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The main task in investigating a plasma is to determine its characteristic parameters: density, potential, temperature, and particle energy distribution. Many methods for investigating plasmas have been developed; involving microwaves, lasers, particle beams, spectroscopy, and probes [1-5].

The microwave method reduces to a study of the interference and absorption of radio waves passing through the plasma. Determination of the electron density n_e is based on the dependence of n_e on the plasma index of refraction. For the probing signal to pass through the plasma, its frequency must be higher than the plasma frequency n _pl = (ne²/p m)^1/2. Developments with millimeter and sub-millimeter waves have permitted measurements of density by this method up to the range 10¹²-10¹³ cm^-3.

The use of lasers has permitted considerable improvement in the methods of optical interferometry of plasmas, and also has made it possible to determine the density and temperature with high spatial resolution by light scattering in the plasma. Laser diagnostics, based on the interaction of powerful coherent radiation with the plasma, permits the measurement of densities on the order of 10¹⁵ cm^-3 (and above) and of temperatures on the order of l0⁴ - 10⁵ ° K.

The microwave, laser, and particle methods do not "contact" the plasma and are employed to diagnose high-temperature plasmas. When investigating low-temperature plasma, and especially TIC plasmas, diagnosis is primarily carried out by spectroscopic and probe methods. This chapter is devoted to a description of these methods. Among other methods, we also mention electron beam diagnostics - a variation of the particle method - used in [6] to determine the potential distribution in the interelectrode space of a TIC at low pressure.

There is no general formula for the current—voltage characteristic of a probe that takes into account all the geometric and physical parameters of the problem. But this is hardly necessary. In view of the inevitable complexity of this type of formula, it would be inconvenient to use in the progress of an experiment. However, there are simple formulas and graphs available, and although they are valid for a limited range of parameters, they are convenient for determining the characteristics of a plasma from the experimental data [1,7].

When the negative bias voltage decreases, fast electrons from the plasma begin to reach the surface of the probe in ever increasing numbers. This gives the mixed flux region II, located about the potential V₀^' of a "floating" probe. With further decrease of the negative probe potential, a rapid increase of electron current (region III) is observed. When the probe potential approaches the plasma potential, the rapid increase of electron flow subsides (region IV). If the probe potential is made increasingly positive, a relatively small increase of electron flow (region V) is observed. But if there is intensive generation in the pre-probe sheath, segments IV and V are distorted and electron "saturation" is not achieved, as is shown in Fig. 7.1 by the dashed line.

The case where the probe radius is small in comparison to the mean free path (a/l « 1) is called the Langmuir case [1, 7]. Here, the particles collected by the probe undergo final collision at a distance where the probe is visible through a small solid angle. Therefore, in the case of a Langmuir probe, the distribution function - of the particles entering the plasma region distorted by the probe - is close to equilibrium, and particle motion along trajectories (disregarding their collisions) need be considered only near the probe. In the opposite case of a dense plasma [8, 9] (a/l » 1), collection of the particles by the probe is determined by the conditions of current passage at distances exceeding the mean free path, where the trajectories of individual particles need not be considered and where such macroscopic characteristics as mobility and diffusion can be used. For a/l » 1, the region near the probe, where perturbation is strong and where more precise calculation of the trajectory of individual particles is required, is relatively narrow and does not play an important role in the boundary conditions presented in §7, Chapter 6.

Finally, the intermediate case (a/l » 1), which is considered in [10, 11], can occur under some experimental conditions.

Another important dimensionless parameter in probe theory is the ratio of the probe radius to the Debye length a/L_D. As was shown in Chapter 6, the main potential drop between the metal electrode and the plasma occurs in the space-charge region, which has a thickness on the order of the Debye length. However, part of the voltage (» kT/e) may occur in the region where, despite the variation of electron and ion density, the plasma remains almost neutral. A quasi-neutral region with a rather strong field is extended to distances considerably greater than the Debye length.

In a dense plasma, where a/l » 1, the charged particle density is usually high, so that L_D « l, and the passage of particles through the accelerating layer of the space charge does not affect the value of the current itself. In the Langmuir case, where a/L_D < ~ 1, an increase of negative voltage leads to an expansion of the region of the electric field which draws the currents to the probe. In this case, for a/L_D < ~ 1, the ion flow (segment I of Fig. 7.1) does not saturate. For a/L_D » 1, expansion of the region of the drift field with an increase of voltage is small compared to the dimensions of the probe and the ion current is essentially saturated.

An important parameter in probe theory is the ratio of electron to ion temperature T_e/T. For T_e/T » 1, as indicated in §2, Chapter 6, one can disregard the initial ion velocities after charge exchange collisions. This considerably simplifies the analysis.

From the viewpoint of TIC plasma diagnostics, the following cases are of interest: 1) the ratio a/L_D » 1; 2) a/l varying from the Langmuir case to a dense plasma; and 3) the ratio T_e/T varying from 1 to 3-5. Moreover, in many cases it is desirable in a TIC, if only approximately, to take into account the proximity of the electrodes. The magnetic field is usually weak and may be disregarded.

Langmuir probe theory. The theory of spherical and cylindrical Langmuir probes can begin with a consideration of the trajectories of charged particles in a potential eV(r) of radial symmetry. From the law of conservation of energy, we have

where E₀ is the particle energy at the limits of the field, v_r and vf are the velocity components, and m is the mass.

where v₀ = (2E₀/m)^1/2 is the initial velocity and b is me impact parameter: (Fig. 7.2). By substituting (1.2) into (1.1), we find for the segment of trajectory where r(t) decreases,

The potential energy eV(r) in (1.3) is an unknown function which should be calculated from Poisson’s equation, taking into account all the possible electron and ion trajectories. However, one can make some qualitative conclusions from (1.3) which are not dependent on the specific form of eV(r).

We first consider the case of a repulsive field (eV(r) > 0). Since eV(r) increases as r decreases, the expression under the radical in (1.3) decreases simultaneously. The particle reaches the probe if r = a when dr/dt £ 0. Therefore, we find that the limiting value of the impact parameter is equal to

where V_p is the probe potential. It is very important that b_m is determined only by the probe potential and is not dependent on the specific type of function eV(r).

where n¥ is the charged particle density in an unperturbed plasma and f(v) is the distribution function. If the particles in the plasma have a Maxwellian distribution, then

where S_p is the total exposed surface of the probe.

Expression (1.8) describes segment III of the probe characteristic. The curve for the dependence of ln F on V_p for a Maxwellian distribution is a straight line whose slope e/kT_e permits easy calculation of electron temperature T_e. Deviations from linearity indicate a non-Maxwellian electron energy distribution. Accordingly, the form of the distribution function can be obtained from the probe characteristic of a spherical probe. According to (1.7), for a spherical probe, the total current is equal to

where f(v)@ = (1/4p )ò W f(v)dW is the mean value of the distribution function at a given velocity, averaged over the total solid angle W in velocity space. Differentiating (1.9) twice with respect to V_p, we find

We now turn to the case of negative potential energy near the probe (segments I and V of the probe characteristic in Fig. 7.1). One of the possible particle trajectories in the case of a collecting potential is shown in Fig. 7.2. We rewrite (1.3) with consideration for the fact that eV(r) < 0:

*This follows directly from the principle of detailed balance. If the probe has the temperature of the plasma, then at V_p = 0 it is in equilibrium with the plasma. The thermionic emission per unit surface of the probe, and consequently, the current to the probe are equal to l/4(en¥ v@ ). An increase of the barrier is equivalent to an increase of the work function by eV_p. In this case it is important that all the emitted electrons for a probe with a convex surface enter the plasma and that the probability of electrons returning to the probe is low.

It is obvious from (1.11) that if eV(r) decreases slowly as distance increases, so that the expression under the radical decreases monotonically as r decreases, then the maximum impact parameter for which the particles reach the probe, the same as for eV > 0, is determined by the condition dr/dt £ 0 as r ® a and is not dependent on the specific form of eV(r). Similar to (1.4),

In this case, current saturation should not be observed in segments I and V (Fig. 7.1): in the case of a cylindrical probe, the current should increase as V_p^1/2 and in the case of a spherical probe, it should increase as V_p.

However, under real conditions, the other case most often occurs, where the variation of potential occurs in a rather narrow sheath, so that the function r²(1 + ï eV(r)ï /E₀) has a minimum somewhere near the probe, at r = r_m. Particles can reach the probe having a small impact parameter b < r_m in which dr/dt = ¬ 0 at b = r_m. For particles with b > r_m, dr/dt approaches zero and changes sign at r > r_m > a, i.e., these particles travel past the probe. The value of the maximum impact parameter at which the particles reach the probe is

Thus, b_m is dependent on V(r_m) and the problem of the potential distribution near the probe must be solved completely to calculate b_m.

It follows from a general consideration of the pre-electrode sheaths (see §2, Chapter 6) that the region of quasi-neutrality has a voltage of about kT/e, where T is the temperature of the particles retarded by the potential barrier. All of the remaining potential drop is contained in the thin layer of space charge near the probe. Therefore, it follows that, when the electron current is collected by the probe (segment V in Fig. 7.1), we have ï eV(r)ï /E₀ » T_i/T_e. When T_i/T_e « 1, parameter b_m satisfies b_m » r_m » a. In this case, good saturation should be observed, at which the current density to the probe should be the same as that at zero probe potential, i.e.,

The opposite case is also relatively simple. Ion current (segment I in Fig. 7.1) is collected by the probe and the temperature T of the collected particles is much less than the temperature T_e of the particles repelled by the probe. In this case, the initial ion velocities can be disregarded and one may simply assume that the ion concentration in the sheath (for a spherical probe) is

where F_is is the saturation ion current to the probe.

Equation (1.17) is rather complicated. A similar equation was solved for the plane case in §2, Chapter 6, where it was shown that the space-charge boundary is located somewhere near the point where the condition of quasi-neutrality n_i » n_e (which is fulfilled at large distances from the probe) leads to the infinite derivative dV/dr. Therefore, instead of solving equation (1.17) completely, we differentiate the equation of quasi-neutrality n_e = n_i. By using (1.15) and (1.16) in this case, we find that dV/dr ® ¥ at V = 1/2(kT_e/e). Therefore, the saturation ion current density is

where the coefficient 0.43 is (2e)^-1/2. A similar formula is found for the ion saturation current density to a cylindrical probe.

The most complicated case is the one where the electron temperature is close to the ion temperature.* Poisson’s equation (1.17) was also solved numerically for this case in [12, 13]. The expression for ion current has the form of formula (1.18) with some coefficient weakly dependent on T/T_e.

If the Debye radius is less than but comparable to the probe radius a, then the ion current in the "saturation" segment increases slightly as voltage increases because of the expansion of the strong field region near the probe. The relationship of ion current to density for this case was determined in [13]. The approximate theory of [7] also yields good results.

Thus, the Langmuir probe permits the calculation of the density of an unperturbed plasma, n¥ , both from the probe electron saturation current and

from the probe ion saturation current. In this case, the electron temperature T_e is obtained from the slope of the semi-logarithmic probe characteristic. The plasma potential is calculated as the point of intersection of the extrapolated linear segment of this characteristic with the electron saturation current. The plasma potential can also be calculated from the value of the floating probe potential V₀^', where the net current to the probe is equal to zero, i.e., F_e = F_i. From (1.8) and (1.18), we have

Probe theory for a dense plasma. Probe technique for a dense plasma, where the probe radius is much greater than the mean free path, is of great practical significance and has been analyzed and checked extensively during the past few years [2, 8, 9, 14, 15]. The electron and ion currents in a dense plasma are described by diffusion equations (4.3.3) and (4.7.2). The diffusion equations are applicable at large distances from the electrode, where the relative variation of plasma parameters over the length of a mean free path is small. Equations (4.3.3) and (4.7.2) should be replaced by more accurate kinetic equations near the electrode, where the plasma parameters change very rapidly. However, as indicated in §7 Chapter 6, the effect of the region near the electrode can be taken into account by introducing the appropriate approximate boundary conditions.

We begin our analysis with the ion saturation current, where a large negative potential is applied to the probe. In this case, we have j_e = 0, and the density is related to the potential distribution near the probe by the Boltzmann factor (1.15). The ion current density to the probe, according to (6.2.3), is

Since div j_i = eG (n, T_e, . . .), where G is a generation function, the equation for the distribution of density near the probe has the form

where Ñ ² is a Laplacian operator.

Equation (1.21) simplifies considerably when ionization in the pre-probe sheath can be disregarded. This is permissible when the ionization length L_i is much larger than the probe radius. In this case. to determine the density, we solve Laplace’s equation

Here n₀ is the value of density at the interface of a quasi-neutral plasma and the space-charge sheath, whose thickness we shall disregard (a » L_D).

where g _i = (v_di)₀/v_i@ = g ₀(Ö p /2)(T_e/T_i)^1/2. The variable (v_di)₀ is the ion drift velocity at the boundary, and g ₀ » 0.76 (see §2, Chapter 6). From (1.25) and (1.26), we obtain, using relation D_i = 1/3(l_iv_i@ ),

therefore, it is obvious that the precise value of the boundary density n₀ is not essential for determining the ion flux to the probe for a » l_i and n₀ « n¥ .

The solution of (1.22) for a cylindrical probe with a finite density near the probe has a logarithmic divergence at infinity. Therefore, the density n¥ far from the probe must be specified at r = R₀, (where R₀ is equal to an order of magnitude to the dimensions of the perturbed region) rather than as r ® ¥ .

The boundary value of the density, n₀, as in the case of a spherical probe, is calculated by equation (1.27), and for a » l_i (when n₀ « n¥ ), the flow to the probe is hardly dependent on n₀.

Due to the fact that R₀ » a and is contained in the logarithm, the concentration n¥ , calculated from the ion saturation current, is weakly dependent on the value of R₀. For a large plasma volume, one may assume that R₀ is approximately equal to the length of the cylindrical probe, i.e., to the distance where the cylindrical current distribution is converted to a spherical distribution. In a TIC, the length of a probe usually exceeds the distance between the electrodes, so in this case, R₀ should be on the order of the distance to the nearest electrode .

R₀ can be calculated more accurately from the solution of equation (1.21), with the corresponding boundary conditions at the electrodes or at infinity. If the ionization length L_i is much less than the distance over which the parameters of the unperturbed plasma vary appreciably (considered briefly at the end of this section) R₀ is on the order of L_i and is weakly dependent on the other plasma parameters (see also [14, 15]).

If a positive potential is applied to the probe and generation in the pre-probe layer can be disregarded, the distribution of density near the probe is described, as previously, by the solution of equation (1.22) with boundary conditions (1.23). The formulas for the saturation density of the electron current to the probe may also be obtained by simply by replacement of subscript "i" (denoting ions) by subscript "e" (which indicates electrons) in formulas (1.25) and (1.29).

where g _e = (v_de)₀/v_e@ and (v_de)₀ is the electron drift velocity at the boundary. It is obvious that formulas (1.30) and (1.31) in the limiting case of a « l_e convert to formula (1.14) for a Langmuir probe, if we assume that g _e = 1/4.

where coefficient r_k takes into account the depletion of the fast electron distribution function because of the current to the probe. The most reliable determination of T_e for eV » kT_e is when (1 - r_k) varies only slightly compared to exp(-eV/kT_e) as the height of the potential barrier increases. However, the collection of fast electrons by the probe may also affect the overall electron temperature near the probe, i.e., when electron cooling is not balanced rapidly by heat conduction through the electron gas. The condition of heat flux continuity for a cylindrical probe has the form

where V is the probe potential with respect to an unperturbed plasma and k _e is the electron component of heat conduction.

To calculate the temperature decrease near the probe, we set V(r) = 0, i.e., we disregard the potential drop in the plasma volume compared to the total potential difference. Then

According to the Wiedemann—Franz law, k _e(r) = 2(k/e)²T_eem _en_e(r). Having substituted the expression for density (1.28) into (1.34) and having carried out the integration from the probe surface (where n = n¥ , and T_e^' = T_e - D T_e) to R₀ (where n = n¥ and T_e is the temperature of an unperturbed plasma), we find that the temperature decrease near the surface of a cylindrical

It is obvious from (1.35) that the temperature decrease near the probe surface is insignificant in the initial probe-characteristic segment for electron flow, when F_e = F_i, and the semi-logarithmic probe characteristic yields the correct value of electron temperature for a wide range of densities in the plasma. If F_e increases, the electron temperature near the probe decreases, the electron current is reduced, and the measured electron temperature is elevated [16].

If the electron part of the probe characteristic has a saturation segment, the potential of an unperturbed plasma may also be calculated from it. At the point of the probe characteristic where V_qn = 0, i.e., where there is no electric field in the pre-probe quasi-neutral layer, the electron flow to the probe is

where j_es is the value of the electron saturation current to the probe.

By using (1.40), it is easy to determine the point of the probe characteristic where V_qn = 0. However, if the electron mean free path differs from the ion mean free path, yet another correction must be introduced at this point for the value of the pre-electrode barrier V_sc^', which for l_e > l_i should retard the ions emerging from the plasma:

Coefficients g _e and g _i may be assumed equal to 1/2 under these conditions.

Probe theory for the intermediate case. The formulas derived above make it possible to calculate the charged particle density, electron temperature, and space potential in a dense plasma (a/l » 1) and in a Knudsen plasma (a/l » 1). The intermediate case (a/l » 1), considered in [10, 11], is more complicated. In these latter investigations, the simplifying assumption T_e/T_i » 1 is made, which makes it possible to disregard diffusion current compared to field current. In the case of a strong electric field, in which the ion mobility is strongly dependent on the ion scattering law, calculations were carried out for the case of ion motion in a parent gas where the main scattering mechanism is resonance charge exchange [17]. Calculations were carried out for two cases: 1) very large values of T_e/T_i, where one may assume for the entire pre-probe sheath that the ions start at zero initial velocity after scattering; and 2) large but finite values of T_e/T_i (for example, T_e/T_i ³ 4). In the first case, one can obtain the solution for the entire pre-probe sheath, but in the second case, one must approximately "sew together" the solution for the different regions of the pre-probe sheath.

Equations (1.42) and (1.43) were solved numerically on a computer. The solution was constructed with a given current F_i from the plasma to the probe and with an asymptotic solution at large distances, which was obtained by the substitution of the approximation f (r) - f (r') = (r - r')d/dr into the differential equation. The solution of equations (1.42) and (1.43) can be constructed only over those values of r where the condition of quasi-neutrality is fulfilled, i.e., where the Laplace potential satisfies

However, the violation of quasi-neutrality essentially coincides with the divergence of dV/dr. This point was assumed to be at the probe radius.

The dependence of the dimensionless ion saturation current density A = F_ie[en¥ S_p(2kT_e/M)^1/2] on the ratio of probe radius to the length of the ion mean free path, a/l_i, for spherical and cylindrical probes is shown in Fig. 7.3 (where, as previously, it is assumed a » L₀). Although one may expect that the method for solving equations (1.42) and (1.43) has the greatest error at small probe radii, both curves yield a maximum value close to A = - 0.43 as a/l_i ® 0, which is obtained for the Langmuir case a: T_e/T_i ® ¥ (see (1.18)). The values of A that are obtained if local mobility is used through the entire pre-probe layer are shown in the same figure by the dashed lines. It is obvious that this assumption yields quite satisfactory results at large values of a/l_i, although the asymptotics of the solution are incorrect as a/l_i ® 0.

Calculation of the potential distribution in the pre-probe sheath by formulas (1.42) and (1.43) shows that the electric field intensity decreases rapidly as the distance from the probe increases, so that the initial ion velocities after scattering may in fact be disregarded only for very small ratios of T_e/T_i.

Calculation of initial velocities requires the integration over different ion trajectories after charge exchange and severely complicates the equation for the potential. Therefore, the plasma in the region of quasi-neutrality was divided into two regions in [11]. The initial ion velocities were taken into account in region III* remote from the probe, but the dependence of current on the field intensity was assumed to be local, i.e., mobility was introduced. The diffusion current in region III was disregarded compared to the field current because T_e/T_i was assumed to be large. The non-local dependence of

Thus, it is assumed that in region III (r > r₀)

and in region II (r < r₀)

s = 1 and B = F_is/2p l_iL in the cylindrical case, and

s = 2 and B = F_is/2p L_i² in the spherical case;

and n(r) = n_e exp(-eV/T_e) in the case of ion saturation current. The first term on the right side of (l.45) is the density of ions which are scattered in region II, and the second term is that of the ions emerging from region III that are not scattered.

Determination of the precise form of the ion distribution function at the interface of region II and III is an independent problem. Moreover, since different assumptions are made in II and III with respect to the initial velocities, function f(v) can be calculated only approximately. However, function f(v) is contained in (l.45) only in the integrand, and therefore, the validity of (l.45) will hardly depend on the exact form of f(v) if the plasma parameters vary smoothly where the solutions join together. Function f(v) was taken in [11] to have the form

Parameters a_s were selected to give continuity of ion current, potential V, field dV/dr and space charge d²V/dr² at r = r₀. Parameter @ was left free, and the independence of the solution from the form of the function f(v) was checked by comparison calculations with reasonable values of C.

One of the potential distribution curves obtained by this method is shown in Fig. 7.4. The boundary r₀ of region II and III and the probe radius a are shown in the same figure. The curves for the variation of the dimensionless ion saturation current A as a function of a/l_i for spherical (a) and cylindrical (b) probes are shown in Fig. 7.5. In this case, the solution for a cylindrical probe was truncated at a distance from the surface of the probe R₀ = 10 a. At this distance the field is already weak and the solution can be easily continued to large distances.

Similar functions, which are obtained if the calculations are carried out with constant mobility up to the surface of the probe using formulas (1.25) and (1.29), are shown by the dashed curves in these same figures. These results were reported in [8, 9]. It is obvious that the difference of the corresponding curves is slight for large values of a/l_i and for ratios of T_e/T_i usual for a TIC. This is related to the fact that the large field region is narrow compared to the probe radius and, therefore, has a weak effect on the value of current.

According to calculation, the potential drop in the region of quasi-neutrality is very sensitive to the ratios a/l_i and T_e/T_i. However, the ratio of drift velocity to thermal velocity at the boundary of the region of quasi-neutrality is very weakly dependent on both these parameters, and the value of g ₀ = (v_di)₀/(2T_e/M)^-1/2 remains within the range 0.6 - 0.8.

Probe theory in a strongly ionized dense plasma. Probe theory for a weakly ionized plasma was outlined in the preceding sections. The expression for ion flow to a probe in a strongly ionized dense plasma (a » l_i) is presented in this section.* Since a strongly ionized plasma is characterized by a comparatively high electron temperature, there will be intensive ion generation near the probe. If the ionization length (the distance over which the ions deflected toward the probe are generated) is large compared to l_i, the following expression can be obtained for the ion saturation current density of a cylindrical probe with radius a [14]:

*The main characteristics of a strongly ionized plasma in a TIC is considered below in §11, Chapter 9.

where D_i0 is the ion diffusion coefficient in a weakly ionized plasma at the interface with the probe; t _e = T_e/T_i; n¥ is the ion density far from the probe, where the ionization is balanced by recombination; the plasma is in a state of local thermodynamic equilibrium (LTE) (see §4, Chapter 9); L₀ = Ö D_i0t ion(1 + t _e)/2@ *; t _ion = [v_e@ s ₀(T_e)N_a]^-1 is the

effective ionization time; and x₀ = a/L₀.

The function Fb _{,t e}(x₀) is dependent on the three dimensionless parameters: x₀, t _e, and b = n¥ /N_a¥ , where N_a¥ is the density of atoms far from the probe in a local thermodynamic equilibrium (LTE) plasma. However, the dependence of b and t _e is very weak, which is illustrated by Fig. 7.6b. Curve 1, which gives the function F_{0,t e}(x₀) for a weakly ionized plasma, is not dependent on t _e and coincides with a similar curve calculated in [15]. For x₀ « 1, where the ionization near the probe is less important, j_is (to a numerical factor of unity) follows from formula (1.29), in which the radius of the perturbed zone is now determined by the ionization length L₀, namely, F_{0,t e}(x₀) = [x₀ ln(1.63/x₀)]^-1. Curves 2, 3 and 4 are for a completely ionized plasma (b = ¥ ) and determine functions Fb _{,t e}(x₀) for values of t _e equal to 2, 3 and 10, respectively. Curves Fb _{,t e}(x₀), which correspond to intermediate values of b (0 < b < ¥ ), are located between the curves for F_{0,t e}(x₀) and F¥ _{,t e}(x₀) in Fig. 7.6b.

Fig. 7.6a corresponds to a » L₀(x₀ ® ¥ ). In this case, Fb _{,t e}(x₀) coincides with that provided by the theory of ion flow to a plane electrode (see §4 and 11, Chapter 9). The values of y (b ,t _e) = Fb _{,t e}(¥ )/F_{0,t e}(¥ ) that characterize the dependence of ion flow on the degree of ionization b , is shown in Fig. 7.6a. It is obvious that y (b ,t _e) hardly differs from unity. The above results belong to the case where L₀ » l_ia. It is possible to show that if l_ia « L₀ the expression for j_is is the same as in (1.18).

Experimental verification of probe theory. The Langmuir theory of probes has been experimentally verified in many investigations. Experience using probes at medium and high pressures is rather limited [8, 9, 18, 62]. The most thorough check of probe theory was carried out in [18], where probe operation in cesium vapor over a wide pressure range was investigated.

The values of dimensionless ion flow density to a probe, A = j_is[en¥ (2kT_e/M)^1/2], as a function of the ratio of probe radius to the length of the ion mean free path are presented in Fig. 7.7. The solid curve is the result of the calculation of [11] and the experimental points are from measurements by probes of different diameters. The electron density was determined by an independent spectral method using the intensity of the recombination 6P-continuum (see §3 of this chapter). The measurements were carried out with a cesium vapor pressure range of P_Cs = 10^-3 - 10 torr and a plasma density range of n = 10¹¹ - 10¹⁴ cm^-3. The dimensionless ion flow in this pressure range was not dependent on either P_Cs or the density n, and was a function only of the ratio a/l_i.@

*L₀ is a factor 1/Ö 2 smaller than the ionization length L_i, calculated in §4, Chapter 9.

@It should be noted that the value of A for constant ratio a/l_i begins to increase as density increases at high pressures (P_Cs > 1 torr) and at high plasma densities (n > 5-6 x 10¹⁴ cm^-3). This effect is related to the onset of ion generation in the pre-probe sheath and to the transition of the plasma to a state of strong ionization.

Under the experimental conditions, the ratio of electron-to-ion temperatures was T_e/T_i » 3 - 4. With these ratios of T_e/T_i, the range of overlapping of regions (1.44) and (1.45) is very narrow, and the results of the calculations in this case are less reliable. However, the experiments showed that the calculated values of A, even in the most unfavorable case, determine rather well the flux to the probe over a wide range of the parameter a/l_i.

Probe operation in a strongly ionized dense plasma was also checked. The measurements showed that, under these conditions, the value of A is sensitive to the relative magnitude of the ion generation length L₀ and the mean free path l_i. For L₀/l_i » 1, the value of ion flow to the probe can be calculated with good accuracy by formula (1.25). As L₀/l_i increases, A also increases (in the limiting case of L₀/l_i « 1) and the ion flux to the probe is the same as in the case of a Langmuir probe (A = 0.43, formula (1.18)).

The densities in an isothermal plasma, calculated from the electron saturation current of the probe characteristic using formula (1.31), are presented in Fig. 7.8a. The values for the space potential found by formulas (1.40) and (1.41), as well as the calculated values of the plasma chemical potential, are compared in Fig. 7.8b. The potential energy of the electrons V = eV₀ + f _p - where V₀ is the plasma potential (in terms of the probe bias) and f _p is the probe work function which was assumed to be equal to 1.7 eV - is plotted as the ordinant? in this graph and in subsequent graphs where the potential distribution is presented.

Calculation of the thermal operating conditions of a probe shows that the probe temperature is mainly determined by the temperature of its supports. Among the factors which lead to additional heating of the probe, the more important ones are cathode radiation and ion flow from the plasma. The probe should have as large a diameter as possible and as short a length possible to give a minimum temperature and minimum thermionic emission. It is desirable to use a material with high heat conduction and a maximum work function in cesium vapor. Molybdenum and tungsten probes 0.10 - 0.15 mm in diameter and 4 - 6 mm long essentially satisfy these requirements.

The probe is usually soldered into a massive copper support, which is maintained at a constant temperature (10 - 20° above the cesium reservoir temperature) by using regulated gas cooling. Since the area of the support is many times greater than the area of the probe working surface, and since part of the holder is sometimes directly in the TIC gap, careful insulation of the support against currents from the plasma and suppression of thermionic emission from the probe support are required. The use of a layered insulation provides the best results [19]. However, in some cases, especially for high plasma densities, the great effort in fabricating this type of probe insulation is not justified, and simply covering the non-operating part of the probe and probe supports yields quite satisfactory results, especially if sapphire or small, low-porosity ceramics are used as insulators.

The type of probe characteristics. In the diffusion mode, where volume ionization is insignificant and electron density does not exceed 10¹¹ - 10¹² cm^-3, a more or less clearly defined section of electron current saturation is observed in the probe characteristics. Good saturation is also observed in the back current. However, experiments have shown that, in this case, the back current of the probe characteristic is often determined by thermionic emission from the probe rather than by the ion flow from the plasma. The ion flow from the plasma begins to exceed thermionic emission of the probe and an undistorted probe characteristic is obtained, only at very high cathode temperatures and high cesium pressures, where the plasma density exceeds 10¹² cm^-3. The typical probe characteristic for the diffusion mode is shown in Fig. 7.10. The probe characteristic, as a semi-logarithmic plot, is presented in Fig. 7.l0b.

Thus, plasma parameters for a diffusion mode plasma can be found, in most cases, only from the electron branch of the probe characteristic. The density is determined from the electron saturation current using formula (1.31), and the potential is determined from the potential V₂ (see Fig. 7.10b) taking into account the retarding barrier V_sc, calculated by formula (1.41). The electron temperature in the diffusion mode is rather difficult to obtain, since the initial segment of ion current is distorted by thermionic emission from the probe. Also for F_e » F_i, the semi-logarithmic probe characteristic of Fig. 7.l0b does not remain linear, because of a decrease of electron temperature near the probe and cause of the dependence of the kinetic reflection coefficient r_C on the voltage of the probe.

The type of probe characteristics in the TIC arc mode is very sensitive to the cesium vapor pressure. The electron flow to the probe is not saturated at high values of P_Cs, so that only the ion section and the initial electron current section of the probe characteristic may be used to determined plasma parameters. Since the density in an arc mode plasma is much greater than that in the diffusion mode, the thermionic emission from the probe is usually much less than the ion flow from the plasma to the probe.

A typical probe characteristic for the arc mode is depicted in Fig. 7.11. By using the value of the ion saturation current, the density of the unperturbed plasma is easily found using formula (1.29). Adding the voltage drop in the pre-probe sheath, V_p (calculated using formula (1.39)), to the potential of the floating probe, the plasma potential is obtained. The electron temperature is obtained from the slope of the semi-logarithmic probe characteristic.

The usual methods of eliminating probe heating by probe current modulation are unsatisfactory for this case, however, since the probe is heated by both electron and ion currents. Therefore, the current through the device rather than that to the probe must be modulated. To do this, square pulses with a length t and repetition period T are applied to the anode of the device. The values of t are selected so that the probe is not heated significantly during the application of the pulse when the discharge is ignited in the TIC gap and ion flow to the probe increases sharply. In this case, also, the pulse length t should be much greater than the time for the relaxation processes in the plasma, so that the mode can be assumed to be in steady state. The pulse method permits a considerable increase in the upper limit of the density that can be measured by probes.

For a dense plasma, the logarithmic divergence of the density near the probe must be terminated at a distance R₀ on the order of the distance to the nearest electrode. If the probe length L is much greater than the interelectrode distance d, and if generation can be disregarded, the distribution of density near the probe is described by the two-dimensional Laplace equation (1.22).

If introduction of the probe does not change the density near the electrodes, then from solution of the Laplace equation [20] we have

where b is the distance from the center of the probe to one of the electrodes.

The condition of constant concentration near the anode is almost always fulfilled, since the value of n is close to zero. The condition of constant concentration near the cathode will be approximately fulfilled if those particles, of which there is an excess near the cathode, are collected by the probe, i.e., for the electron probe saturation current in the under compensated TIC mode and the ion probe current in the overcompensated TIC mode.

If the above condition is not fulfilled, introduction of the probe alters the density near the cathode. If we assume that the probe does not change the current near the cathode surface, then from the solution of the Laplace equation with dn/dx = 0 at x = d

It should be emphasized that the accuracy of determining the density in the middle part of the gap for R₀ » a is rather weakly dependent on the selection of R₀, since R₀ in formulas (1.29) and (1.31) is in the logarithm. However, as can be seen from (2.1) and (2.2), disregarding the dependence of R₀ may lead to an appreciable error in calculating the density distribution through the gap, especially near the electrodes.

The distribution in the gap of electron saturation current to the probe and the distribution of density calculated from it for an isothermal plasma is presented in Fig. 7.12. For selected values of T_C and P_Cs, the chemical potential of the plasma m is greater than the cathode work function f _C and the mode is under compensated. Therefore, the value provided by formula (2.1) was used as the cut-off radius R₀ when calculating the density.

The distribution in the gap of the density calculated from the electron and ion saturation currents for the open-circuit mode* is presented in Fig. 7.13. Formula (2.2) was used to determine R₀ from electron current, and formula (2.1) was used to determine R₀ from ion current for the specific values of T_C and P_Cs (f _C > m ).

The constant density in Fig. 7.12 and the coincidence of the density distributions in Fig. 7.13 obtained by the two methods shows the validity of formulas (2.1) and (2.2) for determining the cut-off radius R₀ when studying a plasma with a TIC geometry.

Additional ion generation in the pre-probe sheath may have an appreciable effect on the value of the ion current. In this case, R₀ should be on the order

Comparison of three methods of determining the plasma density at low cesium pressures - from the intensity of the recombination 6P-continuum (see §3, Chapter 7), and from the ion and electron saturation currents of the probe characteristic - is presented in Fig. 7.14. It is obvious that the calculation of the ion branch by Langmuir probe theory with conditions comparable to those in a confined TIC plasma yields correct values for the density. The proximity of the electrodes has an appreciable effect only on the electron part of the probe characteristic. The independence of the electron saturation current on density for n > ~ 5× 10¹² cm^-3 is related to the fact that the current density of the electrons collected by the probe in the narrow TIC gap exceeds the current density of cathode emission.

Before turning to the description of optical methods of measurement, which are the more widely used when studying low-temperature plasma, particularly TIC plasmas, we note that these problems are considered in more detail in special monographs [1 - 4, 21 - 27].

The main radiation properties of a plasma include the intensity, width, and shape of the spectral lines for emission from excited atoms, and also the intensity of the continuous spectrum for recombination emission. The relationship between the radiation characteristics and plasma parameters was considered briefly, in §7, Chapter 3.

Since continua from other lines, especially the 5D-continuum, whose contribution is approximately 30% [55], are superimposed on the 6P-continuum, this superposition must be taken into account when calculating the true radiation intensity of the 6P-continuum.

254

The absolute radiation intensity of the continuum must accordingly be known to calculate absolute values for the density. To do this, an appropriate calibration of the radiation recording system must be carried out.

Determination of temperature - by emission lines. By measuring the absolute line intensity S_lk from an optically thin plasma and by knowing the oscillator force f_kl@, the total number of atoms in a given excited state can be determined. This follows from formulas (3.7.13)-(3.7.15). Therefore, by knowing the concentration of atoms N_a, the population of the l-level can also be found:

(3.6)

Here, L is the plasma sheath thickness and the factor 4p arises because the radiation intensity is defined per unit solid angle. The population of the l-level can be characterized by a corresponding temperature

(3.7)

Standard continuous spectrum sources are usually employed to calibrate the radiation intensity of a line in units of energy. When possible, the simplest practical method is a measurement at the known radiation temperature of a black body, which is placed in the same space as the plasma being investigated [59]. When selecting the wavelength close to the investigated line, this method eliminates errors related to absorption and reflection of light from the viewing ports through which the radiation emerges.

When a measurement of absolute intensity is made difficult by the self-absorption of a plasma, temperature T_l and the corresponding level population may be calculated by line reversal, i.e., by comparing the plasma radiation and that of a calibrated source, that passes through this plasma [4, 37-39].

When the absolute radiation intensity is unknown, the relative population of two levels l and m can be determined by comparing the relative intensities of the transitions from these levels to the common lower level k:

(3.8)

This relative intensity can be characterized by a corresponding temperature

(3.9)

If the atomic distribution function for the series of levels is close to Boltzmann, then the most accurate value of this temperature can be calculated by constructing the relations

(3.10)

255

for a large number of lines corresponding to transitions of one series from different levels to level k. The degree of deviation of the line from a straight line characterizes the deviation of the distribution of this series of excited levels from the Boltzmann distribution. The method of relative intensities provides satisfactory accuracy if the energy range of the excited levels is no less than 0.5 eV.

If the investigated plasma is in a state of thermal equilibrium, the temperature which determines the population of each of the levels and the temperature T_lm which characterizes the ratio of populations for levels l and m coincide with the electron temperature.

As will be seen later, modes are sometimes produced in thermionic converters where the plasma is not an equilibrium plasma. Under these conditions, comparison of the "excitation temperature" and electron temperature permits some judgment about the degree of deviation from thermal equilibrium.

Determination of temperature - by the continuous spectrum. According to (3.7.19)-(3.7.21), the radiation intensity of the continuum, provided the free electron distribution function is Maxwellian, is calculated by the formula

(3.11)

Here n is the density of free electrons and ions, Q_rec^(ph) is the electron recombination cross section, w _th is the threshold frequency for the edge of the continuum; h@ (w - w _th) is the kinetic energy of the electrons and T_e is electron temperature. A is a constant.

In principle, for an accurate determination of the electron temperature, it is necessary to take into account that not only the exponent but also the pre-exponential factor Q_rec^(ph)h@ (w - w _th) are dependent on frequency in (3.11). However, according to calculations (see Table 7.2) and experimental data [56], the value of Q_rec^(ph)h@ (w - w _th) is essentially constant in the range of wavelengths encompassed by the 6P-continuum in cesium vapor.* Therefore, the temperature T_e can be calculated directly from the slope of the straight line:

?

An optical system of quartz can be used for expanding the range of the Investigated energies to shorter wavelengths.

4. Comparison of the Probe and Spectral Methods

Both probe and spectral methods can be used to determine plasma parameters in the TIC arc mode. No distortions of any kind are introduced into the investigated space during spectral diagnostics of the plasma. However, the calculated parameters for the plasma are averages for the volume of the plasma layer cut by the spectrograph slit, rather than actual local parameters. An important advantage of the probe method is that it makes local measurements. Therefore, the simultaneous use of independent methods increases the reliability of the results obtained.

The probe and spectral methods for determining plasma parameters

_____________

*This is also valid for the 5D-continuum.

256

in the TIC arc mode were compared in [55]. Main attention was given to the measurement of free electron temperature and current carrier density.

Measurements for one of the investigated modes is presented in Fig. 7.18. The current-voltage characteristic of a converter (with the points where plasma measurements were made indicated) is presented in Fig. 7.18a. The density distribution obtained by three methods is shown in Fig. 7.18b. The letters "c, " "1" and "p" refer to measurements using the recombination 6P-continuum, the spectral line broadening caused by the quadratic Stark effect, and the probe method, respectively. The numbers 1 and 2 refer to the indicated operating points in the current-voltage characteristic. The distribution in the gap of the free electron temperature is presented in Fig. 7.18c.

Fig. 7.18

It is obvious from these figures that the plasma parameters determined by the probe and spectral methods are in satisfactory agreement with each other. Differences in absolute values of the density do not exceed a factor 1.4, which is obviously within the range of accuracy of the experiment and theory of both the probe and the spectral measurements. The relative distributions of density, obtained by different methods, agree with each other even better (Fig. 7.19).

Fig. 7.19

The electron temperature, measured by the relative intensity of the continuum, essentially coincides with probe measurements near the cathode, but is 300-400° below the temperature measured by the probes in the pre-anode region. The curves obtained by the spectral method apparently reflect more correctly the variation of the electron temperature through the gap than do the curves obtained with probes. However, it is possible that the abso1ute values of T_e calculated from the continuum, are somewhat underestimated because of some spreading of the plasma beyond the limits of the column bounded by the electrodes. For this same reason, the electron density obtained from the absolute

257

intensity of the continuum is greater than that by other methods of measurement.

Besides the direct determination of the plasma parameters by two independent methods, the plane geometry of the TIC permits a checking of the results by some other methods as well. One of these is calculation of the electron current in different cross sections of the plasma.

Calculation of electron current. A knowledge of the distribution in the gap of density, potential, and electron temperature permits the calculation of the individual components of the electron current (4.3.3): diffusion j_e(dif) = -eD_edn/dx, field j_e(mob) = em _endV/dx, and thermal diffusion j_e(th) = -e(r_e + 1/2)(D_en/T_e)dT_e/dx. Since the ion current in the plasma is usually much less than electron current, the sum of the individual components of electron current is approximately constant for different interelectrode positions and should be equal to the experimental value of the current through the device. Calculation of the current is a way of determining the experimental errors in the calculation of the plasma parameters, since the expressions for current contain practically all the calculated parameters.

Examples of calculation of the electron current are presented in Fig. 7.20. The calculated electron current in an arc mode TIC usually coincides well with experimental values, especially in a well developed arc. The spread of experimental points is within the range of 20-100% and is a maximum near the electrodes, where the errors in determining the plasma parameters increase sharply.

We note that the plasma parameters in the anode part of the gap vary more smoothly than those in the precathode region. This permits them to be extrapolated to the anode surface. At this point, comparison of the experimental value of the pre-anode barrier V_A to values calculated from the condition of the current and energy balance at the anode is an additional check on the correctness of the results.

Calculation of the value of electron current yields good results only in a sufficiently developed arc. In measurements near the point of discharge extinction, agreement of the calculated and experimental values j_e deteriorates sharply. As indicated later, this is a consequence of the large in-homogenates in the radial distribution of plasma parameters near the extinction point.

One of the disadvantages of spectral methods is the absence of information about the variation of potential. However, under TIC conditions, when the plasma is concentrated in the narrow gap between two plane-parallel electrodes, the variation of the potential can be

258

Fig. 7.20

Fig. 7.21

259

easily determined by using either the equation for electron current or the equation for energy flow. In Figure 7.21a, the potential distribution (curve 2) calculated from the equation for electron current (4.3.3) using values of density obtained from spectral measurements is compared with the potential distribution (curve 1) measured directly by probes. The good qualitative agreement is obvious.

Calculation of ion current and the ion generation function. An important characteristic of a gas-discharge plasma is the magnitude and distribution of ion current in the gap.

Direct experimental measurement of the ion current is practically impossible, but the ion current can be easily calculated by equation (4.7.2) if the gradients of density and potential are known. The force of electron-ion friction and the thermal-diffusion current may usually be disregarded in (4.7.2).

With this simplification, the ion current is given by the formula

(4.1)

When using the spectral method, the ion current can be calculated from the equation

(4.2)

which follows from (8.1.13a) and (8.1.13b), removing the necessity in this case of first determining V(x).

The ion current distribution, obtained from a probe (1) and from spectral measurements (2), is presented in Fig. 7.21b.

It should be noted that the plot of ion current distribution through the gap is smoother than the distribution of density and potential. Therefore, it is easier to extrapolate the ion current to the electrodes than to extrapolate density and the potential there.

A knowledge of the distribution of ion current through the gap permits the rate G of ionization-recombination to be determined and to be compared with the theoretical formula (5.3.7):

(4.3)

where t _i = [v_e@ s ₀(T_e)N_a]^-1 is the effective ionization time.

By using experimental values of ion current, electron density, and electron temperature, one can in principle calculate t _I(T_e, N_a) for any point of the plasma. However, large experimental errors in calculating the recombination term (related primarily to inaccuracy in calculating electron temperature) are inevitable during this calculation. Therefore, to calculate the ion generation time and the related cross section, either those modes are used where the recombination term is small or the precathode parts of the plasma are used, where ionization is known to be dominant over recombination (so that the recombination term may again be disregarded).

5. Design of Experimental Devices. Boundary Effects.

Laboratory TICs, with which various types of operating modes are studied and probe and spectral diagnostics of plasmas are carried out, usually have a plane geometry. To achieve conditions in them similar

260

to a practical converter, various systems of shields and protective electrodes are used to reduce boundary effects. Calculations of various plasma characteristics and the comparison of theory and experiment are valid only if the plasma in the gap of the experimental device is sufficiently homogenous. Also, the boundary effects related to the spreading of the plasma and the emission from the lateral surface of the cylindrical cathode must be small.

Analysis of boundary effects permits investigation of the radial distribution of plasma parameters. Such distributions for an arc mode TIC are presented in Fig. 7.22 [60].

Fig. 7.22

The data in this figure were obtained in a device equipped with a cylindrical probe 100 m m in diameter and 1 mm long. The probe was located parallel to the plane of the electrodes and could be shifted from the center of the electrodes to their edge by a bellows connection. The cathode and anode could be shifted as a unit with respect to the probe by changing the distance x from the cathode to the probe. The data shown refer to the point in the gap at x = 1.5 mm. The distribution pattern is similar for the other points of a gap, 0.5 mm < x < 2.0 mm.

261

It is obvious from Fig. 7.22 that the plasma inside the inter-electrode space of a developed discharge (points 4 - 8 of the current-voltage characteristic) is essentially homogeneous in the radial direction; variation of the plasma parameters begins only beyond the edge of the electrode. After that, the electron concentration and temperature decrease monotonically, while the plasma potential increases monotonically (Fig. 7.22c) as the distance from the edge of the electrode increases.

With this character in the radial distribution of the plasma parameters, optical diagnostics, even without corrections for radial in-homogeneity, yield satisfactory accuracy for determining the density in the gap. The radial distribution of the relative intensity of two spectral lines for different points of the interelectrode space (Fig. 7.23a) and the distribution of the intensity of the same line through the gap without data reduction (curve 1) and with calculation by the Abelian function (curve 2) is presented in Fig. 7.23. It is obvious that these curves almost coincide. However, despite the fact that the discharge glow fills the entire area of the electrodes when the discharge is near extinction (points 1 - 3 in Fig. 7.22a), there are significant in-homogenates in the radial distribution of the plasma parameters.

Fig. 7.23

262

The distribution of plasma parameters for a particular point of the current-voltage characteristic is presented in Fig. 7.22c. The electron current component in the radial direction, calculated by the diffusion equation, is small, and the distributions in that direction of the density and potential are well described by the Boltzmann formula. The density distribution calculated by this formula using the experimental value of potential, is shown in the figure by the dashed line.

The ion current in the radial direction is also small (for this mode, about 1 mA/cm² with an ion current along the axis of 10 mA/cm²). For this reason, the electron temperature drop along the discharge periphery is primarily caused by energy losses in radiation rather than in ionization. The observed drop of T_e agrees well with the results of [61], where the spectral method was used for the measurements.

Investigation shows that the character of the radial distribution in the plasma parameters does not vary in the transition from low cesium. pressures (P_Cs » 10^-3 - 10^-2 torr, where the electron mean free path is much greater than the interelectrode distance) to a dense plasma (P_Cs » 1 - 4 torr).

The protective sapphire shields near the electrodes have essentially no effect on the radial distribution of the plasma parameters in the central part of the gap. They do not reduce the effect of the boundary as far as the radial diffusion of the carriers from the gap is concerned. However, the shields considerably reduce the effect of radiation from the lateral surface of a cylindrical cathode. Although the absence of shields has a negligible effect on the accuracy of determining the plasma parameters from spectral and (even more) from probe measurements, with their absence, it is not possible to reliably determine the true density of electron current through the device. This information is very useful for checking the correctness of the results obtained.*

______________

*It should be emphasized that one must carefully insure the absence of spurious discharges to internal parts of the device. The character of the radial distribution in the parameters is disrupted by spurious discharges, and the plasma becomes very inhomogeneous

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