Introduction

Mass and lifetime are the basic properties of atomic nuclei. To date, approximately 3400 nuclides have been identified, of which fewer than 300 are stable nuclei concerning radioactive decay [1]. Theoretical predictions indicate the existence of numerous additional particle-bound nuclei [2, 3]. The discovery of new isotopes and measurements of their mass and decay characteristics require powerful radioactive ion-beam facilities and fast, sensitive detection techniques [4]. Utilizing heavy-ion storage rings coupled to radioactive beam lines, time-resolved Schottky mass spectrometry has been proven to be a powerful tool for measuring the radioactive decay of highly charged ions and investigating exotic decay modes [5-8]. Schottky mass spectrometry is based on nondestructive Schottky detectors mounted on storage rings. Currently, only three operational heavy-ion storage-ring facilities exist worldwide: the GSI-ESR (Germany) [9], HIRFL-CSRe (China) [10], and RIKEN-R3 (Japan) [11]. Schottky mass spectrometry was implemented at all three facilities [12-14] and is planned for future storage-ring facilities under construction: the SRing in Huizhou, China [15] and CR-ring in Darmstadt, Germany [16].

The experimental Cooler Storage Ring (CSRe) was commissioned at the Heavy Ion Research Facility in Lanzhou in 2007 [17]. Since its inception, it has been dedicated to advancing the field of isochronous mass measurements by utilizing time-of-flight (TOF) detectors [18, 19]. Owing to the destructive detection mechanism of TOF detectors, the beam lifetime is significantly reduced. Consequently, lifetime measurements using TOF detectors have been limited to the range of several tens to hundreds of microseconds, and one experiment has been conducted for ^94mRu⁴⁴⁺ [20]. For CSRe Schottky mass spectrometry, two parallel metal-plate Schottky detectors were installed in the CSRe [21]. In 2011, these were replaced by a pillbox resonator-cavity Schottky detector [22], similar to that used in ESR [23], to improve the sensitivity of the CSRe Schottky mass spectrometer [24]. In 2014, the first decay-lifetime measurements of radioactive fully ionized ions, ⁴⁹Cr²⁴⁺ and ⁵³Fe²⁶⁺, which were produced by the projectile fragmentation of the ⁵⁸Ni beam, were performed using the CSRe Schottky resonator [25]. Subsequently, the data-acquisition system of the Schottky detector was upgraded to enable long-term continuous data recording [26]. During this period, three test experiments of beam-lifetime measurements were conducted at the CSRe using ⁵⁸Ni, ⁷⁸Kr, and ³⁶Ar beams. In 2020, a versatile baseline-correction method called BrPLS was developed to subtract the background from the signal peaks in measured Schottky spectra [27]. The results showed that the unstable experimental conditions, such as drifts in the ion-revolution frequency and broadening of the frequency distribution caused by a wide momentum distribution, unstable magnetic fields [28], or unstable electron-cooler voltages, have hindered precise beam-lifetime measurements.

Owing to the nondestructive detection feature of Schottky noise detectors, heavy ions can be measured without disturbing the storage rings [29]. The stored ion species are distinguished by their revolution frequencies, which correspond to their mass-to-charge ratios. This is the foundational principle of storage-ring mass spectrometry. The high mass-resolving power of the storage ring can be enhanced using an electron-cooling device and/or a ring with a specially designed isochronous ion-optical configuration [25, 30-34], which is crucial when the nuclei of interest are among a vast number of other ion species stored in the ring simultaneously. Frequency spectra were measured continuously to obtain lifetime measurements using time-resolved Schottky mass spectrometry. The peak area of the revolution frequency in the spectra was proportional to the signal power induced by the corresponding ions in the Schottky detector [22]. By observing the reduction in the peak area, the decline in ion numbers as a function of storage time can be monitored, and the decay half-life can be deduced [6].

Traditionally, parallel-plate-type capacitive Schottky detectors have been used to detect electromagnetic signals induced by passing ions [35, 36]. Higher sensitivity can be achieved using a resonant cavity [37]. The main advantage of using resonant cavities as Schottky pickups is their increased sensitivity to the characteristic resonance frequency of the cavity; thus, a higher signal-to-noise ratio can be achieved [22, 23, 14]. Moreover, the higher resolution obtained at higher frequencies provides an added benefit [38]. These enhancements make them suitable devices for the fast detection of low-yield exotic isotopes, even for single ions. For example, this detector was used to directly monitor the decay from parent to daughter nuclei in single-ion decay measurements [39].

Individual ion counting is not possible in many-ion decay half-life measurement experiments (see review article [40] and references therein). Therefore, the calibration of the peak area is necessary before the corresponding particle count can be accurately determined. This calibration involves understanding the sensitivity of a Schottky detector system, which varies with frequency and can be characterized by an amplitude-frequency characteristic (AFC) curve. Compared with parallel-plate detectors, the cavity-resonator detector has an enhanced signal from ions and an elevated background-noise level near the resonance frequency. This increased sensitivity results in a more pronounced sensitivity change across the same frequency range.

Typically, an electron-cooling device is used in a storage ring to stabilize the frequency of heavy ions [41-44]. However, the interaction between the stored ions and electron beam from the electron-cooling device introduces additional ion-beam loss, which alters the decay constant of the ions in the storage ring [45]. When electron cooling is absent or deactivated, the signal peak of a particular ion species broadens over a wide revolution-frequency range because of the large momentum spread, and shifts owing to the ongoing energy loss from collisions with the residual gas in the vacuum pipe [25]. Thus, the amplitude-frequency variation of the detector system can distort the ion-peak shapes, leading to peak areas that do not directly correspond to the actual number of ions present.

In this study, a new method is developed to calibrate the AFC curve (sensitivity curve) of the Schottky detector system installed in the CSRe storage ring. This method, combined with the background-noise subtraction technique developed in our previous study [27], has enabled the precise determination of the storage lifetime of stable ⁵⁶Fe²⁶⁺ ions in the ring, despite the broad frequency spread and unexpected frequency drifts. Once the revolution-frequency spectra are normalized to the AFC curve, we can restore the peak shape and determine the peak center with better precision. This method improves the precision and enhances the capability of beam-lifetime measurements using many ions in the ring and is complementary to single-ion decay measurements [39]. When the electron-cooling system is turned off, the ion beam gradually loses energy. By tracking the rate of change in the central frequency of the beam, we determine the effective vacuum level experienced by the ions stored in the storage ring. The methodology developed in this study will be adapted for the future SRing facility in Huizhou, China. It can also be adapted by other heavy-ion storage-ring facilities to improve precision and enhance the capability of beam-lifetime measurements using many ions.

Time-resolved Schottky mass spectrometry at heavy-ion storage ring

A schematic of the Schottky detection system at the CSRe facility is shown in Fig. 1. As the ions pass through the cavity, they induce image charges and deposit energy, creating Schottky noise. This noise was extracted from the cavity using magnetic couplers [23], amplified using a low-noise amplifier, and finally recorded using a spectrum analyzer. The measured data were analyzed online or offline revealing peaks at the ion revolution frequency at each harmonic in the frequency domain (hereafter referred to as the frequency spectrum), obtained using Fourier transformation.

Fig. 1Full-screen View

(Color online) Schematic illustration of the resonator Schottky detector system and its corresponding equivalent RLC circuit [22]. The ion signal is captured by the Schottky cavity, amplified by a low-noise amplifier, filtered by a band-pass filter, and further amplified before being transmitted to the frequency analyzer via a 15-m coaxial cable. The signal is then digitized by the spectrum analyzer and stored by the IQ data recorder

2.1

Reference baseline in the measured frequency spectrum

Within the bandwidth of the detection system, the measured noise power includes both the signal power $P_{\text{ion}}\left(f\right)$ from the circulating ions in the ring and the thermal noise power $P_{\text{thermal}}\left(f\right)$ of the detection system: $P_{\text{total}}\left(f\right)=P_{\text{thermal}}\left(f\right)+P_{\text{ion}}\left(f\right)\mathrm{.}$ (1) Owing to the resonant nature of the cavity, the thermal noise exhibits a Lorentzian-like distribution in the frequency domain. Experimentally, the power-density profile of the thermal noise in the frequency domain can be measured when the beam is off [27]. The result of the averaged spectrum can serve as a benchmark for the reference baseline (blue histogram in Fig. 2) and is subtracted from the frequency spectrum measured when the beam is present. Alternatively, the baseline can be estimated using the method described in Ref. [27] (orange line shown in Fig. 2 and Fig. 3). The advantage of this method is that baseline measurements can be performed in situ when the beam is present, and a smooth reference baseline can be obtained.

Fig. 2Full-screen View

Blue histogram: the measured thermal noise in the frequency domain when no beam is present in the CSRe. It can be used as a reference baseline in the subsequent measurement to extract the signal power induced by the storied ions. Orange line: the estimated baseline determined using the method of [27]. The reference level is set to -50 dBm, data acquisition time is 852.5 ms, frequency resolution is 0.92kHz, and DAQ sampling rate is 3.75 MSamples/s

Fig. 3Full-screen View

(Color online) Top panel: Time-resolved Schottky spectrum of electron-cooled ⁵⁶Fe²⁶⁺ beam with energy of 13.6 MeV/u. The velocity of the ions was shifted in steps such that the revolution-frequency change covers the entire measurement-frequency range of 3 MHz. The time resolution is 86 ms/channel, and the frequency resolution is 0.92 kHz/channel. Middle panel: Single frame from the top panel at time = 0.86 s, including 5 harmonics h = 614, 615, ..., 618. The blue line indicates the original power density of the spectrum. The orange line is the estimated baseline [27]. Bottom panel: The same spectrum as the middle frame after the baseline was subtracted

Calibration of the AFC function

The resonant Schottky cavity detector installed in the CSRe has a pillbox design, as shown in Fig. 1. Owing to the characteristics of the equivalent RLC circuit, an AFC form similar to that of the cavity [23, 22, 48, 49] was used for modeling the AFC function of the entire detector system: $\left|H(f)\right|=\frac{R_{\text{sys}}\gamma \sqrt{\zeta }}{\sqrt{1+Q_{\text{sys}}^2{\left(\frac{f}{f_{\text{sys}}}-\frac{f_{\text{sys}}}{f}\right)}^2}}\mathrm{,}$ (4) where R_sys is the resistance of the equivalent RLC circuit of the entire system, ζ is the loss factor quantifying the energy loss to the wake fields [50], and γ is the relativistic Lorentz factor of the ions. In addition, Q_sys and f_sys are the effective quality factor and resonant frequency of the system, respectively. The AFC curve of the Schottky cavity was measured offline using a network analyzer. In contrast, determining the AFC curve for the entire detection system requires online beam experiments. The resulting AFC curve is influenced not only by the AFCs of the cavity, electronic components, and cables, but also by the AFC of the spectrum analyzer. The AFC of the system can be determined by leveraging the features of multiple peaks of the same ion species that appear simultaneously at several frequency harmonics. If we can simultaneously measure at least two frequency harmonics of the same ion species in the spectrum, we can determine the relative sensitivity ratio between the two harmonic frequencies within the same spectrum.

The optimal method for measuring the AFC curve employs electron-cooled low-energy beams. Low ion energy leads to small frequency intervals between adjacent harmonics. Electron cooling is essential to ensure that the ion peak is sufficiently narrow such that the corresponding AFC region under it can be considered constant. Consequently, multiple revolution-frequency harmonics can be detected simultaneously within the frequency range of the data acquisition (DAQ) system. By adjusting the electron-cooler voltage, we can shift the center frequencies in steps to collect more data points along the AFC curve, thereby enabling the determination of the entire curve within the measured frequency range.

3.1

Calibration measurement

In the experiment, the electron-cooled ⁵⁶Fe²⁶⁺ ions at an energy of 13.6 MeV/u were utilized to measure the AFC curve within the 3-MHz bandwidth around the resonant frequency of the Schottky detector system. The measured time-resolved frequency spectrum is presented in Fig. 3. Up to six harmonics of the revolution frequency were simultaneously covered. By adjusting the voltage of the electron cooler from 126.6 kV to 129.8 kV, the velocity of the ions was altered in 13 increments [51, 52]. Consequently, the center frequency of each harmonic was shifted by approximately 600 kHz.

As the number of stored ions decreases because of inevitable particle losses as $N=N_0{e}^{-{\lambda }_{\text{t}}t}$, the integrated power decreases as a function of time: $P(t)=P_0{e}^{-{\lambda }_{\text{t}}t}\mathrm{,}$ (5) where λ_t is the decay constant, $P_0=2N_0{\left[\left|H\left(h{f}_{\text{rev}}\right)\right|qe{f}_{\text{rev}}\right]}^2$, and N₀ is the initial number of ions. The storage lifetime τ of the ions can be obtained from the decay constant using τ=1/λ. Based on Eq. (3)–(5), the integrated power of the ion beam at each harmonic in the spectrum can be expressed as a function of the center frequency of the harmonic $f=h{f}_{\text{rev}}$ and time t, denoted as $P\left(f\mathrm{,}t|A_0\mathrm{,}{\lambda }_{\text{t}}\mathrm{,}{Q}_{\text{sys}}\mathrm{,}{f}_{\text{sys}}\right)=\frac{A_0}{1+Q_{\text{sys}}^2{\left(\frac{f}{f_{\text{sys}}}-\frac{f_{\text{sys}}}{f}\right)}^2}{e}^{-{\lambda }_{\text{t}}t}\mathrm{,}$ (6) where $A_0=2N_0\zeta {\left(R_{\text{sys}}\gamma qe{f}_{\text{rev}}\right)}^2$ denotes the initial peak area. Parameters A₀ and λ_t are related to the properties of the ion beam, whereas Q_sys and f_sys are uniquely associated with the attributes of the detector system.

The specific data-processing steps are as follows:

1. the reference baseline is estimated and subtracted to generate the background-free spectrum using the method described in Ref. [27]. See Fig. 3.

2. For each spectrum at time tj (j=1, 2, ..., n), the peak area is calculated by simple integration for each peak at different harmonics. Here, n is the number of bins in time during the continued spectrum-recording process. This yields the experimental value $P_{i\mathrm{,}j}^{\text{exp}}$ at frequency f=fi (i=1, ..., 6 harmonic) and time t=tj, see Fig. 4.

Fig. 4Full-screen View

(Color online) Time evolution of the peak of ⁵⁶Fe²⁶⁺ ions near 242.33 MHz (harmonic number h = 616): center frequency (top), peak width (middle), and peak area (bottom). At approximately 63 s, when the center frequency of the harmonic changes abruptly, the peak width is also affected. Therefore, we only utilized the 14 data segments when the center frequency was stable, as indicated by the orange regions

3. Data segmentation. Refer to the next subsection for more details.

4. The function $P\left(f\mathrm{,}t|A_0\mathrm{,}{\lambda }_t\mathrm{,}{Q}_{\text{sys}}\mathrm{,}{f}_{\text{sys}}\right)$ based on Eq. (6) is fitted to the experimental value $P_{i\mathrm{,}j}^{\text{exp}}$ to obtain the parameters A₀, λt, Q_sys, and f_sys.

5. The values of Q_sys and f_sys are implemented into Eq. (4) to obtain the AFC curve of the detection system. These two parameters are independent of the ion beam and can be used throughout the experiment if the detector settings were fixed.

3.2

Error estimation of Q_sys and f_sys

The function $P\left(f\mathrm{,}t|A_0\mathrm{,}{\lambda }_t\mathrm{,}{Q}_{\text{sys}}\mathrm{,}{f}_{\text{sys}}\right)$ in Eq. (5) is a bivariate nonlinear function. In the fitting procedure, the task is to determine the optimized combination of parameters that minimizes the following objective function: $S={\displaystyle \sum _j}{\displaystyle \sum _i}{\left(P_{i\mathrm{,}j}^{\text{exp}}-A_0{e}^{-{\lambda }_{\text{t}}{t}_j}{\left(1+Q_{\text{sys}}^2{\left(\frac{f_i}{f_{\text{sys}}}-\frac{f_{\text{sys}}}{f_i}\right)}^2\right)}^{-1}\right)}^2\mathrm{.}$ (7) Although the processing steps are straightforward, numerous challenges exist. These include:

a. Data segmentation. A₀ in Eq. (7) must be accurately determined. During the measurement, we adjusted the electron cooling 13 times. Figure 4 illustrates the changes in the center frequency, width, and area (i.e., average ion power) of the ion peaks of the 616th harmonic. Upon completion of the electron-cooling adjustment, the center frequency and peak-width fluctuate significantly. Within a few seconds, the velocity of the ions reach equilibrium, and the peak position stabilizes. Consequently, the data are filtered into 14 segments, indicated by the orange regions in Fig. 4, where the equilibrium was reached. Instead of using a single initial peak-area parameter A₀, fourteen independent initial peak areas $A_{\mathrm{0,}k}\left(k=\mathrm{1,}\mathrm{2,}\mathrm{...,}14\right)$ are assigned to the corresponding data segments. The timing tj in each data segment is also readjusted accordingly to the new starting points so that $A_{\mathrm{0,}k}$ is encountered at tk,j=0. Hence, the objective function of the fitting is altered to: $S={\displaystyle \sum _{k=1}^{14}}{\displaystyle \sum _{j=1}^{n_k}}{\displaystyle \sum _{i=1}^{N_k}}{\left(P_{k\mathrm{,}j\mathrm{,}i}^{\text{exp}}-A_{\mathrm{0,}k}{e}^{-{\lambda }_t{t}_{k\mathrm{,}j}}{\left(1+Q_{\text{sys}}^2{\left(\frac{f_{k\mathrm{,}i}}{f_{\text{sys}}}-\frac{f_{\text{sys}}}{f_{k\mathrm{,}i}}\right)}^2\right)}^{-1}\right)}^2\mathrm{,}$ (8) where Nk represents the number of measured harmonics in the spectrum of each data segment.

b. Monte Carlo calculations are used to estimate the value and error of the fitting parameters: Q_sys, f_sys, λ_t, and $A_{\mathrm{0,}k}\left(k=\mathrm{1,}\mathrm{2,}\mathrm{...,}14\right)$. Part of the results is shown in Fig. 5. The estimated value and error of each parameter are the mean and variance of its distribution, respectively.

Fig. 5Full-screen View

Parameter distributions obtained by Monte Carlo calculations: the probability distributions of parameters (a) Q_sys, (b) f_sys, and (c) λ_t. The mean and variance of all the parameters are calculated using the Gaussian distribution with mean value μ and sigma value σ

3.3

Fitting results of the AFC function

The characteristic parameter values of the AFC of the CSRe Schottky system were determined to be Q_sys=374.0±0.9 and the resonant frequency $f_{\text{sys}}=242.3684\pm 0.0007$ MHz, as illustrated in Fig. 5. The resulting peak areas calculated using the obtained AFC parameters are shown as orange dots in Fig. 6. The fitting residuals are uniformly distributed around zero, indicating a good estimation of Q_sys and f_sys. Notably, owing to the larger error in the peak-area determinations, a wider spreading of the residuals is evident at frequencies close to the resonance, near 242.36 MHz.

Fig. 6Full-screen View

Top panel: comparison between the measured (blue dots) and calculated peak areas (orange dots) using fitted AFC function of the CSRe Schottky detector system. A sudden decrease in the data points is observed owing to the frequency range being passed through twice during the measurements, as shown in Fig. 3. Bottom panel: differences between the measured and fitted peak area

The measured Q value of the detection system is smaller than that of the Schottky cavity, Q_cavity = 496.8 [22]. This is reasonable because other components of the system can deteriorate the resonance sensitivities. To achieve a higher Q_sys, and thus, a higher sensitivity of the heavy-ion detection, new Schottky resonant cavities without ceramic beam pipes are under construction for next-generation SRing facilities [15]. Thus, a Q_cavity four times higher can be reached with a stainless-steel cavity and 20 times higher with a copper-coated cavity.

Application of the AFC function to the storage-lifetime measurement of ⁵⁶Fe²⁶⁺ ions at different energies

After the AFC curve of the Schottky detector system was determined, we measured the storage lifetime of the fully ionized ⁵⁶Fe²⁶⁺ ions under the same detector settings. The decay constant λ of the ions was determined from the normalized peak area in the revolution-frequency spectrum, and the storage lifetime was converted from the decay constant by τ=1/λ. The results are summarized in Table 1.

Two beam-energy settings were used, and the electron cooling only had an ON or OFF status. No voltage or current adjustments were made during the ON state, which was different from the measurement procedure used for AFC determination. The first energy was set to 13.7 MeV/u, which was the same as the setting for the AFC curve measurement. The measured decay constant of the electron-cooled ⁵⁶Fe²⁶⁺ ions was $0.0476(1){\text{min}}^{-1}$, which was consistent with ${\lambda }_{\text{t}}=0.054(25){\text{min}}^{-1}$ determined during the AFC-determination measurement (see Fig. 5c). Table 1 shows that the decay constant of the fully ionized ⁵⁶Fe²⁶⁺ is smaller when electron cooling is switched off. This is because the electron beam of the cooler introduces an additional beam-loss mechanism to the stored ion beam.

When the electron cooler was ON, the decay constants of the stored ⁵⁶Fe²⁶⁺ ions at the higher beam energy of 116.4 MeV/u were more than one order of magnitude smaller than those at 13.7 MeV/u. After electron cooling was switched off, the decay constant decreased by another order of magnitude at 116.4 MeV/u. This indicates that at this beam energy, the main contribution to ion loss is from the electron-cooling process. Assuming realistic parameters of the CSRe cooler: an electron-beam vertical temperature of 0.5 eV, electron current of 0.2 A, electron-beam radius of ~4 cm, vacuum of ~2×10^-10 mbar (see Sect. 4.2), and room temperature of 20 ℃ inside the vacuum pipe, the calculated theoretical decay constants agree well with the measured values, both at high energy [53-58] and low beam energy [53, 59, 58], as shown in the last two columns of Table 1.

The importance of applying the AFC curve in lifetime measurements is clearly demonstrated by the set of measurements with a beam energy of 116.4 MeV/u. The subsequent sections utilize the findings from this beam setting to elucidate the indispensability of the AFC curve for many-ion storage-lifetime measurements in the ring.

4.1

Correcting for sudden peak-area changes

Figure 7 illustrates the Schottky spectrum of the electron-cooled ⁵⁶Fe²⁶⁺ ion beam at 116.4 MeV/u. The center frequency of the peak unintentionally underwent five abrupt shifts, as shown in Fig. 7(b). In the original spectra, before applying AFC normalization, this directly resulted in the corresponding rapid change and restoration of both the peak center frequency and peak area within 2 min, as depicted in Fig. 7(b) and (c). Considering the data affected by these five perturbations, the ion-storage lifetime was determined to be τ=144(8) min by using the entire dataset. Excluding the peak area affected by the five perturbations, we deduced τ=145(2) min with the reduced dataset. After the AFC curve was used to normalize the ion-peak areas, as shown in Fig. 7(d), the effects of perturbations were mitigated. The normalized peak area followed an exponential decay trend despite frequency shifts. This indicated that the swift frequency shifts did not introduce an additional loss of ions during the experiment. The ion-storage lifetime derived from the normalized peak area was 145(2) min. This result was consistent with that obtained using the reduced dataset. The accuracy was enhanced by 4%, not only because of the increased statistics in comparison to those derived from the reduced dataset, but also because of the accurate determination of the peak area utilizing the AFC-normalization process. The five observed perturbations can be attributed to the sudden charging and discharging of the high-voltage power supply of the electron cooler. In the case of similar frequency drifts, the AFC curve serves as a key for correctly and precisely determining the storage lifetime of the stored ions.

Fig. 7Full-screen View

(Color online) (a) Time-resolved Schottky spectrum of ⁵⁶Fe²⁶⁺ at beam energy of 116 MeV/u with electron-cooling switched on. The horizontal axis is the observation time, the vertical axis represents frequency, and the power density is represented by color. The frequency resolution is 0.12 kHz/channel, and the time resolution is 6.39 s/bin. The evolutions of the (b) peak center frequency and peak area (c) before and (d) after AFC-normalization as a function of time are also shown. Five jumps are clearly observed during the acquisition period

4.2

Effective vacuum experienced by the stored ions

After the electron cooling is turned off, the energy loss of the beam caused by collisions between ions and the residual gas can no longer be compensated for. Consequently, the momentum of the beam ions decreases, and the momentum spread increases gradually. The absolute rate of frequency change df/dt is proportional to the effective vacuum experienced by the stored ions. Using the AFC normalized spectrum, the peak shape can be restored, and the peak center frequency can be determined with better precision. The effective vacuum can be derived from the obtained df/dt, and the storage lifetime of the ion can be obtained from the normalized peak area, which decreases as a function of storage time.

As shown in Fig. 8, upon deactivating the electron cooling (at approximately 0.8 s), the ion-momentum spread began to increase and the center frequency of the ion peak drifted towards lower frequencies. Based on the measured AFC-curve in Fig. 6, the frequency drifts towards the region where the maximum AFC of the system is situated. This is why the peak area, which is deduced from the spectra before AFC normalization, increases over time, as shown in Fig. 8 (c). In contrast, the decreasing trend was restored, and the storage lifetime was determined to be 1954(139) min from the normalized peak area using the AFC curve of the detector system, as shown in Fig. 8 (d).

Fig. 8Full-screen View

(Color online) (a) Time-resolved Schottky spectrum of ⁵⁶Fe²⁶⁺ at beam energy of 116 MeV/u. Electron cooling is switched off at approximately 0.8 s. The evolutions of the (b) peak center frequency and peak area (c) before and (d) after AFC-normalization as a function of time are also shown. After the electron cooler was switched off, the frequency spread increased and the center frequency gradually drifted toward lower frequencies. The frequency resolution is 0.24 kHz/channel, and the time resolution is 0.32 s/bin

Based on the rate of the peak center-frequency change observed in Fig. 8(b), df/dt≈23 Hz/s, the ion-energy loss rate dE/dt can be deduced using $\text{d}f/f=\left({\gamma }^{-2}-{\gamma }_{\text{t}}^{-2}\right)\text{d}E/E$, where γ_t=2.457 and γ is the Lorentz factor of the ions. This calculation yields a value of dE/dt≈1.89 keV/s. Assuming a room temperature of 20 ℃ and a measured composition of the residual gas (90% H₂, 5% CO, and 5% N₂) inside the vacuum pipe [60], the equivalent vacuum inside the CSRe beam pipe can be estimated using the Lise++ program [61] by considering the energy loss of ions through the material. This was determined to be approximately 2×10^-10 mbar. This value represents the effective vacuum experienced by stored ⁵⁶Fe²⁶⁺ ions. In total, 11 ultrahigh vacuum gauges were used in the experiment and were distributed evenly along the CSRe ring. Most of the gauge readings were at the level of 10^-11 mbar except for that near the internal target area, which was ≈4×10^-10 mbar. The measured effective vacuum experienced by the stored ions was in agreement with the vacuum-gauge readings.

With a derived effective vacuum of 2×10^-10 mbar, the theoretical beam-loss rate owing to ion-gas collisions [58] was calculated to be 3×10^-4 min^-1, corresponding to a storage lifetime of approximately 3333 min. This value was more than two times higher than the measured storage lifetime of 1359(107) min when electron cooling was OFF (see Table 1). Theoretical calculations of the ion-storage lifetimes involve a wide range of parameters for collisions between the ion beam and residual gas inside a vacuum pipe [58]. These parameters change over time. The parameters used in the theoretical calculations, such as the temperature and gas composition inside the vacuum pipe, may differ from the actual parameters of the CSRe ring during experiments [60]. However, the experimental results were consistent in magnitude with the theoretical estimates. This confirms the reliability of our methodology and indicates that the vacuum levels inferred from the ion-frequency drifts accurately reflect the actual vacuum conditions within the storage ring.

Summary and outlook

In this study, we developed a novel method to calibrate the AFC curve of the Schottky detection system at the CSRe storage ring. Following calibration, a significant improvement in the accuracy of the peak position and peak-area determination was observed in the revolution frequency spectra of the many ions stored in the ring. Using the AFC-normalized Schottky frequency spectra, the storage lifetimes (decay constants) of the ⁵⁶Fe²⁶⁺ ions were determined with high precision at 13.7 and 116.4 MeV/u, despite frequency shifts or frequency-spreading increases during the experiment. The experimental-decay constants of the ions stored in the ring aligned with the theoretical calculations. Additionally, when electron cooling was switched off, the effective vacuum experienced by the stored ions was deduced from the frequency-drift rate. The results showed that the vacuum was of the order of 10^-10 mbar, which is consistent with the order of magnitude displayed by the vacuum gauge. The vacuum condition is one of the key parameters that limits the storage lifetime of highly charged ions in the storage ring. Our findings suggest that stronger vacuum pumps are required at the internal target position of the CSRe if a longer beam lifetime is required.

This method serves as a useful tool in storage-ring Schottky frequency-spectrum mass spectrometry to obtain accurate ion lifetimes. When electron cooling is enabled, this method can mitigate the negative effects of the irregularities caused by electron-cooling device instabilities. When the electron cooling is turned off, collisions with the residual gas cause the ions to lose energy. This results in a broadening of the ion-revolution frequency and shift in the frequency center. Accurate and reliable lifetime results were obtained using the method developed in this study. Thus, the method improved precision, increased the capability of beam-lifetime measurements with many ions in the ring, and complemented the single-ion decay-measurement method. This new method extends the lifetime-measurement method to uncooled beams. The validity and usefulness of this method were demonstrated in this study by the precise storage lifetime measurement of a stable ⁵⁶Fe²⁶⁺ ion beam. The method developed at the CSRe can easily be adapted to other running storage-ring facilities, such as the ESR at GSI in Darmstadt, Germany [9] and the R3-Ring at RIKEN in Saitama, Japan [11, 62].

One focus of next-generation Schottky mass spectrometry is the simultaneous measurement of the masses and lifetimes of short-lived ions [40]. Disabling electron cooling can extend the applicability of the technique from the minute range to the tens-of-milliseconds range for the fast broadband detection of exotic nuclei. Moreover, new Schottky detectors with higher particle sensitivities and transverse position measurement capability have been built for next-generation storage rings [16]. In this context, the amplitude normalization of the frequency spectra using the AFCs of the Schottky detector system can play a significant role in precise mass and life measurements using many ions. The methodology developed in this study will serve as a foundational component for the advancement of Schottky mass spectrometry, for example, at SRing [22, 63], which is anticipated to be pivotal for the discovery and precise measurement of exotic nuclei in future endeavors [64].