Transport of Relativistic Electrons Scattered by the Coulomb Force and a Thermionic Energy Converter with a Built-in Discharge Tube

A transport equation of momentum for relativistic electrons scattered isotropically was previously reported. Here, a momentum-transport equation for relativistic electrons “scattered anisotropically” by the Coulomb force is inquired into. An ideal plasma consisting of electrons and deuterons is treated again. Also, to raise a generation-ability of a thermionic energy converter, a means of introducing external electric and magnetic fields within “a converter in which an emitter plate and a collector plate face simply each other” is proposed.


Introduction
In the classical theory (based on the Boltzmann equation with the Fokker-Planck collision term) with respect to the electron transport in an ideal plasma consisting of electrons and deuterons, as a frictional force to suppress unlimited increase of a drift velocity by an external electric field, only the dynamical frictional force coming from the cumulative effect of small angle deflections ceaselessly occurring is generally taken into consideration. However, considering that rare large angle deflections ought to be the scatterings due to the two-body (an electron and a deuteron) collisions, we reported [1] about the evaluation of an effective radius of the Coulomb force of a deuteron. The unexpected result was that a frictional force coming from the two-body collisions is much stronger than the dynamical frictional one (from Equations (18)-(20) of Ref. [1]). So, for simplifi-cation of an analysis in this research, we disregard an effect of the many-body collisions on a drift movement of an electron, compared with one of the two-body collisions. Furthermore, assuming that every electron has a mean thermal velocity υ , we inquire into a transport equation of momentum for relativistic electrons in the ideal plasma.
For changing radiation energy from a huge-sided magnetic mirror reactor into electric energy in future, a thermionic energy converter [2]- [9] is considered to be a promising generator, next to a steam turbine one. We discuss in Section 4 about a way to raise a generation-efficiency of a converter by help of some supplemental equipments.

Momentum Transport Equation
This research is discussed under the presupposition that a deuteron has as effective radius up r P − of the Coulomb force, with respect to the two-body collisions.
Then, a mean collision frequency r ν of an electron is is a deuteron density which is equal to an electron density e n ). A value of up r P − is estimated in after (24), together with r ς appearing later.
We first consider the case where a small electric field E(t) and a magnetic field B are: We use four coordinate systems: Here, ∑ represents summation of the vector quantity of each term over electrons per unit volume. And with respect to electrons scattered by collisions with deuterons per unit volume and per unit time at time t, after P and before P are total momentum of those electrons just after the collisions and just before the collisions, respectively. 1) About after P in (3) The number of electrons scattered by deuterons is e r n ν per unit volume and per unit time. For a velocity distribution of those e r n ν electrons just after the collisions, we assume again such a spherical surface as shown in Figure 1 of Ref. [11]: The quantity in the above bracket is the solid angle element in the direction of ( ) , θ φ , υ is the magnitude of a velocity variable υ of an electron, Here, ˆˆŝin cos sin sin cos Using the following approximation:  The term with products or squares of the drift velocity and the electric field have been neglected in (6) and (7). We will do so also in later calculations.
2) About before P in (3) The number of electrons scattered with velocity magnitudes ~d υ υ υ + in the direction of ( ) , θ φ during the time interval 0 0 per unit volume. Of these electrons, the number of electrons having not collided until time t is at time t after having been scattered with is given after by (9).
These electrons have, at time t, momentum and probability by which these electrons are scattered per unit time at time t is Accordingly, before P is given by We substitute the following three relationships into (8).
c) An Equation (11) below, for a drift velocity of electrons, Furthermore, in order to calculate the second term in the right-hand side (RHS) of (12), we express ( ) 0 t u by the following form: And we use the following relationships where, Next, the 3rd term in RHS of (12) becomes ( ) ( ) e r 1 2  . This can be generalized as: The 3rd term in RHS of (12) Accordingly, the momentum transfer term in the field of the two-body (electron-deuteron) collisions is given by where,

5) About
( ) The above summation is a momentum which e n electrons gain during unit time through the external fields and the collisions. Here, we assume roughly that a velocity distribution of e n electrons at time t is isotropic when it is viewed from the velocity point u(t), similarly in Figure 1 of Ref. [11]. Then, based on the analysis from (4) to (7), we have, as a momentum summation of Thus, we obtain the following momentum transport equation for relativistic electrons: Here, with

ν r , ς r and Drift Velocities
We regard, similarly in Equations (21)-(25) of Ref. [11], that a momentum transfer frequency of relativistic electrons scattered anisotropically in the two-body collisions through the Coulomb force is sin 2 where, x is a deflection angle of an electron with υ and 0 ε is the dielectric

For Efficiency Increase of a Thermionic Energy Converter
It is presumed that a gas (Cs plasma) within general converters will be a weakly ionized plasma. If the gas is replaced with a fully ionized plasma instead of a weakly ionized one, an internal resistance between an emitter plate and a collector plate extremely decreases, and thermionic electrons can save their thermal energies which have been consumed for ionization of Cs atoms. We consider that this replacement of the internal medium will raise a generation-efficiency. Furthermore, if a force field to convey thermionic electrons from the emitter to the collector is given within the converter, the efficiency will rise more compared with the case where electrons cannot but go to the collector for themselves. Under such a consideration, we propose a means adding some equipments, shown Journal of Modern Physics  in Figure 3 and Figure 4, to the converter. The equipments are connected with Converter by Solenoid. In the inner space, Cs gas is enclosed. Discharge tube (shown in Figure 4) is installed as a partner of Converter. Fan makes Cs gas plasma circulate slowly within the closed space. By making a right-circularly polarized wave continue to heat electrons for long time, it is planned that the most part of the closed space is filled with a fully ionized plasma. Even if the electron temperature is not so high, we consider that it is possible to obtain an almost perfectly ionized plasma because the work-function of a Cs atom is very small. Now, let us classify the internal space of Converter into three parts (called space 1, 2, 3), as shown with dotted lines in Figure 3. We assume roughly that, in space 2, an electric field due to a space charge is negligible and also that, only in space 2, a magnetic field B and an external electric field E exist. The magnetic Journal of Modern Physics  field B is supplied by Solenoid and the electric field E is the one in the middle part of the discharge plasma. These external forces convey electrons with the drift velocity z u (in Figure 1) in the direction of −E and with the drift velocity x u (in Figure 2) in the direction of E × B. We set the value of c ω ν to a larger one than 4.0. Then, a loss in an external circuit between Anode and Cathode is sufficiently suppressed and many electrons will try entering from space 2 into space 3. In this situation, however, the following physical condition must be satisfied: "A total number of electrons which can enter within Collector per unit time is equal to a total number (denoted by tot N ) of electrons which jump out of Emitter per unit time." Electrons also combining with ions on Emitter must be counted for tot N (we suppose that neutral atoms produced near the surface of Emitter are soon ionized within space 1). Accordingly, if we design the converter so that electrons flowing into space 3 per unit time may become much more than tot N , a negative potential barrier to suppress the flow of electrons ought to be produced near Collector surface, which is added to the old barrier. The larger the height of a total negative potential barrier becomes, the larger an output voltage becomes, because a potential of Collector lowers more and more as against a potential of Emitter. The convey of electrons by the force E × B makes it possible to lengthen the distance between Emitter and Collector. When the distance c  from the entrance to the exit in the converter of Figure 3 is too long, a distribution of B becomes vague. If it is necessary to lengthen the value of c  , then, we must connect some small-sized converters in series by solenoids.

Conclusion
In the field of the Coulomb force scattering, under the premise that the twobody collisions have much more influence than the many-body collisions on the drift movement of an electron, we have inquired into the transport equation of momentum for relativistic electrons. Also, proposing an idea of introducing a fully ionized plasma and an external magnetic field within the combination-apparatus of the converter and the discharge tube, we have discussed about a means to raise a generation-efficiency of a thermionic energy converter.