2 m ( 1 υ 0 z 2 c ) ] 2 + [ q E t m υ 0 z 2 c ω c t 4 ] 2 } 1 / 2 , ( ω c t 1 ) . (9)

Here, C 0 = C ( t 0 = 0 ) , S 0 = S ( t 0 = 0 ) , and υ x ( t , t 0 ) ( t 0 = 0 ) 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 q E ( t ) υ ( t , t 0 ) / c per unit time. Also, it must be noted that there is some difference between υ r of (9) and an amplitude of a curve of z ^ υ ( t , t 0 ) 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 z ^ υ ( t , t 0 ) .

3. 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 [2] [3] [4] [5] . 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 E 1 and E 2 are supplied within spaces (a) - (b) and (b) - (c), respectively. For an incident angle when a charged particle crosses plane (j), we denote an angle from +y-axis in the y-z plane by θ j ( 90 < θ j < 90 ) and a mere inclination from +y-axis by | θ | j ( 0 | θ | j < 90 ) . A plasma temperature for fusion reaction to continue is assumed to be 4 × 108 K. Then, each mean thermal velocity for electrons and D+ ions is about 1.2 × 108 m/s (= υ ¯ ) and 2 × 106 m/s (= υ ¯ i ), 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 υ ¯ i . Also, we disregard interactions between charged particles 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 ω c or ω i , respectively.

3.1. Reflection of Electrons

First we show in Figure 2 the z-component of velocity, υ z ( t , t 0 ) given in (6), in the time range of ( t 0 t t 0 + 5.9 × 10 8 sec ) for an electron which starts from plane (b) with an initial velocity z ^ υ ¯ sin θ b + y ^ υ ¯ cos θ b ( υ ¯ = 1.2 × 10 8 m / s , θ b = + 5 ) at time t 0 . The numerical conditions are shown in the figure. Curves (a), (b), (c), (d) are for four cases of t 0 . Here, we note the following thing. In the short time range of t 0 ~ 0 (note that t 0 0 ), both the velocity-magnitude and the gyration frequency of the electron hardly vary compared with each initial value at t = t 0 . Therefore, since ω c 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 e z , e x , e z , e x ), starting from plane (b) with initial velocities shown below at time t = 0 (≠ t0, except for curve (a)):

1) z ^ υ ¯ sin θ b + y ^ υ ¯ cos θ b ( θ b > 0 ) for e z ,

2) x ^ υ ¯ sin | θ | b + y ^ υ ¯ cos | θ | b ( | θ | b = | θ b | ) for e x ,

3) z ^ υ ¯ sin θ b + y ^ υ ¯ cos θ b ( θ b < 0 ) for e z , ( θ b = ± 5 in Figure 2)

4) x ^ υ ¯ sin | θ | b + y ^ υ ¯ cos | θ | b ( | θ | b = | θ b | ) for e x ,

(Note that the above four velocities are initial velocities at t = 0 all).

In the characteristics of curve (a) for e z and curve (c) for e z , 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 r ( c ) ( υ r ( c ) 2 + υ ¯ 2 cos 2 θ c ) 1 / 2 > ( B a 4 × 4 B a ) 1 / 2 or υ r ( c ) > 0.26 υ ¯ cos θ c 0.26 υ ¯ cos θ b (10)

(The loss angle of mirror (c) - (d) is 14.5˚).

Here, 4 × 4 B a is the magnetic field in plane (d) and υ r ( c ) is a minimum in velocity-magnitudes perpendicular to B of the four electrons when those cross plane (c). Accordingly, the electric field E 2 ( = z ^ 10 3 cos ω t , ω c / ω = 1.0001 ) in Figure 1 must increase the value of υ r ( c ) from υ ¯ | sin θ b | ( = υ ¯ | sin ± 5 | = 0.105 × 10 8 m / s ) at time t = 0 to 0.26 υ ¯ cos θ b (= 0.31 × 108 m/s). We regard the amplitudes of υ z ( t , t 0 ) in curves (a), (b), (c), (d) to show time-variations of velocity-magnitudes perpendicular to B of e z , e x , e z , e x , respectively. Then, the velocity-magnitudes in curves (a), (b), (c), (d) become larger than 0.26 cos θ b = 0.31 × 10 8 m / s all when t t 0 + 5.9 × 10 8 sec (A necessary 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 | θ | b = 5 ~ 90 will return to the main bottle.

Here, we would note the following thing. We tried obtaining from (9) values corresponding to the maxima of υ z ( t , t 0 ) near t = t 0 + 5.9 × 10 8 5.9 × 10 8 sec (because of π / ω c 10 11 sec ) which are seen in curves (a) and (c) in Figure 2 which is drawn based on (6).

The maxima of υ z ( t , t 0 ) { 0.51 × 10 8 m / s incurve ( a ) , 0.49 × 10 8 m / s in curve ( c ) .

Substituting into (9)

{ c = ( 3 × 10 8 m / s ) 2 , E = 10 3 V / m , B = 2 T , υ ¯ = 1.2 × 10 8 m / s , υ 0 z = υ ¯ sin θ b = { 0.105 × 10 8 m / s θ b = + 5 in curves ( a ) , 0.105 × 10 8 m / s θ b = 5 in curves ( c ) , υ 0 y = υ ¯ cos θ b = 1.192 × 10 8 m / s , q / m = 1.76 × 10 11 × ( 1 ( υ 0 z 2 + υ 0 y 2 ) / c ) 1 / 2 = 1.61 × 10 11 C / kg , ω c = q B / m = 3.22 × 10 11 sec 1 , t = 5.9 × 10 8 sec ,

we obtain

υ r = { 0.57 × 10 8 m / s for curve ( a ) , 0.55 × 10 8 m / s for curve ( c ) .

There are some differences between the maxima in curves (a), (c) and the above values of υ r . Then, considering that the differences have come from some approximation, 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:

1 2 q E m ω c ( 1 υ 0 z 2 c ) S 0 0 , 1 4 q E m υ 0 z 2 c ( 3 S 0 2 ω c ) C 0 0 , } ( ω c t ) .

However, the above approximations are right when ω c t 2 × 10 4 . So, we consider that the cause of “ υ r > the maximum of υ z ( t , t 0 ) ” is in that, of ( υ z r 2 + υ x r 2 ) 1 / 2 in (9), υ x r is not zero when the magnitude of υ z r becomes maximum.

3.2. Reflection of D+ Ions

We aim a D+ ion which starts from plane (a) in Figure 1 with an initial velocity z ^ υ i o z + y ^ υ i o y = z ^ υ ¯ i sin θ a + y ^ υ ¯ i cos θ a at time t 0 and goes to plane (b). An expression (called Equation (11)) for a D+ ion corresponding to (6) is obtained by the following change for the symbols of an electron:

{ q q , E E i , υ 0 z υ i 0 z ( = υ ¯ i sin θ a ) , m e m i ( = 3680 m e ) , m m i i ( = m i / ( 1 ( υ i 0 z 2 + υ i 0 y 2 ) / c ) 1 / 2 ) , ω c ω i ( = q B / m i i ) , υ z ( t , t 0 ) υ i z ( t , t 0 ) . (11)

We show in Figure 3 dependences of υ i z ( t , t 0 ) on four cases of time t 0 , for a D+ ion which starts from plane (a) with an initial velocity z ^ υ ¯ i sin θ a + y ^ υ ¯ i cos θ a ( θ a = + 3 ) at time t 0 . 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 D z + , D x + , D z + , D x + ) starting from plane (a) with initial velocities shown below at time t = 0 (≠ t0, except for curve (a)):

1) z ^ υ ¯ i sin θ a + y ^ υ ¯ i cos θ a ( θ a > 0 ) for D z + ,

2) x ^ υ ¯ i sin | θ | a + y ^ υ ¯ i cos | θ | a ( | θ | a = | θ a | ) for D x + , ( θ a = ± 3 in Figure 3),

3) z ^ υ ¯ i sin θ a + y ^ υ ¯ i cos θ a ( θ a < 0 ) for D z + ,

4) x ^ υ ¯ i sin | θ | a + y ^ υ ¯ i cos | θ | a ( | θ | a = | θ a | ) for D x + ,

(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 ( υ r υ i r ) for a D+ ion corresponding to (9):

υ i r { [ υ i 0 z + q E i t 2 m i i ( 1 υ i 0 z 2 c ) ] 2 + [ q E i t m i i υ i 0 z 2 c ω i t 4 ] 2 } 1 / 2 , ( ω i t 1 ) (12)

Since υ ¯ i 2 / c = 4.4 × 10 5 , (12) is approximated as

υ i r | υ ¯ i sin θ a + q E i t 2 m i | , ( ω i t 1 ) (13)

Since θ a = 3 in curve (c) for D z + in Figure 3, (13) becomes zero when | υ ¯ i sin ( 3 ) | = q E i t / ( 2 m i ) .

Now, we consider again making four D+ ions ( D z + , D x + , D z + , D x + ) reflect by mirror (c) - (d) all. Denoting by υ i r ( b ) a minimum in velocity-magnitudes perpendicular to B of the four D+ ions when those cross plane (b), υ i r ( b ) must satisfy

Figure 3. Dependence of υ i z ( t , t 0 ) on t0, based on Equation (11). The numerical conditions are: An initial velocity z ^ υ i 0 z = z ^ υ ¯ i sin 3 = z ^ 0.105 × 10 6 m / s in plane (a) at time t0, E 1 = + z ^ 10 3 cos ω t V / m , B = 2 T , ω i / ω = 1.0001 , the time variable t = t 0 ~ t 0 + l / ( υ ¯ i cos 3 ) = t 0 ~ t 0 + 52 / ( 2 × 10 6 × cos 3 ) = t 0 ~ t 0 + 2.6 × 10 5 sec .

υ i r ( b ) > 0.26 υ ¯ i cos θ b 0.26 υ ¯ i cos θ a υ r e f l e c t . (14)

Accordingly, the electric field E 1 ( = + z ^ 10 3 cos ω t , ω i / ω = 1.0001 ) in Figure 1 must increase the value of υ i r ( b ) from

υ ¯ i | sin θ a | ( = υ ¯ i | sin ( ± 3 ) | = 0.105 × 10 6 m / s ) at time t = 0 to υ r e f l e c t of (14)

( = 0.26 υ ¯ i cos ( ± 3 ) = 0.52 × 10 6 m / s ). From the characteristics in Figure 3, the condition mentioned above is satisfied when an acceleration time is about 2.6 × 10−5 sec (A necessary length ℓ between planes (a) and (b) is about 52 m) and mirror (c) - (d) can reflect the four D+ ions ( D z + , D x + , D z + , D x + ). However, when we had examined about reflection-characteristics of a D+ ion (in the range of 90 < θ a 0 in the y-z plane) starting from plane (a) at t = 0, we met with a peculiar dependence on θ a . We missed this factor in Ref. [1] . To mention this thing with respect to a D+ ion, we show in Figure 4(a), Figure 4(b) dependences (on θ a ) of the following four quantities in plane (b):

Figure 4. (a) (b) Dependence of four kinds of velocities perpendicular to the magnetic field with respect to reflection of a D+ ion on θ a [ 0 θ a 80 ] . (a) is for ℓ = 52 m and (b) is for ℓ = 110 m, where ℓ is the length between planes (a) and (b) as shown in Figure 1. (1) | υ ¯ i sin θ a | , (2) υ i r of Equation (12), (3) υ i r of Equation (13), (4) υ r e f l e c t of Equation (14).

1) | υ ¯ i sin θ a | , which is the velocity-magnitude perpendicular to B of the initial velocity in plane (a) at t = 0. This value is constant when E 1 = 0 .

2) υ i r of Equation (12), which is the velocity-magnitude perpendicular to B in plane (b) when ω i / ω = 1.0001.

3) υ i r of Equation (13) (= Equation (12) with c ).

4) υ r e f l e c t of Equation (14). This is the minimum of υ i r 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 θ a .

In Figure 4(a), Figure 4(b), when E 1 = 0 , from curves (1) and (4), a D+ ion in the range of 0 θ a > 14.5 are not reflected but those in the range of 14.5 θ a 80 are reflected. On the other hand, when E 1 0 , from curves (2) and (4) in Figure 4(a), a D+ ion in the range of 0 θ a 3 are reflected, but a D+ ion in the range of 3 θ a 36 is not reflected, that is, the acceleration by E 1 has brought about such a disadvantage for reflection of a D+ ion. We missed previously this factor. About relativistic electrons, such a problem does not arise, because the second term increasing together with υ o z 2 within the root of (9) is much more predominant than the first term within the root in magnitude. In Figure 4(a), Figure 4(b), from comparison between curves (2) and (3), relativistic effects are seen from around θ a 50 . In the case where ℓ =110 m (Figure 4(b)), it is seen that a D+ ion starting from plane (a) in the range of 0 θ a 80 are reflected all. Under the assumption that every D+ ion has the velocity-magnitude of υ ¯ i , 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. A schematic diagram of an apparatus for replenishing a large quantity of D+ ions and electrons. Static electric fields ± E S are ones for decreasing the number of charged particles colliding with the metal plate by the forces of ± E S × B . Escaping D+ ions must be sent to a plasma source.

4. 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.

Cite this paper
Nagata, M. and Sawada, K. (2019) Further Improvement of Reflection Efficiency of a Magnetic Mirror and Replenishment against Loss of Escaping Deuteron Ions. Journal of Modern Physics, 10, 145-156. doi: 10.4236/jmp.2019.102011.
References
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[3]   Caspi, R. and Jerby, E. (1999) Physical Review E, 60, 2.

[4]   Savrukhin, P.V. and Shestakov, E.A. (2016) Physics of Plasmas, 23, 112509-1.

[5]   Soldatkina, E., Anikeev, M., Bagryansky, P., Korzhavina, M., Maximov, V., Savkin, V., Yakovlev, D., Yushmanov, P. and Dunaevsky, A. (2017) Physics of Plasmas, 24, 022505-1.

 
 
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