Classical charged particle resonance in induction field

Bhosale, Devesh S.

doi:10.1063/5.0206414

Starting from first principles, the space charge manipulation of charged particles in an induction field in free space based on an unique magnetic field strength and its oscillation frequency relationship is demonstrated numerically and theoretically. With the dispersion relation in ion resonance depending on its frequency of gyration, an alternating current driven electromagnet based particle resonance has been proposed circumventing the use of superconducting permanent magnets. Complete resonance achieved under the proposed conditions results in a sustained, fixed-frequency particle trajectory that is independent of its speed or drift. Such oscillation is visualized in a D-shaped resonant assembly. The amplitude and the wavelength calculations for the trajectory are demonstrated, and the applicability of this unique magnetic field strength and its sinusoidal varying frequency relation in plasma instability is explored.

I. INTRODUCTION

Resonance phenomena involving charged particles are fundamental across diverse scientific disciplines spanning particle accelerators, spectrometry, and resonant antennas. This has prompted a surge in research into the gyromotion of charged particles over the past decade, displayed by strong activity in gyrokinetics.¹ Among the many space charge manipulation that can be imagined, energizing charged ions/electrons and control of plasma profiles have been of great importance, with the development of ICRH and ECRH devices such as the JET ion cyclotron resonance frequency.²

Efforts to intensify highly charged ion beams have predominantly centered around strengthening superconducting magnets³ in recent years. In this study, we propose a new route for attaining resonance using induction field to produce time-varying magnetic field, instead of permanent magnets employed for such resonance. The peak magnetic field strength of this AC electromagnet is comparable to the static magnetic field strength of superconducting magnet, adding a new dimension of viability for the realization of particle resonance.

This paper demonstrates a novel gyromotion of particles using a time varying induction field with its gyrofrequency independent of the particle's velocity. When the natural gyrofrequency (represented by ω₀ in this article) of such charged particles gyrating in a magnetic field matches the frequency of an external electromagnetic wave, resonance occurs. Much attention has been paid to the study of the oscillations when the resonance condition is met, as RF power absorption from the electromagnetic wave can be efficiently transferred to heat the plasma.⁴ The dispersion relation for plasma in traditional systems has direct relationship to its gyrofrequency (ω_ci) during resonance $ω - k_{| |} v_{| |} = n ω_{c i}$ for proton cyclotron resonance⁵ and $k_{| |} v_{A} = ω_{c i}$ for ion cyclotron resonance,⁶ where ω_ci is the cyclotron frequency. As charged particles gain energy from the resonant interaction, they contribute to the excitation of collective plasma modes, leading to increased turbulence, wave–particle interactions, and ultimately, instabilities within the plasma. The production of highly charged ions is strictly governed by this inherent and induced plasma instabilities.⁷

II. BACKGROUND THEORY

The time-varying sinusoidal magnetic field

B_{z} (t) = B_{0} sin (ω_{0} t)

is axially symmetric and applied along the z-axis, while its angular frequency of oscillation being

ω_{0}

⁠. The induced electric field is calculated using Maxwell's equation and Stoke's theorem,

\oint_{P} \vec{E} \cdot d \vec{l} = - \int_{S} \dot{B} \cdot d \vec{S},

(1)

which gives us the required electric field as

\vec{E} = E_{θ} = - \frac{1}{2} r \dot{B_{z}} \hat{e_{θ}},

(2)

along the curl defined by surface S normal to z-axis; refer to Fig. 1. The particle's guiding center may at one point lie on this path P defined along the curl of surface S with radius r. Therefore, in this dynamic situation, the drift of guiding center, which being along the direction defined by vector

\vec{E_{θ}} \times \vec{B}

⁠, is equated to time rate change of r,⁸

\dot{r} = v_{d} = \frac{| \vec{E} \times \vec{B} |}{B^{2}} = \frac{E_{θ}}{B_{z}} = - \frac{r}{2} ω_{0} c o t (ω_{0} t) .

(3)

FIG. 1.

View large Download slide

Trajectory of electron with the magnetic field oscillating in and out of the plane.

Equating and , we obtain

\frac{2}{r} \dot{r} + \frac{\dot{B}}{B} = 0,

(4)

producing the known condition

B r^{2} = constant = G .

(5)

The velocity of the particle

\vec{v}

represented in the cylindrical coordinates is

\frac{d \vec{s}}{d t} = \vec{v} = \dot{r} {\hat{e}}_{r} + r \dot{θ} {\hat{e}}_{θ} + k \hat{z} .

(6)

The particle experiences Lorentz force for non-relativistic case due to the induced electric field E and magnetic field B expressed as

m \dot{\vec{v}} = q (\vec{E} + \vec{v} \times {\vec{B}}_{z}) .

(7)

Since the magnetic field is zero at the beginning, the radius corresponding to it is very large. The induced electric field follows the curl. On equating the radial components of Eq. , we obtain

\ddot{r} - r \dot{θ^{2}} = \frac{q}{m} B_{z} (t) \cdot r \dot{θ},

(8)

while the angular component yields

2 \dot{r} \dot{θ} + r \ddot{θ} = \frac{q}{m} [E_{θ} - \dot{r} B_{z}] .

(9)

Expanding the angular part by using Eq. in Eq. and applying Eq. to it, we immediately arrive at⁸

\frac{d (r^{2} \dot{θ})}{d t} = - \frac{q}{2 m} [\frac{d (B_{z} \cdot r^{2})}{d t}],

(10)

which upon integration produces⁸

\dot{θ} = - \frac{q}{2 m} B_{z} (t) + C / r^{2},

(11)

where C is a real constant. Or,

\dot{θ} = - \frac{q}{2 m} B_{z} (t) + D * B_{0} sin (ω_{0} t),

(12)

where D is the resulting constant by using Eq. in Eq. . Therefore, the integrated form of the above-mentioned equation gives us the angle swept by particle in the said time about the guiding center, thereby laying the groundwork for further derivations,

\int_{θ_{0}}^{θ} \dot{θ} = - \int_{0}^{t} \frac{q}{2 m} B_{0} sin (ω_{0} t) d t + D \int_{0}^{t} B_{0} sin (ω_{0} t) d t,

(13)

where

θ_{0}

is the value of

θ

at t = 0. Upon substituting

\dot{θ}

from Eq. in Eq. , we arrive at

\ddot{r} + r ({[\frac{q}{2 m} B_{z} (t)]}^{2} - {[D \cdot B_{z} (t)]}^{2}) = 0 .

(14)

Integrating , we obtain

θ - θ_{0} = △ θ = \frac{q B_{0}}{2 m ω_{0}} cos (ω_{0} t) - \frac{D B_{0}}{ω_{0}} cos (ω_{0} t) - \frac{q B_{0}}{2 m ω_{0}} + \frac{D B_{0}}{ω_{0}} + A,

(15)

where A is the integration constant. Taking the condition such that when the magnetic field completes its one half cycle, i.e., it completes its positive half cycle in

T_{0} / 2

⁠, where T₀ is the time period of angular frequency ω₀, correspondingly the particle should have traversed from

θ = π

-radians to

θ = 0

as can be seen from Fig. 2. Substituting

T_{0} / 2

for t and

- π

for

△ θ

(since

θ_{0} = π

⁠), in Eq. , we arrive at

- π = - \frac{q B_{0}}{m ω_{0}} + \frac{2 D B_{0}}{ω_{0}} + A .

(16)

FIG. 2.

View large Download slide

Depiction of the continuous beam of electrons entering the D-section containing the oscillating magnetic field. (a) The successive trajectories of electrons with all the particles synchronously oscillating angularly as shown. (b) A 3D model depicting the D-shaped assembly containing the said electromagnet, source, and resonant chamber. (c) The oscillating particles' sectional view is depicted, where the light-colored particles project the subsequent position of the particles.

Similarly, in

T_{0} / 4

time period, the particle should have traversed

- π / 2

radians. Hence, we obtain

- π / 2 = - \frac{q B_{0}}{2 m ω_{0}} + \frac{D B_{0}}{ω_{0}} + A .

(17)

Looking at the above-mentioned two equations, we can immediately see the value of A = 0. Substituting Eq. in Eq. , we arrive at the equation of angular displacement of particle about its guiding center for such conditions as

△ θ = θ - π = - \frac{π}{2} [1 - cos (ω_{0} t)],

(18)

\underline{θ = \frac{π}{2} [1 + cos (ω_{0} t)]} .

(19)

Using , we arrive at

D = \frac{q}{2 m} + \frac{π ω_{0}}{2 B_{0}} .

(20)

Making appropriate substitution in Eq. , we form

\ddot{r} = r {[B_{2} (t)]}^{2} (\frac{q}{m} + \frac{π ω_{0}}{2 B_{0}}) (\frac{π ω_{0}}{2 B_{0}}) .

(21)

Differentiating Eq. , we get an equation for the locus of r. Since the above-mentioned equation holds for all values of r, substituting

t = T_{0} / 4

⁠, we get

\ddot{r} = r \frac{ω_{0}^{2}}{2} = r B_{0}^{2} (\frac{q}{m} + \frac{π ω_{0}}{2 B_{0}}) (\frac{π ω_{0}}{2 B_{0}}) .

(22)

Thereby forming the necessary relationship between the peak magnetic field strength B₀ and the angular frequency ω₀ at which the particle is undulating,

\underline{\frac{π^{2} - 2}{2 π} = \frac{q B_{0}}{m ω_{0}}} .

(23)

As an example, the magnetic field strength for the frequency of 250 MHz comes out to be approximately 0.011 18 T taking electron as our particle. Therefore, our magnetic field function in this case turns out to be

B = 0.01118 * sin (2 π * 250 * 10^{6} t) .

Compared to the cyclotron frequency of 250 MHz, the magnetic field required for cyclotron resonance is $0.008 931 T$ ⁠. The equation above indicates that, for a given frequency (ω₀), the required magnetic field strength is roughly 1.25 times that of traditional cyclotron resonance $(\frac{π^{2} - 2}{2 π} \approx 1.25)$ ⁠. This corresponds to the peak magnetic field strength in the oscillating electromagnet (⁠ $0.011 18 T$ ⁠), in contrast to the static field of a permanent magnet for cyclotron resonance (⁠ $0.008 931 T$ ⁠).

Various methods, such as using tank circuits to reduce the electromagnet's impedance, can be used to make the high frequency oscillation at the given field strength feasible.⁹ Just as with permanent magnets, the magnetic field in electromagnet also has gradient and the field strength is not uniform at all points in the region. We can see that there is a linear relationship between the field strength and frequency at which particle performs resonance with the electromagnet oscillation frequency ω₀.

III. AMPLITUDE AND WAVELENGTH CALCULATION

Equation can be resolved from cylindrical to Cartesian coordinate system,

\vec{v} = \dot{r} [cos (θ) \hat{i} + sin (θ) \hat{j}] + r \dot{θ} [- sin (θ) \hat{i} + cos (θ) \hat{j}] + \vec{v_{z}} .

(24)

We take v_z = 0 for the simplicity of calculations. From Eqs. and to the above-mentioned equation, we form

\vec{v} = r \frac{ω_{0}}{2} ([- cot (ω_{0} t) cos (θ) + π sin (ω_{0} t) sin (θ)] \hat{i} - [cot (ω_{0} t) sin (θ) + π sin (ω_{0} t) cos (θ)] \hat{j}) .

(25)

It is to be noted that that the polar coordinates are defined with respect to the guiding center formed when the particle enters the oscillating magnetic field. Using the constant from Eq. , we write Eq. as

\frac{d \vec{s}}{d t} = \vec{v} = \frac{\sqrt{G} \cdot ω_{0}}{2 \sqrt{B_{0}}} ([- \frac{cos (ω_{0} t)}{sin {(ω_{0} t)}^{3 / 2}} cos (θ) + π \sqrt{sin (ω_{0} t)} sin (θ)] \hat{i} - [\frac{cos (ω_{0} t)}{sin {(ω_{0} t)}^{3 / 2}} sin (θ) + π \sqrt{sin (ω_{0} t)} cos (θ)] \hat{j}) .

(26)

As magnetic field does no work on charged particle, the induced electric field will be the one doing work on it.¹⁰ In order to find the value of constant G, we use the work-energy theorem while taking electron as our particle,

Δ K . E . = e \int \vec{E} \cdot \vec{d s} .

(27)

Using Eqs. , , and , we can form the integral as

e \int_{0}^{T_{0} / 4} \vec{E} \cdot \vec{d s} = e \int_{0}^{T_{0} / 4} \frac{1}{2} r^{2} \dot{B_{z}} \dot{θ} d t = \frac{π}{4} e G ω_{0} .

(28)

While the velocity from Eq. at

t = T_{0} / 4

is

v_{0} = \frac{\sqrt{G} \cdot ω_{0}}{2 \sqrt{B_{0}}} π \hat{i} .

(29)

Applying work-energy theorem for a non-relativistic speed of particle, we form

\frac{π}{4} e G ω_{0} = \frac{1}{2} m v_{0}^{2} - \frac{1}{2} m u^{2},

(30)

where u is the initial velocity with which electron has been projected with and v₀ the velocity at

T_{0} / 4

⁠, finally giving us the value of G as

G = u^{2} / (\frac{ω_{0}^{2} π^{2}}{4 B_{0}} - \frac{e π ω_{0}}{2 m}) .

(31)

Substituting this G in Eq. with the displacement vector

\vec{d s}

summing up from origin and integrating it over to

T_{0} / 4

⁠, we obtain

\vec{S} = \frac{\sqrt{G} \cdot ω_{0}}{2 \sqrt{B_{0}}} (43.0942 \hat{i} + 1.5552 \hat{j}),

(32)

wherein the required amplitude is

\frac{\sqrt{G} \cdot ω_{0}}{\sqrt{B_{0}}} (0.7776)

and the required wavelength is

4 \sqrt{X^{2} + Z^{2}}

⁠, with Z forming an equivalent to the pitch length in ion cyclotron motion; here with v_z = 0, it is simply

\frac{\sqrt{G} \cdot ω_{0}}{\sqrt{B_{0}}} (86.1884)

⁠.

IV. LAGRANGIAN AND CONTINUOUS BEAM

The resulting single particle trajectory is visually similar to that of electrons in Wiggler. When a continuous beam of electrons enters the D-section containing the said magnetic field B_z, the successive particle trajectories have radial distribution as can be seen from Fig. 2. This occurs because particle trajectory begins from the corresponding Larmor orbit of the magnetic field strength at the instant it enters the D-section. The angular displacement θ from Eq. (19) is also the same angular displacement between successive electron travel paths. It is to be noted that there is a synchronous oscillation of all the particles/plasma present in the D-section along its angular direction as indicated in Fig. 2. The frequency of this resonance is ω₀, which draws parallels to the cyclotron frequency ω_ci in permanent magnet based resonance.

The Lagrangian in the polar coordinates for such particle dynamics is

L = \frac{m r^{2}}{2} [\frac{ω_{0}^{2}}{4} {cot}^{2} (ω_{0} t) + \frac{π^{2}}{4} ω_{0}^{2} {sin}^{2} (ω_{0} t)] - e B_{0} ω_{0} cos (ω_{0} t) π r^{2} - \frac{B_{0}^{2} {sin}^{2} (ω_{0} t) * π r^{2} h}{2 μ},

(33)

where h represents the height of the volumetric distribution of magnetic field in the assembly, while

μ

is the effective permeability of the medium. A reproduction of the external magnetic field in the Vlasov equation along with the stated variables and with an appropriate distribution function can help identify the resonance conditions, forming the basis for applications in plasma instability. Coupling of RF power into the plasma using RF antennas, although not displayed in the assembly above and an appropriate accountability for the relativistic velocities, is left to the experts for future discussions.

V. CONCLUSION

In this paper, we have demonstrated a novel resonance condition for charged particles. Particle dynamics generated by such processes has been visualized in the schematics. By extending the model to the case of non-relativistic velocities, we have obtained a simple procedure to achieve a source of RF power, increase the ion kinetic energy, and consequently, increase the plasma temperature for ICRH or ECRH.

The invariance of kinetic energy for the non-relativistic velocities has been displayed in the absence of radiation losses. While in magnetic pumping, heating or cooling, capabilities are limited as the field cannot be ramped up or down indefinitely, it is ideal if particles can be heated or cooled in a periodically varying magnetic field.¹¹ The next-order kinetic energy variation by increasing pumping frequency can be calculated henceforth from this theory. The control of orbit is still limited by maximal magnetic field in the pursuit of higher energy ions,¹¹ and the current method attempts to overcome this by offering higher magnetic field strength employing electromagnets for resonance. Principally noting the objectives of synchrotron in varying the magnetic field and its frequency to generate high energy particles,¹² the present theory addresses the same with a reduction in cost by omitting the use of superconducting magnets. Its applications can be explored in magnetic confinement and sub-harmonic heating and cooling. This approach and the resulting phenomenology may prove valuable in plasma and accelerator physics.¹³

ACKNOWLEDGMENTS

I would like to thank Late Professor Predhiman Kaw from the Institute of Plasma Research, Dr. R. S. Shinde, and Mr. Subrata Das RRCAT, Indore, for valuable comments.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Devesh S. Bhosale: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Project administration (lead); Software (lead); Supervision (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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Classical charged particle resonance in induction field

I. INTRODUCTION

II. BACKGROUND THEORY

III. AMPLITUDE AND WAVELENGTH CALCULATION

IV. LAGRANGIAN AND CONTINUOUS BEAM

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Classical charged particle resonance in induction field Open Access

I. INTRODUCTION

II. BACKGROUND THEORY

III. AMPLITUDE AND WAVELENGTH CALCULATION

IV. LAGRANGIAN AND CONTINUOUS BEAM

V. CONCLUSION

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Classical charged particle resonance in induction field