Plasma and Fusion Research: Regular Articles

Volume 14, 2401035 (2019)

Lyapunov Exponent Analysis for Stochastic Ion Motion in aNon-Adiabatic Trap∗)
Ryo MATSUMOTO, Shuya IWATA, Takahiro URANO, Kojiro SEKIGUCHI, Gaku SAKAI
and Toshiki TAKAHASHI
Gunma University, 1-5-1 Tenjin-cho, Kiryu 376-8515, Japan(Received 30 September 2018 / Accepted 9 January 2019)
The Lyapunov exponent analysis is carried out for ion motions in a non-adiabatic trap where the axial magnetic field from a solenoid is partially cancelled by that from a Helmholtz coil. In particular, relation between theLyapunov exponents and the trapped rate of ions injected in the axial direction is numerically studied. It is foundthat there exist energy minimizing the particle trapped rate, and the Lyapunov exponent for the axial velocity isfound to be minimized with that energy.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: non-adiabatic trap, Helmholtz coil, stochastic motion, particle-tracking calculation, Lyapunov exponent analysis
DOI: 10.1585/pfr.14.2401035

1. Introduction
Development of particle control technologies by applying an open-ended magnetic field is useful not onlyfor fusion development but also for processing and medical technologies. Regarding nuclear fusion, Momota etal. proposed a concept of non-adiabatic trap with theHelmholtz coil inside the solenoid, where the magnetic
field by the Helmholtz coil cancels that from the solenoid
at the center of confinement region [1]. The coil arrangement of the non-adiabatic trap is schematically drawn inFig. 1. By creating a region where the magnetic field becomes zero at the center, it can be expected that the adiabatic invariance in a charged particle motion will be broken. Charged particles, in particular ions, will suﬀer fromcollisionless pitch-angle scatterings [2–4] and will be able
to move all over the accessible region that is specified by
the constant of motion such as the Hamiltonian and thecanonical angular momentum. By connecting multiple
non-adiabatic traps in the axial direction, it becomes possible to extend a net confinement time because of statisticalproperties shown in random walk processes [1].
Also, a particle control using such a special magneticfield structure is worthy to study. It is expected that ionswith smaller axial momentum can be separated from thosewith larger momentum by the magnetic field in the nonadiabatic trap. Therefore, it can be used as a particle separation apparatus. Adachi et al. calculated trajectories ofions in the trap and obtained the trapping ratio [5]. InRef. [5], beam ions were assumed to be spatially dispersed
in a Gaussian distribution when they are drawn out froman ion source. Then, when the magnetic field generated
author’s e-mail: t14304113@gunma-u.ac.jp∗) This article is based on the presentation at the 12th International Conference on Open Magnetic Systems for Plasma Confinement (OS2018).

Fig. 1 Schematic drawing of a non-adiabatic trap.

by the solenoid is completely canceled by that from theHelmholtz coil, it was found that the trapping ratio is afunction of the standard deviation of the beam dispersion
divided by the Larmor radius evaluated by the solenoid
magnetic field and the beam ion energy. This gives theconditions under which the beam ion passes through thedevice. Adachi et al. simultaneously examined the particletrapping in the state where the Helmholtz coil current waschanged and the magnetic field at the center of the device
was not zero. In this case, it has been found that there isan energy band where the trapping rate is relatively low [5]and therefore the ions probably tend to exhibit regular motions. In this study, a Lyapunov exponent analysis [4] iscarried out to investigate chaotic properties of ion motionsin the situation mentioned above.

2. Calculation Model
2.1 Device modelA schematic drawing of a device combining asolenoid, a Helmholtz coil, and a cylindrical structure ma-2401035-1

c 2019 The Japan Society of Plasma
Science and Nuclear Fusion ResearchPlasma and Fusion Research: Regular Articles

Volume 14, 2401035 (2019)

Table 1 Calculation model parameters.

Fig. 2 Magnetic field structure of the non-adiabatic trap. Two
black solid circles indicate the position of the Helmholtz
coil. Here zM is 0.25 m.

terial such as cathode is shown in Fig. 1. Although thecathode is installed inside the Helmholtz coil, we treat it asa simple structure and no voltage is applied to investigateonly influences of the magnetic field. The calculation parameters are shown in Table 1. The current flowing throughthe Helmholtz coil in Table 1 is not large enough to cancel
completely the magnetic field created by the solenoid. The
coordinate system is a two-dimensional cylindrical coordinate system. We carry out a particle-tracking calculation
for ions and study trapping characteristics of particles inthe device shown in Fig. 1. All particles are injected fromthe end of the apparatus in the axial direction. The ionsare classified as trapped and passing particles. The trappedparticles are reflected by the mirror field and passing particles reach the opposite end of device without one mirror
reflection. When particles hit the cathode, it is assumed
that the particles are treated as lost particles. Since we arefocusing on particle trapping rate, no discussion on the lostparticles will be made here.
The azimuthal component of the vector potentialA(r, z) created by the circular current is μ0 Ii  2π
cos χ dχ.
Aθ (r, z) =4π
2022i(z − zi ) + r + ai − 2ai r cos χ(1)Here, µ0 is the vacuum permeability, ai , zi and Ii are the radius, the axial position and the circular current of i-th coil,respectively. The integration on χ in Eq. (1) is done numerically. By substituting the obtained vector potential into thefollowing relation: B = rotA, the magnetic field can be obtained. In this study, the Helmholtz coil current is set to
15 kA. This Helmholtz coil current does not completely
cancel the magnetic field on the midplane and geometric axis generated by the solenoid. Adachi’s research [5]showed that the fraction of trapped particles decreases atspecific energies of ions in this magnetic field. The purpose of this study is to investigate properties of stochastic
motions of ions with energy around the minimum trapping
rate. The magnetic field structure when the Helmholtz coilcurrent is 15 kA is shown in Fig. 2. Here, zM is the half
length of the device and it is 0.25 m. It turns out that the xpoints where the magnetic field is zero exist slightly inside

the Helmholtz coil. Also, it is found that there is the finite
axial field along the geometric axis because of incomplete
cancellation by the magnetic field from the Helmholtz coil.

2.2 Particle-tracking calculation
Deuterium ion trajectories are traced by integrating
the following simultaneous diﬀerential equations:dx= v,
dtdv
q= v × B.
dtm(2)Here, x and v are the position and velocity of ions respectively, and q and m are the charge and mass of ions, respectively. The Runge-Kutta method is employed for numericalintegration. The initial position of all ions are z = −0.25 m
at the left end of the device, and 10,000 particles are arranged at equal intervals in the radial direction. In order tocalculate the trapping rate, it is necessary to give the initial condition of ions with more realistic setting. However,the purpose of our study is to investigate a stochastic behavior of ion motion in this magnetic field structure, and it
is preferable that the relation between the initial conditionand chaoticity is clear. Therefore, although not a realisticmodel, the initial positions of the ions are arranged at equalintervals in the radial direction.

3. Lyapunov Exponent Analysis
As a quantity expressing the degree of chaos, theLyapunov exponent is well-known. In chaotic systems,slightly diﬀerent two solutions move away each other exponentially with time. Therefore, displacements in the position and velocity can be written generally as follows:|Δχi (t)| = |Δχi (0)| exp (λi t) .

(3)

Here, χi is a general description of position or velocity scalar variables, and in the case of three-dimensional
space, i = 1, 2, · · · , 6. Note that χi is not a subscripted
expression of vector variables. The quantity λi in Eq. (3)is the Lyapunov exponent for the i-th variable. Using theequation of motion for ions (2), we can write

2401035-2dχi= Fi (χ j ).
dt(4)Plasma and Fusion Research: Regular Articles

Volume 14, 2401035 (2019)

Fig. 4 Relationship between particle trapped rate and ion energy.

Fig. 3 Example of time evolution of 1/t · ln(|Δz(t)|/|Δz(0)|).

Here, Fi (χ j ) is a function given for each component representing the velocity and acceleration. When an infinitesimal displacement Δχi is given, thend(χi + Δχi )= Fi (χ j + Δχ j ).
(5)dt
Taylor expansion of the right side of Eq. (5) and consideration of Eq. (4) yielddΔχi ∂Fi=Δχ j .
dt∂χ j(6)

By numerically integrating the displacement Δχi using theRunge-Kutta method as in the trajectory calculation ofions, we also obtain the time evolution of Δχi . Using
Eq. (3), we can estimate the Lyapunov exponent for eachvariable in the following way:
λi ≡ lim

1t→∞ tln|Δχi (t)|.
|Δχi (0)|Fig. 5 Relationship between Lyapunov exponents and ion energy.

(7)

The value calculated by Eq. (7) converges to a certain value
over time, and the steady-state value is the Lyapunov exponent. Usually, however, ions are lost away from the confinement device before the conversion. So in the present
study, the final value of the trajectory calculation is adopted
as the Lyapunov exponent. Typical time evolution of
1/t · ln (|Δz(t)| / |Δz(0)|) is shown in Fig. 3, where λz for theaxial position z is calculated and the time t is normalized
by the typical gyration time in the magnetic field generated
by the solenoid (i.e., m/(qBs ) = 0.166 µs and Bs = μ0 nIs
where n and Is are the turns per unit length and the currentof the solenoid, respectively). Although numerical vibration is seen, it is found that λz almost converges to 0.036.

4. Results and Discussion
Ions are injected in the axial direction from the end of
the device. Here, particles moving through the device aredefined as passing particles and particles which are mirrorreflected at least once are defined as trapped particles. Ions
colliding with the cathode shown in Figs. 1 or 2 are excluded for our analysis. Figure 4 shows the energy dependence of the fraction of trapped particles. As in the caseof Adachi et al. [5], it is found that the particle trapped
rate becomes a minimum value at 0.05 keV. That is, the0.05-keV ions are likely to pass through the device without mirror reflection.
Therefore, Lyapunov exponents for six componentsof position and velocity (λr , λθ , λz , λvr , λvθ , and λvz ) arecalculated for ions with energy of 0.05 keV (the minimumtrapped rate), 0.01 keV and 0.07 keV, and the results areshown in Fig. 5. As a result, it is found that the Lyapunov
exponents except λvz increase as energy increases. However, λvz is the smallest at 0.05 keV. In other words, in thecase of 0.05 keV, it is found that the initial value sensitivity
to the deviation of the initial axial velocity of the two ionsis small.
A typical trajectory of (a) a 0.05-keV passing ion and(b) a trapped ion are drawn and projected onto xy plane
as shown in Fig. 6. Here, the axes of coordinate are transformed by x = r cos θ and y = r sin θ from the cylindricalcoordinates shown in Fig. 1. In either case, the trajectory
radius increases as the ion passes through the weak magnetic field region. At the same time, the trajectory deviatesfrom a regular circular motion. However, it can be seenthat the trapped particles are subject to more changes ofthe velocity direction, i.e., collisionless pitch angle scatterings [2, 3], which depend on the gyrophase when ions enter and exit the weak magnetic region. For a small trapped
rate, ions entering the weak magnetic field region are un-2401035-3

Plasma and Fusion Research: Regular Articles

Volume 14, 2401035 (2019)

Fig. 6 Typical 0.05-keV ion trajectory projected onto the xy
plane of (a) passing particle and (b) trapped particle.

likely to cause perturbation of their trajectory and exhibit
regular motions. As a result, changes in the velocity component in the axial direction, that is, in the direction along
the magnetic field are small, and ions tend to pass to theopposite device end. Therefore, the Lyapunov exponent
λvz becomes smaller, and it can be understood that the ion
motion becomes more regular.
The results of the Lyapunov exponents for (a)0.01 keV, (b) 0.05 keV, and (c) 0.07 keV ions as passing
particles and trapped particles are shown in Fig. 7. As described above, an ion trapping by a non-adiabatic trap isdue to a stochastic behavior seen in an ion motion and aconsequent large Lyapunov exponent. In other words, it isclear that statistical behavior of ion motion is strongly related to trapping and passing in non-adiabatic traps. Alsofrom Fig. 7, it can be seen that, which is the maximum
Lyapunov exponent, tends to have diﬀerent tendency fromother Lyapunov exponents. It is found that λθ of trapped
particles is larger than that of passing particles only at
0.05 keV at which the trapped rate becomes minimum.
On the other hand, λθ of trapped particles is smaller at
0.01 keV and 0.07 keV. Here, since the maximum Lyapunov exponent is positive at any energy, the initial value
sensitivity is still noticeable in the ion motion. In the caseof 0.01 keV or 0.07 keV, the initial value sensitivity of passing particles is pronounced, and ions can be easily trapped
due to a slight disorder of orbits, whereas it is found thatthe tendency is relatively small at 0.05 keV.

5. Conclusion
In this study, ion motion in a non-adiabatic trap, which
is a magnetic field structure created by partially cancelinga magnetic field from a solenoid by that from a Helmholtz
coil, has been investigated. In particular, we have focusedon the energy dependence of the trapping rate of beamions injected into the non-adiabatic trap in the axial direction. The Lyapunov exponent analysis has been carried out on ions with energy (0.05 keV in this study) thatminimizes the trapped rate. As a result, it was found that
the Lyapunov exponent for the axial velocity is minimized
at 0.05 keV. This indicates that 0.05-keV ions have relatively regular motion. Furthermore, it was found that theions are diﬃcult to undergo pitch angle scattering whenpassing through the weak magnetic field region. The Lyapunov exponents were compared between passing particles
and trapped particles and it was found that the Lyapunov
exponent of trapped particles was larger. Therefore it was
found that trapping in the device is the result of random
motion.

Fig. 7 Lyapunov exponents of passing particles and trapped particles for (a) 0.01 keV, (b) 0.05 keV and (c) 0.07 keV.

[1] H. Momota et al., J. Fusion Energy 27, 77 (2008).
[2] T. Takahashi et al., Phys. Plasmas 4, 4301 (1997).
[3] T. Takahashi et al., Phys. Plasmas 11, 3131 (2004).
[4] Y. Hayakawa et al., Nucl. Fusion 42, 1075 (2002).
[5] D. Adachi et al., Plasma Fusion Res. 13, 3401069 (2018).

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