PHYSICAL REVIEW VOLUME 80, NUMBER I OCTOBER 1, 1950 A Floating Double Probe Method for Measurements in Gas Discharges E.O. JOHNSON AND L MALTER RCA Laboratories Division, Radio Corporation of America, Princeton, New Jersey Received March 14, 1950) easurements with a Langmuir probe in varying or decaying plasma potential in these cases follows that of the most positive t can contact. The difficulties can be obviated by the use of a pair of probes joined by a variable potential source. The double probe system"floats "with respect to the discharge system. densities can be determined. The method is also applicable to"going discharges where it has the advantage over the single probe of exerting a negligible influence on the discharge. a plot of the logarithm of the electron current in this discharges either of the stationary or time- region us probe potential yields a value for the electron varying type it is generally the case that the elec- temperature Te,(3)A region of electron current only trons present in the plasma regions have a Maxwellian in which the current increases slowly with increasingly distribution. If this is so the concept of temperature can positive potentials be associated with the electrons. The electron tempera In region 2, the current can be expressed as: ture is denoted by Te Knowledge of the electron temperature is of im Nie=-(e/kTe)V +InAjo, portance in the determination of such quantities as the where i, is the electron probe current, jo is the random ambipolar diffusion coefficients. Langmuir and Mott- electron current density, A is the probe area, Te is the Smith have described a single probe technique for electron temperature. measuring electron temperatures as well as of other In Eq. (1)V is the plasma potential measured with quantities such as electron density and wall and space respect to the probe. This infers a knowledge of the potentials. Their method can be used for stationary and plasma potential. In practice this need not be known for certain types of time-varying discharges. However, since it can be replaced by its equivalent Vs-Vp in any case, unless its area is extremely small, the probe where V s is the cathode-plasma potential and vp is the may draw sufficient electron current when operated cathode-probe potential. Equation(1) then becomes lose to space potential to disturb the discharge cor ditions which it is designed to measure. As will be Inie=(e/kTevp+InAjo-ev s/kTe). made clear below, the single probe method(SPM), is Thus it is merely necessary to plot Ini, s.vn quite unsuited for such cases as the decaying which is present following the interruption of to secure a value for Te. An examination of (2) charge. a double probe method (DPM)has some possible difficulties with the SPM. Equa- tion(2)is significant only if Vs, Te, and jo do not developed which exerts a negligible influence on a dis- change with V. In actual practice, as i, increases one charge and which seems to yield accurate temperature often observes changes in the discharge patterns, par- data in all types of discharges, including a decaying ticularly if ie is an appreciable fraction of the discharge plasma. Reifman and Dow have described a double probe method for measurements in the ionosphere. current. Under these circumstances, the probe is dis- intended to measure. It would definitely be more satis II. CONSIDERATIONS REGARDING THE SINGLE fying if measurements could be made in a manner less PROBE M likely to disturb the quantities it is intended to measure In the langmuir As will be seen, in the double probe method the cylindrical, or spherical electrode is immersed in a current to the probe can never exceed the positive ion discharge plasma and the current to it measured as a hundreds of times smaller than the electron current to In this region the current increases slowly with in- this respec, e the DPM appears to be advantageous in are observed:(1)A region of positive ion current only. Langmuir and Mott-Smith have shown how in ad- creasingly negative potential on the probe. (2)A region dition to electron temperature, the probe data can e current passes increases rapidly with increasingly positive potenti yield values for electron and ion densities; space and wall potentials and for random electron currents. The BA Reimo ndH, M. Mott-Smith, Gen Elec. Rev. 27, 449, objections cited above for the use of the SPM in electron in and W. G. Dow, Phys. Rev. 76, 987(1949) temperature determinations are equally applicable here
MEASUREMENTS IN GAS DISCHARGES ANOCE time Il, the grid sheath extends across the 1 mm grie PROBE ARGON AT opening thus isolating the plasmas on both sides of the grid. In Fig 3, t is the time at which the kinks in the decay curves occur. For any value of Ep, E could be set so that at some time (>li, the current to the probe was zero. Under this condition, the probe is at floating CATHODE potential E/. Ep was varied over a range of +16 volts on either side of ground potential and Er was determined FIG. 1. Section th uag e prere en dae twin ora measuring fioating in each case for 1>h The results are plotted III. POTENTIAL PROPERTIES OF A The plasma potential Es, is always slightly positive DECAYING PLASMA with respect to floating potential. The difference Consider a region enclosed by a unipotential be between the two is a function of the electron and ion temperatures. Since(as will be seen) both these tem- of any shape whatsoever and suppose that at peratures are low for I>li, it follows that E, lie close the region is filled with a plasma of arbitrary to Ey and therefore close to Ep, i. e, the space potential strout that, due to their greater velocity, electrons will begin when E,=0, E, appears to be about -0.8 volts Since to pour out into the bounding walls. However, as the contact difference of potentials between the various this occurs, the plasma potential will rise. This will, electrodes is not known, nothing can be said concerning in turn, soon prevent the further loss of all but the the significance of this value fastest electrons,(as determined by the boltzmann In order to determine how rapidly the plasma poten equal that of the rate of loss of positive ions. The plasma potential of the anode was varied by means of steep now decays while retaining its plasma-like character- wave-front signal during the period (>t1. It was found istics In low pressure discharges the losses of charged that equilibrium conditions(as observed on potential will be slightly positive with respect to that Since this is about the rise time of the pulso ishe scope) particles are by the process of diffusion. The plasma were always re-established within about i es employed of the surrounding envelope which will be at the wall it is impossible to estimate the time for equilibrium to potential. As will be seen later the difference in potential be established. Very likely the times involved are deter between the walls and the decaying plasma will not be more than of the order of tenths of a volt Now let some portion of the envelope be increased in potential with respect to the remainder. Then excess electrons will pour into that portion ing the poten- tial of the plasma to rise until once again the balance between loss of electrons and positive ions is attained The plasma will now assume a potential slightly positive with respect to the most positive electrode with which it"makes contact. "[An exception to this occurs when the area of the most positive electrode is so small that the normal electron diffusion current to FIG, 2, Circuit for measuring floating potential of probe it does not exceed the positive ion diffusion current to the entire boundary. This property of decaying plasmas vas demonstrated in a very simple fashion a tube was built in the form shown in Fig. 1. The ling, was argon at 2504. It was connected in a circuit as shown in Fig. 2. The tube was fired by impressing a negative pulse on the cathode. The current to the probe as a function of time and Eb, following the inter- rup: ion of the discharge was measured by means of the scope across Rs(Eb is the probe battery potential Two typical oscilloscope traces are shown in Fig. 3 The lower curve is for a case in which the probe is FIG 3. Decay currents to probe in tube of Fig. 1 slightly positive with respect to the anode; i.e upper curve is for the case in which Es-Ep o,(Ep is the anode supply potential). The
E. O. JOHNSON AND L. MALTER mined primarily by electron mobility considerations current id must be zero since no net potential acts in and are thus less than 10- sec. for this particular tube. the current loop. This condition corresponds to point o on the curve of fig. 7 IV. THE DOUBLE PROBE METHOD (DPM The double probe method makes use of two probes, (b)Va=small negative voltage( Fig. 6b) each similar to the single probe of the SPM. They are The probe potentials with respect to the plasma must interconnected as shown in the circuit of Fig. 5. The adjust themselves so that the basic current relations are potential Va is termed the differential voltage, and its still satisfied. The consideration of a few possibilities associated current, id the circuit current. The positive will show that the only way in which the system can sense of these quantities is established by the arrow satisfy all conditions is that it assume the potentials directions where we define positive current as the rate of shown in Fig. 6b. Probe No. 1 moves closer to plasma flow of positive charge. Unless otherwise noted the potential and collects more electrons, and probe No. 2 circuit of Fig. 5 and its polarity convention will apply moves away from plasma potential and collects fewer to all of the discussion which follows. In brief, the elec- electrons. The extra electrons flowing to probe No. 1 tron temperature will be determined from the way in pass through the circuit to make up the deficiency at which ig varies with v probe No. 2. All conditions are again satisfied and the As with the sPm the dpm is based on the boltzmann Dint b on Fig. 7. lation and the plasma-sheath properties of a gas discharge. In addition it is based on an application of application of (c)va=somewhat larger negative voltage(Fig. 6c) Kirchhoff's current law which requires in this case that Probe No. 1 moves still closer to space potential and electny instant the total net current of positive ions and collects the entire electron current to the system since ons flowing to the system from the plasma must probe 2 is now so highly negative with respect to he plasma that no electrons can reach it. Half of the Qualitative Treatment electrons reaching probe No. 1 now pass through the external circuit to probe No. 2. All conditions are satis- As an aid in understanding the mathematical formu- fied and the system locates at some point y on Fig. 7 lation, let us consider qualitatively how the system acts for several different values of vd. For sim- plicity, let us first assume that both probes are equal in plasma potential from point to point exist. Further- more we assume that v a has no effect on the ion current system.This is very closely approximated in a)Va=0(Fig 6a) plasma and will ride at the same floating potential. The SCoPE FOR R OScTIAL Each probe will collect zero net current from the FIG. 5. Ba ble probe cir Further increase in the negative value of Va can cause lo further change in the current distributions because probe No. 1 already collects a sufficient electron current to balance the entire positive ion current flowing to the system. Consequently probe No. 1 remains fixed with respect to the plasma and probe No. 2 goes negative along with V d. We can speak of the latter probe as EP VOLTS being saturated with respect to positive ions as the system moves along the fat porte of Fig. 7. In practice one finds that this fat portion has a slight lope as shown by the dotted portion yx. This slow ncrease is due to an expansion of sheath thickness as the probe goes increasingly negative with respect to the plasma. This will be discussed in greater detail FIG. 4. Probe floating potential f anode potential The symmetry of the system will cause it to reverse the previous results when V d is positive, giving the
MEASUREMENTS IN GAS DISCHARGES ortion ozw or ozw. The flat portion ww or sw corre- sponds to positive ion saturation current to probe No. 1 The total positive ion current to the system is simply the sum of the positive ion currents to both probes and so can be found by adding the magnitudes of the cur rents at y and z, as symbolized by ip and ipr The electron current which flows from the plasma to probe No. 2 is simply the difference between the total space current and the positive ion current to this probe. Thus the electron current ie, to probe No. 2 is given by The value of ie, which corresponds to a voltage V d is illustrated graphically in the same figure FIG. 7. Ve aracteristic of the double where he potential diagram of Fig. 8 yields VI+vo=V2+vd or V1=V2+Va-Ve.(5) (b) Substituting (5)into(4)and rearranging, we obtain In[(2ip/ie-1]=-pVa+In o=In I,(6) where T=(Eip/ie)-1 (A1jo,/Azjo ) et Thus the plot of In r against Va should straight line whose slope is a measure of the d"FAIRLY LAR temperature. This equation is seen to be EGATIVE VOLT form to that used in the SPM except that in(6)one uses r instead of the electron current It is to be noted that the slope of (6) is essentially unaffected by any of the factors included in electron random current densities, difference in plasma potential FIG. 6. Sample potential diagrams of the double probe method. between probes, and contact potentials. For an unam biguous determination of Te, the random current den V. TEMPERATURE DETERMINATION sities should not change with probe current. This is A. Logarithmic Plot Method much more likely to be the case with the DPM than The generalized potential diagram for the system with the SPM since the current drain can be hundreds of Fig. 5 is shown in Fig 8. The potentials Vi and v, of times smaller in the former case. Another important represent the voltages of the surrounding plasmas with difference between the two cases is seen from Eas respect to the corresponding probes. The potential Ve and( 8). We note that the constant term of the latter is represents any small difference in plasma potential free from any restricting dependence on the plasma which may exist between the regions surrounding the potential. Thus we see that the DPM is inherently a probes, plus the total contact potentials acting in the more general method. It can be used during or after the system. The other symbols are defined in the figu discharge and even when the plasma potential varies Since the net current to the system must be zere with time Possible errors in this method require discussion. The ralues of ipi and ip, used were those at which the curve Substituting the equivalents for ie, and i, in terms of of Fig. 7 broke away from the saturated regions Boltzmann relation we obtain of error present. In the first place, there is Σip=A1jiof1+Ajig∈p, (4) tainty in the choice of points y and z. To see how serious
E. 0. JOHNSON AND L. MALTER this can be, trial points for y and s were chosen ex- B. Equivalent Resistance Method tending over a considerable range away from the obvious"break"points. It was found that when this Plotting Eq.(6)involves some laborious computa- as done, the end points of the log plot would deviate tion. It turns out that this can be avoided and Te deter- from the straight line defined by the central points. If mined very quickly from id as. Va plots. Equation(6) one used the slope of the still well-defined straight can central portion of the plot for determining the tem- perature, the values obtained did not vary significantl as the chosen values of ip, and ip, were moved around. Taking the derivative of ie, with respect to va and The reason for this is quite simple. Any change in the evaluating at va=0, one obtains selected value of the i ' s, introduces a change (in the [die, avail (10) same direction)of the estimated value of ier It turns out that(except near y and s)the significant quantity Solving for T. and substituting for dva/dig, its prac- Zip/iex)-1] is inappreciably affected. Thus, one can tical equivalent, dv a/did, one obtains safely say that this method is insensitive to small varia tions in the choice of ip, and ipr =11,600 e, a second possible source of error lies in the fact, that (1+o)判 dia jva=0 ren in the region between y and z of Fig. 7, the ion current of each of the probes varies slightly due to From (6)we can obtain [(Σin/ia)-1]p For convenience we introduce the factor g such that /(1+a). (12) This factor G, which can be obtained directly fre current-voltage characteristic, obviates the need for calculating o from(8). Substitution of(12) into(11) yields Te=11, 600(G-G)[2ipdva/dialy =11,600(G-G)R,Σi(13) where I- Electron space current in the plasma Ro=Ldva/dia]vd=o Va=se tex to plasma potential probe No. 2 The factor Ro is denoted as the equivalent resistance This gives the method its name. The simple relation q.(13)provides a rapid and convenient means of mall changes in sheath thickness. 4 Strictl obtaining the electron temperature directly from the one should compute the positive ion current Vaid characteristic. The use of both( 8)and(13)will the probes over the range ys and use the sum be illustrated in the sets of experimental data which in the log plot. This appears quite unnecessary and is, will follow in fact, inconsequential, for the following reasons Since, in the equivalent resistance method one make (1)Unless the slope of the saturated ion current regions use of the slope of the V a-id curve at only one point is considerable, the value of >ip can hardly change viz. where Vd=0, it would be desirable to use the value appreciably over the range ys. In any case any actual of Zip corresponding to this point when computing t. and accepted change must be accompanied by a cor- from Eq.(13). A simple analysis (presented in the responding value for ie. In that case for the reason Appendix) indicates that one is usually justified in assuming that the rate of change of ion current alor (>ip/is)will be small, since both Ei, and ie change in the sloping saturated portions of Fig. 7 is maintained the same direction in roughly the same ratio, (2)As in the regions between the knees. Manipulation of the one moves through the region between the knees, i.e., basic equations of the DPM then yields the necessary between y and s, a change in ip, tends to be compensated relation between probe-space potential and va which of Eip remains approximately constant. region between the knees. It is found that for all prac- tical purposes, if the Va-id characteristic is reasonably afe in extending the lines x'y and estion of sheath thick wz(see Fig. 7)0.8 of the way into the line through id=0 and then horizontally the rest of the way. The
MEASUREMENTS IN GAS DISCHARGES T. The result is -OLT METER T=1160va-V4 (16) In the case of the intercept method it is found tI nless vd and vd'' are chosen too close to the knees TERENTLAL he value of T. determined is in excellent agreement TYPE 52。 SCILLOSCO with that obtained by the other methods. In summary we can say that any error in the choice of Zip tends to be compensated with the result that the error registered in T. will usually be 5 percent or less. It is obvious that the most certain values of T, are obtained in the cases where the flat portion slopes are a COLLECTING AREA minimum. Comparison of corresponding temperatures IM ARGON FILLING obtained with different sets of probes, each set having different values of slopes in the flat portions, indicates 日) EXPERIMENTAL TUBE that the 5 percent estimate is a reasonable one This is ARC CURRENT DURING DISCHARGE also borne out by the close correspondence of tem peratures as obtained from,(1)the logarithmic plot method based on Eip evaluated at the knees, and(2)the C)OPERATING CONDITIONS other two methods where 2ip is computed from the FIG 9. Tube and circuit for double probe studies of interpolated values of ip, and i It will be noticed that the mathematical treatment of the dPm is based on the potential diagram of Fig. 8 choice of this value is explained in the Appendix. From which represents an ideal case in which uniform electron these extended curves one can obtain the value of Eip densities and probe-plasma potentials exist along the corresponding to Va=0. (An illustration of this pro- probe surfaces. Such an ideal situation is not achieved edure is given in Fig. 11)Experience has shown that in practice. However, it can be shown in a perfectly if one fails to make this correction for Si, when em- general and rigorous manner that non-uniform electron ploying the equivalent resistance method, but instead densities and probe-plasma potentials introduce no make of the positive ion currents corresponding to errors into the temperature measurements, Such non- points y and z of Fig. 7, then the values for T. come out uniformities, even when quite large, only introduce to be too large, but virtually never by more than 5 percent. Thus for rapid and approximate temperature determinations one can simply make use of ip, and i as determined from Fig. 7. C. The Intercept Method In some cases the equivalent resistance method ot be used owing to the fact that when vd=0, one is operating in a region of positive ion saturation to one of the probes(i.e, in region xy or ww of Fig. 7). In that case another rapid method is possible which does not require the laborious computations of the logarithmic FIG 10. Double nt decay curves be of Fig 9 plot method small corrections in the value of o as determined by By simple algebra Eq(6)can be transformed into: Eq (7). This deduction is consistent with the observa- ion that the value of o determined by Eq. (7)is usual (15) in close agreement with its value determined from either the log plot or the factor G. In the case of probes used in tubes with oxide Now let Va be the value of Va which corresponds to cathodes it has been found at times the deposition of i/ieD, and let Vd be the value of Vd which cor- barium onto the glass probe insulation(with subsequent responds to Eip/ie= F. D and F are chosen arbitrarily. leakage)causes the flat portions to have considerably use of the fact that p =e/kTe, one can solve readily for the probe leads seems to burn out the g Tesla coil to Then, substitution of these values in (15)and making increased slopes. a short application of a
E.O. JOHNSON AND L. MALTER 142 (4004S AFTER DISCHARGE 15 1600av WHERE AV, IS THE VOLTAGE CHANGE IN THE RATIO 2 73: 1 FIG. 11. Double probe current-voltage characteristic. DIFFERENTIAL LTs) nating the leakage. Interestingly enough, the DPM FIG. 12. Double probe temperature determination plot seems to be sufficiently insensitive to error that the following the interruption of the discharge are plotted thods outlined above) within a few percent The values of v a are corrected for the voltage drops with those taken after the leakage has been eliminated. in the resistor R. The values selected for the computa- VI ILLUSTRATIONS OF THE USE OF THE DPM tion of ip are indicated in the same figure. The plot of the function [(2ip -1)/ie, against V a is presented in The presentation of the data which follow is mainly Fig. 12. The slope yields an electron temperature of intended for the purpose of illustrating the use of the 950K. The temperature was also computed by means DPM and not as a detailed study of any particular of Eq(13)(the equivalent resistance method). The phenomena which take place in the afterglow period. factor G is found directly from Fig. 11 and is The tests were carried out in the cylindrical triode shown in Fig. 9(B). Two sets of double probes were G=[in2/ip]va=0=112/243=0.463 employed, one in the cathode-grid region and the other The value of 0. 142 volts per ua for Ro is computed in the anode -grid region. The measuring set-up is shown from the slope of the characteristic of Fig. 6 at Va=0 in Fig. 9(A).(Only the cathode-grid probes are illus- The extensions of the fat portions of the characteristic trated.) give the value of ip at va=0, as (1.15+1.28)or The tube(containing Ar at 1 mm pressure)was fired 2.43 wamp. Substituting the above values into(13) by the simultaneous application of 8 usec. pulses to get; mp grid and anode. The probe current was determined fro T=11600(0.249)(243)(0.142)=1000°K deflections on a Tektronix No. 512 scope which has a This agrees quite well with the 950%K determined from the semilog presented in Fig Various cases illustrating the The value of o computed from Eq.(12)and the applicability of the DPM are presented below above value of G is 1.16. This agrees very well with the Case 1. Typical probe current decay curves for various value of 1.15 determined by insp ralues of Va are shown in Fig. 10. From these curves the point where Vd=0. We can ompute o Irom he current voltage characteristic can be obtained for any time. As an illustrative case, the data for 400 use The reason for this choice is discussed in the Appendix
MEASUREMENTS IN GAS DISCHARGES VII. DISCUSSION OF EXPERIMENTAL RESULTS njo/yi3a-oec=(1)(1.28/1.12)(1.1)=1.24 A serious question arises regarding whether or not temperature determinations have a basis in reality. The We assume that jo /jo, ip/ip, on the basis of the concept of an electron temperature is permissible only charge neutrality of the plasma. eVe=1.1 since if the electrons have a Maxwellian distribution. The V0.01andφ=11.6(see 1). This value of a method here described samples only a small fraction of agrees reasonably well with the other two the electrons present, the electrons collected by the Case 2. As another illustration of the use of Eq. probes being only those which have velocities sufficient (8)and(13), for a case wherein the system is more to overcome the ever present retarding fields at the dissymmetrical than in the previous case, we consider probes. The range of electrons sampled can be extended set of data taken from a simple diode filled with 250u by making tubes with probes of dissimilar of argon. The probes in this tube were equal in area and It is of interest to determine what fraction of the identical in size with those of the preceding case. electrons are sampled in one of the cases studied above The ar current was 200 ma and all data were col- Consider Case 1, whose results are presented in Figs. 11 lected at 100 usec. after cessation of the discharge. The and 12. An approximate expression for floating potential voltage characteristic is plotted in Fig. 13. V,(with respect to space al) is given by The values of the various factors are found from Fig V,=-(kTe/2e)In(Tem/TpM) and ar hen for the case of Figs. 11 and 12, T=950.K.F ∑ip=10.7a,[Eina=0=9.95a, M we use the mass of the argon atom. We assume that R=36.700ohms.G=0.401 is the gas temperature ut350°Kfor Then this case. Then from Eq.(17)V,=0.50 volts. Since ipraipy the probes must be close to V, when id=0 T=11,600(0.240)(995)(0.0367)=1015°K Then from the Boltzmann relation, the ratio of electron The log plot, show Fig. 14 yields a concurrent value of 1055K. It is interesting to note that the values of o again check, their values being(a)1.55 from log plot, (6)1.50 from G, (c)1.51 from Eq. (7) BEH4437-24 (10o4S AFTER DISCHARGE (DDIFFERENTIAL PROBE OLTAGE VOLTS T=目6ooo·1s5然 DIFFERENTIA L, B. Loeb, Fundamental Processes Gases ohn Wiley and Sons, Inc, New York, 1939), first FIG. 13. Double probe current-voltage characteristic p.242
E.O.JOHNSON AND L. MALTER This is not the case for the SPM. The probe or elec- trode potentials for this case can affect the plasma Ya tanh X potential in such a way as to alter the quantities being measured. If a hot cathode is present, then the use of a positive probe may give rise to oscillations during the afterglow period. This is an undesirable state of affairs the measuring device affects the quantity it is designed to measure. If a hot cathode or other copious electron source is not t(as in the case of a co 30 cathode discharge, or for the isolated anode-grid region plasma of case 3 above), then it is possible to use a single probe combined with the other electrodes as a IX. ANALYTICAL EXPRESSIONS FOR id aNd i Let us consider a symmetrical double probe system in which:(1)the probes are identical, (2) the randor current density at both probes is equal, (3)the contact FIG. 15. Ideal double probe current-voltage characteristic mf between the probes is potential near both probes is the same. Then from Eq current density at the probe to the random current density in the plasma near the probe(when the probe tential is v jd/0=e(mkT)=e-(1.619)0=0.0021 From Fig. 11 we see that the measurements extend to points where the current to either probe becomes about double the above value. Thus the maximum value assumed by ja/jo in this case is about 0.005. Thus less Y than the top 1 percent of the electrons are sampled in this case In other tubes than those described here, up to 5 ercent of the electrons were sampled by means of probes with area ratios of 25. In these cases, too, the results always corresponded to Maxwellian distribu tions. It may obviously represent a dangerous extra polation to conclude from the properties of this small Y:In IR sample that the electron velocity distribution is com pletely Maxwellian VIII. FURTHER CONSIDERATIONS OF THE SPM AND THE DPM It may appear that the single probe can be employe i gthe pfasbioas of tt doubl e perot doubles dering the Fic. 10. Theoretical and ie prime mel hoset on arent plot o and the remainder of the system as constituting the other probe of the pair. This is actually possible enly in (8)o=1. Then from Eq(6) ertain special cases. From the earlier description of the DPM we see For this symmetrical case, ip" Ip:=Ip. Then: Sip=2t py potential of the probes)the space potential in the where ip is the positive ion current to either probe plasma is unaffected by changes in va. The plasma is Making use of this relation and the fact that ie, ia +ip, dominant in determining its own potential as well as we can transform Eq.(18)into that of the floating probes. The potential of the latter id/ip=tanh(φVd/2). is determined by that of the plasma and the condition A plot of this function is given in Fig. 15. It is seen to of equality of electron and ion currents to the probe have just the shape of the observed data. ( See Figs. 11 and 13
MEASUREMENTS IN GAS DISCHARGES quation(19)above can be written as Maxwellian distribution. Then exp(ivA)/cosh(lov a). where j, is the random ion current density, but p)=iova-ln cosh (pv d).(20) A plot of Eq(20)is given in Fig. 16. The values of where ip is positive ion current to In(iea/ip) were computed for a set of experimental data sheath area and the points plotted on Fig. 16. The agreement Estimates made by the method of Langmuir and between theory and experiment is agreeably good M1ott-Smith! indicate that at the points y and a of Reifman and Dow have plotted the quantity In(ie/ip) Fig. 4, the positive ion current to the probe is space- for measurements in the ionosphere. Their double probe charge limited but that the sheath area may be appre consisted of the nose and a portion of the body of a ciably larger than the probe area rocket. From their curve they conclude that the ob- Making the substitution Cp=1.87X10-(Tp/M) we served electron distribution is not Maxwellian. Since obtain the shape of their observed curve is similar to ours, it is np=(1.34×10/A)ip(M/Tp) (24) of the In(ie/ip) should not be altered qualitatively even Let us apply this to the case of Fig. 11. For that case ,=1.35×106 amp, T X. DETERMINATION OF ELECTRON AND ION =350°K.Then DENSITIES AND OF WALL POTENTIAL p=(8.0×10/4)ions/cm3 Neither the SPM nor DPM are suited for the deter- To determine 4, we make use of the space-charge- mination of the electron density ne in decaying plasmas. limited current equation for cylindrical diodes This arises from the fact that the plasma potentia ip=14.66×10(m/M)(Lv/r6), Gi. e, in a positive direction) in potential. As a con- where v is the difference in potential between the sequence, ent to the probe unless 35 SPM it is the saturated electron 82=162V current (corresponding to the bend in the current To obtain V, we recall first that when the probe is at voltage characteristic) which is used to cor apute the wall potential(ia=o) The situation is not completely hopeless, however. varies from zero in Fig. 11 to the saturation value of kno p, it is merely necessary ipu, ien changes by a factor of about 2(this is readily to set a value on one unknown, the positive ion tem- obtained from Fig 12). The probe-space potential must perature Tp. This is an exceedingly fortunate situation, then decrease by an amount determined from the doubtedly very close to Te, the gas temperature This Boltzmann relation: nce the value of the decaying un- follows from the fact that even though the electro 0.5=exp[(11,600/950°K)△V], mperature may still (in some cases)be considerably or above gas temperature, the kinetics of the impacts of △V=-(950/11,600×0.093≤-0 the ions with electrons and gas molecules is such that Then it is the temperature of the latter which will dominate V=(0.50-0.06)=0.44volt will be seen, e and n, vary as the square root of t Thus, errors in selecting a value for Tp will hav much smaller effect on the values of n, and n. We set where ca is the average drift velocity of the ions. In the decaying plasma, where the space-charge fields extremely small, CA must be due almost entirely to the outward motion from the plasma into the sheath arising from the random motion of the ions that case CA=tCp, where Cp is the ion velocity averaged over a re figures very close those obtained by the methods here described FIG. 17. Idealized double probe characteristic
