INTRODUCTION

The pseudopotential approach is a useful tool for a qualitative description of the motion of ions in inhomogeneous high-frequency electric fields [1–15]. In particular, a correct understanding of these processes is important in analyzing the operation of radio-frequency quadrupole mass filters [15–18], radio-frequency traps [19–24], radio-frequency transport systems [25–33], etc. However, in considering the motion of ions in high-frequency electric fields with close frequencies, a problem arises, because the mathematical description of such systems can be performed in many ways, leading, generally speaking, to different mathematical models of motion, which are incompatible with each other.

Let us consider the one-dimensional motion of an ion in an electric field with the potential

$$\begin{gathered} U\left( {x,t} \right) = {{U}_{0}} \\ \times \,\,\left[ {\sin \left( {x/L} \right)\cos \omega t - \cos \left( {x/L} \right)\sin \omega t} \right]\sin \Omega t, \\ \end{gathered} $$
((1))

which corresponds to a periodic high-frequency transporting system with annular diaphragms and Archimedean high-frequency electric fields [29–31]. The general theory of transporting systems with Archimedean electric fields was considered in [34–40]; examples of their practical use were given in [41–46]. Here, U0 determines the scale of the electric potential, L is the characteristic distance between the electrodes creating the field (1), Ω is the circular frequency of the high-frequency voltages applied to the electrodes, and ω is the circular frequency of the amplitude modulation of high-frequency voltages. The three-dimensional potential of an axisymmetric system with axial distribution (1), satisfying the Laplace equation, has the form [30, 31]

$$\begin{gathered} U\left( {x,y,z,t} \right) = {{U}_{0}}{{I}_{0}}\left( {\frac{{\sqrt {{{y}^{2}} + {{z}^{2}}} }}{L}} \right) \\ \times \,\,\left[ {\sin \left( {\frac{x}{L}} \right)\cos \omega t - \cos \left( {\frac{x}{L}} \right)\sin \omega t} \right]\sin \Omega t, \\ \end{gathered} $$
((2))

where I0 is the modified Bessel function of the zeroth order. As the radial components of the electric field (2), which deflect the ion from the axis, are equal to zero on the system axis at y = z = 0, using the model distribution (1) and the one-dimensional movement of the ion along the axis as a test is allowable.

For ω ⪡ Ω, the amplitude of the sinusoidal function (1) slowly changes in time compared to high-frequency voltage oscillations, so that, in accordance with the pseudopotential model [1–3, 8, 10, 12–15], the actual motion of an ion with the mass m and charge e in a high-frequency electric field (1) can be replaced by “averaged” motion in a quasi-stationary electric field with the pseudopotential

$${{\bar {U}}^{{rf}}}\left( {x,t} \right) = \frac{{e{{{({{U}_{0}}/L)}}^{2}}}}{{4m{{\Omega }^{2}}}}{{\cos }^{2}}\left( {x/L - \omega t} \right).$$
((3))

Pseudopotential (3) is a sine wave with alternating maxima and minima, which captures charged particles at the minimum points of the pseudopotential, forms compact packets of them, and transports them with a single group velocity from input to output, regardless of charges (both the magnitude and sign of the charge), masses, and initial energies of ions [29–31]. This behavior of the ions entirely coincides with both the results of mathematical simulation and the observed experimental data.

However, the electric potential (1) can be written in the equivalent form

$$\begin{gathered} U\left( {x,t} \right) = \frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right)\cos \left( {\Omega + \omega } \right)t \\ + \,\,\frac{1}{2}{{U}_{0}}\sin \left( {x/L} \right)\sin \left( {\Omega + \omega } \right)t \\ - \,\,\frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right)\cos \left( {\Omega - \omega } \right)t \\ + \,\,\frac{1}{2}{{U}_{0}}\sin \left( {x/L} \right)\sin \left( {\Omega - \omega } \right)t, \\ \end{gathered} $$
((4))

that is, as the sum of four independent sinusoids with constant amplitudes in time (depending, however, on the x coordinate). The formal expression for the pseudopotential of the high-frequency field, written in the form (4), is

$${{\bar {U}}^{{rf}}}\left( {x,t} \right) = \frac{{e{{{({{U}_{0}}/L)}}^{2}}\left( {{{\Omega }^{2}} + {{\omega }^{2}}} \right)}}{{8m{{{\left( {{{\Omega }^{2}} - {{\omega }^{2}}} \right)}}^{2}}}}.$$
((5))

In this form, it is not a traveling wave transporting ions from input to output; moreover, it does not have alternating maxima and minima responsible for capturing ions and converting the initial distribution of ions into compact discrete packets located at fixed points in space.

The model of a vortex drift [12], developed to describe the motion of charged particles in high-frequency electric fields of the form (4) with close frequencies (see “Model of Vortex Forces and Vortex Drift,” Eqs. (37), (38), and (35)), predicts that for the high-frequency electric field (2), the averaged motion of ions is controlled by the pseudopotential

$${{\bar {U}}^{{rf}}}\left( {x,t} \right) = \frac{{e{{{({{U}_{0}}/L)}}^{2}}}}{{4m}}\frac{{\left( {{{\Omega }^{2}} + {{\omega }^{2}}} \right)}}{{{{{\left( {{{\Omega }^{2}} - {{\omega }^{2}}} \right)}}^{2}}}}{{\cos }^{2}}\left( {x/L - \omega t} \right)$$
((6))

(the vortex part of vortex drift equations (37) and (38) for the high-frequency electric field (2) is equal to zero in the whole space, including the axis of the transporting system). Equation (6) differs from Eq. (3) only by a factor; therefore, it qualitatively describes the same effects. It also predicts that, with an increase in frequency ω, that is, with an increase in the transportation speed, the height of the walls of local pseudopotential wells, responsible for the formation of compact ion packets and their transport, increases. This effect is confirmed neither by a numerical simulation of ion motion in a high-frequency electric field (Eqs. (1) and (4)) nor by an in-depth theoretical study of the equations of motion (see “Discussion of Results”).

Finally, for the degenerate case of ω = Ω, the high-frequency electric potential (1) takes the form of

$$\begin{gathered} U\left( {x,t} \right) = - \frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right) + \frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right)\cos 2\Omega t \\ + \,\,\frac{1}{2}{{U}_{0}}\sin \left( {x/L} \right)\sin 2\Omega t, \\ \end{gathered} $$
((7))

and the corresponding full pseudopotential (also taking into account the static component of the electric field (10)) is

$${{\bar {U}}^{{rf}}}\left( {x,t} \right) = - \frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right) + \frac{{e{{{({{U}_{0}}/L)}}^{2}}}}{{64m{{\Omega }^{2}}}}$$
((8))

and is a stationary sine wave. For the case close to degenerate, when ω = Ω – δ, where δ/Ω ⪡ 1, the high-frequency electric field (4) is written as

$$\begin{gathered} U\left( {x,t} \right) \\ = \frac{1}{2}{{U}_{0}}\left( {\cos \left( {x/L} \right)\cos \delta t - \sin \left( {x/L} \right)\sin \delta t} \right)\cos 2\Omega t \\ + \,\,\frac{1}{2}{{U}_{0}}\left( {\sin \left( {x/L} \right)\cos \delta t + \cos \left( {x/L} \right)\sin \delta t} \right)\sin 2\Omega t \\ - \,\,\frac{1}{2}{{U}_{0}}\cos \left( {x/L} \right)\cos \delta t + \frac{1}{2}{{U}_{0}}\sin \left( {x/L} \right)\sin 2\delta t, \\ \end{gathered} $$
((9))

where “fast” and “slow” frequencies are clearly identified, leads to a pseudopotential

$${{\bar {U}}^{{rf}}}\left( {x,t} \right) = - \frac{1}{2}{{U}_{0}}\cos \left( {x/L + \delta t} \right) + \frac{{e{{{({{U}_{0}}/L)}}^{2}}}}{{64m{{\Omega }^{2}}}}.$$
((10))

This expression is described a quasi-static electric potential wave, running in the opposite direction, on which a high-frequency electric field is superimposed; this field, on average, does not affect the movement of ions.

The obtained theoretical models cannot be correct simultaneously. Of course, it is possible to determine which of the listed models is correct for a particular system with specific parameters using a numerical experiment. The task, however, is to find out what should be the method for analyzing ion motion in high-frequency electric fields in order to lead to a correct answer from the very beginning.

FAST AND SLOW TIME SIGNALS

An arbitrary periodic time signal f(t) with a period of T0 = 2π/ω0 can be represented as a Fourier series, that is,

$$f\left( t \right) = {{p}_{0}} + \sum\limits_{k = 1,\infty } {\left( {{{p}_{k}}\cos k{{\omega }_{0}}t + {{q}_{k}}\sin k{{\omega }_{0}}t} \right)} ,$$
((11))

where ωk = kω0 are circular frequencies of the Fourier harmonics, and pk and qk are constants of the Fourier series expansion. A generalization of periodic time signals is provided by almost periodic time signals [47–49], which can be represented as

$$f\left( t \right) = {{p}_{0}} + \sum\limits_{k = 1,\infty } {\left( {{{p}_{k}}\cos {{\omega }_{k}}t + {{q}_{k}}\sin {{\omega }_{k}}t} \right)} ,$$
((12))

where ωk are arbitrary circular harmonics frequencies, renumbered appropriately. In particular, the whole set of rational numbers can be used as a set of harmonics. The next level in the hierarchy of generalization of signals of the form of Eq. (11) is trigonometric time signals with amplitudes slowly varying in time, which can be represented as

$$f\left( t \right) = {{p}_{0}}\left( t \right) + \sum\limits_{k = 1,\infty } {\left( {{{p}_{k}}\left( t \right)\cos {{\omega }_{k}}t + {{q}_{k}}\left( t \right)\sin {{\omega }_{k}}t} \right)} .$$
((13))

Here, it is assumed that the functions pk(t) and qk(t) are “slow” in comparison with the “fast” sinusoidal functions cos ωkt and sin ωkt. A characteristic feature of temporary signals of the form (13) is that they are characterized by two different time scales: “slow” and “fast.” Probably, we could introduce time signals with three or more time scales, by analogy with Eq. (13); however, this is beyond the goals of this work.

Equation (13) cannot be used as a formal definition of time signals with two time scales: the same time signal can be represented by completely different trigonometric functions, some of which are not of the form (13). For example,

$$\begin{gathered} {{f}_{0}}\left( t \right) = {{U}_{0}}\cos \left( {\Omega t} \right)\cos \omega t \\ = \frac{1}{2}{{U}_{0}}\cos \left( {\omega - \Omega } \right)t + \frac{1}{2}{{U}_{0}}\cos \left( {\omega + \Omega } \right)t, \\ \end{gathered} $$
((14))

can be represented both as an amplitude-modulated sinusoidal signal, in which two time scales are clearly distinguished, and as a sum of purely sinusoidal signals with two different close frequencies, in which the presence of two time scales is veiled. More complicated cases may occur, in which the fact that two completely different mathematical expressions are actually identically equal to each other requires nontrivial calculations and is not apparent.

The solution is the transition from formal expressions (13) to a description of characteristic features of the Fourier spectrum, which is an objective characteristic of time signals and is independent of their representation as a specific mathematical equation. The Fourier spectrum (Fig. 1) for signals that can be shown in the form (13) consists of the intervals (ωk – Ω0) ≤ ω ≤ (ωk + Ω0) with a relatively small width Ω0 = 2π/T0, on which the spectrum (continuous or discrete) is only nonzero. These intervals are located at a relatively large distance from each other: |ωk – ωj| ⪢ Ω0. Here, T0 is the time scale for the “slow” functions, and T = 2π/ω0 ⪡ T0 is the time scale for “fast” functions, that is, for sinusoidal harmonics with frequencies ωk ~ ω0.

Fig. 1.
figure 1

Fourier-transform spectrum for a time signal with two characteristic time scales.

Full size image

For the self-consistent theory proposed here, it is essential that time signals with the Fourier spectrum divided into narrow islands with nonzero values (Fig. 1) can be represented (among other equivalent mathematical expressions) in the form of Eq. (13) with slow functions pk(t), qk(t) and fast frequencies ωk, which are far from each other. Namely, for a single narrow “island” of the full spectrum, in shifting it to the beginning of the spectral axis ω ≈ 0, there is a prototype pk(t) + iqk(t) in the time domain, where pk(t), qk(t) are slow functions. The prototype is a complex-valued function, because the separated single section of the spectrum shifted to the origin most likely does not satisfy the symmetry requirements necessary for real time signals. To bring the spectral island back to its original position, we must multiply pk(t) + iqk(t) by exp(±iωkt), where ωk is the center of the spectral island under consideration. Summing up the individual elements and simplifying the result lead to a representation similar to Eq. (13), where the result is always real because of the symmetry of the initial Fourier spectrum.

The theory of time signals with two time scales is considered in more detail in [36, 40]. Apparently, the time signals with two time scales described above are the most common case of high-frequency electric fields, in which a pseudopotential formalism can be applied. It is further assumed that the high-frequency voltages applied to the electrodes are related to the time signals of this type, while the time signals (voltage at the electrodes) are already presented in canonical form (13) with far-reaching fast frequencies ωk.

PSEUDOPOTENTIAL MODEL WITH SEVERAL HIGH-FREQUENCY HARMONICS

Let us consider the movement of ions in a high-frequency electric field with an electric potential represented as

$$\begin{gathered} U\left( {x,y,z,t} \right) = {{U}^{0}}\left( {x,y,z,t} \right) \\ + \sum\limits_k {\left( {{{V}^{k}}\left( {x,y,z,t} \right)\cos {{\omega }_{k}}t + {{W}^{k}}\left( {x,y,z,t} \right)\sin {{\omega }_{k}}t} \right)} . \\ \end{gathered} $$
((15))

Here, ωk are circular frequencies that are fast and located quite far from each other: \(\forall k,j:\,\,\,\left| {{{\omega }_{k}} - {{\omega }_{j}}} \right| \gg 2\pi /{{T}_{0}},\)\(\forall k:\,{{\omega }_{k}} \gg 2\pi /{{T}_{0}},\) where T0 is the characteristic time of ion motion, U0(x, y, z, t) is the potential of a quasi-static electric field, V k(x, y, z, t) and W k(x, y, z, t) are the quasi-static (slow) amplitudes of the corresponding electric components of the high-frequency field with a characteristic time scale of ~T0 or slower. A high-frequency electric field similar to (15) can be created by applying time-dependent voltages to the electrodes of the system with two characteristic time scales, which were determined in the previous section, provided that frequencies ωk are small enough to neglect the Maxwell effects,Footnote 1 like electromagnetic waves. Apparently, Eq. (15) is the most general representation for time-dependent almost periodic electric fields, when the pseudopotential method can be successfully used.

It is assumed that the functions U0(x, y, z, t), V k(x, y, z, t), and W k(x, y, z, t) are slow in comparison with sinusoidal signals with fast circular frequencies: \(\left| {\partial {{U}^{0}}/\partial t} \right|\) ≤ |U0|/T0, \(\left| {\partial {{V}^{k}}/\partial t} \right|\) ≤ |V k|/T0, \(\left| {\partial {{W}^{k}}/\partial t} \right|\) ≤ |Wk|/T0, ωk ⪢ 1/T0. Since frequencies ωk are small enough so that it is not necessary to take into account the Maxwell effects, and the charges, currents, and dielectric materials are outside the region of ion motion, the Laplace equation must be satisfied for the functions U0, V k, and W k, that is

$$\begin{gathered} U_{{xx}}^{0} + U_{{yy}}^{0} + U_{{zz}}^{0} = 0,\,\,\,\,V_{{xx}}^{k} + V_{{yy}}^{k} + V_{{zz}}^{k} = 0, \\ W_{{xx}}^{k} + W_{{yy}}^{k} + W_{{zz}}^{k} = 0. \\ \end{gathered} $$
((16))

The subscripts hereafter denote partial derivatives with respect to the corresponding variables.

The Newtonian equations of motion for an ion with mass m and charge e in an electric field (15) have the form of

$$\begin{gathered} \left( {m/e} \right)\ddot {x} = - U_{x}^{0} - \sum\limits_k {\left( {V_{x}^{k}\cos {{\omega }_{k}}t + W_{x}^{k}\sin {{\omega }_{k}}t} \right),} \\ \left( {m/e} \right)\ddot {y} = - U_{y}^{0} - \sum\limits_k {\left( {V_{y}^{k}\cos {{\omega }_{k}}t + W_{y}^{k}\sin {{\omega }_{k}}t} \right),} \\ \left( {m/e} \right)\ddot {z} = - U_{z}^{0} - \sum\limits_k {\left( {V_{z}^{k}\cos {{\omega }_{k}}t + W_{z}^{k}\sin {{\omega }_{k}}t} \right).} \\ \end{gathered} $$
((17))

The pseudopotential approach to the description of the motion of ions suggests that the solution of Eqs. (17) can be represented as

$$\begin{gathered} x\left( t \right) = {{x}_{0}}\left( t \right) + \delta x\left( t \right), \\ \delta x\left( t \right) \approx \sum\limits_k {x_{k}^{c}\left( t \right)\cos {{\omega }_{k}}t} + \sum\limits_k {x_{k}^{s}\left( t \right)\sin {{\omega }_{k}}t} \\ + \,\,\sum\limits_k {x_{k}^{{2c}}\left( t \right)\cos 2{{\omega }_{k}}t} + \sum\limits_k {x_{k}^{{2s}}\left( t \right)\sin 2{{\omega }_{k}}t} \\ + \,\,\sum\limits_{k > j} {x_{{k,j}}^{{cp}}\left( t \right)\cos \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right)t} + \sum\limits_{k > j} {x_{{k,j}}^{{sp}}\left( t \right)\sin \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right){\kern 1pt} t} \\ + \,\sum\limits_{k > j} {x_{{k,j}}^{{cm}}\left( t \right){\kern 1pt} \cos {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t} \, + \,\sum\limits_{k > j} {x_{{k,j}}^{{sm}}{\kern 1pt} \left( t \right){\kern 1pt} \sin {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t\, + \, \cdots ,} \\ \end{gathered} $$
((18))
$$\begin{gathered} y\left( t \right) = {{y}_{0}}\left( t \right) + \delta y\left( t \right), \\ \delta y\left( t \right) \approx \sum\limits_k {y_{k}^{c}\left( t \right)\cos {{\omega }_{k}}t} + \sum\limits_k {y_{k}^{s}\left( t \right)\sin {{\omega }_{k}}t} \\ + \,\,\sum\limits_k {y_{k}^{{2c}}\left( t \right)\cos 2{{\omega }_{k}}t} + \sum\limits_k {y_{k}^{{2s}}\left( t \right)\sin 2{{\omega }_{k}}t} \\ + \,\,\sum\limits_{k > j} {y_{{k,j}}^{{cp}}\left( t \right)\cos \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right)t} + \sum\limits_{k > j} {y_{{k,j}}^{{sp}}\left( t \right)\sin \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right){\kern 1pt} t} \\ + \,\sum\limits_{k > j} {y_{{k,j}}^{{cm}}\left( t \right){\kern 1pt} \cos {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t} \, + \,\sum\limits_{k > j} {y_{{k,j}}^{{sm}}{\kern 1pt} \left( t \right){\kern 1pt} \sin {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t\, + \, \cdots ,} \\ \end{gathered} $$
((19))
$$\begin{gathered} z\left( t \right) = {{z}_{0}}\left( t \right) + \delta z\left( t \right), \\ \delta z\left( t \right) \approx \sum\limits_k {z_{k}^{c}\left( t \right)\cos {{\omega }_{k}}t} + \sum\limits_k {z_{k}^{s}\left( t \right)\sin {{\omega }_{k}}t} \\ + \,\,\sum\limits_k {z_{k}^{{2c}}\left( t \right)\cos 2{{\omega }_{k}}t} + \sum\limits_k {z_{k}^{{2s}}\left( t \right)\sin 2{{\omega }_{k}}t} \\ + \,\,\sum\limits_{k > j} {z_{{k,j}}^{{cp}}\left( t \right)\cos \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right)t} + \sum\limits_{k > j} {z_{{k,j}}^{{sp}}\left( t \right)\sin \left( {{{\omega }_{k}} + {{\omega }_{j}}} \right){\kern 1pt} t} \\ + \,\sum\limits_{k > j} {z_{{k,j}}^{{cm}}\left( t \right){\kern 1pt} \cos {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t} \, + \,\sum\limits_{k > j} {z_{{k,j}}^{{sm}}{\kern 1pt} \left( t \right){\kern 1pt} \sin {\kern 1pt} \left( {{{\omega }_{k}}\, - \,{{\omega }_{j}}} \right){\kern 1pt} t\, + \, \cdots ,} \\ \end{gathered} $$
((20))

where the high-frequency corrections δx(t), δy(t), and δz(t) are sufficiently small compared to x0(t), y0(t), and z0(t), and the functions x0(t), \(x_{k}^{c}\left( t \right),x_{k}^{s}\left( t \right),x_{k}^{{2c}}\left( t \right),x_{k}^{{2s}}\left( t \right)\), etc. are slow in comparison with the rapidly oscillating functions cos ωkt and sin ωkt.

Let the frequencies of sinusoidal signals be presented in the form of ωk = Ωkω, where Ωk are dimensionless factors of ~1, and parameter ω ⪢ 1/T0 characterizes the scale of the “speed” of sinusoidal oscillations (T0 is the characteristic time scale for the slow part of x0(t), y0(t), and z0(t) of ion trajectories). It is natural to assume that at ω → ∞, the high-frequency correcting factors δx(t), δy(t), and δz(t) tend to zero because of the inertia of the motion of ions with nonzero masses (at very high frequencies of the electric field, the ions vibrate in one place, not having time to shift one way or another). This means not only that each of the functions \(x_{k}^{c}\left( t \right),x_{k}^{s}\left( t \right),x_{k}^{{2c}}\left( t \right),x_{k}^{{2s}}\left( t \right)\), … tends to zero as ω → ∞, but also that these functions can be expanded in a Taylor series in a neighborhood of the point ω = ∞ in powers of the parameter 1/ω, more precisely, in powers of the small dimensionless parameter 1/(ω T0), as

$$\begin{gathered} x_{k}^{c}\left( t \right) \approx \frac{1}{\omega }x_{k}^{{\left( {c,1} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}x_{k}^{{\left( {c,2} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}x_{k}^{{\left( {c,3} \right)}}\left( t \right) \\ + \,\,\frac{1}{{{{\omega }^{4}}}}x_{k}^{{\left( {c,4} \right)}}\left( t \right) + \cdots , \\ x_{k}^{s}\left( t \right) \approx \frac{1}{\omega }x_{k}^{{\left( {s,1} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}x_{k}^{{\left( {s,2} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}x_{k}^{{\left( {s,3} \right)}}\left( t \right) \\ + \,\,\frac{1}{{{{\omega }^{4}}}}x_{k}^{{\left( {s,4} \right)}}\left( t \right) + \cdots , \\ x_{k}^{{2c}}\left( t \right) \approx \frac{1}{\omega }x_{k}^{{\left( {2c,1} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}x_{k}^{{\left( {2c,2} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}x_{k}^{{\left( {2c,3} \right)}}\left( t \right) \\ + \,\,\frac{1}{{{{\omega }^{4}}}}x_{k}^{{\left( {2c,4} \right)}}\left( t \right) + \cdots . \\ x_{k}^{{2s}}\left( t \right) \approx \frac{1}{\omega }x_{k}^{{\left( {2s,1} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}x_{k}^{{\left( {2s,2} \right)}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}x_{k}^{{\left( {2s,3} \right)}}\left( t \right) \\ + \,\,\frac{1}{{{{\omega }^{4}}}}x_{k}^{{\left( {2s,4} \right)}}\left( t \right) + \cdots . \\ \end{gathered} $$
((21))

Similar decompositions are used for \(y_{k}^{c}\left( t \right),\)\(y_{k}^{s}\left( t \right),\)\(y_{k}^{{2c}}\left( t \right),\)\(y_{k}^{{2s}}\left( t \right),\) … and \(z_{k}^{c}\left( t \right),\)\(z_{k}^{s}\left( t \right),\)\(z_{k}^{{2c}}\left( t \right),\)\(z_{k}^{{2s}}\left( t \right),\) …. However, upon twofold differentiation of Eqs. (18)–(20) with respect to t, the values on the right-hand side must be of the order of unity in accordance with Eqs. (17); therefore, the first nonzero term of expansions (21)–(Q14) is of the order of at least 1/ω2. Since the corrective factors δx(t), δy(t), and δz(t) are considered small in comparison with the main values of x0(t), y0(t), and z0(t), it is assumed that in Eqs. (17), we can replace functions \(U_{x}^{0},U_{y}^{0},U_{z}^{0},V_{x}^{k},V_{y}^{k},V_{z}^{k},W_{x}^{k},W_{y}^{k},W_{z}^{k}\) with truncated Taylor series in a neighborhood of point x0(t), y0(t), z0(t), that is,

$$\begin{gathered} U_{x}^{0}\left( {x,y,z,t} \right) \approx U_{x}^{0}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta xU_{{xx}}^{0}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\delta yU_{{xy}}^{0}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta zU_{{xz}}^{0}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\left( {\delta {{x}^{2}}/2} \right){\kern 1pt} U_{{xxx}}^{0}{\kern 1pt} \left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right)\, + \,\delta x\delta yU_{{xxy}}^{0}{\kern 1pt} \left( {{{x}_{0}},\,{{y}_{0}},{{z}_{0}},t} \right)\, + \ldots , \\ V_{x}^{k}\left( {x,y,z,t} \right) \approx V_{x}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta xV_{{xx}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\delta yV_{{xy}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta zV_{{xz}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\left( {\delta {{x}^{2}}/2} \right){\kern 1pt} V_{{xxx}}^{k}{\kern 1pt} \left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right)\, + \,\delta x\delta yV_{{xxy}}^{k}{\kern 1pt} \left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \ldots , \\ W_{x}^{k}\left( {x,y,z,t} \right) \approx W_{x}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta xW_{{xx}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\delta yW_{{xy}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) + \delta zW_{{xz}}^{k}\left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right) \\ + \,\,\left( {\delta {{x}^{2}}/2} \right){\kern 1pt} W_{{xxx}}^{k}{\kern 1pt} \left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right)\, + \,\delta x\delta y{\kern 1pt} W_{{xxy}}^{k}{\kern 1pt} \left( {{{x}_{0}},{{y}_{0}},{{z}_{0}},t} \right)\, + \, \ldots , \\ \ldots . \\ \end{gathered} $$
((22))

Equations (17)–(22) are sufficient if pseudopotential expansion in powers of at most 1/ω2 is required for Eqs. (17). However, if correct expressions of higher order are needed, one more mathematical trick is required. Based on general assumptions, self-sufficient differential equations for the functions x0(t), y0(t), z0(t) should be obtained,

$$\begin{gathered} x_{0}^{{''}}\left( t \right) \approx {{X}_{0}}\left( t \right) + \frac{1}{\omega }{{X}_{1}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}{{X}_{2}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}{{X}_{3}}\left( t \right) \cdots , \\ y_{0}^{{''}}\left( t \right) \approx {{Y}_{0}}\left( t \right) + \frac{1}{\omega }{{Y}_{1}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}{{Y}_{2}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}{{Y}_{3}}\left( t \right) \cdots , \\ z_{0}^{{''}}\left( t \right) \approx {{Z}_{0}}\left( t \right) + \frac{1}{\omega }{{Z}_{1}}\left( t \right) + \frac{1}{{{{\omega }^{2}}}}{{Z}_{2}}\left( t \right) + \frac{1}{{{{\omega }^{3}}}}{{Z}_{3}}\left( t \right) \cdots , \\ \end{gathered} $$
((23))

where the functions Xk(t), Yk(t), Zk(t) are expressed through the coordinates x0(t), y0(t), z0(t), velocities \(x_{0}^{'}\left( t \right)\), \(y_{0}^{'}\left( t \right)\), \(z_{0}^{'}\left( t \right)\), and characteristics of the electric field. Therefore, Eqs. (23) can be written in an indefinite form from the very beginning, and with their help, the highest time derivatives \(x_{0}^{{''}}\left( t \right)\), \(y_{0}^{{''}}\left( t \right)\), \(z_{0}^{{''}}\left( t \right)\), \(x_{0}^{{'''}}\left( t \right)\), \(y_{0}^{{'''}}\left( t \right)\), \(z_{0}^{{'''}}\left( t \right)\), etc. can be eliminated from the resulting equations. Only then we can find the real form functions Xk(t), Yk(t), Zk(t) from the obtained equations. Fortunately, there are no cyclic references of functions to themselves in this way, so that unknown functions (17)–(23) can be expressed step-by-step in an algebraic form in an explicit form through known data. In order to avoid mistakes, especially in calculating high-order terms, it is essential to consider the values \(x_{k}^{c},x_{k}^{s},x_{k}^{{2c}},x_{k}^{{2s}},\) …, Xk, Yk, Zk as unknown slow functions that depend on seven variables x0(t), y0(t), z0(t), \(x_{0}^{'}\left( t \right)\), \(y_{0}^{'}\left( t \right)\), \(z_{0}^{'}\left( t \right)\), and t. The reason is that in differentiating Eqs. (18)–(20) with respect to t and eliminating the highest derivatives \(x_{0}^{{''}}\left( t \right)\), \(y_{0}^{{''}}\left( t \right)\), \(z_{0}^{{''}}\left( t \right)\), \(x_{0}^{{'''}}\left( t \right)\), \(y_{0}^{{'''}}\left( t \right)\), \(z_{0}^{{'''}}\left( t \right)\) with the help of Eqs. (23), additional degrees of the form 1/ωk arise, which must be correctly grouped with each other.

Now we can combine Eqs. (17)–(23) together. Using the program for algebraic calculations of the type [50], we can determine the coefficients at the linearly independent trigonometric terms with different frequencies and different powers of ω without errors. Moreover, the products and degrees of trigonometric functions are replaced by linear combinations of trigonometric functions according to well-known trigonometric equations, and trigonometric functions of the form cos ωkt, sin ωkt, cos 2ωkt, sin 2ωkt, cos(ωk ± ωj)t, sin(ωk ± ωj)t, … are considered linearly independent; although, this is not true, for example, for a degenerate case of Fourier series ωk = kω. Here, the logic of reasoning is as follows: when small perturbations are introduced into the frequencies of the Fourier series ωk = kω, where all ωk = kω + δωk, are small and arbitrary, the indicated trigonometric terms are really linearly independent. After that, the limit transition of δωk → 0 can be used for the resulting equations, having obtained fully functional equations for the degenerate case from the correct and only possible equations, even though these equations could be written in other forms. Nonzero coefficients in the expansions (18)–(21) are

$$\begin{gathered} x_{k}^{{\left( {c,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}V_{x}^{k},\,\,\,\,x_{k}^{{\left( {s,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}W_{x}^{k}, \\ x_{k}^{{\left( {c,3} \right)}}\left( t \right) = \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {W_{{xt}}^{k} + x_{0}^{'}\left( t \right)W_{{xx}}^{k} + y_{0}^{'}\left( t \right)W_{{xy}}^{k} + z_{0}^{'}\left( t \right)W_{{xz}}^{k}} \right), \\ x_{k}^{{\left( {s,3} \right)}}\left( t \right) = - \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {V_{{xt}}^{k} + x_{0}^{'}\left( t \right)V_{{xx}}^{k} + y_{0}^{'}\left( t \right)V_{{xy}}^{k} + z_{0}^{'}\left( t \right)V_{{xz}}^{k}} \right), \\ \end{gathered} $$
((24))
$$\begin{gathered} y_{k}^{{\left( {c,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}V_{y}^{k},\,\,\,\,y_{k}^{{\left( {s,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}W_{y}^{k}, \\ y_{k}^{{\left( {c,3} \right)}}\left( t \right) = \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {W_{{yt}}^{k} + x_{0}^{'}\left( t \right)W_{{xy}}^{k} + y_{0}^{'}\left( t \right)W_{{yy}}^{k} + z_{0}^{'}\left( t \right)W_{{yz}}^{k}} \right), \\ y_{k}^{{\left( {s,3} \right)}}\left( t \right) = - \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {V_{{yt}}^{k} + x_{0}^{'}\left( t \right)V_{{xy}}^{k} + y_{0}^{'}\left( t \right)V_{{yy}}^{k} + z_{0}^{'}\left( t \right)V_{{yz}}^{k}} \right), \\ \end{gathered} $$
((25))
$$\begin{gathered} z_{k}^{{\left( {c,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}V_{z}^{k},\,\,\,\,z_{k}^{{\left( {s,2} \right)}}\left( t \right) = \frac{e}{{m\Omega _{k}^{2}}}W_{z}^{k}, \\ z_{k}^{{\left( {c,3} \right)}}\left( t \right) = \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {W_{{zt}}^{k} + x_{0}^{'}\left( t \right)W_{{xz}}^{k} + y_{0}^{'}\left( t \right)W_{{yz}}^{k} + z_{0}^{'}\left( t \right)W_{{zz}}^{k}} \right), \\ z_{k}^{{\left( {s,3} \right)}}\left( t \right) = - \frac{{2e}}{{m\Omega _{k}^{3}}} \\ \times \,\,\left( {V_{{zt}}^{k} + x_{0}^{'}\left( t \right)V_{{zy}}^{k} + y_{0}^{'}\left( t \right)V_{{zy}}^{k} + z_{0}^{'}\left( t \right)V_{{zz}}^{k}} \right), \\ \end{gathered} $$
((26))

where functions x0(t), y0(t), z0(t) replace arguments x, y, z for electric potentials and their partial derivatives.

Functions Xk(t), Yk(t), Zk(t), which are used in the “averaged” equations (23), for degrees not higher than 1/ω2 have the form of

$${{X}_{0}} = - \frac{e}{m}U_{x}^{0},\,\,\,\,{{Y}_{0}} = - \frac{e}{m}U_{y}^{0},\,\,\,\,{{Z}_{0}} = - \frac{e}{m}U_{z}^{0},$$
((27))
$$\frac{1}{\omega }{{X}_{1}} = 0,\,\,\,\,\frac{1}{\omega }{{Y}_{1}} = 0,\,\,\,\,\frac{1}{\omega }{{Z}_{1}} = 0,$$
((28))
$$\begin{gathered} \frac{1}{{{{\omega }^{2}}}}{{X}_{2}} = - \sum\limits_k {\frac{{{{e}^{2}}}}{{2{{m}^{2}}\omega _{k}^{2}}}} \left( {V_{x}^{k}V_{{xx}}^{k} + V_{y}^{k}V_{{xy}}^{k}} \right. + V_{z}^{k}V_{{xz}}^{k} \\ + \,\,\left. {W_{x}^{k}W_{{xx}}^{k} + W_{y}^{k}W_{{xy}}^{k} + W_{z}^{k}W_{{xz}}^{k}} \right), \\ \frac{1}{{{{\omega }^{2}}}}{{Y}_{2}} = - \sum\limits_k {\frac{{{{e}^{2}}}}{{2{{m}^{2}}\omega _{k}^{2}}}} \left( {V_{x}^{k}V_{{xy}}^{k} + V_{y}^{k}V_{{yy}}^{k}} \right. + V_{z}^{k}V_{{yz}}^{k} \\ + \,\,\left. {W_{x}^{k}W_{{xy}}^{k} + W_{y}^{k}W_{{yy}}^{k} + W_{z}^{k}W_{{yz}}^{k}} \right), \\ \frac{1}{{{{\omega }^{2}}}}{{Z}_{2}} = - \sum\limits_k {\frac{{{{e}^{2}}}}{{2{{m}^{2}}\omega _{k}^{2}}}} \left( {V_{x}^{k}V_{{xz}}^{k} + V_{y}^{k}V_{{yz}}^{k}} \right. + V_{z}^{k}V_{{zz}}^{k} \\ + \,\,\left. {W_{x}^{k}W_{{xz}}^{k} + W_{y}^{k}W_{{yz}}^{k} + W_{z}^{k}W_{{zz}}^{k}} \right). \\ \end{gathered} $$
((29))

Functions (27) are the quasi-static part of Eqs. (17). Functions (28), proportional to 1/ω, are zero. Functions (29), proportional to 1/ω2 can be represented as

$$\begin{gathered} \frac{1}{{{{\omega }^{2}}}}{{X}_{2}} = - \frac{e}{m}\frac{{\partial {{{\bar {U}}}^{{rf}}}\left( {{{x}_{0}}\left( t \right),{{y}_{0}}\left( t \right),{{z}_{0}}\left( t \right),t} \right)}}{{\partial x}}, \\ \frac{1}{{{{\omega }^{2}}}}{{Y}_{2}} = - \frac{e}{m}\frac{{\partial {{{\bar {U}}}^{{rf}}}\left( {{{x}_{0}}\left( t \right),{{y}_{0}}\left( t \right),{{z}_{0}}\left( t \right),t} \right)}}{{\partial y}}, \\ \frac{1}{{{{\omega }^{2}}}}{{Z}_{2}} = - \frac{e}{m}\frac{{\partial {{{\bar {U}}}^{{rf}}}\left( {{{x}_{0}}\left( t \right),{{y}_{0}}\left( t \right),{{z}_{0}}\left( t \right),t} \right)}}{{\partial z}}, \\ \end{gathered} $$
((30))

where \({{\bar {U}}^{{rf}}}\left( {x,y,z,t} \right)\) is given by the equation

$$\begin{gathered} {{{\bar {U}}}^{{rf}}} = \sum\limits_k {\frac{e}{{4m\omega _{k}^{2}}}\left( {{{{\left( {V_{k}^{x}} \right)}}^{2}} + {{{\left( {V_{y}^{k}} \right)}}^{2}} + {{{\left( {V_{z}^{k}} \right)}}^{2}}} \right.} \\ + \,\,\left. {{{{\left( {W_{x}^{k}} \right)}}^{2}} + {{{\left( {W_{y}^{k}} \right)}}^{2}} + {{{\left( {W_{z}^{k}} \right)}}^{2}}} \right). \\ \end{gathered} $$
((31))

This is the conventional equation for the pseudopotential of the electric field [1–3, 8, 10, 12–15], but here, it is applied to the general case of multicomponent near-periodic high-frequency fields with slowly changing amplitudes of the form (15). Checking that Eq. (31) still works under new conditions is an important result; it could be expected that because of the interaction of multiple harmonics of the Fourier series and/or different harmonics of almost periodic high-frequency fields, additional terms for X2, Y2, Z2 may appear. In particular, in averaging the equations of motion with the conservation of terms of the order of 1/ω4, this happens: the result of the combined action of several harmonics of the high-frequency field is not the same as the sum of the effects of each harmonic separately.

Equations (23), (27)–(29) are self-sufficient differential equations that contain only x0(t), y0(t), z0(t) as unknown functions. After substituting the solutions of Eqs. (23) into Eqs. (18)–(20) and (24)–(26), we can calculate the correcting high-frequency factors δx(t), δy(t), δz(t) and, as a result, approximate trajectories x(t) = x0(t) + δx(t), y(t) = y0(t) + δy(t), z(t) = z0(t) + δz(t) for the movement of an ion in a high-frequency electric field. Thus, the pseudopotential model consists of two equal parts: the averaged equations (23), (27)–(31) and Eqs. (18)–(20) and (24)–(26) for approximate ion trajectories, expressed through averaged movement. Although, in practice, they are often limited to analyzing only the averaged part, neglecting the high-frequency corrections δx(t), δy(t), δz(t).

In deriving the equations of the pseudopotential model, it is assumed that the harmonics cos(ωk ± ωj)t and sin(ωk ± ωj)t are fast functions (that is, that the fast frequencies ωk are far apart from each other), and the quasi-static part U0(x, y, z, t) of the electric field and amplitudes V k(x, y, z, t), W k(x, y, z, t) of the high-frequency harmonics of the electric field are “slow” functions. If these conditions are violated, the equations of the pseudopotential model cease to work, despite the fact that, for example, denominators of the form ωk ± ωj, explicitly indicating the divergence of the model when the frequencies of fast harmonics approach each other, are absent in the equations for the averaged equations of motion (although they appear, for example, in taking into account corrections of the type of 1/ω4 in the equations for δx(t), δy(t), δz(t), which leads to the divergence of a pseudopotential solution.

VORTEX FORCE AND VORTEX DRIFT MODEL

Let us consider how the result correlates with the vortex drift model [12, ch. 2, sect. 2] in a high-frequency electric field, where an alternative model of the motion of charged particles in high-frequency electric fields with close frequencies is proposed. Let the electric field be represented as

$$\begin{gathered} \vec {E}\left( {\vec {r},t} \right) = - \nabla {{U}^{0}}\left( {\vec {r},t} \right) - \sum\limits_k {\left( {\nabla {{V}^{k}}\left( {\vec {r},t} \right)\cos {{\omega }_{k}}t} \right.} \\ + \,\,\left. {\nabla {{W}^{k}}\left( {\vec {r},t} \right)\sin {{\omega }_{k}}t} \right) = {{{\vec {E}}}_{0}}\left( {\vec {r},t} \right) + {{{\vec {E}}}_{{rf}}}\left( {\vec {r},t} \right), \\ \end{gathered} $$
((32))

where \({{U}^{0}}\left( {\vec {r},t} \right),{{V}^{k}}\left( {\vec {r},t} \right),{{W}^{k}}\left( {\vec {r},t} \right)\) are the slow potentials satisfying the three-dimensional Laplace equation, frequencies ωk of sinusoidal harmonics are fast, but no restrictions are imposed on the relative position of ωk frequencies, in contrast to the previous case. Following [12], the Newtonian equations of motion \(\ddot {\vec {r}} = \left( {e/m} \right)\vec {E}\left( {\vec {r},t} \right)\) for an ion with mass m and charge e can be approximately written up to terms of the form of 1/\(\omega _{k}^{4}\) as

$$\ddot {\vec {r}} + \ddot {\vec {\rho }} \approx \frac{e}{m}\vec {E}\left( {{{{\vec {r}}}_{0}},t} \right) + \frac{e}{m}\left( {\vec {\rho } \cdot \vec {\nabla }} \right)\vec {E}\left( {{{{\vec {r}}}_{0}},t} \right) + \cdots ,$$
((33))

where \({{\vec {r}}_{0}}\left( t \right)\) is the slow (main) part of the ion trajectory, \(\vec {\rho }\left( t \right)\sim 1/\omega _{k}^{2}\) is the fast-but-small (with respect to \({{\vec {r}}_{0}}\left( t \right)\)) additive correction to the main component of the ion trajectory. The strength \(\vec {E}\left( {\vec {r},t} \right)\) of the high-frequency electric field is presented as a truncated Taylor series at point \({{\vec {r}}_{0}}\left( t \right)\) with the linear terms of relatively small increment \(\vec {\rho }\left( t \right)\) remaining the same.

    We can assume that the high-frequency part \({{\vec {E}}_{{rf}}}\left( {\vec {r},t} \right)\) of the equations of motion is proportional to some parameter ε, so that at ε → 0, when the high-frequency component of the field disappears, the averaged equations of motion go over to the equations of quasi-static motion \({{\ddot {\vec {r}}}_{0}} = \left( {e/m} \right){{\vec {E}}_{0}}\left( {{{{\vec {r}}}_{0}},t} \right)\). Then, \(\vec {\rho }\left( t \right)\) can be represented as a power series with respect to parameter ε, where the main part of \(\vec {\rho }\left( t \right)\) is proportional to ε and, therefore, must satisfy the equation

$$\begin{gathered} \ddot {\vec {\rho }} \approx - \frac{e}{m}\left( {\vec {\rho } \cdot \vec {\nabla }} \right)\left( {\nabla {{U}^{0}}\left( {{{{\vec {r}}}_{0}},t} \right)} \right) \\ - \,\,\frac{e}{m}\sum\limits_k {\left( {\nabla {{V}^{k}}\left( {{{{\vec {r}}}_{0}},t} \right)\cos {{\omega }_{k}}t + \nabla {{W}^{k}}\left( {{{{\vec {r}}}_{0}},t} \right)\sin {{\omega }_{k}}t} \right)} . \\ \end{gathered} $$
((34))

Equation (34) generates an approximate, rapidly oscillating solution

$$\begin{gathered} \vec {\rho }\left( {{{{\vec {r}}}_{0}},t} \right) \approx \frac{e}{m}\sum\limits_k {\frac{1}{{\omega _{k}^{2}}}} \\ \times \,\,\left( {\nabla {{V}^{k}}\left( {{{{\vec {r}}}_{0}},t} \right)\cos {{\omega }_{k}}t + \nabla {{W}^{k}}\left( {{{{\vec {r}}}_{0}},t} \right)\sin {{\omega }_{k}}t} \right) + \cdots , \\ \end{gathered} $$
((35))

constructed in the form of a series in powers of \(1/\omega _{k}^{n}\) and trigonometric functions, where only quadratic (principal) terms of the form \(1/\omega _{k}^{2}\) remain. Equations (32) and (35) give the relations div \(\vec {E}\) = 0, rot \(\vec {E}\) = 0, div \(\vec {\rho }\) = 0, rot \(\vec {\rho }\) = 0 (in accordance with Eqs. (18)–(21) and (24)–(26), the conditions div \(\vec {\rho }\) = 0, rot \(\vec {\rho }\) = 0 are satisfied if we take into account not only the quadratic terms \(1/\omega _{k}^{2}\) for \(\vec {\rho }\)(t) but also the cubic terms \(1/\omega _{k}^{3}\), when \(\vec {\rho }\) and \(\vec {E}\) are no longer proportional to each other). These relations together with the identity

$$\begin{gathered} \left( {\vec {\rho } \cdot \vec {\nabla }} \right)\vec {E} \equiv \frac{1}{2}{\text{grad}}\left( {\vec {E} \cdot \vec {\rho }} \right) + \frac{1}{2}{\text{rot}}\left[ {\vec {E} \times \vec {\rho }} \right] + \frac{1}{2}\left( {{\text{div}}\vec {E}} \right)\vec {\rho } \\ - \,\,\frac{1}{2}\left( {{\text{div}}\vec {\rho }} \right)\vec {E} - \frac{1}{2}\left[ {\vec {E} \times {\text{rot}}\vec {\rho }} \right] - \frac{1}{2}\left[ {\vec {\rho } \times {\text{rot}}\vec {E}} \right] \\ \end{gathered} $$
((36))

after its substitution into Eq. (33) and averaging over the fast frequencies, give the averaged equations of motion for the slow part \({{\vec {r}}_{0}}\left( t \right)\) of the ion trajectory, that is,

$${{\ddot {\vec {r}}}_{0}} \approx - \frac{e}{m}\left( {\nabla {{U}^{0}}\left( {{{{\vec {r}}}_{0}},t} \right) + \nabla \Phi \left( {{{{\vec {r}}}_{0}},t} \right) + {\text{rot}}\vec {A}\left( {{{{\vec {r}}}_{0}},t} \right)} \right),$$
((37))
$$\begin{gathered} \Phi \left( {\vec {r},t} \right) = - \frac{1}{2}\overline {\left( {{{{\vec {E}}}_{{rf}}}\left( {\vec {r},t} \right) \cdot \vec {\rho }\left( {\vec {r},t} \right)} \right)} , \\ \vec {A}\left( {\vec {r},t} \right) = - \frac{1}{2}\overline {\left[ {{{{\vec {E}}}_{{rf}}}\left( {\vec {r},t} \right) \times \vec {\rho }\left( {\vec {r},t} \right)} \right]} , \\ \end{gathered} $$
((38))

where the horizontal stroke at the top indicates averaging over high-frequency oscillations. Equation (37) predicts the emergence of a slow motion r0(t) of nonpotential (vortex) forces in addition to the usual pseudopotential forces if rot \(\vec {A}\) ≠ 0. This effect was called vortex drift [12].

Let us suggest that all frequencies ωk ± ωj are fast. Then, after averaging over fast frequencies, Eqs. (38) take the form of

$$\begin{gathered} \Phi = \sum\limits_k {\frac{e}{{4m\omega _{k}^{2}}}\left( {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{k}}} \right) + \left( {\nabla {{W}^{k}} \cdot \nabla {{W}^{k}}} \right)} \right)} , \\ \vec {A} = \sum\limits_k {\frac{e}{{4m\omega _{k}^{2}}}\left( {\left[ {\nabla {{V}^{k}} \times \nabla {{V}^{k}}} \right] + \left[ {\nabla {{W}^{k}} \times \nabla {{W}^{k}}} \right]} \right)} = 0. \\ \end{gathered} $$
((39))

The scalar potential Φ in Eq. (39) coincides with the pseudopotential \({{\bar {U}}^{{rf}}}\) from Eq. (31), and the vector potential \(\vec {A}\), which is responsible for the vortex part of the averaged equations of motion (37), becomes zero. In such high-frequency electric fields, the effect of vortex drift is absent [12].

Let \({{\omega }_{k}} = \omega + {{\delta }_{k}}\), where ω is the fast frequency and δk are the slow additives. Then, after averaging over fast frequencies, Eqs. (38) take the form of

$$\begin{gathered} \Phi = \sum\limits_k {\frac{e}{{4m{{{\left( {\omega + {{\delta }_{k}}} \right)}}^{2}}}}\left[ {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{k}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{k}}} \right)} \right.} \\ + \,\,\sum\limits_{k \ne j} {\frac{e}{{4m{{{\left( {\omega + {{\delta }_{k}}} \right)}}^{2}}}}\left[ {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{j}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{j}}} \right)} \right.} \\ \times \,\,\cos \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t + \left( {\nabla {{V}^{k}} \cdot \nabla {{W}^{j}} - \nabla {{V}^{j}} \cdot \nabla {{W}^{k}}} \right) \\ \left. {^{{}} \times \,\,\sin \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t} \right], \\ \end{gathered} $$
((40))
$$\begin{gathered} \vec {A} = \sum\limits_{k \ne j} {\frac{e}{{4m{{{\left( {\omega + {{\delta }_{k}}} \right)}}^{2}}}}\left[ {\left( {\nabla {{V}^{k}} \times \nabla {{V}^{j}} + \nabla {{W}^{k}} \times \nabla {{W}^{j}}} \right)} \right.} \\ \times \,\,\cos \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t + \left( {\nabla {{V}^{k}} \times \nabla {{W}^{j}} + \nabla {{V}^{j}} \cdot \nabla {{W}^{k}}} \right) \\ \left. {^{{}} \times \,\,\sin \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t} \right]. \\ \end{gathered} $$
((41))

If a high-frequency electric field has several fast frequencies, then the scalar potential Φ and the vector potential \(\vec {A}\) are the sums of equations of the form of Eqs. (40) and (41) over all fast frequencies. Equations (40) and (41) can be reduced to an elegant symmetrized form if summed separately over k > j and k < j (with transposition of the notation k → j, j → k of the summation indices for k > j).

However, the electric field presented in the form

$$\begin{gathered} \vec {E}\left( {\vec {r},t} \right) = - \nabla {{U}^{0}} - \sum\limits_k {\left( {\nabla {{V}^{k}}\cos \left( {\omega + {{\delta }_{k}}} \right)t} \right.} \\ + \,\,\left. {\nabla {{W}^{k}}\sin {{{\left( {\omega + \delta } \right)}}_{k}}t} \right), \\ \end{gathered} $$
((42))

can be written in a mathematically equivalent form as

$$\begin{gathered} \vec {E}\left( {\vec {r},t} \right) = - \nabla U - \cos \omega t\sum\limits_k {\left( {\nabla {{V}^{k}}\cos {{\delta }_{k}}t} \right.} \\ + \,\,\left. {\nabla {{W}^{k}}\sin {{\delta }_{k}}t} \right) - \sin \omega t\sum\limits_k {\left( { - \nabla {{V}^{k}}\sin {{\delta }_{k}}t} \right.} \\ + \,\,\left. {\nabla {{W}^{k}}\cos {{\delta }_{k}}t} \right), \\ \end{gathered} $$
((43))

where δk are slow frequencies. Using Eq. (43) to construct solution (35), after averaging Eqs. (38) over the fast frequency ω, we obtain that \(\vec {A}\) = 0 and

$$\begin{gathered} \Phi = \frac{e}{{4m{{\omega }^{2}}}}\sum\limits_k {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{k}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{k}}} \right)} \\ + \,\,\frac{e}{{4m{{\omega }^{2}}}}\sum\limits_{k < j} {\left[ {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{j}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{j}}} \right)} \right.} \\ \times \,\,\cos \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t + \left( {\nabla {{V}^{k}} \cdot \nabla {{W}^{j}} - \nabla {{V}^{j}} \cdot \nabla {{W}^{k}}} \right) \\ \left. {^{{}} \times \,\,\sin \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t} \right]. \\ \end{gathered} $$
((44))

Since δk ⪡ ω, the difference between (44) and (40), (41) has the form of

$$\begin{gathered} \delta \Phi \approx \frac{e}{{2m{{\omega }^{2}}}}\sum\limits_k {\frac{{{{\delta }_{k}}}}{\omega }\left[ {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{k}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{k}}} \right)} \right.} \\ + \,\,\frac{e}{{2m{{\omega }^{2}}}}\sum\limits_{k < j} {\frac{{\left( {{{\delta }_{k}} + {{\delta }_{j}}} \right)}}{\omega }\left[ {\left( {\nabla {{V}^{k}} \cdot \nabla {{V}^{j}} + \nabla {{W}^{k}} \cdot \nabla {{W}^{j}}} \right)} \right.} \\ \times \,\,\cos \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t + \left( {\nabla {{V}^{k}} \cdot \nabla {{W}^{j}} - \nabla {{V}^{j}} \cdot \nabla {{W}^{k}}} \right) \\ \left. {^{{}} \times \,\,\sin \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t} \right]. \\ \end{gathered} $$
((45))
$$\begin{gathered} \delta \vec {A} \approx - \frac{e}{{2m{{\omega }^{2}}}}\sum\limits_{k < j} {\frac{{\left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)}}{\omega }} \\ \times \,\,\left[ {\left( {\nabla {{V}^{k}} \times \nabla {{V}^{j}} + \nabla {{W}^{k}} \times \nabla {{W}^{j}}} \right)} \right.\cos \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t \\ \left. { + \,\,\left( {\nabla {{V}^{k}} \times \nabla {{W}^{j}} + \nabla {{V}^{j}} \times \nabla {{W}^{k}}} \right)\sin \left( {{{\delta }_{j}} - {{\delta }_{k}}} \right)t} \right]. \\ \end{gathered} $$
((46))

The following conclusion that the effect of potential forces with a scalar pseudopotential (45) can be balanced by the action of vortex forces with a vector pseudopotential (46) looks implausible.

As far as one can judge, the motion of ions in pseudopotential (44) slowly varying in time is a more realistic description of the physical process under consideration than the addition of vortex forces (also changing in time) to the averaged equation of motion. Doubts are also caused by the fact that the balance between potential and vortex forces in Eq. (37) depends on the mathematical representation of the field, and in using different mathematical expressions for the same high-frequency electric field, different models of ion motion are obtained at the output. It follows from each other’s mathematical equivalence of Eqs. (42) and (43) that the vortex drift effect in high-frequency electric fields with close frequencies, which is described in [12], does not actually exist, because it can be replaced by the movement of ions in a pseudopotential that varies slowly over time. Accordingly, the high-frequency harmonics of a high-frequency electric field with close frequencies must first be replaced by equivalent amplitude-modulated harmonics with the same frequency in accordance with Eqs. (43). The theory of transporting systems with Archimedean high-frequency fields [29–31, 34–40] and the results by Chirkov [10, 11] confirm the correctness and reliability of this approach.

FEATURES OF PARAMETERIZATION OF FAST AND SLOW TIME SIGNALS

When deriving expansion (13), the centers ωk of the spectral islands are determined with an error of δωk ~ Ω0/2, where Ω0 = 2π/T0, and T0 is the characteristic time of the slow motion. Recalculation to the new center frequency for time signals corresponding to the voltages applied to the electrodes is performed using the identity

$$\begin{gathered} {{p}_{{ik}}}\left( t \right)\cos {{\omega }_{k}}t + {{q}_{{ik}}}\left( t \right)\sin {{\omega }_{k}}t \\ \equiv \left[ {{{p}_{{ik}}}\left( t \right)\cos \delta {{\omega }_{k}}t\, - \,{{q}_{{ik}}}\left( t \right){\kern 1pt} \sin \delta {{\omega }_{k}}t} \right]{\kern 1pt} \cos {\kern 1pt} \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right)t \\ + \,\,\left[ {{{p}_{{ik}}}\left( t \right)\sin \delta {{\omega }_{k}}t\, + \,{{q}_{{ik}}}\left( t \right){\kern 1pt} \cos \delta {{\omega }_{k}}t} \right]\sin \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right)t. \\ \end{gathered} $$
((47))

For the problem of how to select the shifts δωk for the centers ωk of the isolated parts of the spectrum in order to obtain the optimal equation of the form (13) (and what exactly should be understood by the optimality of the equation), a solution has not yet been found.

In the framework of the quasi-static model, when the Maxwell effects can be neglected (that is, in using frequencies not exceeding several megahertz and at distances between electrodes not exceeding tens of centimeters), the electric potential for harmonics of a high-frequency electric field with a common base frequency ωk is determined by the equation

$$\begin{gathered} {{V}^{k}}\left( {x,y,z,t} \right)\cos {{\omega }_{k}}t + {{W}^{k}}\left( {x,y,z,t} \right)\sin {{\omega }_{k}}t \\ = \sum\limits_i {{{V}^{{ik}}}\left( {x,y,z} \right)} \left[ {{{p}_{{ik}}}\left( t \right)\cos {{\omega }_{k}}t} \right] \\ + \,\,\sum\limits_i {{{W}^{{ik}}}\left( {x,y,z} \right)} \left[ {{{q}_{{ik}}}\left( t \right)\sin {{\omega }_{k}}t} \right] \\ = \left[ {\sum\limits_i {{{p}_{{ik}}}\left( t \right){{V}^{{ik}}}\left( {x,y,z} \right)} } \right]\cos {{\omega }_{k}}t \\ + \,\,\left[ {\sum\limits_i {{{q}_{{ik}}}\left( t \right){{W}^{{ik}}}\left( {x,y,z} \right)} } \right]\sin {{\omega }_{k}}t, \\ \end{gathered} $$
((48))

where \({{p}_{{ik}}}\left( t \right)\cos {{\omega }_{k}}t\) and sin \({{q}_{{ik}}}\left( t \right)\sin {{\omega }_{k}}t\) specify the time variation of the high-frequency voltages applied to the electrodes, and \({{V}^{{ik}}}\left( {x,y,z} \right)\) and \({{W}^{{ik}}}\left( {x,y,z} \right)\) are the electric potentials at constant voltages at the corresponding electrodes. The contribution to the pseudopotential function (31) for the high-frequency potential (48) is

$$\begin{gathered} \bar {U}_{k}^{{rf}}\, = \,\frac{e}{{4m\omega _{k}^{2}}}{\kern 1pt} \left( {\sum\limits_{i,j} {{{p}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} {{p}_{{jk}}}\left( t \right)} {\kern 1pt} \left( {V_{x}^{{ik}}V_{x}^{{jk}}\, + \,V_{y}^{{ik}}V_{y}^{{jk}}\, + \,V_{z}^{{ik}}V_{z}^{{jk}}} \right)} \right. \\ \left. { + \,\,\sum\limits_{i,j} {{{q}_{{ik}}}\left( t \right){{q}_{{jk}}}\left( t \right)} \left( {W_{x}^{{ik}}W_{x}^{{jk}}\, + \,W_{y}^{{ik}}W_{y}^{{jk}}\, + \,W_{z}^{{ik}}W_{z}^{{jk}}} \right)} \right).\,\,\, \\ \end{gathered} $$
((49))

In recalculating expansion (47) to a new center frequency, Eq. (48) for a fragment of the pseudopotential of a high-frequency electric field takes the form of

$$\begin{gathered} {{{\tilde {V}}}^{k}}\left( {x,y,z,t} \right)\cos \left( {{{\omega }_{k}} + \delta {{\omega }_{k}}} \right)t \\ + \,\,{{{\tilde {W}}}^{k}}\left( {x,y,z,t} \right)\sin \left( {{{\omega }_{k}} + \delta {{\omega }_{k}}} \right)t = \sum\limits_i {{{V}^{{ik}}}\left( {x,y,z} \right)} \\ \left[ {{{p}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} \left( \begin{gathered} \cos \left( {\delta {{\omega }_{k}}t} \right){\kern 1pt} \cos \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right)t \hfill \\ + \,\,\sin {\kern 1pt} \left( {\delta {{\omega }_{k}}t} \right){\kern 1pt} \sin {\kern 1pt} \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right){\kern 1pt} t \hfill \\ \end{gathered} \right)} \right]\, + \,\sum\limits_i {{{W}^{{ik}}}{\kern 1pt} \left( {x,y,z} \right)} \\ \times \,\,{\kern 1pt} \left[ {{{q}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} \left( \begin{gathered} \cos \left( {\delta {{\omega }_{k}}t} \right)\sin \left( {{{\omega }_{k}} + \delta {{\omega }_{k}}} \right)t \hfill \\ - \,\,\sin \left( {\delta {{\omega }_{k}}t} \right)\cos \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right){\kern 1pt} t \hfill \\ \end{gathered} \right)} \right] \\ = \left[ {\sum\limits_i {\left( \begin{gathered} {{p}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} \cos {\kern 1pt} \left( {\delta {{\omega }_{k}}t} \right){{V}^{{ik}}}\left( {x,y,z} \right) \hfill \\ - \,\,{{q}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} \sin {\kern 1pt} \left( {\delta {{\omega }_{k}}t} \right){\kern 1pt} {{W}^{{ik}}}\left( {x,y,z} \right) \hfill \\ \end{gathered} \right)} } \right]{\kern 1pt} \cos {\kern 1pt} \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right){\kern 1pt} t \\ + \,\,\left[ {\sum\limits_i {\left( \begin{gathered} {{p}_{{ik}}}\left( t \right)\sin \left( {\delta {{\omega }_{k}}t} \right){{V}^{{ik}}}\left( {x,y,z} \right) \hfill \\ + \,\,{{q}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} \cos {\kern 1pt} \left( {\delta {{\omega }_{k}}t} \right){\kern 1pt} {{W}^{{ik}}}{\kern 1pt} \left( {x,{\kern 1pt} y,{\kern 1pt} z} \right) \hfill \\ \end{gathered} \right)} } \right]{\kern 1pt} \sin {\kern 1pt} \left( {{{\omega }_{k}}\, + \,\delta {{\omega }_{k}}} \right){\kern 1pt} t. \\ \end{gathered} $$
((50))

Accordingly, the contribution of Eq. (50) to the pseudopotential function (31) is

$$\begin{gathered} \bar {U}_{k}^{{rf}}\, = \,\frac{e}{{4m{{{\left( {{{\omega }_{k}} + \delta {{\omega }_{k}}} \right)}}^{2}}}}{\kern 1pt} \left( {\sum\limits_{i,j} {{{p}_{{ik}}}{\kern 1pt} \left( t \right){\kern 1pt} {{p}_{{jk}}}\left( t \right)} {\kern 1pt} } \right. \\ \times \,\,\left( {V_{x}^{{ik}}V_{x}^{{jk}}\, + \,V_{y}^{{ik}}V_{y}^{{jk}}\, + \,V_{z}^{{ik}}V_{z}^{{jk}}} \right) \\ \left. { + \,\,\sum\limits_{i,j} {{{q}_{{ik}}}\left( t \right){{q}_{{jk}}}\left( t \right)} \left( {W_{x}^{{ik}}W_{x}^{{jk}}\, + \,W_{y}^{{ik}}W_{y}^{{jk}}\, + \,W_{z}^{{ik}}W_{z}^{{jk}}} \right)} \right). \\ \end{gathered} $$
((51))

The numerators in Eqs. (51) and (49) are the same, but there are differences in the denominators. Therefore, pseudopotential expressions (31) of the order of \(\sim \,\left( {1/\omega _{k}^{2}} \right)\) are actually determined up to the implicit correction \(\sim {\kern 1pt} \left( {\delta {{\omega }_{k}}/\omega _{k}^{3}} \right)\). In particular, the presence of such uncertainty means that if the averaged equations of motion for a high-frequency electric field generated by electric voltages of the form (13) contain nonzero second-order terms ~1/ω2, then allowance for the corrections above the third-order terms 1/ω3 in the averaged equations does not make sense. Naturally, in the case when the second-order terms for the averaged equations of motion are equal to zero in any direction, the role of the third and fourth orders becomes dominant.

RESULTS AND DISCUSSION

Now, we can analyze the reasons for the discrepancy between different models of ion motion for the example discussed in Introduction. Equation (3) correctly describes the process of ion motion in the high-frequency potential (1) at ω ⪡ Ω, which is confirmed by coincidence between the theory and numerical results for transporting systems with Archimedean high-frequency fields [29–31, 34–40]. Equation (5) is categorically incorrect, because the basic assumption that the frequencies ωk are far apart from each other, on the basis of which Eq. (31) is obtained for the pseudopotential of a high-frequency field, does not hold for trigonometric expansion (4). Equation (6), obtained on the basis of Eqs. (37) and (38) for vortex drift, is subject to the same doubts as to its correctness, which are analyzed in the section Model of Vortex Forces and Vortex Drift. In particular, the effect predicted by this equation that with increasing frequency ω (that is, transport speed), the height of the walls of local pseudopotential wells responsible for the formation of compact ion packets and their transport increases, is definitely a mathematical artifact. Namely, when passing in the equations of motion \(\left( {m/e} \right)\ddot {x} = - {{U}_{x}}\left( {x,t} \right)\) for potential (1) to the dimensionless variable y = π/2 – x/L + ωt and dimensionless time τ = Ωt, we obtain the nonlinear nonstationary equation y'' = R sin(y)sin(τ) with dimensionless parameter R = \((e{{U}_{0}}/m{{L}^{2}}{{\Omega }^{2}})\), whence it follows that for any adequate pseudopotential model of motion, the height of the pseudopotential barrier at local pseudopotential wells should not depend on the ion transport rate, which is determined by parameter ω. Finally, Eq. (10) and its special case, Eq. (8), describe correctly the averaged movement of ions in the corresponding high-frequency electric field, but only at ω ~ Ω (that is, when |δ| = |Ω – ω| ⪡ Ω). Then, Eq. (3) obtained under the assumption that ω ⪡ Ω, in turn, becomes invalid.

CONCLUSIONS

The pseudopotential model of ion motion in high-frequency electric fields with many harmonics seems to be the most appropriate tool for a qualitative consideration of the corresponding physical effects.

In using the pseudopotential model of ion motion in a radio-frequency electric field, it is necessary to check carefully the basic assumptions about the nature of the high-frequency electric field, under which the equations of the pseudopotential model of motion are obtained.

When a high-frequency electric field is decomposed into the sum of radio-frequency harmonics, the harmonics with close frequencies must be replaced by equivalent amplitude-modulated harmonics with the same carrier frequency in accordance with Eqs. (42) and (43).

In describing the motion of charged particles in radio-frequency electric fields with close frequencies, the vortex drift model [12] can be replaced by a pseudopotential model of ion motion with pseudopotentials that change slowly with time.

High-frequency electric fields characterized by the Fourier spectrum in the form of isolated narrow and far-spaced intervals with nonzero spectral values can always be reduced to the form (15), which is required to use the pseudopotential model of motion.

To verify that the pseudopotential motion model correctly describes the averaged motion of ions in a high-frequency electric field, it is useful to verify that the third-order corrections (or, in the case when they vanish, the fourth-order corrections) for the averaged equations of motion are much smaller than the pseudopotential terms of the second order [10].

Because of the ambiguity of the representation of the high-frequency electric field in the form (15) (see Features of Parameterization…), it does not make sense to use corrections of the form 1/ω3 and 1/ω4 for the averaged equations of motion, if the pseudopotential terms of the form 1/ω2 are not zero.

If for averaged equations of motion, terms of the form 1/ω2 turn to zero (the pseudopotential function turns out to be constant along some coordinate), one pseudopotential model is not enough to describe ion motion in a high-frequency electric field adequately. In such cases, it is necessary to use the averaged equations of motion with the corrections of the next order [10] (1/ω3 or, if they also vanish, 1/ω4) or some other model of ion motion that does not use the pseudopotential approach.

The pseudopotential model of motion is qualitative and somewhat approximate. Because of this, it may incorrectly describe certain features of ion motion in high-frequency electric fields. For example, for quadrupole radio-frequency fields, it cannot predict the existence of a second boundary near the main stability zone, as well as the existence of an infinite set of isolated stability zones [15–17, 23, 24]. The results of a qualitative analysis of the behavior of ions in a radio-frequency mass spectrometric device, which is carried out using a pseudopotential approach, is useful to compare with the results obtained using more laborious and expensive, but also more reliable methods (for example, using numerical simulation of ion trajectories).

The paper discusses in detail typical mathematical errors that generate plausible, but incorrect theoretical conclusions and proposes a correct theory of ion motion in high-frequency fields based on the pseudopotential approach, which enables one to make a qualitative estimate in complex cases quickly. In order to remove the inevitable questions, how much can one trust the results, and whether the analysis of this subtle question, proposed here, turns out to be no better than its predecessors, in developing the theory, “intuitively obvious” empirical and heuristic considerations are used to a minimum extent, and formal considerations are used to the maximum, that is, formal and laborious methods reliably substantiated in serious mathematical publications.

Despite the high degree of mathematization of the work, it seems to be significant and practically useful for the mass spectrometry community; although initially, it was intended for a less broad audience. As a result, the text is partly overloaded with mathematical expressions and contains slang phrases typical of theoretical physicists. In particular, the formulation “Maxwell effects” in the context of this work has the following meaning. In the Maxwell equations for the electromagnetic field, the equations for the electric field contain corrections associated with the change of the magnetic field in time, and the equations for the magnetic field contain corrections for changes of the electric field in time. If the rate of the change of these fields in time is zero or small, these corrections (“Maxwell effects”) can be neglected in comparison with the other terms of the equations. In this case, Maxwell’s equations can be correctly decomposed into a system of equations for quasi-static electric field and the system of equations for a quasi-static magnetic field, which are in no way dependent on each other.

The criterion for the fact that the equations for a high-frequency electromagnetic field can be considered in a similar quasi-static approximation is the following simple consideration. For a typical device, the characteristic propagation time of the electromagnetic disturbance through the electrode system, which is caused by a change in time of the voltage at the electrodes and currents in the coils (equal to the characteristic size of the device divided by the speed of light), is much less than the characteristic time for which the electrical and/or magnetic field created in the volume of the device changes noticeably. For the high-frequency fields commonly used in ion optics, the indicated times differ by three orders of magnitude (with characteristic device sizes of the order of tens of centimeters and characteristic voltage frequencies of the order of several megahertz). This gives reason to neglect the Maxwell effects in the equations, which obey the time-varying electric and magnetic fields that control the motion of ions.