1 Introduction

Electrostatic multi-reflection ion traps [1–3] are one of the most recent developments for ion trapping. Several of these devices have been constructed that cover a wide spectrum of applications ranging from cluster physics [4–9], molecular physics [10–13], electron collisions [14], high-resolution mass separation [15–17] to precision mass spectrometry like recent direct mass determinations in nuclear-physics experiments [18, 19].

The two latter applications exploit the mass-spectrometric advantages of multi-reflection devices, i.e., a mass resolving power \( R = m/\Updelta m = t/2\Updelta t \) increasing with the time-of-flight (ToF) t of stored ion bunches with a bunch length Δt. If several ion species with individual mass-to-charge ratios m/q are trapped simultaneously, the increasing differences in ToF while performing multiple reflections (revolution time: \( T \propto \sqrt {m/q} \)) allow for mass separation with high resolving power, which then facilitates an effective purification of ions for subsequent experimental steps. With these multi-reflection time-of-flight mass separators (MR-ToF MS), resolving powers in excess of R = 105 can be reached within time scales in the order of ten milliseconds [13, 15–19].

Recently, a particularly high interest came from nuclear-physics experiments at radioactive ion beam facilities, where a fast and efficient separation of the ions of interest from simultaneously delivered isobaric contaminants is required. However, cleaning at high resolving powers means, in turn, a high sensitivity to Coulomb interactions between the trapped ions. This leads to a limitation with respect to the maximum number of ions that can be processed simultaneously [20]. If this number is exceeded, the purification is not effective, i.e., the ions of interest cannot be separated. On the other hand, decreasing the total number of ions at a given time is inefficient, in particular if the experimental steps after the purification are much more time-consuming than the purification steps. Only very few ions of interest can then be investigated during the limited time of an on-line experiment.

For example, precision mass measurements of short-lived nuclides in Penning traps [21, 22] suffer from such contamination scenarios. At the ISOLTRAP mass spectrometer at ISOLDE/CERN [23, 24], beam purification methods have been developed and studied since more than two decades [25–28]. The ISOLTRAP setup employs a linear radio-frequency-quadrupole (RFQ) ion trap for bunching of the quasi-continuous on-line ISOLDE ion beam [29], a recently installed MR-ToF MS [17], and a preparation Penning trap for mass-selective buffer-gas cooling [25] as devices for beam preparation and purification. The actual mass spectrometry is performed in a precision Penning trap by use of the time-of-flight ion-cyclotron-resonance (ToF-ICR) method [30, 31]. Recently, the MR-ToF MS at the ISOLTRAP setup has reduced the ion-separation time by an order of magnitude for resolving powers similar to that of the preparation Penning trap which, previously, was the purification device of choice.

In the following, a method is introduced, where the ion purification phase, using the RFQ and MR-ToF MS, is decoupled from all further processes of the ToF-ICR measurement cycle. In this new experimental sequence multiple ion bunches, each purified by the MR-ToF device, are accumulated and cooled in the preparation Penning trap, comparable to the use of an accumulation RFQ, as previously suggested [15]. In contrast to an RFQ approach, however, the present method allows to use the Penning trap as a second high-resolution cleaning stage for further suppression of contaminations. This new approach improves the performance of the ISOLTRAP setup significantly, in particular in the cases where the ions of interest have half-lives larger than the duration required for the ion separation in the MR-ToF MS, but where the ions come at low production rates relative to a high amount of accompanying contaminations. The fast ion separation in the MR-ToF MS is repeated several times and acts as a pulsed ion source for the preparation Penning trap, to which the purified ion bunches are transferred for accumulation (stacking). This procedure multiplies the number of ions of interest available for the subsequent ToF-ICR detection by the number of applied accumulation steps (stacking loops), while the total duration of the ToF-ICR measurement cycle is only increased by a small fraction of this factor.

In the 1990s, the accumulation and buffer-gas cooling of charged-particle bunches in a Penning trap have been demonstrated, e.g., in case of gold clusters [32], with rubidium ions in the preparation Penning trap of the ISOLTRAP setup [26], and, more recently, such accumulations have been employed at JYFLTRAP [33]. However, in the present investigation, we purify the ion bunches in an MR-ToF MS before they enter the Penning trap to be accumulated.

In the following, the ISOLTRAP setup and event sequence for the new stacking method are described and demonstrated with off-line and on-line measurements. In addition, the realization of the high repetition rate of the stacking loop is described in detail. Finally, the optimal number of stacking loops for mass measurements of short-lived nuclei as a function of the ions’ half-life, release time from the production target, and durations of the experimental procedures is investigated.

2 Experimental setup and procedure

All measurements have been performed with the ISOLTRAP setup at ISOLDE/CERN. The components relevant for the present study are shown in Fig. 1 and are described briefly in the following.

Fig. 1
figure 1

Sketch of the horizontal beamline and the preparation Penning trap of the ISOLTRAP setup

Full size image

At ISOLDE, radioactive atoms are produced by impact of 1.4-GeV protons on a thick target, released by thermal diffusion, ionized, accelerated, and delivered as a 30–60-keV quasi-continuous ion beam for durations controlled by the “ISOLDE beamgate” [34]. When reaching the ISOLTRAP setup, the ions are electrostatically decelerated to a few eV and injected into a radiofrequency quadrupole (RFQ) for accumulation, buffer-gas cooling, and bunching. After the ejection of ion pulses from the RFQ toward the MR-ToF MS, the (potential) energy of the ions is adjusted from the HV-potential of the RFQ to the ground potential of the ISOLTRAP setup by a pulsed HV drift tube. The ions are then captured in the MR-ToF MS by use of an in-trap lift [35] and trapped until a sufficient separation between the ions of interest and the contaminants is achieved. Subsequently, the ions are ejected toward a Bradbury–Nielsen gate (BNG) [17, 36], which deflects the contaminating ions in closed mode, whereas the ions of interest are transmitted toward the preparation Penning trap in open mode. In the preparation Penning trap, mass-selective resonant buffer-gas cooling can be applied (duration typically 100–500 ms) to remove, e.g., residual contaminants and to cool and center the ions, before they are transported to the precision Penning trap (not shown in Fig. 1). The mass measurement is then performed via determination of the cyclotron frequency with the ToF-ICR technique (duration typically 100 ms to 1 s). In general, ToF-ICR resonance measurements consist of several (tens of) cycles where different excitation frequencies of a quadrupolar RF field are used to probe the influence on the motion of the ions in the measurement Penning trap. The resonance curve consists of the mean time-of-flight of the ions after ejection from the trap toward a micro-channel plate detector as a function of the excitation frequency [31, 37].

The new event sequence of the experimental cycles is characterized by a repetitive structure with the duration t loop (see Fig. 2). This loop comprises the RFQ bunching, MR-ToF-MS + BNG purification, and the capture in the preparation Penning trap for accumulating the mass-selected species by lowering the voltage at the injection-side end-cap electrode (Fig. 3a). After the end-cap electrode is activated again, the ions thermalize due to buffer-gas collision at typically p ≈ 10−4 mbar, resulting in a reduction of the axial ion motion amplitudes (Fig. 3b) while performing the next stacking cycle. Therefore, ions can be injected again by lowering the end-cap electrode without losing the previously captured ion bunches (Fig. 3c) and are cooled as well (Fig. 3d). In case of medium-mass ions of m ≈ 100 u, the time required for a sufficient reduction of their kinetic energy is on the order of 10 ms. This allows a repetition rate of up to 100 Hz for the stacking loop structure. After repeating the loop n times (duration: n × t loop), the accumulated purified ions undergo a final high-resolution cleaning step (mass selective buffer-gas cooling) and the subsequent ToF-ICR detection, where the duration of both last steps will be called t meas. The additional cleaning is applied to remove contaminants that may have survived the earlier cleaning stage (limited suppression factor of the BNG [17]) or that may be created during the stacking by charge transfer reactions or in-trap decay [38]. A high repetition rate of the loop sets special demands on the pulsing capability of the HV-pulsed drift tube. Thus, the corresponding switching system had to be upgraded (for details see Sect. 4).

Fig. 2
figure 2

Flow diagram of one ToF-ICR measurement cycle, which consists of several stacking loops and the further processing of ions in the Penning traps: a final cleaning in the preparation Penning trap and ToF-ICR detection using the precision Penning trap

Full size image
Fig. 3
figure 3

Potential along the axis of the preparation Penning trap (solid lines). a capture mode during arrival of the first ion bunch: trap (partially) opened for injection of purified ions, b trapping mode: storage of ions and cooling of ion motion by buffer gas (indicated by dashed line) between capture events, c capture mode during the arrival of another ion bunch, and d cooling of next ion bunch

Full size image

3 Experimental results

In the following, several results from both off-line tests and on-line applications of the MR-ToF MS/Penning trap stacking method are described. In particular, initial tests with the existing hardware (Sect. 3.1) showed that a faster high-voltage switch for the drift tube behind the RFQ was essential for the implementation of the stacking method. The new switch improved the performance significantly (Sect. 3.2).

3.1 Tests with the old HV switch system

The first tests of ion stacking at the ISOLTRAP setup were performed with 133Cs+ ions from an off-line alkali reference ion source (see Fig. 1). The performance of the previous HV switch that was connected to the pulsed drift tube was tested with up to ten stacking loops. The number of ions accumulated in the preparation Penning trap as a function of the number of stacking loops was measured by ejecting the ions toward a micro-channel plate (MCP) detector. For a substantially higher throughput of purified ions as compared to a single injection in the preparation Penning trap, the stacking loop duration should be significantly shorter than the typical storage times of ions in the preparation Penning trap for mass-selective buffer-gas cooling. However, when the loop length was decreased, already a duration of 100 ms (for 10 pulses per second) turned out to be too fast for an efficient transport of ions due to the recovery time of the HV switch (see Fig. 4). This leads to inadequate transport energies of the ions on their way through the system, which causes ion losses.

Fig. 4
figure 4

Average number of detected ions after ejection from the preparation Penning trap as a function of the number of stacking loops. Solid line linear fit to the data. Dashed line expected increase in ions for ideal transfer, capture, cooling, and storage in the preparation Penning trap

Full size image

As a result, the number of ions obtained when adding additional stacking loops is not the product of loop number times the number of ions obtained in the first loop as marked by the dashed line. Instead, the average ion number as a function of number of loops increases only by a fraction of the initial ion bunch (solid line).

3.2 Performance with the new HV switch system

The performance of the stacking scheme after exchanging the HV switch was tested with an on-line beam of 179Lu+ ions (half-life of t 1/2 = 4.59 h) from ISOLDE, contaminated with stable 163Dy16O+ ions. The contaminants were more abundant than the ions of interest, 179Lu+, by a factor of 1,300. The ion losses due to decay were negligible because of the long half-life. In every stacking loop, a mixture of about 1,200 ions (calculated with 30 % detection efficiency)—of both 179Lu+ and 163Dy16O+—was injected in the MR-ToF MS for purification. Note that this means that on average in every loop only about one single ion of interest is leaving the MR-ToF MS. The overall length of the stacking loop was t loop = 52 ms, where the major part was used as waiting time of 30 ms in order to slow down the repetition rate to roughly 20 Hz. In principle, the stacking loop can run with a repetition frequency of several hundred Hertz; however, the ISOLDE beamgate limited the repetition rate in this case. The opening time of the ISOLDE beamgate was 1 ms, the cooling and bunching time in the RFQ was set to 5 ms, and 16 ms were chosen for the ion bunch separation in the MR-ToF MS, by far exceeding the needed resolving power of m/Δm ≈ 8,000. The average number of detected ions as a function of the number of stacking loops is shown in the top part of Fig. 5.

Fig. 5
figure 5

Top Average number of detected ions after ejection from the preparation Penning trap as a function of the number of stacking loops. Solid line to guide the eye; dashed line linear extension from the data at low loop numbers. Bottom ToF-ICR resonance of 179Lu+ recorded by use of the new stacking scheme. Solid line fit to the resonance curve

Full size image

Up to n = 20 injections, i.e., for about one second, the count rate increases linearly with the number of accumulations. For more than 20 stacking loops, the increase in the number of ions slows down (solid line). This change of the slope may be due to radial losses of trapped ions whose motion is damped by buffer-gas collisions including an increase in the magnetron motion. This ion loss could be avoided with an adequate quadrupolar RF-excitation during the sequence of stacking loops. However, even without such a recentering [32], the ion number available for the ToF-ICR detection was increased by a factor of 40 for 50 accumulations, where 12 average counts (about 40 ions) per ejection from the preparation Penning trap have been achieved compared to 0.3 counts with only one stacking loop, i.e., a single capture event per measurement cycle. A ToF-ICR resonance of 179Lu+ was recorded (see bottom part of Fig. 5) by use of the new accumulation scheme with 50 stacking loops per ToF-ICR measurement cycle. The mass-selective buffer-gas cooling and the ToF-ICR detection lasted t meas = 890 ms including further excitation and cooling times in the preparation Penning trap, transfer times, and 400 ms of quadrupolar RF-excitation in the precision Penning trap. Thus, the duration of one ToF-ICR measurement cycle including the 50 stacking loops was 50 t loop + t meas = 3.49 s. For comparison, the time needed for an equivalent measurement without ion stacking (using 50 measurement cycles) would have been 50 (t loop + t meas) = 47.6 s. Thus, the reduction in measurement time due to the stacking method was a factor of about 13. The total time of data taking for the measurement of the resonance (40 frequency steps) with sufficient statistics was 35 min. If the measurement had been performed in the conventional way without stacking, more than 7 h would have been required.

4 Realization of the fast-loop repetition

For an effective application of the stacking method, all electrically switched beamline elements have to be operated at switching frequencies of several tens or hundreds of Hertz. In general, this requirement poses no problem for low-voltage modules requiring only moderate precision, e.g., for the RFQ extraction pulse or the capture pulse of the preparation Penning trap. However, the switching module of the HV drift tube (see Fig. 1) had to be replaced to implement the new accumulation method.

This pulsed drift tube transfers the ions from the RFQ on the U RFQ = 60 kV high-voltage platform to considerably lower kinetic energies in the ground-potential beamline. To this end, the “high potential” U H of the drift tube has a value only somewhat (about 3.1 kV) lower than the high-voltage platform when the ions are ejected. Once the ions are inside, the voltage on the drift tube is switched to ground potential. Thus, when the ions leave the tube, their remaining energy is \( q\;U_{\text{transfer}} = q\;(U_{\text{RFQ}} - U_{\text{H}} ) \), i.e., around 3.1 keV, which is the transfer energy to the MR-ToF MS and the preparation Penning trap.

The module used so far for the switching of the drift tube (combination of two Behlke HTS MOSFET 650) provides either only a fast (~10 to 100 ns) rising or only a fast falling edge, while the corresponding other edge is considerably slower. Due to the need for a load resistor R L (see Fig. 6) in the range of some 100 MΩ, sufficiently recharging the load capacitance C L (in the range of some 100 pF) takes some 100 ms. A significant reduction of the load resistor is impossible due to power-dissipation considerations and the current provided by the high-voltage supplies.

Fig. 6
figure 6

Simplified scheme of the previous switching circuit for the pulsed drift tube

Full size image

To overcome this limitation, the switching system has been replaced by a push–pull module. The power to be provided by the voltage supply and therefore to be dissipated by the resistances in the switching system is \( P = f\;C_{\text{L}} \;U_{\text{H}}^{2} \), where f is the repetition rate. At a nominal voltage of \( U_{\text{H}} = U_{\text{RFQ}} - U_{\text{transfer}} = 56.9\;{\text{kV}} \), a total load capacitance of about C L = 300 pF and a switching rate of f = 100 Hz, the required power dissipation is P ≈ 100 W, which cannot be achieved by air-convection cooling without increasing the temperature of the switch considerably.

To avoid the excessive heating, a liquid-cooled switch (Behlke HTS 651-10-GSM, 65 kV, 100 A peak current) has been installed (see Fig. 7). The high-voltage side is buffered by a C B = 500 nF oil-filled capacitor, keeping the voltage drop below 0.1 % of the nominal value. A series resistor of R SH = 5 kΩ on the high-voltage side reduces the current in case of a discharge or failure on the output while it is connected to the high side. Furthermore, it reduces the \( \partial U/\partial t \) transient speed of the positive edge, but which does not become critical for the application. On the low-voltage (i.e., ground) side, the impedance is matched to the load by integrating a R SL = 175 Ω series resistor. This value was chosen to achieve critical damping. A fast but over-shoot and ringing-free edge transient is essential because any voltage fluctuation will modulate the energy of the ions when they leave the pulse drift tube. Furthermore, an over-shoot increases the voltage across the switching module, which can cause damage if the maximum operating voltage is exceeded. The time constant of the falling edge has been determined to τ fall ≈ 21 ns and the time 10 τ fall ≈ 210 ns to discharge the pulsed drift tube to less than 5 × 10−5 U H < 5 V is short enough for the required passage time of the lightest ions investigated at the ISOLTRAP setup. The power supply connected to the switching circuitry (Heinzinger PNChp ± 60 kV, 10 mA) has a specified stability of \( \Updelta U_{\text{drift}} < 10^{ - 5} U_{\text{H}} / 8\;{\text{h}} \), a peak-to-peak ripple of \( \Updelta U_{\text{pp}} < 10^{ - 5} U_{\text{H}} , \) and a temperature coefficient of \( \Updelta U_{\text{temp}} < 10^{ - 5} U_{\text{H}} /{\text{K}} \). No noticeable voltage drift was observed, which would have resulted in a change of the capture and transport efficiency of the preparation Penning trap. The complete high-voltage switching circuitry is installed in a radio-frequency-sealed grounded copper cage. The system has been operated with voltages up to 50 kV and frequencies of several hundred Hertz. With the available heat exchange rate of the liquid-cooling system, no significant temperature increase could be observed even at high switching voltages and rates.

Fig. 7
figure 7

Sketch of the new switching circuit for the pulsed drift tube

Full size image

5 Optimization of the stacking parameters

For a particular measurement, the optimal number of stacking loops depends on the properties of the ions of interest and of the contaminants. The lifetime \( \tau = t_{1/2} /{ \ln }(2) \) of the ions of interest is the most important parameter and competes with the storage times in the ion traps. Short half-lives, for example, exclude long accumulation times, since the number of ions stagnates quickly with increasing number of stacking loops. This number has to be chosen such that the final number of purified ions in a given measurement time is maximal.

To calculate the number of ions per ToF-ICR measurement cycle, the number of ions which reach the preparation Penning trap after each stacking loop is integrated over the opening time of the ISOLDE beamgate (t bg) including the decay of the radioactive ions. The ions of the next stacking loop are added by use of the same integral to the ions already stored taking into account the decay loss of the latter during the last loop period t loop. After n loops, the measurement is continued by mass-selective buffer-gas cooling and ToF-ICR detection which together last for t meas. The number of ions at the end of a ToF-ICR measurement cycle with n repetitions of the stacking loop can be expressed by:

$$ \begin{aligned} N(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) & = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} \mathop \sum \limits_{k = 1}^{n} {\text{e}}^{{\frac{{ - (n - k)t_{\text{loop}} }}{\tau }}} \mathop \int \limits_{{(k - 1)t_{\text{loop}} }}^{{(k - 1)t_{\text{loop}} + t_{\text{bg}} }} I(t){\text{e}}^{{\frac{{ - (kt_{\text{loop}} - t)}}{\tau }}} {\text{d}}t, \\ & = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} \mathop \sum \limits_{k = 1}^{n} {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} \mathop \int \limits_{{(k - 1)t_{\text{loop}} }}^{{(k - 1)t_{\text{loop}} + t_{\text{bg}} }} I(t){\text{e}}^{{\frac{t}{\tau }}} {\text{d}}t, \\ & = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{{\left( {k - 1} \right)t_{\text{loop}} }}^{{(k - 1)t_{\text{loop}} + t_{\text{bg}} }} I(t){\text{e}}^{{\frac{t}{\tau }}} {\text{d}}t. \\ \end{aligned} $$
(1)

The ion number per time interval I(t) includes all ion losses within the ISOLTRAP setup. This may also include time dependent ion losses independent of the delivered beam.

If one to a few events per ToF-ICR measurement cycle can be achieved, a further increase in the number of stacking loops is not required. In more challenging cases, where the latter cannot be achieved, one needs to optimize the number of ions per cycle length:

$$ Q(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) = \frac{{N(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} )}}{{nt_{\text{loop}} + t_{\text{meas}} }}. $$
(2)

So far no approximations have been introduced, as all further complications can be described by I(t). In the following, the same experimental conditions are assumed for each stacking loop. Two cases will be discussed in more detail, which are of practical interest for the operation of the ISOLTRAP setup at ISOLDE, where radionuclides are produced by subjecting targets to pulsed proton beams.

5.1 Short half-lives and synchronization with the proton pulses

In case the half-life of the ions of interest is considerably shorter than the time between two proton pulses, it is favorable to synchronize the experiment with the proton impact on the target. I(t) is correlated to the proton impact and contains the time-dependent release characteristics of the ions of interest from the target (see e.g. [39, 40]). A typical release curve has a fast rise compared to a slower falling component, where the corresponding time constants are determined by the chemical properties of the element in question, as well as by the half-life of the ions of interest. It is convenient to set a delay after each proton impact before the start of the cycle to hinder the intake of contaminating ions released faster from the target than the ions of interest. In this situation, one usually probes the falling side of the release curve, which can be approximated with a single time constant as:

$$ I(t) = I_{0} {\text{e}}^{{\frac{ - t}{{\tau_{\text{r}} }}}} . $$
(3)

The quantity τ r, the time constant of the release, is in general different from the lifetime of the ion species. Thus, in addition to the exponential function of the radioactive decay also that of the ion release enters as factor in the argument of the integral. The ion number of a cycle is:

$$ \begin{aligned} N(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) & = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{{(k - 1)t_{\text{loop}} }}^{{(k - 1)t_{\text{loop}} + t_{\text{bg}} }} \left( {I_{0} {\text{e}}^{{\frac{ - t}{{\tau_{\text{r}} }}}} } \right){\text{e}}^{{\frac{t}{\tau }}} {\text{d}}t \\ & = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} I_{0} \mathop \sum \limits_{k = 1}^{n} \mathop \int \limits_{{(k - 1)t_{\text{loop}} }}^{{(k - 1)t_{\text{loop}} + t_{\text{bg}} }} {\text{e}}^{{ - t\left( {\frac{1}{{\tau_{r} }} - \frac{1}{\tau }} \right)}} {\text{d}}t. \\ \end{aligned} $$
(4)

The integral has a closed solution and Eq. (4) simplifies thus to:

$$ N(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) = {\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} I_{0} \frac{{\tau \tau_{\text{r}} }}{{\tau_{\text{r}} - \tau }}\left[ {{\text{e}}^{{t_{\text{bg}} \left( {\frac{1}{\tau } - \frac{1}{{\tau_{\text{r}} }}} \right)}} - 1} \right]\mathop \sum \limits_{k = 0}^{n - 1} {\text{e}}^{{kt_{\text{loop}} \left( {\frac{1}{\tau } - \frac{1}{{\tau_{r} }}} \right)}} . $$
(5a)

As the sum over the terms depending on k is reduced to a geometric series, the number of ions per cycle length (Eq. 2) can be written as:

$$ Q(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) = \frac{{\tau \tau_{\text{r}} }}{{\tau_{\text{r}} - \tau }}I_{0} \left[ {{\text{e}}^{{t_{\text{bg}} \left( {\frac{1}{\tau } - \frac{1}{{\tau_{\text{r}} }}} \right)}} - 1} \right]\frac{{{\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} }}{{{\text{e}}^{{t_{\text{loop}} \left( {\frac{1}{\tau } - \frac{1}{{\tau_{\text{r}} }}} \right)}} - 1}}\frac{{{\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{{\tau_{r} }}}} - {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} }}{{nt_{\text{loop}} + t_{\text{meas}} }}. $$
(5b)

Considering only the terms that are functions of the number of stacking loops n, the optimal number of loops is the solution of the equation:

$$ \begin{aligned} \frac{\text{d}}{{{\text{d}}n}}\left( {\frac{{{\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{{\tau_{r} }}}} - {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} }}{{nt_{\text{loop}} + t_{\text{meas}} }}} \right) & = 0 \\ {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} \left( {\frac{{nt_{\text{loop}} + t_{\text{meas}} }}{\tau } + 1} \right) - {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{{\tau_{\text{r}} }}}} \left( {\frac{{nt_{\text{loop}} + t_{\text{meas}} }}{{\tau_{\text{r}} }} + 1} \right) & = 0 , \\ \end{aligned} $$
(6)

which can be solved numerically. The optimal number of stacking loops is a function of the lifetime of the ions of interest, as well as of the lengths of the loop time, the time for the Penning trap processes, and the release time.

5.2 Long half-lives and no synchronization with the proton pulses

Consider now that the release time of the ions of interest is in the order of or even longer than the time between two proton pulses on the ISOLDE target, which is in the order of a second. In this case, there is no advantage in synchronizing the loop with the proton pulses and I(t) can be regarded as being a statistical distribution, i.e., a mean beam current I 0. In Eq. (3), this is represented by an infinite release time (\( \tau_{r} \to \infty \)). In this case, Eqs. (5a, 5b) thus simplifies to:

$$ Q(n,t_{\text{bg}} ,t_{\text{loop}} ,t_{\text{meas}} ) = I_{0} \tau \left[ {{\text{e}}^{{\frac{{t_{\text{bg}} }}{\tau }}} - 1} \right]\frac{{{\text{e}}^{{\frac{{ - t_{\text{meas}} }}{\tau }}} }}{{{\text{e}}^{{\frac{{ t_{\text{loop}} }}{\tau }}} - 1}}\frac{{1 - {\text{e}}^{{\frac{{ - nt_{\text{loop}} }}{\tau }}} }}{{nt_{\text{loop}} + t_{\text{meas}} }}. $$
(7)

The analog of Eq. (6) then reads:

$$ \frac{{nt_{\text{loop}} + t_{\text{meas}} }}{\tau } + 1 = {\text{e}}^{{\frac{{nt_{\text{loop}} }}{\tau }}} , $$
(8)

which, again, can be solved numerically.

To provide a numerical example, a beam is considered which allows a maximum beamgate period of t bg = 1 ms due to a high yield of contaminating ions. In addition, times of 9 ms in the RFQ and 10 ms in the MR-ToF MS are assumed which sum up to t loop = 20 ms. The time t meas is set to 500 ms (200 ms cooling in preparation Penning trap and 300 ms in precision Penning trap), and the release-time constant is assumed to be τ r = 300 ms. The evolution of Eqs. (5a, b) for a lifetime of τ = 100 ms (domination of radioactive decay) and τ = 500 ms (stronger influence of chemical release) is shown in the left part of Fig. 8. Note that the case τ < τ r implies a further production of the nuclide of interest by radioactive decay of other species in the target after the proton impact (e.g., by alpha decay of heavy nuclides). In the right part of Fig. 8, an example is shown using the same conditions but assuming a (hypothetically) higher repetition rate of the proton impact on the ISOLDE target, so that the production rate of the nuclides is quasi-continuous and Eq. 7 can be used.

Fig. 8
figure 8

Ion number per cycle length (normalized to the maximum of each curve; maxima vary by orders of magnitude) as a function of the number of stacking loops for two different life times of the ions of interest. Left τr = 300 ms, right continuous production

Full size image

6 Conclusion

A new purification/accumulation scheme has been introduced at the ISOLTRAP setup, which combines the MR-ToF MS and the preparation Penning trap. The fast separation of ions provided by the MR-ToF MS reduces the on-line beam time for measurements of heavily contaminated ion beams significantly by repetitive stacking of purified ions in the preparation Penning trap. Depending on the ions of interest, the required experimental time can be reduced by more than an order of magnitude as compared to the use of cycles without accumulation loops. In order to lower the ion energy from 30 or 60 keV as delivered by ISOLDE to the transfer energy of about 3 keV between the ISOLTRAP components at repetition rates of up to several 100 Hz, a new high-power switch has been installed and tested. The new method is most efficient for cases of nuclides with half-lives above several seconds produced with only low yields but accompanied by high numbers of contaminating ions.