Stacked-Ring Ion Guide for Cooling and Bunching Rare Isotopes

Without loss of generality, we assume that $w(t)$ is a 2-state rectangular function with a variable duty cycle—the proportion of the high state in a period—slightly different from $50\%$, as specified by a small deviation $\delta$. For one period it can be written as

Due to the imbalance between the positive and negative duration, the wiggling motion, which is originally determined by Eq. (1), will effectively receive a gentle kick in every cycle. These kicks add up coherently and result in a slow drift. Within the context of superposition, this drifting component should belong to the secular motion rather than the wiggling motion. As a result, the rectified wiggling motion is determined by a shifted rectangular function $\tilde{w}$:

while the rectified secular motion is additionally influenced by an electrostatic field $\tilde{\mathbf{E}}=\delta\mathbf{E}$.

After integration of $\tilde{w}$ with respect to $t$ and proper selection of integral constants, $\tilde{u}$ is obtained, to a first-order approximation, to be

Plugging Eqs. (7) and (8) into Eqs. (3) and (5) and neglecting higher order terms give rise to the first-order approximations of the adiabaticity and pseudopotential for the digital RF potential of the rectangular waveform:

where $f=1/\tau$ is the RF frequency. The $\delta$-independent formulas in Eqs. (9) and (10) reveal that for a small deviation of the duty cycle from $50\%$, the rectified wiggling motion is almost unaffected while the rectified secular motion is subject to a proportionally small electrostatic field $\delta\mathbf{E}$ apart from the pseudopotential $\Phi_{0}$. This observation entails that by adjusting the duty cycle of the digital RF, the DC level of the RF potential is accordingly biased.

Owing to the rotational symmetry of stacked rings, the interior electrostatic potential $\Phi$ is preferably expressed in cylindrical coordinates $(r,\phi,z)$, where the $z$-direction aligns with the central axis of the concentric rings. In general, the expression involves multiple terms, also known as modes, which are determined by a boundary condition [46]. Nevertheless, the simplest fundamental mode can be written as

where $A$ and $k$ are scaling factors, and $I_{0}$ is the modified Bessel function of the first kind $I_{\nu}$ with $\nu=0$.

A color-coded map of $\Phi(r,z)$ is plotted on the lower panel of Fig. 2, together with equipotential lines to better illustrate the trend. In particular, null-potential lines lie in the radial direction, as shown by the dash-dotted lines. Such a potential can be induced by aligning the surfaces of charged rings to certain equipotential lines, such as the bright ones on the lower panel, or equivalently, the corresponding ones on the upper panel. Note that the voltages of adjacent rings have the same magnitude $V$ but opposite polarities. As a result, the potential in the inner volume can be represented by a small fraction in a subvolume—the potential-unique region as shown by the hatched box on the upper panel. It is trivial to derive the potential elsewhere by charge conjugation and/or parity inversion. Also labeled on the upper panel are the radius $a$ of the aperture and the pitch $p$ of the rings, where the latter is the distance between two adjacent null-potential lines. Hence, the scaling factors $A$ and $k$ in Eq. (11) can be written as $V/I_{0}(ka)$ and $\pi/p$, respectively.

The electric field $\mathbf{E}$ inside stacked rings can be obtained by taking the gradient of the potential $\Phi$:

The adiabaticity $\eta$ and the pseudopotential $\Phi_{0}$ can be obtained by plugging Eq. (12) into Eqs. (9) and (10), respectively. Color-coded maps of $\eta(r,z)$ and $\Phi_{0}(r,z)$ are plotted in Fig. 3, together with their respective contours.

Recalling the constraint on the adiabaticity, it follows that $\max(\eta)=\eta(a,0)$ should satisfy the following inequality:

To confine an ion in the radial direction, the pseudopotential barrier on the boundary, which is represented by $\Phi_{0}(a,p/2)$, must be higher than its initial kinetic energy $E_{i}$:

As the ion loses its energy via ion-neutral collisions, it will move towards the central axis and may even be confined in one of the pseudopotential buckets, which are enclosed by the contours corresponding to the value of $\Phi_{0}(0,0)$ on the lower panel of Fig. 3. This scenario forms a basis for ion bunching, which entails that the final kinetic energy $E_{f}$ is constrained by

Eqs. (13) to (15) have set ground rules for determining the parameters of a stacked-ring ion guide. Although they are derived based on a constant radius $a$ over the entire structure, the equations can still facilitate parameter selections in case of varying apertures, which is useful in practice for the manipulation of the radial beam size. Furthermore, the pitch $p$, or equivalently $k$, is varied accordingly such that the pseudopotential barrier, being the right hand side of Eq. (14), stays constant on the edges of the varying apertures. By this means, the electric boundary matches the geometric one, thereby maximizing the confining volume and reducing ion losses.

The ideal shape of the ring electrodes as shown in Fig. 2 turns out to be unrealistic, since its radial dimension is not finitely bounded and adjacent rings would in practice touch each other at large radii. Therefore, we have adopted a simpler ring shape, of which an illustration is given in Section 4.1 (see Fig. 5). All rings are annular plates with a fixed outer radius, while the inner radii vary between $2$ mm and $7$ mm. Moreover, the sharp inner edges are radiused at $0.5$ mm to reduce risks of electric discharge. Apart from the pitch $p$, the thickness $l$ of the plate also needs to be determined for a given inner radius $a$. Because of limited manufacturing precisions, these geometric parameters can only take certain rounded values in practice. Specifically, $p$ and $l$ are rounded to one-tenth of a millimeter, while $a$ is controlled with a quarter-millimeter precision.

The search for the optimal $p$ corresponding to a given $a$ while respecting the dimensional constraints is graphically demonstrated on the left panel of Fig. 4, where practically feasible $(a,p)$ pairs must locate at grid points. The procedure begins with initial values $(a_{n},p_{n})=(2,2.1)$ mm, which are taken as the narrowest aperture and marked by a square dot at the bottom left corner of the plot. An equi-adiabaticity line of $\bar{\eta}$, derived from Eq. (13), is drawn across the square dot as a dash-dotted curve on the plot, with a “$+$” (“$-$”) sign denoting the ascending (descending) side where the adiabaticity at any point is greater (less) than $\bar{\eta}$. The inequality in Eq. (13) for any aperture is then translated to finding a solution on the descending side of $\bar{\eta}$. Likewise, an equi-final-energy line of $E_{f}$, derived from Eq. (15), is drawn across the square dot as a dotted curve with associate “$+$” and “$-$” signs. That ions are only bunched at the narrowest aperture requires a solution to be limited to the descending side of $E_{f}$. Due to the common scaling factor on the right hand sides of Eqs. (14) and (15), an equi-initial-energy line of $E_{i}$ is accordingly determined, which is drawn as a dashed curve on the plot with associate “$+$” and “$-$” signs. To let the electric boundary match the geometric one while fulfilling the inequality in Eq. (14), the solution of $(a,p)$ pairs is eventually found to be the closest grid points to the $E_{i}$-line on the ascending side, which are marked by round dots on the plot. Note that the square dot is also on the ascending side, which guarantees radial confinement for the narrowest aperture.

The optimal thickness $l$ corresponding to a given $(a,p)$ pair determined above is found by minimizing the root-mean-square deviation of an actual electrostatic potential $\Phi$ to the ideal one over the potential-unique region (see the hatched box in Fig. 2), where the former is numerically solved by the SIMION^® software version $8.1$ and the latter is calculated from Eq. (11). Interestingly, the optimal $l$ does not explicitly depend on $a$. Its dependency on $p$ is illustrated on the right panel of Fig. 4. Note that with the optimal $l$, $\Phi$ is merely distorted by a few parts per thousand, which supports the pragmatic simplification of the ideal ring shape. As an example, the root-mean-square relative deviation of $\Phi$ is $3\times 10^{-3}$ for $a=7$ mm, $p=2.5$ mm, and $l=1.3$ mm. Furthermore, the null-potential lines (see the dash-dotted lines in Fig. 2) remain intact due to the reflection symmetry of $\Phi$ in the axial direction.

It can be concluded from the left panel of Fig. 4 that the adiabatic approximation is valid for all the determined $(a,q)$ pairs if Eq. (13) holds for $(a_{n},q_{n})$. Moreover, to avoid imposing unnecessary obstacles to ion transport in the axial direction, only $(a_{n},q_{n})$ should fulfill Eq. (15). Therefore, two electric parameters, namely the frequency $f$ and the voltage $V$ of a digital RF, can be constrained by substituting $a_{n}$ and $k_{n}=\pi/p_{n}$ in Eqs. (13) and (15). Canceling out $V$ in both equations results in the constraint on $f$:

Plugging this constraint in Eq. (15) gives rise to

As an example, to cool and bunch ${}^{203}\mathrm{Ir}^{+}$ ions in the NEXT experiments, the minimum frequency and voltage of the digital RF are $3.82$ MHz and $82$ V, respectively.

Eqs. (17) and (18) reveal scaling laws of electric parameters with respect to ion properties. For ions extracted from a gas cell, which have a common charge state of $1+$, or $2+$ for superheavy elements, the voltage is independent of the particular ion species. The frequency, however, is inversely proportional to the square root of the ion mass. Therefore, a stacked-ring ion guide designed for given initial and final kinetic energies can serve as a high-pass mass filter with fixed frequency and voltage.

The ion guide consists of $78$ concentrically stacked rings of varying apertures. The layout is schematically shown in Fig. 5, where every constituent ring is numbered sequentially from entrance to exit, and quantitatively listed in Table 2. Additionally, a three-dimensional model of the ion guide is shown in Fig. 6. To restrain the field distortion due to the aperture change, all the rings are laid out in such a way that the null-potential lines of a ring overlap with those of its neighbors. Recalling the definition of the pitch in Section 2.2, the total length of the ion guide, which amounts to $190.8$ mm, is thus the sum of all the pitches except for the rings at both ends. According to the right panel of Fig. 4, the thickness of a ring is always less than its pitch. Therefore, isolating gaps can naturally arise in between.

The specific shape of the ion guide mainly stems from the optimization of the transmission efficiency. With the helium buffer gas at a pressure of $10^{-3}$ mbar, the mean-free-path of test ions is around $5$ mm. An ion’s trajectory within the ion guide is predominantly determined by electric fields while ion-neutral collisions only impose a light damping effect. Consequently, incoming ions tend to be reflected right away by contracting apertures, which are necessary for the transition from the wide entrance to the narrow exit. As a remedy, a spacious thermalization section is included in the ion guide to sufficiently cool the ions before they are transported towards the exit. The entire structure is functionally divided into four parts, namely for focusing, thermalization, refocusing, and bunching. The divergent ions coming from left first confront the focusing section and are focused to the center of the last ring (№ $8$) of this section with a-few-millimeter-wide aberration. Afterwards, the ions enter the thermalization section, which consists of $60$ identical rings with the largest inner radius of $7$ mm. Here, the ions will reach the thermal equilibrium with the buffer gas after sufficient ion-neutral collisions. The thermalized ions will then be collimated by the refocusing section into the last bunching section, where the smallest inner radius of $2$ mm helps store the ions in a pseudopotential bucket to form an ion bunch.

After ions are fully thermalized in the thermalization section, the diffusion in the buffer gas can greatly delay their arrivals at the bunching section. Thus, a traveling wave of bias voltages is employed to speed up this process. Similar ideas have been applied in ion mobility separation [47, 48, 49, 50] and ion transport in a gas cell [51, 52, 53, 54]. Specifically, the wave is running from the first ring (№ $9$) of the thermalization section to the first ring (№ $74$) of the bunching section. The schematic of all four phases of the wave is shown in Fig. 7. At a given moment, e.g. $t=0$, the DC level of every fourth ring is raised to a wave voltage $V_{w}$ while the DC levels of the others remain intact, as illustrated by the dots in Fig. 7(a). This configuration creates a potential well in the middle of each segment, as marked by the triangles in the figure. The ions will be attracted to, and eventually stored in, these potential wells. At a later moment $t=t_{w}$, the configuration advances towards the exit by one ring, as illustrated in Fig. 7(b). The thermalized ions will be pushed forward by the potential switch, but will still be confined in the same segments bouncing back and forth until they are recooled by the buffer gas towards the new potential wells, as revealed by simulations. Consequently, the ions “ride” the traveling wave to the bunching section. The optimal values of $V_{w}$ and $t_{w}$ are found by simulations to be $2.5$ V and $750$ $\mu$s, respectively. They are compiled in Table 3, which lists the electric parameters of the ion guide.

Due to the force exerted by the traveling wave, ions entering the bunching section will initially be more energetic than any pseudopotential bucket in the section. To avoid their immediate escapes from the exit, the DC level of the last ring (№ $78$) of the bunching section is raised to a gating voltage $V_{g}$. The value of $V_{g}$ is set to be that of $V_{w}$ such that when the traveling wave propagates furthest to the ring № $74$, a potential well is formed in the middle, i.e. at the ring № $76$. The ions will be attracted to, and eventually stored in, this potential well. Even after the DC level of the ring № $74$ is restored to zero, the gating voltage on the ring № $78$ can impose a negligible perturbation on the ions. They will still be stored in the pseudopotential bucket at the ring № $76$. Consequently, the incoming ions will continually be transported to and accumulated at this location, where they form an ion bunch.

The gating voltage at the exit can also help prevent direct passages of paraxially incoming ions through the ion guide. Moreover, based on simulations, an incoming ion can sometimes bounce back in the refocusing section. To reflect such backwards flying ions that are still to be thermalized, the DC level of the last ring (№ $8$) of the focusing section is raised to a reflection voltage $V_{r}$. The optimal value of $V_{r}$ without hampering ion injections into the thermalization section is found by simulations to be $1.3$ V. Although it is less than the value of $V_{w}$ (see Table 3), ion reflections caused by $V_{r}$ is enhanced by a smaller aperture of the ring № $8$ than that of the rings in the thermalization section.

When an ion bunch is ready for ejection, the last four rings (№ $75$ to № $78$) of the ion guide will be disconnected from the digital RF and independently charged to ejection voltages. The voltage change can be completed typically in less than $40$ ns with a fast switch such that the ion bunch will virtually experience no loss of confinement before it is accelerated by the ejection gradient. Special attention ought to be paid to the timing of the ejection. It should occur right before the DC level of the ring № $74$ is restored to zero. As a result, the ion bunch is already in the thermal equilibrium, and the least perturbation is imposed on other ions that are being transported. The ensemble of the ejection rings acts as an ion-optical lens system, which can flexibly focus the ejected ion bunch on the axis towards a given point with corresponding ejection voltages. This is particularly useful when the ion bunch needs to pass through a tiny orifice to the downstream higher vacuum stage. Table 4 gives an example of optimal ejection voltages to focus the ion bunch $12$ mm behind the ion guide.

The realization of the entire process described above critically depends on the digital RF that is applied to the ion guide. It is typically generated by two Power Supply Units (PSUs) with the same magnitude $V$ but opposite polarities and two fast switches. The latter switch between the positive and negative PSU at the RF frequency $f$, and feed two alternating voltages respectively to even and odd rings.

Such a scheme has readily been applied to digitally driven RFQs and ion funnels. However, for the present ion guide, the DC levels of certain rings need to be individually raised to bias voltages, as listed in Table 3. According to Section 2.1, such a voltage, e.g. $\delta V$, can effectively be induced by deviating the duty cycle of the digital RF from $50\%$ by an amount of $\delta$. The resultant time spectra for even and odd rings in case of a positive deviation $\delta$ are shown in Fig. 8. The induced bias voltage can easily be adjusted by reprogramming the timing pattern for the fast switch without any modifications to the hardware.

We note that Fig. 8 dose not necessarily imply the same spectrum being applied to all the even (or odd) rings at the same time. During the operation of the ion guide, the duty cycle of the digital RF changes according to the phase sequence in Fig. 7 for the participating rings in the traveling wave, while it remains constant for the others. Fig. 9 shows the timing pattern of the deviation $\delta$ of the duty cycle for every constituent ring of the ion guide. The value of $\delta$ is calculated as the ratio of the bias voltage to the PSU voltage, and listed in Table 3 for all possible scenarios. Based on the timing diagram in Fig. 9, rings sharing the same pattern and polarity can directly be wired, thus notably simplifying the circuitry complexity.

After sequentially injecting the aforementioned $300$ test particles into the ion guide, the full trajectory of each ion is tracked in the simulation until a terminating condition is met. The statistics of ion terminations are listed in Table 6. Only a fraction of ions are lost, mainly due to slipping off the traveling wave. Most ions are successfully stored in the pseudopotential bucket at the ring № $76$ for ion bunching. The simulated transmission efficiency is $80\%$. To estimate its uncertainty, the process has been repeated $10$ times. The standard deviation of the $10$ resultant transmission efficiencies is calculated to be $3\%$, which is also included in Table 5.

Owing to the stochastic nature of ion-neutral collisions, the thermalized ions possess no “memories” of their initial conditions. Therefore, the ion bunch at the ring № $76$ can equivalently evolve from standstill ions in the corresponding pseudopotential bucket. To randomly draw a sample from the ion bunch, a standstill ion is released from the center of the pseudopotential bucket and its kinetic state is recorded after a random duration, which is sufficiently long for thermalization. This process has been repeated $300$ times.

The distribution of the $300$ samples in a six-dimensional phase space is illustrated in Fig. 10(a)–(c) with projections onto two-dimensional planes corresponding to three mutually orthogonal $x$-, $y$-, and $z$-directions, where the $z$-direction aligns with the axis of the ion guide. Owing to the narrowest aperture in the bunching section, the ion bunch can loosely be confined in a ($2\times 2\times 2$)-mm³ cubic volume of the configuration subspace, with a slightly stronger compression in the axial direction. In the velocity subspace, the ion bunch manifests no preferences towards any direction as it is all tightly bounded within $1$-mm/$\mu$s range in each dimension. Moreover, the energy distribution of the ion bunch shown in Fig. 10(d) agrees well with the theoretical prediction given by Eq. (16), which is drawn as the solid line in the figure.

After the ion bunch is ejected by applying the ejection voltages in Table 4, it is profiled on the focal plane $12$-mm behind the ion guide. Its distribution in a four-dimensional transverse trace space—a cousin of the phase space with the velocity subspace replaced by the angle subspace with respect to the beam direction—is illustrated in Fig. 11(a)–(c) with three two-dimensional projections. Fig. 11(a) reveals a well focused beam spot with an about $1$-mm radial size. The root-mean-square (RMS) transverse beam emittances $\epsilon_{2\sigma,x}$ and $\epsilon_{2\sigma,y}$ at twice standard deviation ($2\sigma$) are calculated to be $20.8$ $\pi$ mm mrad and $21.0$ $\pi$ mm mrad, respectively. These values are shown as the areas of the ellipses in Fig. 11(b) and (c). Moreover, the distribution of the ion bunch in a two-dimensional longitudinal phase space is shown in Fig. 11(d), where $E_{e}$ is the ejection energy and $t_{\mathrm{tof}}$ is the time of flight of an ion to the focal plane. The $2\sigma$ RMS longitudinal beam emittance $\epsilon_{2\sigma,z}$ is calculated to be $0.9$ $\pi$ $\mu$s eV and is likewise illustrated in the figure.

The uncertainties of the beam emittances are obtained from $10$ repeated simulation runs in a similar manner as above. The values corresponding to the two transverse directions both are $0.5$ $\pi$ mm mrad, while the value for the longitudinal direction is $0.1$ $\pi$ $\mu$s eV (see also Table 5). It is thus found that $\epsilon_{2\sigma,x}$ is the same as $\epsilon_{2\sigma,y}$ up to the uncertainty, which aligns with the rotational symmetry of the structure.