Further Improvement of Reflection Efficiency of a Magnetic Mirror and Replenishment against Loss of Escaping Deuteron Ions

We reported previously the idea to improve reflection-ability of a magnetic mirror by installing a cyclotron resonance space in the front part of the mirror. However, since the previous analysis was insufficient from the examination after that, we complement the following two things in this work: 1) A simpler procedure of design to make a supplemental magnetic mirror with the simplest magnetic configuration, compared with the procedure reported previously. 2) A peculiar characteristic arising only in reflection of a nonrelativistic charged particle (a deuteron ion).


Introduction
We found the term seen in the rear of (9) to accelerate only a relativistic charged particle rapidly at a cyclotron resonance point and reported previously [1] applying a work of the term to decreasing the half-vertical angle (called the loss angle) of the loss cone of a magnetic mirror by installing a cyclotron resonance space within.However, based on the examination after that, we would complement two things here.One is about the simplest supplemental magnetic mirror.
The procedure of design is mentioned in §3.1 together with Figure 1.Another thing is that we missed in Ref. [1] a factor which must be taken into consideration with respect to reflection of a nonrelativistic particle (a deuteron ion).The factor is that an electric field being installed in the front part of a magnetic mir-

Relativistic Cyclotron Resonance
We first describe theoretical expressions to be necessary for after analysis.The relativistic equation of motion  (1) below by the help of the following two relationships: 1) The time derivative of energy with respect to a relativistic electron ( ) , e e m t t t t t m c t t t c t t c q t t t 2) The integration of the above equation from 0 t to t ( ) , In the above four equations, e m is the rest mass of an electron, −q is the elec- tron charge, c is the speed of light, E(t) is an electric field, B is a magnetic field, ( ) .
Since a solution of (1), ( ) , is to be given by a form of is a quantity which consists of components of , we neglect terms including products and squares with respect to components of ( ) First, we note the following problem which arises due to the linearization.It is that, in Figure 2 Accordingly, we consider that the primary cause giving rise to this problem is underestimation in the linearization for the total mass ( ) ( ) and under the following initial conditions at 0 t t = which are given by ( ( ) .
The results are given in ( 5) and (6) below, ) Here, (The speed c of light appears always in the form of square in after analysis.Then, we have introduced the symbol c′′ for 2 c , because we would like to use c as the symbol for cos t ω ), and c qB m ω = . Since the second term of ( 5) is extremely small compared with the first term 0 y υ in after analysis un- der the conditions of c ω ω  and ( ) Here, Here, ( ) is given in Equation (15) of Ref. [1].It should be noted that the second term within the root in (9) works greatly only for a relativistic charged particle but the relativistic work of the second term will reduce more and more than the estimation in (9) as time t passes, because of the underestimation in (2) for the mass increment ( ) ( ) 0 , q t t t c′′ − ⋅ E υ per unit time.Also, it must be noted that there is some difference between r υ of (9) and an amplitude of a curve of ( ) 0 ˆ, z t t ⋅υ seen in Figure 2 (discussed in §3.1).
However, we try designing a magnetic mirror for electrons based on a time-variation of an amplitude of ( ) 0 ˆ, z t t ⋅υ .

Design of a Supplemental Magnetic Mirror
In order to return back deuteron ions (called D + ions) and electrons escaping from a main bottle as many as possible, we consider installing (at the exit of the main bottle) a supplemental magnetic mirror which has a cyclotron resonance space within, as shown in Figure 1.We intend to reduce a loss angle of the supplemental mirror by increasing magnitudes of velocities perpendicular to a magnetic field B of the escaping particles within the cyclotron resonance space . The supplemental mirror is divided to three spaces by plane (j) (j = a, b, c, d).We define the x, y, z coordinate-system, as shown in plane (a).Also, we assume the y-coordinate and a strength B of a magnetic field B in each plane as shown in the figure.The magnetic field is regarded to be only in the +y-direction.Electric fields 1 E and 2 E are supplied within spaces (a) -(b) and (b) -(c), re- spectively.For an incident angle when a charged particle crosses plane (j), we denote an angle from +y-axis in the y-z plane by j θ ( ) and a mere inclination from +y-axis by j θ ( ) A plasma temperature for fu- sion reaction to continue is assumed to be 4 × 10 8 K.Then, each mean thermal velocity for electrons and D + ions is about 1.2 × 10 8 m/s (= υ ) and 2 × 10 6 m/s (= i υ ), respectively.To simplify after discussion, we assume that every electron and every D + ion are flying about within the main bottle, with each mean thermal velocity υ and i υ .Also, we disregard interactions between charged par- ticles through Coulomb force within the supplemental mirror.Also in Figure 1, an electron or a D + ion is regarded to actually interact only with the electric field having c ω or i ω , respectively.

Reflection of Electrons
First we show in Figure 2 the z-component of velocity, given in (6), in the time range of ( t .Here, we note the following thing.In the short time range of 0 ~0 t (note that 0 0 t ≤ ), both the velocity-magnitude and the gyration frequency of the electron hardly vary compared with each initial value at 0 t t = .Therefore, since c ω ω  , curves (a), (b), (c), (d) can be regarded to show time-variations of the z-components of velocities perpendicular to B for four electrons (called , , , In the characteristics of curve (a) for z e and curve (c) for z e − , the difference is hardly seen, which is due to that the second term within the root in Equation ( 9) is much more predominant than the first term within the root in magnitude.We consider making these four electrons reflect by mirror (c) -(d) all.For this purpose, the four electrons must satisfy the following reflection condition: υ is a minimum in velocity-magnitudes perpendicular to B of the four electrons when those cross plane (c).Accordingly, the electric field ( ) in Figure 1 must increase the value of ( ) 5.9 10 sec, There are some differences between the maxima in curves (a), (c) and the above values of r υ .Then, considering that the differences have come from some ap- proximation, we re-examined the calculation process from (15) of Ref. [1] and (6) to ( 7) and (8).However, there were no mathematical approximations.On the other hand, in the process from ( 7) and ( 8) to (9), there were two approximations below: ( ) However, the above approximations are right when . So, we consider that the cause of " r υ > the maximum of ( ) " is in that, of ( ) υ is not zero when the magnitude of zr υ becomes maximum.

Reflection of D + Ions
We aim a D + ion which starts from plane (a) in Figure 1 We show in Figure 3  ŝin cos (Note that the above four velocities are initial velocities at t = 0 all).
The amplitude in curve (c) changes from decrease into increase halfway.This variation can be explained based on an expression ( r ir υ υ → ) for a D + ion cor- responding to (9): ( )

Conclusion
We have made it clear that, in order to reclaim escaping D + ions whose incident directions make small angles for a direction of B by a magnetic mirror of a linear type, a very long cyclotron resonance space is necessary.Though the loss angle, about 5˚, of the mirror (a) -(d) designed in Section 3 is presumed to be still too large from the viewpoint of plasma confinement, the mirror (a) -(d) shown in Figure 1 is a sufficiently too long apparatus.Therefore, we consider that, for shortening a length of acceleration space, a powerful means replenishing a large quantity of D + ions and electrons from the outside ought to be introduced.We show with Figure 5 an idea about a means for replenishment.This apparatus must be protected from heating and damage of the metal surface due to collisions of escaping D + ions.

Figure 1 .
Figure 1.A supplemental magnetic mirror (a)-(d) for reclaiming charged particles to escape from the exit (plane (a)) of the main bottle.The half-vertical angle of the loss cone of mirror (c)-(d) is 14.5˚.It is tried to decrease the loss angle of mirror (a)-(d) from 14.5˚ to about 5˚ with the help of the electric fields E 1 and E 2 .E 1 is the electric field to accelerate deuteron ions, E 2 is the electric field to accelerate electrons, i ω is an ion cyclotron frequency and cω is an electron cyclotron frequency.

Figure 2
Figure 2. Dependence of ( ) 0 , z t t υ on t 0 , based on Equation (6).The numerical conditions are: an initial velocity 0 Next let us obtain velocities perpendicular to the magnetic field at the resonance point c ω ω → , in the simple case where 0 0 t = : ) at time 0 t .The numerical conditions are shown in the figure.Curves (a), (b), (c), (d) are for four cases of 0 starting from plane (b) with initial velocities shown below at time t = 0 (≠ t 0 , except for curve the above four velocities are initial velocities at t = 0 all).

υ
in curves (a), (b), (c), (d) to show time-variations of velocity-magnitudes perpendicular to B of , , , .Then, the velocity-magnitudes in curves (a), (b), (c), (d) become larger than length between planes (b) and (c) is about 7m).If the four electrons are reflected by mirror (c) -(d), we estimate that the most of electrons with 5 ~90 b θ =  will return to the main bottle.Here, we would note the following thing.We tried obtaining from (9) values corresponding to the maxima of seen in curves (a) and (c) in Figure2which is drawn based on (6).
10 m s for curve a , 0.55 10 m s for curve c . r t and goes to plane (b).An ex- pression (called Equation (11)) for a D + ion corresponding to (6) is obtained by the following change for the symbols of an electron:(

υ
on four cases of time 0 t , for a D + ion which starts from plane (a) with an initial velocity at time 0 t .The numerical conditions are shown in the figure.Based on the same consideration with in §3.1, the amplitudes in curves (a), (b), (c), (d) in Figure 3 are regarded to show time-variations of velocity-magnitudes perpendicular to B of four D + ions (called from plane (a) with initial velocities shown below at time t = 0 (≠ t 0 , except for curve (a)): 1)
4) reflect υ of Equation (14).This is the minimum of ir υ in plane (b) which is required for a D + ion to be reflected by mirror (c) -(d).It was a peculiar variation that, in curves (2) and (3) of Figure 4(a), minima had appeared at two points of a 80 a θ ≥ ≥ −   are reflected all.Under the assumption that every D + ion has the velocity-magnitude of i υ , the loss angle of mirror (a) -(d) becomes nearly zero in the case of ℓ = 110 m.But in the case of ℓ = 52 m, it is presumed that the loss angle of mirror (a) -(d) will be larger than 3˚, due to the minus factor mentioned above with respect to a D + ion escaping from plane (a) with a velocity-component in the direction of -z near the y-z plane.

Figure 5 .
Figure 5.A schematic diagram of an apparatus for replenishing a large quantity of D + ions and electrons.Static electric fields S ±E are ones for decreasing the number of charged particles colliding with the metal plate by the forces of