Principle

In conventional IMS, the ion-revolution times T_exp are measured using a single TOF detector [36, 37]. The mass-to-charge ratios of the ions, m/q, are derived according to [19, 20]: $\frac{m}{q}=\frac{B\rho }{\gamma v}=B\rho \sqrt{\frac{T_{\text{exp}}^2}{C^2}-\frac{1}{v_{\text{c}}^2}}\mathrm{,}$ (1) where Bρ, v, γ, and C are the magnetic rigidity, velocity, Lorenzt factor, and orbit length of the stored ions, respectively. v_c is the speed of light. Bρ and C are assumed to be the same for all ions and the masses of aimed ions can be deduced using ions with known masses as calibrants.

In real experiments, the momentum acceptance of the storage ring should be considered, and the dispersed Bρ and C of the stored ions should follow the Bρ(C) function. The relative slope of Bρ(C) $\frac{\Delta B\rho /B\rho }{\Delta C/C}={\gamma }_{\text{t}}^2\mathrm{,}$ (2) introduces a new quantity, γ_t, called the transition point of the storage ring [38]. γ_t is determined solely by the machine lattice settings and is independent of the ion species.

Owing to the momentum dispersion of the stored ions, the variation in the revolution times is as follows: $\frac{\Delta T_{\text{exp}}}{T_{\text{exp}}}=\frac{\Delta C}{C}-\frac{\Delta v}{v}=\left(\frac{1}{{\gamma }_{\text{t}}^2}-\frac{1}{{\gamma }^2}\right)\frac{\Delta B\rho }{B\rho }\mathrm{.}$ (3) In the experiments, the storage ring was tuned to be isochronous, that is, γ～γ_t, for the ions of interest, and a high mass-resolving power could be achieved in a small region of the m/q spectrum. For other ions with $\gamma \ne {\gamma }_{\text{t}}$, the mass-resolving power decreased rapidly.

To further reduce the influence of momentum dispersion and achieve broadband high mass-resolving power, two TOF detectors are installed in one of the straight sections of the ring such that both the revolution times T_exp and velocities v_exp of the stored ions [39] can be measured simultaneously. Consequently, the orbital length of each ion C_exp=T_exp×v_exp can be deduced directly.

Using ions with known masses as calibrants, a one-to-one correspondence of $B\rho \sim C$ is obtained, providing more information on the momentum dispersion. By fitting the $B\rho \sim C$ data, we obtain the $B{\rho }_{\text{fit}}\left(C_{\text{exp}}\right)$ function, and the m/q value of any ion can be deduced from the fitted function as follows: $\frac{m}{q}=\frac{B{\rho }_{\text{fit}}\left(C_{\text{exp}}\right)}{{\left(\gamma v\right)}_{\text{exp}}}\mathrm{.}$ (4) In the simulations, the uncertainties of m/q values are mainly due to the uncertainties of the velocity measurements and instability of γt [40-42]. A data-analysis method was developed to achieve Bρ-defined IMS at the storage ring CSRe in Lanzhou.

Technique details

The experiments were conducted at the accelerator complex of the IMP in Lanzhou, China. The nuclides of interest were produced by fragmenting the primary beams at a speed of >0.7× v_c on 10∼15 mm ⁹Be production targets. The produced nuclides were selected using the in-flight fragment separator, the second Radiactive Isotope Beam Line in Lanzhou (RIBLL2) [43]. They were then injected into and stored in the CSRe.

Figure 2 presents a schematic of the CSRe with the three TOF detectors noted in this figure. The detector TOF0 was used in the conventional IMS. The detectors TOF1 and TOF2 were used in Bρ-defined IMS. Each detector consists of a thin carbon foil (ϕ=40 mm, 18 μg/cm² thick) and a set of microchannel plates (MCP) [36, 37].

Fig. 2Full-screen View

(Color online) Schematic of CSRe with the arrangement of three TOF detectors [34]

When an ion passes through the carbon foil, secondary electrons are released from the foil surface and guided to the MCP by perpendicularly arranged electric and magnetic fields. Compared to the previous detector TOF0, the electric-field strengths in TOF1 and TOF2 increased from 130 to 180 V/mm to improve the time resolution of the detector, as shown in Fig. 3 (a). A time resolution of 18.5±2 ps was achieved in the offline tests of TOF1 and TOF2 [37]. Fast timing signals from the two MCPs were recorded using a single oscilloscope to avoid the jitter effects of two or more electronic instruments [45]. The sampling rate was set to 50 GHz.

Fig. 3Full-screen View

(Color online) (a) Time resolution of the TOF detector as a function of the electric-field strength. [37]. (b) Dispersion functions used in conventional IMS (black solid line) and Bρ-defined IMS (red dashed line) as a function of the longitudinal position s of CSRe [44]. The positions of the three TOF detectors are denoted using dotted lines. (c) The transition energy γ_t as a function of orbital length due to isochronous corrections of quadrupole magnets and sextupole magnets [44]

To achieve Bρ-defined IMS, a new isochronous mode setting was specially designed at the CSRe [44]. Figure 3 (b) shows the dispersion function $D_x=\frac{x}{\Delta p/p}$ as a function of the longitudinal position s of the CSRe. The dispersion function at the detector positions was designed to be close to 0 m for a high transition efficiency of the ions owing to the limited size of the carbon foils (ϕ=40 mm). To ensure the accuracy of the detector installation, a calibration system using a short-pulsed ultraviolet laser was also applied at the CSRe [46].

In real experiments, γ_t is not constant for all orbit lengths owing to high-order magnetic fields from imperfect storage-ring facilities [47, 44, 48]. A minor modification of the quadrupole and sextupole fields was performed to reduce the variation in γ_t as a function of the orbit length C [44], as shown in Fig 3 (c). However, the variation in γ_t within the momentum acceptance cannot be eliminated completely in a real experiment. The residual nonlinear effects in the γ_t(C) curve should be considered in data analysis.

Velocity measurement of stored ions

Figure 4 shows the passage times t_exp of a single ⁴⁵V²³⁺ ion as a function of revolution number N for TOF1 and TOF2 [39]. The slope $\partial t_{\text{exp}}/\partial N$ is the revolution time T_exp, and the distance between the two curves is the time of flight (TOF) Δt_exp between the two detectors. The mean velocity of an ion circulating in the ring v is obtained by $v=\frac{L}{\Delta t_{\text{exp}}-\Delta t_{\text{d}}}\mathrm{,}$ (5) where L is the distance between the two TOF detectors in the straight section of the CSRe and $\Delta t_{\text{d}}=t_{\text{d}}^{\text{TOF2}}-t_{\text{d}}^{\text{TOF1}}$ is the time-delay difference between the two timing signals transmitted from the detectors to the oscilloscope. The values of L and Δt_d were measured in a dedicated study using a laser-calibration system [46] and were redetermined in each experiment using stored ions with known masses as calibrants [35]. The precise velocity measurement of a stored ion critically depends on the precision determination of Δt_exp between TOF1 and TOF2.

Fig. 4Full-screen View

(Color online) The passage times t_exp of a single ⁴⁵V²³⁺ measured by detectors TOF1 and TOF2 as a function of the revolution number N [39]. For illustration purposes, the time stamps from TOF2 are shifted upwards by 10 μs. The inset illustrates an enlarged region at 299≤N≤309 showing the missing time stamps in some revolutions. The error bar of each data point is within the symbol size

Data analyses of conventional IMS usually show that the flight time of an ion is a smooth function of the revolution number. Considering the energy loss in a thin carbon foil, t_exp(N) can be described by a third-order polynomial function: $t_{\text{fit}}\left(N\right)=a_0+a_1\times N+a_2\times N^2+a_3\times N^3\mathrm{.}$ (6) In addition, the influence of the betatron oscillations of the ions should be considered [39], and the fit function of t_exp(N) curves should be modified as follows: $\begin{array}{c}{t}_{\text{fit}}\left(N\right)=a_0+a_1\times N+a_2\times N^2+a_3\times N^3\\ +A_x\times \left[2\pi \left(Q_{x0}\times N+Q_{x1}\times N^2\right)+{\varphi }_x\right]\\ +A_y\times \left[2\pi \left(Q_{y0}\times N+Q_{y1}\times N^2\right)+{\varphi }_y\right]\mathrm{.}\end{array}$ (7) The polynomial function in Eq. (7) describes ion motion with a mean orbital length, whereas the sinelike terms describe the periodic time fluctuations owing to betatron oscillations.

A straightforward method to deduce Δt_exp is to use the existing coincident timestamps from both TOF detectors in the same revolution. Alternatively, a fitting procedure using Eq. (6) and Eq. (7) have been performed, and Fig. 5 shows the average uncertainties of Δt_exp as a function of the atomic number Z using these three methods.

Fig. 5Full-screen View

(Color online) Average uncertainty of Δt_exp as a function of atomic number Z, extracted from coincident time stamps (black filled circles) and by using the third-order polynomial fit function with [Eq. (7)] (blue filled triangles) and without [Eq. (6)] (red opened circles) the sinelike terms

The smallest mean value of σ(Δt_exp) has been obtained using Eq. (7). This indicates that considering and addressing the effects of betatron oscillations can significantly improve the precision of the Δt_exp values as well as the ion velocities. The mean uncertainties exhibited a decreasing trend from lighter to heavier ions. This can be understood if the Z-dependence of the detection efficiency of the detectors is considered; that is, a higher detection efficiency yields lower statistical errors in Δt_exp. The lowest mean uncertainties of Δt_exp were within the range of 2.0-6.4 ps, corresponding to a relative precision of (2.2-7.2)×10^-5 with respect to the average TOF between the two TOF detectors of $\Delta t_{\text{exp}}\approx 89\text{ns}$. Compared with the momentum acceptance ±0.2% of the CSRe [49], the precision of Δt_exp is sufficient for new IMS development.

Determination of mass values

Using ions with known masses and the measured revolution times T_exp and velocities v_exp, the magnetic rigidity Bρ_exp=m/q× (γ v)_exp and orbit length $C_{\text{exp}}=T_{\text{exp}}\times v_{\text{exp}}$ can be easily extracted. Fig. 6 shows a scatterplot of Bρ_exp versus C_exp [34, 35].

Fig. 6Full-screen View

(Color online) Plot of Bρ_exp versus C_exp with the fit results (solid line) [35]. The fitted expression is noted in this figure

For an ideal storage ring with a completely constant γ_t within the momentum accpetance, the one-to-one correspondence of Bρ～C should follow the simple form: $\begin{array}{l}B\rho \left(C\right)=B{\rho }_0{\left(\frac{C}{C_0}\right)}^{{\gamma }_{\text{t}}^2}\mathrm{,}\hfill \end{array}$ (8) where Bρ₀ and C₀ are reference parameters.

In real experiments, the nonlinear effects in the γt(C) curve shown in Fig. 3 (c) should be considered. Because these residual nonlinear effects may differ in each experiment, Eq. (8) is modified by including additional terms. For example, $\begin{array}{l}B\rho \left(C\right)=B{\rho }_0{\left(\frac{C}{C_0}\right)}^K+a_1{e}^{-a_2\left(C-C_0\right)}\hfill \end{array}$ (9) was used in the experiments described in Refs. [34, 35]. The fit result of Eq. (9) is shown in Fig. 6. The m/q spectrum was deduced from the fitted results.

The instability of magnetic fields is one of the most significant challenges in precision mass measurements at the CSRe storage ring. Over the past decade, extensive efforts have been dedicated to both hardware improvements [50] and data-analysis enhancements [51, 52], aimed at mitigating the impact of magnetic-field drift. Using data from the two TOF detectors, we developed a more precise correction method [35].

The changes in the dipole magnetic fields led to vertical drifts in the measured Bρ(C) curve. This phenomenon can be clearly observed in Fig. 7 (a), where the fit residuals $B{\rho }_{\text{exp}}^i-B\rho \left(C_{\text{exp}}^i\right)$ are presented as functions of the injection number. To achieve the high mass-resolving power, the originally determined quantities ｛$C_{\text{exp}}^i\mathrm{,}{T}_{\text{exp}}^i\mathrm{,}{v}_{\text{exp}}^i$, i=1,2,...,Ns｝ need to be corrected to ｛$C_{\text{cor}}^i\mathrm{,}{T}_{\text{cor}}^i\mathrm{,}{v}_{\text{cor}}^i$, i=1,2,...,Ns｝ corresponding to a reference magnetic-field setting with B₀ assuming the magnetic field is stable. To obtain the latter, $C_{\text{exp}}^i=C_{\text{cor}}^i$ must be confined (equivalent to a common radius ${\rho }_{\text{exp}}^i={\rho }_{\text{cor}}^i$). Consequently, the magnetic rigidity and velocity of each ion in the same injection are scaled by a constant ratio: $\begin{array}{l}B{\rho }_{\text{cor}}^i=\frac{1}{R_M}B{\rho }_{\text{exp}}^i\text{or}{\left(\gamma v\right)}_{\text{cor}}^i=\frac{1}{R_M}{\left(\gamma v\right)}_{\text{exp}}^i\mathrm{,}\hfill \end{array}$ (10) where $\begin{array}{l}{R}_M={\left(\frac{B}{B_0}\right)}_{\text{CSRe}}={\left(\frac{B{\rho }_{\text{exp}}^i}{B\rho \left(C_{\text{exp}}^i\right)}\right)}_{\text{ion}}\mathrm{.}\hfill \end{array}$ (11) The ratio RM can be deduced from the experimental data of reference ions with known masses for each injection. Figure 7 (b) shows that the fitted residuals after correction produce $B{\rho }_{\text{cor}}^i-B\rho \left(C_{\text{exp}}^i\right)$ as a function of the injection number. The slow variations in the magnetic-field drifts were almost completely removed. After the field-drift correction, the obtained values ｛$B{\rho }_{\text{cor}}^i\mathrm{,}{v}_{\text{cor}}^i,C_{\text{exp}}^i$｝ are available for all ion species. Their m/q values were then calculated from the $B{\rho }_{\text{cor}}\sim C_{\text{exp}}$ curve, using Eq. (4) following the same procedure as that shown in Fig. 6.

Fig. 7Full-screen View

(Color online) (a) Scatter plot of fit residuals, $B{\rho }_{\text{exp}}^i-B\rho \left(C_{\text{exp}}^i\right)$, as a function of injection number illustrating slow variations of the magnetic fields of the CSRe [35]. (b) Similar to (a), but with the corrected Bρ_cor

Improvements compared with the conventional IMS

Figure 8 shows part of the m/q spectrum of Ref. [53]. The peaks of the nuclei with nearly identical m/q ratios may overlap. To obtain the m/q values from the overlapping peaks, we introduce a Z-dependent parameter, $U=\epsilon \overline{H}$, extracted from the timing signals of the TOF detectors [54, 55]. Here, ε is the detection efficiency for a specific ion and $\overline{H}$ is the average signal amplitude of that ion.

Fig. 8Full-screen View

(a) Part of the m/q spectrum and (b) the corresponding scatter plot of U versus m/q in Ref. [53]. The ion species are also indicated. Note that the unresolved ⁶²Ge³²⁺, ⁶⁶Se³⁴⁺, and ⁷⁰Kr³⁶⁺ in the m/q spectrum are completely separated from ³¹S¹⁶⁺, ³³Cl³⁷⁺, and ³⁵Ar¹⁸⁺, respectively, in the plot of U versus m/q

To demonstrate the increasing resolving power of the new approach compared to the conventional IMS, we transformed the m/q spectrum into a new revolution-time spectrum [34, 35], T_fix, at a fixed magnetic rigidity Bρ_fix = 5.4758 Tm (the corresponding orbit length is C_fix = 128.86 m), as follows: $\begin{array}{l}{T}_{\text{fix}}=C_{\text{fix}}\sqrt{\frac{1}{{\left(B\rho \right)}_{\text{fix}}^2}{\left(\frac{m}{q}\right)}^2+{\left(\frac{1}{v_c}\right)}^2}\mathrm{.}\hfill \end{array}$ (12) For example, scatter plots of T_exp and T_fix versus C_exp for ²⁴Al¹³⁺ ions are shown in Fig. 9 [34, 35].

Fig. 9Full-screen View

Scatter plots of T_exp and T_fix versus C_exp for ²⁴Al¹³⁺ ions [34, 35]

Clearly, the two states in ²⁴Al, separated by a mass difference of 425.8(1) keV [9], cannot be resolved in the T_exp spectrum, whereas the two peaks can be separated in the T_fix spectrum. The standard deviations of the T peaks derived from the corresponding spectra are shown in Fig. 10.

Fig. 10Full-screen View

Standard deviations of the T peaks (left scale) derived from the original revolution-time spectrum (black filled squares) and from the newly constructed spectrum (blue circles). The corresponding absolute accuracies of mass-to-charge ratios are given on the right scale. [34, 35]

The standard deviations, σ_T, of the T peaks in the original T_exp spectrum exhibit a parabolic dependence on m/q. σ_T approaches a minimum at approximately 2 ps for a limited number of nuclides (isochronicity window). In the spectrum obtained using the new Bρ-defined IMS technique, σT=0.5 ps was achieved in the isochronicity window, corresponding to a mass-resolving power of 3.3 × 10⁵ (FWHM). At the edges of the spectrum, a mass-resolving power of 1.3× 10⁵ (FWHM) can be achieved, which is an improvement by a factor of approximately 8 compared to the conventional method. We emphasize that this was done without reducing the Bρ acceptance of either the ring or transfer line. The right scale in Fig. 10 shows the corresponding absolute m/q precision. We emphasize that an m/q precision of 5 keV is obtained for a single stored ion.

Using the Tz=-1/2 nuclides to calibrate the spectrum, the redetermined masses of the Tz=-1 nuclides are compared with well-known literature values in Fig. 11[35]. The results obtained using T_exp as in conventional IMS are shown in Fig. 11 (a), where the systematic deviations for the Tz=-1 nuclides can be clearly observed. Such systematic deviations are caused mainly by the different momentum distributions and energy losses of the two series of nuclides with Tz=-1/2 and -1 [49]. Not only did the statistical uncertainties significantly decrease, but the systematic deviations demonstrated in Fig. 11 (a) are completely removed over a wide range of revolution times.

Fig. 11Full-screen View

Comparison of redetermined mass excesses of Tz=-1 nuclides (filled red squares), deduced from (a) T_exp and (b) T_fix, with literature values [9] using the Tz=-1/2 nuclides (filled black circles) as calibrants. Masses are determined from the revolution-time spectrum following the procedures described in [56] and χ_n is the normalized Chi-square for the reference nuclides used in the calibration [34, 35]

GS 1826-24 X-ray Burst and Neutron Stars

Type-I X-ray bursts occur on the surfaces of neutron stars, accreting hydrogen- and helium-rich matter from companion stars in a stellar binary system [59]. The burst is powered by a sequence of nuclear reactions, termed the rapid-proton-capture nucleosynthesis process (rp process) [60], which is a sequence of proton captures, and β decays along the proton drip line. Figure 13 shows a nuclear chart around the rp-process waiting point ⁶⁴Ge. Measurements of neutron-deficient ⁷⁸Kr projectile fragments at the CSRe provided the masses for ⁶³Ge, ^64,65As, and ^66,67Se using Bρ-defined IMS, and all the relevant separation energies around the waiting point ⁶⁴Ge were determined [57].

Fig. 13Full-screen View

Nuclear chart around the rp-process waiting point ⁶⁴Ge [57]. Nuclides whose masses were obtained from AME2020 [9], whose masses were experimentally determined, or whose mass uncertainties were improved in Ref. [57] are indicated with black, red, and blue, respectively. The one- (Sp) and two-proton (S_2p) separation energies (values expressed in keV) follow the same color code. The pathway of rp-process nucleosynthesis is shown with black arrows. Refer to the legend for more details

The impact of these new masses was investigated using state-of-the-art multizone X-ray burst simulations. Simulated X-ray burst light curves are shown in Fig. 14. The new nuclear masses, particularly the less-bound ⁶⁵As and more-bound ⁶⁶Se, result in a stalled rp-process at the ⁶⁴Ge waiting point. The new distributions of the elements produced through the rp-nucleosynthesis (“ashes”) are also modified, as shown in the inset of Fig. 14. The ash abundances of Urca nuclides are particularly notable [66, 67]. A 17% increase in the A=64 ash mass fraction results in increased electron-capture heating, and a 14% decrease in the A=65 ash mass fraction resulted in reduced Urca cooling, implying a somewhat warmer accreted neutron-star crust.

Fig. 14Full-screen View

(Color online) Calculated X-ray-burst light curves [57]. The baseline nuclear-physics input is labelled as AME2020 (dashed-dotted line and gray band). The result including the nuclear masses determined in Ref. [57] is marked as Updated (solid-line and red band). The light-curve bands correspond to 68% confidence intervals. The inset shows the calculated mass fractions X(A) as a function of mass number

The X-ray-burst luminosity measured by a telescope is directly proportional to the emitted light flux and inversely proportional to the square of the distance d² and surface gravitational redshift (1+z)² of the burst corrected by the electromagnetic-wave transport efficiency [68-70]. Our new light curve enables the setting of new constraints on the optimal d and (1+z) parameters that fit the observational data [57]. The results are presented in Fig. 15(a). The increased peak luminosity requires distance d to be increased by ∼0.4 kpc from approximately 5.8 kpc to approximately 6.2 kpc. The rp-process stalled at ⁶⁴Ge leads to extended hydrogen burning in the light curve, thereby extending the burst tail. Consequently, the modelled light curve must be less time-dilated, thereby reducing (1+z).

Fig. 15Full-screen View

(Color online) (a) Surface gravitational redshift versus the distance of the source GS 1826-24 obtained through a comparison of the modelled light curves shown in Fig. 14 and the observational data from the year 2007 bursting epoch [57]. The colors indicate 68% (red), 90% (yellow), and 95% (gray) confidence intervals. Two regions correspond to results using previously known masses (AME2020) and those including our new masses (Updated). (b) Neutron-star mass (M_NS) versus radius (R_NS) constraints calculated from the (1+z) 95% confidence intervals shown in (a) [57]. Limits for neutron-star compactness determined using other astrophysical observables [61-65] are also shown in this figure

The constraints on (1+z) can be further converted into limits on the neutron-star compactness (M_NS/R_NS) following the approach introduced in [71]. The general relativistic neutron-star mass M_GR and radius R_GR are determined using $\left(1+z\right)=1/\sqrt{1-2G{M}_{\text{GR}}/\left(R_{\text{GR}}{v}_c^2\right)}$, where G is the gravitational constant. If the Newtonian mass is equal to the general relativistic mass M_NS=M_GR, then $R_{\text{GR}}=\sqrt{\left(1+z\right)}{R}_{\text{NS}}$. The compactness constraints corresponding to the 95% confidence intervals shown in Fig. 15(a) are shown in Fig. 15(b) along with the mass and radius constraints from other observational probes and theoretical limits.

Recently, the newly determined compactness $\xi =M_{\text{NS}}/R_{\text{NS}}$ was incorporated into a comprehensive Bayesian statistical framework to investigate its impact on the equation of state (EOS) of supradense neutron-rich matter and the spin frequency required for GW 190814’s minor m₂ with mass $2.59\pm 0.05M_{\odot }$ to be a rotationally stable pulsar. The EOS of high-density symmetric nuclear matter must be softened significantly, while the symmetry energy at supersaturation densities is stiffened compared to our prior knowledge from earlier analyses using data from both astrophysical observations and terrestrial nuclear experiments [72, 73].

Thermonuclear reaction rate of ⁵⁷Cu(p,γ)⁵⁸Zn in the rp process

In addition to research around the waiting-point ⁶⁴Ge, the newly determined mass of the neutron-deficient nuclide ⁵⁸Zn was used to re-evaluate the thermonuclear rate of the ⁵⁷Cu(p,γ)⁵⁸Zn reaction around the waiting-point ⁵⁶Ni [74]. The re-evaluated thermonuclear rate was higher than the most recently published rate by a factor of up to three in the temperature range of 0.2 GK≤T≤1.5 GK. One-zone post-processing type-I X-ray-burst calculations were performed. Figure 16 shows the fractional difference in abundance as a function of mass number. The updated rate and new mass of ⁵⁸Zn resulted in noticeable abundance variations for nuclei with A = 56-59 and a reduction in A = 57 abundance by up to 20.7%.

Fig. 16Full-screen View

Fractional difference of abundance by mass number of Ref. [74] (present) compared to those using the Langer et al. rate [75] (red filled circles) and Lam et al. [70] rates (blue open triangles). The black dashed line represents the result with initial mass fractions [76]

Residual Proton–Neutron Interactions in the N=Z Nuclei

The binding energy of a nucleus, B(Z,N), is derived directly from the atomic masses and embodies the sum of the overall interactions inside the nucleus. Binding-energy differences can isolate specific classes of interactions and provide insights into nuclear structural modifications [1, 15]. An important mass filter, the double binding-energy difference denoted as δV_pn, has been used to isolate the residual proton-neutron (pn) interactions [77-79] and to sensitively probe a variety of structure phenomena, such as the onset of collectivity and deformation [80-83], changes to the underlying shell structure [84], and phase-transitional behavior [81, 85]. Conventionally, δV_pn values are derived as follows [79]: $\begin{array}{c}\delta V_{\text{pn}}^{\text{ee}}\left(Z\mathrm{,}N\right)=\frac{1}{4}[B\left(Z\mathrm{,}N\right)-B(Z\mathrm{,}N-2)\\ -B(Z-\mathrm{2,}N)+B(Z-\mathrm{2,}N-2)]\mathrm{,}\end{array}$ (13) $\begin{array}{c}\delta V_{pn}^{\text{oo}}\left(Z\mathrm{,}N\right)=[B(Z\mathrm{,}N)-B(Z\mathrm{,}N-1)\\ -B(Z-\mathrm{1,}N)+B(Z-\mathrm{1,}N-1)],\end{array}$ (14) $\begin{array}{c}\delta V_{\text{pn}}^{\text{oe}}\left(Z\mathrm{,}N\right)=\frac{1}{2}[B(Z\mathrm{,}N)-B(Z\mathrm{,}N-2)\\ -B(Z-\mathrm{1,}N)+B(Z-\mathrm{1,}N-2)],\end{array}$ (15) $\begin{array}{c}\delta V_{\text{pn}}^{\text{eo}}\left(Z\mathrm{,}N\right)=\frac{1}{2}[B(Z\mathrm{,}N)-B(Z\mathrm{,}N-1)\\ -B(Z-\mathrm{2,}N)+B(Z-\mathrm{2,}N-1)],\end{array}$ (16) and are termed the empirical residual proton–neutron interactions. Equations (13)–(16) are suitable for nuclei with (Z, N) =even-even, odd-even, even-odd, and odd-odd, respectively.

Measurements of neutron-deficient ⁷⁸Kr projectile fragments at the CSRe provided masses for ⁵⁸Zn, ⁶⁰Ga, ⁶²Ge, ⁶⁴As, ⁶⁶Se, and ⁷⁰Kr Tz=-1 nuclides as well as ⁶¹Ga, ⁶³Ge, ⁶⁵As, ⁶⁷Se, ⁷¹Kr, and ⁷⁵Sr Tz=-1 nuclides [53]. The newly measured masses provide the δV_pn values using Eq. (13) for nuclei with $N=Z=\text{even}\left(\delta V_{\text{pn}}^{\text{ee}}\right)$ and Eq. (14) for nuclei with $N=Z=\text{odd}\left(\delta V_{\text{pn}}^{\text{oo}}\right)$. The results are presented in Fig. 17 (a) together with the δV_pn values extracted using the currently available mass models [86-94].

Fig. 17Full-screen View

(a) Experimental δV_pn for N=Z nuclei beyond A=56 and comparison with different mass-model predictions [86-94]. Red lines are included to guide the eye. Red symbols indicate that one of the new masses in Ref. [53] was used. δV_pn of ⁵⁸Cu using the binding energy of the T=1, J^π=0⁺ excited state [9] is marked with open diamond shapes. ME(⁷⁰Br)=-52030(80) keV is obtained from [95, 96]. δV_pn values obtained using the extrapolated masses in AME2020 [9, 9] are connected by the black solid line. (b) and (c) show experimental δV_pn values for N=Z and N=Z+2 nuclei beyond A=56 together with the ab initio calculations [53]. Data uncertainties are within the size of the symbols. δV_pn values from ab initio calculations using 2NF+3NF and 2NF only are plotted as red and blue lines (solid lines for even-even and dashed lines for odd-odd), respectively

Our results show that an increasing trend in $\delta V_{\text{pn}}^{\text{oo}}$ beyond Z=28 (red dashed line) is established, which has been suggested as an indication of the restoration of the pseudo-SU(4) symmetry in the fp shell [97, 98]. $\delta V_{\text{pn}}^{\text{ee}}$ follows a decreasing trend (red solid line), as reported in the lower-mass region [78, 99]. This contrary to the expectation of pseudo-SU(4) symmetry [79, 100]. We extracted the δV_pn values using the predicted masses of frequently used mass models [86-94] and observed that none of the models can reproduce the bifurcation of δV_pn (see Fig. 17(a)).

To understand the bifurcation of $\delta V_{pn}$, we performed an ab initio valence-space in-medium similarity renormalization group (VS-IMSRG) calculation using a chiral interaction with a two-nucleon force (2NF) at N³LO and a three-nucleon force (3NF) at N²LO, named EM1.8/2.0 [101].

As shown in Fig. 17(c), our calculations with 3NF excellently reproduced the experimental δV_pn with N = Z + 2 nuclei, demonstrating the capability of the ab initio approach. For N = Z nuclei, the calculations reproduced the $\delta V_{\text{pn}}^{\text{ee}}$ data from ⁵⁶Ni to ⁷²Kr, which decreased marginally with increasing A. As both T = 0 and T = 1 pn correlations are naturally included in the ab initio calculation, reasonable agreement with the experimental $\delta V_{\text{pn}}^{\text{oo}}$ values is also obtained from ⁵⁸Cu to ⁷⁰Br. In particular, the increasing trend of $\delta V_{\text{pn}}^{\text{oo}}$ with changing A is well reproduced.

Recently, a shell-model-like approach based on relativistic density-functional theory was established [102] by simultaneously treating the neutron-neutron, proton-neutron, and proton-proton pairing correlations both microscopically and self-consistently. The calculated δV_pn values reproduced the observed bifurcation in Fig. 17 well. The mechanism for this abnormal bifurcation was owing to the enhanced proton–neutron pairing correlations in the odd-odd N=Z nuclei compared with the even-even ones.

Mirror Symmetry of Residual Proton–Neutron Interactions

In addition to the δV_pn of N = Z nuclei, mirror symmetry of the residual p-n interactions on both sides of the Z = N line [56] were observed. Such mirror symmetry was observed many years ago by Jänecke [103] and was recently used by Zong et al. [104] to make high-precision mass predictions for neutron-deficient nuclei.

Figure 18 shows the energy differences were defined as $\Delta \left(\delta V_{\text{pn}}\right)=\delta V_{\text{pn}}\left(T_z^{<}\mathrm{,}A\right)-\delta V_{\text{pn}}\left(T_z^{>}\mathrm{,}A\right)$ [35]. Here, $T_z^{<}/T_z^{>}$ represents the negative/positive value of Tz for the mirror nuclei. The δV_pn values are calculated using Eqs. (13)–(16), using mass data from AME2020 [9]. Masses of ⁴⁹Fe and ⁵⁵Cu Tz=-3/2 nuclides as well as ⁵⁴Ni Tz=-1 deduced from measurements of neutron-deficient ⁵⁸Ni projectile fragments at the CSRe are also used in these calculations, as shown by the blue dots.

Fig. 18Full-screen View

(Color online) Differences of δV_pn for mirror nuclei [35]. The gray shadow indicates an error band of 50 keV. The neutron-deficient partners are indicated for the mirror pairs with A≤20

Δ(δV_pn) values scatter around zero within an error band of ±50 keV for the A > 20 mirror pairs, indicating that the mirror symmetry of δV_pn holds well in this mass region. Notably, the mass of ⁵⁵Cu was redetermined with significantly improved precision, and its value was 172 keV more bound than the previous value [105]. Using this new mass, the calculated δV_pn value of ⁵⁶Cu was approximately equal to that of ⁵⁶Co. Consequently, $\Delta \left(\delta V_{\text{pn}}\right)=\delta V_{\text{pn}}\left({}^{56}\text{Cu}\right)-\delta V_{\text{pn}}\left({}^{56}\text{Co}\right)$ fits general systematics well with a high level of accuracy (see Fig. 18). This result further confirmed the reliability of the new mass value for ⁵⁵Cu.

In the lighter-mass region with $A\le 20$, the so-called mirror symmetry of δV_pn is broken into a few mirror pairs (see Fig. 18). That is, the δV_pn values of the neutron-deficient nuclei are systematically smaller than those of the corresponding neutron-rich partners. Such mirror-symmetry breaking is simply attributed to the binding-energy effect [103]. Here, we note that one of the four nuclei is particle-unbound, which may lead to the mirror-symmetry breaking of δV_pn. Thus, the mirror symmetry of δV_pn cannot be used to predict masses with $A\le 20$.

Ground-state mass of ²²Al and test of ab initio calculations

The level structure of mirror nuclei is commonly addressed and discussed based on the isospin symmetry, which is a basic assumption in the fields of particle and nuclear physics. However, this symmetry is approximate and the corresponding deviation is called isospin symmetry breaking (ISB) [106-110]. Studies on mirror nuclei offer profound insights into the origin of ISB and further information about nuclear structures, as well as facilitate the evaluation of nuclear models [106-113]. A key quantity in the investigation of ISB in mirror nuclei is the mirror-energy difference (MED) [114, 106, 115], which is defined as follows: $\begin{array}{l}MED=E_x\left(J\mathrm{,}T\mathrm{,}{T}_z=-T\right)-E_x\left(J\mathrm{,}T\mathrm{,}{T}_z=T\right)\mathrm{,}\hfill \end{array}$ (17) where $E_x\left(J\mathrm{,}T\mathrm{,}{T}_z\right)$ denotes the excitation energy of a state with spin J, isospin T, and z-projection Tz.

²²Al is the lightest-bound Al isotope with Tz=-2. Two low-lying 1⁺ states in odd-odd ²²Al were identified via β-delayed one-proton emissions from ²²Si [112]. However, their excitation energies have not been determined precisely because of the lack of an experimental ground-state mass of ²²Al. Recently, the ground-state mass excess of ²²Al was measured for the first time using Bρ-defined IMS as 18103(10) keV [58], which is 97(400) keV smaller than the extrapolated ME=18200(400) keV in AME2020 [9]. The new mass excess value allowed the determination of the excitation energies of the two low-lying states, $E_x\left(1_1^{+}\right)=1002(51)$ keV and $E_x\left(1_2^{+}\right)=2242(51)$ keV in ²²Al, with a significantly reduced uncertainty of 51 keV. The experimental MEDs of $1_{\mathrm{1,2}}^{+}$ states in ²²Al-²²F mirror nuclei are shown in Fig. 19 [58].

Fig. 19Full-screen View

(color online) MEDs of the $1_{\mathrm{1,2}}^{+}$ states in ²²Al/²²F mirror nuclei by employing ab initio VS-IMSRG calculations with four sets of nuclear interactions. The experimental MED values are plotted with error bars, whereas the calculated values are not

The MEDs of ²²Al-²²F mirror nuclei were calculated using an ab initio VS-IMSRG approach, employing several sets of nuclear forces derived from chiral effective-field theory [58]. The large uncertainties in the MEDs obtained in Ref. [112] prevented the benchmarking of the theoretical calculations. The uncertainties of the MEDs obtained in this study are significantly reduced, and the results show that the MED of the $1_1^{+}$ states is larger than that of the $1_2^{+}$ states in ²²Al-²²F mirror nuclei. Based on the ab initio calculations, we observe that the MEDs calculated with 1.8/2.0(EM), NNLO_opt, and NN+3N(lnl) interactions are in good agreement with the experimental data, especially the result from NNLO_opt, whereas the predicted MED values with NNLO_sat for the $1_{\mathrm{1,2}}^{+}$ states are inverted when compared with the experimental data. Therefore, the MED of the mirror nuclei serves as a more sensitive observable tool for testing theoretical calculations when comparing the excitation spectra. Moreover, mirror systems ²²Al-²²F were investigated using the ab initio Gamow shell model based on the same EM1.8/2.0 interaction [116]. Within the Gamow shell-model calculations, the effective Hamiltonian is derived through many-body perturbation theory, and continuum coupling is well accounted for using the Berggren basis. The results obtained from ab initio GSM calculations were consistent with those obtained from the VS-IMSRG calculations based on the EM1.8/2.0 interaction.

Test of Isospin Multiplet Mass Equation

The mass of a set of isobaric analog states (IASs) can be described by the well-known quadratic isospin multiplet mass equation (IMME) [117]: $\begin{array}{l}ME\left(A\mathrm{,}T\mathrm{,}{T}_z\right)=a(A\mathrm{,}T)+b(A\mathrm{,}T)\times T_z+c(A\mathrm{,}T)\times T_z^2\mathrm{,}\hfill \end{array}$ (18) where the MEs are the mass excesses of the IASs of a multiplet with a fixed mass number A and total isospin T. T is equal to or greater than the projection of T, $T_z=(N-Z)/2$, for a specific nucleus. The coefficients a, b, and c depend on A, T, and other quantum numbers, such as spin and parity Jπ, but are independent of Tz. The quadratic form of the IMME, that is, Eq. (18), is commonly considered accurate within uncertainties of a few tens of keV. In this context, precise mass measurements can be used to test its validity (see Ref. [118] and references therein). Typically, we add extra terms to Eq. (18) such as $d{T}_z^3$ and/or $e{T}_z^4$, which provide a measure of the breakdown of the quadratic form of the IMME. Numerous measurements were performed to investigate the validity of the IMME. Reviews and compilations of existing data can be found in Refs. [119, 120] and the references cited therein.

Measurements of neutron-deficient ⁵⁸Ni projectile fragments at the CSRe provided the masses for ⁴¹Ti, ⁴⁵Cr, ⁴⁹Fe, ⁵³Ni, and ⁵⁵Cu using Bρ-defined IMS [35]. Using these new masses, the four experimental masses of the T = 3/2 IASs were completed; thus, the validity of the quadratic form of the IMME could be tested, reaching the heaviest A = 55 isospin quartet. The mass data were fitted to the cubic form of the IMME by adding the $d{T}_z^3$ term to Eq. (18) and the fitted d coefficients are shown in Fig. 20.

Fig. 20Full-screen View

　d coefficients of the cubic form of IMME for the T = 3/2, A = 41, 45, 49, 53, and 55 isospin quartets [35]. The solid line connects the predicted d values from theoretical calculations [121]

Compared with previous results in Ref. [118], we conclude that the trend of a gradual increase in d with A in the fp shell [118] was not confirmed, at least at the present level of accuracy. Given that all the extracted d coefficients are compatible with zero, the quadratic form of the IMME is valid for the cases investigated here. Figure 20 shows the d values obtained from the theoretical calculations [121]. The predicted nonzero d coefficients for these T = 3/2 isospin quartets cannot be discarded owing to large experimental uncertainties.

Ground-state mass of ⁷⁰Br

The masses of the Tz=-1 nuclides ⁵⁸Zn, ⁶⁰Ga, ⁶²Ge, ⁶⁴As, ⁶⁶Se, and ⁷⁰Kr deduced from measurements of neutron-deficient ⁷⁸Kr projectile fragments at the CSRe are used to investigate the systematics of b and c coefficients of the quadratic form of the IMME up to the upper fp-shell nuclei [122].

Using the new mass results reported in Ref. [53] and the literature values in the latest atomic-mass evaluation (AME2020) [9], the b and c coefficients are deduced according to Eqs. (18) up to A = 70, as shown in Fig. 21 [122].

Fig. 21Full-screen View

　b and c coefficients of the quadratic IMME [122] extracted using the AME2020 mass data [9] (black circles) and the new masses from the CSRe [53] (red circles). The c coefficient using ME(⁷⁰Br =-51934(16) keV is indicated by the blue triangle [122]. The solid lines denote a semiempirical formula. Refer to details in Ref. [122]

Figure 21 (a) shows that the deduced b coefficients follow the smooth trend established in the lighter-mass region. The obtained c coefficients exhibit regular zig-zag staggering with changes in A, as shown in Fig. 21 (b). However, the regular staggering trend breaks at A = 70 and the derived c coefficient becomes negative. This anomaly was analyzed and attributed to the presently adopted mass of ⁷⁰Br in the latest atomic-mass evaluation (AME2020).

The smooth trend of the b coefficients and the regular staggering pattern of the c coefficients can be deduced using the mass formula given in Ref. [123] for the members of an isobaric multiplet characterized by A, T, and Tz: $M\left(A\mathrm{,}T\mathrm{,}{T}_z\right)=M_0\left(A\mathrm{,}T\right)+E_{\text{c}}\left(A\mathrm{,}T\mathrm{,}{T}_z\right)+T_z\times \Delta m_{\text{nH}}\mathrm{,}$ (19) where $E_{\text{c}}\left(A\mathrm{,}T\mathrm{,}{T}_z\right)$ represents the total charge-dependent energy in the nucleus and Δm_nH is the neutron–hydrogen mass difference. The semi-classical approach [124, 125], $E_{\text{c}}=｛0.6Z^2-0.46Z^{4/3}-0.15\left[1-{\left(-1\right)}^Z\right]｝\frac{e^2}{r_0{A}^{1/3}}\mathrm{,}$ (20) can be used to calculate the theoretical values of parameters b, c. The calculated b and c coefficients are shown in Fig. 21 as solid lines.

As reported in Refs. [126, 127], a combination of the theoretical calculations of Coulomb-energy differences (CEDs) and experimental masses of neutron-rich nuclei provides reliable masses for proton-rich nuclei. The CED between the Tz = 0 and Tz =+1 isobaric pairs was extracted using $\text{CED}=M\left(A\mathrm{,}{T}_z=0\right)-M\left(A\mathrm{,}{T}_z=+1\right)+\Delta m_{\text{nH}}\mathrm{.}$ (21) Figure 22 shows the CDE difference $\Delta \text{CED}={\text{CED}}_{\text{exp}}-{\text{CED}}_{\text{th}}$ as a function of the mass number A [122].

Fig. 22Full-screen View

Plot of $\Delta {\text{CED=CED}}_{\text{exp}}-{\text{CED}}_{\text{th}}$ and the least-squares fits with linear functions [122]. $\Delta \text{CED}=-152(16)$ keV at A = 70 is thereby deduced. Data uncertainties are within the size of the symbols. Orbits for major shells and subshells are indicated

The ΔCEDs follow very good linear behavior within each shell closure, while exhibiting a large deviation at A = 70 between A = 58 and A = 74.

Under the assumption of a smooth variation in the CED with changing mass number A, the ground-state mass excess of ⁷⁰Br is deduced to be -51934(16) keV, which is 508(22) keV more bound than the adopted value [122]. Using the new ME of ⁷⁰Br, the recalculated c coefficient is obtained using Eq. (18), and the results are shown in Fig. 21 (b). The new c coefficient now fits well into the systematics and is consistent with a simple theoretical estimate. This supports our recommended ME value of ⁷⁰Br obtained from the systematics of the Coulomb-energy differences.