Laser spectroscopy in low-pressure sodium-neon discharges
Citation for published version (APA):
Cornelissen, H. J. (1986). Laser spectroscopy in low-pressure sodium-neon discharges. Technische Hogeschool Eindhoven. https://doi.org/10.6100/IR242355
DOI:
10.6100/IR242355
Document status and date: Published: 01/01/1986 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
LASER SPECTROSCOPY IN LOW-PRESSURE
SODIUM-NEON DISCHARGES
•
LASER SPECTROSCOPY IN LOW-PRESSURE
SODIUM-NEON DISCHARGES
LASER SPECTROSCOPY IN LOW-PRESSURE
SODIUM-NEON DISCHARGES
PROEFSCHRIFT
ter verkrijging van de graad van doctor in de technische wetenschappen aan de Technische Hogeschool Eindhoven, op gezag van de rector magnificus, prof. dr. F.N. Hooge, voor een commissie aangewezen door
het college van decanen in het openbaar te verdedigen op dinsdag 18 februari 1986 te 16.00 uur
door
HUGO JOHAN CORNELISSEN geboren te Geldrop
Dit proefschrift is goedgekeurd door de promotoren Prof. Dr. Ir. D.C. Schram
en
Prof. Dr. Ir. H.L. Hagedoorn Copromotor Dr. Q.H.F. Vrehen
The work described in this thesis was performed in the Gaseous electronics group of the Philips Research Laboratories in Eindhoven
Aan Marina Aan mijn ouders
CONTENTS
1. Introduetion 1
1. The low-pressure Na-Ne discharge 1
2. The problem 4
3. Laser absorption experiment 6
4. Doppler-free two-photon absorption experiment 7
5. Remarks to guide the reader 8
2. Measurements of the diffusion coefficient of sodium in
neon at 530 K 10
1. Introduetion 10
2. Experiments 11
1. Principle of the metbod 11
2. The discharge 11
3. Experimental set-up 12
3. Experimental results 12
1. Interpretation 12
2. Results 14
4. Comparison with molecular theory 16
5. Summary 19
Appendix 2 21
3. Langmuir probe experiments 24
1. Introduetion 24
2. Principle of the metbod 25
3. Experimental set-up 26
4. Results 27
5. Electron energy distribution function 29
6. Discussion and conclusions 32
4. Electron density in low-pressure Na-Ne discharges deduced
from laser absorption experiments 34
1. Introduetion 34 2. Electron mobility 36 3. Radial profiles 38 4. Experiments 41 1. Laser absorption 41 2. Langmuir probes 42 5. Results 43
6. Coaxial bar model 49
7. Discussion and conclusions 50
5. Measurement of the electron density in a low-pressure Na-Ne
discharge using Doppler-free two-photon spectroscopy 55
1. Introduetion 55
2. Experimental set-up. 57
3. Stark parameters 61
4. Results 65
5. Discussion and conclusions 72
Appendix 5 77 6. Concluding remuks 80 Summary 82 Samenvatting 85 Dankwoord 88 Levensloop 89
Parts ofthe investigations described in this thesis have beenpresentedat the following conferences:
Chapter 2:
Cornelissen H J 1983
XVI ICPIG Düsseldorf p52-3 Chapter 2:
Cornelissen H J 1984
37th GEC Boutder A-5 Chapter 4:
Cornelissen H J and Merks-Eppingbroek H J H 1984
37th GEC Boutder GD-3 Chapter 4:
Cornelissen H J and Merks-Eppingbroek H J H 1985
XVII ICPIG Budapest p1067-9 Chapter 5:
Cornelissen H J and Burgmans AL J 1982
35th GEC Dallas MB-1
The text of Chapter 2 is literally taken from: Cornelissen H J 1985
J. Phys. B: At. Mol. Phys. 18 3445-55
Chapter 4 is the slightly typographically modified text of the paper: Cornelissen H J and Merks-Eppingbroek H J H 1985
J. Appl. Phys. accepted for publication
A first account of the work described in Chapter 5 has been publisbed in: Cornelissen H J and Burgmans A L J 1982
Opt. Comm. 41187-91
Cover: Laser induced fluorescence on a spontaneous emission back-ground (artist impression). See also Fig. 5.1.
1. Introduetion
t.t. The low-pressure Na-Ne discharge
The low-pressure Na-Ne discharge finds wide application in lamps for road and safety lighting (Waymouth 1971, Elenbaas 1972, Denneman 1981). A schematic representation of the low-pressure Na lamp as it is presently constructed is given in Fig. 1.1.
0
lnfrared reflecting coating
Electrades Discharge tube
Outer bulb (evacuated}
Dimples fitled with sodium
Fig.1.1 Schematic drawing of the low-pressure Na lamp (66W version).
Ll...l....LLJ
0 50mm
The discharge is contained in a glass tube of circular cross section with electrades at both ends. The tube is bent into a U-shape. A second, evacuated tube surrounds the folded discharge tube and is provided with an infrared reflecting coating to reduce the heat losses and to rnaintaio the wall temperature ofthe discharge at a value around 530 K. In the discharge tube liquid sodium drops reside in regularly spaeed dimples in the glass wall to minimize axial Na density gradients.
At the time of the introduetion of the low-pressure Na lamp in the nineteen-thirties it was the most efficient light souree available. A constant research and development eff ort over the years has paid off to rnaintaio this leading position as is illustrated in Fig. 1.2. In this figure the luminous efficacy, defined as the ratio ofthe visible light output over theelectric input power, expressed in lm/W, of different light sourees are compared as a function of time (after Denneman 1981).
>. u 0 .!:::!
--
Q) 200 Low-pressur sodium VI 100 :;:J 0 c .Ë :;:J tungsten ~ha logen ineendescent 0~--L---~---L--~----~--~ 1875 1900 1925 1950 1975 2000----year
Fig.1.2 The development of the luminous efficacy of several types of lamps is given as a function of time (schematically).
lm/W, and was improved to the presently reported values of 225 lm/W (De Groot et al. 1985). There are still considerable possibilities for further improvements since, if all the input power could he converted to light emission at the 589.0/589.6 nm lines, the lumineus efficacy amounts to 520 lm/W.
In Table 1.1 somecharacteristic dimensions and parametervalues oftypical examples of a low-pressure Na lamp, a low-pressure Hg lamp and a high-pressure Na lamp are compared. In all systems noble gas is added to decrease the loss of the charged particles to the vessel walls and/ or to facilitate the ignition of the discharge. In the ionization balance of the burning discharges the noble gas hardly plays a role. The high-pressure
system is characterized by the fact that it is in local thermodynamical equilibrium, that is, the kinetic temperatures of all particles are equal. Furthermore, the population of the excited states is according to the Saha-Boltzmann equations (Mitchner and Kruger 1973). In the two low-pressure systems, which are quite similar to each other, the electron temperature is much higher than the temperature of the ions and of the neutral partiel es. Equilibrium relations such as the Boltzmann excited state density distribution ortheSaha equation for the electron density cannot be used. Instead, if one wants to obtain a more than qualitative understanding of the behaviour of the low-pressure discharge a model has to be set up in which the dominant microscopie processes are included. For the low-pres-sure Na discharge such a model, based on the works of Cayless ( 1963) and Waymouth and Bitter (1956), has been developed by Van Tongeren (1975) and was later extended by Vriens et al. (1983). The model includes a
description of elementary processes such as electron impact ionization and excitation, radiative decay, radiation trapping and partiele transport.
low-pressure high-pressure
Na Hg Na
lamp wattage [W] 131 36 400
tube radius [mm] 9.5 12 3.75
current density [A/m2] 2000 800 9 104
power density [W/m3] 2 105 7 104 1 108 wall temperature [K] 530 325 1500 gas temperature at axis [K] 570 335 4000 electron temperature at axis [K] 104 1.4 104 4000
noble gas Ne Ar/Kr Xe
filling pressure [Pa] 730 200 4000
densities in operating lamp
Na [m-3] 6 1019 1.8 1023
Hg [m-3] 2 1020 1.41024
electrans [m-3] 5 1018 41017 11022
noble gas [m-3] 1.8 1023 51022 11024
Table 1.1. Some characteristic dimensions and parameter values for typical examples of a low-pressure Na lamp, a low-pressure Hg lamp, and a high-pressure Na lamp (after Van Vliet and De Groot 1981).
An important phenomenon to take into account in a model for the low-pressure Na-Ne discharge is the strongly non-uniform radial Na distribution. Because of the presence of liquid sodium drops at the wall of the discharge, the sodium vapour pressure near the wall is fixed and is determined by the wall temperature. At the tube axis the sodium density is depleted by radial transport processes. Electroos and ions are produced in the volume of the discharge, while recombination only takes place at the wall. The electroos and ions drift to the wall in an ambipolar field and ground state sodium diffuses back into the discharge. The ambipolar diffusion coefficient is typically a factor of ten to twenty larger than the neutral diffusion coefficient. In a steady state condition, large neutral density gradients are needed to balance the ion flux. Since the neutral density at the wall is fixed this leads to a significant decrease of the Na density at the axis of the discharge.
With increasing discharge current this radial Na depletion increases. At a eertaio current the Na density at the tube axis has decreased to such a value that also the neon gas is excited and ionized. At lower currents the neon acts only as a buffer gas to slow down the loss of the electroos and ions to the wall. The behaviour of the discharge changes drastically if neon ionization and excitation becomes ofimportance (Van Tongeren 1975). In the present study the low-pressure Na-Ne discharge is operated in the low current regime, where Ne ionization and excitation may be neglected.
1.2 The problem
Comparison of the model calculations with experiments using Langmuir probes shows a reasonable agreement (10-20%) for the electron temperatu-re, Te, and for the axial electric field strength, E, as a function of the discharge current (Vriens et al. 1983). For the electron density at the tube
axis, ne(O), the model calculations exceed the probe values by a factor of about 2 (Van Tongeren 1975). However, an inconsistency in the probe measurements for ne(O) is observed which can be statedas follows. In the low-pressure Na-Ne discharge the current I can be expressed as:
R
I = 211: eE
f
ne(r)J.Le(r, Te) r dr,0 (1.1)
where Ris the tube radius, eis the elementary charge and J.Le(r, Te) is the electron mobility, which is a function of the local partiele densities and of
radial position: the tube axis. Ifthe probe-measured n/0) and electric fields are substituted in Eq. 1.1 and a parabolic or Bessel-type radial profile for ne(r) is assumed, the calculated current is approximately a factor of two lower than the measured discharge current. Or, stated alternatively, the electron density as expected from the discharge current, the measured electric field and the calculated electron mobility exceeds the probe measured ne(O) by a factor of two approximately.
This discrepancy may be caused by (i) an error in the calculated electron mobilities or by (ii) an error in the probe-measured ne(O). Other possible effects will be of minor influence. The accuracy of the measurement of the axial electric field is estimated to be better than ten percent. The radial electron density profile does not deviate strongly from the assumed Bessel-type profile; only for discharge currents close to the current where neon excitation and ionization plays a role such deviations become of importance (Van Tongeren 197 5). In the present study the discharge current is always well below this value.
(i) In the calculation of the electron mobility collisions with Ne buffer gas atoms are dominant. But also collisions with the Na ground state atoms and with the ions play a role. In Chapter 4 the results of calculations of the mobility using different mixture rul es (Schirmer 1955, Frost 1961) are given. The results are compared to the more refined Chapman-Enskog calcula-tions for the electron mobility in which also electron-electron interaccalcula-tions are taken into account (Mitchner and Kruger 1973). It is shown that for the present plasma composition the mixture rules are adequate.
(ii) It is widely known in literature that probe experiments suffer from several systematical errors (see the probe review articles referred to in Chapters 3 and 4). Therefore it was conjectured (Van Tongeren 1975) that electron reflection on the probe surface or perturbation of the plasma by the probe might account for the differences between the model calculations and the probe experiments. From comparative studies of probe and microwave techniques (see for instanee Kinderdijk and Van Eek 1972) it bas been found that the probe-measured values are systematically below the values deduced from the microwave experiments. However, the discrepan-cy observed in the microwave studies amounted 'only' toabout 30%, soit would resolve the observed inconsistency in the probe experiment only partly.
To prove the presence of large systematic errors in the probe experiments, this thesis describes two laser spectroscopie experiments that have been set up to yield the electron density in the low-pressure Na-Ne discharge in an independent way, and yield this value without the need to perturb the plasma as is done by the intrusion of a probe.
1.3 Laser absorption experiment
The first of the two laser experiments is described in Chapter 4. The radial Na ground state density distribution, n0(r), is measured with a laser absorption technique. The radial electron density distribution, ne(r), is obtained by combining the results with a partiele balance: it is assumed that the Na ion flux to the wall of the discharge is balanced by the return flux of neutral Na from the wall towards the axis of the tube. The radial profile of the electron density can then be straightforwardly related to the radial profile of the Na ground state atoms (Waszink and Polman 1969):
(1.2)
The absolute value for the electron density depends on the ratio of the neutral diffusion coefficient, D0, and the ambipolar diffusion coefficient,
Da. The value of D0 was not known accurately enough at the temperature of interest (530 K). Therefore it was determined in a separate experiment, which is described in Chapter 2 (Cornelissen 1985).
The profile n0(r) is measured with dye laser absorption on the Na D11ine ( wavelength 589.6 nm ). Since the laser beam is propagating perpendicularly to the tube axis, an Abel inversion is applied. The value of Da in Eq. 1.2 depends on the electron temperature. For Te the probe-measured value is taken, deduced from the probe characteristics as described in Chapter 3. In principle however, it is not necessary to use the probe-measured value for Te, since from the laser absorption experiment the electron temperature can be deduced too. This can be shown as follows. Substituting Eq. 1.2 for ne(r)
into Eq. 1.1, arelation results f or the discharge current as a function of the ground state density profile, the axial electric field and the electron temperature. Since n0(r), E and I are experimentally determined, Te can be solved from the resulting (implicit) equation. The value thus found for Te appears to agree remarkably well with the probe-measured value. This indicates that the electron density as deduced from Eq. 1.2 using the probe-measured electron temperature yields a consistent value for the discharge current if it is substituted in Eq. 1.1 ( whereas the probe-measured value for ne does not).
Moreover, exploiting the spatial flexibility of the laser absorption technique it is observed that the radial Na ground state density profile is disturbed in the vicinity of the probeover length scales of mm's. This indicates that the ionization-recombination equilibrium in the discharge is disturbed by the presences of the probe, and it partly explains the observed discrepancy in
ne(O). In an Appendix to Chapter 4 the spatial extension of a perturbation in the partiele densities as caused by the presence of the probeis estimated. It is shown that not the probe wire with its diameter of 22 ~-tm, but the glass support pin (diameter 1mm) must probably be held responsible for the observed disturbances.
1.4 Doppler-free two-photon absorption experiment
In the second laser spectroscopie experiment, described in Chapter 5, the electron density is determined from the plasma shift (a lso called Stark shift) of the Na 3S-4D transition as measured with Doppler-free two-photon spectroscopy. In a plasma, the atoms that are emitting or absorbing radiation are perturbed by collisions with the charged partiel es. The micro electric field induced in the atoms during the collision process causes a Stark shift and splitting of the spectrallines. As a result of these collisions a shifted and broadened spectralline is observed (Griem 1974).
In low-pressure discharges, with electron densities in the order of 1018-1019
m·3, the expected Stark shift and width is one ortwoorders of magnitude
below the Doppier width. It is therefore essential that a Doppler-free spectroscopie technique be used. Only fairly recently the feasibility ofusing a Doppler-free technique to measure Stark shifts and widths was demon-strated (Weber and Humpert 1981, Cornelissen and Burgmans 1982, Van Veldhuizen 1983). The principle of the Doppler-free spectroscopie metbod as applied in Chapter 5 is that the effect of Doppier broadening for a two-photon absorption line can be eliminated by using two counter-propa-gating laser beams (Grynberg and Cagnac 1977). The (first order) Doppier shifts that are experienced by the thermally moving absorbers have opposite signs for the two laser beams and hence cancel if the two photons are absorbed from different beams.
Special attention bas to be given to the line broadening theory since the existing theories have only been applied to plasmas with high electron densities (ne 1022 m·3
). In such high density plasmas the Debyeshielding
can markedly influence the calculation of the Stark parameters and the results obtained for these plasmas cannot directly be used in the present circumstances. Therefore, a calculation of the Stark parameters at low electron densities is presented.
The electron density as deduced from the measured Stark shift and the calculated shift parameter exceeds the probe-measured value withafactor of more than two and this confirms the results of the laser absorption experiment as described in Chapter 4. Measuring within mm's from the
probe tip reveals an increase in the line shift which possibly points at the existence of large electric field strengtbs close to the probe.
1.5 Remarks to guide the reader
Chapter 2 describes the experiments performed todetermine the diffusion coefficient of Na in Ne at 530 K. A comparison is made between experimental values and values as calculated using different Na-Ne interaction potentials. Chapter 2 is a reprint of a paper publisbed in J. Phys.
B(Cornelissen 1985). In an Appendix some new results are presented which were calculated after the publication of the paper.
In Chapter 3, details ofthe Langmuir probe experiments are given and some results for E, ne(O) and Te are listed. Additionally, the electron energy distribution function is derived from the probe characteristic and is compared with results of a two-electron-group model.
In Chapters 4 and 5, the central problem as sketched in Section 1.2 is tackled. Chapter 4 describes the laser absorption experiments where the electron density is deduced from the measured radial Na ground state density profile.
In Chapter 5, the Doppler-free spectroscopie experiments are described that are performed to deduce the electron density from the Stark shift ofthe Na 3S-4D two-photon transition.
Finally, in Chapter 6 concluding remarks are made.
References
Cayless MA 1963 Brit. J. Appl. Phys. 14 863-9
Cornelissen H J 1985 J. Phys. B: At. Mol. Phys. 18 3445-55 Cornelissen H J and Burgmans AL J 1982 Opt. Comm. 41 187-91
De Groot J J, Jack AG and Coenen H 1984 Joumal of JES 188-210
Denneman J W JEE Proc. A 128 397-414
Dimitrijevié M S and Sahal-Bréchot S 1985 J. Quant. Spectrosc. Radiat. Transfer 34 149-61 Etenbaas W 1972 Light Sourees (London: McMillan)
Frost L S 1961 J. Appl. Phys. 32 2029-36
Griem H R 1974 Speetral Line Broadening by Plasmas (New York: Academie) Grynberg G and Cagnac B 1977 Rep. Prog. Phys. 40 791-841
Kinderdijk H M J and Van Eek J 1972 Physica 59 257-84
Mitchner Mand Kruger C H 1973 Partially Jonized Gases (New Vork: Wiley) Schirmer H 1955 Z. Phys. 142 1-13
Van Tongeren H 1975 Philips Res. Rep. Suppl. 3
Van Vliet JA J Mand De Groot J J 1981 lEE Proc. A 128 415-41
Vriens L, Smeets A H M and Cornelissen H J 1983 Eleccrical Breakdown and Discharges in
Gases. Part B, ed E E Kunhardt and L H Luessen (New York: Plenum) pp 65-117 Waszink J Hand Polman J 1969 J. Appl. Phys. 40 2403-8
Waymouth J F 1971 Electric Discharge Lamps (Cambridge: MIT Press) Chapter 7 Waymouth J F and Bitter F 1956 J. Appl. Phys. 27 122-31
2. Measurement of the ditrusion coefficient of sodium in neon at 530 K
H J Cornelissen
Philips Research Laboratories, PO Box 80.000, 5600 JA Eindhoven, The Netherlands
Received 28 January 1985
Abstract. The diflusion coefficient D of sodium in neon bas been measured at 530 K. In
a cylindrical discharge tube tilled with neon at densities N of 1023 to I 024 m _, a strong radial sodium density gradient is induced by means of a discharge current. In the afterglow of the discharge the decay rate of the fundamental diflusion mode is measured using dye
laserabsorptionallhesodium Dlline. Thevalue DN = (1.43±0.18) Xl021
m-• ç' is found. The diflusion coefficient is calculated for temperatures between 300 and 600 K using
Lennard-Jones (n-6) interaction potentials wîth n = 12, 8 and 7, and usîng a Bom-Mayer
potential. Potential parameters are taken from the literature. Best agreement with experi-ment is found for the Bom-Mayer potential and for the Lennard-Jones (7-6) potentiaL
l. Introduetion
Alkali metal vapours are applied in various plasma devices such as MHD generators
(Mitchner and Kruger 1973) and discharges for lighting purposes (Van Vliet and De Groot 1981). Knowledge of the alkali ditfusion coefficient is necessary to descrîbe the transport processes occurring in these devîces. For example, in the rnadelling of the low-pressure sodium-neon discharge (Van Tongeren and Heuvelmans 1974, Vrîens et al 1983) a collisional radiative model is extended to include ambipolar ditfusion and ground-state ditfusion since these processes are very important in the partiele balances. The working temperature of the discharge is 530 K but experimental values for the ditfusion coefficient of sodium in neon are only reported in the temperature range of 300 to 453 K (Coolen and Hagedoorn 1975, Franz and Sieradzan 1984, Anderson and Ramsey 1963, Bicchi et al 1978).
Given the interaction potential of the alkali with the noble gas, a calculation of the binary ditfusion coefficient in termsof molecular Chapman-Enskog theory (Hirsch-felder et al 1964) can be performed. For the Lennard-Jones (12-6) potential and the modified Buckingham potential, results of the numerically intricate calculations have been tabulated. Results for a large number of potentials can be obtained using the workof Red'ko (1983). Data of Na-Ne scattering experiments from which the potential is deduced are scarce (Carter et al 1975, Van den Berg et al 1983) but an extensive laser spectroscopie study of the NaNe molecule has been performed (Lapatovich et al 1980). Na-Ne has also been the subject of modeland pseudopotential calculations (Masnou-Seeuws et a/1978, Peach 1982) the results of which oompare excellently with the laser spectroscopie experiments. All these investigations show that Na-Ne has only a weakly bound ground state ( well depth e = 11.5 K) which implies that the ditfusion process at 530 K is almost entirely governed by the strongly repulsive part
ofthe interaction potential (Red'ko 1983). For Na-Ne a Lennard-Jones (8-6) potentlal is found to describe the potential well region, but unfortunately neither the laser spectroscopie experiments nor the scattering experiments give very accurate informa-tion on the required repulsive part of the potential. For this part of the potential an exponentlal (Born-Mayer) potential is a good approximation as results from ab initio self-consistent-field theory (Tang and Toennies 1977).
For the experimental determination of the ground-state Na diffusion coefficient in Ne two approaches are found in the literature, both based on the creation of a spatlal density gradient and the study of the temporal evolution of the density distribution towards a steady state. One is used in spin relaxation experiments (Balling 1975) where an oriented Na vapour is produced by optica! pumping. The diffusion coefficient ofthe Na atom is a (more prosaic) byproduct ofsuch experiments. The other approach is to use a 20 Me V proton beam to produce 20Na atoms by nuclear reaction in neon gas (Coolen and Ragedoorn 1975). The technique offers interesting possibilities but it is not widely available nor is it easily used. This paper presents a different metbod to create a spatial gradient in a sodium ground-state density distribution, by using a discharge current.
The principle of the metbod and the experiments are described in some detail in the next section. After an interpretation and a discussion ofthe results a fourth section contains a comparison with molecular theory.
2. Experimeots
2.1. Principle of the method
In the experiments use is made of the strong radial dependenee of the sodium ground-state density in a low-pressure sodium-neon discharge. Because ofthe presence of liquid sodium drops at the wal! of the discharge, the sodium vapour pressure near the wall is determined by the wall temperature. At the axis the sodium density is depleted which is caused by radial transport processes. Electrons and ions are produced in the volume of the discharge, while recombination only takes place at the wall. The electrons and ions drift to the wall in an ambipolar field and ground-state sodium diffuses back into the discharge. Because the ambipolar diffusion coefficient is typically a factor of ten to twenty larger than the neutral diffusion coefficient and the neutral flux bas to balance the ion flux in a steady-state condition, large neutral gradients result, teading to a significant depletion at the axis of the discharge.
Next, the discharge current is switched off and the sodium density profile evolves to a homogeneaus distribution. The density profile obeys the diffusion equation and after some time in the afterglow the fundamental diffusion mode remains. In the cylindrical geometry studled this is a zerotb-order Besset function J0 with argument
j0 •1r/ R. Here j0 , 1 ( =2.405) is the first zero of J0 , r is the radial coordinate and R is the tube radius. The time dependenee of the fundamental mode is a single exponentlal with time constant r1 R2/i~.
1D. This time constant is measured with a laserabsorption set-up as described further on.
2.2. The discharge
The discharge consists of a cylindrical glass tube with electrodes at both ends (Denne-man 1981). Thetube is bentinto a U-shape. A second cylindrical tube surrounds the
discharge and is provided with an infrared reftecting coating to reduce heat losses and minimise axial gradients. In the discharge tube liquid sodium drops reside in regularly spaeed dimples in the glass wal!. The internal diameter of the tube is 14 mm and the tube is tilled with neon. Several tubes are used with neon densities ranging from about 1023 m -3 to 1024 m -3
. Discharge currents are ofthe order of300 mA. The heat generaled by the discharge maintains the wall temperature at 530 ± 5 K, being the working temperature of a commercially available low-pressure sodium lamp. Neon serves as a buffer gas and is not ionised.
2.3. Experimental set-up
The beam of a frequency-stabilised ring dye laser (Spectra Physics 3800) is attenuated to a bout 100 IL W to avoid saturation effects and focused to a bout 100 ~J-m diameter into the discharge ( tigure 1 ). Part of the beam is split off and directed through a reference cell which is tilled with sodium and with neon at a density of 1.8 x 1023
m -3 • The temperature of the cell is kept at 523 K. The laser transmission through the cell is used for electronically locking the laser wavelength in the blue wing of the sodium Dl Jine (589.6 nm).
The discharge can be moved perpendicular to the axis and to the laser beam to allow a radial scan to be made. This is defined as the y direction (figure 1). The discharge current is essentially oe. lt is switched off within a few microsceconds for a duration of about tive times the fundamental time constant T 1 after which the polarity is reversed to prevent cataphoresis. The resulting radial ditfusion is measured by recording the transmitted laser intensity normalised to the laser power. The logarithm of the transmitted intensity is proportional to the chord-integrated sodium ground-state density. Digital signa! averaging over a hundred afterglow periods is applied to improve the signa! to noise ratio. A typical result is given in tigure 2 where the difference between the chord integrated density at t ~ oo and the measured chord integrated density is plotted on a logarithmic scale as a function of time. The density at t ~ oo is found from a non-linear least-squares analysis as described in a following section.
Figure 1. Schematic drawing of the experimental set-up. ND, neutral density filters; BS, beam splitter; L, lens; PD, photodetector; F, libre opties; ,..c, microcomputer.
3. Experimental results
3.1. Interpretation
The spatial and temporal behaviour of the sodium ground-state density n(r, t) in the afterglow can be found from the ditfusion equation in cylindrical coordinates. Axial
t !msl
Figure 2. The dîfference between the steady·state value ii(y, t-> co) and iï(y, t) on a logarithmic scale as a function of time. The curve drawn is the result of the curve fitting. The broken line is an extrapolation of the behaviour of the fundamental mode towards the onset of the afterglow. y = 0.57 R., R = 7 mm, PNe 1130 Pa.
dependenee is neglected, as is the influence of electron-ion volume recombination. The ditfusion coefficient is taken to be radially independent, since the neon density and the gas temperature will be homogeneaus to within a few per cent as can be estimated from the energy balance ofthe discharge (Vriens et al 1983). The influence of Na- Na+ charge transfer collisions is estimated to be negligible. The wall density is assumed to be constant, so that n ( r = R, t) = nw. n ( r, t) can be written as a summa ti on of 'modes'
00
n(r,t) nw-LckRdr)exp(-t/rk)· (1)
l
Rk(r) is the kthradial mode:
Rk(r) loUo,kr/ R) (2)
where Jo.k is the kth zero of 10 • The modes decay exponentially with time constants
Tk R2/i~.kD. (3)
ck is the kth decomposition coefficient of the initial profile n(r, t = 0):
2
JR
ck =~(. ) rRk(r)(nw n(r, t = O)) dr.
}I }O,k 0
(4) Here 11 is the tirst-order Besset function. With the absorption set-up (figure 1) the
line integral over the viewlength is measured, denoted by iï(y, t)
n(y, t) (5)
The time dependenee is the same as for n ( r, t); the dependenee as a function of the lateral displacement y can be numerically found fora given profile. Conversely, n(r, t) follows from n(y, t) by Abel inversion. The time constants rk have fixed relative values, e.g. T1 = 5.268T2 and r1 12.95T3 , which means that the higher modes decay much faster
five times T 1• The signals are analysed startingat t T2 to get rid of most of the higher modes. A least-squares fit is performed with five parameters,
(6) in which a0 is directly proportional to nw, and a1(a2 ) is proportional to the decomposi-tion coefficient c1 ( c2 ) ( compare equation (1 )).
3.2. Results
Fora range of neon filling pressures (400-4200 Pa at 298 K) the lowest-order decay time constant T 1 was measured. The result is shown in tigure 3. The indicated error
bar in T 1 is the 95% reliability interval as estimated from the curve fitting. The straight line in tigure 3 is the weighted least-squares line that is calculated using all the points with the restrietion that it should pass through the origin. The slope of the line is (1.44±0.18) ms kPa-1
from which the value DN = (1.43 ±0.18) x 1021 m-1 is derived using equation (3) with k = 1.
4
j+
/~
I
l/
0 pikPolFigure 3. The measured time constant of the fundamental dillusion mode as a function of the filling pressure. The straight line is the weighted least-squares line through all the experimental points. R 7 mm.
The measured decay time at the highest tilling pressure lies significantly below the line. To check whether an error in the assumed tilling pressure could account for this deviation some of the tilling pressures were verified by laser spectroscopie line-broadening measurements. Below 1 kPa the homogeneaus linewidth of the Doppler-free 3S-4D two-photon transition was determined with an experimental set-up analogous to one described elsewhere (Cornelissen and Burgmans 1982). For the linewidth a value of w/N (l43±5)xt0-23MHzm3 at T=510K was found, which is in reasonable agreement with the value expected (Biraben et al 1977), wf N
=
(132±9) x 10-23 MHz m3• At pressures above 2 kPa the linewidths of the Doppler-free transitions become comparable with the Doppier widths (1.75 GHz) and no resolution wi11 be gained with the Doppler-free technique.
Therefore the Dl absorption line profile was analysed for filling pressures above 2 kPa. The tubes were placed in an oven at 440± 10K. The total absorption profile of the Dl line was measured by scanning the laser approximately 20 GHz. To account for the hyperfine splitting of the 3S ground state, the absorption profile is assumed to consist of two Voigt functions with relative amplitudes 5:3, spaeed at 1836 MHz and with known Doppier broadening. A non-linear least-squares analysis yielded the Lorentzian width, consisting of natura! broadening and collisional broadening. Collisional broadening was in accordance with the expected value using the mean of the broadening parameters listed in Allard and Kielkopf (1982) except at the highest pressure where a correction of -10°/o in p had to be applied to get agreement. The indicated error bars in p reflect the uncertainties in the pressure checking measurements. The reason for the remaining discrepancy in tigure 3 is unknown.
In deducing (1) it is assumed that the wal! density is constant in the afterglow. This can be verified by performing a lateral scan. The density in the discharge n(y, t = 0) is measured for 0 ";; y < R and is normalised by dividing it by the density in the afterglow
n(y, t~oo). Small variations in the laser wavelengthare thus eliminated. The result
of a scan is given in the right-hand side of tigure 4.
It is seen that approaching the wall (y ~ R), the density in the discharge (t = 0) is approximately equal to the density in the afterglow in the limit t ~ oo. This can be quantified by Abel-inverting the measured profile n(y, t): the initia! density profile is assumed to have the form
(7) The free parameters p1 and p2 can be found from a fit of the resulting fim(Y, t O)
to the measured values. This is the brok en curve in tigure 4. For the profile of tigure 4thevalues p, = 1.1 ±0.1 P2 = 8.8± 1.1 8 1.0
'
~ <>: <: ;; 0.5 " ~ <.:: "'·----...
---""'
.
.
I ,~<'"''e
• 1.0 f.
.
.
:
/ ;; 0.5 ~"'
~~~~~~~~~~~~~~o 0 0.5 1.0\
0 1.0 0.5 r/R y/RFlgure 4. Result of a measurement of the radial sodîum densîty dîstribution. The points on the right-hand side of the tigure are the measured chord integrated densities in the
discharge ( 1 = 0) as a function of the displacement y. They are normalised to the measured
densities in the aftergluw. The broken curve represents the least-squares fitted function using the assumption as given in equation (7). On the left-hand si de of the tigure the Abel inverted profile is shown, normalised to the measured wall density in the afterglow.
are found, resulting in a strongly depleted profile as drawn on the left-hand side of tigure 4. From this Abel inversion the quantitative condusion can be drawn that the wall density in the discharge is equal to 1.1 ±0.1 times the density in the afterglow and the assumption of constant wall density is correct.
The given interpretation implies that the time constants of the lowest and higher modes have fixed relative values. This can also be verified independently. From a numerical simulation of the ditfusion process it appears that for profiles as in tigure 4 the second mode ( r2 ) is most clearly se en for 0.3 < y IR< 0.6. For smaller y the
higher modes appear to cancel each other and the lowest one is dominantly present. Closer to the wall the signals show only small time dependenee ( compare tigure 4 where n(y, t)l n(y, t ~ oo) approaches one for y ~ R) which prevents accurate analysis. For the mentioned range it is found that
r11 r2 = 5.5 ± 0.5
which compares well with the expected value of 5.268. This also confirms the interpretation given.
4 Comparison with molecular theory
In the frameworkof molecular Chapman-Enskog theory (Hirschfelder et a/1964) the ditfusion coefficient for a binary gas mixture is given by the expression
1.932 X 1019 [T(M1 + M2)12M1M2]112
D= N (0"12ffl\~·l)* (8)t
where D is given in m2
s-1
, T and N are the temperature and gas density of the
mixture, expressed in K and m-3
respectively, M1 and M2 are the atomie weights of
particles of species 1 and 2, and CT12 is the internuclear distance in 11m at which the interaction potential has the value zero. Finally, the ditfusion collisional integral fl g.l)* can be interpreted as a correction factor on the geometrical cross section 7r(CT12)2, and depends on the kinetics of the collisional process. It is of order unity and can be calculated once the interaction potential U(R;n) is known (R;n is the internuclear distance). The ditfusion collisional integral is a function of the reduced temperature T* =kT Ie, where e is the depth of the 'well' in the potential curve.
For Lennard-Jones (12-6) and modified Boekingham (exp-6) potentials the ditfusion collisional integrals have been tabulated (Hirschfelder et al 1964). For Lennard-Jones ( n- m) potentials with 8.;; n .;; 12 and 3.;; m.;; 6 the results of Red'ko can be used (Red'ko 1983) by reducing the calculation to the following. An etfective ditfusion cross section Qd and radius Rd are defined
Qd = 7T( 0"1z)2fl \~·!)*
= 7TR~( T). Next, Rd is solved from
U(R;n) = f( T*)kT.
(9a)
(9b)
(10)
t Note added in reprint: in the original paper the power of ten in equation 8 was erroneously given as 1021
The function f as given by Red'ko is rather insensitive to the exact form of the interaction potential. f varies from -1.0 at T* = 1 toa saturation value of approximately 1.8 at T* > 50 (Red'ko 1983). From equations (9) and (8) DN then follows. Equation (10) also holds for the calculation of the effective diffusion cross section fora purely repulsive potential. The factorfis then a constant, e.g.f = 1.70 fora potential varying as (R;n)-6
andf= 1.85 for (R;n)-12
(Hirschfelder et al1964).
From an extensive laser spectrosopic study of the diatomic van der Waals molecule Na Ne (Lapatovich et al1980) and a detailed analysis of the data (Gottscho et al1981) accurate values for e and rm (position of the well) are found:
E = 11.5 ± 0.4 K r m = 0.53 ± 0.01 nm.
These potential parameters agree well with model potential calculations (Masnou-Seeuws et al 1978). Calculations made by Peach (1982) show a slightly deeper well
(e = 13.7 K). At 530 K the reduced temperature T* has the value of 46.1.
In view of the large reduced temperature the procedure for the calculation of D proposed by Red'ko seems particularly suitable since the function f varies only slowly with T* for T*::;. 30. The large reduced temperature also implies that in fact the repulsive part of the interaction potential dominantly determines D (equation (10)). For the shape of the potential Gottscho et al titted a Lennard-Jones (8-6) potential totheir experiments. In table 1 the effective diffusion cross section has been calculated:
Table 1. Calculation of efiective diffusion cross sections, using Lennard-Jones (n-6)
potential shapes with n = 12, 8, 7 (u) and using a Born-Mayer potential (BW)
Qd (lo-2o m2) T(K) T* f L-J 12-6 8-6 7-6 BM 300 26.1 1.70 44.4 36.7 34.5 36.2 400 34.8 1.74 42.6 34.5 32.0 32.6 500 43.5 1.77 41.4 32.8 30.5 30.0 600 52.2 1.78 40.0 31.6 29.5 28.0
(i) using the laser spectroscopie potential parameters and Lennard-Jones (n-6) potential shapes with n = 12, 8 and 7;
(ii) using a Born-Mayer potential, U(R;n) =A exp( -bR;n), with parameters (Tang and Toennies 1977)
A = 2.24 x 105
K b = 17.92 nm-1
and with the same values for f as are used for the Lennard-Jones potential.
The various experimental values for DN are listed in table 2 and they are indicated in tigure 5. The curves drawn in tigure 5 are the calculated values deduced from table
1 and equation (8). It is seen from tigure 5 that the (12-6) potential is much too repulsive, yielding diffusion coefficients that are signiticantly lower than according to the experiments (approximately by a factor of 1.5). The (8-6) and (7-6) potentials give better agreement with experiment but are probably still too repulsive. The Born-Mayer potential gives results almost identical to the Lennard-Jones (7-6) poten-tial, yielding values that are a factor 1.2-1.3 below experiment.
Table 2. Experirnental diffusion coefficients for Na in Ne in the ternperature range of
interest rneasured with different techniques as indicated. PB, proton bearn; OP, optica!
purnping; nc, discharge current.
DN T(K} (1019 rn-• ç') 300 84±20 388 93±7 428 86±29 453 109±8 530 143 ± 18
• Coolen and Hagedoom (1975).
b Franz and Sieradzan (1984).
< Andersen and Rarnsey (1963). d Bicci et al (1978 ). •This work. 160 140 40 Metbod PB" OPb oP< OPd oc• 20 OL-~30~0~~4~0~0~~5~0~0~~6~00~ T(Kl
Figure 5. Comparison of ex perimental values of DN with calculations. The open syrnbol gives the result of the experiment described in this paper. References for the other experiments can be found in table 2. The full curves represent the calculations for the Lennard-Jones (n-6} potentials. The dotted curve is for the Bom-Mayer potential.
Here a totally different result is reached than by Van den Berg et al (1983). They performed a differential scattering experiment at high angular resolution and at three different relative collision energies to test several NaNe interaction potentials. Best agreement with experiment was found using the laser spectroscopie values for e and rm and the empirica! potential of Buck and Pauly (1968) with a repulsive branch of (R;n)-20, much steeper than the model potential curves of Masnou-Seeuws or of Peach, and much steeper than the potential used in the laser spectroscopie study. A condusion must be that the scattering experiments are sensitive to a different region of the repulsive branch than the ditfusion experiments, probably much closer to the potential well. A final condusion is that a disagreement of a factor 1.2 to 1.3 remains between calculations and experiments using Lennard-Jones (n-6) potential shapes and potential parameters given by Gottscho et al (1981). Using a purely repulsive exponential potential with parameters given by Tang and Toennies does not change this picture.
5. Summary
The diffusion coefficient of sodium in neon has been measured by studying the time evolution of a radial sodium density gradient towards equilibrium. The density gradient was induced by a discharge current and the density was measured with dye laser absorption in the afterglow of the discharge. The solution of the diffusion equation has been given and the boundary condition was experimentally checked. The exponen-tial decay of the fundamental diffusion mode has yielded the diffusion coefficient. It should be possible to apply this method toother binary systems and in other temperature ranges.
A comparison of other experimental data in the literature with molecular theory has been made in the temperature range of 300 to 530 K. In view of the large relative kinetic energies it is appropriate to use the (purely repulsive) Born-Mayer potential. With parameters given by Tang and Toenniesthere is a disagreement between calcula-tions and experiments of about a factor 1.3. This is of the same order as was found in drift tube experiments for Na in He, Ar and N2 (Silver 1984). Using a Lennard-Jones
(n-6) potentlal with potentlal parameters given by Gottscho et al there is a clear tendency that the agreement between calculation and experiment improves going from the steeply repulsive n 12 to the softer n = 7 potential.
Extracting better potentlal parameters from the results as given in figure 5 would be unrealistic in view of the limited number of experiments and the large uncertainties in some of them. However, it is feit that the results of this paper, in addition to scattering and spectroscopie data, might lead to a better approximation of the inter-molecular Na-Ne forces.
Acknowledgment
The author would like to thank Dr Q H F Vrehen for stimulating discussions.
Heferences
Allard N and Kielkopf J 1982 Rev. Mod. Phys. 54 1103-82 Anderson LW and Ramsey AT 1963 Phys. Rev. 132 712-23
Balling L C 1975 Advances in Quanturn E/ectronics vol 3 Optica/ Pumping ed D W Goodwin (London: Academie) pp 2-167
Bicchi P, Moi L, Savino Pand Zambon B 1978 Nuovo Cim. B 55 395-8
Biraben F, Cagnac B, Giacobino E and Grynberg G 1977 J. Phys. 8: At. Mol. Phys. 10 2369-74 Buck U and Pauly H 1968 Z. Phys. 208 390-417
Carter G M, Pritchard D E, KapJan M and Ducas T W 1975 Phys. Rev. Lett. 35 1144-7 Coolen F C Mand Hagedoorn H L 1975 Physica C 79 402-8
Cornelissen H J and Burgmans A L J 1982 Opt. Commun. 41 187-90 Denneman J W 1981 JEE Proc. A 128 397-414
Franz FA and Sieradzan A 1984 Phys. Rev. A 29 1599-601
Gottscho RA, Ahmad-Bitar R, Lapatovich W P, Renhom I and Pritchard DE 1981 J. Chem. Phys. 75 2546-59 Hirschfe1der J, Curtis C and Bird R 1964 Molecular Theory of Oases and Liquids (New York: Wiley) Lapatovich W P, Ahmad-Bîtar R, Moskowitz P E, Renhom I, Gottscho RA and Pritchard D E 1980 J.
Chem. Phys. 73 5419-31
Masnou-Seeuws F, PhilippeMand Valiron P 1978 Phys. Rev. Lelt. 41 395-8 Mitchner Mand Kruger C H 1973 Partially Ionized Oases (New York: Wiley) Peach G 1982 Comment. At. Mol. Phys. U 101-18
Red'ko T P 1983 Sov. Phys.- Tech. Phys. 28 1065-9 Silver JA 1984 J. Chem. Phys. 81 5125-30
Tang KT and Toennies J P 1977 J. Chem. Phys. 66 1496-506
Van den Berg FA, Morgenstem R, Van der Valk F and Alkemade C Th J 1983 Z. Phys. A 312 271-6 Van TongerenHand Heuvelmans J 1974 J Appl. Phys. 45 3844-50
Van Vliet JA J Mand De Groot J J 1981 lEE Proc. A 128 415-41
Vriens L, Smeets A H M and Comelissen H J 1983 Electrîcal Breakdown and Discharges in Gases Part B, edE E Kunhardtand LH Luessen (New York: Plenum) pp65-117
Appendix 2
After completion and publication of Chapter 2, the workof Havey et al.
(1981) was brought to the attention of the author by Woerdman (1985). Havey et al. present the experimentally determined potentials for the ground state and an excited state of the Na Ne van der Waals molecule. The potentials are generated from the temperature dependenee of the intensity in the far-red wing of the Na resonance line perturbed by neon gas. The shape of the ground-state potential is found to be purely repulsive within the range of internuclear distances for which their experiment is sensitive, i.e. 0.2-0.35 nm. The diffusion coefficient in the temperature range 300-600 K is also determined by the potential at these internuclear distances. In the experiments of Havey et al. the scale factor for the internuclear distance is fixed by model-potential calculations by Philippe et al. (1979). These model-potential calculations are in excellent agreement with the laser spectroscopie data of Gottscho et al. (1981) (see also Chapter 2) for the potential well region. In the following it is shown that the model-potential calculations are presumably quite reliable also in the mentioned internucle-ar distance range where the potential is purely repulsive.
The ground-state potential given by Havey et al. is described by a Born Mayer potential, U(Rin)
=
A exp (-b Ri0), with parameters:A (8.9 ± 0.7) 104 K,
b
=
(16.1 ± 0.38)nm·1.The value of D Nis calculated along the lines given in Chapter 2. First, the effective diffusion radius Rd is calculated from Eq. 10, using the Born-Mayer potential and using val u es for
f
as given in Table 1:Rd = -fIn
(~).
(2.A.1)Rd varies from 0.321 nm at 300 K to 0.275 nm at 600 K. Next, the resulting effective cross section Qd (Eq. 9b) is substituted in Eq. 8 with the result:
(2.A.2)
Ne and Na respectively. DN is in m-1s-t, Tand A are expressed in K, and
bis expressed in nm-1 •
The resulting calculated values f or D Nagree m u eh better with ex perimental values than with the potentials assumed in Chapter 2. This is illustrated in Fig. 2.A.1 where the calculated and the experimental values for DN are compared. For the curve marked with H the parameters given by Havey et al. areused, for the curve marked with TTthe parameters are given by Tang and Toennies(1977). This last curve is thesameas the dotted curve in Figure 5 of Chapter 2. The mean discrepancy between the experiments and the calculations is now reduced to
tt%.
160
!
H140
120
~TT
100
I~~
111 I80
E 0> 0 ...60
...z
ei40
20
0
300
400
500
600
T[K]Fig.2.A.1 Comparison of ex perimental values of D N with calculations. References for the experimentscan be found in Table 2 of Chapter 2. The full curves represent the calculations for the Born-Mayer potential. H: parameters given by Havey et al. TT: parameters given by Tang and Toennies.
From Eq. 2.A.2 the dependenee of the calculated diffusion coefficient on the two potential parameters A and b can be seen. The dependenee on the potential parameter A is only slight since for the temperatures of interest
jT I A < 1 so the logarithmic term in Eq. 2.A.2 does not change much with
A. The calculated DN is proportional to the square of the potential parameter b. This parameter is given by Havey et al. with an experimental
uncertainty of 2.4% (see above) which will thus result in an uncertainty of at most 5% in DN.
As mentioned above, only the shape of the repulsive part of the Na-Ne ground-state potentialis determined in the experiment of Havey et al., not the absolute value. They observe an excellent agreement with the results of model-potential calculations by Philippe et al. ( 1979). The absolute value of the internuclear distance scale is fixed with the aid of the result of this model-potential calculation at Rin 0.251 nm ( 4. 7 5 times the Bohr radius). This is close to the range of Rd values resulting from Eq. 2.A.2, so an extrapolation of the potential to this range is allo wed. Hence, the calcula-tion of D N is a verification of the absolute value of the model poten ti al. From the good agreement observed in Fig. 2.A.1 it can thus be concluded that the model potential represents quite well the repulsive part of the Na-Ne potential in the internuclear distance range 0.275 nm < Rin < 0.32 nm.
Relerences
Gottscho RA, Ahmad-Bitar R, Lapatovich W P, Renhom I and Pritchard DE 1981 J. Chem. Phys. 15 2546-59
Havey MD, Frolking SE, Wright J J and Balling L C 1981 Phys. Rev. A 24 3105-10 Philippe M, Masnou-Seeuws F and Valiron P 1979 J. Phys. B: At. Mol. Phys. 12 2493-510 Tang KT and Toennies J P 1977 J. Chem. Phys. 66 1496-506
3. Langmuir probe experiments
Abstract
In this chapter details are given of the Langmuir probe experiments that are performed to measure the axial field strength, E, the electron density at the tube axis, ne, and the electron temperature, Te. The electron energy dis tribution function (EED F) is also derived from the probe characterîstics. A comparison is made with results of a two-electron-group model.
3.1 Introduetion
Probably the first technique to derive fundamental local properties from gas discharges was the Langmuir probe technique (Mott-Smith and Langmuir 1926). It has grown into one of the most commonly used techniques, deriving its popularity from the relative ease with which it can experimentally he applied. Moreover, it can give an indication of the local properties of the plasma, both in space and in time.
Since the pioneering work of Langmuir considerable progress has been made in the interpretation of the probe characteristics and the modern pro he theories cover a wide range of partiele densities and temperatures. It would not he useful to give a complete survey of the literature on probes here, and instead only references tosome of the more recent reviews and monographs are given (Cherrington 1982, Loeb 1973, Swift and Schwar 1970, Chen 1965).
In the low pressure sodium-neon discharges with a neon filling pressure of 7 30 Pa the mean free path À.e of the electroos is approximately 500 !J.m. The sheath surrounding the pro he is of the order of the Debye shielding length
À.n. With electron densities of the order of 1018 m 3 and electron
temperatures around 8000 K the Debye length is 6 IJ.m which is small compared to the mean free path of the electrons. Probe theory is fairly simpte in this paramèter regime: the sheath surrounding the probe is collisionless. The cylindrical probes are made from tungsten wire with a radius rP of 111J.m (for constructional details see below). Since this is of the same order of magnitude as the dimensions of the sheath, orbital motion theory can he applied.
3.2 Principle of the method
Measurement of a probe characteristic is performed according to the following procedure. A bias voltage is applied to the probe in ordertobring the probeat a potential, V, that is negative with respect to plasma potential. In this way the probe surface is cleaned by ion bombardment. Then the probe is brought at a potential close to the local plasma potential, VP. A triangular voltagepulseis applied with an amplitude of approximately 10 voltand a total duration of 2.5 ms, and the probe current, I pr• is measured.
Sweeping the voltage more slowly reveals hysteresis in the /pr - V
characteristic due to heating of the probe wire. Sometimes it is necessary to repeat the ion bombardment cleaning or to apply addîtional electron bombardment to obtain hysteresis-free probe characteristics.
Taking as a reference the plasma potential, two regimes exist:
(i) For negative probe potentials electrons are retarded and ions are attracted to the probe. For the measured electron current the following relation holds in case of a Maxwellian EEDF:
-1 Te
V< VP: Je=- nee 0 - - -exp
4 1rme (3.1)
Here 0 is the probe area 2nr Pl, with I the length of the probe, e is the elementary charge and me is the electron mass. The ion current to the probe is much smaller than the electron current and an estimate can be made from the Bohm criterion (Bohm 1949): to forma stabie sheath the ions should enter the sheath with a velocity at least equal to VkTel Mi where Mi is the ion mass. This velocity corresponds to an ion current of 20 IJA (Te = 8000 K, ne = 1018 m-3 , I = 1 mm). Cf. the electron current at V= VP would
be 1.5 mA. The ion current is approximated by the constant value measured at strongly negative V. The ln(Je) vs. V plot is approximated by a straight line and from the slope the electron temperature is obtained.
(ii) For positive potentials the electrons are attracted and from the orbital motion theory the expression for the electron current (squared) reads:
202e3n2 (kT ) V> VP: /~ = e _e + V- VP .
n2m e e
(3.2) The /~ vs. V plot shows a straight line and from the slope the electron density is calculated. The intersection of the ( extrapolated) line with the V-axis gives the value of VP kTele. Substituting the Te value as derived from the electron retarding part, the plasma potentialis determined. Once the plasma potential is known the electron density can be determined in a second way. Substituting VP in Eq. 3.1 the electron density is obtained from
the absolute value of the measured probe current in the electron retarding regime. This has the advantage that the currents drawn from the plasma are smaller than in the orbital motion regime, but a disadvantage is that Eq. 3.1 depends exponentially on V/ Te so that errors in VP are strongly amplified.
3.3 Experimental set-up
The probe construction is schematically drawn in Fig. 3.1. Following Verweij (1961) a glass capillary tube shields the probe. A spaeer coil is used to prevent contact ofthe probe with material that is possibly sputtered from
TUBE WALL TUBE tXIS
Qmm
I
1mm 1lmm
(r4~~~~~~~~~-~
..
F;~w~P21.7pm
Spaeer coili
Pt!ll 0.2mm Glass capillary O.O. 1mm
I
Fig.3.1 Schematic diagram of the probe construction.
~0--7calibr.
T gate
switch
pul se
the probe. The length of the probe protruding from the capillary is measured with a telescope. The plasma is probed at the tube axis.
A block diagram of the measuring circuit is given in Fig. 3.2. The probe current is measured over a resistance of 100 Q with a digital signal averager (Channel A). The applied voltage putse is also recorded (Channel B). Typically 128 pulses are averaged to improve the signal-to-noise ratio. The system is calibrated with a separate current source. The bias voltage is read with a digital voltmeter. Since the polarity of the discharge current is inversed ten times per second to prevent cataphoretic effects, a gate switch is used to synchronously conneet the probe to the measuring resistor.
discharge polarity: gate switch: voltage pul se: closed open
-1ov-o
-2.5msFig.3.3 Discharge circuitry and timing diagram.
In Fig. 3.3 details ofthe discharge circuitry are given. The electrodes ofthe discharge are pre-heated and additional discharges are applied between the electrodes and separate anode buses. This is extremely helpfut to suppress instahilities arising in the anode region of the discharge. Probe characteris-tics are measured with the anode as the reference electrode.
3.4 Results
as function of the probe potential fora discharge current of 130 mA. The straight lines are the approximations as discussed in Section 3.2.
The electron temperature is derived from the electron retarding regime. Since the probe curves are analysed for currents above a threshold of 50 IJ.A the procedure foliowed to correct for the ion current bas little influence on the value of Te. In theory the probe characteristic shows a sharp "knee" at the plasma potential which is the break point between the electron retarding and the orbital motion regime. In practice this is never found which is attributed in literature to non-uniformities in work function along the length ofthe probe or to the relatively large current that bas to be supplied by the plasma to the probe.
In Table 3.1 the experimental results obtained in a discharge with a neon filling pressure at room temperature PNe of 730 Pa, internal tube radius R of 9.5 mm and wall temperature T w of 523 K are compiled as function of the discharge current I (see also Chapter 4 where these measurements are compared with laser absorption experiments).
3 9 -6 Vp ln
IIel
t
.//·~·--····--···
-7 -N-
2 6 ~ <( E .§-8'"
... Ql NQI ... Ie c ... t-4 , .... / ..~····'_...
... -·· -9 1 3 /-·
,".~".
.f' ....
-10 .• .... .' 0 -11 0.0 1.0 2.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 PRO BE VOLTAGE [V]Fig.3.4 Measured Ie, I~ and In (IJ as a function ofthe probeto plasma potential. Thepotential is shifted by an arbitrary amount. The discharge current is 130 mA. The plasma poten ti al VP is indicated. The straight !in es are the approximations as discussed inSection 3.2. The straight line in the In (I.)-plot is extrapolated (dashed) below the current threshold of 50 !J,A. The straight line in the /~-plot is extrapolated (dashed) below plasma potential.
The plasma potential as deduced from the orbital motion regime using the observed values for Te is compared with values derived from the voltage at which the second derivative of the probe current is zero (see Section 3.5). Typically an agreement within 0.1 volt is found. From the difference in plasma poten ti al of two neighbouring probes and their axial separation the axial electric field, E, is determined.
Finally, in the tabletheelectron density as deduced from the orbital motion regime is compared with the value deduced from the electron retarding part, substituting the value of the plasma potential as deduced from the orbital motion regime in Eq. 3.1. As mentioned, the latter value for the electron density depends exponentially on the plasma potential and the experimen-tal scatter tends to be larger. Still, agreement with the orbiexperimen-tal motion values typically is within 1
o%.
I Te ne
vp
vp
ne Eelec.ret. orb.mot. orb.mot. Jlf e 0 elec.ret.
[mA] [K] [10t7m-3] [10t7m-3] [V/m] 130 9000 3.0 19.02 19.15 2.9 143 200 8400 5.2 13.00 12.97 4.5 110 300 8100 8.0 9.25 9.12 6.9 90 400 8400 10.5 8.61 8.88 8.9 85 500 8600 12.2 9.53 9.46 10.9 87 600 10000 12.8 11.96 11.94 12.5 101
Table 3.1 Results of probe measurements. Tw 523 K; PNe 730 Pa; R = 9.5 mm. VP
indicates the difference between the anode potential and the plasma potential at the probe and is given in volt.
3.5 Electron energy distribution function
In Eq. 3.1 and 3.2 a Maxwellian EEDF was assumed. This is not strictly necessary since in principle the number of electronsper unit volume/(&) in an energy range & to &
