ity creating a magnetic barrier, which inhibits the ionizingelectrons from escaping. Due to the alternating polarity ofthe magnet rows, the magnetic field has substantial strengthonly very close to the walls, so that the main plasma regionis essentially magnetic field free.

A typical Langmuir probe I-V characteristic obtained in

the multidipole device in the University of Iowas under-graduate Advanced Physics Lab is shown in Fig. 5. Thischaracteristic was taken with a 6 mm diameter planar diskprobe in an argon plasma at a pressure of 0.5 mTorr. The topcurve Fig. 5a is the positive probe current due to electroncollection. As discussed in Sec. II see Fig. 4, the electroncurrent continues to increase slightly with increasing voltageabove the plasma potential due to the sheath expansion ef-fect. The determination of the electron saturation current andplasma potential is facilitated by replotting the current on asemilog scale as shown in Fig. 5b. The break point occursat VP4 V, with Ies= 1005 mA. The slope of the down-ward portion of the line on the semilog plot gives Te1.5 eV. The negative ion current is shown on an ex-panded scale in Fig. 5c. Again we see the sheath expansioneffect as the negative probe voltage increases. The ion satu-ration current is estimated by extrapolating the linear portionof the ion current to the plasma potential, where IiVP= 0.850.05 mA. The floating potential is also found fromFig. 5c as VfI=05 V. The ion and electron densitiescan now be calculated using Eqs. 2 and 4 with Te=1.5 eV. We find that ni= 8.30.51016 m3, and ne= 5.50.551016 m3. Even taking into account the uncer-tainties involved in measuring the saturation currents fromthe plots, there remains a 25% difference between theplasma density obtained from the ion and electron currents.This difference is a typical occurrence with Langmuir probesmeasurements. In a magnetized plasma, the discrepancy inthe densities obtained from the electron and ion saturation

Fig. 4. Model Langmuir probe I-V characteristic including the effect ofsheath expansion, computed with VP=4 V, Te=4 eV, Ti=0.1 eV, andIes / Iis=200. a Total current. The dotted curve depicts the rounding of theknee due to plasma noise or averaging effects. b log IVB versus VB. Theintersection of the horizontal and vertical dotted lines occurs at the coordi-nates VP , Ies. The electron temperature is obtained from the slope of thelinear part of the downward sloping portion of this curve. c Expanded viewof the ion current. The sloping dotted line is a linear fit to the ion current.The ion saturation current is found by extrapolating this line to the plasmapotential.

Fig. 5. Langmuir probe I-V characteristic obtained in a multidipole plasmain argon at a pressure of 0.5 mTorr. a Electron current. b log IVB versusVB. The semilog plot of the electron current provides a clear demarcation ofthe plasma potential and electron saturation current. Te is found from theslope of the exponentially decreasing portion. c Expanded scale view ofthe ion current used to find Iis.

1082 1082Am. J. Phys., Vol. 75, No. 12, December 2007 Robert L. Merlino

currents is considerably larger, but expected. The gyroradiusof the electrons is typically much smaller than that of theions so that the collection of electrons is affected more thanthe collection of ions. In that case, measurements of theplasma density using the ion saturation current are more re-liable.

It is interesting to calculate the fraction of the neutral ar-gon atoms that are actually ionized in the plasma using themeasured value of the plasma density. This fraction is knownas the percent ionization or ionization fraction. The densityof the neutral argon atoms is na= P /kTg, where Tg is thetemperature of the neutral gas, and P is the neutral gas pres-sure. For P=5104 Torr and Tgas300 K, na=1.651019 m3. With ni=81016 m3, we obtain ni /na=0.005.Thus, only 0.5% of the neutral atoms are ionized.

The fact that the neutral density is roughly 1000 times theplasma density might lead one to wonder about the role ofthese neutral atoms on the plasma and probe measurements.To access the possible effects of collisions of the ions andelectrons with neutrals, we need to estimate a few typicalcollision mean free paths, = na1, where is the crosssection22 for the particular process considered. For ioniza-tion, ionz81020 m2 for 50 eV electrons on argon, soionz75 cm. Thus ionz is on the order of the dimensions ofa typical laboratory plasma device. The relatively long ion-ization mean free path explains, in part, the relatively lowvalue of the ionization percentageelectrons that are ener-getically capable of ionizing atoms are more likely to make itto the wall before ionizing an atom.

The purpose of the magnets on the walls of the multidi-pole device is to reflect the ionizing electrons back into theplasma, thus increasing their chances of having an ionizingcollision. Electrons can also make elastic collisions with neu-tral atoms; a typical cross section in this case is en1020 m2, giving en6 m. For collisions between theions and neutral atoms, the most important process to con-sider is charge exchange, Ar++ArAr+Ar+, in which anargon ion exchanges an electron with an argon atom, result-ing in the production of very slow argon ions and argonatoms with an energy practically equal to the initial energy ofthe ions. The cross section for this process is in51019 m2, giving in12 cm.

We note that for all of the processes considered the meanfree paths are much greater than the probe size and theshielding distance or sheath size, so that even though theneutral density far exceeds the plasma density, the neutralgas atoms produce negligible effects on the probe measure-ments.

B. A positive ion/negative ion plasma in a Q machineFigure 3 is an example of a Langmuir probe I-V charac-

teristic in a plasma in which the positive and negative par-ticles have the same mass. This example might appear to beexotic, but it is not difficult to produce a plasma havingalmost equal numbers of positive and negative ions of com-parable mass. We have produced positive ion/negative ionplasmas also known as electronegative plasmas in a devicecalled a Q machine.23 In a Q machine the plasma is producedby surface ionization, an effect discovered by Langmuir andKingdon in 1923.24 They found that cesium atoms that comeinto contact with a tungsten filament heated to 1200 Kemerge as cesium ions. The reason is that the ionizing poten-tial of cesium is 3.89 eV, and the work function for tungsten

is 4.52 eV. Surface ionization is exploited in a Q machine23by directing an atomic beam of cesium or potassium atomsonto a hot 2000 K tungsten or tantalum plate, usuallyseveral centimeters in diameter. Both positive ions and ther-mionic electrons emerge from the plate forming a nearlyfully ionized plasma that is confined by a strong0.10.5 T longitudinal magnetic field. The relativelygood thermal contact between the plasma and the hot plateresults in a plasma in which both the electrons and positiveions are at roughly the plate temperature, typically 0.2 eV.The Q machine has been used mainly for studying the basicproperties of magnetized plasmas, and in particular plasmawaves. The Q designation refers to the expectation that athermally produced plasma would be quiescent, that is, rela-tively free of low frequency plasma instabilities.

Negative ions are readily formed in a Q machine plasmaby leaking into the vacuum chamber sulfur hexafluoride SF6at a pressure 105104 Torr. Electrons attach to SF6forming SF6

negative ions.25 The cross section for electronattachment to SF6 is energy dependent and peaks in the en-ergy range that coincides closely with that of the Q machineelectrons. Under these circumstances it is possible to produceplasmas in which the ratio of electron density to positive iondensity is ne /n+103. A Langmuir probe I-V characteristicobtained in such a K+ /SF6

m

/m+=3.7 is shown in Fig. 6.Note that the negative ion positive current and positive ionnegative current saturation currents are comparable. Withsuch a characteristic the plasma potential is most easilydetermined as the voltage at which the first derivative ofthe characteristic is a maximum. In this case we see thatVP1 V. The characteristic is roughly symmetric about I=0, with a floating potential VfVP, a result that is to beexpected in a plasma with n+n and nen+. When thenegative ion and positive ion densities are comparable, itmay even be possible to extract both the negative and posi-tive ion temperatures from the Langmuir characteristic.

Fig. 6. Langmuir probe I-V characteristic obtained in a singly ionized po-tassium plasma produced in a Q machine. SF6 gas was introduced into theplasma to form a negative ion plasma by electron attachment. A substantialfraction of the electrons became attached to the heavy SF6 molecules result-ing in a nearly symmetric probe characteristic with I+s Is. The lower curveis the derivative of the probe current, dI /dVB. The plasma potential is thevalue of the VB for which dI /dVB is a maximum.

1083 1083Am. J. Phys., Vol. 75, No. 12, December 2007 Robert L. Merlino

IV. COMMENTS

A Langmuir probe I-V characteristic becomes less confus-ing once we are able to see the individual current contribu-tions as well as the total probe current. The procedure forconstructing an I-V characteristic given an appropriate set ofplasma input parameters has been presented. A MAPLE pro-gram that creates the I-V characteristic is available onEPAPS16 and is also available on the authors website.

The inclusion of plasma physics experiments in upperlevel advanced laboratory courses for physics majors canprovide students with much exposure to many important top-ics and methods in experimental physics including basicvacuum techniques, vacuum measurement methods, solder-ing, spot welding, brazing, electronic circuit design and fab-rication, data acquisition methods, curve fitting techniques,and instrument design and construction building a Langmuirprobe. Students also experience using basic concepts in thekinetic theory of gases.

Plasma physics experiments also provide ideal researchtopics for undergraduate thesis projects. For instructors con-templating the inclusion of plasma experiments in advancedlaboratory courses, my suggestion is to start with the basicmultidipole plasma.1921 This device is relatively simple andinexpensive, with the most costly component being thevacuum pumping system. If money is not a concern, it ispossible to purchase fully operational vacuum systems thatare easily adaptable for plasma production. Although it isnow possible to purchase off the shelf Langmuir probe sys-tems, complete with probe and associated electronics, theexperience of constructing probes from scratch is a valuableone that should not be avoided. Building a probe is often thefirst instance in which students are required to use theirhands to create an experimental instrument. Far too oftenstudents are left with the impression that everything neededto perform a measurement can be found at manufacturersweb sites.

ACKNOWLEDGMENT

This work was supported by the U.S. Department ofEnergy Grant DE-FG02-04ER54795.

APPENDIX: SUGGESTED PROBLEMSFOR FURTHER STUDY

The following two problems are intended to extend thebasic probe theory to include some other important effectsoften encountered in using Langmuir probes in realistic plas-mas.

Problem 1. It is not uncommon to find in low pressureplasma discharges that there are two distinct Maxwellian dis-tributions of electronsa cold and hot distribution with tem-peratures Tec and Teh, respectively. Extend the analysis ofSec. II to include a two-temperature electron distribution. Inthis case the electron probe current is written as IeVB= IecVB+ IehVB. Take the respective densities of the coldand hot components to be nec and neh with ne=nec+neh. Tosimplify the analysis, introduce the parameter fehneh /ne asthe fraction of hot electrons, so that nec /ne=1 feh. An inter-esting issue arises as to what value of Te to use in calculatingthe Bohm ion current. It was shown26 that the appropriate Teis the harmonic average of Tec and Teh:

1Te

= necne

1Tec + nehne 1Teh . A1

After you have produced a Langmuir I-V plot, replot theelectron current as a semilog plot to see more clearly theeffect of the two-temperature electron distribution.

Problem 2. In plasmas produced in hot-filament dis-charges, the effect of the ionizing primary electrons on theprobe I-V trace can be observed, particularly at neutral pres-sures below 104 Torr. Extend the probe analysis to in-clude the presence of these energetic primary electrons,which can be modeled as an isotropic monoenergetic distri-bution. Express the total electron current as IetVB= IeVB+ IepVB, where IeVB is the contribution from the bulk elec-trons, and IepVB is the primary electron contribution, whichfor an isotropic monoenergetic distribution is3

Iep

=Iep

* 14

enepvepAprobe, VB VP,

Iep* 1 2eVP VB

mevep2 , Vp mevep22e VB VP,

0, VB VP mevep22e , A2

where nep is the density of primary electrons, and vep=2Ep /me is the speed of the primary electrons with energyEp. To produce an I-V plot, assume that the primary electronsare accelerated through a potential drop 5060 V, and thedensity is in the range of 0.0010.1ne.

aElectronic mail: robert-merlino@uiowa.edu1It is common in plasma physics to give temperatures in equivalent energyunits eV. For example, we say that Te=2 eV, which means we are reallygiving kTe converted to electron volts. The actual temperature corre-sponding to 1 eV is 11,600 K.

2I. Langmuir and H. Mott-Smith, The theory of collectors in gaseousdischarges, Phys. Rev. 28, 727763 1926.

3I suggest that Langmuir probe novices start by reading Noah Hershkow-itzs article, How Langmuir probes work, in Plasma Diagnostics, Dis-charge Parameters and Chemistry, edited by O. Auciello and D. L.Flamm Academic, Boston, 1989, Vol. 1, Chap. 3.

4B. E. Cherrington, The use of Langmuir probes for plasma diagnostics:A review, Plasma Chem. Plasma Process. 2, 113140 1982.

5F. F. Chen, Electric Probes, in Plasma Diagnostic Techniques, edited byR. H. Huddlestone and S. L. Leonard Academic, New York, 1965,Chap. 4. A concise summary of Langmuir probe techniques by F. F. Chen,Lecture notes on Langmuir probe diagnostics is available atwww.ee.ucla.edu/~ffchen/Publs/Chen210R.pdf.

6L. Schott, Electrical probes, in Plasma Diagnostics, edited by W.Lochte-Holtgreven North-Holland, Amsterdam, 1968, Chap. 11.

7J. D. Swift and M. J. R. Schwar, Electrical Probes for Plasma Diagnos-tics American Elsevier, New York, 1969.

8I. H. Hutchinson, Principles of Plasma Diagnostics, 2nd ed. CambridgeU.P., Cambridge, 2002, Chap. 3.

9J. G. Laframboise, Theory of spherical and cylindrical Langmuir probesin a collisionless, Maxwellian plasma, Univ. Toronto Aerospace StudiesReport No. 11 1966.

10Reference 3, p. 118.11 It is common in discharge plasmas to have TiTe due to the fact that the

ions are created from neutral atoms at room temperature, while the elec-trons are considerable hotter by a factor of about 100 because they must

1084 1084Am. J. Phys., Vol. 75, No. 12, December 2007 Robert L. Merlino

be energized to ionization energies to maintain the discharge. Energytransfer between the massive ions and light electrons is inefficient, so theions remain relatively cold.

12Reference 3, p. 125; see also the review article by K.-U. Riemann, TheBohn sheath criterion and sheath formation, J. Phys. D 24, 4935181991, and the recent article by G. D. Severn, A note on the plasmasheath and the Bohm criterion, Am. J. Phys. 75, 9294 2007.

13F. F. Chen, Introduction to Plasma Physics and Controlled Fusion, 2nded. Plenum, New York, 1984, Vol. 1, p. 290.

14Reference 13, p. 8.15M. A. Lieberman and A. J. Lichtenberg, Principles of Plasma Discharges

and Materials Processing, 2nd ed. Wiley, New York, 2005, Chap. 6.16See EPAPS Document No. E-AJPIAS-75-009710 for a MAPLE program

that can be used to produce Langmuir I-V curves. This document can bereached through a direct link in the online articles HTML referencesection or via the EPAPS homepage http://www.aip.org/pubservs/epaps.html. This program can also be accessed fromwww.physics.uiowa.edu/~rmerlino/.

17See, for example, T. Sunn Pedersen, A. H. Boozer, W. Dorland, J. P.Kremer, and R. Schmitt, Prospects for the creation of positron-electronplasmas in a non-neutral stellarator, J. Phys. B 36, 10291039 2003.

18D. Lee and N. Hershkowitz, Ion collection by planar Langmuir probes:

Sheridans model and its verification, Phys. Plasmas 14, 033507-142007.

19R. Limpaecher and K. R. MacKenzie, Magnetic multipole confinementof large uniform collisionless quiescent plasmas, Rev. Sci. Instrum. 44,726731 1973.

20A. Lang and N. Hershkowitz, Multidipole plasma density, J. Appl.Phys. 49, 47074710 1978.

21R. A. Bosch and R. L. Merlino, Multidipole confinement of argon andpotassium plasmas, Rev. Sci. Instrum. 57, 29402950 1986.

22A good source for cross sections relevant to plasma physics is S. C.Brown, Basic Data of Plasma Physics, 2nd ed. MIT Press, Cambridge,MA, 1967.

23R. W. Motley, Q Machines Academic, New York, 1975.24I. Langmuir and K. H. Kingdon, Thermionic effects caused by alkali

vapors in vacuum tubes, Science 57, 5860 1923.25Bin Song, D. Suszcynsky, N. DAngelo, and R. L. Merlino, Electrostatic

ion-cyclotron waves in a plasma with negative ions, Phys. Fluids B 1,23162318 1989.

26S. B. Song, C. S. Chang, and Duk-In Choi, Effect of two-temperatureelectron distribution on the Bohm sheath criterion, Phys. Rev. E 55,12131216 1997.

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