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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).

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 [

We first describe theoretical expressions to be necessary for after analysis. The relativistic equation of motion

∂ ∂ t m e υ ( t , t 0 ) [ 1 − υ ( t , t 0 ) 2 / c 2 ] 1 / 2 = m e ∂ υ ( t , t 0 ) / ∂ t [ 1 − υ ( t , t 0 ) 2 / c 2 ] 1 / 2 + [ m e υ ( t , t 0 ) / c 2 ] υ ( t , t 0 ) ∂ υ ( t , t 0 ) / ∂ t [ 1 − υ ( t , t 0 ) 2 / c 2 ] 3 / 2 = − q E ( t ) − q υ ( t , t 0 ) × B

is changed into (1) below by the help of the following two relationships:

1) The time derivative of energy with respect to a relativistic electron

∂ ∂ t m e c 2 [ 1 − υ ( t , t 0 ) 2 / c 2 ] 1 / 2 = m e υ ( t , t 0 ) ∂ υ ( t , t 0 ) / ∂ t [ 1 − υ ( t , t 0 ) 2 / c 2 ] 3 / 2 = − q E ( t ) ⋅ υ ( t , t 0 )

2) The integration of the above equation from t 0 to t

m e c 2 [ 1 − υ ( t , t 0 ) 2 / c 2 ] 1 / 2 − m e c 2 [ 1 − υ 0 2 / c 2 ] 1 / 2 = ∫ t 0 t − q E ( t ) ⋅ υ ( t , t 0 ) d t

{ m e [ 1 − υ 0 2 / c 2 ] 1 / 2 + ∫ t 0 t − q E ( t ) ⋅ υ ( t , t 0 ) c 2 d t } ∂ υ ( t , t 0 ) ∂ t + υ ( t , t 0 ) ( − q E ( t ) ⋅ υ ( t , t 0 ) c 2 ) = − q E ( t ) − q υ ( t , t 0 ) × B (1)

In the above four equations, m e is the rest mass of an electron, −q is the electron charge, c is the speed of light, E(t) is an electric field, B is a magnetic field, υ ( t , t 0 ) (= x ^ υ x ( t , t 0 ) + y ^ υ y ( t , t 0 ) + z ^ υ z ( t , t 0 ) ) is the velocity of an electron at time t after start with an initial velocity υ 0 (= x ^ υ 0 x + y ^ υ 0 y + z ^ υ 0 z ) at time t 0 , and υ ( t , t 0 ) = | υ ( t , t 0 ) | .

Since a solution of (1), υ ( t , t 0 ) , is to be given by a form of

υ ( t , t 0 ) = υ t + o { υ t = υ ( t , t 0 ) ( E ( t ) = 0 ) , o isaquantitywhichconsistsofcomponentsof E ( t ) ,

we neglect terms including products and squares with respect to components of E ( t ) in (1) and linearize (1) as

m e ( 1 − υ 0 2 c 2 ) 1 2 ∂ υ ( t , t 0 ) ∂ t + ( ∫ t 0 t − q E ( t ) ⋅ υ t c 2 d t ) ∂ υ t ∂ t + υ t ( − q E ( t ) ⋅ υ t c 2 ) = − q E ( t ) − q υ ( t , t 0 ) × B (2)

First, we note the following problem which arises due to the linearization. It is that, in

{ m e [ 1 − υ 0 2 / c 2 ] 1 / 2 + ∫ t 0 t − q E ( t ) ⋅ υ ( t , t 0 ) c 2 d t } at time t . A more rapid increase of the

total mass ought to reduce variation of | υ ( t , t 0 ) | to zero before | υ ( t , t 0 ) | arrives at the speed c of light.

We solve (2) for υ z ( t , t 0 ) and υ y ( t , t 0 ) under the following external force fields:

{ E ( t ) = − z ^ E cos ω t , B = y ^ B , (3)

and under the following initial conditions at t = t 0 which are given by

{ υ ( t , t 0 ) ( t = t 0 ) = z ^ υ 0 z + y ^ υ 0 y , ( ∂ υ z ( t , t 0 ) ∂ t ) ( t = t 0 ) = q E m cos ω t 0 − q E m c 2 υ 0 z 2 cos ω t 0 , ( where m = m e ( 1 − υ 0 2 / c 2 ) − 1 / 2 ) . (4)

The results are given in (5) and (6) below,

υ y ( t , t 0 ) = υ 0 y + υ 0 y υ 0 z q E m c ″ { sin [ ( ω c + ω ) ( t − t 0 ) ] 2 ( ω c + ω ) + sin [ ( ω c − ω ) ( t − t 0 ) ] 2 ( ω c − ω ) } . (5)

Here, c ″ = c 2 (The speed c of light appears always in the form of square in after analysis. Then, we have introduced the symbol c ″ for c 2 , because we would like to use c as the symbol for cos ω t ), and ω c = q B / m . Since the second term of (5) is extremely small compared with the first term υ 0 y in after analysis under the conditions of ω ≃ ω c and ω c ( t − t 0 ) ≫ 1 , υ y ( t , t 0 ) is regarded to be constant υ 0 y in this work.

υ z ( t , t 0 ) = a 1 s + ( a 2 s ) S 2 + ( a 3 s ) C 2 + ( a 4 c ) S C + a 5 t S + C 1 C + C 2 S , (6)

(We have used υ t ( = x ^ υ 0 z sin ω c ( t − t 0 ) + y ^ υ 0 y + z ^ υ 0 z cos ω c ( t − t 0 ) ) ).

Here,

C = cos ω c ( t − t 0 ) , S = sin ω c ( t − t 0 ) , c = cos ω t , s = sin ω t ,

a 1 = q E m − ω ω c 2 − ω 2 ,

a 2 = q E m c ″ 1 ( ω c 2 − ω 2 ) 2 [ − 2 ω c 2 ω υ 0 z 2 ] ,

a 3 = q E m c ″ 1 ( ω c 2 − ω 2 ) 2 [ ( ω c 2 ω − ω 3 ) υ 0 z 2 ] ,

a 4 = q E m c ″ 2 ω c 3 ( ω c 2 − ω 2 ) 2 ( − υ 0 z 2 ) ,

a 5 = q E m c ″ ω c ω c 2 − ω 2 ( υ 0 z 2 ω sin ω t 0 ) ,

C 1 = υ 0 z − a 1 sin ω t 0 − a 3 sin ω t 0 ,

C 2 = q E m ω c cos ω t 0 − q E m c ″ 1 ω c υ 0 z 2 cos ω t 0 − a 1 ω ω c cos ω t 0 − a 3 ω ω c cos ω t 0 − a 4 cos ω t 0 − a 5 t 0 .

Next let us obtain velocities perpendicular to the magnetic field at the resonance point ω → ω c , in the simple case where t 0 = 0 :

υ z r ≡ lim ω → ω c υ z ( t , t 0 ) ( t 0 = 0 ) = lim ω → ω c ( 6 ) ( t 0 = 0 ) = υ 0 z C 0 + q E m ( 1 − υ 0 z 2 c ″ ) 1 ω c S 0 + q E m ( 1 − 2 ω c ( S 0 − ω c t C 0 ) ) + q E m υ 0 z 2 c ″ ( 1 8 ω c 2 ( 2 ω c 3 t 2 S 0 − 4 ω c 2 t C 0 + 4 ω c S 0 + 6 ω c S 0 3 ) ) , (7)

υ x r ≡ lim ω → ω c υ x ( t , t 0 ) ( t 0 = 0 ) = υ 0 z S 0 + 1 2 q E m t S 0 + q E m υ 0 z 2 c ″ ( 1 8 ω c 2 ( − 4 ω c 2 t S 0 − 2 ω c 3 t 2 C 0 − 6 ω c S 0 2 C 0 ) ) , (8)

υ r ≡ ( υ z r 2 + υ x r 2 ) 1 2 ≃ { [ υ 0 z + q E t 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. [

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 ^{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 (= 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 ^{+} ion is regarded to actually interact only with the electric field having ω c or ω i , respectively.

First we show in _{0}, 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

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 ^{8} 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

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 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.

We aim a D^{+} ion which starts from plane (a) in ^{+} 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 ^{+} 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 ^{+} ions (called D z + , D x + , D − z + , D − x + ) starting from plane (a) with initial velocities shown below at time t = 0 (≠ t_{0}, 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

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

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

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

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 ^{−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. [^{+} ion, we show in

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

In ^{+} 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 ^{+} 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 ^{+} 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.

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 ^{+} ions and electrons from the outside ought to be introduced. We show with ^{+} ions.

The authors declare no conflicts of interest regarding the publication of 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. https://doi.org/10.4236/jmp.2019.102011