Introduction

Mass is a fundamental property of atomic nuclei and plays a crucial role in studies of nuclear structure and nuclear astrophysics [1, 2]. Significant efforts has been devoted to developing mass-measurement technologies, particularly for short-lived nuclides [3, 4]. Isochronous mass spectrometry (IMS) [5, 6] on heavy-ion storage rings has proven to be a powerful tool, providing numerous new mass values for short-lived nuclides [7-12].

In conventional IMS equipped with a single time-of-flight (TOF) detector, nuclear masses are determined by measuring the revolution times (T) of ions stored in the ring. Given that a storage ring operates within a certain acceptance of magnetic rigidity Bρ (or momentum p), stored ions of the same species exhibit a velocity spread. Although ions with higher velocities are expected to circulate in the ring along larger orbital paths and vice versa, the revolution time of ions in different orbits will still differ if the velocity variation is not perfectly compensated by variations in orbital length. This velocity spread results in a broad distribution of T. The dependence of T on velocity can be effectively removed for ions that satisfy isochronous conditions [5]. However, for most ions, the velocity dependence of T leads to significant deterioration in mass precision and may introduce systematic errors [13] in the measured masses.

To reduce the effect of velocity spread on the T distribution and enhance the performance of IMS, several methods have been proposed. One such method, known as Bρ-tagging IMS [14], employs a mechanical slit at the dispersive focal plane of the beam line or in the ring to restrict the Bρ acceptance, thereby narrowing the T distribution [14, 15]. However, this approach reduces transmission efficiency and is unfavorable for mass measurements of exotic nuclei with extremely low production yields. Another method, referred to as Bρ-defined IMS or Bρ-IMS [16, 13], employs two TOF detectors to measure the velocity of ions directly and determine Bρ values [17]. With the measured Bρ and velocity, the mass-to-charge ratio m/q of the ions is calculated, resulting in improved mass-resolving power [16, 13]. However, compared to conventional IMS utilizing only one TOF detector, Bρ-IMS introduces double the energy loss for ions during each revolution. Since energy loss increases with proton number, heavier ions experience more pronounced losses. Repeated traversal through the carbon foils of the two detectors can cause the ion momentum to exceed the storage ring’s acceptance range, leading to ion loss and a significant reduction in storage time, complicating the precise determination of ion velocity and Bρ. Additionally, the installation of two TOF detectors in a straight section of the ring presents practical challenges, such as the precise determination of the distance between detectors and managing time delay differences [18]. Furthermore, a previous study revealed that using a single oscilloscope to record flight time signals is necessary to avoid jitter at the start times of two oscilloscopes [19]. However, using only one oscilloscope introduces other issues, such as signal attenuation due to longer cables and excessive writing times caused by the large volume of data being recorded.

In this paper, we introduce a new method called Tune-IMS to improve conventional IMS. The motion of a charged particle with nonzero emittance in a storage ring exhibits betatron oscillations around a closed orbit [20, 21]. The number of oscillations per revolution, known as the tune (Q) value, is directly related to the ions’ Bρ and their revolution time T [21]. A previous study [22] proposed a method for measuring Q using a single TOF detector. By using the correlation between T and Q for each ion species, the measured T can be transformed into a new revolution time, TQ, corresponding to a reference closed orbit with a fixed Q. As a result, for ions that do not satisfy the isochronous condition, the new TQ distribution is narrower, leading to improved mass-resolving power.

In this study, we briefly present the principle of this method and apply it to an IMS experiment [23, 24] conducted at the experimental cooling storage ring (CSRe) of the Heavy Ion Research Facility in Lanzhou (HIRFL-CSR). The re-determined mass values of ⁶³Ge, ⁶⁵As, ⁶⁷Se and ⁷¹Kr with higher precision were compared with those from a recent Bρ-IMS experiment [25, 26]. The Coulomb displacement energies (CDEs) extracted from these new mass values were also compared with theoretical estimates.

Principle of Tune-IMS

Under ideal conditions in a stable magnetic field, the magnetic rigidity Bρ, revolution time T, and tune Q are uniquely determined for a specific ion circling a closed orbit. If the closed orbit of the same ion is slightly altered, the corresponding variation in T with Bρ (denoted as Δ T and Δ Bρ, respectively) is described by [27]: $\frac{\text{Δ}T}{T}=\left(\frac{1}{{\gamma }_{\text{t}}^2}-\frac{1}{{\gamma }^2}\right)\frac{\text{Δ}B\rho }{B\rho }=-\eta \frac{\text{Δ}B\rho }{B\rho }\mathrm{,}$ (1) where ${\gamma }_{\text{t}}$ represents the transition point of the ring, γ is the Lorentz factor, and $\eta \equiv 1/{\gamma }^2-1/{\gamma }_{\text{t}}^2$ is the phase-slip factor. As shown in Eq. (1), for ions satisfying the isochronicity condition where $\gamma \approx {\gamma }_{\text{t}}$ (or η≈ 0), Δ T is independent of Δ Bρ, achieving a relatively high resolution of T. However, for most ions where η≠ 0, variations in Bρ results in corresponding changes in T. This leads to a deterioration in T which may introduce systematic deviations in the obtained mass values [13].

In contrast, the tune number Q represents the betatron oscillations around the closed orbit. In a linear storage ring equipped solely with dipoles and quadrupoles, the ratio of variations in Q (denoted as Δ Q) to variations in Δ Bρ/Bρ (or equivalently, Δ p/p), is defined as the natural chromaticity ξ [28]: $\xi \equiv \frac{\text{Δ}Q}{\text{Δ}B\rho /B\rho }\mathrm{.}$ (2) According to Eqs. (1) and (2), the correlation between T and Q can be expressed as follows: $\text{Δ}T=-\eta T\frac{\text{Δ}Q}{\xi }\mathrm{.}$ (3) Thus, each (T, Q) set corresponding to the closed orbit can be approximated as (TQ, Q₀), where TQ corresponds to a reference closed orbit with a fixed tune value, Q₀: $T_Q=T-\text{Δ}T=T\left(1+\eta \frac{\text{Δ}Q}{\xi }\right)\mathrm{.}$ (4) Thus, if Δ Q, ξ and η are properly determined, the effect of Bρ spread on T can be corrected.

Experimental results

3.1

Measurement of T

The new tune-IMS method was applied to an early IMS experiment conducted at the HIRFL-CSR [23, 24]. In the experiment, a ⁷⁸Kr²⁸⁺ beam was accelerated to 483 MeV/u and bombarded with a 15-mm thick beryllium target. The fragments, predominantly bare ions, were selected using the second Radioactive Ion Beam Line at Lanzhou (RIBLL2) and injected into the CSRe (see Fig. 1). The magnetic rigidity Bρ of the RIBLL2-CSRe system was set to 5.9464 Tm, and the transition point ${\gamma }_{\text{t}}$ of CSRe was set to 1.406.

Fig. 1Full-screen View

(Color online) Layout of RIBLL2 and CSRe, along with a schematic diagram of the TOF detector [11]

A TOF detector equipped with a 19-μg/cm² thin carbon foil was installed in the CSRe [29]. Each ion passes through the detector, releasing secondary electrons from the foil, which are then transmitted isochronously through perpendicularly arranged electric and magnetic fields to a microchannel plate (MCP) counter. The electrical signal from the MCP anode was directed to a fast digital oscilloscope for full-waveform sampling without amplification. The detection efficiency of the signal at each revolution turn mainly depends on the atomic number of the ions. After each injection, a measurement time of 200 μs was set, corresponding to approximately 300 turns for the ions stored in the CSRe. For each ion, a time sequence t(N) as a function of turn number N was obtained.

To account for energy loss, a third-order polynomial was fitted to the t(N) sequence. In this case, T is the first derivative of t(N) at a specific N, such as the middle of the ion’s full storage period.

In reality, the magnetic field instability can induce drifts in T. This study used the method reported in Refs. [30] to correct T drifts caused by unstable magnetic fields. Compared to the previous approach [24], this method can handle injections with only a few stored ions. A portion of the obtained T-spectrum is shown in Fig. 2.

Fig. 2Full-screen View

(Color online) Revolution time spectrum within the range of 605 ns ≤ T ≤ 629 ns. Different colours indicate nuclei with different isospin projections T_z=(N-Z)/2. The N=Z nuclei with T ~ 620 ns are not shown due to difficulties in particle identification

3.2

Determination of Q and ξ

The method for measuring Q and ξ using a single TOF detector is described in detail in Refs. [22]. The remainder of this paper is organized as follows:

Due to betatron oscillations, the time sequence t(N) recorded by the TOF detector includes an oscillation term [17]. Using a ⁵¹Fe²⁶⁺ ion measured in this experiment as an example, Fig. 3 shows the results obtained by applying a Digital Fourier Transform (DFT) [31, 22] to the residuals of t(N) fitted with a third-order polynomial. Three significant peaks, labeled A, B, and C, were identified, indicating three periodic components within the time sequence. This analysis was applied to all recorded time sequences, consistently revealing a three-peak pattern.

Fig. 3Full-screen View

Result of the Fourier transform applied to the fitting residuals of the time sequences for a ⁵¹Fe²⁶⁺ ion is shown. Three significant peaks, labeled A, B, and C, were identified within the frequency range $0\le f\le 0.5$. Since the time sequence is sampled once per revolution, the DFT analysis can only detect periodic components within the frequency range of 0 to 0.5, according to the Nyquist-Shannon sampling theorem [32]

A detailed study of the origin of these peaks can be found in Ref. [22]. To accurately model the three periodic components, a comprehensive fitting function [17] was employed to refit the passing time sequences as follows: $\begin{array}{c}{t}_{\text{fit}}\left(N\right)=a_0+a_1\cdot N+a_2\cdot N^2+a_3\cdot N^3\\ +A_{\text{A}}\cdot \sin \left[2\pi \cdot \left(Q_{\text{A}0}\cdot N+Q_{\text{A}1}\cdot N^2\right)+{\varphi }_{\text{A}}\right]\\ +A_{\text{B}}\cdot \sin \left[2\pi \cdot \left(Q_{\text{B0}}\cdot N+Q_{\text{B}1}\cdot N^2\right)+{\varphi }_{\text{B}}\right]\\ +A_{\text{C}}\cdot \sin \left[2\pi \cdot \left(Q_{\text{C}0}\cdot N+Q_{\text{C}1}\cdot N^2\right)+{\varphi }_{\text{C}}\right]\mathrm{.}\end{array}$ (5) Here, sine-like terms account for the oscillations, where Ai and ϕi (i corresponds to A, B, C) represent the amplitudes and initial phases of the periodic components, respectively. The first derivative of each phase term, $Q_i=Q_{i0}+2Q_{i1}\cdot N\mathrm{,}$ (6) corresponds to the fractional tune at each N [17], where $Q_{i0}$ is the fractional tune value and $Q_{i1}$ is a factor accounting for energy loss as the ion passes through the carbon foil of the TOF detector [17]. Similar to the determination of T, the middle turn number was used to determine Qi.

The accuracy of the measured Qi value depends on the amplitude Ai of the oscillation component, which is proportional to the square root of the emittance [22]. For ions with low emittance, the amplitude of the betatron oscillation is small, resulting in low peaks in the P(f)-spectrum. Moreover, for lighter ions with lower charge numbers, the detection efficiency of the time signals was significantly reduced, increasing the background noise in the P(f) spectrum. We defined the signal-to-noise ratio as the ratio of the peak amplitude to the mean amplitude of the background noise. Obviously, a low signal-to-noise ratio can lead to inaccurate determination of the Q value. Therefore, only the (T,Q_i) datasets with a signal-to-noise ratio exceeding 7 were used for further analyses.

Using all ⁵¹Fe²⁶⁺ ions as an example, Fig. 4 presents scatter plots of the selected Qi against T, revealing a clear correlation. As shown in the figure, QA exhibits the highest counts and Pearson correlation coefficients (PCC) [33]. Consequently, QA was selected for subsequent analyses.

Fig. 4Full-screen View

(Color online) Scatter plots of T against Qi (where i spans A, B, C) for all ⁵¹Fe²⁶⁺ ions are shown. Each subplot includes the data entries and the PCC [33]. The red lines represent the results of linear fitting

According to Eq.(3), we obtain $-\eta T=\xi \frac{\text{Δ}T}{\text{Δ}Q}\mathrm{.}$ (7) As ξ is a machine parameter of the storage ring and is independent of the ion species, ξ is the slope if -ηT is plotted as a linear function of Δ T/Δ Q for different ion species: 1) Δ T/Δ Q, the rate of change of T with respect to Q, can be estimated as the slope of the linear function fitted to T vs QA (see the red fitting line in Fig. 4 (a) for ⁵¹Fe²⁶⁺, for example). 2) η is calculated as $1/{\gamma }^2-1/{\gamma }_{\text{t}}^2$, with ${\gamma }_{\text{t}}=1.406$. To obtain γ, the velocity was estimated as $\overline{C}/\overline{T}$, where $\overline{C}$ is the average orbit length determined using the method introduced in Ref. [30] and $\overline{T}$ is the average T value for each ion species. Figure 5 shows a plot of -ηT as a function of Δ T/Δ Q. The average chromaticity ξ, extracted from the slope of the linear fit shown in Fig. 5, is 4.87(3).

Fig. 5Full-screen View

Plot of -ηT against ΔT/ΔQ for all ion species is shown. Each point marker represents one ion species. The linear fitting function is presented as a red solid line. The slope of this line, which represents the average chromaticity ξ according to Eq.(3), is also noted

3.3

The T transformation and the mass determination

Using Eq.(4), the measured (T,Q_A) values for each ion were transformed into (T_Q,Q_A0) at a fixed Q_A0=0.298, which is the mean of all QA values.

Figure 6 illustrates the standard deviation (σ) of T and TQ for each ion species, corresponding to the revolution times before and after the transformation, respectively. As shown in Fig. 6, the farther an ion species is from the one with the minimum σ(T), the larger the ratio of σ(T)/σ(TQ). For ion species with $T\approx 629$ ns, σ(T) decreases by up to a factor of 2.5 to σ(TQ). Consequently, the mass-resolving power of these ion species was improved by the same factor. However, for ion species with η values close to zero, the decrease from σ(T) to σ(TQ) is almost negligible (see Eq. (3)).

Fig. 6Full-screen View

Standard deviation of the revolution time before and after transformation using Eq.(4)is denoted as σ(T) (black filled squares) and σ(TQ) (red filled triangles), respectively

Using both T and TQ, the mass of each nuclide was determined following the procedure outlined in Ref. [35]. Figure 7 shows the differences between the redetermined MEs and the literature values [34] for all reference nuclides. To assess the agreement, the χn value is defined as ${\chi }_n=\sqrt{\frac{1}{N_{\text{c}}}{\displaystyle \sum _{i=1}^{N_{\text{c}}}\frac{{\left(M{E}_i-M{E}_{\text{AME}\mathrm{20,}i}\right)}^2}{{\sigma }_i^2+{\sigma }_{\text{AME}\mathrm{20,}i}^2}}}\mathrm{,}$ (8) Here, N_c=27 represents the number of calibrants, and ${\sigma }_i$ and ${\sigma }_{\text{AME}\mathrm{20,}i}$ denote the 1σ uncertainties of $M{E}_i$ and $M{E}_{\text{AME}\mathrm{20,}i}$, respectively.

Fig. 7Full-screen View

Comparison of the re-determined mass excess (ME) values with the literature values (ME AME20) [34] for the reference nuclides is shown. Panels (a) and (b) represent the results using T and TQ, respectively. The calculated χn values are noted on the plots

For the re-determined MEs using T, the calculated χn=1.35 exceeds the 1σ confidence level of ${\chi }_n=1\pm 1/\sqrt{2N_{\text{c}}}=1\pm 0.14$, indicating that an additional systematic error of 13.1 keV should be considered. In contrast, when using TQ, χn value is reduced to 0.97(14), suggesting that no additional systematic errors are necessary. Table 1 summarizes the final ME values and counts determined for ⁶³Ge, ⁶⁵As, ⁶⁷Se, and ⁷¹Kr using T and TQ. The table also includes results from a previous IMS study (CSRe2011) [23] and the most recent $B\rho$-IMS experiment (CSRe2023) [26, 25].

Table 1Full-screen View

ME values and counts for the nuclides using T and TQ are provided

	CSRe2011		Using T		Using TQ		CSRe2023		Final
Atom	ME (keV)	Counts	ME (keV)	Counts	ME (keV)	Counts	ME (keV)	Counts	ME (keV)
63Ge	-46921(37)	212	-46887(26)	391	-46927(18)	125	-46978(15)	279	-46957(25)
65As	-46937(85)	37	-46798(61)	56	-46803(53)	15	-46806(42)	33	-46805(33)
67Se	-46580(67)	67	-46482(43)	138	-46534(32)	46	-46549(20)	174	-46545(17)
71Kr	-46320(141)	17	-46150(92)	30	-46038(72)	11	-46056(24)	148	-46054(23)

Show more

Results from prior IMS work based on the same experiment (CSRe2011) [23, 24] and a recent Bρ-IMS experiment (CSRe2023) [26, 25] are also included. The last column presents the weighted average ME values derived from TQ and CSRe2023.

For comparison, ME values are shown in Fig. 8, with recent CSRe2023 results serving as a benchmark. Compared to the results from CSRe2011 and those using T, the results using TQ not only align more closely with CSRe2023 but also exhibit higher precision. The improvements in accuracy and precision are mainly due to the significantly reduced spread of TQ, as shown in Fig. 6, despite the lower statistics reported in Table 1. Notably, the mass precision using TQ is comparable to that achieved with the advanced Bρ-IMS method used in CSRe2023.

Fig. 8Full-screen View

For ⁶³Ge, ⁶⁵As, ⁶⁷Se, and ⁷¹Kr, the ME differences between the values obtained from CSRe2011, using T and TQ, respectively, and those from CSRe2023 are shown. The gray shadow indicates the mass uncertainty of CSRe2023

Notably, for ⁶³Ge, the ME value obtained using T exhibits a significant deviation compared to CSRe2023. Although the discrepancy is reduced with TQ, it remains as large as 2.2σ. Therefore, as shown in the last column of Table 1, the final ME value for ⁶³Ge was adjusted, and the weighted average uncertainty was increased to 25 keV, based on a birth ratio [36] of 2.17.

Discussion

The newly obtained ME values facilitate the calculation of CDE [37], which is the binding-energy difference between mirror nuclei. The CDE arises from differing Coulomb interactions in mirror nuclei due to their varying proton numbers and provides valuable insight into nuclear forces and symmetries[38].

For mirror pairs with $T_z=\pm 1/2$, the CDE can be extracted using the following equation: $CDE=M{E}_{>}-M{E}_{<}+{\text{Δ}}_{\text{nH}}\mathrm{,}$ (9) where ME_> and ME_< represent the ME values of the proton-rich nucleus and its mirror partner, respectively, and ΔnH=782.3470 keV represents the mass difference between the neutron and ¹H.

Given that the CDE can be expressed as the CED between mirror nuclei [39, 40], a semi-classical approach to the Coulomb energy [39] can be used to estimate the CDE. $E_{\text{c}}=\left\{0.60Z^2-0.46Z^{4/3}-0.15\left[1-{\left(-1\right)}^Z\right]\right\}\frac{e^2}{r_0{A}^{1/3}}\mathrm{,}$ (10) here, r₀=1.2 fm and e²=1.44 MeV fm.

In Fig. 9, the CDEs obtained using experimental ME data from AME’20 [41], CSRe2011 [23] and the present study (using TQ) are represented by different symbols. The CDEs calculated based on the semi-classical approach are depicted as dashed lines. The CDEs from this work align more closely with the semi-classical approach compared to those from CSRe2011 and indicate a more regular change behavior.

Fig. 9Full-screen View

Comparison of CDEs for $T_z=\pm 1/2$ mirror pairs, obtained using experimental masses from AME20 [41], CSRe2011 [23] and this work (using TQ). The dotted line denotes the result from a semi-classical approach (see text)

To clearly illustrate the variation in CDEs, the quantity $\text{Δ}CDE(A)=CDE(A)-CDE(A-2)$ (11) was introduced to measure the difference in CDE between nuclei with mass number A and A-2.

In Fig. 10, experimental ΔCDEs are displayed alongside typical theoretical results. The zig-zag pattern in the theoretical Δ CDE is evident. Experimental ΔCDEs align well with various theoretical values for $A\ge 63$, except for those from CSRe2011. Deviations at A=71 and A=73 in CSRe2011 suggest abnormal staggering in Δ CDE, which was previously noted in Ref. [44] and attributed to a possible incorrect mass assignment for ⁶⁹Br due to its 574 keV isomer. However, using the updated mass value of ⁷¹Kr obtained from this work (also CSRe2023), these deviations at A=71 and A=73 are removed, and the normal staggering pattern is restored.

Fig. 10Full-screen View

Comparison of ΔCDEs. Experimental CDEs are indicted by symbols, while theoretical CDEs are presented as lines. For clarity, the CSRe2023 results (purple stars) are slightly right-shifted. Theoretical values are also presented, including those deduced from the semi-classical approach [39], and studies by Brown [42], Kaneko [38], and Ma [43]

While the ME values of ⁶³Ge from this work and CSRe2023 do not align well, the ΔCDEs at A=63 (and 65) are consistent with each other and agree with theoretical calculations (Fig. 10). Therefore, future mass measurements of ⁶³Ge are required to resolve the mass discrepancy.

Conclusion and outlook

This study introduces Tune-IMS, a novel technique designed to enhance the performance of IMS. By effectively utilizing measured Q values, Tune-IMS was successfully implemented in IMS experiment at HIRFL-CSR. This approach led to a narrower revolution time spread, particularly for nuclides not satisfying the isochronous condition. Consequently, more precise mass values for fp shell nuclei ⁶³Ge, ⁶⁵As, ⁶⁷Se, and ⁷¹Kr were re-determined, with validation supported by recent Bρ-IMS experiments. These redefined masses facilitated the calculation of CDEs, clarifying anomalies observed with previous mass values.

In Tune-IMS, the ion statistics are reduced due to the selection of Q values. To improve efficiency, a new type of TOF detector with positional sensitivity is currently under development [45]. This detector will provide additional position information, improving Q measurement performance and data utilization efficiency. Overall, Tune-IMS serves as a valuable complement to newly developed Bρ-defined IMS techniques and is particularly beneficial for conventional IMS systems equipped with a single TOF detector.