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Experimental and theoretical studies of the radial distribution function of the electron temperature (RDFT) in a low-density plasma and weakly ionized gas for the abnormal glow region are presented. Experimentally, the electron temperatures and densities are measured by a Langmuir probe moved radially from the center to the edge of the cathode electrode for helium gas at different pressures in the low-pressure glow discharge. The comparison of the final experimental data for the radial distribution of electron temperatures and densities for different low pressures ranging from 0.2 to 1.2 torr, with the final proved equation of RDFT confirms that the electron temperatures decrease with increasing product of radial distance and gas pressures, showing a radial decrement dependence of the electron temperature from the center to the edge of the electrode. This is attributed to the increase of the number of electron-atom collisions at higher gas pressures and consequently of the rate of ionization. For the axial distance (
*L*) from the tip of the probe to cathode electrode and the cathode electrode radius (
*R*), a theoretical and experimental comparison for the two conditions
*L* <
*R* and
*L *>
*R*, for both cases the produced plasma temperatures decrease and densities increase. It is concluded that the RDFT accurately shows a dramatic decrease for
*L* <
*R* by 60% less than RDFT values for
*L* >
*R* similar as for conditions of magnetized and unmagnetized effect for DC plasma. This means that the rate of plasma loss by diffusion decreased for
*L* <
*R*, agrees well with the applied of magnetic field behavior

Plasma is the fourth state of matter, considered to be a quasi-neutral medium. However, when a diagnostic such as probe (a small metallic electrode) is inserted into a weakly ionized plasma, a very thin sheath is formed around the conducting surface of the probe due to the redistribution of charges. The amplitude and the polarity of the probe potential control the motion of electrons and ions near the probe. When the probe potential is sufficiently negative, only the ions can reach the probe surface. The probe current is thus equal to the ion random current (I_{ri}) [

Ambipolar diffusion is the most important process taking place within a weakly ionized plasma and considered as a vital process for the distribution of the plasma parameters. Assuming a Maxwellian velocity distribution [

The experimental and numerical radial distribution of electron temperature and density from the axis of the tube up to the tube wall have been amply investigated [

In the present work, an experimental study of the radial dependence of the electron temperature in a low-density plasma using a single probe as diagnostic technique was performed. Theoretically, it was proven that the ambipolar diffusion is an important factor in the weakly-ionized gas. Furthermore, a theoretical physical model for the radial distribution of the temperature was derived. The theoretical considerations are compatible with the experimental data of the temperatures at different pressures in a low-pressure glow discharge of a DC (cold cathode) magnetron sputtering unit.

_{P} is applied to the probe.

The potential distribution was measured at discharge currents of 10 mA and gas pressure of (0.2 - 1.2 torr) for He as shown in

Parameters | Values |
---|---|

Discharge Current (I_{a}) | 4 - 30 mA |

Gas Pressure (P) | 0.1 - 1.5 torr |

Discharge Voltage (V_{a}) | 200 - 1200 V DC |

Current Density | 2 - 15 mA/m^{2} |

Working Gas | He |

distribution can be divided into three regions. In region I (AB) (cathode fall), the potential increases sharply within a small discharge length. Whenever, gas breakdown takes place in the tube, a rapid growth in the rate of ionization, near the cathode is detected. Meanwhile the electrons transfer in the electric field much more faster than positive ions (due to their masses), electrons are swept rapidly towards the anode leaving a dense positive space charge near the cathode. Thus, the electric field is distorted and most of the applied potential is dropped across a narrow space in front of the cathode. Region II (BC) (negative glow) indicates that the potential decreases slightly and hence the electric field will be weak, since this region contains many free electrons. In region III (CD) (positive column), the potential distribution is nearly constant and linear since the positive and negative carriers densities are closely, equal. The positive column can be extended to any length to fill the remaining space between the end of the negative glow and the anode.

Values of the electric field distribution are obtained by differentiating the measured potential distribution of

cathode. This is related to the intense positive space charge which lies in front of the cathode. This acts as an accelerator for the electrons towards anode. Thus, the electrons emitted from the cathode are then accelerated away, until they reach the negative glow region where the electric field becomes weak (zero and sometimes it reaches a negative value). In this region, the gained kinetic energy of the electrons is dissipated in collisions with the atoms of the gas and thus secondary electrons would be produced. In the positive column region constant and linear electric field is needed to maintain discharge along the large length of the column which is required to carry the discharge current.

The region investigated in the present work is the abnormal region. The radial dependence of the electron temperature in low-density plasma at edge and center of the cathode electrode has been studied for different He pressures.

The plasma parameters like the electron temperature and electron density can be determined from the current-voltage (I-V) characteristic curve of the single Langmuir probe, based on the theory and the fundamental technique discussed in detail in many articles [

_{e} is decreased and n_{e} increased due to the general trend that values of T_{e} and n_{e}_{ }are inversely proportional [

Increasing the helium working pressures from 0.2 to 1.2 torr, the temperature T_{e} decreases at the center from 9.7 eV to 5.5 eV and at the edge from 4.1 to 1.15 eV. Values of densities n_{e} increased at center n_{e} from 0.6 × 10^{9} cm^{−3} to 1.3 × 10^{9} cm^{−3} and at the edge from 2.5 to 3.95 × 10^{9} cm^{−3}.

Conductivity at the tube axis, and charge density of plasma according to the following expression [

D s = D a [ 1 − μ e ρ σ ] (1)

where D_{a} is the ambipolar diffusion coefficient as follows:

The flow of ions and electrons are the same, hence

Γ = Γ i = Γ e

(Congruence approximation). If there is no external electric field, the fluxes of ions and electrons (drift-diffusion model) can be expressed as:

Γ i = − D i ∇ n i + μ i n i E D

Γ e = − D e ∇ n e + μ e n e E D

where E_{D} follows Poisson’s equation. If we multiply the first equation by μ e n e and the second one by μ i n i and then subtract them, we get:

μ i n i Γ e − μ e n e Γ i = μ e n e D i ∇ n i − μ e n e μ i n i E D − μ i n i D e ∇ n e + μ i n i μ e n e E D

Since Γ = Γ i = Γ e , the terms with E_{D} cancel each other out and using n_{e} = n_{i}, the expression for the flow of particles becomes:

Γ = − ( D e μ i − D i μ e ) μ i − μ e ∇ n i

With the ambipolar diffusion coefficient being

D a = D e μ + − D + μ e μ + − μ e (2)

and ρ is the charge density in the plasma given by:

ρ = e [ n + − n e ] (3)

and σ is the conductivity at the tube axis given by:

σ = e [ μ + n + − μ e n e ] (4)

Considering that diffusion coefficient D given by:

D = k T m ν m / s 2 (5)

where μ ε , + are the mobility of electrons (e) and ions (+), kT (ev) is the particle temperature, ν is the collision frequency between electrons and neutral atoms in Hz. By substituting with Equations (2, 3 and 4) into Equation (1), then we get

D s = D e μ + − D + μ e μ + − μ e [ 1 − μ e n + − n e μ + n + − μ e n e ] (6)

But by substituting by the Einstein relation [

μ = ⌈ q ⌉ D k T (7)

into Equation (6) by the value of D, then we get

D s = μ e μ + K T e − μ e μ + K T + q ( μ + − μ e ) [ 1 − μ e n + − n e μ + n + − μ e n e ] (8)

or

D s = μ e μ + ( K T e − K T + ) e ( μ + − μ e ) [ μ + n + − μ e n e − μ e n + + μ e n e μ + n + − μ e n e ] (9)

Neglecting k T + , where k T e ≫ k T + finally results in

Then

D s = n + μ e μ + ( k T e ) e ( μ + − μ e ) [ μ + − μ e μ + n + − μ e n e ] (10)

D s = n + μ e μ + ( k T e ) e [ 1 μ + n + − μ e n e ]

Then

D s = k T e e [ μ e μ + μ + − μ e n e n + ] (11)

or

D s = k T e e [ μ e 1 − μ e μ + n e n + ] (12)

Substituting μ e = e m e ν e and μ i = e m i ν i from Equations (5) and (7) into Equation (12) results in

D S = k T e e [ ( e m e ν e ) 1 − ( e m e ν e ) n e ( e m i ν i ) n + ] (13)

or

D S = k T e [ 1 m e ν e 1 − m i ν i n e m e ν e n + ] (14)

or

D S = k T e [ n + m e ν e n + − m i ν i n e ] (15)

D S ν i = k T e m e ν e ν i − m i ν i 2 n e n + (16)

Experimentally, due to the diffusion process during plasma formation an interesting process occur in the plasma formation stage of the basil discharge. Theoretically, basil discharge process related with the recurrence relation for Bessel condition as shown in appendix A, where from the recurrence relation for the Bessel condition, the Schottky condition can be derived and proofed [

D S ν i = ( R 2.405 ) 2 (17)

A proof of the Schottky condition will be discussed briefly by two methods in Appendix A, where (R cm) represents the radial distance from the center to the edge through the cathode electrode measured by the Langmuir probe moved radially, and by substituting Equation (17) into (16), then

( R 2.4 ) 2 = k T e m e ν e ν i − m i ν i 2 n e n + (18)

Substituting the following equation into Equation (18)

ν e = N n 〈 Q e - n υ e - n 〉 (19a)

ν e = ( 3.55 × 10 16 P ) 〈 Q e - n 〉 ( 2 k T e m e ) 1 / 2 (19b)

ν i = ( 3.55 × 10 16 P ) 〈 Q i - n 〉 ( 2 k T i m i ) 1 / 2 (19c)

where P in torr, Q e - n and Q i - n are the cross sections of electron-neutral and ion-neutral collisions, respectively results in

( R 2.4 ) 2 = k T e [ m e ( 2 k T e m e ) 1 / 2 ( 2 k T i m i ) 1 / 2 ( 3.55 × 10 16 P ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + m i [ ( 3.55 × 10 16 P ) 2 〈 Q i 〉 ] 2 ( 2 k T i m i ) ] (20)

then

( R P 2.4 ) 2 = 1 [ ( 4 m e k T i m i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 2 k T i m i k T e ) ] (21)

But from [

Or

m i ≅ ( 1 7.64 μ e μ i ) 2 (22a)

Then substituting (7) into (22a) gives

1 m i ≅ ( 7.64 k T i k T e ) 2 (22b)

Moreover, substituting (22b), m i m e ≅ 2000 and ν i ν e ≅ 10 − 2 into (21) gives

( R P 2.4 ) 2 = 1 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (23)

( R P ) 2 = 2.4 2 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (24)

Substituting the values of the cross sections, masses of ions and electrons into (24) gives

( R P ) 2 = 2.4 2 ( 3.55 ) 2 ( 7.64 ) ( 2 ) [ [ ( k T i k T e ) ( m e k T i k T e ) 1 / 2 ] − [ n e n + 7.64 ( k T i k T e ) 2 ( k T i k T e ) ] ] (25)

or

( R P ) 2 = 2.4 2 ( 3.55 ) 2 ( 7.64 ) ( 2 ) ( k T i k T e ) 3 / 2 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] (26)

Then finally

R P = 0.1729 1 ( k T i k T e ) 3 / 4 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] 1 / 2 (27)

Using the data of the electron temperatures and densities shown in

n + = I + 0.6 ( k T e m i ) 1 / 2 A P (28)

Knowing T_{e}, ( A p = 4 π r p 2 ) area of spherical probe) and calculating positive ion current I_{+} from the I-V characteristic curve of the spherical single probe [_{+ }can be determined.

_{e} at the edge of the electrode are more pronounced at the center due to edge effect [

D S ν i = ( R 2.405 ) 2 + ( L π ) 2 (29)

and by substituting Equation (29) into (16), then

( R 2.4 ) 2 + ( L π ) 2 = k T e m e ν e ν i − m i ν i 2 n e n + (30)

Substituting by Equations (19a, b and c) into Equation (30), respectively results in

( R 2.4 ) 2 + ( L π ) 2 = k T e [ m e ( 2 k T e m e ) 1 / 2 ( 2 k T i m i ) 1 / 2 ( 3.55 × 10 16 P ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + m i [ ( 3.55 × 10 16 P ) 2 〈 Q i 〉 ] 2 ( 2 k T i m i ) ] (31)

then

( R 2.4 ) 2 + ( L π ) 2 = 1 [ ( 4 m e k T i m i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 2 k T i m i k T e ) ] (32)

Substitution from (22a and b) into (32) gives

( R P 2.4 ) 2 + ( L π ) 2 = 1 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] (33)

( R P ) 2 = 2.4 2 [ ( 7.64 k T i k T e ) ( 4 m e k T i k T e ) 1 / 2 ( 3.55 × 10 16 ) 2 〈 Q i 〉 〈 Q e 〉 ] − [ n e n + ( 3.55 × 10 16 〈 Q i 〉 ) 2 ( 7.64 k T i k T e ) 2 ( 2 k T i k T e ) ] − 2.4 2 ( L π ) 2 (34)

Substituting the values of the cross sections, masses of ions and electrons into (34) gives finally

R P = ( 0.0299 ( k T i k T e ) 3 / 2 [ m e 1 / 2 − 7.64 n e n + ( k T i k T e ) 3 / 2 ] − 5.76 ( L π ) 2 ) (35)

For 1.2 torr of applied pressure,

The reasons for the radial distribution of the electron temperatures for both cases.

L < R and L > R are the following:

a) The temperature decreases for increasing pressure may be due to the relation between the temperature and the pressure given by [

| T e | = 6.5 | P − 0.066 | (36)

b) As the pressure increase leads to a further increase in the breakdown voltage, a sharp increase in electron density and a higher electron-electron collision frequency, leads to the decrease of the electron temperatures [

c) The present theoretical and experimental results are compatible and agree fairly well with the behavior of the data of von Engel [

For L < R, RDFT accurately shows a dramatic decrease by 60% less than RDFT values for L > R. This may be attributed to:

The cathode fall, negative glow and positive column regions are compressed when L < R, i.e. the cathode fall region is compressed in thickness and henceforth higher potentials are expected. Thus, strong electric field is produced and therefore ions would accelerate, and more efficient sputtering processes take place. The produced plasma temperatures decrease, and densities increases; this means that the rate of plasma loss by diffusion decreased similarly as in the applied of magnetized DC plasma [

Theoretically, using the Einstein relation of ambipolar diffusion, charge density of the plasma, conductivity at the axis of the tube, the Schottky condition, and the cross sections of ion-neutral and electron-neutral collisions, the decrease of the radial distribution function of the electron temperature with increasing product of the radial distance and gas pressures can be determined. The theoretical study agrees well with experimental data of electron and ion temperatures and gives the radial decrement dependence of the electron temperatures from the center to the edge of the electrode.

GVO acknowledges for the partial financial support from MEPhI and NRU MPEI in the framework of the Russian Academic Excellence Project.

The authors declare no conflicts of interest regarding the publication of this paper.

Galaly, A.R. and Van Oost, G. (2019) Comparison between Theoretical and Experimental Radial Electron Temperature Profiles in a Low Density Weakly Ionized Plasma. Journal of Modern Physics, 10, 699-716. https://doi.org/10.4236/jmp.2019.107050

There are two classes of Schottky condition models using.

Appendix A: the recurrence relation for the Bessel condition

Appendix B: the boundary conditions

D S ν i = K T e m i ν i 2 n e n + + m e ν e ν i (36)

Due to the diffusion process during plasma formation an interesting process occur in the plasma formation stage of the basil discharge. The Schottky condition stating that

D S ν i = ( R 2.4 ) 2 can be demonstrated as follows:

Appendix A: From the Recurrence Relation for Bessel Condition [2 J ′ n ( x ) = J n − 1 ( x ) − J n + 1 ( x ) (37)

and

2 n x J n ( x ) = J n − 1 ( x ) + J n + 1 ( x ) (38)

Subtracting equation (37) from (38), gives

2 n x J n ( x ) − 2 J ′ n ( x ) = J n − 1 ( x ) + J n + 1 ( x ) − J n − 1 ( x ) + J n + 1 ( x ) (39)

Or

n x J n ( x ) − J ′ n ( x ) = J n + 1 ( x )

Or

n J n ( x ) − x J ′ n ( x ) = x J n + 1 ( x )

This results in

x J ′ n ( x ) = n J n ( x ) − x J n + 1 ( x ) (40)

Differentiating (40) with respect to x, gives:

x J ″ n ( x ) + J ′ n ( x ) = n J ′ n ( x ) − x J ′ n + 1 ( x ) − J n + 1 ( x ) (41)

Multiplying Equation (40) in n/x gives:

n J ′ n ( x ) = n 2 x J n ( x ) − n J n + 1 ( x ) (42)

Adding Equations (37) and (38) gives

x J ′ n ( x ) = x J n − 1 ( x ) − n J n ( x ) (43)

Putting n = n + 1 in Equation (43) gives

x J ′ n + 1 ( x ) = x J n ( x ) − ( n + 1 ) J n + 1 ( x ) (44)

Substituting (39) and (40) into (38), results in

x J ″ n ( x ) + J ′ n ( x ) = n 2 x J n ( x ) − x J n ( x )

Then

J ″ n ( x ) + 1 x J ′ n ( x ) + ( 1 − n 2 x 2 ) J n ( x ) = 0 (45)

Finally

J ″ + 1 x J ′ + ( λ − n 2 x 2 ) J = 0 (46)

For n = 0

J ″ + 1 x J ′ + ( λ ) J = 0 (47)

For the Bessel condition in cylindrical coordinate

x = K r = ( 1 D a τ ) 1 / 2 R = ( λ D a ) 1 / 2 R

with the solution J n ( x ) = 2 π cos ( x − n π 2 − π 4 ) , and J 0 = 0 for n = 0, gives

0 = cos ( x − π 4 ) with the solution x = 3 π 4 ≈ 2.4

x = K r = R ( D τ ) 1 / 2 ≈ 2.4 or R 2.4 = ( D τ ) 1 / 2

Then

R 2.4 = ( D S ν i ) 1 / 2

or

D S ν i = ( R 2.4 ) 2

Appendix B: From the Boundary Conditions [n e υ e = − D S ∇ n e

Then

D S = − n e υ e ∇ n e (48)

But

ν i = ∇ ⋅ n e υ e n e

Then

D S ν i = − ( n e ∇ n e ) 2 (49)

For boundary conditions r = R , n e = n e 0 , and J 0 = 2.4 , then

∇ n e = − 2.4 n e R (50)

Substituting from (50) into (49) then we get

D S ν i = ( R 2.4 ) 2