Introduction

Mass spectrometry of nuclei or ions has wide applications in nuclear physics [1] and other areas [2, 3]. Isochronous mass spectrometry (IMS) conducted at the Cooler Storage Ring for experiments (CSRe) in Lanzhou, China, has emerged as a pivotal technology for nuclear mass determination. This method has yielded numerous new mass values [4-10], demonstrating itself as an essential tool in nuclear mass spectrometry [11-13].

Early IMS experiments relied on the measurements of revolution times (T) of stored ions, and high resolving power was achieved only for the ion species of limited mass-to-charge ratios (m/q), fulfilling the so-called isochronous condition of γ=γ_t [14]. Here, γ is the relativistic Lorentz factor of the ion, and γ_t is the transition point, which is an ion-optical quantity of the ring. However, for the majority of ion species with , the time resolutions inevitably deteriorate depending on how far the γ’s differ from γ_t [15].

To make the IMS broadband, great efforts have been made [16, 17], and an additional velocity (v) measurement at the straight part of the ring has also been proposed [15, 18]. Velocity measurements were finally realized at the CSRe [19, 20], which allowed for the determination of both the orbit length, C=vT, and magnetic rigidity, Bρ=m/qγ v, for reference ions with well-known masses. Consequently, the universal calibration function, , is constructed, enabling the Bρ determination for non-reference ions using their measured C values-thereby realizing the Bρ-defined IMS (or Bρ-IMS) [21, 22]. Using this technique, mass measurements with remarkable precision down to 5 keV have been achieved [23] and numerous new mass values have been determined for the first time [24-26]. However, to obtain an accurate Bρ(C) function, one must eliminate the effect of magnetic field drifts during the experiment, which can be larger than ~10^-5. Thus, a complicated correction procedure has been developed, which requires multiple reference ions to correct for the change in the magnetic field for each injection [23]. This requirement is unfavorable for the mass measurement of exotic nuclei, as the number of reference ions in each injection can be quite low. Furthermore, the correction process may potentially introduce improper mass uncertainty assignments for both reference and non-reference ions (similar examples can be found in Ref. [27]).

To address these challenges, a new approach to Bρ-IMS is needed. Given the short storage time of ions in the ring (a few hundred microseconds) per injection, magnetic field drift becomes negligible over this short duration. Therefore, if the Bρ value of non-reference ions can be directly determined via a single reference ion within the same injection, the need for correcting magnetic field drifts using multiple reference ions would be eliminated.

According to the definition of γ_t [14]: (1)γ_t connects the Bρ and C differences between two ions within the same injection, thereby enabling the Bρ determination of a non-reference ion based on a single reference ion.

In this paper, we propose a novel method for Bρ-IMS that utilizes well-characterized γ_t values [28] for Bρ and mass determination. This method demonstrates two key advantages: (1) it requires only a single reference ion per injection, and (2) it naturally reduces the effects of magnetic field drifts, thereby obviating the need for extra correction procedures. In Sect. 2, we outline the principle of this method. In Sect. 3, we exemplify its performance through an experiment [21, 23] using ⁵⁸Ni¹⁹⁺ as the primary beam. In Sect. 4, the discussion is presented, followed by a summary and outlook in Sect. 5.

Principle of the new method

According to the fundamental equation , the mass-to-charge ratio m/q of a stored ion can be expressed as: (2)where v is the velocity of the ion, and c is the speed of light in vacuum. Because v is measured directly by the double time-of-flight (TOF) detectors installed in the straight section of the storage ring [20], the challenge in determining the unknown mass lies in the Bρ determination.

In previous Bρ-IMS studies [21, 23], Bρ and C were assumed to follow a Bρ(C) curve characterized by an analytic function, which was used to determine the Bρ values of non-reference ions based on their measured C values. However, the Bρ(C) curve varies with time because of magnetic field drifts. To obtain an accurate Bρ(C) function, a complicated method was utilized to mitigate the effect of the field drifts, as detailed in Ref. [23].

In this study, we propose a more straightforward approach to determine the Bρ value of non-reference ions in each injection. For ions circulating in the ring, according to Eq. (1), the relative changes in Bρ and C satisfy the following equation: (3)where the parameter γ_t serves as a bridge connecting the relative variations in Bρ and C. Although γ_t is often considered constant with respect to C, it actually varies with C, forming a γ_t(C) curve [29].

For simplicity, consider two ions simultaneously stored in the ring: one is a reference ion with a known mass, and the other is a non-reference ion with an unknown mass. Let Bρ₀ and C₀ represent the magnetic rigidity and orbit length of the reference ion, respectively, and Bρx and Cx denote those of the non-reference ion. Note that Bρ₀, C₀ and Cx can be determined by measuring T and v for both ions. According to Eq. (3), one has: (4)By defining (5)which can be calculated through numerical integration as long as γ_t(C) is known (details will be introduced later), Bρx can be derived as: (6)Through Eq. (6), Bρx of the non-reference ion is obtained via a single reference ion in the same injection, and the mass value is directly determined via Eq. (2). Consequently, the effect of magnetic field drift between injections is expected to be effectively eliminated.

The γt, Bρ and mass determination

Several methods have been developed to measure the γ_t values [28, 30], and their results are in excellent agreement. In this work, we employed the method using energy loss to calculate the γ_t(C) values owing to its simplicity. Details of this method can be found in Ref. [28].

Figure 1 illustrates a scatter plot of γt values obtained from all ions measured in the experiment using ⁵⁸Ni¹⁹⁺ as the primary beam [21, 23]. The unreasonable scattered points are primarily attributed to the low detection efficiency of the TOF detector for light ions [28]. To obtain more accurate averaged γ_t values, only ions with mass number A greater than 18 were adopted. The averaged γt values within approximately 3 mm intervals are presented as red curves. The γ_t value at any C within the specified range can be determined using linear interpolation.

Fig. 1Full-screen View

(Color online) γ_t(C) values obtained from all the measured ions in the experiment using ⁵⁸Ni¹⁹⁺ as primary beam. The red line represents the averaged γ_t(C) values within a small orbit interval of approximately 3 mm

Using the γ_t(C) curve, the parameter K defined in Eq. (5) is obtained through the following numerical integration: (7)Here, and ΔC_s=Cx-C₀)/n, where n represents the number of intervals. By choosing a large n, or equivalently, a sufficiently small Δ C_s (e.g., 0.3 mm), the γ_t(C) value within each interval can be regarded as constant during integration.

With the calculated K value, Bρx was determined according to Eq. (6) based on the Bρ₀ value of the reference ion. If there are multiple reference ions in the same injection, the same number of Bρx values will be obtained. All these Bρx values were averaged to obtain the mean Bρx value. Then, the m/q value is obtained using Eq. (2).

In this study, ions with mass uncertainties below 50 keV were selected as references to determine the mass values of non-reference ions. However, to validate the mass determination, the mass of each reference ion is first assumed to be unknown and then re-determined using the other reference ions in the same injection. As shown in Fig. 2, all the obtained m/q values were combined to form distinct m/q peaks.

Fig. 2Full-screen View

(Color online) The m/q peaks obtained from this work. Different colors represent the series of nuclides with a constant isospin projection , as shown in the legend

Figure 3 compares the standard deviation σ_(m/q) of m/q peaks (in units of keV/e) from three methods: this work, previous Bρ-IMS, and transformed from the original T peaks without any post-process procedure, such as the magnetic field correction procedure. Here, the transformed m/q peaks were obtained using Eq. (2), where v=C/T, and Bρ and C are fixed at 128.86 m and 5.4758 Tm, respectively (see Ref. [23] for more details). It can be observed that the σm/q from this work is significantly lower than that transformed from the original T peaks but is comparable to those obtained from the previous Bρ-IMS. This indicates that the impact of magnetic field drifts on mass determination has been effectively eliminated by this new method. We note that this was achieved without any additional magnetic field correction procedures. Nevertheless, the σm/q obtained in this study is slightly (approximately 1 keV/e) larger than that obtained from the previous Bρ-IMS results. This may be attributed to the variation of the γ_t(C) curve caused by the magnetic field drifts during the experiment. Further discussion is provided in Sect. 4.

Fig. 3Full-screen View

(Color online) Standard deviations of the m/q peaks derived from the original T measurement (black squares), the previous Bρ-IMS method (blue triangles) and this work (red dots)

Assuming that each m/q value contributes equally, the final determined m/q and its uncertainty were calculated as follows: (8)where N is the number of counts.

For each nuclide, the comparison of the mass excess (ME) with the literature value (e.g., the previous Bρ-IMS result [23]) is illustrated in Fig. 4, showing a good agreement between them.

Fig. 4Full-screen View

The mass excess (ME) difference between this work and previous Bρ-IMS result of the same experiment [21, 23], for the reference (a) and non-reference nuclides (b). The demonstrated error bars are from this study, whereas the shaded areas represent the error bars from the previous Bρ-IMS ME results

To further quantitatively evaluate the agreement, the normalized χn defined as (9)is employed. Here, N_c is the total number of nuclides for comparison, and are the mass excesses determined in this study and from the literature, respectively, and σ_exp and σ_lit represent the corresponding mass uncertainties.

The obtained χn values of 0.80 and 0.53 for the reference and non-reference nuclides, respectively, indicate that the mass results from the two methods are in good agreement. However, the mass uncertainties in this study are slightly larger than those from the previous Bρ-IMS work, consistent with the slightly larger σm/q observed in this study, as shown in Fig. 3.

We note that the two parameters L and (see Ref. [23] for details) used to determine the velocity of each ion were optimized to minimize the χ² value. The literature ME values used for this optimization were taken from Ref. [31]. The obtained optimal values of L=18.046 m and ps agree well with those (L=18.046 m and ps) from the previous Bρ-IMS study [23].

Discussion

4.1

Effects of magnetic field (or γ_t) drifts

According to the study presented in Ref. [29], the γ_t(C) curve is affected by magnetic field drifts. For example, it can be shifted horizontally, vertically, or rotated by varying the dipole, quadrupole, and sextupole magnetic fields. Given that the utilized γ_t(C) value in this method is an average value over the entire experiment [30, 28], for each injection, the γ_t(C) value may vary from this average value owing to magnetic field drifts, introducing potential uncertainties in the Bρ and mass determination.

To quantitatively evaluate the effect of γ_t variation, we first assume that γ_t is independent of C. Considering that is much smaller than Cx or C₀, the parameter K can be approximated as: (10)Because γ_t is close to one, the K value is on the same order of magnitude as , which is significantly less than one. Consequently, according to Eq. (6), Bρx can be estimated as: (11)Assuming that the magnetic field drift induces a variation δγ_t in γ_t, the corresponding variation in the calculated Bρx is denoted as . From Eq.(11), we obtain: (12)Combining Eq. (2) and Eq. (12), one yields: (13)Equation (13) clearly indicates that the effect of γ_t variation on the final mass value is significantly reduced by the small factor , which is generally on the order of 10^-4 in Bρ-IMS. In this experiment, the average ΔC/C ratio was approximately .

To further support this conclusion, we present a specific example from the ⁵⁸Ni experiment. The variation in the (dipole) magnetic fields with respect to the injection numbers is illustrated in Fig. 5(a) (one injection every 25 s; see Fig. 7(a) and the accompanying text in Ref. [23] for further details).

Fig. 5Full-screen View

(Color online) (a) The drifts of (dipole) magnetic field of CSRe as a function of injection number (see Fig. 7(a) in Ref. [23] for details). (b) The γt(C) curve corresponding to the injections marked in (a) with red and blue colors, illustrating a horizontal shift caused by the magnetic field variation. The maximum relative change in the γ_t value at a certain C is on the order of 10^-3. (c) Similar to Fig. 3, but the green points represent the σm/q values derived from the artificially downward-shifted γ(C) curve in Fig. 1 by 10^-3. See the text for further details of the study. (d) The differences between the ME values resulting from the γ_t(C) curve with and without the artificial shift. Most differences are within ±5 keV, as indicated by the shaded region

First, to examine the effect of magnetic field drifts on the γ_t(C) curve, two injection groups marked in red and blue in Fig. 5(a) were selected. The corresponding γ_t(C) curves are depicted in Fig. 5(b), demonstrating horizontal shifts between the two groups, which is consistent with the conclusion drawn from Ref. [29]. The largest relative deviation occurred at the left part of the curve and reached approximately 10^-3.

Second, to quantify the impact of such γ_t(C) variation, we examined a simplified scenario in which all γ_t(C) values were systematically decreased by 10^-3, corresponding to a downward shift of the γ_t(C) curve (Fig. 1) by this amount. The resulting σm/q values are represented by triangles in Fig. 5(c). These values are significantly smaller than the original values (squares), but are still approximately 1 keV/e larger than the normal values (circles) obtained using the unshifted γ_t(C) curve. Notably, this magnitude of increase matches the discrepancy observed between our normal results (this work) and previous Bρ-IMS results (see Fig. 3). This suggests that variations in the γ_t(C) curve, caused by the (dipole) magnetic field drifts, may be responsible for the observed discrepancy in σm/q between the two methods.

Equation (13) indicates that a 10^-3 shift in the γ_t(C) curve introduces a relative m/q uncertainty: . This corresponds to an expected additional uncertainty of ~ 4 keV/e in m/q, leading to an approximate 1 keV/e increase in σm/q. This estimation is in agreement with the observed 1 keV/e increase in σm/q in Fig. 5(c), thereby confirming the validity of Eq. (13).

Finally, to demonstrate the effect of the γ_t(C) curve with and without the artificial shift on the final mass determination, the differences in the resulting ME values are presented in Fig. 5(d). Most of these differences lie within a narrow range of ±5 keV (see the shaded area in Fig. 5(d)), supporting the robustness of this method.

4.2

The advantage of requiring only one reference ion in each injection

Currently, nuclides with well-known masses have been extended to the quite exotic region, characterized by short half-lives and very low production yields. When measuring the mass values of mass-unknown nuclides using the Bρ-IMS technique, the number of stored ions per injection can be remarkably low.

Figure 6 presents an example of the number of stored ions per injection in a Bρ-IMS experiment using ³⁶Ar as the primary beam. In this experiment, the magnetic rigidity of the beam lines was optimized to maximize the transport efficiency for the extremely exotic ion of ²²Si¹⁴⁺. Consequently, the most frequently observed number of stored ions per injection was reduced to approximately three. Under such conditions, the previous Bρ-IMS method, which requires as many reference ions as possible for magnetic field correction, encounters significant challenges. One potential solution is to discard injections containing fewer than, for example, three reference ions to ensure successful magnetic field correction. However, this approach leads to a significant loss of statistics (28%) and may introduce biases in the mass uncertainty assignments for both reference and non-reference ions. Alternatively, one could use the Bρ-C curve without magnetic field correction, but this would allow magnetic field fluctuations between injections to directly affect mass measurements at the 10^-5 level, significantly degrading mass precision.

Fig. 6Full-screen View

The statistics of the numbers of ions per injection for the Bρ-IMS experiment using ³⁶Ar as primary beam, which aims to measure the mass value of very exotic nuclide ²²Si

In such challenging scenarios, the method proposed in this study effectively overcomes all these challenges. It requires only one reference ion to determine unknown mass values and achieves high precision without the need for magnetic field correction. Using this method, the re-determined ME value of ²³Si, which is 23365(16) keV [32], is fully confirmed by the LEBIT Penning trap result of 23362.9(5.8) keV [33], demonstrating the efficiency and reliability of this important improvement for Bρ-IMS.

Summary and outlook

In this study, we propose a novel method relying on the γ_t(C) curve for Bρ-IMS to determine nuclear mass values. This method significantly reduces the need to correct the effects of magnetic field drifts and requires at least only one reference ion with a known mass value for the mass determination of the ions of interest. Remarkably, the achieved mass precision without any correction procedure is still comparable to that of the previous Bρ-IMS method. Consequently, it not only simplifies the data analysis procedure for Bρ-IMS but is also suitable for future IMS experiments involving exotic nuclei with extremely low yields.

Nonetheless, further improvements to this method are possible. First, while this method significantly reduces the influence of variations in the γ_t(C) curve (or magnetic field), residual effects cannot be entirely neglected, especially as the mass precision of Bρ-IMS is expected to continue improving. From this perspective, a more stable magnetic field environment remains essential. However, given that absolutely stable magnetic fields are technically unachievable, and considering that the drifts of the dipole magnetic field induce only a horizontal shift in the γ_t(C) curve, a flatter γ_t(C) curve would further minimize the impact of dipole magnetic field drifts on the final mass determinations, and thus is highly desirable. Second, the current mass determination of this method relies on averaging all obtained m/q values under the assumption of equal contribution. Actually, each individual obtained m/q value has a different uncertainty, and the simple assumption may introduce deviations, particularly when the statistic is limited. Future refinements should incorporate uncertainty quantification for the individually obtained m/q values of each nuclide to enable a weighted mean value and uncertainty.