1 Introduction

Masses of atomic nuclei are needed in the study of a plethora of phenomena in nuclear structure and nuclear astrophysics, as well as for testing fundamental interactions and symmetries [1,2,3,4]. Masses of about 2550 nuclides are known experimentally today [5] while about 7000 nuclides are expected to exist [6]. Of present interest are the masses of nuclides lying far away from the valley of \(\beta \)-stability [7]. Especially masses of very neutron-rich nuclides are needed for modelling the r-process nucleosynthesis [8]. Such nuclides are inevitably short-lived and have tiny production yields, making their mass measurements extremely challenging. Hence, measurement techniques capable of determining precisely the mass from a single, short-lived nuclide are especially demanded.

The isochronous mass spectrometry (IMS) at heavy-ion storage rings is an efficient and fast experimental technique [9, 10] well suited for mass measurements of exotic nuclei with short lifetimes down to several tens of microseconds. Since the pioneering experiments conducted at the ESR in GSI, Darmstadt [9, 10], IMS has been established at the experimental cooler storage ring (CSRe) in IMP, Lanzhou, and at the Rare RI Ring (R3) at RIKEN Nishina Center in Wako/Japan [11, 12].

To conduct the isochronous mass measurements, the beam energy has to be specially tuned in order to fulfill the isochronous condition. Thereby, the high mass resolving power can be achieved for the nuclides of interest [9]. Usually, the isochronous condition is fulfilled only for the ion species in a limited range of mass-to-charge (m/q) ratios, while the resolving powers are declining for the majority of ion species not fulfilling the isochronous condition. In addition, the momentum distributions of different ion species injected into the ring have different shapes, often asymmetric due to nuclear production mechanisms and limited acceptance of the transfer line and storage ring. This leads not only to large statistical uncertainties, but may also cause systematic deviations in the mass determination.

In order to improve the mass resolving power as well as to reduce the systematic deviation in a broad m/q range, the magnetic-rigidity tagging (\(B\rho \)-tagging) IMS was realized at the FRS-ESR facility at GSI [13,14,15] by inserting metal slits at the second dispersive focal plane of FRS [16]. In CSRe, an in-ring slit was used to restrict the magnetic-rigidity (\(B\rho \)) acceptance of stored fragments [17]. By using these two techniques, the mass resolving powers have been improved significantly and the systematic deviations have been largely reduced [13,14,15, 18,19,20,21]. However, both approaches have a dramatic drawback of losing transmission efficiency, which is not tolerable for the mass measurements of exotic nuclei with very low production yields. Therefore, a novel idea was conceived to measure the velocity of each stored ion [13,14,15, 22]. Recently we have installed two identical TOF detectors 18 m apart in one of the straight sections of CSRe [23]. This enabled simultaneous precision measurements of revolution time T as well as velocity v of each stored ion. Using these two parameters, a novel mass measurement technique, \(B\rho \)-defined IMS, has been established, which–in a limiting case of just a single event–is capable of determining the mass-to-charge ratio, m/q, with an unprecedented precision of about 5 keV.

In this paper, the new scheme of IMS using the two measured quantities, revolutions times and velocities of the stored ions, is described. Experiment and data processing are briefly introduced in Sect. 2. We describe the details of the \(B\rho \)-defined IMS in Sect. 3, focussing on the construction of the \(B\rho =B\rho (C)\) function and the mass determination. In Sect. 4, the re-determined masses for some fp-shell nuclides are reported and used to investigate the mirror symmetry of empirical residual proton-neutron (pn) interactions and to test the validity of the isospin multiplet mass equation. The conclusion and outlook are given in Sect. 5.

2 Experiment and revolution time spectrum

The nuclides of interest were produced by fragmenting 440 MeV/u \(^{58}\)Ni\(^{19+}\) primary projectiles on 15 mm thick \(^9\)Be target. They were selected with the in-flight fragment separator RIBLL2 [24, 25]. Every 25 s, a cocktail beam including the nuclides of interest was injected into and stored in CSRe. The duration of each measurement was merely 400 \(\mathrm {\mu s}\). Figure 1 presents the schematic view of the storage ring CSRe having a circumference of 128.8 m on the central orbit. CSRe was tuned into the isochronous mode with expected \(\gamma _t=1.365\). The whole \(B\rho \)-acceptance was about \(\pm 0.2\%\). The RIBLL2-CSRe system was set to a fixed central magnetic rigidity of \(B\rho =5.471\) Tm. At the employed relativisitc energies, the produced fragments were fully stripped of bound electrons. The primary beam energy was selected according to the LISE++ simulations [26] such that the \(^{55}\)Cu\(^{29+}\) ions had the most probable velocity corresponding to \(\gamma \sim \gamma _t\).

Fig. 1
figure 1

Schematic view of the CSRe with the arrangement of the TOF detectors

Two identical TOF detectors were installed 18 m apart in the straight section of CSRe (see Fig. 1). Each detector consists of a thin carbon foil (\(\phi \)40 mm, thickness 18 \(\mathrm \mu g/cm^2\)) and a set of micro-channel plates (MCP) [27]. When an ion passed through the carbon foil, secondary electrons were released from the foil surface and guided to MCP by perpendicularly arranged electric and magnetic fields. Fast timing signals from the two MCPs were recorded by a digital oscilloscope at a sampling rate of 50 GHz. The time resolutions of the TOF detectors were determined off-line to be \(\sigma =20\) - 40 ps by using \(^{241}\)Am \(\alpha \) source. Details on the detector performance can be found in Refs. [27, 28].

For each particle circulating in the ring, two time sequences, i.e. the time stamps when passing the two TOF detectors, \(t_\mathrm{TOF1}(N)\) and \(t_\mathrm{TOF2}(N)\), as a function of the revolution number, N, were extracted from the recorded signals. In order to have more statistics and keep sufficient precision, particles stored for more than 230 \(\mathrm {\mu s}\) and inducing more than 50 signals are used in the data analysis, making the IMS, in principle, a background free technique. The revolution times, \(T_\mathrm{TOF1}\) and \(T_\mathrm{TOF2}\), of each stored ion were deduced independently following the procedures described in Ref. [29]. We define \((T_\mathrm{TOF1}+T_\mathrm{TOF2} )/2\) at the central revolution number to be the revolution time. All individual revolution times were plotted in a histogram forming an integrated time spectrum as shown in Fig. 2. Particle identification was made following the procedures described in Ref. [29]. Three series of nuclides with \(T_z=-1/2,-1\), and \(-3/2\) can be seen in Fig. 2.

Fig. 2
figure 2

Revolution-time spectrum extracted from the raw data acquired by the two time-of-flight detectors. Unambiguous particle identification has been made as described in Ref. [29]. Different colours indicate the series of nuclides with a constant isospin projection \(T_z=(N-Z)/2\), see legend

Nuclear masses can be deduced from the revolution time spectrum, see e.g. our previous work [19]. Using the \(T_z=-1/2\) nuclides to calibrate the spectrum, the re-determined masses of the \(T_z=-1\) nuclides are compared with the well-known literature values in Fig. 3. One observes systematic deviations for the \(T_z=-1\) nuclides. Such systematic deviations are caused mainly by different momentum distributions and energy losses of the two series of nuclides with \(T_z=-1/2\) and \(-1\). This phenomenon was also observed in our previous experiments, and could be reduced by limiting the \(B\rho \) acceptance of the ring at a cost of a dramatic loss of the transmission efficiency. In the following, we will show that, by using the velocity information of stored ions, not only can the systematic deviations be removed, but also the mass resolving powers are improved significantly over a wide range of revolution times.

Fig. 3
figure 3

Comparison of re-determined mass excesses of \(T_z=-1\) nuclides (filled red squares) with literature values [30] using the \(T_z=-1/2\) nuclides (filled black circles) as calibrants. Masses are determined from the revolution time spectrum following the procedures described in [19] and \(\chi _n \) is the normalized Chi-square for the reference nuclides used in the calibration

3 \(B\rho \)-defined IMS

3.1 Principle

The revolution time, T, of an ion circulating in a storage ring is given by

$$\begin{aligned} T=\frac{C}{v}, \end{aligned}$$
(1)

with C and v being the orbit length and velocity of the ion, respectively. Magnetic rigidity, \(B\rho \), is defined as

$$\begin{aligned} B\rho =\frac{m}{q}\gamma v, \end{aligned}$$
(2)

where the Lorentz factor \(\gamma =1/\sqrt{1-\beta ^2}\) with \(\beta \) being the velocity in units of the speed of light in vacuum \(v_c\). Stored ions with identical \(B\rho \) values move on the same mean orbits in the storage ring. By combining Eqs. (1) and (2), one obtains

$$\begin{aligned} \frac{m}{q}=\frac{B\rho }{\gamma v}=B\rho \sqrt{\Bigg (\frac{T}{C}\Bigg )^2-\Bigg (\frac{1}{v_c}\Bigg )^2}. \end{aligned}$$
(3)

Equation (3) is the basic formula for any kind of \(B\rho \)-Time-of-Flight mass spectrometry [31]. According to this equation, m/q values can be determined if the quantities \((B\rho ,v)\) or equivalently \((B\rho ,T,C)\) are measured with high precision. However, it is difficult to precisely measure the absolute \(B\rho \) values. In the following, a method is decribed to obtain the relation of \(B\rho \) versus C using the directly measured quantities in the experiment.

3.2 Functional form of the \(B\rho (C)\) curve

If the transition point of the storage ring, \(\gamma _t\), defined as [9]

$$\begin{aligned} \frac{1}{\gamma _t^2}=\frac{\Delta C /C}{\Delta B\rho /B\rho }, \end{aligned}$$
(4)

is constant within the entire \(B\rho \)-acceptance of the ring, the \(B\rho (C)\) function should have a simple form

$$\begin{aligned} B\rho (C)=B\rho _0\left( \frac{C}{C_0}\right) ^{\gamma _t^2}, \end{aligned}$$
(5)

with \(B\rho _0\) and \(C_0\) being the reference parameters. In reality, the experimental \(B\rho _\mathrm{exp}\) vs \(C_\mathrm{exp}\) relation cannot be well described by Eq. (5) because \(\gamma _t\) is not constant for all orbit lengths [32,33,34]. Thus Eq. (5) was extended by adding an extra term

$$\begin{aligned} B\rho (C)=B\rho _0\left( \frac{C}{C_0}\right) ^{K}+a_1e^{-a_2(C-C_0)}. \end{aligned}$$
(6)

The free parameters \(B\rho _0\), \(C_0\), \(a_1\), \(a_2\), and K can be determined via fitting the experimental data. We emphasize that Eq. (6) characterizes the ion’s motion in the ring, i.e., all experimental \(B\rho _\mathrm{exp}\) and \(C_\mathrm{exp}\) data should fall onto an identical curve in the \(B\rho \) - C plane which can be described by the \(B\rho (C)\) function. The second term of Eq. (6) is chosen to describe the effect of the variable \(\gamma _t\), and its expression depends on the specific beam-optical setting of the ring.

3.3 Determination of the \(B\rho (C)\) function

\(B\rho (C)\) function can be determined using the measured quantities of the known-mass nuclides. In the present experiment, on average \(\sim \)15 ions were stored in one injection in CSRe. Each of them is easily identified in the recorded time stamp sequences. The velocity of each ion was determined according to

$$\begin{aligned} v=\frac{L}{\Delta t_\mathrm{TOF}}=\frac{L}{\Delta t_\mathrm{exp}-\Delta t_\mathrm{d}}, \end{aligned}$$
(7)

where L is the distance between the two TOF detectors, \(\Delta t_\mathrm{TOF}=\Delta t_\mathrm{exp}-\Delta t_\mathrm{d}\) is the flight time of the particle from TOF1 to TOF2 at the central revolution number, and \(\Delta t_\mathrm{d}\) is the time delay difference of the timing signals from the two detectors.

The \(\Delta t_\mathrm{exp}\) values were determined from the time sequences measured by the two TOF detectors [35] with the mean uncertainties of 2.0–6.4 ps for different ion species corresponding to the relative uncertainties of (2.2 - 7.2)\(\times 10^{-5}\). L and \(\Delta t_\mathrm{d}\) were measured offline by using a dedicated laser setup [36]. In this experiment, L and \(\Delta t_\mathrm{d}\) were determined from experimental data to be \(L=18.046\) m and \(\Delta t_\mathrm{d}=-146.83\) ps, see Appendix A.

The ion species with well-known masses (mass uncertainties less than 5 keV) and having more than 100 events were used to establish the \(B\rho (C)\) function. Using the redetermined values of L and \(\Delta t_\mathrm{d}\), the velocities for all stored ions were obtained. The orbit length of the \(i^\mathrm{th}\) ion was deduced via \(C^i_\mathrm{{exp}}=v^i_\mathrm{{exp}} T^i_\mathrm{{exp}}\). The corresponding magnetic rigidity was calculated with Eq. (2). A part of the \((B\rho _\mathrm{{exp}}^i,C_\mathrm{{exp}}^i)\) dataset, obtained under relatively stable field conditions of CSRe, are plotted in Fig. 4a.

Fig. 4
figure 4

a Plot of \(B\rho _\mathrm{exp}\) versus \(C_\mathrm{exp}\) with the fit results (solid line) according to Eq. (6). b Scatter plot of the fit residuals, \(R_{B\rho }= B\rho _\mathrm{exp}-B\rho (C_\mathrm{exp})\), as a function of \(C_\mathrm{exp}\) and its projected spectrum. Only a part of the injections under relatively stable magnetic field condition were used

Fit to the \((B\rho _\mathrm{{exp}}^i,C_\mathrm{{exp}}^i)\) data using Eq. (6) is shown with red solid line in Fig. 4a. One sees that the fitted K equals approximately to \(\gamma _t^2\) and the second term is a small correction. The fit residuals, \(R_{B\rho }= B\rho _\mathrm{exp}-B\rho (C_\mathrm{exp})\), exhibit a Gaussian-like distribution with a mean \(R_{B\rho }\) value at zero (see Fig. 4b), indicating that Eq. (6) is a good approximation of the expected \(B\rho (C)\) function. Once the \(B\rho (C)\) function is established, the mass-to-charge ratio of any stored ion can be directly obtained according to Eq. (3).

3.4 Uncertainty of the \(B\rho (C)\) function

Since the mass-to-charge ratio of an ion species is determined event-by-event according to Eq. (3), its uncertainty should be given simultaneously. In our data analysis, the uncertainties of velocity and revolution time are transformed into \(B\rho (C)\) which defines the final uncertainty of m/q. We note that the individual uncertainties of velocity and revolution time are in the order of \(10^{-5}\) and \(10^{-8}\) [35], respectively, therefore, only the former is considered in extracting the uncertainty of \(B\rho (C)\).

To schematically show the uncertainty of \(B\rho (C)\), the expected \(B\rho (C)\) curve is given in Fig. 5. The slope of the \(B\rho (C)\) curve gives \(\gamma _t\) values which vary as a function of orbital length C, see Eq. (4).

Let the measured velocity for a given ion deviate from the true value v by \(\delta v\). Then the obtained \(B\rho \) deviates from the true value by \(\delta (B\rho )=B\rho \gamma ^2\delta v/v\), and analogously \(\delta C=C\delta v/v\) by ignoring the uncertainty of T. On the other hand, along the expected \(B\rho (C)\) curve, one has \(\delta (B\rho )=B\rho \gamma _t^2\delta C/C\approx B\rho \gamma _t^2\delta v/v\) using Eqs. (1) and (4). Therefore, the two \(B\rho \) values differ by \(\delta (B\rho )=B\rho (\gamma ^2-\gamma _t^2)\delta v/v\), see Fig. 5. This means that the experimental \(B\rho \sim C\) data locate within an error band smaller than the relative velocity uncertainty of the order of \(\sim 10^{-5}\).

Fig. 5
figure 5

Schematic plot of the \(B\rho (C)\) relation. The parameters for an example ion with a true velocity v are indicated by the filled red circle. Due to the limited measurement accuracy, the velocity of this ion was determined as \(v+\delta v\). The corresponding deviations of the magnetic rigidity \(\delta B\rho \), and orbit length \(\delta C\) values are shown

To deduce the uncertainty of \(B\rho (C)\) directly from the information of reference nuclides, the fit residuals shown in Fig. 4b, are replotted in Fig. 6a as a function of the deviation from the isochronicity window, \(\gamma ^2-\gamma _t^2\). Determined standard deviations, denoted as \(\sigma _{B\rho }(\gamma ,\gamma _t)= \sigma (R_{B\rho })\), are given in Fig. 6b. One sees that \(\sigma _{B\rho }(\gamma ,\gamma _t)\) is small at regions of \(\gamma ^2-\gamma _t^2\approx 0\), while it becomes larger with \(\gamma \ne \gamma _t\). The variation of \(\sigma _{B\rho }(\gamma ,\gamma _t)\) versus \(\gamma ^2-\gamma _t^2\) is fitted by using a parabolic function

$$\begin{aligned} {\sigma _{B\rho }(\gamma ,\gamma _t)}=b_0+b_1(\gamma ^2-\gamma _t^2)+b_2(\gamma ^2-\gamma _t^2)^2, \end{aligned}$$
(8)

where \(\gamma _t\) is deduced from the obtained \(B\rho (C)\) function according to

$$\begin{aligned} \gamma _t^2=\frac{dB\rho /B\rho }{dC/C}=\frac{C}{B\rho (C)}\frac{dB\rho (C)}{dC}. \end{aligned}$$
(9)
Fig. 6
figure 6

a Scatter plot of fit residuals, \(R_{B\rho }= B\rho _\mathrm{exp}-B\rho (C_\mathrm{exp})\) versus \(\gamma ^2-\gamma _t^2\). Only nuclides with mass uncertainties below 5 keV and with more than 100 detected ions are considered. b Variation of standard deviations \(\sigma _{B\rho }(\gamma ,\gamma _t)\) as a function of \(\gamma ^2-\gamma _t^2\). The solid line is the fit result according to Eq. (8). Only a part of the injections with stable magnetic field were used

The fit result is shown with the solid line in Fig. 6b. The obtained uncertainty \(\sigma _{B\rho }(\gamma ,\gamma _t)\) can be applied to all stored ions. Consequently, the relative uncertainty of each \((m/q)_\mathrm{exp}^i\) value, including the ion species with unknown masses, can be calculated in the event-by-event analysis via the following expression

$$\begin{aligned} \frac{\sigma [(m/q)_\mathrm{exp}^i]}{(m/q)_\mathrm{exp}^i}=\frac{\sigma _{B\rho }(\gamma _\mathrm{exp}^i,\gamma _t)}{B\rho (C_\mathrm{exp}^i)},~~i=1,~2,~3,~\ldots . \end{aligned}$$
(10)

In principle, the resulting \(B\rho (C)\) function can be directly employed to calculate the m/q values according to Eq. (3). However, in the present experiment, the magnetic fields of CSRe varied in time which led to large fluctuations of measured quantities. Therefore, the values such as \(B\rho \) and v derived directly from the experiment should be corrected in order to achieve high accuracy in the mass determination.

3.5 Correction for the magnetic field drifts

Fig. 7
figure 7

a Scatter plot of fit residuals,\(R_{B\rho }^i= B\rho _\mathrm{{cor}}^i-B\rho (C_\mathrm{{exp}}^i)\), as a function of injection number illustrating slow variations of the magnetic fields of the CSRe. b \(B\rho (C)\) plot for two magnetic field values. The points shown in red and blue colors correspond to the regions marked with the same color in the left panel. c Same as a but with the corrected \( B\rho _\mathrm{{cor}}\). d Standard deviations, \(\sigma _{B\rho }(\gamma ,\gamma _t)\), as a function of \(\gamma ^2-\gamma _t^2\) before (black squares) and after (red circles) the correction procedure. The solid lines are the fits according to Eq. (8)

In a constant magnetic field \(B_0\), the magnetic rigidities and orbit lengths of all ions must fall onto an identical \(B\rho (C)\) curve. However, the magnetic fields of CSRe vary in time which lead to strong deviations of \(B\rho (C)\) curve from the expected one. There were several methods developed for correction of the magnetic field drifts in the CSRe experiments [29, 37, 38]. In the present work, we adopt a much more precise correction method by using the data from the two TOF detectors.

The changes of the dipole magnetic fields lead to an up-down shifts of the \(B\rho (C)\) curve. This is clearly seen in Fig. 7a which shows the fit residuals, \(R_{B\rho }= B\rho _\mathrm{exp}-B\rho (C_\mathrm{exp})\), as a function of injection number. To achieve the high mass resolving power, the effects due to magnetic field drifts are corrected as follows.

Let there be \(N_s\) ions stored in an individual injection which is characterized by the magnetic field B. The determined quantities \(\{C_\mathrm{exp}^i,T_\mathrm{exp}^i,v_\mathrm{exp}^i,~i=1,2,\ldots ,N_s\}\) need to be corrected to \(\{C_\mathrm{cor}^i,T_\mathrm{cor}^i,v_\mathrm{cor}^i,~i=1,2,\ldots ,N_s\}\) corresponding to a reference setting with \(B_0\). To obtain the latter ones, \(C_\mathrm{exp}^i=C_\mathrm{cor}^i\) should be confined (equivalent to a common radius \(\rho _\mathrm{exp}^i=\rho _\mathrm{cor}^i\)). Then the ratio of the magnetic fields, M, can be deduced from the experimental data according to

$$\begin{aligned} M=\left( \frac{B}{B_0}\right) _\mathrm{CSRe}=\left( \frac{B\rho _\mathrm{{exp}}^i}{B\rho _\mathrm{{cor}}^i}\right) _\mathrm{ion},~~i=1,~2,~\ldots ,~N_s. \nonumber \\ \end{aligned}$$
(11)

For a specific injection, the ratio M is constant for all \(N_s\) stored ions. Hence,

$$\begin{aligned} B\rho _\mathrm{{cor}}^i= & {} \frac{1}{M}B\rho _\mathrm{{exp}}^i,~~i=1,~2,~\ldots ,~N_s, \end{aligned}$$
(12)
$$\begin{aligned} (\gamma v)_\mathrm{{cor}}^i= & {} \frac{1}{M}(\gamma v)_\mathrm{{exp}}^i,~~i=1,~2,~\ldots ,~N_s. \end{aligned}$$
(13)

By using the measured velocities, \(B\rho \) values can be precisely determined for nuclides with well-known masses. For this purpose, nuclides with tabulated mass uncertainties less than 5 keV and recorded for more than 100 counts have been utilized.

The procedure was done iteratively. First, the data \(\left\{ B\rho _\mathrm{{exp}},C_\mathrm{{exp}}\right\} \) were fitted with Eq. (6), giving an initial \(B\rho (C)\) function. By analyzing the distribution of the fit residuals \(B\rho _\mathrm{{exp}}-B\rho (C_\mathrm{{exp}})\), the estimated uncertainty \(\sigma _{B\rho }\) was obtained for each ion as a function of \(\gamma ^2-\gamma _t^2\). Then, the expected M value for an individual injection with \(N_p\) known-mass nuclides (note \(N_p\le N_s\)) can be deduced from experimental data according to

$$\begin{aligned} M = \frac{1}{\sum _i^{N_p}{w_i}}{\sum _i^{N_p}{w_i\left[ \frac{B\rho _\mathrm{{exp}}^i}{B\rho (C_\mathrm{{exp}}^i)}\right] }},~~i=1,~2,~\ldots ,~N_p, \end{aligned}$$
(14)

where

$$\begin{aligned} \frac{1}{w_i} = \frac{[\sigma _{B\rho }(\gamma _\mathrm{exp}^i,\gamma _t^i)]^2}{[B\rho (C_\mathrm{exp}^i)]^2},~~i=1,~2,~\ldots ,~N_p. \end{aligned}$$
(15)

The obtained M value provides the corrected \(B\rho _\mathrm{{cor}}^i\) and \((\gamma v)_\mathrm{{cor}}^i\) via Eqs. (12) and (13). The corrected data \(\{B\rho _\mathrm{{cor}}^i,C_\mathrm{{exp}}^i\}\) for the ions from all injections were used to establish a new \(B\rho '(C)\) function and a new \(\sigma '_{B\rho }\) estimation, which were used in the next iteration. The iterations were repeated until the convergence was reached.

Figure 7c shows the fit residuals after the correction procedure, \(R_{B\rho }^i=B\rho _\mathrm{{cor}}^i-B\rho (C_\mathrm{{exp}}^i)\), as a function of the injection number. The slow variations of the magnetic field drifts have been nearly completely removed. Figure 7d illustrates the reduced standard deviations \(\sigma _{B\rho }(\gamma ,\gamma _t)\) as a function of \(\gamma ^2-\gamma _t^2\) obtained by analyzing all injections.

After the field-drift correction, the obtained values \(\{B\rho _\mathrm{cor}^i,v_\mathrm{cor}^i\}\) were available for all ion species. Their m/q values and the corresponding uncertainties were then calculated through Eqs. (3), (8), and (10).

3.6 Improvements in respect to the conventional IMS

Using the \(B\rho \)-defined IMS, the m/q values of nuclides have been determined in the event-by-event analysis according to Eq. (3). To demonstrate the power of the new approach in comparison with the conventional IMS, Fig. 8 shows a comparison of the re-determined masses of the \(T_z=-1\) nuclides with the well-known literature values. Note that the \(T_z=-1/2\) nuclides were used as calibrants in the analysis, and the masses of the \(T_z=-1 \) nuclides were assumed to be unknown. One sees that not only the statistical uncertainties significantly decreased, but also the systematic deviations demonstrated in Fig. 3 have been nearly completely removed over a wide range of revolution times.

Fig. 8
figure 8

Comparison of re-determined mass excesses of \(T_z=-1\) nuclides (filled red squares) with literature values [30] using the \(T_z=-1/2\) nuclides (filled black circles) as calibrants. Masses are determined with the \(B\rho \)-defined IMS and \(\chi _n \) is the normalized Chi-square for the reference nuclides used in the calibration [19]

Fig. 9
figure 9

Scatter plots of \(T_\mathrm{exp}\) and \(T_\mathrm{fix}\) versus \(C_\mathrm{exp}\) for \(^{24}\)Al ions. The excitation energy of the isomeric state \(^{24m}\)Al is 425.8(1) keV [5]

To demonstrate the increasing resolving power of the new approach as compared to the conventional IMS, we transformed the m/q spectrum into a new revolution time spectrum, \(T_\mathrm{fix}\), at a fixed magnetic rigidity, \(B\rho _\mathrm{fix}=5.4758\) Tm (the corresponding orbit length is \(C_\mathrm{fix}=128.86\) m), according to

$$\begin{aligned} T_\mathrm{fix}^i = C_\mathrm{fix}\sqrt{\frac{1}{B\rho _\mathrm{fix}^2}\left( \frac{m}{q}\right) _i^2 + \left( \frac{1}{v_c}\right) ^2},~~i=1,~2,~3,~\ldots \end{aligned}$$
(16)

where m/q is the mass-to-charge value determined in the \(B\rho \)-IMS without need of its particle identification. The scatter plots of \(T_\mathrm{exp}\) and \(T_\mathrm{fix}\) versus \(C_\mathrm{exp}\) for \(^{24}\mathrm{Al}^{13+}\) ions are shown in Fig. 9. Note that the revolution time peak of \(^{24}\mathrm{Al}^{13+}\) ions lies at the edge of the revolution time spectrum at about 622.5 ns, see Fig. 2. It is obvious that the two states of \(^{24}\)Al, separated by the mass difference of 425.8(1) keV [5], cannot be resolved in the \(T_\mathrm{exp}\) spectrum, while the two peaks can be clearly separated in the \(T_\mathrm{fix}\) spectrum in Fig. 9 (right).

The standard deviations of the TOF peaks derived from the \(T_\mathrm{exp}\) and \(T_\mathrm{fix}\) spectra are shown in Fig. 10. The standard deviations of the TOF peaks in the original TOF spectrum, \(\sigma _T\), have a parabolic dependence versus m/q. \(\sigma _T\) approaches minimum at approximately 2 ps only for a limited number of nuclides (isochronicity window). In the re-constructed \(T_\mathrm{fix}\) spectrum without the field drift correction, \(\sigma _T \approx 2\) ps has been achieved for basically all nuclides in the entire m/q-range.

Fig. 10
figure 10

Standard deviations of the time peaks (left scale) extracted from original revolution time spectrum (filled black squares), and from newly-constructed \(T_\mathrm{fix}\) spectrum with (blue circles) and without (red triangles) magnetic field drift correction. The absolute accuracies of mass-to-charge ratios are given on the right scale

We emphasize that this was done without reducing the \(B\rho \)-acceptance of neither the ring nor the transfer line. Here the resolving power is limited by the magnetic field drifts. After the field drift correction, \(\sigma _T=0.5\) ps is achieved in the isochronicity window, corresponding to the mass resolving power of \(3.3\times 10^5\) (FWHM). The mass resolving power at the edges of the spectrum has been improved by a factor of about 8. The right scale in Fig. 10 shows the corresponding absolute mass precisions for m/q values. It is emphasized that the mass precision of merely \(5\cdot q\) keV can be achieved for just a single stored ion.

Fig. 11
figure 11

m/q spectra of \(^{24,24m}\)Al, \(^{44,44m}\)V, and \(^{52,52m}\)Co obtained from the \(B\rho \)-defined IMS. Solid lines are fit results using two Gaussian functions of identical standard deviation

Table 1 Experimental mass excesses (MEs) obtained from this work (third column), from an earlier CSRe measurement (fourth column) [19, 39,40,41] and from the literatures (sixth column). Also the recent Penning-trap measurements for \(^{44g,44m}\)V [42], \(^{52g,52m}\)Co [43], \(^{56}\)Cu [44], \(^{51}\)Fe [45] and AME2016 for \(^{43}\)Ti [46] are included. All ME units are in keV

4 Mass results and discussions

4.1 Re-determined masses

Each individual m/q value and the uncertainty were obtained in the event-by-event analysis. The m/q values were plotted in a histogram forming the m/q spectrum. For the ion species with well-separated peaks in the m/q spectrum, the weighted-average m/q was directly derived and converted [19] into atomic mass excess (ME). For the nuclides with a low-lying isomer, two peaks are overlapped such as the cases of \(^{24,24m}\)Al, \(^{44,44m}\)V, and \(^{52,52m}\)Co. Therefore, the mean m/q values were deduced by fitting the m/q spectrum using two Gaussian functions with an identical standard deviation. The fit results are given in Fig. 11. The normalized \(\chi _n\) values are all close to unity, indicating that the peak shapes can be well described by the proposed fitting functions.

Table 1 presents the newly determined MEs for the nuclides that were not used in the precedures of \(B\rho (C)\) construction and field drift correction. Therefore, these MEs should be considered to be from independent measurements. Since the previous masses obtained by using the conventional IMS at CSRe have already been included into AME2016 and AME2020, the present results are compared directly to the previous CSRe results and to the available recent PT data. The re-determined masses for the \(T_z=-1\) nuclides have been reported in Ref. [47] emphasizing that the \(B\rho \)-defined IMS can provide masses of neutron deficient nuclides with the precisions comparable to those from the conventional Penning-trap mass spectrometer.

The mass excess of \(^{43}\)Ti obtained in a previous CSRe experiment utilizing a single TOF detector [19] was 15 keV larger than the AME2016 value [46]. This small deviation was attributed to a possible mixture with a low-lying isomer at \(E_x=313\) keV. However, this isomer has a half-life of only 11.9 \(\mathrm{\mu }\)s [48], which can not survive in the ring for more than 230 \(\mu \)s and hardly affect our mass measurements. Our new results confirm the previous CSRe result, with the mass being 19 keV larger than the AME2016 value. The latter was determined from two energy measurements in transfer reactions [49, 50].

Fig. 12
figure 12

Comparison of the re-determined masses with literature values. The grey shadow represents the mass uncertainties in the latest atomic mass evaluation [30]. The blue diamonds indicate the nuclides not used in constructing the \(B\rho (C)\) function and in the correction procedure of the magnetic field drifts. The filled black circles stands for the nuclides used in constructing the \(B\rho (C)\) function and in the correction procedure of the magnetic field drifts. \(\chi _n \) is the normalized Chi-square [19]

A comparison of newly-determined masses with literature values is shown in Fig. 12. The filled black circles represent the nuclides that were used in constructing the \(B\rho (C)\) function and in the field drift correction. Although they are the references in the mass determination, their values can be re-determined. For this purpose, each one is assumed to have an unknown mass and is redetermined using the remaining reference masses. The normalized \(\chi _n=0.74\) for the re-determined reference masses indicates that the quoted errors are conservative and no additional systematic errors need to be considered.

The re-determined masses for the \(T_z=-3/2\) nuclides are compared with literature ones in Fig. 12. Except for \(^{49}\)Fe, the present mass excesses are in good agreement with the previously-measured values by using the conventional IMS [39,40,41]. The mass accuracies are significantly increased allowing us to investigate the mirror symmetry of empirical residual proton-neutron interactions (p-n interaction), as well as to test the isobaric multiplet mass equation (IMME).

4.2 Mirror symmetry of residual p-n interactions

The binding energy of a nucleus B(ZN), derived directly from atomic masses, embodies the sum of overall interactions inside the nucleus. Differences of masses or binding energies isolate specific nucleonic interactions and provide signatures and different features of the nuclear structures. For example, the nucleon pairing correlations are clearly exhibited by the zig-zag pattern of one-nucleon separation energies with changing N and Z, and the shell closures can be revealed by kinks in smooth trends of two-nucleon separation energies. Apart from the one- and two-nucleon separation energies, a double difference of masses or binding energies, denoted as \(\delta V_{pn}\), has been used to isolate the average interaction strength between the last proton and the last neutron. Conventionally, \(\delta V_{pn}\) values are derived according to [51]

$$\begin{aligned} \delta V_{pn}^{ee}(Z,N)= & {} \frac{1}{4}[B(Z,N)-B(Z,N-2) -B(Z-2,N)\nonumber \\&+B(Z-2,N-2)], \end{aligned}$$
(17)
$$\begin{aligned} \delta V_{pn}^{oe}(Z,N)= & {} \frac{1}{2}[B(Z,N)-B(Z,N-2) -B(Z-1,N)\nonumber \\&+B(Z-1,N-2)], \end{aligned}$$
(18)
$$\begin{aligned} \delta V_{pn}^{eo}(Z,N)= & {} \frac{1}{2}[B(Z,N)-B(Z,N-1) -B(Z-2,N)\nonumber \\&+B(Z-2,N-1)], \end{aligned}$$
(19)
$$\begin{aligned} \delta V_{pn}^{oo}(Z,N)= & {} [B(Z,N)-B(Z,N-1)-B(Z-1,N)\nonumber \\&+B(Z-1,N-1)], \end{aligned}$$
(20)

and termed as the empirical residual proton-neutron interactions. Equations (17) through (20) are suited for nuclei with \((Z,N)=\) even-even, odd-even, even-odd, and odd-odd, respectively.

It has been recognized that the p-n interactions are closely related to many nuclear structure phenomena such as the onset of collectivity and deformation [52, 53], changes of underlying shell structure [54], and phase transitions in nuclei [54,55,56]. From systematic investigations of \(\delta V_{pn}\) throughout the mass surface, strong orbital dependence of \(\delta V_{pn}\) has been revealed for the self-conjugate nuclei and in the regions of doubly shell closures [57, 58].

Inspecting the available \(\delta V_{pn}\) data of the nuclides on both sides of the \(Z=N\) line, one observed a mirror symmetry of the residual p-n interactions [19]. Indeed, such a mirror symmetry was pointed out many years ago by Jänecke [59], and recently was used by Zong et al. [60] to make high-precision mass predictions for the very neutron-deficient nuclei.

Fig. 13
figure 13

Differences of \(\delta V_{pn}\) of mirror nuclei. The grey shadow indicates an error band of 50 keV. The neutron-deficient partners are indicated for the mirror pairs with \(A\le 20\)

Table 2 ME values of \(T=3/2\), \(A=55\) isobaric quartet. The data with symbol “#” are from AME2020 [30], and the ME value of \(^{55}\)Cu is from this work. The excitation energies of the IASs in \(^{55}\)Ni and \(^{55}\)Co are shifted by \(-5.5\) keV and 10.3 keV, respectively. See text for details

We present in Fig. 13 the energy differences, defined as \(\Delta (\delta V_{pn})=\delta V_{pn}(T_z^{<},A)-\delta V_{pn}(T_z^{>},A)\). Here \(T_z^{<}\)/\(T_z^{>}\) represents the negative/positive value of \(T_z\) of the mirror nuclei. The \(\delta V_{pn}\) values are calculated according to Eqs. (17) through (20) using the mass data in AME2020 [30] and the new masses in Table 1. It can be seen that \(\Delta (\delta V_{pn})\) values scatter around zero within an error band of \(\pm 50\) keV for the \(A>20\) mirror pairs, indicating that the mirror symmetry of \(\delta V_{pn}\) holds well in this mass region. It is worthwhile to note that the mass of \(^{55}\)Cu is re-determined with significantly-improved precision, and its value is 172 keV more bound than the previous one [40]. Using this new mass, the calculated \(\delta V_{pn}\) of \(^{56}\)Cu is approximately equal to that of \(^{56}\)Co. Consequently, \(\Delta (\delta V_{pn})=\delta V_{pn}(^{56}\mathrm{Cu})-\delta V_{pn}(^{56}\mathrm{Co})\) fits well into the general systematics at a high level of accuracy (see Fig. 13). In turn this result provides a further confirmation of the reliability of the new mass value of \(^{55}\)Cu.

In the lighter mass region with \(A\le 20\), the so-called mirror symmetry of \(\delta V_{pn}\) is apparently broken in a few mirror pairs (see Fig. 13), i.e., the \(\delta V_{pn}\) values of the neutron-deficient nuclei are systematically smaller than those of the corresponding neutron-rich partners. Such a mirror-symmetry breaking was simply attributed to a binding energy effect [59]. Here we note that one of the four nuclei is particle-unbound which may lead to the mirror-symmetry breaking of \(\delta V_{pn}\). In this sense, the mirror symmetry of \(\delta V_{pn}\) cannot be used to predict masses with \(Z\le 20\). Any further interpretation of this deviation would require detailed theoretical study.

4.3 Validity of isospin multiplet mass equation

The masses of a set of isobaric analog states (IASs) can be described by the well-known quadratic isospin multiplet mass equation (IMME) [61],

$$\begin{aligned} \mathrm{ME}(A,T,T_{z}) =a(A,T)+b(A,T)T_{z}+c(A,T)T_{z}^{2}, \end{aligned}$$
(21)

where MEs are mass excesses of IASs of a multiplet with fixed mass number A and total isospin T. T is equal to or is larger than the projection of T, \(T_z=(N-Z)/2\), for a specific nucleus. The coefficients ab, and c depend on A,  T, and other quantum numbers such as the spin and parity \(J^{\pi }\), but are independent of \(T_z\). The quadratic form of the IMME, i.e., Eq. (21), is commonly considered to be accurate within uncertainties of a few tens of keV. In this context, precision mass measurements can be used for testing its validity (see Ref. [39] and references therein). Typically one adds to Eq. (21) extra terms such as \(dT_z^3\) or/and \(eT_z^4\), which provide a measure of the breakdown of the quadratic form of the IMME. Numerous measurements have been performed investigating the validity of the IMME. Reviews and compilations of existing data can be found in Refs. [62, 63] and references cited therein.

The re-determined ME of \(^{55}\)Cu is 172 keV smaller than the previous value [40]. Using this new mass, four experimental masses of the \(T=3/2\) IASs are complete, and thus the validity of the quadratic form of the IMME can be tested reaching to the heaviest \(A=55\) isospin quartet. Table 2 collects the relevant data for the \(A=55\), \(T=3/2\) isospin quartet.

It is noted that the \(T=3/2\) IASs in \(^{55}\)Ni \((T_z=-1/2)\) and in \(^{55}\)Co \((T_z=+1/2)\) have strong isospin mixing with the close-lying \(T=1/2\) states [64, 65], leading to energy shifts for the pure IASs. Their isospin mixing matrix elements were inferred from experiments to be 9(1) keV and 13(4) keV [64], respectively. Consequently, the energy splitting between the two unperturbed \(T=3/2\) and \(T=1/2\) levels have been extracted according to Ref. [64], and the excitation energies of the corresponding IASs in NUBASE2020 [48] have been shifted by \(-5.5\) keV and 10.3 keV, respectively. The modified values are listed in Table 2.

The mass data were fitted using Eq. (21) with the normalized \(\chi _n\) = 0.26. The obtained a,  b, and c coefficients are given in Table 2 which are consistent with systematics in Refs. [62, 63]. In addition, the mass data were also fitted by the cubic form of IMME by adding the \(dT_z^3\) term to Eq. (21). The obtained d coefficient is \(-1.2(47)\) keV and is compatible with zero. These results indicate that Eq. (21) can well describe the mass data, i.e., the quadratic form of the IMME holds well for the \(A=55\), \(T=3/2\) isospin quartet.

Similar procedure has been applied to the \(T=3/2\), \(A=41,~45,~49\), and 53 isospin quartets using the new mass data in Table 1. The obtained d coefficients are shown in Fig. 14. Comparing with the previous results of Fig. 2 in Ref. [39], one concludes that the trend of a gradual increase of d with A in the fp shell [39] is not confirmed, at least at the present level of accuracy. Given the fact that all extracted d coefficients are compatible with zero, the quadratic form of the IMME is valid for the cases investigated here. Figure 14 also shows the d values from theoretical calculations [66]. The predicted non-zero d coefficients for these \(T=3/2\) isospin quartets can not be yet ruled out due to large experimental uncertainties. More precise mass measurements for the associated IASs are still needed to draw a definite conclusion.

Fig. 14
figure 14

d coefficients of the cubic form of IMME for the \(T=3/2\), \(A=41,~45,~49,~53\), and 55 isospin quartets. The solid line connects the predicted d values from theoretical calculations [66]

5 Summary and outlook

A novel isochronous mass spectrometry, the \(B\rho \)-defined IMS, has been developed at the experimental cooler-storage ring CSRe. The measurements of both the revolution time and the velocity of every stored ion enabled us to construct the \(B\rho (C)\) function, which is a universal mass calibration curve for all simultaneously stored ions. The method for constructing the \(B\rho (C)\) function was described in detail by using the experimental data in the mass measurements of \(^{58}\)Ni projectile fragments. The high mass resolving power over the whole \(B\rho \)-acceptance of the storage ring was realized and an uncertainty band as small as \(\approx 5\cdot q\) keV was achieved. The latter indicates that a single short-lived (\(T_{1/2} > rsim 100~\mu \)s) ion is now sufficient for its mass determination with \(\approx 5\cdot q\) keV precision.

Masses of several \(T_z=-3/2\) fp-shell nuclides were re-determined with high accuracy. The precision mass value of \(^{55}\)Cu complete the four masses of the \(A=55,\) \(T=3/2\) isospin quartet, and thus the isospin multiplet mass equation have been validated up to the heaviest quartet with \(A=55\). The new masses were also used to investigate the mirror symmetry of the empirical residual p-n interactions, which could be employed, in turn, to test the reliability of newly-measured masses, or to predict the masses of very neutron-deficient nuclei. It was also pointed out that the mirror-symmetry breaking of \(\delta V_{pn}\) in the \(A\le 20\) region may be due to a proton-unbound nucleus involved in the extraction of \(\delta V_{pn}\). Further experimental as well as theoretical studies are needed to address this issue.

The merits of the novel IMS are demonstrated by the dramatically increased sensitivity and accuracy of the measurements. Next-generation storage rings [67] are under construction, including the Spectrometer Ring (SRing) [68] at the high-intensity Heavy-Ion Accelerator Facility (HIAF) [69, 70] in China, and the Collector Ring (CR) [71, 72] at the Facility for Antiproton and Ion Research (FAIR) [22, 73,74,75] in Germany. Mass measurements of exotic nuclei are among the research programs of these storage rings [31]. Owing to the high resolving power, high accuracy, ultimate sensitivity, short measurement time, broadband, and background-free characteristics, the \(B\rho \)-defined IMS is the technique of choice for the future storage-ring-based mass spectrometers. A further improvement of the \(B\rho \)-defined IMS is to work on the ion-optical setting of the ring to achieve a constant \(\gamma _t\) over the full momentum acceptance, which will make the \(B\rho (C)\) function well defined with a simple expression.