### I. Introduction

*n*= 2, where the higher-order terms after

*n*= 3 are truncated due to the small amplitude of oscillation assumption.

### II. Electrostatic Theory

### 1. Method for an Electrostatic Potential Function

*ϕ*(

*r*,

*θ*) denotes the electrostatic potential function created by the volumetric charge density distribution function

*ρ*(

*r′*,

*θ′*),

*r*and

*θ*respectively signify a radial distance and an inclination angle in the polar spherical coordinate at the location under consideration, while

*r′*and

*θ′*, respectively represent a radial distance and an inclination angle in the polar spherical coordinate (pointing to the location of a charge distribution), and

*dv′*denotes the infinitesimal volumetric body of a charge distribution. To express 1/||

*r*

**-**

*r′*|| in the algebraic form to facilitate the upcoming derivation process the definition of the vector norm is employed.

*r*> represents the maximum magnitude between ||

*r*|| and ||

*r′*||, whereas

*r*< denotes the minimum magnitude between these two variables, namely

*r*> =

*max*(||

*r*||,||

*r′*||) and

*r*< =

*min*(||

*r*||,||

*r′*||). Therefore, Eq. (2) can be rewritten, as shown in Eq. (3):

*r*< is less than

*r*>; in other words,

*r*< /

*r*> is literally less than unity;

*r*</

*r*> < 1, which leads to (r < /

*r*>)

^{2}– 2(

*r*.

*r′*)/(

*r*>)

^{2}and is also less than unity, according to the binomial series expansion definition, can be rewritten, as shown in Eq. (4):

*r*<

*r′*condition is satisfied, the potential field relation provided by Eq. (4) is finally obtained under an azimuthally symmetrical charge distribution speculation [23–25]. This can be expressed by an integral form, as shown in Eq. (5):

*ϕ*(

*r*,

*θ*) represents the electrostatic potential generated by a distributed charge at

*r*and

*θ*coordinate,

*p*

*(*

_{n}*cosθ*) is a Legendre polynomial of order n as a function of cos

*θ*parameter,

*ρ*(

*r′*,

*θ′*) denotes a density function of azimuthally symmetrical charge distribution

*r′*and

*θ′*coordinate,

*p*

*(*

_{n}*cosθ′*) is a Legendre polynomial of order

*n*[29] corresponding to

*cosθ′*parameter, and

*v′*signifies the volume of an azimuthally symmetrical charge distribution. The numerical orders of the Legendre polynomial function are provided in Table 1.

### 2. Determination of Electrostatic Potential Function

*z*-axis direction, where the position under consideration at radial distance

*r*is confined within the

*c*value to ascertain that the

*r*<

*r′*condition has been met. The azimuthally symmetrical charge distribution density function of a single ionic ring, per Fig. 2 in the polar spherical coordinate adapted from Hassani [34] and Wang [35], is given via Eq. (6) below,

*K*is a constant that we need to determine, and

*δ*(

*u*) represents a Dirac-delta function [29, 34, 35]. To find

*K*, we integrate Eq. (6) over an ionic ring body to ensure that its integral value is equal to total charge

*Q*via Eq. (7):

##### (7)

*ϕ*

*(*

_{ring}*r*,

*θ*) represents the electrostatic potential of the ionic ring shown in Fig. 2 at

*r*,

*θ*coordinate,

*Q*denotes the total charge of an ionic ring under the uniform charge distribution assumption, and

*ɛ*

_{0}signifies the vacuum electric permittivity parameter, which can be rewritten, as shown in Eq. (9):

*c*value,

*r*<

*c*, as shown in Fig. 2, which enables us to conveniently analyze and study the electrostatic potential field near the origin’s location. In the next section, we develop a one-dimensional equation of motion for a trapped ion confined in a non-uniform electrostatic field generated by two ionic rings, based on Eq. (9), to investigate how a trapped ion behaves in this field when restricted to one dimension.

### III. Motion of a Trapped Ion in a Non-uniform Electrostatic Field

*z*

^{1}direction, whereas that of ring 2 points to the

*z*

^{2}direction, as shown in Fig. 3(b). Nonetheless, the electrostatic field generated by each ring is still based on Eq. (9), with a different parameter setting. The electrostatic functional form of each ionic ring can be provided using Eq. (10):

*θ*

*represent the electrostatic field function and the inclination angle of the ring, respectively. According to Fig. 3(b),*

^{i}*θ*

^{1}= 0

*and θ*

^{2}=

*π*if the positive sign pointing to the

*z*

^{1}direction is specified.

*n*= 0, 1, 2 orders, where all expansion terms after

*n*= 3 are not considered, Eq. (13) becomes Eq. (14):

*n*= 2 are truncated. It is important to note that

*P*

_{0}(

*x*) = 1,

*P*

_{2}(

*x*) = (3

*x*

^{2}– 1)

*/*2 per Table 1, and

*P*

_{1}(1) +

*P*

_{1}(1) = 0 condition have been employed to obtain Eq. (14), based on Eq. (13). To develop the equation of motion of an ion particle in the

*z*direction in the non-uniform electrostatic field using Eq. (13), we assume the idea of a hypothetical tube, an imaginary tube shown in Fig. 3(a), to limit the motion of an ion particle in one dimension and implement the Lagrangian definition [27, 28], where

*z*direction, per Lagrangian mechanics, as shown in Eq. (15). Therefore, the equation of motion of this charged particle in a non-uniform electrostatic field generated by two ionic rings becomes Eq. (16):

*ɛ*

_{0}= 8.854 × 10

^{–12}F/m, the electric permittivity in a vacuum in the SI unit, and

*q*= 1.6 × 10

^{−19}C, the electrical charge of an ion in the SI unit. Eq. (16) represents the linear second-order differential equation that can exhibit a harmonic motion if appropriate parameters have been set up to ensure that the stability condition has been satisfied, that is,

*l*> 2

*a*. If the stability condition is met, the time interval of harmonic motion based on Eq. (16) is provided using Eq. (17):

### IV. Methodology

*z*(0) = 0.018 m, and initial velocity

*ż*(0) = 0 m/s are used to run our simulation. We specifically examine how the magnitude of the electrical charge of each ring, represented by

*ζ*=

*Q*/

*q*parameter, significantly influences ion particle displacement responses. Additionally, the effect of an ion ring size, represented by ring radius

*a*, on the time interval of the oscillation has been thoroughly studied. All parameters used to run the simulation in this part are provided in Table 2. In the second part of our investigation, we focus on studying the impact of

*ζ*=

*Q*/

*q*parameter, the magnitude of the electrical charge of each ring, on velocity response, as well as how its maximum velocity amplitudes are influenced by the ring size, represented by ring radius

*a*. All parameter settings in this part are also listed in Table 2.

### V. Results and Discussion

### 1. Displacement Responses and Time Interval

*ζ*= 1, 5, and 10. To verify the stability of these periodic motions, we establish the phase trajectory plots, as illustrated in Fig. 5, where all setting parameters are provided in Table 2, to ascertain whether the system has stable periodic motions. Based on the phase portrait, we discover that all phase trajectories,

*ζ*= 1, 5, and 10, exhibit an elliptical motion around the equilibrium position at the point of origin without asymptotically stable and unstable evidence, indicating that all trajectories respond in a neutrally stable manner. These simulation results confirm the prospect of confining an ion particle in a non-uniform electrostatic field generated by two ionic rings, based on a one-dimensional speculation.

*T*, highly depends on the magnitude of the electrical charge, represented by

*ζ*parameter. In Fig. 4(a), it is noticeable that the time interval of oscillation of an ion particle tends to decrease when the

*ζ*parameter value increases, indicating that the system will oscillate at a higher frequency rate if the electrical charge in each ring has greater concentration. We investigate this observation in detail to determine how significantly the electrical charge concentration in an ionic ring influences the time interval of oscillation, per Fig. 4(b), where

*ζ*parameter ranges from 1 to 10. As a result, we observe that the time interval of oscillation,

*T*, is remarkably influenced by

*ζ*parameter, the magnitude of electrical charge in each ring. Specifically, the time interval of oscillation decreases when

*ζ*parameter tends to increase, as expected, which confirms our belief that a trapped ion will respond at a higher frequency at an elevated electrical charge concentration. Lastly, we also find that the time interval of oscillation is considerably dependent on the ring size, represented by ring radius

*a*—that is, an ion particle oscillates at a higher frequency and a shorter time interval when the ring size becomes smaller, as shown in Fig. 4(b).

### 2. Velocity Responses based on Changes in Charge Ratio

*ζ*parameter, the magnitude of the electrical charge of each ionic ring, has been altered, per the data provided in Table 2. The simulation results discussed in this part are shown in Fig. 6. In Fig. 6(a), we observe that the velocity responses of a trapped ion highly depend on

*ζ*parameter, where

*ζ*= 1, 5, and 10 are used in our study, per the information in Table 2. Regarding the simulation results, it is quite clear that not only does

*ζ*parameter affect the time interval of oscillation of a trapped ion, as discussed earlier, but also has a considerable effect on the amplitude of velocity responses—that is, the amplitude will significantly rise if the magnitude of the electrical charge in each ring becomes higher. To validate the observation in detail, it can be confirmed through another numerical experiment, as shown in Fig. 6(b), where the velocity amplitude of ion oscillation noticeably increases when

*ζ*parameter rises, based on the latter value ranging from 1 to 10. This indicates that an ion particle in a non-uniform electrostatic field is likely to oscillate at a higher velocity when the magnitude of the electrical charge in each ring becomes more concentrated. Another interesting observation is the effect of the ring size, represented by ring radius

*a*, on the velocity amplitude of oscillation, which tends to increase if ring radius

*a*becomes smaller, as illustrated in Fig. 6(b).

### VI. Conclusion

*n*= 3 are negligible due to the small amplitude oscillation speculation. Then, the equation of motion of an ion particle is developed by means of the Lagrangian formulation. According to the numerical study, the displacement responses of an ionic particle can exhibit a stable periodic motion, verified by displacement responses and phase trajectory plots, which cements the prospect to trap an ion by a proposed electrostatic field within a stability parametric setting,

*l*> 2

*a*. Additionally, we find that the magnitude of the electrical charge of each ring and the ring radius, representing the ring size, significantly influence the frequency and velocity amplitude of the ion oscillation, where an ion particle tends to oscillate at a higher frequency if the charge concentration is greater, but the ring size becomes smaller. We also notice that the velocity amplitude of ion particle oscillation tends to rise when the charge concentration increases, while the ring radius decreases. In future work, we will extend our study to a three-dimensional case to gain more insights and discover the realistic behavior of a trapped ion in a non-uniform electrostatic field, which might lead to the development of the experimental prototyping.