF A PROBE TO PLASMA WAVES Ernest G. Fontheim Walter R. Hoegy Madhoo Kanal Andrew F. Nagy ORA Project 06106 under contract with: NATIONAL AERONAUTICS AND SPACE ADMINISTRATION GRANT NO. NsG-525 WASHINGTON, D.C. administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR February 1965

TABLE OF CONT'ENTS Page o INTRODUCTION 1 IIo PHOTOELECTRON ENERGY DISTRIBUTION IN TEE LOWER F-REGION 3 2o1 General Considerations 3 2.2 The Photoelectron Distribution Function 5 IIIo THEORY OF CURRENT COLLECTION BY MOVING CYLINDRICAL PROBES IN THE PRESENCE OF A MAGNETIC FIELD 15 53, Equations of Motion 15 3,2 General Expression, for the Current 22 303 Accelerated Current (qV < O) 25 354 Retarded Current (qV > 0) 32 355 Discussion 35 IVO THEORY OF THE PLASMA WAVE PROBE 37 4l1 General Expression for the AC Response 37 402 Accelerating Potential 42 4o3 Retarding Potential 47 4o4 Discussion 50 V. DISCUSSION 53 REFERENCES 57 APPENDIX A 59 APPENDIX B 61 iii

LIST OF ILLUSTRATIONS Table Page I lTransitions Taken into Account in Deriving the Energy Distribution of Photoelectrons 8 Figure 2.1 Energy loss rates vs, energy0 9 2.2 Total energy loss rates vs. energy. 10 2,3 Accumulated production rate vs, energy. 11 2~4 Photoelectron energy distribution f(E) vs, energy E at the three altitudes 150, 200, and 250 km. 12 2.5 Total energy distribution f + fm vso energy at 200 km for electron temperatures of 1000~K, 2000~K, and 4000~K. 13 3ol Cylindrical probe in a magnetic field, 15 352 Domain of integration for the current Eqo (3.19) for the above cases, 20 353 Domain of integration for the current Eqo (3519) for the case qV > 0 (retarding potential) with qV > (a2-r2)m 2 2qV 0a r2) 21 r -L- m 4 3.4(a) Domain of integration for the case shown in Fige 352(a) indicating the three subregions into which the domain has been decomposed for purposes of integration, 25 354(b) Domain of integration for the cases shown in Fig. 352(b) indicating the two subregions into which the domain has been decomposed for purposes of integration. 30 4,1 Perpendicular cross section of the sheath edge showing relation of velocity components to wave vector ko 41 4,2 Region of integration Q for the case of an accelerating potential (Vo < 0) 4

LIST OF ILLUSTRATIONS (Concluded) Figure Page 4,3 Region of integration for an accelerating potential for the case ] < T T-Vo 45 4,4. Region of integration for an accelerating potential for the case q > T _-Voo 45 4,5 Region of integration for retarding potential (Vo > 0). 48 4.6 Region of integration for the case r < 1Voo 49 4.7 Region of integration for the case r >A ~Vo- 49 vi

Io INTRODUCTION This report describes the work done under NASA Grant No. NsG-525 during the period 1 October 1963 to 30 September 1964o As the title of the grant indicates, the research effort was directed toward a better general understanding in the following areas: Ao the presence of plasma instabilities in the ionosphere and their effect on the thermal structure9 Bo characteristics of probes in the ionosphere considering magnetic field effects and plasma waves. 1,2,3 Some of this work relating to Part A has been reported earlier, 3 and, therefore, that material will be outlined only briefly. The initial work in the area of plasma instabilities consisted of the treatment of a one-dimensional Maxwellian plasma with a superposed high energy hump in the distribution functiono High momentum transfer electronneutral collisions were included in the treatment by means of a relaxation term in the kinetic equation This work has shown that the temperature of certain plasmas, which are initially unstable against growing plasma waves, increases toward a temperature value for which the system is stable o In order to extend this work to ionospheric problems it became necessary to formulate a realistic model of the high energy tail of the electron energy distribution in the ionosphereo A preliminary calculation involving only one of the many possible photoionization processes has been presented 1

in the semiannual report 3 The complete results involving all photoionization processes of importance are reported in Section II below. The current collection equation of a stationary cylindrical probe in the presence of a magnetic field, parallel to its axis, was given in the semiannual reporti3 This work has been extended to the more general case of moving probes as given in Section IIIo The results from the Alouette topside sounder satellite increased the interest for a better understanding of the excitation and detection mechanisms of space charge waves~ Besides the fundamental interest in the possible existence of naturally occurring space charge waves, artificially excited waves could provide an excellent tool for the measurement of ambient electron densities. In such a technique a relatively large volume about the probe would be "sampled," therefore, if sufficiently good "coupling" can be achieved this approach might be especially valuable at high altitudes, where other direct techniques of sampling much smaller effective volumes become sensitivity limited. Theoretical work to predict the response of a cylindrical. probe to such space charge waves has been started and is described in Section IVO 2

II. PHOTOELECTRON ENERGY DISTRIBUTION IN THE LOWER F-REGION 2.1 GENERAL CONSIDERATIONS The state of the electrons in the ionosphere differs from an equilibrium state because of various energy and particle sources and sinks. The most significant as well as best known of these energy sources is the electromagnetic solar radiation,'A considerable amount of this incoming energy first goes into the production of photoelectrons which, in turn, share their energy with neutrals, ambient electrons, and ions. Hanson4 and more recently, Dalgarno et alo, have calculated the steady-state electron temperature, considering a number of atmospheric conditions. In these calculations solar electromagnetic radiation was considered to be the only heat source, and only binary collisions were included, These authors also assumed that all energy given up by the photons is deposited locally, Recently, Geisler and Bowhill6 carried out detailed calculations using information from the work of Dalgarno et alio5 in which they also included the effect of nonlocal heating, as well as conduction by ambient electrons, The calculations, however, do not consider long range electrostatic interactions (plasma oscillations).o It has been suggested that such an energy transfer mechanism may also be important because energy could then be transferred by relatively high energy photoelectrons to ambient electrons before they lose a considerable portion of their energy through binary collisions with neutrals and ions, In order to study such an effect, we have proceeded in two stepso First the distribution of photoelectrons has been calculated assuming that the 5

photoelectrons interact with the ambient gas by binary collisions only. The resultant electron distribution is then assumed to be the sum of a Maxwellian distribution fm at some temperature Te plus the computed photoelectron energy distribution fo Considering this resultant distribution as an "initial" distribution the effects of long range interactions are included in the second step by using the self-consistent field method of Vlasov~ Such a calculation will give a new electron distribution function modifying both the temperature of the ambient electron gas and the shape of the photoelectron distribution, So far, the calculations involving the first step have been completed and are reported in this section. In the calculations it has been assumed that only photoelectrons contribute to the non-Maxwellian shape of the energy distribution. The effect of corpuscular radiation is not included since little reliable data are currently available. However, the calculations can be modified to include such an effecto The assumption of the independence of the two component distribution functions should be elaboratedo With fm and f independent, f can be computed by fixing the ambient electron dis5tribution fSm and calculating the energy loss of the energic photoel-ectrons as they traverse the ambient plasma,, In other words, the energy loss is computed by considering the photoelectrons as test particles which are effected by the ambient particles but which themselves do not alter the ambient distributiono The assumption of independence faiLs, however, because the photoelectrons are continually interacting with the ambient electrons, ions, and neutralso In fact, after a photoelectron 4

has suffered several collisions, it cannot be distinguished from an ambient electron-there is only one distribution function for all the electrons present. However, when the initial energy of the photoelectron is much higher than the average energy of the ambient electrons, it is possible to distinguish photoelectron from ambient electron, Then, to the extent that the photoelectron energy exceeds the average electron energy, the test particle assumption is valid and the two distributions f and fm can be considered independent. A more exact solution would be found by solving the Boltzmann equation with a source term representing the production of photoelectrons and a sink term representing all processes which remove electrons. In this way, a steady-state electron distribution function would be obtained whose general appearance would be Maxwellian near the average kinetic energy with an increased high energy region similar to the curves in Fig. 2O.5 2,2 THE PHOTOELECTRON DISTRIBUTION FUNCTION Let f(e) be the photoelectron energy distribution function, such that f(e)de gives the number of photoelectrons per unit volume in the energy interval e to e + de, Integrating this function over the interval E to oo gives the total number of photoelectrons per unit volume N(E) having energy greater than E, or 00 N(E) = f(e)de. (2,1) E Let q(E') be the photoelectron production rate, such that q(E')dE' is the rate at which photoelectrons are produced in the energy interval E' to 5

E' + dE', in units of number of electrons per see per unit volume, Upon suffering collisions with ambients, the electrons lose energy and are spread into the interval E to ET(E<ET) in a time T(E,E') given by, E7 de T(E,E') = / -r(e) (2.2) where r(E) = dE/dt is the energy loss rate. Thus the steady-state number of photoelectrons N(E,E'), residing in the energy interval E to E' due to production at energy ET is, N(EE') = T(E,ET) q(E)dE' (2.3) The total number of photoelectrons having energy greater than E is given by the integral of expression (2.3) over all production energies E' greater than E, N(E) = / E(E~E)dE' (2o4) E Equating expressions (2,l) and (2.4) and making use of (2.2) and (2.3) it is found that, after changing the order of integration, 00 00 00 f(e)de = de- q,(E')dE' (2.5) -r(e) t oel tfn riE E e Since the lower limit of integration E is arbitrary, the integrands must be equal, viz, 6

f(E) - Q(E), (2~6) -r(E) where Q(E), the accumulated production rate, is defined by, 00 Q(E) = q(E')dE' (2j7) Thus the distribution function is the ratio of the accumulated production rate divided by the energy loss rate, and its properties can be discussed separately in terms of the two functions r(E) and Q(E)c. The energy loss rate r(E) is computed from binary collisions between energetic photoelectrons and ambient ions, electrons, and neutrals, The collision processes are many in number and the available cross section data are by no means complete, To avoid repetition of the calculations of Dalgarno et alo,, their energy loss computations are employed without change, The collision processes considered by Dalgarno are listed in Table I, In Figo 2ol the spatial energy loss rates for electron-neutral and electron-electron collisions are plotted separately, The altitude dependence of the loss rates results from the altitude dependence of number density and composition. Considering first the rN curves (electron-neutral collisions), the dominant contribution in the 5 eV to 15 eV range is from electronic excitation of 0, 02, and N2o Below 5 eV this energy loss becomes insignificant, In the range from about 1,5 eV to 5 eV the predominant energy loss process is the vibrational excitation of N2o Below 1o5 eV this process becomes negligible compared to the electron-electron loss, given by the re curves0 The 7

TABLE I TRANSITIONS TAKEN INTO ACCOUNT IN DERIVING THE ENERGY DISTRIBUTION OF PHOTOELECTRONS Atomic Oxygen Transitions Electron removed Resulting state of 0+ Threshold energy in eV 2p 4s 15.6 2p 2D 1,6.9 2p 2p 18,7 2s 4p 28.5 2s 2p 4oo0 Molecular Oxygen Transitions Electron removed Resulting state of 02O Threshold energy in eV rtg2 2 2 12 1 tu2p 4 tu 16.2 r~u2p 2,u 17. O Cg2p 4L- 18 2 ou2s 2 u 28 Cg2s 2 40 2Cg Molecular nitrogen transitions Electron removed Resulting state of N't Threshold energy in eV g2p 2,+ 15 6 g u.2p 2Iu2 16o7 (u2s 2 F 18.8 ag2s 27 355 g 8

re loss curves were computed analytically by Butler and Buckingham.7 -5 rN 150 km rN 200 km 6 rN 250 km 10 E 10Z - I jI / \~ f~re 250 km rN: electron- neutral collisions re: electon -electron collisions -8 80 I I I I I 0 2 4 6 8 10 12 14 E(eV) Fig. 2.1. Energy loss rates vs. energy. In Fig. 2.2 the total time rates of energy loss are given for the three altitudes 150, 200, and 250 km. The minimum at around 5 eV is connected with the rapid decrease in both the vibrational excitation of N2 and the electronic excitation of 0, 02, and N2. The maximum at around 3 eV is connected with the high rate of energy loss in vibrational excitation of N2. The increase in loss rate below 1.5 eV is connected with the high probability of electron-electron collisions. It is to be noted that the general behavior of the energy dis9

tribution curve will follow closely the inverse of the loss curve. 10000 1_50 km 1000 200 km 250 km | Z \ / / /250 km 10 10 I I I I I I I 0 2 4 6 8 10 12 14 E (eV) Fig. 2.2. Total energy loss rates vs. energy. The accumulated production rate Q(E), found by adding all the contributions to the production rate q at energies greater than E, is computed from the most recent data on photon flux and photoionization cross sections.8 The photoelectron production rate is the product of neutral particle density, photoionization cross section, and photon flux. Photoionization of 0, 02, and N2 is included. The result of these computations is given in Fig. 2.3. The altitude dependence of the accumulated production rate follows the altitude variation 10

of neutral particle density and solar transmission coefficients. Since the photoelectron distribution function is directly proportional to the accumulated production rate Q, it is observed from Fig. 2.3 that the Q curves only slightly modify the distribution until an energy of about 15 eV where they rapidly cut off the distribution function. 10000 150 km \"~~200 km 1000 | \ ^^^^50 km 0 10I 0 2 4 6 8 10 12 14 E (eV) Fig. 2.3. Accumulated production rate vs. energy. The energy distribution function f, product of the Q curves and the l/r curves, is presented in Fig. 2.4. For energies above 10 eV the number of photoelectrons decreases very rapidly due to the cut-off behavior of the Q curves, A hump or maximum occurs at around 5 eV as a result of the simultaneous 11

decrease in vibrational and electronic excitation energy losses which "trap" the photoelectrons in this region. The trough at around 5 eV is connected with the peak in energy loss to vibrational excitation of N2. As a result of this peak, fewer electrons are allowed to remain at this energy. The distribution function increases below 3 eV to a second hump at around 1.5 eV corresponding to the decrease in vibrational energy loss. Below this energy the dominant electron-electron energy loss increases, causing a rapid decrease in the number of photoelectrons toward lower energy values. 100 "E I 0 200 km 15k50 km 0 2 4 6 8 10 E (eV) Fig. 2.4. Photoelectron energy distribution f(E) vs. energy E at the three altitudes 150, 200, and 250 km. 12

The total electron energy distribution function is the sum of the above photoelectron energy distribution function and an appropriately chosen Maxwellian distribution. Fig. 2.5 illustrates the total energy distribution for I000 - 100 EV \1 940000K 0 - \ \20000~K 000~K\ 100 2 4 6 8 10 E (eV) Fig. 2.5. Total energy distribution f + fm vs. energy at 200 km for electron temperatures of 1000~K, 2000~K, and 4000~K. the three temperatures, 1000, 2000, and 4000~K at an altitude of 200 km. The character of the high-energy hump is highly dependent on the ambient electron temperature chosen. At the two lower temperatures the hump is clearly evident. At some temperature just below 4000~K, the hump disappears, and for all higher temperatures there is no hump at all, but merely a raised high energy tail. 15

The daytime temperature at an altitude of 200 km is about 2500~Ko Hence, we conclude that in the absence of long-range interactions the electron energy distribution displays a hump on its high energy tailo l4

III. THEORY OF CURRENT COLLECTION BY MOVING CYLINDRICAL PROBES IN THE PRESENCE OF A MAGNETIC FIELD 3.1 EQUATIONS OF MOTION The work leading to the equations of current collection by a moving cylindrical probe in the presence of an axial magnetic field is outlined in this section. In the general case a uniform magnetic field of intensity B is oriented in a direction which makes an arbitrary angle a with the axis of the cylindrical collector, as shown in Fig. 3.1. Let a denote the radius of the z / = / 4m Fig. 3.1. Cylindrical probe in a magnetic field. sheath, rc the radius of the collector, I its length ( ~rc), and $ the potential at any point in the sheath with respect to the neutral plasma. If $ is a function of r only, with r2 = x2 + y2, then the electric field intensity E = -dO/dr. 15

The force F experienced by a charged particle of mass m and charge q moving with velocity u in the sheath in the presence of an electric field E and a magnetic field B is given by the Lorentz equation. F = q [ E+uxB] (31) The components of the acceleration along the x, y, and z axes, respectively, are X -= q [Ex-yBz+zBy], m y -= [Ey-zBx++xBz], (3.2) Z = q [.yBx-xBy] m where dots represent the time derivatives and Bx, By, Bz are the components of B along the corresponding axes. The system of equations given in (3.2) can be easily solved if Bx = = =. in that case let Bz = B and reduce the system to; x~ = q E-ExyrB] m y L= [Ey+xB], (3.3) m y = 0 To obtain the relation between the angular monmenta of a particle at the sheath edge and inside the sheath, multiply the first equation in (53 3) by 16

y, the second equation by x and subtract. Note the fact that yEx - xEy = 0, since E is a radial field. Thus, d cu d d (y -xy) = - (2+y2) dt 2 dt This gives us, yx - xy D (x2+y2) + c,4) 2 where w = qB/m is the cyclotron frequency, and cl is the constant of integration to be determined by the initial values of the position and the velocity of the particle at the sheath edge. If we transform Eq. (3.4) from the cartesian rectangular coordinate system to polar coordinates by putting x = r cosG, y = r sinG we obtain - r d = - r2 + cL Then employing the initial values, i.e. at r = a, a dG/dt = ut, where ut is the tangential velocity component of the particle at the sheath edge, we get ru = (a2-r2) + aut, (5) dt 2 where uA is the tangential velocity component of the particle inside the sheath. The corresponding relation between the energy of the particle at the sheath edge and that inside the sheath may be obtained from (335) by multiplying the first equation by x and the second equation by yo Then on adding the 17

two equations, we get x~x + yy = q [xEx+yEy] m In polar coordinates this becomes d (12,2V2 2q dr d ( dt r t m dt dt where -u1 = dr/dt is the radial velocity of the particle inside the sheath. u{ has been defined in such a way that it is positive for a particle traveling toward the probe Integration of the above equation with respect to t yields u12 + i42 2 c r u m where c2 is the constant of integration to be determined from the initial values. If at r = a, u = 0O ur ur and ut = u, then we get, U,2 + 2 u 2q + + 2 (36) u: +u =+ to (3~6) r L m r u In order to obtain the condition of collection we require that u, be real and positive at the collector surfaceo Hence, at the collector surface Eqs. (355) and (3o6) become, after setting ur> 0, r P o a-2_r2 a..T.. + —u. (537) tc 2 r r u s a 2qV utc < ur + u (3.8) 18

where utc is the tangential velocity component at the collector surface and V = ((rc) is the probe potential, ieo,, the voltage of the collector with respect to the plasmao Elimination of ulc between (3 7) and (538) yields _a Ut a ) a -rc < u2 + uf - 2qV rc 2 rc - m or 2 \a2-rc m 4 C. - v ) +. < 2? a (ur a m 4 (39) Equation (539) defines the range of ut for which a particle with a given ur is collected. Graphically this range can be represented by the area bounded by the two branches of the hyperbola wna | Tr. 2 2 2qVT 2__ 2 \a2a (Ur m 4 ( ur > O) ut = - - L ~ ~(U>) as shown by Figs. 3.2(a) and 3,2(b) for the case qV < 0, i,e., for a particle which is accelerated by the collector potential. The vertices p1 and P2 of the hyperbola are C= a o22 - a2r2 2qV Pl'2 \|j4 cm 22 2 c a re rc 2qV_ 2 aa-rm 0 PZi ~ 2 1 19

.t Ut UrX (a) qV < (b) qV>< 0 (a) qV<0 (accelerating potential) with (b) qV<0 (accelerating potential) with 2- \| 1 4 22 a coa dr~ r~ 2qYV (Da rC 2qV 2 4 a-rc m 2 4 a-rc m c m and qV > 0 (retarding potential) with c2(a2-r 2) M 2 qV< < U2 > 0 8 Figo 3.2 Domain of integration for the current Eq. (3,19) for the above cases. If a < P. i 2qV (3.11a) 2 \ 4 a2 r2 m c the area is shown by Fig. 3.2(a), while for 22 2 V 2> 2_\^^i 4 2 m(3.11b) 2 \ 4 a2-rc m the area is shown by Fig. 352(b). If qV > 0, i.eo, if the particle is retarded by the collector potential, either of the following two conditions must be 20

satisfied in order for the expression under the square root in Eq. (2.9) to be non-negative, qV <, Ur > (3.12a) qV > o!, Ur > (3.12b) qV > co(a2-r))m 8 4 8^ ^,' ur m. (5.l2b) In the case of condition (5.12a) the range of allowed values is again represented by the shaded area between the two hyperbolas shown in Fig. 3.2(b). Since 2 2n2 r r 2qV + JD < _y < pa a2 r2 m 4 - 2 2 it follows that pi > 0 always. If, on the other hand, the probe potential satisfies the inequality (53.12b), then the range of values of (ur,ut), for which a particle will be collected, is given by the area bounded by the conjugate hyperbola shown in Fig. 3.3. Thus, under the conditions specified Ut Ur Fig. 5.5. Domain of integration for the current Eq. (5.19) for the case qV > 0 (retarding potential) with 2/ a 2qV o o2( a2-r2) qV > X (a r2)m u2 > 2qV (a2-r2) 8 Ur > m 4 21

above, FigSo 302(a), 352(b), and 353 show the domains of permissible velocities of the particles at the sheath edge which will reach the collector and contribute to the total current to the probe. At this stage it is desirable to change the variables in Eqs. (357) and (3.8) by setting ur = u cosG, ut = u sinG. Here 9 is the angle which the particle velocity at the sheath edge makes with the radius vector. This is done because, for the case of the moving probe, the subsequent integrations to obtain the corresponding current equations for q4 > 0 and qV < 0 are considerably simplified when the variables of integration are u and G0 Thus in terms of u and 9 the range defined by relation (359) is given by 2- 2 sin-1 wfi a- - a u2- 2qV < L2u a au m J sin-1 a a2-. 2 2qV < sin-l r a i + ~a U -, (5135) L2u a au m where for uniqueness G has to be chosen such that < < 2- 2 Now we will set up a general expression in the integral form which will represent the current of either sign to the probe and then integrate it over the domains for qV < 0 and qV > 0 (as given above) to obtain the corresponding current equationso 3 2 GENERAL EXPRESSION FOR THE CURRENT The plasma is assumed to have a Maxwellian velocity distribution at the sheath edgeo If the probe is stationary, this distribution is given by 22

21: C L C r -t) F(UroUtoyuzo) = N exp - (Uro+Ut2+Uz), (3514) where N is the number density of the particles under consideration and c is their most probable velocity, i e., c = ~2kT/m; where k is Boltzmann's constant, T is the -temperature of the medium, and m is the mass of the particle, Uro, uto and Uz0 are the components of the particle velocity in a frame of reference fixed in space When the probe is moving, let ur, ut, and uz denote the components of the particle velocity in a probe-fixed coordinate system with z-axis along the probe axis. Then Uor = Ur - W sing cosP Uto = ut - W sink sinp uZo = UZ - W coso where W is the probe velocity, g is the angle between W and the axis of the cylinder, and 3 is the angle between the projection of W on the plane perpendicular to the z-axis and the radius vector ro In terms of the new coordinates (3514) becomes f(urutUz) = ~ exp c (-(ur-W sin cos)2 + (ut-W sinu sinp)2 + (uz-W cos )2) j (5315) The number of particles crossing an infinitesimal strip of area ladp of the sheath surface per unit time is urdN = aurf( ur, ut,Uz) durdutduzdg, (3o 16) where ~ is the length of the cylinder~ On multiplying this expression with 23

the charge q of the particle and integrating over the desired limits, the following equation for the current to the probe is obtained. Nqla U e 2 = c^7U J Ur Ut r eup - I[(ur-W sing cosp) c ur ut Uz 2 21 + (ut-W sing sinp) + (uz-W cos) 2 du]dutduzdp (3517) In both cases of current collection (qV < 0 and qV > 0) the limits of 3 and uz are 0 < B < 23j and -oo < uz < oo (since ~~>a), respectively. The integration of (3517) with respect to these two variables and rearrangement of the terms yields, I = 2Nqrac'2e Ur exp [-. (Ur+Ut) I (-1. tU durdut Ur ut (3.18) where K = (W sin~)/c and Io(x) is the modified Bessel function of order zero. Integration of (3518) is greatly simplified by setting ur = u cosG, ut = u sinGo Thus 2 a J e / i K2 I - e u e uucose eu2/Io (2 )du dG, (5.19) -c re cr 3 el 31 Ca ~ where J = k/kT/2mi- Nq2.jrc is the random current to the probe. Equation (3,19) is the general expression for the current collected by a moving cylindrical probe of length I and radius rc surrounded by a sheath of radius a. The area of integration is determined by the different cases of interest (ioeo., accelerated current, retarded current, etco), as discussed in the preceding section. 24

3.3 ACCELERATED CURRENT (qV < 0) We have already seen that when the particles are accelerated by the collector potential, the domain of integration is given by Eq. (5.13) with qV < 0 and is represented graphically by Figs. 3.2(a) and 5.2(b) in association with the inequalities (5.11a) and (3.11b), respectively. Let us, first, consider the situation where the inequality (3.11a) holds, i.e., _.a r2 2qV < 2 2 (3 11a) 2 )4 a2-rc m Upon dividing both sides of (3.11a) by c = — 2kT/m and letting o/2c = a, 2qV/mc2 = qV/kT = Vo and T2 = r2/(a2-r2), relation (3.11a) becomes aa < Cro-T7Vo. (3.20) In order to integrate (3.19), the shaded region in Fig. 5.2(a) is divided into three regions R1, R2, and R3 as shown in Fig. 3.4(a). cating the three subregions into which the domain has been decomposed for 25

(j = JJ G cose du dG, R where G = u2e U2/c 12 - and R = R1UR2UR3. Thus = f = 77/+,7+ +7 R R1 R2 R3 flP|1 T/2 P2 It/2 00 ~ 2 ~= / G cos~ du d9 + G cosG du dG + G cosG du dG o - T/2 IP ile =1 2 9 where = s-1 2a-rc r- 2qV G1 = sin (u u ] - and ~2u ~ a' / au m -1 1 ta2-r2g r2c, qV sin L2uK a + au - m and pi and P2 are given by (3.10). After integrating with respect to 9 we get, pitl P2 00 = 2 G du + G(-sindu + G si )d + G(sisin-sinG1)du o IplI P2 Pitl P2 P2 00 = / G du + G du. - G sin9ldu + G(sinG2-sinG1)du 0 0 iP1I P2 Taking each integral separately and integrating one obtains 26

pi ~P -u 2/c2 o u) A = G = du u2eK 0 0 o o 3, K2n Pil l(u2n+2 _u2/c2 du (n ()o Cc n=o 002 222 U ~C n=o where 7(Clx) =J e t dt is the incomplete Gamma function. Similarily, 2 22 B~- = Gd du 00 c3 (nK2n 2 +2j.2 2 2 c2 n=o where 7(a~x) = / e t dt is the incomplete Gamma function. Similarily, P2 c = Gsi du lPn I 27 n=o C = / G sin9i du 1Pil 27

2 2, P2 e2/c 2 P2 -27C2 — u e I n du - u e Io, L- 2 Ka a oU01 I a d Also 00 00 D = G(sinG2-sinG1) = 2 u e-u2/c (2 u u2 2 du -C+.D = - P eJ 2/Ia 2 ) m P2 P2 + 1 + u e u /CIf2 d 2 a 0I/PlU e I + La +7 u e V I J - - du a L1 J m Here 00 c^P2 u2/C2/ 2 2 K2 2 2~ u Ie I0(K )du c )2 P2 I IP~ 2 - lC(n 2 and 00 00 + +l u e-u!c I _? 2qV du P11~I P2'e2/iQz!j~/ - m du +pj - V +P 00 3o n o 2 2 c where 28

) oo r(v,x) e t ldt x This last integration has been carried out in detail in an earlier paper,~0 Substitution of the results A, B, -C+D, into Eq, (3514) yields the expression for the accelerated current for the case oa <;a2r2-T2or Ial j _=a a r,2nY + 2' C2 2' c-.) (a(YP.) 00 (- ~ 7+ Vo2 2 n2 2 - yn+l ac + e no na(- L-3n/2K +{( 2' c2 o) P n-+ -- +l + rn +, Vl)2 Jn(2 ),' (5 21) where n — o 3... _ r n 3 P Pa ='a - r TVo0 P2 era + /nc2r-2-TV0o) c c and Ja = vkT/2m~jNqa2jrrc is the random current to the probe of the accelerated particles of charge qao Equation (3o21) reduces to the following form when K- = 0, which occurs either when the probe is stationary (w = 0) or when G = 0~ la -=ja Lferf -rcc-T Vo-a) + erf( - +aa) K=0 2 r2+c + e -Voerfc( o2a2(l+T2) VO -crc) + erfc(gJo2a2(l+T2)Vo +rc) 31, (3.22) 29

where pl/c = oaa2-r2-T2V has been replaced by -pl/c since the limit of integration is |p1l, and pl/c < 0. Equation (3.22) has been reported earlier.3 Several other cases of interest which arise by setting c = 0 (i.e., B = 0) in Eqs. (3.21) and (3.22) have been discussed in full detail earlier.~ When the inequality (3.lib) holds, i.e., if aa > Ja2r2-T2V (3.23) c o then the domain of integration of (3.19) is represented graphically by Fig. 3.2(b). As before, the region is divided into two regions R1 and R2 as shown in Fig. 3. 4(b) Fig. 3.4(b). Domain of integration for the cases shown in Fig. 3.2(b) indicating the two subregions into which the domain has been decomposed for purposes of integration. Thus, 50

= G cosG du dG+ G cosG du dG B1 B2 P2 Tr/2 ~P G2 = G G cos9 du d9 + G cos9 du dG Pi 91 P2 1 P2 00 =, G(l-sinG9)du + G( sinG2-sinGl) du P1 P2 All the integrations have been performed above. The final result given below then represents the accelerated current under the condition of inequality given in (3o23) r- - 1 2n2 2 2 2 I = en + 2 P2 n + P +1 - ~{e~ K ~ n + 2' c/ ar'n+l, Lcn^ r^ Zo(n'V2 b 2' c2^)- 2' +) - Lr yn ^\~^ -or - 7(n+l, ^) e- + o^U &, n- -V_ V)(n + 32 - + 2 Vo1 x -J~ n=o - Jn(2 o) j (3524) For K = 0 it can be easily shown that Eq, (3.24) reduces to Eq. (3.22). The negative sign in front of the function y(n+3/2, pI/c2) in Eq. (3.24) is taken care of because the argument of the first error function in Eq. (3,22) is -pi/c, as has been pointed out above, while the corresponding error function obtained by setting K = 0 in Eqo (s324) has the argument pl/co Since, however, erf(-pl/c) = -erf(pl/c), the correct sign in front of the error function is automatically assured0 Therefore, if K = O, Eq0 (3,22) is the expression for 51

the accelerated current for both cases Pi > 0 and pl < 0, 3. 4 RETARDED CURRENT (qV > 0) o If the particles entering the sheath are repelled by the collector potential, Eqs. (3.9) or (3013) define the domain for integrating (3519) in association with the ranges of V given by the inequalities (3,12a) and (3512b). Let us consider each case separately. If Vo < a2r2/T2 the domain of integration of (3,19) is represented by the dashed region enclosed between the two branches of the hyperbola shown in Fig, 3o2(b). However, this is the same domain as in the case of the current Ia2o Thus the solution is of the same form as Eq, (3.24). Denoting by I the retarded current for the case V0 < o2r2/T2, we have, therefore, I 00 I (nil, +e *eo2 2n c9 7Y + 2 V c P2 )-'T cf nn=oo n I2(2K 0) n 25225) Y +1 - +e-VO r + -Y V +r + - VO _ where Jr = 2jrckT/m NqMrrc is the random current to the probe of the retarded particles of charge qr and where the modified Bessel function In(z) is defined by In( ) 1 iJn(iz) For K = 0, Eqo (5325) reduces to the following form 32

"- c Tr | s=o = 2r Lacerf(~ 2r -T2Vo-~a)+erf(~ 2r -T2V0 ~oa)3 + eVo(erfc(N/2a2-(l+T2) Vo -rc)+erfc(lcy2a2-(1+T2)Vo +cyrc) (3526) which is of the same form as Eq, (3,22) for the accelerated current. It is evident from Eqs. (3525) and (3.26) that in the presence of a magnetic field and VO < 2rC2/T2 the current to the probe due to the retarded particles involves the parameter a/rC which would not be the case in the absence of the magnetic field -or when Vo > Cy2r2/T2 as shown below. If V0 > ~ 2r2/T2, the domain of integration of Eq. (3519) is shown in Fig. 3535 To integrate (3519), we first need to find the minimum value of u. This may be done as follows: The minimum value of u occurs when G9 = 92, i.oeo where 91 = sin -L ---- 1 a - U2 2qVu [2u. a au m J = sin- ( r- c + E u2 2qV = G2 L2u\ a - au m J This gives umin = 2qV-/mo If Ir2 denotes the retarded current for the condition V0 > a2r2/T2 then we write 33

Ir = - a r i e2 du de u cosG eu /c Io 2K r2 - r r c3 Ge\ c2qV 9~ \ m SeK u^u2 _ 2qV eu/C2 du (27) c3 m. 2qV m Equation (3527) is clearly independent of c and a/rC From this we conclude that the retarded current to the probe is unaffected by the presence of the magnetic field whenever V0 is larger than or equal to a2r2/T2. The integral in Eqo (3027) has been evaluated by Kanal,0 The result is 00 Ir2 = Jr e~Vo -2 (2n+l)n 1/2 (.28) 1 -j e IIr.,,/ In(2KVo ) 3,28 n=o For values of Vo = o2 rc/T2 Eqs. (3525) and (3528) reduce to the following form: 00 n 2 2lV (2n+1)AT 2 r I = J exp - - K2 ( 2n+l)C >_ ( (_ o29) n~o For K = 0, Eq, (5329) becomes: Jlr~o - JI. exp~ C -r' (3o30) r2IKO = Jr exp< r (5r50) r2 K=O r T 2 Equations (3529) and (3530) represent the current of the retarded particles at the point of transition, ioeO, when Vo = 2 rc/T2o 5^

3 5 DISCUSSION All the equations we have derived are based on the assumption that the particles have a Maxwellian velocity distribution at the sheath edge and that the mean free path of the particles is sufficiently large to ensure a collisionless sheath. We have ignored the effect of the magnetic field on this distribution. Then in conjunction with a given sheath model (of Langmuir type) the current equations were derived for accelerating and retarding probe potentials. Thus for the accelerating potential we obtained Eq, (3,21), valid through the domain prescribed by the inequality (3520), and Eq. (3,24) valid for the corresponding inequality given by (3.23)o For K = 0 both Eqs. (5321) and (3524) yield Eq, (3522) For the retarding potential, Eq, (3525) was obtained under the condition that Vo < o2rc/T2 For K = 0, this equation reduced to Eqo (3026) For Vo > o2r2/T2, we concluded that the magnetic field had no effect on the current as exhibited by Eq. (3528). For Vo = 2r2/T2 both Eqs (3o25) and (5328) reduced to Eqo (3 29) which, then, was specialized for K = 0 to obtain Eq. (5330) In plotting the current characteristics one is usually faced with the problem of defining the sheath dimensions explicitly in relation to the probe potentials0 In a plasma without a magnetic field one need consider the sheath dimensions only insofar as the accelerated current is concerned0 This, however, is not the case when the magnetic field is present, since then, the retarded current for Vo < o2rr/T2, also exhibits dependence on the sheath dimension as is clear from Eqo (5325)o Thus for a meaningful plot of the current characteristics it is imperative to seek a proper relat-ion be 55

tween Vo and a/rc for both modes of current collection, The degree of accuracy of such a relation is of critical importance in avoiding the spurious behavior of the current characteristics in the acceleration region, as has been discussed in detail in the semiannual report 3 One must obviously solve the sheath problem to obtain the above mentioned equation in order to be able to affix any physical meaning to the equations so far derived. On account of the absence of such a relation no volt-ampere characteristics can be included in this report. 36

IVo THEORY OF THE PLASMA WAVE PROBE 4.1 GENERAL EXPRESSION FOR THE AC RESPONSE Work has been started on the derivation of the expression for the response of a probe to a plasma wave, The two cases of a low frequency wave and a high frequency wave will be treated separately, In this connection the low frequency range is defined as w << tr, where tr is the time of reflection of an electron from the negative probe; and conversely the high frequency is the range where X >> tr. So far the low frequency case has been solved. Work on the high frequency case will be carried out in the future. If u << tr then the phase of the wave remains essentially unchanged during the time the electron spends inside the sheath. Therefore, steady state conditions can be applied, We consider a Maxwellian electron distribution with a superposed longitudinal plasma wave,, t) = fo(v) + f (r,vt) (4.1) fo(v) = 73 ee (4.2) where the velocity v is expressed in units of the thermal velocity c and No = average electron number density 2 2KT C = — m 37

T = electron temperature m = electron mass K = Boltzmann's constant For a plasma described by the linearized Vlasov equation the perturbation fi, proportional to exp[i(kor-cut) ], is given by the expression11 f = A e-V2 (453) k vk-q where >vk = component of v in the direction k, n = ck ck' and A(r,t) is assumed to be proportional to exp[i(kor-ct) ] The response of a cylindrical probe to a perturbation of the above form for the case where the probe makes an angle of 90~ with the propagation vector k has been calculated. It is assumed that the diameter of the sheath surrounding the cylinder is small compared with the wavelength of the plasma waveo In that case all points on the sheath edge can be considered in phase -> -> so that the factor ei kor will be approximately uniform throughout the sheath. The parameter A is therefore taken to be of the form Be,it where B is constant inside the sheatho In addition any diffraction of the wave by the probe is neglected; that is the wave is considered to be undisturbed by the probe. The total current collected by the probe is then 38

I = acq dz d9 dvr dvt dvzvrf(zr, Gvrvtvz) 0 o O, V -VVt -o (4)4) where a = sheath radius, ~ = length of the cylinder, q = charge of the particles under considerations z = coordinate axis along axis of cylinder, G = azimuth angle in the plane perpendicular to the axis of the cylinder, Vr = radial velocity component in the plane perpendicular to the axis of the cylinder, vt = tangential velocity component in the plane perpendicular to the axis of the cylinder z = velocity component along the z-axis, The limits of the vr and vt integrations are determined from conservation of energy and angular momentum, namely t = T-Vro v = Vo where r = \ a2-rj qV v~ KTI V = potential of the collector with respect to the sheath edge, rc = radius of collector. 39

The collector potential is taken positive when the collector attracts the electrons. The lower limit of the vr integration is 0 when Vo < 0 (accelerating potnetial) and vl when Vo > 0 (retarding potential). End effects due to the finite length of the cylinder are neglected. Substitution of Eqs, (4,1), (4o2), and (4-3) into Eq. (4.4) gives an expression of the form I = IDC + IAC where IDC is the term involving the equilibrium distribution function fo, and IAC represents the response of the probe to the plasma wave fro In the development which follows we shall be concerned only with the AC component of the collector current. The DC component has been calculated previously by many authors, We have then a v vr vk -(00 VV ) IAC = Q d dvr dvt v) e, Vk v-B 0 O,V1 -Vt1 where Q is given. by 00 ac Bq jicit ez e e z dv -00 ac~Bq 4 e-iot (4 45) k Vk can be expressed in terms of v and vt, see Fig, 41, vk = vrcos9+vtsin9o 40

Fig. 4.1. Perpendicular cross section of the sheath edge showing relation of velocity components to wave vector k. Hence IAC becomes 2IAC = Q/2 / vtl vrcosG+vtsin -(v 2+v 2 IAC = Q dd d vr dvtvr v........ e r+t) O V1 -Vt r cos+vtsinG-Ti O o,Vi -vt l r The above integrand can be separated into two parts as follows vrcos+vt sinG 1+ + i vrcosG+vtsinG-r vrcos9+vtsinG9Substituting those two terms into the expression for IAC one obtains IAC = Q dG dvr dvtvr e(Vr+ t) L0o ovl1 -vtl /2, poo 0 vtl. -(vr+vt) 7 + d9 dv. dvt re o~v -vdt ov1 vt vrcos9+vtsin9-G i 0 O(4.6) where Q is given by Eq. (4.5). Equation (4.6) is the general expression for 41

the AC response of a probe to a plasma wave under the assumptions stated above, Integration of the first term is elementary. The result is 2Tr t 1 2 2 J1 - dO dvr dvtv e r+vt) 0 0,V1 -Vt1 Jt5/2erf(T -VO) + C- e verfc( -VO(l+T2)) if VO < 0 a (4 7) 53/2. e-Vo if Vo > 0 a where x erf(x) = 2 e-2dy /o and erfc(x) = 1 - erf(x) 4.2 ACCELERATING POTENTIAL In carrying out the integration of the second term, the case of the accelerating potential (Vo < 0) will be considered first, After the following transformation of the variables of integration Vr = U coso Vt = u sinz dvrdvt = u du dI 42

the integral becomes u2cosp -U2 J2 = d du dd. (4.8) ucos(9G-)-Tr 0 Q' is the region bounded by the two hyperbolas vtl = +T v-Vo, as shown by the shaded area in Fig. 4.2. Vt Vr Fig. 4.2. Region of integration ~ for the case of an accelerating potential (Vo < 0). In terms of the variables u,, the region of integration is given by the expressions - - <- for O<u< < T N2- -2 - - -~1 < < 1 for T -V0 < u < where = sin lFIT Ju-V] 435

As shown in Appendix A the G-integration gives the following result 0 u > T) 1 2 dG9 (4. 9) o cos(-S ) - O< u 2_ i< TIU2 U < Hence, after substitution of Eq, (4,9) into Eq. (4o8) the following result is obtained, o -*T/2 2 u2 -2rr du d~ u~e- 2 ~ < T (4o1Oa) J2= r pT\/T " p p -/u2 Peu- cosp PJ u2 c -2 / ~ du do/ + du do^ue CO =-r2 2 2 > TVJ-v (4,10b) where h = sin rL a uo v43 The regions of integration for the two above cases are shown in Fig4s 445 and 4o4, 44

Vt ~7 potential for the case /j < TVt Vr -77 Fig. 4.4. Region of integration for an accelerating potential for the case r< > T /-Vo. The case r < T V-V0 will be considered first. The v-integration introduces a factor 2. Hence, J21 = - 1 - - 45

The above integral is evaluated in Appendix B. The result is, according to Eq, (B-2), Ja = X22 e~-2/2 V- I o('| q < -V (4.11) This is the solution of the integral in Eq. (4.8) for the case where < < T Vo If'r > T -Vo, then J2 is a sum of two integrals [see Eq. (4olOb)], After performing the integration with respect to ~ this becomes T-Nr Uh 2 -U -u 2 -u2 J22 = 4j it du e + T du u-N e-u VO V- V 2-U22 The solution of the first of these two integrals is obtained by setting a = 0, b = T N/-VO and c = 0 in Eq. (B-l) of Appendix B while the solution of the second is obtained by setting a = T -Vo, b = q, c = -Vo. Hence, JC22 becomes 24=,F2 ly +C 3+ -T2V +T 2e 2+ m=o m m=o T1 r 5 2 5 VI 2 Fi>3 1i Q4 L2 ( [V) r(f + 2' r-V) - 2+. 2' -Vo(T +i)0. (4,12) (ha2Vo)m +, j m+ - \J That part of the above sum containing the function (m +, 2 -2Vj can be carried out by making use of Eq6 (B-3), according to which

00 — 1 — 1- 2n + ( 2V - 2 -72Vo -n (2_v)y 2. -v - n=o _ ( n ) e-( e -V/2 /a1( IoV ( 2 o (4.13) 4 2 2 Eq. (4.12) may therefore be rewritten in the following way, (21 -T2Vc - T e"O 20In j m=o +2 m=o (~ _Vo)m 7[ + -V(2, J_ ( 2 ~( 1l > 2 1 3 TVo)e e 2,L' - Io > o >T V (4.14) Upon substitution of Eqs. (4.7), (4.11), and (4.14) into Eq. (4*6) the accelerated current finally becomes acBq I e-it5 /2[ erf(T V) + e erfc(Vol+T))+ T I Mk a 2 a M=...._ O 2 (4 15) where Ji and J22 are given by Eqs, (4o11) and (4,14) respectively, 403 RETARDING POTENTIAL In the case of a retarding potential (V0 > 0) the lower limit of integration with respect to vr in Eq, (4,6) is vl = JVo. The first integral appearing in. Eq. (46) has been solved for the case of V0 > 0. The solution 47

is given in Eq. (4.7). It is + / d dv / tdvtv e-( = V/2 t eo./ (4.16) j~jdvrjd a 0 O -vt1 The second integral appearing in Eq. (4.6) is + 2 2 = f d du d u2cos eu4.17) * *Jv- ~ucos(9-O)-r1 where the variables of integration have again been transformed to polar coordinates and where the region of integration Q is given by the area bounded by the hyperbola vtz = IT l/vr-Vo t as shown in Fig. 4.5. The integration Vt Vr Fig. 4.5. Region of integration for retarding potential (Vo > 0). with respect to 9 has already been carried out in the section on the accelerating potential. The result is given by Eq. (4.9). In the integration over u and 0 the two cases 9 < goVo and Tr > \Vo have to be distinguished. The integral in Eq. (4.17) is non-vanishing only over the region of intersection of the circle of radius [ with the area Q+, shown in Fig's. 4.6 and 4.7 for the two cases r] < 1N and r1 > V0. 48

Vt Vt vrV Fig. 4.6. Region of integration Fig. 4.7o Region of integration for the case r < Vo. for the case r > VO. The region of intersection is shown shaded. It is seen that J2 vanishes when r, < fVo. The rest of the treatment will therefore be devoted to the case rl > 1Vo. In that case Eq. (4.17) can be written in the form + _ 2 1u2, s J2 = - 2 du d ue cos (4.18) where as before -1/ T u2_Vo z1 = sin1 JT 2After the -Rintegration this becomes J+ = 4t -H- 1 eT2 2 J2 — 4So -+. e rl_ — du The above integral has been evaluated in Appendix B. Putting a = AVo, b = q, c = -Vo in Eq. (B-i) we obtain 49

00 1 + e" _ 1 2' J2 = - 2r -T e —- 1Yn + 12 - L ~hn 2-v n' o ) 2 SlTT;2 V^~~-V n=o According to Eq. (4.13) this is equal to J2 = 2(a e n.I<, (4,19) ~ v+T 2 2 Hence, the retarded current is given by the following expression I rcc~Bq e e < iwt e e-Ve <V0 k + I IAC S (4.20) ac~Bq e-iwt 2 e e Vo rs J21 > k k i a where J2 is given by Eq. (4,19). 4.4 DISCUSSION Equations (4.15) and (4.20) are the expressions for the accelerated and retarded AC currents, respectively. Both of these currents are proportional to e-it. We conclude therefore that the current collected by the probe consists of the usual DC component plus a superposed AC component which oscillates with the frequency of the plasma wave, A detailed analysis of the amplitude of the AC response as a function of probe potential must await further calculations, especially the summing of the expression J22 given by Eq, (4..14) This work is being carried out at the present time. In addition the important question of whether or not the amplitude of the response is large enough for detection will be investigated. Of particular interest is the limiting value of the response as the ratio rc/a goes to zero. This limit depends, of course, on the value of the 50

potential V0. In ionospheric applications, however, the probe radius will in general be small as compared with the Debye length. If the probe is at its equilibrium potential (i.e,, if it is not driven), the sheath thickness is of the order of the Debye length XD. Therefore, if rc/D << 1, it follows that r,/a << 1. As the probe is driven more negative than its equilibrium poten-tial, the sheath radius increases and hence rc/a decreases further. Taking the limit as rc/a O one obtains for the AC current the following expressions Lim - c~Bq 1e ri +[r-Vo +e erfc(Vo) ] Lim IAC e 0 = k V rc ~ 1. + ^4r () X A m <( + 2 v Lr 2(m-0)-o 2 m==o -. T 5/2(n2 V^ e-, ve / I,( Io _ > rc~vo (4.15'I a i,~.T2 0 r cTBqTr e-V'o e - ic7t Tole e _- rentf th cs < V / Vo Lim 1AC 0 = (4.20') _ k f' 1'o 2 2^ V I> The limit of the accelerated current for the case T) < rc/a/ v-V0 has not been computed since the phase velocity of the plasma wave always exceeds the 51

electron thermal. velocity (ioe0, e > 1), while C/a) -Vo << 1 has been assumed in taking the above limit, Another limiting value of interest is the one as a/rc -+ 1 This limit occurs, when the probe potential approaches the plasma potential, and gives rise to the following expressions, 2 2j 1( 1 Lim IA =e + e - a -— < 1 | c_ i -Vo- iwt Lim IAC rc cI RBqt.e t-irc eVo +aq 7'2-Vo) e(21Vo/ k L LTK-T Q ]v~ n> (Vo20) [I<2 )I rl > 2; (Lso ) By varying the probe potential the experimenter can control whether the probe operates in the region where a/rc - oo or a/rc + l1 The investigation is continuing with a view toward obtaining a better understanding of the general behavior of the plasma wave probe in all frequency regions so that it can be used for meaningful ionospheric measurements. 52

Vo DISCUSSION The study of the electron distribution as described in Section II shows that, considering only solar electromagnetic radiation as energy input and binary collisions as electron energy loss mechanism, a high energy hump in the energy distribution results. Drummond et al.,9 have studied the question of stability of certain isotropic velocity distributions assuming a hump in the energy distribution. This work has shown that if the ambient electron temperature is low enough, the presence of such a hump may give rise to growing plasma waves. Such an instability would modify both the ambient electron distribution (ioeo, the temperature) and the high energy hump in such a way as to quench the instability. The investigation of the stability conditions for spherical plasma waves has been continued and is near completion. The result of this work will be described in a future report, The electron energy distribution calculated in Section II is also being used to study the importance of the contribution of electron impact excitation to the total 63000 A red line emission in the atmosphere. 2 The existence of a hump in the energy distribution may also be important in obtaining theoretical estimates of space craft equilibrium potentialo Calculations to establish the significance of this hump on the equilibrium potential are being carried out, It has been pointed out at the end of Section III that in order to obtain a meaningful voltage-current relation it is necessary to have an equation 53

which relates the sheath dimension to the probe potential. Approximate relations of this type have been derived for the case of no magnetic field; however the use of such a relation causes spurious results as was shown in the semiannual report.3 For this reason a study of the sheath problem in the presence of a magnetic field has been started. A brief account of two methods thus employed is as follows: Taking the cylindrical collector as an example (although the analysis holds equally well for spherical geometry), the first method13 relies on the fact that in the absence of collisions the total currents carried by each kind of particles across two coaxial cylinders of arbitrary radii are equal. This enables one to determine the velocity distribution of the particles inside the sheath in terms of that in the undisturbed zone. Then, on obtaining the region of permissible velocities inside the sheath by means of Eqs. (3.5) and (3.6), the equation for the density of the particles may be derived by integrating the distribution function over that region of phase space. This method has the advantage of mathematical simplicity. The other method for obtaining the density distribution function relies on the fact that in the absence of collisions the general solution of Boltzmann's equation can be obtained by the method of characteristics. The general solution is an arbitrary function of the energy and angular momentum, and the precise function is then exactly determined by the boundary conditions. After the density distribut tion is evaluated by means of one of these methods, Poisson's equation serves to yield the potential distribution

inside-the sheath~ Both of these methods have been profitably employed in the case of zero magnetic fieldo 3 Work is in progress at this time to incorporate the effect of the magnetic field on the sheath structure, The work described in Section IV is just the first step in what is expected to be a detailed study of the interaction between a probe and longitudinal plasma waves. At the present time the equations derived in Section IV are being analyzed further. The next step planned in this study is the investigation of the response of a probe to waves whose period is comparable or small compared to the time an electron spends inside the sheath.

REFERENCES 1o Nagy, Ao F,, Fontheim, Eo Go, and Dow, Wo Go, "Space Charge Waves in the Ionosphere and their Effect on the Heating of the Atmosphere," TransO Am0 Geophyso Union 43, 439, 1962, 2o Fontheim, E, Go, Establishment of Stability by Collective Interactions in a Plasma with Collisions, Scientific Report 04613, o6lo61I-S, Univ. of Mich,, Space Physics Research Laboratory (March 1964)o 35 Fontheim, E. G,, Hoegy, W. Ro, and Kanal, Mo, A Theoretical Study of the Effect of Collective Interactions on the Electron Temperature in the Ionopshere and of the Lanqmuir Probe Characteristics in the Presence of a Magnetic Field, Semiannual Report No. i, 06o06-2-P, Univo of Mich,, Space Physics Research Laboratory (May 1964). 4, Hanson, Wo Bo, "Electron Temperatures in the Upper Atmosphere," Space Research III (ed, W. Priester) 282, North Holland Publishing Co,, Amsterdam, 19635 5. Dalgarno, A, Mo, McElroy, Mo Bo, and Moffett, Ro J,, "Electron Temperatures in the Ionosphere," Planetary Space Scio, 11, 463 (1963), 6, Geisler, Jo Eo, and Bowhill, So A., "Ionospheric Temperatures at Sunspot Minimum " J Atmospheric Terresto Phs (to be published)..; 7o Butler, So To, and Buckingham, Ma J,, "Energy Loss of a Fast Ion in a Plasma, " Physo Rev0 126, 1 (1962), 80 Watanabe, K0, and Hinteregger, H. Eo, "Photoionization Rates in the E and F Regions," J. Geophyso Res 7 67, 999 (1962) 9.Drummond, Jo Eo, Nelson, Do Jo, and Hirshfield, Jo Lo, Collision-Induced Instability in a Plasma with an Isotropic Velocity-Space Distribution, Boeing Scientific Laboratories, Report Dl-82-0273(1963) 10o Kanal, Mo, "Theory of Current Collection of Moving Cylindrical Probes, J~ Appl0 Physo 35, 1697 (1964). llo Jackson, J. Do, "Longitudinal Plasma Oscillations," Plasma Physics (Journal of Nuclear Energy, Part C) 1, 171 (1960) 0 12o Walker, J0Co Go, and Dalgarno, Ao, "The Red Line of Atomic Oxygen in the Day Airglow, " Jo Atmosg. Scio 21, 463 (1964) 57

REFERENCES (Concluded) 135 Mott-Smith, H, Mo, and Langmuir, I., "The Theory of Collectors in Gaseous Discharges," Phys. Rev. 28, 727 (1926). 14. Bernstein, I. Bo, and Rabinowitz, I. N., "Theory of Electrostatic Probes in a Low-Density Plasma, " Phys, Fluids 2, 112 (1959) 58

APPENDIX A Consider the integral 2r dGI do cos(G- ) -q cosa-q 0 0 (a) q<l In that case the integrand has a pole, and the integral has meaning only in the principal value sense, It is sufficient to evaluate the integral between the limits 0 and jco Then ~ do da Lim d + d' cosa-q cosa-q +0 o o,/' where ao = cos q /liq2 tan -q+l = Lim log 2 a log!q c ao'~!"q2 J q tan 2 +Q q-1 1 2 q o \1l-q a tan ~+- -q+l 1! =lq 1 2 + - log - 1 log vy~l-ql l-q2 l-q2 tan 4, +q-1 = 0 (b) q > Let z = e; then de = dz/iz. The integral is then 59

2 dz iT z2-2qz+l where C is the unit circle, The roots of the denominater are at Z1 = q -'2l and Z2 = q + q2 Since q > 1, the two roots are real and z1 lies inside the unit circle, Hence, using Cauchy's theorem we obtain 2, dz 2~ i z2-2qz+l - 60

APPENDIX B We consider an integral of the following form: b u 2 H1 = duu u ---- e a < b, b < *2-u2 a I N Putting u2 = x-c, u du = 1/2 dx, one obtains 2 c b +c H1 = dx e 2 2C -x+c C b2+ +c c 1e _x_ e-dx 2 a 2+ c e\ 1 Tj+ \ 1 +C _ c \- \! b +c 1 e0 NII n b +C n+l/2 -x 1 ec: k J n 1 +b 2- n (' l2+c)n J x e dx eTaj2+c - n' (a+cn'a2+c n=o where the factorial function () n is defined by (a)n = n'r (a+kl), k=l 1 eC QI_ 1 3 c __n_' ( T+c) n + b - 2' + a, (B-1) no L 2 b2 2 + n=o where y(a,x) is the incomplete gamma function defined by y(a,x) = tl et dto o Of special interest is the case a = c = o, b = r 61

1 2- u2e-U2 H2 = - du rl s 1F 7 \ r Let u2 = sr22, u du = (Tr2/2) ds. Then H2 = 2 qJI Ll/2 2 H2 = ds s-/ e-s 2 2 0 Let s = cos2G, ds = -2 sinG cosG dG. Then <T/2 2 2 2 ~/2 2 = 2 dG cos2G e- 2cos2 2e- 2/2 rd/2 e_-2/2)cos29 + e-T2/ 2 29 e2/2) cos2 2i o o 0 0 Let 2G = o, dG = 1/2 da. Then 72 2/ Tc -2/2) cosa _ fa/2) cosUi H2 = e'' da e + da cosa e 40 0e ^ /no w Lz e < o i/2Io K- )-T c(ti) o (B-2) 4 2 where In(z) is the modified Bessel function of the first kind of index n defined by In(z) = i-nJn(iz) and Jn(z) is the Bessel function of the first kind of index n. This result can be used to sum an infinite series of incomplete CGamma functions~ Setting a = c = o, b = = in Eq. (B-l), equating with H2, and 62

using Eq. (B-2) one obtains 00 i\m 1 n _ 3 2= L e o/2 o(I ()] n=o - - 65

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