DIFFUSIONMODE

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267

Chapter 8

THE DIFFUSION MODE OF THERMIONIC CONVERTERS

When classifying the modes of thermionic converters, it was pointed out that the diffusion mode occurs when the motion of current carriers is by diffusion (because of the high density of atoms) and when the main mechanism of ion formation (required to compensate the electron space charge) is by surface ionization at a hot cathode. In this mode, the electron and ion mean free paths, l_e and l_i, are much less than the interelectrode distance d, and ionization and recombination within the volume are essentially absent. Investigation of the diffusion mode leads to qualitative understanding of the main characteristics of current passage through a dense TIC plasma. These characteristics are important to the arc mode as well. The first theoretical investigation of the diffusion mode was reported in [1].

Subsequent theoretical investigations were devoted both to more accurate calculation of current-voltage characteristics and to different methods for solving the kinetic equation in order to calculate the kinetic coefficients of the plasma appropriate to thermionic converters [2-9]. These investigations permitted the evaluation of the effect of Coulomb scattering in various modes, permitted an evaluation of the energy exchange for collisions between cesium ions and atoms, etc. On the whole, the results obtained in these investigations do not differ essentially from the results of [1]. The diffusion mode was investigated experimentally in a number of investigations [10-13].

1. The Theory of the Diffusion Mode

Equations of the diffusion mode. The values of electron and ion current, j_e and j_i, in the diffusion mode are dependent only on the electric field intensity E = dV/dx and the gradients of the density n and temperature T. For the one dimensional case, these currents are given by the diffusion equation (4. 3.18) and take the form; for the electron current,

(1.1)

and, for the ion current,

(1.2)

In equation (1.1), expression (4.3.15) was used for the thermal—diffusion ratio K_e^(T). And in equation (1.2) the friction force R_ie, which is usually small at the degree of plasma ionization typical for the diffusion mode, is omitted in contrast to (4.7.2).

The distributions in the plasma of temperatures T_e and T are calculated with the heat conduction equation. Since the degree of ionization in the diffusion mode is usually low, only the heat fluxes transferred by the electrons,

268

(1.3)

(see (4.3.19)) and by the atoms,

need. be taken into account. Expression (4.3.15) for K_e^(T) and the relation k _e = (r_e + 2)D_en_ek, which follows from (4.3.14), (4.3.16), and (4.3.20), were used in (1.3).

If the current carrier density is sufficiently high that the inter-electrode distance is much greater than the Debye length L_D = (kT/4p ne²)^1/2 (and this is the case of greatest interest for thermionic conversion), then in the entire volume, with the exception of narrow boundary regions with thicknesses on the order of the Debye length, the condition of quasi-neutrality is fulfilled

The interelectrode space may then be divided into three regions the region of quasi-neutrality, where the equations (1.1)-(l.3) described above are valid, and two narrow pre-electrode regions of space charge near the cathode and anode. The thicknesses of these regions (i.e., L_D) are usually much less than the electron mean free path. In this case, the boundary conditions derived in §7, Chapter 6 can be used to describe the effect of the precathode and pre-anode sheaths. We note that the boundary conditions (6.7.28)-(6.7.33) are applicable only in those cases where the potential distribution in the pre-electrode region is monotonic.

According to formulas (6.7.28)—(6.7.33), the boundary conditions near the cathode may be described in the following form: if the potential barrier accelerates the electrons emitted by the cathode (f _C > m ), then

(1.4)

(1.5)

In the opposite case, when (f _C > m ),

(1.6)

(1.7)

In (1.4)-(1.7)

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T_eC is the electron temperature in the precathode region of the plasma, eV_TC = ï f _C - m _Cï/e is the equilibrium potential barrier near the plasma-electrode interface (Fig. 8.1), D V_C is the change in the absolute value of the potential barrier during the passage of current, n(T_C) is the equilibrium density of electrons near the boundary, which, according to (5.1.13) is equal to

(1.8)

and n_C is the density of charged particles in the plasma directly in front of the potential barrier. Function f₀(V), which takes into account the asymmetry of the electron and ion distribution functions in the plasma in front of the decelerating potential barrier, is given by formula (6.7.13) For eV_TC » kT, g = 1/4 and for eV_TC « kT, g = 1/2.

We note that similar boundary conditions may be written by using equation (6.7.30) and (6.7.33) for the energy flow S_eC transferred by the electrons at the plasma boundary.

Fig. 8.1

If the net current is low compared to the current flow to and from the cathode, the boundary conditions simplify considerably and reduce to the conditions

(1.9)

Formulas (1.9) are valid for a mode which is nearly in thermal equilibrium, i.e., for ï f _C - m _Cï/kT <~ 1.

The boundary conditions near the anode are strongly dependent on the extent of anode emission. At anode temperatures below 1100-1200° K, and at sufficiently high cesium vapor pressures, the anode work function f _A has a value considerably less than the chemical potential of the plasma, m _A. Therefore, the degree of ionization of cesium atoms on the anode surface is low, and the ion current from the anode may be disregarded. If there is a barrier V_A near the anode which retards

270

ions emitted by the anode and electrons travelling from the plasma, the boundary conditions, according to (6.7.31) and (6.7.32), have the form

(1.10)

(1.11)

where j_sA is the anode electron emission and n(T_A) is the equilibrium electron density near the anode, which corresponds to the density of cesium atoms N_A near the anode at T = T_A:

(1.12)

(the second term in (1.10) takes into account the back emission from the anode).

In the opposite case, where the anode barrier retards the electrons from the plasma, the boundary conditions have a form similar to equations (l.4) and (1.5). For boundary conditions (1.4)-(1.7) and (1.10)-(1.11), the direction of electron motion from the cathode to anode and of ions from the anode to the cathode, as with equations (1.1) and (1.2), is taken as the forward direction for the currents.

The precise solution of the system of equations (1.l)-(1.3) is possible only by numerical methods; therefore, an approximate solution is obtained below, and then the errors that occur in this approach are evaluated.

The general form of the current-voltage characteristics and the approximate value of the saturation current (for a cold-anode) can be obtained from the following simple concepts. It is obvious from (1.1) and (1.2) that in the plasma there is a field component of current j(mob) proportional to the electric field E = dV/dx, a diffusion component j(dif) proportional to the density gradient, and a thermal-diffusion component j(th) proportional to the temperature gradient. However, the thermal-diffusion current is usually small, and if it is disregarded, equation (1.1) and (1.2) can be rewritten in simpler form:

(1.13a)

(1.13b)

In the open-circuit mode, where electron current is essentially equal to zero, an electric field is established in the volume which compensates the electron diffusion current from the cathode to the anode:

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D_edn/dx = nm _edV/dx (Fig. 8.2a). With decreased load voltage and increased current, the plasma electric field decreases, changes sign, and eventually reaches a value where field ion current is equal and opposite to diffusion ion current (Fig. 8.2b), giving zero net ion current. At this point, the electron field current and electron diffusion current are also equal, but in the same direction. There can be no further decrease in the plasma field (and plasma drop V_p) since this would give a negative ion current - which cannot be as long as ions are produced only at the cathode. Were such a field to occur, ion flow from the plasma to the cathode would cause an electron space charge which would remove the excess negative field.

With a further decrease of output voltage V_L, the entire additional potential difference D V shows up in the pre-anode space charge sheath (Fig. 8.2c), and the saturation current through the converter j_es remains equal to twice the diffusion current. If the variation in the diffusion coefficient and the mobility across the gap are disregarded, the density varies linearly across the gap, i.e., dn/dx » (n_C - n_A)/d » n_C/d, and since n_A « n_C, the diffusion current becomes j(dif) » D_een_C/d, and the saturation current is approximately equal to twice the diffusion current:

(1.14)

Approximate solution of the diffusion equations. If we rewrite the transport equations (1.1)-(1.3) by introducing the dimensionless variables

where D_e,i@ and m _e,i@ are the average values of D_e,i and m _e,i, which will be calculated below. Then, from (1.1)-(1.3), we obtain

(1.15)

(1.16)

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(1.17)

Consider the case where f _C - m _C £ kT and the boundary conditions in dimensionless variables have the form

(1.18)

The boundary conditions near the anode with a barrier which decelerates the electrons emitted by the anode has the form

(1.19)

(1.20)

where

(1.21)

(l_e,i is the electron or ion mean free path). In this case we assume that D_e,i = (l/3)(lv_e,i@ )(T_e,i). In these boundary conditions, we disregard the slight difference between v_e(T_e)@ and v_e(T_A)@.

To solve equations (1.15)-(1.17), it is necessary to know how the ion temperature T_i and electron temperature T_e vary across the gap. Since there is a rapid energy exchange between the ions and atoms at high cesium vapor pressure, the ion temperature should be essentially equal to the gas temperature T. If the temperature drops are not very great the dependence of thermal conduction on temperature can be disregarded and one can assume that T varies linearly across the gap, i.e.,

(1.22)

The electron temperature distribution across the gap is calculated from the rate of transfer of their energy to the atoms. Since the electrons impart only a fraction of their energy - on the order of 2m/M (for cesium 2m/M = 0.8x10^-5) - to atoms during elastic collision, the energy relaxation length L_E exceeds the momentum relaxation length L_p by a factor of (M/2m)^1/2. (This assumes that the electron temperature is not very high so that the probability of atomic excitation is very low.) If L_E exceeds the interelectrode distance d, the electron temperature is essentially determined by the condition of constant heat flux, calculated by (1.3) and (1.17). In this case, the electron temperature is sensitive to the electric field distribution in the volume.

However, as indicated by the analyses in [10], the contributions of thermal-diffusion electron and ion currents compensate for each other to a significant degree and, on the whole, do not exceed 5% of saturation current. Therefore, we disregard these inputs. The dependence of the

273

diffusion coefficient on temperature is more significant. However, for simplification, we first assume that D_e = D_e@ and D_i = D_i@ are constants. and then calculate the mean diffusion coefficients D_e@ and D_i@ to give the least error in the current saturation mode. By adding (1.15) and (1.16), with the above simplifications, we obtain

(1.23)

Having integrated equation (1.23) and taking (1.18) into account, we calculate the density distribution across the gap:

(1.24)

By deriving (1.16) and (1.15), we obtain

(1.25)

By substituting v(x ) from (1.24) into (1.25) and taking (1.18) into account, we obtain

(1.26)

and consequently, the voltage drop in the plasma is equal to

(1.27)

and the relative density near the anode, according to (1.24) is equal to

(1.28)

Consequently, by knowing the anode temperature T_A (and its work function), the cathode temperature T_C, and the density of atoms N_a, we can obtain t , m _A and n(T_C). From the system of four equations (1.19), (1.20), (1.27), and (1.28), given one parameter, for example, I_e, we can obtain I_i, v_A, V_p, and D V_A; and at the same time, can calculate the total voltage on the load:

(1.29)

The saturation current can be calculated immediately from (1.19) and (1.28). At high anode sheath voltages V_A, the ion current in the plasma, as well as the electron current from the anode, approaches zero, and consequently, we have

274

(1.30)

(1.31)

or in dimensionless variables*

(1.32)

We can now calculate D_e@. First we take into account the dependence of the diffusion coefficient D_e on the density of atoms N_a. If electron-atom scattering dominates, we have D_e ~ 1/N_a, and if there is a linear temperature distribution T(x), according to (1.22),

(1.33)

where t = T_A/T_C, and consequently, when extracting saturation current (I_i = 0), according to (1.16),

(1.34)

The electron current j_e (disregarding thermal-diffusion current), according to (1.15) and (1.18) is equal to

Therefore, we can immediately obtain j_e:

(1.35)

In this case, we assume that near the anode n (1) = I_e/g _e » 0. It is obvious from the comparison of (1.35) and (1.32) that the mean diffusion coefficient, introduced above in the derivation of the saturation current, is equal to

(1.36)

___________

*Formula (1.32), although derived for the case of l « d, remains valid in the opposite limiting case of the Knudsen mode, when l » d. Of course, in this case, n_C should be calculated from equations for the vacuum mode (Chapter 10), while D_e = lv_e(T_e)/3. Therefore, expression (1.32) can be used as an extrapolation formula in the intermediate range as well [14].

275

In the approximate calculations below, this value of D_e@ is used at all voltages.

A plot of D_e@/D_ec = (1 - t )/2 ln((1 +t )/2t ) is shown in Fig. 8.3. The relation D_e@/D_eA = I_es(T_A)/I_es(T_A = T_C) is shown by the dashed line. It is obvious from Fig. 8.3 that the variation of D_e in the range T_A > 0.25T_C can be easily described by replacing the "mean" diffusion coefficient D_e@, calculated by (1.36), by the expression

(1.37)

Fig. 8.2

Fig. 8.3

The straight line corresponding to this approximation is shown in Fig. 8.3 by the thin line. For T_A » (0.3-0.5) T_C, the "mean" diffusion coefficient exceeds D_eA by 25-50%.

In the calculations above, the variation of electron temperature (which also affects the diffusion coefficient) was not taken into account. However, corresponding analyses [10] show that the contribution of this effect to the saturation current is approximately 2% and partially compensates for the thermal-diffusion current. These analyses show that neglect of thermal-diffusion electron and ion currents, and also electron heating, yields an error in the value of saturation current which does not exceed 5%. Consequently, if one uses the correct value of the "mean" diffusion coefficient, formula (1.32) has an accuracy which is undoubtedly adequate for all practical calculations, that is, for retarding voltages in the plasma near the anode which are not very high, where collected currents are not less than 0.3-0.4 of saturation current.

276

With a hot anode, when the current from the anode is more than 0.3 of saturation current, large retarding voltages generally correspond to negative current. Thus, these equations are suitable over the entire operating range with a hot anode.

As already noted, it was assumed in the calculation above that the electron density near the cathode is an equilibrium density, equal to n(T_C). We can now take into account the deviation of the density near the cathode from equilibrium. For the case f _C < m _C, in dimensionless variables, the boundary conditions (1.6) and (1.7) have the form

(1.38a)

(in this case the slight difference between T_eC and T_C is neglected)

(l.38b)

Here, D V_C = D f _C/kT_C is the change in the precathode barrier with the passage of current (see Fig. 8.1). When saturation current is being collected, i.e., when I_i = 0,

(1.39)

Since, according to (1.31), I_es = 2/(1 + 2/b _e), then

and the electron saturation current is

(1.40)

Here, l_eC and l_eA are the electron mean free paths for electron-atom scattering near the cathode and anode, respectively.

For f _C > m _C, the boundary conditions have the form

(1.41a)

(1.41b)

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Here b _e' = 3g d/l_e and b _i' = (3/2)(d/l_I). Again, assuming that I_i = 0 and I_ee = 2/(1 + 2/b _e), we have exp(-D V_C) = n_C/n(T_C),

Therefore,

and the electron saturation current is then calculated by the expression

(1.42)

where s = en(T_C)D/j_sC = 4/3(l_e exp(f _C - m _C)/kT_C), and j_sC is the emission current. For d » l, this complicated function is extrapolated rather well by the simple formula

(1.43)

In the limiting cases of d/2s » 1, formulas (1.42) and (1.43) transform to (1.14) at n_C » n(T_C), and for d/2s « 1,

(1.44)

i.e., in this case, the collected current is given by the emission current.* Since b _e/I_e » 1 and b _i/I_i » 1, then, according to (1.38a) and (1.41b),

(1.45)

Thus, the density near the boundary decreases in both cases with the passage of current, while the height of the barrier V_C + D V_C increases. It follows from (1.38a), (1.41a), and (l.45) that

(1.46)

i.e., the deviation of electron density from equilibrium depends not only on the relative value of the work function and the chemical potential of the plasma but also on the value of the ratio d/l, i.e., on parameter pd. This problem is considered in more detail in the next section.

_______________

*Expression (1.44) for j_sC is obtained from (6.7.24), if one takes into account that n(T_C) = N_e exp(-m _C/kT_C) and eV_T = f _C - m _C.

278

Besides the saturation current and voltage on the load, the main output parameters of a converter include the output power, efficiency and open circuit voltage.

According to (1.9), the load voltage for f _C £ m _C is given by

(1.47)

Consequently, the output power is equal to

(1.48)

In the concern for energy, maximum power P_max, when dP/dV = 0, may be of greatest interest. One may assume the approximation

(1.49)

because usually m _C - f _A considerably exceed the voltage drop in the plasma and in the pre-anode region.

The efficiency is given by the expression

(1.50)

Here S_eA = j_ee {2kT_e/e + V_A^* + f _A/e} is the heat transferred to the anode, where V_A^* = V_TA + D V_A with an accelerating barrier, and V_A = 0 with a decelerating barrier. Also, T_eA is the electron temperature near the anode, S_A is the conduction heat flux, and S_rad is the radiant flux. The open-circuit efficiency, where the total current is j_es + j_is = 0 (Fig. 8.2a), is given by the expression

(1.51)

Since D V_C = 0, it is necessary to calculate e(V_p + V_TA + D V_A). It is easy to show that in this case

(1.52)

also, the chemical potential in equation (1.51) can be replaced by the ionization potential m = E_ion/2 + kT_e ln(N_e/N_a)^1/2 (where N_e is the density of states and N_a is the density of atoms), so that

(1.53)

In the case of a cold anode and f _A » l.7-l.8 eV, the first two terms in expression (1.53) yield 0.1-0.2 eV, i.e., the main contribution to the open-circuit voltage is the last term. We note that the ion-atom scattering cross section can be calculated from the open-circuit voltage.

279

2. The Characteristics and Properties of a Diffusion Plasmas

Solutions of the diffusion equations were given in the preceding section. The current-voltage characteristics of a TIC were calculated and general relations were obtained which link the so-called output parameters of the TIC (saturation current and short-circuit current, open-circuit voltage, etc.) with external conditions. By using these calculations, let us analyze the dependence of the output parameters on the cesium vapor pressure, the interelectrode spacing, and the electron temperature. And let us compare theoretical calculations with experimental results.*

Fig. 8.4

Current-voltage characteristics. A typical calculated current-voltage characteristic is shown in Pig. 8.4. The experimental characteristic obtained for the same pressure, spacing, and cathode temperature is also shown in the figure. These characteristics were obtained with no emission from the anode. As can be seen, the current-voltage characteristics actually have a clearly defined section of saturation current, the absolute value of saturation current being determined in this case by (1.14): j_es = 2D_e@ n(T_C)/d, where n(T_C) is the equilibrium density near the cathode, calculated by (1.8). Comparison of the experimental and calculated values of j_es shows that their difference does not exceed 20%. The thermal efficiency of a thermionic converter is determined primarily by the difference between the plasma chemical potential near the cathode m _C and the anode work function f _A.

The maximum power P_max, collected from the converter, can be interpreted according to (1.49). The values of P_max in the diffusion mode are comparatively small and do not exceed 2-1.5 W/cm². The efficiency, even optimized with a gap on the order of 1 mm, does not exceed 5-7%, because the radiation losses are excessive. Development of the cathode surface does not yield an appreciable advantage in the output characteristics [15].

Dependence of saturation current on cathode temperature. According to (1.14), saturation current at sufficiently low temperatures (where f _C < ~ m _C and where electron-atom scattering dominates) should depend exponentially on temperature with the dependence lnj_es = A - E_ion/2kT_C. The slope of the curve, therefore, should correspond to the ionization potential of the cesium atoms. As noted in the preceding section, in the cathode temperature range where the current calculated by (1.14) exceeds the cathode emission current j_sc = (1/4)(en)(T_C)v_e@, the collected current j_es is determined by the emission current. Semi-logarithmic plots of j_es = f(1/T_C) obtained at different cesium vapor pressures are presented in Fig. 8.5

______________

*Comparison is usually carried out with the experimental results from converters with molybdenum electrodes.

280

Fig. 8.5

Fig. 8.6

as an illustration of this dependence. Similar investigations have been carried out with rubidium vapor. The slope of the straight lines j_es = f(1/T_C) obtained in this case correspond to the ionization potential of rubidium atoms to within an accuracy of ±0.05 eV (Fig. 8.6).

The slopes of the experimental curves are somewhat less than the theoretical slopes with small gaps where the ratio d/l is not very large.

At high temperatures, when j_es » j_sC the current decreases briefly as temperature increases. This is typical for emission current (S-curves) and is related to the increase of the work function with the decrease of cathode coverage. This is shown in Fig. 8.7, where the segments of the corresponding emission S-curves for tungsten and molybdenum in cesium vapor (P_Cs = 1 torr) are plotted along with the maximum current, for the diffusion mode (for d = 1 mm). The thick lines indicate the values of the current which can be passed in the diffusion mode under these conditions.

Dependence on cesium vapor pressure. According to (1.14), the saturation current j_es should be proportional to P_Cs^-1/2, since D ~ P_Cs^-1 and n ~ P_Cs^1/2 . This dependence is observed in the case of f _C < m _C and also f _C > m _C, with sufficiently large gaps (Fig. 8.8).

In the under-compensated mode (f _C < m _C), where the cathode work function is "shielded" by the potential barrier, the function j_esP_Cs^1/2 = f(1/T_C) is universal for different cathode materials and is a straight line with a slope of E_ion/2. It follows from (1.14) that the ratio of the anode current j_es in the diffusion mode to the total emission current j_sC at f _C = m _C is given by the expression j_es/j_sC » 8/3(l_e/d). A further decrease of cesium vapor pressure increases the mean free path, which leads to an increase of current until the increase of the work function, i.e., decrease of emission current, leads to a decrease of current according to (1.42). It is obvious from this formula that the

281

current is independent of f _C until

(2.1)

The range of values for the work function (and consequently, for the cesium vapor pressures) in which current is not dependent on f _C is broader, the greater the interelectrode distance d.

Fig. 8.7

Fig. 8.8

If condition (2.1) is fulfilled, a decrease of pressure actually leads to an increase in the anode saturation current. At large values (f _C - m _C)/kT_C » ln((3/4)d/l_e), despite the strong scattering (l » d), the total emission current will be collected. Therefore, at these work functions, the cesium vapor pressure can always be increased to reduce the work function, to increase emission current, and consequently, to increase the anode current as well. The optimum value of cesium pressure corresponds to (f _C - m _C)/kT_C » ln((3/4)d/l_e), i.e., it is dependent on the cathode temperature and on the value of the interelectrode spacing. An increase of cathode temperature leads to an increase of the optimum pressure, because the cathode cesium coverage decreases as T_C increases and an increase of P_Cs is required to maintain the work function at the previous value.

Dependence on interelectrode distance. According to (1.14), saturation current for f _C < ~ m _C is inversely proportional to the inter-electrode distance: jes ~ 1/d. This function has been confirmed experimentally (Fig. 8.9).

282

If spacing d is very small, terms on the order of l/d must be taken into account. i.e., equations (1.32) and (1.40) must be used. According to (1.32), for f _C < ~ m _C.

The theoretical dependence of saturation current on spacing for the case of f _C - m _C » kT_C has a more complicated form and is

given by (1.42).

The experimental data agree well both qualitatively and quantitatively with these theoretical results. The straight lines in Fig. 8.9, taken at T_C = 1575 and 1650° K, are for a case of f _C < m _C. It is obvious that these straight lines intersect the x-axis at d = -140 m m, whereas the mean free path, found experimentally at P_Cs = 0.44 torr is l_e = 100 m m. The condition f _C » m _C is approximately fulfilled for T_C = 1725° K and the straight line for this temperature intersects the x-axis at approximately twice the value of the above case. The straight line corresponding to T_C = 1810° K is known to be for a case of f _C > m _C, and its extrapolation to d = 0, according to (1.43), allows the calculation of the cathode emission current and, accordingly, the cathode work function f _C.

Fig. 8.9

The effect of anode temperature. Anode heating does not lead to a significant deviation from the common type current-voltage characteristics, with the exception of the appearance of back emission.

Characteristics for converters with high temperature anodes have clearly defined saturation segments. There is a shifting to the characteristics, however, caused primarily by changes in the anode work function. The work function passes through a minimum as temperature increases (at a pressure of P_Cs = 1 torr located at approximately T_A(min) = 900° K). Accordingly, for T_A < T_A(min), the characteristic is shifted to the left, and at T_A > T_A(min)), it is shifted to the right.

Therefore, the output voltage of the converter decreases as anode temperature increases. The higher the anode temperature, the steeper the slope is of the current-voltage characteristic in the range of negative currents, and the closer to zero is the point of intersection of the characteristic with the voltage axis. The theoretical and experimental current-voltage characteristics, normalized to saturation current, are presented in Fig. 8.10. As can be seen, the shape of the curves, the variation of the characteristics as a function of the anode temperature, the absolute values of short-circuit currents, and the open circuit voltages are all in good agreement with the theory. The slight differences, i.e., the shifting somewhat to the left of the experimental characteristics compared to the theoretical characteristics, and the greater slope in the range of negative currents, should be due primarily to an error in the calculation of f _A.

283

Fig. 8.10

The relation j_es(T_A) is presented in Fig. 8.11. It is obvious that if the anode temperatures are not very high, j increases linearly as anode temperature increases. This result is related primarily to an increase of the mean electron diffusion coefficient as the anode temperature increases, as a result of the decrease in cesium atom density near the anode. *

According to (1.37) and Fig. 8.3, the saturation current is linear in the anode temperature over a rather wide range of T_A. This linear

____________

*For high values of T_A, j_es should also increase as a result of the emission of excited atoms, ionized near electrodes, and as a result of ion emission from the anode, because each ion formed near the anode permits the passage to the anode of D_e/D_i electrons. Ion emission from the anode becomes appreciable for T_A > l400-1500° K. It is possible that the formation of additional ions near the anode can be partially explained by the faster increase of saturation current compared to D_e@, observed at high values of T_A.

284

Fig. 8.11

dependence of the curve I(T_A)/I(T_A = T_C) = f(T_A/T_C) intersecting 0.2 on the axis T_A = 0 is in good agreement with experimental observations of I(T_A = 0)/I(T_A = T_C) = 0.24 (Fig. 8.11).

We also note that, despite the increase of saturation current as T_A increases, the short-circuit current at high values of T_A decreases to the right because of the shift of the characteristics.

Plasma properties in the diffusion mode. Data on the plasma parameters in the diffusion mode have been obtained primarily from probe measurements, because the radiation from this plasma is too weak in most cases to permit the use of optical methods. But comparison of probe measurements and current-voltage characteristics permits a qualitative check of the theory.

The probe method, as already noted, does not always permit sufficiently accurate judgment about the electron temperature in the diffusion mode, because the absence of thermal equilibrium between the electrons in the pre-probe sheath affects the measurement. Nevertheless, at high values of T_C, when ion current to the probe exceeds its thermionic emission, electron temperature can be determined quite reliably by the initial segment of the electron branch of the probe characteristic. This treatment of the probe characteristics shows that the electron temperature, both in the short-circuit and in the open-circuit modes, is approximately equal to the cathode temperature and varies negligibly across the converter gap. Therefore, when calculating the density and space potential in the diffusion mode for f _C > m _C, we may assume that T_e = T_C.

Having determined experimentally the current j_e through the device, we can, by using equations (1.1) and (1.2), calculate the distribution of the density n(x) and potential V(x) in the interelectrode space.

It follows from equations (1.13) that at j_i = 0

(2.2)

According to (1.10), the electron density near a cold anode is n_A = j_e/((1/2)ev_e(TeA)@ ). Equations (2.2) are integrated numerically taking into account the dependence of the diffusion coefficient on distance (as a result of variation of the atom density through the gap). Typical inter-electrode distributions of density and space potential for a short-circuited thermionic converter are shown in Fig. 8.12 for three cesium vapor pressures. The experimental values are given as points; the solid lines are calculated. As already indicated, the density near the cathode

285

is close to the thermodynamic value n(T), and the density in the center of the gap is approximately 0.5 of the equilibrium value.

Fig. 8.12

The space potential in the quasi-neutral region of the diffusion mode is V₀ = m _C/e + V_p (Fig. 8.1), where m _C is the chemical potential of the plasma and V_p is the voltage drop across the plasma. Since V_p usually small, one may assume to a first approximation that V₀ ~ m _C/e. This is also in good agreement with experimental data.

The distribution of density and potential at typical points of the current-voltage curve - the open-circuit mode (dark points), short-circuit mode (light circles), and the pre-discharge mode, when glow associated with the beginning of volume ionization (crosses) is observed in the pre-anode sheath - is presented in Fig. 8.13. The dashed curves were plotted from experimental data. The theoretical distributions n(x) and V(x), calculated by (1.1) and (1.2) for the short-circuit mode, is shown by the solid curve.

The variation of the distribution of plasma parameters over the entire current-voltage characteristic is shown in Fig. 8.14 as a function of the voltage on the converter. The distribution of the density across the gap for all points of the current-voltage characteristic, upon transition from the open-circuit to the pre-discharge modes, remains practically constant. However, the potential distribution varies appreciably. Effectively, the space potential near the cathode does not vary for any mode where f _C < ~ m _C, and is close to the value of the chemical potential of the plasma, m _C, in absolute value, as already noted.

For retarding potentials ï Vï > ï V_ocï, the field in the volume retards motion of electrons from the cathode to the anode, and in this case, the total voltage drop V_p in the plasma is essentially independent of the converter voltage. All the excess voltage occurs in the pre-anode barrier. For ï Vï < ï V_ocï, there is a gradual decrease of the space potential, so that the electric field inside the plasma is essentially

286

Fig. 8.13

Fig. 8.14

287

absent at j_e » j_es/2 and V₀(x) » m _C/e (Fig. 8.13c). With further increase of current, there is a change of sign of V_p and a field is established in the volume which retards motion of ions from the anode to the cathode. The space potential becomes constant after the current reaches saturation, and again, all the excess voltage appears in the anode barrier.

As we can see, the experimental data are in good agreement with the general pattern of the diffusion mode outlined above and correspond completely with the conclusions of the theory.

Since diffusion theory makes it possible to obtain sufficiently accurate formulas to calculate the current in the saturation mode, variation of current in this mode can be used to find the kinetic coefficients, and consequently, the electron-atom and ion-atom scattering cross sections [11, 13].

3. The Effect of Magnetic Fields on the Diffusion Mode

The magnetic fields have two effects in the diffusion mode. First, the magnetic field reduces electron mobility and, consequently, the diffusion coefficient. Second, the magnetic field creates ponderomotive forces which lead to a pressure gradient in the plasma or motion of the plasma.

The problem of the effect of a magnetic field on mobility was considered above in Chapter 4. Variation of the diffusion coefficient D or of the mobility in a magnetic field with atom scattering where the mean free path is not dependent on energy, according to (4.4.17), is equal to

(3.1)

Function I(q ) is presented in Fig. 4.5. Variation of the diffusion coefficient in weak magnetic fields is calculated by the expression*

(3.2)

With electron-atom scattering, when we can assume that the mean free path is independent of energy, d = 1.77; and with electron-ion scattering d = 5.9.

According to (1.14), variation of saturation current in the diffusion mode is

(3.3)

where D_e is the mean diffusion coefficient calculated by (1.37). Equation (3.2) is valid when the effect of the pre-electrode barriers is small. If the barriers are large, the current drop in the magnetic field will be slower than follows from (3.3), and if the precathode barrier is large, the current will be determined by the emission current and will be essentially independent of the magnetic field.

The magnetic field created by currents passing through a converter depends on the specific geometry. For a simple geometry (as with plane

_____________

*The effect of a strong magnetic field is considered in [16].

288

or cylindrical electrodes), this field can be calculated by using the expression

(3.4)

The integral in the left side is taken over an arbitrary closed loop, where dl is the loop element, and H_l is the projection of the field on the tangent to this loop. The integral on the right side is taken over an arbitrary surface enclosed by this loop. The current component j_n is normal to the surface at a given point. This integral yields the total current passing through the loop and is independent of the specific shape of the surface.

If the electrodes are flat disks of radius R₀, and the current density j® from the converter is uniform, then

(3.5)

If j is expressed in amp/cm², r is expressed in cm, and H is expressed in Oe, then

(3.6)

The maximum value of the field is obtained near the edge of the electrode:

(3.7)

where R₀ is the electrode radius.

With cylindrical electrodes, which are of greatest interest for practical TICs, the field is

(3.8)

(current density j varies along the cathode length z) while total current is

(3.9)

where r₀ is the cathode radius. It is assumed in this case that the electrode is connected at point z = 0, and therefore, the current along the cathode at point z = L is equal to zero. If the interelectrode distance fulfills d « r@, where r@ = r₀ + d/2 is the mean electrode radius, then r₀/r » 1 and

(3.10)

289

If the current density along the cathode length is constant, then

(3.11)

By expressing j in amp/cm², L and z in cm, and H in Oe, we obtain

(3.12)

The maximum field is obtained at the point of cathode connection, i.e., at point z = 0:

(3.13)

As shown by analyses, because of the self magnetic field from converter currents, a decrease in mobility becomes appreciable with sufficiently large electrodes and not very high pressures. Thus, at P_Cs = 1.0 torr, T_C » 1500° K and T_A = 700° K, a magnetic field of approximately 80 Oe is required to reduce mobility and, consequently, saturation current by 25%.

With cylindrical electrodes 10 cm long, this field is obtained at the end of the cathode with a current density of 6.4 amp/cm². However, the mean diffusion coefficient, which determines the current, varies by only 8%. If we take into account that a decrease of current leads to a decrease in the magnetic field, the resulting variation of current will be somewhat less. This effect is difficult to calculate by using formula (3.3), which gives the dependence of current on the magnetic field. An experimental check of the effect of the magnetic field was carried out in a number of investigations [10, 17]. These confirm the concepts outlined above.

The effect of a magnetic field on the diffusion coefficient is not the only cause of current variation over the length of the converter. It was mentioned above that a magnetic field leads to a ponderomotive force in the plasma, which is calculated by the expression

(3.14)

These forces create a pressure gradient in the plasma, and in those cases where converter design permits it, they induce plasma motion.

A pressure drop in the plasma and the velocity of its motion under the effects of the external force are calculated by the equation

(3.15)

where h is viscosity and v@ is plasma velocity. In the case of a plane electrode system, force f@, according to (3.14), is directed toward the center of the disks, and if there are no openings in the electrodes, it will not induce any kind of plasma motion.

Consequently, in this case, according to (3.15) and (3.5), the pressure distribution is given by

(3.16)

290

where

If the pressure drop is small, so that the dependence of current on pressure j(p) can be disregarded, and if the dependence of current on H may also be disregarded, the pressure drop is

(3.17)

Here p(R₀) = p₀ is the pressure outside the electrodes. The greatest pressure p(0) will be in the center, The maximum drop is equal to

(3.18)

where p is in torr, j is in amp/cm²and R₀ is in cm. For example, at j = 10 amp/cm²and R₀ = 2 cm, D p = 10^-2 torr.

If the converter design is cylindrical, v = 0 (if the device is closed on one of the ends), and pressure distribution for d « r@ is calculated by the equation

(3.19)

where I'(z) = ò _r^Lj(z)dz = F/2p r₀ is the current through the cathode, divided by its circumference 2p r₀. Since dI' = -j(z)dz, then, by integrating (3.19), we find that

(3.20)

If the current density along the length of the cathode is constant, the pressure distribution is calculated by the equation

(3.21)

The greatest pressure will be near the end of the cathode to which current is applied and the maximum drop is equal to

(3.22)

where F is measured in amp/cm. Therefore, it is obvious that the pressure drop with any function j(z) is determined only by the value of total current F.

For example, if, F = 100 amp/cm, then D p = 0.5 torr. If the end where current is applied is also the end where the pressure is defined by the cesium reservoir, the pressure on the other end will be higher by a value D p. If, on the other hand, current is applied to the other end, a closed end, the pressure near it will be D p lower than that on the cesium reservoir end.

If a cylindrical converter is open on both ends, the ponderomotive forces will cause plasma motion. The velocity distribution across the gap is calculated by (3.22):

291

(3.23)

where x is the distance from the cathode. Assuming that p and f are not dependent on x, and that v = 0 at x = 0 and x = d, on the boundaries we obtain

(3.24)

where v@ = (1/d)ò ₀^dv(x)dx is the mean velocity. According to (3.23),

(3.25)

If the compressibility of the plasma is disregarded, the mean velocity should not be dependent on z. By assuming that v is constant, from (3.24) we can find the pressure distribution. If current density is j = const, then f is calculated by (3.14) and (3.11); therefore,

(3.26)

In this case, since the pressure on both ends is p = p₀, the pressure distribution has the form

(3.27)

By substituting (3.27) into (3.26), we find A and v@: A = (2p /c)j², i.e.,

(3.28)

and

(3.29)

The plus sign indicates that the current is directed toward z = L, i.e., to the end from which current is supplied.

In this case, it is obvious from (3.28) that the internal pressure will be higher than the edge pressure, and moreover, the maximum pressure will be in the middle

(3.30)

i.e., in this case, the pressure drop will be four times less than that of the closed end. The rate of plasma motion can be quite large: for example, at T_C = 1500° K, an ionization rate of n/N_a = 0.01, T_eA = 1.3 T_C, T_A = 700° K, and 4L = 10 cm, the rate v@ has the value 4.5 in/sec.

We note in conclusion that an additional field d E = v@ H/c = 10^-8vH V/cm occurs during the motion of the plasma, and the voltage on the converter decreases by d E· d. However, this effect is usually very small and may be disregarded.

292

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