We present a theoretical basis and simple experimental realization of a multipole radio-frequency trap consisting of four equal cylindrical electrodes, where all of the bars have an in-phase applied voltage. An effective potential, which describes three additional stable quasi-equilibrium points, is obtained, and an electrostatic distribution is calculated using the method of image charges. We construct an experimental setup and localize a group of charged silicate microspheres at normal pressure. The experimental results agree well with the proposed analytical model. A strong dependence on modulation of the radio-frequency field and effective potential is confirmed.

## I. INTRODUCTION

Multipole ion traps are widely used as precision devices for ion transport and localization.^{1,2} In particular, octupole ion guides have found application in particle accelerators^{3–5} and the Orbitrap mass analyzer.^{6} The multipole field is formed in a standard way as in a quadrupole Paul trap design, but with additional electrodes. In both quadrupole and multipole traps, voltages are applied with different phases on neighboring electrodes: cos(Ω*t*) and cos(Ω*t* + *π*), respectively (an experimental setup where half of the electrodes are connected to ground is equivalent to the one described).^{7,8} It should be noted that the setup with different phases on neighboring electrodes is a universal one, and a description of an electrostatic field in the trap in terms of harmonic polynomials applies to all of its manifestations.^{9–11} At the same time, the electrostatic field depends on many factors, for example, the electrode geometry,^{12–17} the modulation of the radio-frequency voltage applied to the electrodes,^{18–20} and the spatial positions of the electrodes.^{21–23} For example, even a small change in the voltage of the neighboring electrodes from cos(Ω*t*) and cos(Ω*t* + *π*) to the in-phase mode cos(Ω*t*) and cos(Ω*t*) can lead to a significant deformation of the quadratic form of an effective potential. In this case, we still retain the condition of a sign-alternating voltage, so that the equations of motion will satisfy Earnshaw’s theorem, but the stability of localization is no longer obvious, and additional research on this issue is required.

In this paper, we show that a real in-phase trap creates a quasi-octupole effective potential. The existence of three stability points is confirmed. We have designed and constructed a prototype of the proposed trap, and the experimental results of localization confirm our theoretical justification.

## II. GENERAL CONCEPT

The spatial distribution of the potential is usually given by^{24,25}

where *r*_{0} is the trap radius, *e* is the charge of the localized particle, *U*_{0} cos(Ω*t*) is the applied voltage with frequency Ω, and *n*_{m} is the order of the multipole, which is equal to twice the number of electrodes.

Therefore, a quadrupole field (*n*_{m} = 2) is formed by four electrodes [Fig. 1(a)], and an octupole (*n*_{m} = 4) is formed by eight electrodes [Fig. 1(b)]. To obtain Eq. (1), we should apply different phases 0 and *π* on neighboring electrodes, which means that electrodes have applied voltages of *U*_{0} cos(Ω*t*) and *U*_{0} cos(Ω*t* + *π*), accordingly, or, in other words, ±*U*_{0} cos(Ω*t*).^{7,24,26} An experimental setup where half of the electrodes are connected to ground, in this case, voltages +*U*_{0} cos(Ω*t*), 0, +*U*_{0} cos(Ω*t*), and 0, correspondingly, is also often used. Both setups give a quadrupole field.

Therefore, the potential in Eq. (1) does indeed provide a good description of the field in the center of a trap with hyperbolic electrodes. The common description tells about the quadrupole trap in four-bar traps, and multipole one for traps with more than four electrodes.

However, Eq. (1) cannot be applied when there are equal phases on neighboring electrodes, and the voltages are given by +*U*_{0} cos(Ω*t*), +*U*_{0} cos(Ω*t*), +*U*_{0} cos(Ω*t*), and +*U*_{0} cos(Ω*t*)—they are the same for all electrodes [Fig. 1(c)].

This kind of trap consists of four electrodes and will not describe the quadrupole field in the usual sense. To demonstrate the non-quadrupole (multipole) field in a system of four electrodes, we consider an in-phase trap shown in [Fig. 2(a)]. The proposed configuration differs from [Fig. 1(c)] with the surface of zero potential at the distance *L* from the center of the trap. To determine the correct field for the four-bar in-phase trap [Fig. 2(a)], we solve the electrostatic problem using the method of image charges in Appendix A.^{27}

Figure 2(b) shows an (*x*, *y*) section of the equipotential surfaces for *r*_{e} = 3 mm, *r*_{0} = 17 mm, *L* = 1200 mm, and *U*_{0} = 2700 V at a fixed moment of time, *t* = 0. We can observe an asymmetry of the equipotential lines about *y* = 0. This asymmetry is caused by the one-sided position of the surface of zero potential. Indeed, even though *L* ≫ *r*_{e}, *r*_{0}, the value of *L* is finite. Therefore, we can see that the distribution of the potential for a four-bar trap with equal phases [Fig. 2(a)] is not consistent with the quadrupole distribution for the trap with different phases, Eq. (1). However, with the appropriate boundary conditions on the surface of the electrodes and with the Laplace equation being satisfied at every point, Δ_{x,y}*U*(*x*, *y*) = 0 ∀ (*x*, *y*), we find that localization is indeed possible in practice for this case.

Charged particle motion in such traps depends on the calculated potential and the field of the end-caps ( Appendix A). The field of four bars does not influence the particle motion around the axis as long as the bars do not deviate from their ideal parallel positions. In contrast, the field of the end-cap electrodes will always influence the radial motion. Since the end-caps have weak effect on each other, we can calculate the field of end-caps as a Dirichlet problem for a half-plane. In our design, we used flat circle end-cap electrodes in Appendix B.

To determine the stable equilibrium points in the obtained field, we calculate the effective potential. The equations of damped motion of a charged particle in the (*x*, *y*) plane take the following form:

where *e* and *m* are the particle charge and mass, *U*(*x*, *y*) is the field from the cylindrical electrodes with the same phase voltages, $Uend\xb1$ is the field from end-cap electrodes, and *β* = 6*πμr*_{p} is the Stokes damping parameter (where *μ* is the coefficient of dynamical viscosity and *r*_{p} is the radius of a spherical particle). Although the potential $U(x,y)+Uend\xb1(x,y,z)$ does not satisfy Eq. (1), we still can describe the effective potential using the averaging method.^{25} We can derive this effective potential from the average kinetic energy of the fast oscillations.^{24,28} The result is clearly different from the cylindrical effective potential and takes the form of Eq. (4).

The effective potential [Eq. (4)] comprises the complete “slow” components describing the averaging ($\cdots \u2009$) of radio-frequency terms and “direct” terms $Uend\xb1$ from the end-cap electrodes,

## III. RESULTS AND DISCUSSION

It is clear that linear damping does not influence the form of the effective potential,^{29,30} and Eq. (4) takes the usual form.^{25} However, the form of Eq. (4) is correct for linear damping only. For large applied voltages, we can observe trapped particle oscillation period doubling as a result of a strong contribution from nonlinear damping.^{31–33} The critical voltage for the change to non-linear damping depends on the nature of the particles and buffer gases. In our case (20 *μ*m silicate spheres in air), this voltage has been determined experimentally to be above *U*_{0crit} = 3200 V.

The effective potential in the trap contains three stable quasi-equilibrium points [shown as black stars in Fig. 3(a)], instead of the one as in the quadrupole trap. These quasi-equilibrium points appear in the absence of a DC voltage, in contrast to multipole Paul-type traps, where a DC voltage is a necessary condition for the creation of additional stable points (in the ideal case, although, in practice, displacements of the electrodes can force symmetry breaking to occur).^{23,34}

Indeed, the potential shown in Fig. 2(b) and the corresponding effective potential shown in Figs. 3(a) and 3(b) seem very similar to the octupole field. At the same time, since there is a permanent asymmetry around the *y* axis, we will obtain three stable points. We note that the observed symmetry breaking can be eliminated by one outer cylindrical zero potential surface [instead of the flat on the distance *L*, such as in Fig. 2(a)]. In this case, the pseudopotential without DC voltage will depend on the following spatial distribution:

which exactly matches the distribution model [Eq. (1)] for *n*_{m} = 4 (octupole field). At the same time, the presence of three minima for asymmetric zero potential surface well agree with the theory of symmetry breaking.^{34}

In our single-phase four-bar trap, we can achieve stable localization at points A, B, and C for a wide range of voltage and mass/charge parameters. Figure 3(b) shows sections of the equipotential surfaces in the (*x*, *z*) plane under the influence of a DC voltage on the end-cap electrodes. It can be seen that the additional voltage does not change the spatial positions of the quasi-equilibrium points. In addition, the amplitude of the AC voltage and the mass/charge ratio affect only the depth of the quasi-equilibrium points. The results shown in Figs. 3(a) and 3(b) are fair for the chosen trap geometry with arbitrary masses and charges of particles.

In order to test the theoretical hypothesis, we constructed a multipole linear radio-frequency trap, the schematic diagram of which is shown in Fig. 4. Four equal stainless steel cylindrical (power) electrodes, with a radius of *r*_{e} = 3 mm and a length of *l* = 75 mm, are fixed in a dielectric casing so that the distance between diametrically opposite electrodes is equal to *r*_{0} = 17 mm. Two flat end-cap electrodes with a radius of *r*_{E} = 3 mm are located at an equal distance from the axis of each power electrode, and the distance between the end-cap electrodes themselves is 55 mm. The multipole trap is mounted on an optical table so that the axis of the trap is at a distance of *L* = 1200 mm from the grounded horizontal plate [Fig. 2(a)].

Four power electrodes are switched together and connected to the output of a high-voltage transformer, capable of producing 6 kV at a frequency of 50 Hz. The end-cap electrodes, in turn, are connected to the output of a DC power supply unit, capable of supporting voltages up to 300 V. As object of localization, microspheres of borosilicate glass with a radius of 20 *μ*m were used, and such particles are a smooth glass shell with an air pocket in the center. The experimental localization of microspheres was held under normal conditions, where the voltage of the cylindrical electrodes was 2659 V and the voltage at the end-cap electrodes was 260 V.

The demonstration of the experimental localization process of a group of charged microspheres that scatter the light of a green laser, directed horizontally from the side of the trap, is shown in [Fig. 5(a)]. The obtained localization clearly shows the splitting of the group of charged particles into three regions [A, B, and C by analogy with Fig. 3(b)]. Figure 5(b) shows the same group of localized particles from above, perpendicular to the axis of the linear trap. The experimental implementation shows a distinct isolation of localized particles in regions A, B, and C, respectively.

The obtained distribution of particles, with constant parameters of the power supply of the trap, is stable with time. We were able to observe such effect for more than 20 min without localized particles changing their position. At lower voltages on four power electrodes, particles drift under a gravity force from point A to points B and C. Therefore, with a voltage less than 2200 V, we observed particles localized at only two lower points (B and C).

Thus, we have achieved an effective potential split in the case of a trap where all four bars have in-phase radiofrequency supply with an asymmetric zero potential surface.

## IV. CONCLUSION

In this paper, we tried to prove an important idea: It is possible to achieve an octupole field on the basis of a well-known four-electrode trap.

The in-phase four-bar trap we introduced, shown in Fig. 2(a), is not a quadrupole one, which means that the localization process cannot be described by the formalism of the stability diagram by analogy with the Strutt–Ince diagram for Mathieu equations. We obtain a potential distribution for this trap similar to octupole. The difference from the octupole field is due to the asymmetric position of zero potential surfaces.

As a result, splitting of the effective potential was observed in full accordance with the symmetry breaking in the octupole traps. At the same time, using the outer cylindrical zero potential surface, we can obtain an ideal octupole field [Eq. (5)]. Therefore, by hot-switching the electrode scheme from Figs. 1(a)–1(c), we can easily obtain an octupole mode out of a quadrupole one.

The trap proposed in [Fig. 2(a)] can find an application in the field of microparticle study. We note that the spatial position of three stable quasi-equilibrium points for a given mass to charge ratio *m*/*e* depends on the voltage, the power, and the end-cap electrodes [Eq. (4)]. By increasing DC voltage on the end-caps, it is possible to achieve linear selection from “light” to “heavy” charged microparticles, since the coordinates of quasiequilibrium points for light particles will exceed the radius of a trap. The existence of three potential minima may find an application as a base for a non-destructive method of measuring the physical characteristics (charge, mass, and size) of trapped microparticles.^{33,35} Moreover, the correlation between the location of stable equilibrium positions for a certain particle with a fixed physical characteristics can potentially help us to determine these characteristics directly in the trapping process—with no need for physical interaction of the studied particle with the detector.

## AUTHORS’ CONTRIBUTIONS

All authors contributed equally to this work.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## ACKNOWLEDGMENTS

This study was supported by the Theoretical Physics and Mathematics Advancement Foundation BASIS (Grant No. 19-1-5-136-1) and the Ministry of Education and Science of the Russian Federation (Grant No. 074-U01).

### APPENDIX A: THE ELECTROSTATIC PROBLEM FOR IN-PHASE TRAP

To do this, we place into a center of each electrode a charge *q*_{0}, equal to a full charge on the cylindrical surface and defined by the applied potential. Then, we reflect these charges into the other electrodes. The distances between the image charge and the given charge in the center of the electrode for one-sided and diagonal electrodes are given by

where *r*_{e} is the radius of the cylindrical electrode and *r*_{0} is the trap radius [Fig. 1(a)].

To impose equipotential conditions, we also add an additional charge to the center of each electrode. The system of charges is defined as it is shown in Fig. 1(a). For an additional approximation, we impose the following recurrence relations for the one-side charges and diagonal charges, accordingly:

The zero–potential plane is at a distance *L* from the center of the trap. Since *L* ≫ *r*_{0}, *r*_{e}, we can reflect all the additional uncompensated charges 4*q*_{0} [for the case in Fig. 1(a)] in this plane at 2*L*. It should be noted that the correct conditions on the potential will be obtained when *n* → ∞ in Eq. (A2). After introducing coordinates with origin at the center of the trap, we can easily determine the potential at an arbitrary point (*x*, *y*) outside the cylindrical electrodes as

where (*x*_{i}, *y*_{i}) are the Cartesian coordinates of charge *q*_{0i}.

We note that Eq. (A3) is correct for all points *on* the electrode surface. Therefore, we can equate the potential at the point (*r*_{0}, *r*_{0} − *r*_{e}) to the applied voltage,

where *U*_{0} is the amplitude of the AC voltage with frequency Ω. Solving Eq. (A3) for *q*_{0i} with *L* ≫ *r*_{0}, *r*_{e}, we have Eq. (A5).

After substituting the obtained value into Eq. (A3), we obtain the potential distribution as a function of the geometry of the electrodes (*r*_{0}, *r*_{e}) and the applied voltage *U*_{0} cos(Ω*t*),

### APPENDIX B: THE ELECTROSTATIC PROBLEM FOR END-CAPS

For each of the end-cap electrodes, upper (−*l*/2, ∞) and lower (−∞, *l*/2) half-planes were taken, where *l* is the length of the trap. Taking *V*_{end} as a DC voltage on the end-cap electrode and integrating over the surface one, we obtain the final expression for the end-cap field [Eq. (B5)],^{32} where we substitute

where $\Pi ({n1\xb1,n2\xb1},C)$ are complete elliptic integrals of the third kind, and *r*_{E} is the radius of end-cap electrode. We note that the available part of space is *z* ∈ [−*l*/2, *l*/2], and the signs “+” and “−” correspond to the left and right end-cap electrodes, respectively. It could be noticed that this field contains both axial and radial terms, describing confinement in the (*R*, *z*) plane, where $R=x2+y2$,