Study of glow discharge positive column by electrostatic probes: Double probe
Contents
1 Symmetric double probe at floating potential 2
2 Operation of double probe 2
2.1 A) Zero voltage Vd = 0.............................. 2
2.2 B) Small negative voltage Vd < 0........................ 3
2.3 C) High negative voltage     <C 0........................ 3
3 Theory of the operation of double probe 3
4 Calculation of plasma parameters from I-V characteristic of double probe 5
4.1 Electron temperature: method of currents ratio................ 5
4.2 Electron temperature: resistance method ................... 5
4.3 Electron density in plasma............................ 6
4.4 Ion and electron density in plasma....................... 7
5 Experimental set-up and measurement 8
1
Introduction
Diagnostics of positive column of glow discharge using electrostatic Langmuir probes can be used to determine several plasma parameters. There are different constructions of the probe. One of them is double probe which consists of two electrodes of the same sizes with distance sufficient for not overlapping of their sheaths.
There are several advantages in comparison with single Langmuir probe. Double probe system is floating and there is always steep slope around zero voltage in current voltage (I-V) characteristic. Saturated ion current limits total current in the probe circuit therefore probes does not disturb plasma that much.
1    Symmetric double probe at floating potential
Double probe system consists of two same probes located in the equipotential surface in the plasma. They are not connected to electrodes and without influence from external potential end up at floating potential Vg. We can study plasma using double probe by applying small voltage Vd between two probes and measuring current Id in probe circuit. The schematic representation of double probe is shown in fig. 1. Double probe is suitable for the study of high frequency plasma and plasma decay.
Figure 1: The schematic representation of double probe.
2   Operation of double probe
We need to discuss operation regimes for different voltage Vd to understand how the measurement with double probe works. Lets assume same surface areas of the probes, no contact potentials and that they are located in spot with same plasma potential Vp. Voltage between probes Vd does not affect ion current.
2.1    A) Zero voltage Vd = 0
Both probes collect same electron and ion current and are at same floating potential Vfl. The current in the probe circuit Id = 0 because there is not any electromotive force in this circuit. This point is shown as zero of the I-V characteristic in fig. 2. Distribution of the potential at probes is shown in fig. 3a.
2
X
Y
A
pl
Figure 2: Current-voltage characteristic of ideal double probe.
2.2 B) Small negative voltage Vd < 0
Probe potential in reference to plasma is determined by equilibrium condition ^2 Ip + ^2 Ie = 0. Distribution of the potential is shown in fig. 3b. Potential of first probe is close to plasma potential and this probe collects more electrons. Potential of the second probe is lower than floating potential and flow of electrons is decreasing. Electrons from first probe compensate decrease of electron current of the second probe. The sum of currents is zero and system is located in point B of I-V characteristic.
2.3 C) High negative voltage Vd <C 0
In this case one of the probes collects electron current while the other is strongly negative in respect to plasma potential, therefore electrons can't reach her. Half of the electrons of first probe flow to the second probe trough external circuit. System is described by point C in I-V characteristic in fig. 2. Distribution of the potential is shown in fig. 3c. Further change of the Vd does not affect probe current due to effect of current flow between probes balancing ion flow at the probes, in case of decreasing Vd, first probe stays at the value close to the plasma potential, meanwhile second probe becomes more negative. Ion current of the second probe is saturated and probe current in external circuit Id remains constant: region X-Y in fig 2.
Total ion current is given by sum of saturated current at first probe Ip± and at second probe ip2 in region X and Y V-I characteristic.
Electron flow at the second probe is given by the difference of total current in external circuit and ion current Ip2 at the probe as shown in fig. 2.
3   Theory of the operation of double probe
We can see generalised diagram of the double probe potential in fig. 4. Potential of the probes are V\ and V2 in respect to the plasma. Vc is constant potential or small difference in plasma potential in probes location. Using Kirhoff Law we can write for total current
3
vd=o
a)
Figure 3: Distribution of the potential in double probe system.
J01 J02
Figure 4: Generalised diagram of the potential of double probe.
in probe circuit as:
7P = 7Pl + 7P2 = !el +h2=0
using Boltzmann relation for electron current we get:
2_^h= ^1 Joi exp-^- + 62J02 exp-
The diagram of potential in fig. 4 shows:
V1 + VC = V2 + VA^V1 = V2 + VA- vc Using last two equations we get:
In
'e2
1
eVd
a = —— exp
<->2 J02
eVc
+ In a.
kTP
(2)
(3)
(4)
where joi a jq2 are electron current densities at probes at potential equal to plasma potential. Electron temperature Te is given by the slope of straight line G = ^\Ip/Ie2\ — 1 of the plotted graph of lnG = f{Vd) according to 4.
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4   Calculation of plasma parameters from I-V characteristic of double probe
4.1    Electron temperature: method of currents ratio
Evaluation of the data obtained by double probe is similar to simple Langmuir probe analysis. From the plot of I-V characteristic is possible to determine current of both probes Ip\ and Ip2- It is shown in fig. 5. Then we subtract electron current Ie2 and plot lnG = /(Vd). Electron temperature Te can be calculated from the slope according to equation 4 in case of Maxwellian velocity distribution. Coefficient a depends on probe and sheath sizes. If they are same for both probes a = 0.
Figure 5: Determination of Rq and g from I-V, currents IPl and IP2 for     = 0.
4.2   Electron temperature: resistance method
Equation 4 can be written as:
'e2
crexp
eVd
+ 1
(5)
Derivation of Ie2 with respect to Vj, we get for     = 0
d/e2
dFd
J2h ae
yd=o
(6)
If we replace 4^- = dV^d/j, we get for electron temperature
e a
k(l+a)
'd/d.
(7)
vd=o
where a can be calculated using equation 4
'e2
We can simplify eq.7 using G:
G
and by replacing a using G in eq. 7, we get:
vd=o
Ie2
(G - G1
d/d
vd=o
(8)
(9)
(10)
where Rq is so called equivalent resistance of double probe
~dVd
Ro
d/d
11
yd=o
Electron temperature can be calculated easily from I-V curve using equation 10. We need to determine Rq,^2Ip and G first. The slope of the middle part of I-V in point = 0 gives us value of Rq.
Ion currents Ip± and Ip2 in case of = 0 can be determined using asymptotes of saturated currents. We prolong them to y axis. Then we divide distance MN into 5 parts. There is a point a located in distance equal to | MN from y axis - that is value of Ipi or Ip2 f°r Vd = 0. This graphical method can is shown in fig. 5. Electron current at second probe is given by:
Ie2 — \Ip2 \ + Id
and can be determined directly from I-V according to fig. 5. For calculation of G:
G
Ie2 £/p
(12)
(13)
vd=o
it is important to use Ie2 for Vd = 0 and ^2 Ip = Ip\ and Ip2 for a
4.3   Electron density in plasma
According to Kir hoff law the current in probe circuit is:
7P = 7Pl + 7P2 = lei +h2=0
e Vi 1     T      T [ eV2
ho + ho exp 1 — exp
+ ho + Ieo exp + 11- exp
kT,
eV2 kTP
(14)
(15)
(16)
(17)
6
Current in probe circuit is / = Ip± = IP2 and therefore:
/ = iiO   1 - exp
Using:
eV1
T¥P
tanhx
Ii0   1 - exp
eV2 kTP
we can rewrite 18 as:
/ = I[q tanh
e2x _ 1
e2x + 1 eVd
kTP
(18)
(19)
(20)
where V<i = V2 — V±.
By differentiation of previous equation we get:
dl
dVd
vd=o
2k TP
(21)
Electron temperature can be determined from the slope of I-V in point Vd. Then we can determine electron density from the equation of saturated ion current:
ho = aneApvB
(22)
Vk¥e
(23)
where M is mass of ions.
0MnPA
kTP M
(24)
4.4   Ion and electron density in plasma
np can be determined from
Electron density nP and ion density np in case of nP saturated ion current of the probe. The problem for np is that we do not know ion temperature Tp. There are some cases when Tp is equal to gas temperature, such as decaying plasma. Then we can use equation:
jp = npe(vP).
(25)
Small error of Tp does not affect calculation of np because there is Tp already in equation. Average drift speed of electrons (ve) is given (in case of plasma decay) by flow on ions to probe sheath which is dependant on thermal movement of ions. In case of Maxwellian velocity distribution (vP) = \{vp), where (vp) is average ion speed.
Ion density is np
4JP
e(vp)
and by substituting ion current density jp = -# we get:
4/n
S e (vp)'
(26)
I 8kT \
where (vp) = I J . Tp is temperature in Kelvins, M mass of ions, Ip current in Amp. For cylindrical probe is area S bigger than just surface area of the probe and depends on probe potential. There is need to make a correction for np.
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5   Experimental set-up and measurement
Double probe measurements are realised using experimental set-up described in fig. 7.
• Connect double probe according to circuit in fig. 6.
• Discharge tube is evacuated using a rotary vane pump to 3-5 Pa.
• We are using argon as working gas which flows in a continuous flow at rate 80 seem.
• Pressure is measured by Pirani manometer. Adjust the value of gas flow using flow controller to achieve pressure of 100 Pa.
• Set up the high voltage power source to 500 V and ignite the discharge.
• You can adjust the value of current in the discharge using adjustable current stabiliser next to the power source. Set up the value to 10 mA, 30 mA and 50 mA.
• Measurement is realised by ammeter and voltmeter in probe circuit connected to computer for semi-automated measurements.
• Measure I-V characteristics of double probe for different values of discharge current (10 mA, 30 mA and 50 mA).
• Determine sizes of the probes using photograph of discharge tube if you know its diameter.
• Plot measured I-V characteristics.
• Calculate electron temperature and electron density using mentioned methods.
400-800 V DC
0-24V DC + O
+/- 12V
0-80 V DC
3
probes
PC
Figure 6: Scheme of the electrical connections of discharge (above) and double probe (below).
References
[1] Tálský A., Janča J. Speciální praktikum z vysokofrekvenční elektroniky a fyziky plazmatu Přírodovědecká fakulta UJEP, Brno 1975.
[2] Smíd R. Návod k úloze Měření rozdělovači funkce energie elektronů pomocí vf kompenzované Langmuirovy sondy.
8
gas flow regulation
vacuum meter
gas flow controller
power source (probe)
power source (discharge)
rotary vane
function generator
discharge tube
U, I in probe circuit
ammeter voltmeter meassurement of I, U in discharge
Figure 7: Experimental set-up.
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