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 (Iri)    .
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 plasma parameters especially electron temperature and electron density can be determined from the current-voltage characteristic curves of the probe  . Diffusion phenomena occur in a plasma when a spatial gradient of the charged species is present  . Furthermore, it is caused by the differently charged species having different diffusivities, hence there is a loss of neutrality  due to plasma rapidly diffusion for some of charged species more than the others. Furthermore, a minor loss of neutrality, however, induces an ambipolar electric field which, if the Debye length is sufficiently small, slows down the fast-diffusing species and speeds up the slow-diffusing species in such a way that the plasma remains quasi-neutral  .
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   , but not the theoretical derivation in terms of the effect of Schottky condition and ambipolar diffusion due to the recurrence relation of the Bessel condition.
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.
2. Experimental Study
2.1. Experimental Set-Up
Figure 1 shows the experimental setup of DC (cold cathode) magnetron sputtering unit to generate a glow discharge in a glass tube between two circular, parallel and movable metallic discs acting as electrodes. Two parallel electrodes made of aluminum, are enclosed in the discharge cell, one of the two electrodes is grounded represented cathode electrode and the other is movable to change the axial distance represented the anode electrode, where the axial distance (L) between tip of the probe and the cathode electrode, 5 cm and with 3 cm in radius (R) taking into account that we will deal with two conditions R > L and R < L. The discharge unit is evacuated using a rotary pump to a base pressure of 7 mtorr. A pressure gauge is connected to the discharge tube to measure the inside gas pressure. A stationary DC-glow discharge was generated between two both electrodes, for different parameters as shown in Table 1.
Figure 1 also shows the schematic diagram of the spherical single Langmuir probe circuit. The probe made of molybdenum wire (diameter 3.0-mm and length 0.5 mm) and the tip of a probe is inside the glow discharge plasma. The single probe (between the cathode fixed at the ground potential, and the anode) move axially to measure the axial potential distribution between the two electrodes. Also, it moves radially from the center to the edge of the cathode electrode, and a potential VP is applied to the probe.
2.2. Experimental Study of Axial Potential and Electric Field Distribution Measurements
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 Figure 2, shows that the potential
Figure 1. Schematic diagram of the experimental setup and the single Langmuir probe circuit.
Figure 2. Axial potential distribution at constant current (I = 10 mA).
Table 1. Shows the values of operating parameters.
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 Figure 2. Figure 3 and Figure 4 show the electric field distribution for He discharge. In the cathode fall region at edge and center, high electric field is observed which is decreased sharply away from the
Figure 3. I-V curves of the single probe for cathode fall region at different He pressures at the edge.
Figure 4. Measured I-V curves of the single probe for cathode fall region at different He pressures at the center.
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.
2.3. Experimental Study of Radial Dependence of the Electron Temperature
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   .
Figure 3 and Figure 4 show I-v characteristic curves of the single probe at different He pressures at edge and center of the cathode electrode for the abnormal glow region  . The probe was moved to investigate the radial distribution from the center to the edge of the cathode, in the direction perpendicular to the direction of the electric field lines. Figure 5 and Figure 6 show the radial temperatures and densities distributions at different pressures from center to the edge. Te is decreased and ne increased due to the general trend that values of Te and ne are inversely proportional  .
Increasing the helium working pressures from 0.2 to 1.2 torr, the temperature Te 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 ne increased at center ne from 0.6 × 109 cm−3 to 1.3 × 109 cm−3 and at the edge from 2.5 to 3.95 × 109 cm−3.
3. Theoretical Consideration of the Radial Dependence of the Electron Temperature
Conductivity at the tube axis, and charge density of plasma according to the following expression  
where Da is the ambipolar diffusion coefficient as follows:
The flow of ions and electrons are the same, hence
Figure 5. The Experimental radial temperature distribution of abnormal region for helium at different pressures.
Figure 6. Measured radial density distribution of abnormal region for helium at different pressures.
(Congruence approximation). If there is no external electric field, the fluxes of ions and electrons (drift-diffusion model) can be expressed as:
where ED follows Poisson’s equation. If we multiply the first equation by and the second one by and then subtract them, we get:
Since , the terms with ED cancel each other out and using ne = ni, the expression for the flow of particles becomes:
With the ambipolar diffusion coefficient being
and is the charge density in the plasma given by:
and is the conductivity at the tube axis given by:
Considering that diffusion coefficient D given by:
where are the mobility of electrons (e) and ions (+), kT (ev) is the particle temperature, 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
But by substituting by the Einstein relation   :
into Equation (6) by the value of D, then we get
Neglecting , where finally results in
Substituting and from Equations (5) and (7) into Equation (12) results in
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   stating that:
3.1. In the Case of L (Axial Distance) > R (Radial Distance)
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
Substituting the following equation into Equation (18)
where P in torr, and are the cross sections of electron-neutral and ion-neutral collisions, respectively results in
But from  
Then substituting (7) into (22a) gives
Moreover, substituting (22b), and into (21) gives
Substituting the values of the cross sections, masses of ions and electrons into (24) gives
Using the data of the electron temperatures and densities shown in Figure 4 and Figure 5 respectively, into the Equation (27), taking into account that ( ),  and for the spherical probe, the ion density is given by  :
Knowing Te, ( ) area of spherical probe) and calculating positive ion current I+ from the I-V characteristic curve of the spherical single probe  , n+ can be determined.
Figure 7 shows the radial distribution of the electron temperature (RDFT) theoretically using Equation (27), in the abnormal cathode fall region from the center to the edge of the electrode as a function of He as an inert gas. The RDFT from the theoretical method presented above accurately shows a dramatic radial reduction of the electron temperature for any region (cathode fall or negative glow or positive column) and for any applied pressure. Furthermore, the decrements of Te at the edge of the electrode are more pronounced at the center due to edge effect  .
3.2. In the Case of L (Axial Distance) < R (Radial Distance)
and by substituting Equation (29) into (16), then
Substituting by Equations (19a, b and c) into Equation (30), respectively results in
Figure 7. Theoretical radial distribution of the electron temperature (RDFT).
Substitution from (22a and b) into (32) gives
Substituting the values of the cross sections, masses of ions and electrons into (34) gives finally
For 1.2 torr of applied pressure, Figure 8 shows comparison between the radial distribution of the electron temperature (RDFT) for L < R using Equation (35) and for L > R using Equation (27), in the abnormal cathode fall region from the center to the edge of the electrode as a function of He as an inert gas.
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  :
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 the radial distribution of the electron temperature.
For L < R, RDFT accurately shows a dramatic decrease by 60% less than RDFT values for L > R. This may be attributed to:
Figure 8. Comparison between the radial distribution of the electron temperature (RDFT) for L < R and for L > R, at P = 1.2 torr.
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  , therefore the current and current density are increased.
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.
Appendix: Proofing of the Schottky Condition
There are two classes of Schottky condition models using.
Appendix A: the recurrence relation for the Bessel condition
Appendix B: the boundary conditions
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
can be demonstrated as follows:
Appendix A: From the Recurrence Relation for Bessel Condition   
Subtracting equation (37) from (38), gives
This results in
Differentiating (40) with respect to x, gives:
Multiplying Equation (40) in n/x gives:
Adding Equations (37) and (38) gives
Putting n = n + 1 in Equation (43) gives
Substituting (39) and (40) into (38), results in
For n = 0
For the Bessel condition in cylindrical coordinate
with the solution , and for n = 0, gives
with the solution
Appendix B: From the Boundary Conditions  
For boundary conditions , , and , then
Substituting from (50) into (49) then we get