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377

The diffusion and arc TIC operating modes considered in the preceding chapters were characterized by interelectrode spacings much greater than electron and ion mean free paths, so that the motion of charged particles was described by diffusion equations. A TIC operating mode is considered in this chapter where the mean free path is greater than or on the order of the interelectrode spacing. As before, the Debye screening length L_D for converter currents of practical interest is small compared to the interelectrode distance d. Thus, in this mode also, a plasma is formed in the gap which is separated from the electrodes by potential barriers, i.e., by regions of uncompensated space charge with widths the order of L_D. Such plasmas where the charged particles do not undergo collisions, is in this monograph called a Knudsen plasma.

The sign of the precathode potential barrier V_C is determined primarily by the ratio of electron to ion emission from the cathode. However, unlike the diffusion operating mode, where the plasma in the precathode region is close to thermodynamic equilibrium with the cathode a Knudsen plasma is significantly non-equilibrium. Therefore, the precise value V_C is different from its equilibrium value V_TC = ï f _C - m ï /e@ (see §1, Chapter 6).

A Knudsen plasma is separated from the anode by a potential barrier in the Debye sheath, the value of the barrier being the anode potential V_A relative to the plasma. A typical potential distribution V(x) for a TIC operating in the direct-flight (Knudsen) mode is shown in Fig. 10.1.

When the converter current is equal to cathode emission current and there is no emission from the anode, the maximum output power is extracted (with anode work function f _A) when the cathode work function is given by f _C = f _A + kT_C [l]. In this case, V_A = V_A = 0, and the output voltage of the converter is V_L = kT_C/e@. To have enough ions from the cathode to compensate the electron space charge, the chemical potential m of the plasma should be approximately equal to the cathode work function f _C. However, in reality, this mode cannot be realized, even in the presence of a cathode with the required work function, because the cesium pressure must be limited to a value the order of several tenths of torr for practical gap widths, d > @ 100 m m. In this case, if T_C > @ 2200° K and f _A = 1.8 eV, the chemical potential of the plasma greatly exceeds the value of f _A + kT_C. That is, the rate of surface

ionization is inadequate to compensate the electron space charge and to allow passage of the total electron emission of the cathode through the TIC gap. A further increase of P_Cs requires impracticably narrow gaps to produce the direct-flight mode. Also, a decrease of the cathode work function compared to the chemical potential leads to an appreciable increase in the converter current and nominal power. But, the cathode temperature must be increased to give this increase of current and power. Therefore, the Knudsen mode of TIC operation can be achieved only at high cathode temperatures and this hinders its practical use.

The theory of Knudsen mode TIC operation was initially based on the hypothesis of ideal compensation of the electron space charge by ions generated at the cathode. Different TIC operating modes have been analyzed with this hypothesis, and the power and efficiency of the converter has been calculated [1-6]. A number of theoretical [7, 11, 50] and experimental [11-15, 51] investigations have appeared during the past decade in which the potential distributions in the TIC gap were analyzed and measured. The oscillations which frequently occur in the Knudsen mode have also been investigated [16-19].

where n_i(x) and n_e(x) are the electron and ion densities and eV(x) is the electron potential energy. The cathode surface potential is taken as zero potential (see Fig. 10.1). The voltage drop V_d across the converter gap is assumed to be positive if it accelerates the electrons toward the anode (V_C and V_A are the absolute values of the potential barriers at the boundaries of the plasma and the electrodes).

The particle distribution functions over velocities, f_i(v,x) and f_e(v,x), must be found in order to calculate the densities n_i(x) and n_e(x). These distribution functions are calculated from the kinetic equation, in which collisions may be disregarded because of the smallness of the interelectrode distance d compared to the electron and ion mean free path, l_e and l_i. In this case, assuming in (4.1.9) that ¶ f/¶ t = 0, I{f} = 0, and F = ± eE, we obtain the following equations to solve for f_i(v@ ,x) and f_e(v@ ,x):

Since the coefficients of equations (1.2) and (1.3) are not dependent on the transverse velocity components v_y and v_z, it is convenient to integrate the distribution functions f_i(v@ ,x) and f_e(v@ ,x) over v_y and v_z. As a result, we obtain equations for the distribution functions which are dependent only on v_x and x. These distribution functions are denoted by f_i(v_x ,x) and f_e(v_x ,x), respectively. Having calculated f_i and f_e from (1.2) and (1.3), we obtain the densities n_i and n_e by the formulas

and then, having substituted n_i and n_e into Poisson’s equation (1.1), we obtain the potential distribution V(x).

These solutions of equations (1.2) and (1.3) are functions only of the total ion and electron energy in the electric field, i.e., f_i = f_i(Mv_x²/2 - eV(x)) and f_e = f_e(Mv_x²/2 - eV(x)). Of this we can be certain by substituting these functions into (1.2) and (1.3). To determine the explicit form of the dependence of f_i and f_e on their own arguments, the boundary conditions must be used. The boundary conditions for equations (1.2) and (1.3) are expressions for the ion and electron distribution functions at the electrode surfaces. The emitted particles at the cathode have a Maxwellian velocity distribution, i.e.

(j_is and j_s are the ion and electron emission currents from the cathode, @ is the positive electronic charge, and T_C is the cathode temperature). There is no ion or electron emission from the anode if its temperature is sufficiently low. Then,

Boundary conditions (l.5)-(l.7) are given at both ends of the interelectrode space. However, they can be simply formulated so that the distribution function is completely calculated (both for v_x > 0 and for v_x < 0) on one end, for example, at x = 0. Actually, with the potential distribution shown in Fig. 10.1, all the electrons emitted by the cathode reach the anode. Therefore, f_e(v_x ,x) = 0 for v_x < 0 and for any value of x. In the case of ions, part of them return to

the cathode, namely, all those for which Mv_x²/2 < eV_d, where V_d is the voltage across the converter gap. Therefore, for ï v_xï < (2eV_d/M)^1/2 the ion distribution function at the cathode surface is symmetrical over velocities. At v_x < - (2eV_d/M)^1/2, f_i(v_x ,0) = 0, and for v_x > (2eV_d/M)^1/2, f_i(v_x ,0) coincides with (1.5).

where the function q (z) is calculated so that q (z) = 1 for z > 0 and q (z) = 0 for z < 0. Since f_i = f_i(Mv_x²/2 - eV(x)) then, to find the expressions for f_i for x > 0, it is necessary to substitute Mv_x²/2 for Mv_x²/2 - eV(x) in (1.8). Therefore, @

Similarly, by substituting Mv_x²/2 for Mv_x²/2 - eV(x) in (1.9), we obtain the expression for the electron distribution function

*Boundary conditions (1.8) and (1.9) are similar to the conditions for the vacuum mode (see Chapter 2, page 49).

@ With this zero of potential, the value of V(x) is everywhere negative ( - V_d £ V(x) £ 0).

(a is the compensation parameter introduced above (see §7, Chapter 9)). At a = 1 and for zero voltage across the converter gap (V_d = 0), the space charges of the electrons and ions are mutually compensated.*

In the undercompensated mode, i.e., for a < 1, the number of ions is inadequate to compensate for the electron space charge, and a potential barrier occurs near the cathode which retards the cathode emission electrons. In the overcompensated mode, i.e., for a > l, on the other hand, a potential barrier develops at the cathode which retards the cathode ion emission. We note that the value of L_D' is close to the Debye screening length L_D (see 6.1.3)), since the ratio j_s/ev_e determines, to an order of magnitude, the charged particle density n in the gap.

Tables of functions h ± (z) are presented in [20]. As a result of the second integration, by taking into account (1.14a), we obtain

The integration constant y (0) in (1.15) is determined from condition (1.14b), according to which

where x _d = d/L_D'.

Having calculated y (0) from (1.18) and having substituted the derived expression into (1.17), we obtain the desired dependence of x _D on V or V on x _D. The difficulty in solving the problem is in the fact that integrals (1.17) and (1.18) cannot be calculated analytically. However, for L_D’ « d and x _d » 1, the main results can be obtained by

*In this case, the electron and ion emission currents are inversely proportional to their directed velocities at the cathode surface j_is/j_s = v_i/v_e, and in the absence of an electric field in the gap, the charged particle densities are equal n_i(x) = n_e(x).

a rather simple method without a direct calculation of integrals (1.17) and (1.18) (see [7]). We note that the space charge occupies only a small part of the gap near its boundaries, while the electric field is close to zero in the main volume. This is immediately obvious from equation (1.18). Since V_d is close (by an order of magnitude) to unity, the condition x _d » 1 can be satisfied only for a very high value of the integrand in (1.18), i.e., for small values of y . In this case, we may assume to a first approximation that x _d » ¥ , i.e., that the integral on the right side of (1.18) diverges. To do this, there must exist a point V = - V_C, in which

where y ' = dy /dV. In this case, expansion of the function y (V) into a Taylor series near the point @ V = - V_C begins with terms quadratic in V + V_C, and therefore, the integral in (1.18) diverges logarithmically.

Conditions (1.19) indicate that the value of y , i.e., the square of the field intensity, approaches zero at V = - V_C. They also indicate that the value of y '(V), which is equal to the right side of (1.13) and therefore is proportional to the space charge density at the point with potential -V_C, also approaches zero simultaneously. As a result, the Knudsen plasma - an extended region with a compensated space charge, in which there is no electric field - occurs at V = -V_C. Unlike a diffusion plasma, there are no collisions in a Knudsen plasma, and consequently, there are also no friction forces which accompany the motion of charged particles in the plasma. Thus, a moving charged particle does not lose its momentum and can move through the plasma even in the absence of an electric field.

As can be seen from Fig. 10.1, V_C is equal to the height of the potential barrier in the Debye sheath, the potential difference between the plasma and the cathode. Similarly, V_A = V_d - V_C is the pre-anode potential barrier in the Debye sheath. The value of V_C is calculated from the second condition of (1.19). The first condition of (1.19) is used to find the integration constant y (0). It is necessary that function y (V), which is under the radical sign in (1.17) and (1.18), be negative for all values of V in the range zero to V_d in order for this potential distribution to exist. Since the function y (V) is dependent on V'_d and a , the range of values of V'_d and a within which all the conditions formulated above are fulfilled must be found in order to define the regions for the desired mode. To do this, we fix the voltage across the gap at V_d > 0 and we vary the compensation parameter a .

At sufficiently small values of a , the mode will be undercompensated, i.e., a maximum potential energy eV_m > 0, which limits current passage from the cathode to the anode, will occur in the gap. At some value of a = a ₁, this maximum approaches the cathode and disappears. The point of transition to the monotonic mode corresponds to having the maximum V at the cathode surface, so that dV/dxï _x = 0? = 0. Thus, one must assume thaty (0) = 0 in (1.15), and then (1.15) must be substituted into (1.19) to calculate a ₁. As a result, we obtain

Having eliminated V_C from equations (1.20) and (1.21), we obtain the desired dependence of a ₁ on V_d.* This dependence is shown by curve I in Fig. 10.2.

With a further increase of a , the electric field intensity at the cathode, and along with it, the value of y (0), increases in absolute value. The ion space charge increases in the precathode Debye sheath. The point V = - V_C, where the ion space charge is compensated by electron charge, is shifted to greater values of ï Vï , where there are fewer ions. At some value of a = a ₂, the potential of the Knudsen plasma with respect to the cathode is equal to the total voltage drop across the gap. The point V_C = V_d limits the monotonic mode in the direction of greater values for the compensation parameter.

The desired existence boundary for the monotonic mode can be found by requiring the second condition of (1.19) to be fulfilled at V_C = V_d. Since y '(V) is equal to the right side of (1.13), then, having set the right side of (1.13) equal to zero at V_C = -V_d, we obtain

The dependence of a ₂ on V_d is depicted by curve II in Fig. 10.2. The region for existence of the monotonic potential distribution is located between curves I and II.

*When solving the system (1.20) and (1.21), it is necessary to first eliminate a ₁ and to find the dependence of V_C on V_d graphically, and then to calculate a ₁ for the known values of V_d and V_C.

Let us follow how the pre-electrode potential barriers V_C and V_A vary as a function of the voltage applied across the gap, i.e., as a function of V_d. For example, consider the overcompensated mode (a < 1). If the voltage on the gap is comparatively small, V_d = V_C and the pre-anode potential barrier is V_C = 0. The value of the precathode potential barrier V_C is calculated from equation (1.22). If V_d increases, the pre-anode barrier V_a increases (V_A = V_d - V_C, see Fig. 10.1).

One may assume in (1.13) that erfÖ V_d -V_C@ » 1 in order to calculate the value of V_d at large values of V_d. Then, by setting the right side of (1.13) equal to zero, we obtain the equation

Since the value of a remains constant as voltage changes, the right sides of (1.22) and (1.23) should be identical. But since the function exp(2z)(1 - erfÖ z@) increases rapidly as the argument increases, the value of V_C changes comparatively little.

We note that when the pre-anode potential barrier V_A » 1, i.e., eV_A » kT_C, the precathode barrier ceases to depend on the anode voltage, so that all the additional voltage applied to the gap appears on the pre-anode potential barrier.

Consider now the monotonic mode with negative voltage across the gap. To do this, it is necessary that the electrons and ions in all the formulas given above exchange places, after which, the same curves I and II are obtained which limit the existence region for the monotonic mode. Therefore, by replacing V_d by -V_d and a by l/a , we can immediately calculate the existence region of the monotonic mode for negative values of V_d V_d (see Fig. 10.2).

There is no need to calculate the potential jumps V_C and V_A in order to construct the current-voltage characteristic of a converter with a monotonic potential distribution. The expression for electron current to the anode, j_e, can be written immediately. With a negative potential on the anode,

and for a positive value of V_d, the total cathode emission current is collected.

number of collisions increases, and as a result, particles begin scattering out of the potential well. The well will continue to fill until the arrival of particles to the well are balanced by the departure from the well. Although the probabilities for particles entering and emerging from the potential well are proportional to the small value of d/l (d is the interelectrode distance and l is the charged particle mean free path) in the Knudsen mode, the concentration of particles captured by the hole remains finite as d/l ® 0. The particles enter the potential well as a result of collision. If mv_x²/2 < eV' after collision, where V' is the least of the pre-electrode potential barriers, then the charged particle is captured by the potential well.

Of greatest interest for TIC theory is the undercompensated mode, where the compensation parameter has the value a < 1, since the greatest output power is obtained from this mode (see below). In this case, a potential hump for electrons, i.e., a potential well for the ions, forms in the gap and the current of the device, j_e, is less than the cathode emission current j_s. The potential well is filled with ions, which compensate for the electron space charge in the main part of the interelectrode space. The corresponding potential distribution is depicted, in Fig. 10.3. The ions captured in the potential well have approximately a Boltzmann distribution with respect to the potential variation. Therefore, the density of captured ions decreases sharply in the pre-electrode regions, where the potential - V(x) for the ions increases. Electron space charge is predominant in these regions. A characteristic feature of the precathode region of the TIC is the presence of a non-monotonic potential distribution with a maximum V_m at the plasma boundary.

gap for ions, and when a positive voltage is across the gap V_A > V_C (see Fig. 10.3a).

The equation for the distribution function of the particles captured in the potential well. When finding the ion distribution function in the gap, the ions may be subdivided into two groups as a function of the velocity component v_x. For V(x) > 0 and ï v_xï < (2eV(x)/M)^1/2, the ions are captured by the potential well. The trajectory of motion for such an ion in phase space, with coordinates v_x and x, is a closed curve (see curves 1 in Fig. 10.4). For ï v_xï < (2eV(x)/M)^1/2, the ion can overcome the potential barrier at the plasma boundary and reach the electrode. If (2eV(x)/M)^1/2 < ï v_xï < (2e[V(x) + V_d]/M)^1/2, then the ion reflected from the pre-anode potential barrier returns to the cathode (curves 2 in Fig. 10.4). If ï v_xï > (2e[V(x) + V_d]/M)^1/2, then the ion reaches the anode (curve 3 in Fig. 10.14). The trajectories of the second and third types above, which correspond to uncaptured ions, begin and end on the electrodes.

On these trajectories, the ion distribution functions (henceforth denoted as f_i2(v@ ,x) and f_i3(v@ ,x)) may be obtained, as in the cases considered above, by integrating the kinetic equation with the necessary boundary conditions at the electrodes. In this case, for f_i2(v@ ,x) and f_i3(v@ ,x) an expression is obtained similar to (1.10):

Here v is the absolute value of ion velocity. For (2eV(x)/M)^1/2 < ï v_xï < (2e[V(x) + V_d]/M)^1/2, formula (2.1) yields the function f_i2(v@ ,x), and for v_x > (2e[V(x) + V_d]/M)^1/2, expression (2.1) yields f_i3(v@ ,x). For the remaining values of v_x, the distribution function of the uncaptured ions is equal to zero.

Consider now the distribution function f_i1(v@ ,x) for the ions captured in the potential well. When calculating f_i1(v@ ,x), to a first approximation, the collisions can be disregarded and f_i1(v@ ,x) can be calculated from equation (1.2). It is then found, as before, that the distribution function of the captured ions is dependent only on the total ion energy, i.e., f_i1 = f_i1(Mv²/2 - eV(x)). Collisions must be taken into account to find the explicit form of dependence of the distribution function f_i1 on this argument. In this case, it is convenient to write the kinetic equation for f_i1(v@ ,x) in the following form:

where I_i is the collision term, which is a function of the ion distribution function. In (2.2), d f_i1/dt denotes the complete time

If collisions are disregarded, when I_i = 0, f_i1(v@ ,x) = const, i.e., the number of particles on the trajectory in the phase space does not vary with time. Thus, it is obvious that f_i1(v@ ,x) is an even function of the velocity v@. If there are collisions, the ion distribution function is dependent on time. However, in steady state (in view of the periodicity of motion), it should return within a particular period to its own initial value. The latter requirement means that

where T is the period of motion along the trajectory.

Returning to the variables v_x and x, we can write the above condition in the following form:

where v_x = ± (v_x1 + 2e[V(x) - V(x₁)]/M)^1/2 is the ion velocity on the trajectory in phase space (v_x1 and V(x₁) the velocity and potential at the turning point), and the integral is taken along the closed trajectory in phase space. The explicit form of the distribution function of captured ions f_i1(Mv²/2 - eV(x)) can be calculated from the integral equation (2.3). @

Filling of the potential hole with ions during charge exchange. If the mechanism for filling the potential well is arbitrary, equation (2.3) cannot be solved analytically. Based on the results of [10], we present a solution of the problem for one of the simplest cases, where the potential well is filled with ions in the undercompensated mode during charge exchange of ions with neutral atoms. This filling mechanism is predominant in a weakly ionized plasma, where collisions of ions with neutral atoms occur more frequently than Coulomb collisions.

Consider equation (2.3). The integral on the left side of (2.3) can be divided into two parts, which correspond to the motion of the ion captured in the well in moving from the cathode to the anode (v_x > 0) and in the opposite direction (v_x < 0) (Fig. 10.4). Denote the collision term corresponding to v_x > 0 by I_i⁺{ f_i1) and the collision term for v_x < 0 by I_I^-{ f_i1). Since the ion, most of the time, moves within the plasma, where the potential and other parameters do not vary, the values of I_i⁺ and I_I^- may be assumed to be independent of x.

*The kinetic equation in the form (2.2) is completely equivalent to expression (4.1.9). One can be certain of this if the complete time derivative is represented in the form df(v@ , r@, t)/dt + ¶ f/¶ t + (dr@ /dt)× grad_rf + (dv@ /dt)× grad_v and if one takes into account that v@ = dr@ /dt and F = mdv/dt. The only difference is that v and r in (4.1.9) are independent variables, whereas v@ and r@ in (2.2) are regarded as time functions v(t)@ and r(t)@.

@Equation (2.3) was obtained in [8], where the necessity of taking into account filling of the potential wells in the Knudsen mode was first indicated. A similar result can also be found from equations (17) of [21].

Then equation (2.3) is written in the form [I_i⁺{ f_i1) + I_I^-{ f_i1)]T/2 = 0, where T/2 is the ion flight time from the precathode to the pre-anode potential barrier. Thus, the distribution function of the captured ions should satisfy the condition

Generally speaking, (2.4) is an integral equation, as before, because the collision term is expressed by the integral of the unknown ion distribution function. However, if the probability of charge exchange 1/t ^ia is not dependent on the relative velocity of the ion and atom, the expression for the collision term is considerably simplified (see formula (4.7.12)). In this case, it follows from (2.4) that

where f_a(v@ ) is the distribution function of neutral atoms and n_i and N_a are the total densities of ions and neutral atoms in the gap. If we take into account that the distribution function of the captured ions should be an even function of velocity, i.e., f_i1(v@ ) = f_i1(- v@ ), then the following expression is obtained from (2.5) for the distribution function of the ions captured in the potential well:

The ion density n_i can be represented in the form n_i = n_i1 + n_i', where n_i1 and n_i' are the densities of the captured and uncaptured ions, respectively. In order to eliminate the yet unknown value of n_i1 from the final expression, we integrate (2.6) over all the velocities of the captured particles. As a result, we find that n_i1 = (n_i/N_a)N_a1, where N_a1 is the density of those atoms in which the velocity v_x after charge exchange and conversion to an ion is insufficient for the ion to emerge from the potential well. For the potential distribution shown in Fig. 10.3a, this means that ï v_xï < (2eV(x)/M)^1/2.

The density of atoms in the gap can also be represented in the form N_a = N_a1 + N_a', where N_a' is the density of atoms which are not captured by the potential well after charge exchange and conversion to ions. It then follows from the relation n_a1/N_a1 = n_a/N_a that n_a'/N_a' = n_a/N_a. As a result, the following expression is obtained for the distribution function of the captured ions:

(ï v_xï < (2eV(x)/M)^1/2). The numerator of (2.7) contains the unknown total density n_i ' = n_i2 + n_i3 of uncaptured ions with trajectories of the second and third types, and the denominator contains the concentration of atoms N_a', in which ï v_xï > (2eV(x)/M)^1/2. It is obvious that the numerator of (2.7) is proportional to the transition probability of the ion to the captured state, while the denominator is proportional to the probability that the ion will emerge from this state after charge exchange.

The expressions for the distribution functions of uncaptured ions (see (2.1)) and electrons are obtained by integrating the kinetic equations in a way similar to that done for a monotonic potential distribution in the gap (see [8, 10]). For reference, we include the expressions obtained in this case for the densities of the uncaptured particles. The dependence of ion density on potential has the following form:

Here V = eV/kT_C, V_m = eV_m/kT_C, and V_d = eV_d/kT_C, where the voltage drop V_d across the gap, as previously, is assumed positive if it accelerates the electrons from the cathode to the anode (see Fig. 10.3a).

The expressions for electron density on the left (x ³ x_m) and on the right (x £ x_m) of the maximum potential energy eV_m in the gap are given by the formulas

Here i_a is the flow of atoms evaporated from each electrode (since there is no transfer of matter between the electrodes in steady state, the flows of atoms desorbed from the two electrodes are identical). It follows from (2.12) that the number of atoms N_a', for which ï v_xï > (2eV(x)/M)^1/2, is equal to

where t = T_A/T_C. Now we integrate expression (2.7) over v to obtain the density of captured ions n_i1(V). In this case, the integration variable v_x should run from - (2eV(x)/M)^1/2 to + (2eV(x)/M)^1/2. By using (2.9) and (2.13) and by adding the density n_i'(V) of the uncaptured ions to the result obtained, we derive an expression for the total density of captured and uncaptured ions:

The electron density n_e(V) is given by formulas (2.10) and (2.11). Expressions (2.8), (2.14), (2.16), (2.10), and (2.11) give the density of ions and electrons in the TIC gap as a function of the potential V at the particular point of space.

The typical potential distribution in the gap, obtained as a result of solving Poisson’s equation (1.1), is shown in Fig. 10.3. It is obvious that a Knudsen plasma is formed to the right of the potential maximum and that the potential distribution near the cathode is non-monotonic. This result is general: the potential distribution where the plasma forms on the bottom of the potential well, i.e., at V = V_m, cannot occur in the Knudsen mode. Actually, if the electron and ion space charges at the bottom of the well were mutually compensated (n_i(V_m) = n_e(V_m)), then, since dn_e/dV = ¥ at V = V_m,* the ion space charge should be predominant to the right of the maximum at V < V_m, which is impossible. As a result, the plasma should be formed at V < V_m, where dn_e/dV < dn_i/dV.

The plasma potential with respect to the cathode is equal to V_C = eV_C/kT_C. The values of V_C and V_m = eV_m/kT_C are calculated from the condition that the space charge and field intensity in a Knudsen plasma are equal to zero at V = V_C (compare with (1.19)). The first of these conditions, as can be seen from (2.11), (2.14), and (2.16), is written in the following form:

*The dependence of n_e(V) on the right of the maximum potential is expressed by a formula similar to (6.2.16), where V_B is substituted for V_m. It is then obvious from (6.2.16) that actually dn_e/dVï _V = Vm = ¥ .

to zero means that the total space charge, integrated over V from V = V_m (at V = V_m, dV/dx _D = 0) to V = V_C, is equal to zero. By using the tabulated functions of (1.16), this condition can be written in the form

After solving the system of equations (2.17)-(2.20) for V_C and V_m the, electron current to the anode is calculated by the formula

Consider now the boundaries of the existence region for the above potential distribution. As can be seen from Fig. 10.2, for a > 0.41,* the non-monotonic potential distribution changes to monotonic (at the point dV/dxï _x = 0 = 0) as the voltage on the gap increases. For a < 0.41, the transition to a monotonic potential distribution does not occur, but the collected current is less than the cathode emission current. However, there is current saturation (Fig. 10.5) for this case also, because V_m and V_C cease to vary after the pre-anode potential barrier V_A increases so much that all the ions emitted by the cathode are reflected back to the cathode. We then have erf Ö V_d + V@ » 1, and the solution of the system of equations (2.17)-(2.20) is no longer dependent on V_d, and all the additional voltage appears at the pre-anode barrier V_A, the same as in the diffusion mode.

When V_A decreases to V_A = 0, the above mode disappears, along with the above mechanism for filling the potential well by ions. That is, the mode disappears when the potential of the Knudsen plasma is equal to the anode potential. This occurs with negative values of voltage V_d. Line AB, which limits the existence region of this TIC operating mode in the range of negative voltages across the converter gap, is shown by the dashed line in Fig. 10.2.

Filling of potential wells by elastic scattering of charged particles from neutral atoms. With an arbitrary mechanism of charged particle (ion and electron) scattering from neutral atoms, it is not possible to find an analytical expression for the distribution function and density of particles captured in the well. To find an approximate solution in [10], the neutral atom distribution function was approximated by a Maxwellian distribution f_a(v) = N_a(M/2p kT_C)^3/2exp(-Mv²/2kT_C) with a temperature equal to the cathode temperature.

With this approximation the distribution function of particles captured in the well were determined for two cases: for zero voltage across the gap and for the case where the gap voltage satisfies eï V_dï » kT_C and has the polarity that it reflects back to the cathode those charged particles which are captured in the potential well. The particles

*As V_d ® ¥ , curve I in Fig. 10.2 asymptotically approaches the value of a » 0.41 [7].

A Maxwell-Boltzmann distribution of captured ions is explained by the fact that a Maxwell-Boltzmann distribution of uncaptured ions is established as a result of ion reflection from the pre-anode potential barrier for eV_d » kT_C (see (2.23)). The arrival of ions to the well becomes balanced by their departure, and the two distributions come into equilibrium. At eV_d = 0 (see (2.22), un-captured ion density decreases by exactly one-half, which leads to a corresponding decrease in the number of captured ions.

This energy loss must be appreciable for electrons to fill the well. It is obvious that, with TIC conditions, electron filling of states of the second type will occur mainly by electron-electron collisions, where there is appreciable change of energy even with a comparatively low degree of plasma ionization. Electron collisions with atoms are accompanied by only a very slight (on the order of m/M) change of electron energy. The importance of electron-electron scattering complicates the analysis of modes in which a potential well for electrons is formed.

We note that filling of states of the first type is important primarily with small potential wells (eV_m « kT_C), while filling of states of the second type is important with deep wells (eV_m » kT_C) [10].

Modes with shallow potential wells, filled primarily by changes in the directions of the velocities of charged particles, are of two types: modes with a potential well for ions and modes with a potential well for electrons. They can be treated together. The boundary of the mode with a single potential extremum (an electron well) is shown by Curve III in Fig. 10.2 [10]. It is obvious from Fig. 10.2 that the boundaries of modes both with monotonic and with a single potential extremum converge fan-like at the point of total compensation of space charge, a = 1 and V_d = 0.

Filling of potential wells by Coulomb scattering. With large current densities in the Knudsen mode, Coulomb collisions occur more frequently than collisions of charged particles with neutral atoms. Filling of potential wells by Coulomb scattering was considered in [8], where the distribution functions of the particles captured in the well were found for the case of - zero voltage drop across the gap and of a deep potential well (V_d = 0, ï eV_mï » kT_C), and for the case of a large decelerating pre-anode potential barrier (eï V_dï » kT_C). These distribution functions are given by formulas (2.22) and (2.23), respectively, for ions. The electron distribution functions are obtained by substituting j_s for j_is, m for M and - V for V.

Oscillating potential distribution. As can be seen from Fig. 10.2, a region in which the potential distribution is characterized by more than one extremum should be located between curves II and III, on the diagram of TIC operating modes (this region is cross-hatched in the drawing). Potential distributions which oscillate in the gap should occur. However, calculation of the oscillating potential distributions and accounting for the filling of the potential wells with charged particles is a very complicated problem. This problem was considered in [24], where the potential distributions were considered which occur with transition from the monotonic to the oscillating mode (the modes in the cross—hatched zone near curve II in Fig. 10.2). It was shown in [24] that periodic potential distributions may occur in the cross-hatched zone if specific conditions are fulfilled. The amplitude and wavelength of these periodic distributions were calculated.

Here i₀ is the flux density of atoms and ions from the plasma to the cathode, equal to the flux density of the ions and atoms desorbed from the cathode; f _C is the cathode work function; g_a = 2 and g_i = 1 are the statistical weights of the atomic and ionic states, respectively; and w_a and w_i are the probabilities of desorption of the heavy particle in the form of an atom or in the form of an ion, respectively. By using (2.24) and (2.1.7) and by assuming that r@ = 0, we obtain the expression for the compensation parameter:

If there is a low degree of ionization in the particles desorbed from the cathode, where w_i « w_a, expression (2.25) can be simplified by disregarding unity in comparison with 2exp[(E_ion - f _C))/kT_C].

If the potential distribution in the gap is such that the ions emitted by the cathode are not returned (see, for example, Fig. 10.3b), then i = i_a is the flow of atoms from the anode.* If part of the ion emission is returned to the cathode, then

where V’ is the potential barrier for ions in the gap. For example, with the potential distribution depicted in Fig. 10.1, V’ = V_d. Thus, if j_is/e is an appreciable fraction of i_a, then a is dependent on the potential distribution in the gap, i.e., on the operating mode of the device. Expressions (2.25) and (2.26) determine a , if f _C and i_a are known.

To calculate i_a, it is necessary to consider the balance of particles between the end of the TIC working gap and the external gas volume (Fig. 10.6). For clarity, consider the case where w_i « w_a and where the flow of particles desorbed from the cathode is weakly ionized. One may then easily show that the flux density of atoms from the gap to the external gas is equal to i_a. @ Since the emergence of the particles from the gap is compensated by their arrival to the gap in the steady state, the flux density of atoms from the gas to the gap is also equal to i_a.

If both electrodes (cathode and anode) have identical temperature T_C, then i_a = i'_a, where

Actually, the state of the gas at the boundary with the gap can differ appreciably from equilibrium, since T_C ¹ T_A, so that, strictly speaking, expression (2.27) is inapplicable.

For a more accurate estimate of the value of i_a, we circumscribe in the space the cylindrical surface ABC with a radius equal to the atomic mean free path i_a, as shown in Fig. 10.6. If we assume that the gas atoms have an equilibrium Maxwellian velocity distribution at some temperature T(q ) at a distance on the order of l_a from the gap-gas interface, then i_a = i'_a', where

@If w_i » w_a, then the electric fields at the gap-gas interface must be taken into account when calculating the flow of particles from the gap.

(Cosq under the sign of the integral takes into account the angle of incidence of the particles from the gas to the boundary (see Fig. 10.6), and N_a(q ) = P_Cs/kT(q ) and v_a(q )@ = (8kT(q )/p M)^1/2 are the density and random velocity of the atoms, respectively, at points of the semicircle ABC). In this case the temperature of the atoms T(q ) is calculated from the equation of heat conduction for the gas.

For d « l_a, the heat conduction equation has the following form

and should be solved with boundary conditions T(- p /2) = T_A and T(p /2) = T_C. Since the atoms arriving normal to the boundary make the main contribution to (2.28), and since the heat conduction of the gas k _a is weakly dependent on T, one may assume in (2.28) that v_a@ = (8kT_m/p M)^1/2 and N = P_Cs/kT_m, where T_m = (T_C + T_A)/2. In this case,

Equation (2.29) may be solved for a more accurate estimate. In particular, if the thermal conductivity of the gas has the dependence k _a ~ T^1/2, then

Comparison of formulas (2.27), (2.28), and (2.30) shows that the real flow of atoms to the cathode is greater than the value i'_a. This circumstance can be very important if f _C is determined by the cesium coverage, because even slight variation of the flow of atoms to the cathode significantly alters the values of f _C and a . Expression (2.21) is most often used for approximate calculations. In this case, the condition f _C = m (T_C) corresponds to the mode of total space charge compensation (a = 1), while the emission currents j_is and j_s coincide with expressions (6.7.24)-(6.7.27), obtained previously on the assumption of thermodynamic equilibrium between the plasma and the cathode.

The useful power in the Knudsen mode. Useful power P is calculated most easily for the overcompensated mode (a > 1), where the potential distribution has a potential well for electrons (Fig. 10.7). If the voltage on the load satisfies V_L £ (f _C - f _A)/e, i.e., if the voltage drop is positive (V_d ³ 0), then P = j_sV_L. If the voltage drop is negative (V_d £ 0), i.e., V_L ³ (f _C - f _A)/e, then

If f _C - f _A ³ kT_C, then the maximum useful power is achieved at zero voltage drop, where V_d = 0.

It is much more complicated to calculate the dependence of P in the undercompensated mode (a < 1), where a potential well for ions forms in the gap and the collected current satisfies j_e < j_s (see Fig. 10.3). However, calculations show that even in this case the maximum value of P is achieved at V_d » 0. This is obvious from Fig. 10.5, where the dependence of power and current on the voltage drop V_d is shown. This dependence is obtained by the numerical solution of equations (2.17)-(2.20). It was assumed for these equations that the potential well is filled by ions from resonance charge exchange. For a « 1, the potential well is deep, and the asymptotic representation 1 - erf Ö X » exp(- x)/Ö p x@ can be used in (2.15), (2.17), and (2.19). For t < 1, the second term in the numerator in (2.15) is small, so that p(x) » Ö p x@ (1 + 1/Ö t )exp(x). Then, it follows from (2.17) and (2.19) that*

Here coefficient k = 1 at V_d = 0 and k = 2 for V_d » 1. It is obvious from (2.32) and (2.33) that the values of V_C and V_m at V_d = 0 should exceed the corresponding values for V_d » 1 by the factor lnÖ 2. In this case, as follows from (2.21), the saturation current j_s' on the current-voltage characteristic exceeds by a factor Ö 2 the current j_e0, which corresponds to zero voltage drop. The same result is also obtained with other mechanisms for filling of the potential well considered above.

Since an increase of voltage drop V_d does not lead to an appreciable increase of current j_e, and at the same time is accompanied by a decrease of output voltage V_L - (f _C - f _A)/e = V_d, an increase of V_d leads mainly to a decrease of useful power P = j_eV_L. At the same time, a decrease of V_d is accompanied by the emergence of ions from the potential well and by an increase of the precathode potential barrier V_m. This in turn retards electron emission from the cathode, which leads to an appreciable decrease of current j_e through the converter. As a result, the maximum useful power corresponds to a small positive voltage drop V_d, while power P₀ is close to maximum with zero voltage drop across the gap.

Consider now the dependence of useful power P in the Knudsen mode on the TIC parameters P_Cs, f _C, f _A, and T_C. As indicated above, the value of P is limited in the Knudsen mode mainly by the rate of ion

If the value of ka (1 + 1/Ö t )(exp2V_m) is eliminated from (2.32) and (2.33), a transcendental equation is obtained to calculate V_m - V_C, which yields a value of V_m - V_C » 0.35. Thus, it is obvious that the values of V_m and V_C are comparatively close to each other.

formation by surface ionization at the cathode. Thus, the cathode work function f _C must be increased by reducing cathode emission current j_s to provide the necessary level of surface ionization.

The pressure P_Cs in the gap is selected as high as possible to provide adequate ionization, with the limitation that there be no scattering of the charged particles in the gap. If the converter anode is cold, the anode work function f _A is close to the cesium work function (f _A » 1.7 eV). Therefore, the dependence of P on f _C and T_C is much more important. We first determine the dependence of P on f _C at constant T_C. With given values of P_Cs and T_C, the compensation parameter a , according to (2.25), is itself dependence on the value of f _C.

For w_i « w_a, it follows from (2.25) that a = exp(2D f _C/kT_C) where D f _C = f _C - f _C0, and where f _C0 is the cathode work function corresponding to a = 1. By using the relationship between a and D f _C, we can plot the dependence of converter current j_e on D f _C. This dependence is presented in Fig. 10.8. The dependence shown corresponds to zero voltage drop, i.e., approximately to the mode of maximum useful power. It is obvious from the figure that the highest current in the converter is collected in the undercompensated mode, when a < 1. The result obtained is explained by the potential diagram presented in Fig. 10.9, where the potential distributions in the mode of total space charge compensation and in the undercompensated mode are compared. It is obvious that the effective cathode work function f _C + eV_m decreases in the undercompensated mode because of the filling of the potential well with ions. This leads to an increase of current j_e through the converter.

The dependence of the converter power P₀ (with zero voltage drop) on the value of f _C is presented in Fig. 10.10. Here f _A was assumed equal to 1.8 eV, and the flow of atoms to the cathode was calculated by formula (2.27). It was assumed that the well is filled by ions from resonance charge exchange. It is obvious from Fig. 10.10 that there is maximum output power in the undercompensated mode, where in reality, the power can even be increased somewhat because of an increase of voltage drop V_d (see Fig. 10.5).

At the same time, the difference between maximum power and power P₀', which corresponds to the mode of total space charge compensation (a = 1 and V_d = 0), is comparatively small. Therefore, for approximate calculations, one can calculate the maximum power as that obtained in the mode of total space charge compensation. If it is assumed in this case that f _C = m (T_C), then P₀' = (1/4n(T_C)v_e@ x[m (T_C) - f _A], where n(T_C) = N_eexp(-m /kT_C) is the equilibrium electron

density in the plasma for a given pressure P_Cs and for a given cathode temperature T_C.

The results of this calculation for two values of cesium pressure are presented in Fig. 10.11. The dependence of optimum work function f _C = m (T_C) on temperature T_C at P_Cs = 0.1 torr is also shown here. It is obvious that a rather high cathode temperature T_C > @ 2100° K is required to obtain power P₀' > @ 10 watt/cm². When ion scattering is added to the electron scattering from neutral atoms, comparatively short interelectrode distances d > @ 100 m m are required for the realization of a practical Knudsen mode [27].

We note in conclusion that, with anode emission, the voltage V_d across the gap must be increased to obtain maximum power. Electrons emitted by the anode are then reflected by the pre-anode potential barrier, so that current does not decrease appreciably (see Fig. 10.3a). The converter power decreases, however, primarily because of a decrease in voltage V_L.

The effect of TIC geometry on the nature of potential distribution. Modes with unfilled potential wells. It was assumed everywhere above that TIC electrodes are two infinite plane-parallel plates. The finiteness of the electrode dimensions can affect the potential distribution not only at the boundary but also far from the edges of the gap, especially if this distribution in non-monotonic. With a non-monotonic distribution, a decrease of the electrode dimensions increases the probability that particles will emerge from the potential well laterally. As a result, the density of particles captured in the well decreases, and this alters the potential distribution in the gap. The effect becomes appreciable at sufficiently low pressure, where the mean free path l of those particles captured in the potential well is comparable to or exceeds the electrode dimensions L.

At low pressures, this lateral escape can deplete the potential wells, so that in determining the potential distribution, the results obtained by disregarding the filling of the potential wells can be used [11, 14, 20, 22, 23, 25, 26, 47, 48]. The possible types of

potential distribution in this case remain the same as previously (in particular, a Knudsen plasma develops even with unfilled potential wells [25, 26]), but the boundaries of the existence regions for the modes with the non-monotonic potential distribution change [14, 23]. The existence region for modes with a single potential extremum (region 2) is constricted in Fig. 10.2. If the potential wells are unfilled,* curve III in Fig. 10.2 is shifted to the region V_d > 0 for a < 1 and V_d < 0 for a > 1.

If the potential wells are unfilled, as with the case of filled wells, modes with periodic potential distributions in the gap may exist. These modes were recently investigated in [11, 47], where the different types of periodic potential distributions were considered theoretically, and the amplitude of the potential variation in the gap was calculated as a function of the precathode potential jump and of the compensation parameter a . The current-voltage characteristic of a TIC in the mode with periodic potential distributions was considered in [47]. With these modes, there are many segments on the current-voltage characteristic, with negative internal resistance, which indicates that these modes are unstable in many cases.

The current-voltage characteristics of a converter in the Knudsen mode. According to the theory outlined above, the saturation current in the overcompensated mode is equal to the emission current, and the saturation current j_s' in the undercompensated mode is less than the cathode emission current j_s. However, passing to saturation in the current-voltage characteristic is usually accompanied, in experiment, by the appearance of intensive current oscillations, which leads to a decrease of the average current j_s' through the device.

The family of current-voltage characteristics [17], obtained with a rather high temperature tantalum cathode (T_C = 2350° K) and low cesium pressures (P_Cs = 8x10^-7 - 10^-2 torr) is presented in Fig. 10.12 as an example. In this family, the cathode work function does not decrease by adsorption of cesium. Curve 1 in Fig. 10.12 corresponds to a very low cesium pressure, where the rate of surface ionization is inadequate, not only for compensation of electron space charge near the cathode, but even for the formation of a neutral plasma in the gap. Therefore, the corresponding current-voltage characteristic is close to the characteristic of a vacuum diode.

A plateau forms on the characteristic at higher cesium pressures (curves 2 and 3), where the ions formed at the cathode partially compensate

*However, this shift is small for values of a which do not differ significantly from 1.

for the electron space charge in the gap. However, in going to the plateau, the converter current decreases from the value j_s' to j_s''. In this region, current oscillations occur in the circuit, whose amplitude is just equal to the difference j_s' - j_s''.

With a further increase of voltage, the plateau on the current-voltage characteristic disappears, and a rise to the true cathode saturation current j_s begins.* The current oscillations also disappear. Curves 2 and 3 correspond to the undercompensated mode, where j_s' < j_s. Curve 4, taken at a higher cesium pressure, corresponds to the over-compensated mode where the output current, in the absence of oscillations, should be equal to the cathode emission current j_s. However, the current is actually less than the emission current by the value of the amplitude of the variable component of the current.

The first measurements of the potential distribution, in the gap used the probe method. It was apparent even in the first investigations [12, 13] that a Knudsen plasma is formed in the gap. However, the probe method did not allow the determination of all parts of the potential distribution, and in particular, it did not allow the demonstration of the non-monotonic potential distribution in the precathode space charge sheath and the presence of fluctuating potential distributions. An electron beam deflection method was used in later investigations [14] to study a Knudsen plasma. The beam was in the shape of a ribbon, inclined toward the electrodes at an angle of 45° (Fig. 10.13). Rotation of the ribbon beam from position 3 to position 4 during

passage of the beam to the screen was achieved by special focusing. Line 4 was the reference line on the screen in the absence of an electric field in the gap. If there was a field, the electrons were deflected in a direction parallel to the axis of the device, the value of deflection being proportional to the field. Since the electrons in the ribbon beam passed through the gap at different distances from the cathode, a pattern of the electric field in the entire gap was obtained on the screen. The experimental distribution of field intensity E = dV/dx for different values of collector potential V with respect to the emitter are shown in Fig. 10.14 (We recall that V = -V_L, so that V is negative in the TIC operating mode.)

The results of Fig. 10.14 are for an undercompensated mode (a = 0.7). As can be seen from the diagram of TIC operating modes (Fig. 10.2), monotonic potential distribution should correspond to high negative voltages, where E > 0. This is observed experimentally (Fig. 10.14a and b). When V increases, the space charge layer at the potential barrier V_A near the anode decreases, whereas the precathode potential barrier V_C remains approximately constant.

A further increase of V leads to the region of oscillating potential distributions (Fig. 10.14c - e). This region is cross-hatched in Fig. 10.2 and is located between curves II and III for V_d < 0. With still further increase of V, according to Fig. 10.2, after passing through the exceptionally narrow region for existence of modes with a single potential extremum, we enter the region of monotonic modes, where E < 0 in the entire gap. Curve E(x) in Fig. 10.14f corresponds to this monotonic mode. The current-voltage characteristic goes to saturation. However, in this case, the development of oscillations in time accompanies further increase of the voltage.

The experimental distributions E(x) for a strongly overcompensated mode (a = 9) are presented in Fig. 10.15. In this case, the potential distribution with a single extremum corresponds to high negative voltages across the gap (see Fig. 10.2), which is also observed in experiment (see Fig. 10.15a - c). With overcompensation, according to theory, the potential distribution in the precathode space charge sheath is non-monotonic, and E changes sign near the cathode.

With a further measure of V, there is a transition to the regions of oscillating and then of monotonic distributions (Fig. 10.15d and e). This same result is also obtained in theory (see Fig. 10.2). The subsequent increase of voltage leads to development of instabilities and oscillations.

At the same time, we note that the experiments in [14] were carried out at very low cesium pressure, where apparently the ion mean free path l_i satisfies l_i > L (where L is the characteristic dimension of the electrode, see Fig. 10.13). With this inequality satisfied, the potential wells are not filled by charged particles. With a value of a = 0.7, as in Fig. 10.14, the absence of well filling does not alter the ranges of V_d values for which different TIC modes exist. At a = 9, as in Fig. 10.15, the absence of well filling leads to a decrease of region 2 and to a corresponding increase of region 3, within the limits of which should be observed fluctuating potential distribution.

The effect on the Knudsen mode that filling of the potential well with charged particles has on the TIC characteristics was studied experimentally in [11, 47]. In particular, to determine the nature of potential distribution in the TIC gap, the authors investigated the effect of a transverse magnetic field on the value of current through the device. It was shown that, with insufficiently low cesium pressure in the undercompensated mode, a virtual cathode with a deep minimum potential, typical for a potential well not filled with ions, can be formed in front of the emitter.

As indicated above, intensive current fluctuations are observed in a number of Knudsen modes. These current fluctuations have a period on the order of the flight time of ions from the cathode to the pre-anode potential barrier. At these frequencies, the flight times for electrons in the interelectrode gap are sufficiently less than the period of oscillations that the electron current is determined by the instantaneous potential distribution in the gap. Thus, current fluctuations should be related to periodic formation of a minimum potential (maximum V) in the gap, which limits the electron current. This conclusion has

been confirmed both by numerical calculations [18] and by the most recent experimental investigations [19].

When large currents pass through a TIC, considerable transverse magnetic fields occur which have an appreciable effect on TIC operation. Theoretical [28, 29, 30] and experimental [31, 32, 33] investigations of the effect of a transverse magnetic field on TIC operation in the direct-flight mode show that these magnetic fields can lead to considerable decrease of current.

In the direct-flight mode, the equations of electron motion in the interelectrode space under the effects of a permanent magnetic field directed along the z-axis and of an arbitrary electric field directed along the x-axis perpendicular to the electrodes have the form

Using the relation dt = dx/v_x, we express dt by dx, so that equations (4.1) assume the form

It follows from equations (4.2) that the component v_z does not vary across the gap and remains equal to the initial value of v_0z, while the components v_y and v_z are calculated from the following equations:

Here v_0x, v_0y, and v_0z are the electron velocity components for flight from the cathode. Those electrons for which the right side of (4.4) approaches zero at any point in the gap, having reached this point, return to the cathode and do not contribute to the current. The electrons for which there is no turning point reach the anode.

In order to calculate the current and electron density at any point with a given magnetic and electric field, we divide the electron emission current into beams, in each of which are included the electrons which have approximately equal velocities and which move along identical trajectories. For each beam, the electron current D j_e remains constant at each point and is equal to

where f(v_0x, v_0y, and v_0z) - [m²j_s/2p (kT_C)²e] exp(- mv₀²/kT_C) is the Maxwellian velocity distribution of electrons emitted by the cathode with temperature T_C, while the element dv_0xdv_0ydv_0z is determined by

The electron density in the beam is dependent on x and is equal to

We divide the half-space of initial velocities v_0x ³ 0 into two regions W ₁ and W ₂ [29, 30], which correspond to two types of trajectories: those reaching the anode, i.e., without a turning point, and those which do not reach the anode, i.e., those which contain a turning point. We also note at each point x the region W (x), which is part of W ₂ and which corresponds to the trajectories which reach point x, but do not reach the anode. Points with initial velocities of W (x) pass through point x twice, and therefore, the contribution of (4.6) from these beams should be doubled when determining the density at point x. As a result, the expression for electron density n_e(x) has the form

The total current is calculated by only the integral over the region W ₁:

The boundary of region W ₁ is calculated most simply if the right side of (4.4) does not have a minimum in the interelectrode space. In particular, this occurs if the following inequality is satisfied at any point of the gap:

In this case, as the initial velocity component v_0x decreases, the turning point initially appears on the anode and then shifts to the cathode. Therefore, the boundary of region W ₁ is determined by the condition mv_x²/2ï _x = d = 0. The components v_0x and v_0x on the boundary of region W ₁ are linked by the relation

where V_d = V(0) - V(d) is the potential difference between the anode and cathode.

Condition (4.9) is fulfilled if the space charge V''(x)/4p is equal to zero (curve 1 in Fig. 10.16a) or is greater than zero (curve 2 in Fig. 10.16a). If domain W ₁ is calculated by condition (4.10), then, after integration of (4.8), we obtain

where F (x) = (1/Ö p )ò _-¥ ^xexp(- x²)dx is the probability integral, and r_c = c(8mkT_C)^1/2/eH is the effective Larmor radius for an electron with velocity (8mkT_C/m)^1/2. The dependence of the ratio j_e/j_s on eV_d/kT_C which follows from (4.11) for different values of d/r_c, is shown in Fig. 10.17.

With the potential distribution indicated in Fig. 10.16b - d, there are regions with a large electron space charge where condition (4.9) is not fulfilled. Then, the boundary of region W ₁ and, consequently, the current are dependent on the potentials inside the plasma. If the

radius is much less than r_c, the bending of the trajectories in the space-charge regions, caused by the magnetic field, can be disregarded. Then all the electrons which reach point 2 in Fig. 10.16b and 10.16e, reach the anode (i.e., the accelerating pre-anode barrier), and the value of the current will be the same as if the anode were located at point 2, with a potential equal to the plasma potential at this point.

The effect of the retarding precathode barrier in Fig. 10.16c and 10.16d can be taken into account by introducing a virtual cathode located at point 1, whose potential is equal to the space potential of this point, instead of the potential of the real cathode. The emission current is equal to j_sexp(- eV_C/kT_C). With this approximation, the coordinates of points 1 and 2 must be assumed coincident with the coordinates of the cathode and anode.

where D V is the potential difference of points 1 and 2.

To construct the current-voltage characteristic by expressions (4.12)-(4.14), it is necessary to calculate the values of potential in the volume, which themselves may be dependent on the magnetic field. The smallness of the Debye radius with respect to r_c, as in the absence of a magnetic field, permits, in the gap outside the pre-electrode regions, the use of the condition of quasi-neutrality n_i(x) = n_e(x) for the calculation of the electric field, instead of Poisson’s equation.

This problem is considered in [30] for the potential distribution shown in Fig. 10.16b. In this case, the current I is dependent on V_d and on the plasma potential V₂ at point 2 (see (4.12)). The equation which calculates V₂ is the condition of quasi-neutrality for point 2. The value of electron density at point 2 can be obtained by using the general expression (4.7). This is accomplished by substituting v_x from (4.4) into (4.7), assuming that V(0) is the cathode potential and that V(x) is the potential of point 2.

where n_is is the concentration of ions emitted from the cathode. By setting n_e and n_i equal at point 2, we obtain the equation which relates V_d and V₂, for different compensation parameters a . Its solution for d/r_c = 1/2 and for different values of a is shown in Fig. 10.18.

By using the dependence of V₂ on V_d presented in Fig. 10.18 for V_d > ï V_2ï , we can plot the calculated current-voltage characteristics for the overcompensated modes (a > 1) [30]. These characteristics, for two values of d/r_c and for different values of a , are presented in Fig. 10.19. The characteristics for different values of a coincide until there is a well for electrons in the gap, i.e., until the potential distribution corresponds to curve 2 at point 10.16a and the current is not dependent on the potentials in the plasma.

As V_d increases, the potential distribution changes to that indicated in Fig. 10.16b and current begins to be dependent on a . This dependence allows the plasma potential to be determined from the current-voltage characteristics [29]. In this case, the decrease of saturation current in the magnetic field compared to emission current is greater, the smaller the value of a .

The experimental dependence of j_e/j_s on magnetic field B, with zero potential difference between the cathode and anode, is shown in Fig. 10.20 [32]. The solid lines indicate the experimental data for different potentials on the guard electrode, while the dashed line is plotted by using formula (4.11).

The authors of [33] experimentally checked this conclusion of a decrease of saturation current in a magnetic field with the potential distribution shown in Fig. 10.16b. The characteristics, obtained by them for a TIC operating on a Ba-Cs mixture, is shown in Fig. 10.21 for different magnetic fields. A very slight slope is observed on the segment of the characteristic where, according to theory, there should be saturation. This effect is related to the expansion of the pre-anode space charge region as V_d increases, which was not taken into account by the theory.

pressure (l_ea > d) where the volume ionization is predominant over surface ionization at the cathode. Transition to the arc mode with a Knudsen plasma is difficult compared to a dense plasma, since the excited atoms and the ions are generated more slowly, and dissipation of them to the electrodes occurs more rapidly.* However, with large current densities and high emitter temperatures, the Knudsen arc forms with the current-voltage characteristic in the power quadrant and permits an appreciably increased current output, i.e., when the emitter is operating in the undercompensated mode.

Although the arc mode now used for the TIC is that with a dense plasma, the Knudsen arc may be of practical interest with the binary vapor systems (Cs-Ba, Cs-CsF, etc.) and with cathodes having a low work function (metal alloys, lanthanum hexaboride [34], etc.).

The theory of the Knudsen arc has not been as well developed as the theory of the dense plasma arc and is usually based on some prior assumptions gleaned from the analysis of experimental data. The Knudsen arc has been investigated experimentally in great detail. The experiments are usually carried out at comparatively low cathode temperatures, because the cathode temperature does not play an important role in the developed arc. The electrons gather their main energy at the large precathode potential drop. Transition to large values of T_C (with the same cathode emission current) leads simply to a shifting of the current-voltage characteristic by the extent of the variation of the contact potential difference.

Distribution of plasma parameters. The distributions of density n, electron potential energy V and electron temperature T_e, obtained by using a Langmuir probe for different points of the current-voltage characteristic of the Knudsen arc, are presented in Fig. 10.22. Similar distributions have also been obtained for other modes. It is obvious that the density distribution across the gap has a small maximum in the middle of the gap. The absolute values of n are dependent mainly on cathode emission current j_s and on the voltage

*Argon atoms are introduced into the interelectrode gap at a pressure for which electron scattering in argon atoms is low. The purpose is to reduce the losses of excited atoms and ions to the electrodes [35]. However, the ion density in the plasma increases so much when argon is introduced that there is intensive electron scattering from the cesium ions [45]. Therefore, the beneficial effect from the introduction of argon is not very great.

output V.

The condition of the hydrodynamic equilibrium for a strongly ionized plasma is the equality of saturated vapor pressure above cesium reservoir and of the total pressure in the gap - created by the atoms, ions, and electrons. As ionization increases, the electron gas pressure increases and, consequently, the total density of atoms and ions must decrease. If the degree of ionization approaches 100%, the ion density should approach saturation as voltage on the converter increases, and then should even decrease somewhat because of some increase in the electron temperature T_e. This trend is seen in the experimental data shown in Fig. 10.23. The absolute values of the measured densities, which

The main variations of potential in the Knudsen arc occur in the narrow sheath near the cathode. Variations of the potential in the volume of the plasma are small and do not exceed 0.2 - 0.4 eV . There is a small potential barrier near the anode which retards the flow of electrons from the plasma to the anode V_A » l.5 - 2kT_e. As the voltage on the device increases, all the additional drop goes to increase the precathode barrier V_C, the value of which usually exceeds the first excitation potential of the cesium atom (V_C ³ 1.4 V).

The temperature T_e of the Maxwellized electrons is essentially constant across the gap. As voltage on the device increases, T_e increases. With a constant value of V_L, T_e increases as emission current j_s decreases and as cesium pressure decreases. The temperature T_e is within the range of (4-8)x10³ ° K in all the investigated modes.

The current-voltage characteristics of a Knudsen arc. The current-voltage characteristics of a Knudsen arc have the same general features as the corresponding characteristics of an arc in a dense plasma. At comparatively low cathode temperatures, the current-voltage characteristics have hysteresis (the ignition potential V_ig exceeds the extinction potential V_ex). As T_C increases, the difference V_ig - V_ex decreases, and at high values of T_C and large emission currents j_s, the transition to the arc mode is smooth [35].

When the plasma becomes strongly ionized, the electric field intensity at the cathode (6.2.17) E_C = (16p j_iC)^1/2(M/2e)^1/2V_C^1/4 (where j_iC is the ion current to the cathode and V_C is the cathode potential drop) increases weakly as the applied voltage increases. This is because j_iC reaches saturation, and then even decreases somewhat because of the increase of electron partial pressure. Therefore, the Schottky effect is much more weakly expressed in a Knudsen arc. Moreover, because the total flow of cesium atoms and ions to the cathode decreases in a strongly ionized plasma (because of the increase of electron partial pressure in the gap) the degree of cathode cesium coverage and the emission current decrease. All these effects combine to give the observed current saturation on the current-voltage characteristics.

The Maxwellized electrons do not play an important role in the total converter electron current at low plasma densities (n £ 10¹² cm^-3). In this case, the current is transported by the cathode electron beam, and corresponds to cathode emission current (j_s £ 0.3-0.4 A/cm²). However, at current densities 3-5 amp/cm², the charged particle densities in the plasma are so high (n ³ (3-5)10¹³ cm^-3) that the cathode beam manages to spread. In this case, the current in most of the gap is carried by the Maxwellized electrons. And the measured values of electric field intensity in the plasma are close to the calculated values, i.e., if the value of s for a completely ionized gas is substituted into the formula E = j/s . Problems of cathode beam relaxation in a Knudsen arc were studied in [36-39, 47-48] using probe diagnostics.*

If the plasma contains an electron beam moving in one direction W ₀, then the electron distribution function for the beam has the form f(v) = f(ï vï )d (W - W ₀). With such a distribution, the probe characteristic

*In principle, the spectroscopic methods can also be used to investigate the cathode beam [52].

of a cylindrical probe can be used to determine some of the beam parameters. A semi-logarithmic probe characteristic typical for these conditions, is presented in Fig. 10.25. The deviation from the Maxwellian distribution function (segment abcd) occurs because of the beam of electrons which is emitted by the cathode and accelerated through the precathode potential drop.

If the retarding potential of the probe is greater than the electron energy in the beam, and the electron distribution function of the beam f(ï vï ) is a Maxwellian function, then the electron current to the probe is exponentially dependent on the probe potential V₀ with an electron temperature in the beam of T₀ (segment ab).

The values of potential, at which deviations from a linear function begin, correspond approximately to the electron energy of the beam E₀. For eV₀ < E, electrons whose radial velocity component is sufficient to overcome the pre-probe barrier will impinge on the probe (segment cd). By using (7.1.6), we obtain the average electron current density to the cylindrical probe perpendicular to the velocity direction W ₀ of the beam:

where j_b is the beam current density.

It is obvious from (5.1) that the function j2 = f(V₀) is a straight line. By extrapolating this line to the plasma potential, where V₀ = 0, one may obtain the electron current density in the beam j_b.

For an electron beam, the second derivative of current to the cylindrical probe does not yield the distribution function averaged over directions. As can be seen from (5.1), on segment cd of the probe characteristic we have d²j/dV² < 0. Nevertheless, the value of d²j/dV² may serve as a sensitive, qualitative indicator of the presence of a beam and the variation of the electron distribution function in the beam.

The electron part of the current-voltage characteristics, for a probe located parallel to the electrodes and at different plasma concentrations from 5x10¹⁰ to 2x10¹³ cm^-3, is shown in Fig. 10.26. The density n was varied by changing the cathode temperature, and thus the current through the device, with a constant external voltage across the electrodes. It is obvious that at low plasma concentration (n » 10¹¹ cm^-3) the beam passes through the interelectrode gap without being deformed (curves 1). This is also obvious in Fig. 10.26 on the right, where the second derivative of the probe current is plotted. At high densities (n » 10¹³ cm^-3), the beam, moving toward the anode, relaxes completely to a Maxwellian distribution (curves III). At intermediate concentrations (n » 10¹¹ - 10¹³ cm^-3), the beam manages to relax appreciably while moving toward the anode, but the fast electron distribution function is far from Maxwellian (curves II) even near the anode.

where E is electron energy in eV, n is the plasma density in cm^-3, and lnL is the Coulomb logarithm (approximately equal to 10). It is obvious from (5.2) that l_E < 1 mm for E » 2 eV and n > 2x10¹² cm^-3. This is in good agreement with the experimental data presented in Fig. 10.26 (curves III).

At average values of density, n » 10¹¹ - 10¹² cm^-3, and at typical spacings, we have l_E » d. However, this still does not mean that the beam remains unchanged in velocity space. For l_E » d, the Coulomb collisions may lead to energy spreading in the narrow beam, with the average beam energy remaining unchanged. According to [40], the mean-square-deviation of electron energy in a beam on length l, as a result of Coulomb collisions, is equal to

where E is expressed in eV, n is expressed in cm^-3 and l is expressed in cm. For n » 10¹² cm^-3, E » 2 eV, kT_e » 0.5 eV, and l » 1-2 mm, as can be seen from (5.3), the variation of electron energy in the beam is very significant.

Thus, the observed relaxation in a Knudsen arc with comparatively low cathode beam energies (E » 1.5 - 2 volt) may be explained within the framework of the theory of Coulomb collisions. That is, for high plasma densities, there is a total

deceleration of the beam and the distribution function in the main part of the gap is Maxwellian. And on the other hand, the observed variations in the probe characteristics for n » 10¹¹ - 10¹² cm^-3 indicate only a spreading of the beam, with the average electron energy in the beam remaining unchanged. It should be emphasized that cathode drops no greater than only l.5 - 2 eV are of interest for the conversion of thermal to electric energy.

For beam energies E ³ 3 eV, the Coulomb collisions may not explain all the available experimental data, and beam instabilities in the plasma are responsible to a significant degree for beam relaxation [36, 39].

Elastic collisions of the beam electrons with atoms and ions produce scattered fast electrons. These electrons may be reflected from the anode (at grazing angles) and be in the gap for a rather long time. If the probability of electron scattering from the beam at an angle of about 90° or more during flight across the gap is denoted by w, then the density of the scattered fast electrons is

where t _p is the lifetime of a scattered fast electron in the gap, t ₀ is the flight time of a beam electron, D is the electrode diameter, and d is the interelectrode spacing.

Scattered-fast-electron density has been measured [38] in devices using two cylindrical probes. One of the probes was parallel to the planes of the electrodes and could trap electrons with different velocity directions. The other probe was perpendicular to the electrodes and trapped only those fast electrons which changed velocity directions as a result of collision.

The probe characteristics of both probes, taken at the same point of space, are shown in Fig. 10.27. It is obvious that an explicit difference in the characteristics, which indicates asymmetry in the distribution function, is observed in the high-energy range. The number of fast electrons moving in the cathode-to-anode direction is greater than the number of electrons moving in the perpendicular direction. However, as indicated by the experiments, the density of scattered fast electrons, collected by the probe perpendicular to the electrode surface, is rather high and in some cases may reach a value of about 30% of the total number of fast electrons. Since the time that the scattered fast electron is in the gap considerably exceeds the flight time of the cathode beam, the probability that this electron will collide with the Maxwellized electrons of the plasma or the probability of any elastic collision of this electron with a cesium atom is much higher than that for a beam electron. Therefore, Maxwellization of the scattered fast electrons begins at lower values of charged and neutral particle densities.

However, experiments show that the density and temperature of the Maxwellized electrons, and the potential in the well, hardly vary across the gap. Therefore, an attempt can be made to obtain the main plasma characteristics from the integral relations for the entire volume of plasma, such as the laws of conservation for current, energy, etc. [39, 41].

Calculation of the rate G of multi-step ionization in the presence of a fast beam and slow electrons in the well is a special problem [41, 42]. The important role of the beam is indicated by the experimental fact that the arc can be maintained at low cesium pressures if the precathode drop exceeds the first excitation potential of the cesium atom ( » 1.4 eV), whereas the arc is ignited at even lower values of V_C in a dense plasma.

Uniform ion distribution in the volume in the absence of an electric field can be obtained if one assumes that there is some probability, constant across the gap, of ionization of atoms desorbed from the electrodes. Then the density of ions flying from the cathode to the anode is @ and that from the anode to the cathode is n² = k(d - x), where k is some coefficient proportional to the ionization probability, and for this case

The electron current of Maxwellized electrons from the well through the anode barrier should be equal to the ion current (5.5) and should also compensate for the electrons knocked out of the beam and captured in the potential well as a result of inelastic collisions with atoms and as a result of electron-electron scattering (j_ee):

where r_C is some number (r_C < 1), which takes into account the deviation of the distribution function of the electrons at the anode barrier from a Maxwellian distribution [43].

where a is the part of ionization energy expended by the Maxwellized electrons and D E@ is the average energy of the electrons captured in the well (referenced to the bottom of the well). For example, if an electron enters the well because of excitation of a cesium atom from the ground level to the first excited level, then D E@ = eV_C - E₀₁.

Equations (5.5)-(5.9) form a complete system of equations for calculating the plasma parameters n, T_e, j_i, V_A, and V_C as a function of the applied voltage V_d. To them should also be added the equation for particle balance at the outer edge of the interelectrode gap, which, for a dense plasma, replaces the condition of pressure equality between the reservoir and the gap.

If necessary, one may also take into account the Schottky effect at the cathode by the usual formulas [44].

However, to solve the system (5.5)-(5.9), one must also know the value of G , the number of electrons captured in the potential well j_ee, and also the coefficients a and r_C, which is a separate and rather cumbersome problem. Therefore, at the present time, there is no solution for the system (5.5)-(5.9).