High-resolution, accurate multiple-reflection time-of-flight mass spectrometry for short-lived, exotic nuclei of a few events in their ground and low-lying isomeric states

Mass measurements of fission and projectile fragments, produced via ${}^{238}\mathrm{U}$ and ${}^{124}\mathrm{Xe}$ primary beams, have been performed with the multiple-reflection time-of-flight mass spectrometer (MR-TOF-MS) of the Fragment Separator (FRS) Ion Catcher with a mass resolving power (FWHM) of up to 410 000 and an uncertainty of down to $6\times {10}^{-8}$. The nuclides were produced and separated in flight with the fragment separator FRS at 300 to 1000 MeV/u and thermalized in a cryogenic stopping cell. The data-analysis procedure was developed to determine with highest accuracy the mass values and the corresponding uncertainties for the most challenging conditions: down to a few events in a spectrum and overlapping distributions, which can be distinguished from a single peak only by a broader peak shape. With this procedure, the resolution of low-lying isomers is increased by a factor of up to 3 compared to standard data analysis. The ground-state masses of 31 short-lived nuclides of 15 different elements with half-lives of down to 17.9 ms and count rates as low as 11 events per nuclide were determined. This is the first direct mass measurement for seven nuclides. The excitation energies and the isomer-to-ground-state ratios of six isomeric states with excitation energies of as little as 280 keV were measured. For nuclides with known mass values, the average relative deviation from the literature values is $(4.5\pm 5.3)\times {10}^{-8}$. The measured two-neutron separation energies and their slopes near and at the $N=126$ and $Z=82$ shell closures indicate a strong element-dependent binding energy of the first neutron above the closed proton shell $Z=82$. The experimental results deviate strongly from the theoretical predictions, especially for $N=126$ and $N=127$.

Figure 1

Schematic mass-to-charge spectra illustrating the four classes of close-lying peaks as defined in the text. The distances between the mass distributions are 10, 2, 1, and 0.3 FWHM for classes A to D, respectively. The peak shape and the abundance ratio of 1 to 10 is selected to be identical for all classes. The thin black curves show the sum of both distributions.

Figure 2

Schematic figure of the experimental setup, including the FRS and the FRS Ion Catcher, which consists of the cryogenic stopping cell (CSC), the RFQ beamline, and the MR-TOF-MS.

Figure 3

Schematic figure of the multiple-reflection time-of-flight mass spectrometer. The ions enter from the CSC in gas-filled radio-frequency quadrupoles. They are mixed with ions from the calibration sources and are cooled and bunched in a system of linear RF traps. From here, they are injected in the TOF analyzer.

Figure 4

Mass resolving power (FWHM) obtained with the MR-TOF-MS for ${}^{39}\mathrm{K}^{+}$ ions as a function of the time of flight and of the number of turns. The kinetic energy of the ions in the drift tubes was 1300 eV and the repetition rate was 50 Hz. The measured data are shown as full squares. The solid line represents a fit of Eq. (8) to the data. The dashed line indicates the asymptote obtained from the fit for long flight times. The maximum mass resolving power (FWHM) achieved is 620 000; the asymptote is 870 000. Note that the flight time was limited to 20 ms due to the repetition frequency of the experiment.

Figure 5

The flow diagram of the data-analysis procedure. Only the main steps are shown. LS is a least-squares fit to determine the peak shape and wMLE is the weighted maximum likelihood estimation of the peak position and area.

Figure 6

Example for the time-resolved calibration (TRC) of a mass measurement of ${}^{39}\mathrm{K}^{+}$ ions that performed 980 isochronous turns in the MR-TOF-MS, corresponding to a total time-of-flight of 18.92 ms. The top diagrams show the peak positions as a function of the experiment time without TRC (a) and with TRC (b). The TRC was performed every 5 s. In the bottom part of the figure, the corresponding mass-to-charge spectra are shown. Because of TRC, an increase in the mass resolving power is obtained; the mass-resolving power (FWHM) after the TRC amounts to 510 000. Note that all horizontal axes have the same scale for easier comparison.

Figure 7

Measured mass spectrum of ${}^{211}\mathrm{Pb}^{+}$ ions, which was used as a calibrant in the mass measurement of ${}^{213}\mathrm{Rn}^{+}$ ions. The peak shape requires one exponential tail on each side [Hyper-EMG(1,1)] and additionally a Gaussian distribution (red full line) on the left side. The Gaussian distribution results from ion-optical effects. In addition, a Gaussian function (orange long-dashed line) and a Hyper-EMG(1,0) without a Gaussian distribution (purple dotted line), a Hyper-EMG(1,0) plus a Gaussian distribution (green dash-dotted line), and a Hyper-EMG(1,1) (blue dashed line) are shown. The mass spectrum is shown in linear scale in the inset. A mass resolving power (FWHM) of 200 000 was achieved for the time of flight of 5.8 ms, corresponding to 128 isochronous turns.

Figure 8

Measured mass spectrum of ${}^{211}\mathrm{Pb}^{+}$ ions, which was used as a calibrant in the mass measurement of ${}^{212}\mathrm{At}$ ions. The peak shape requires one exponential tail on each side [Hyper-EMG(1,1)] (red full line). In addition, a Gaussian function (orange long-dashed line), a Hyper-EMG(1,0) (purple dotted line), and a Hyper-EMG(0,1) (gray dash-dotted line) are shown. The mass spectrum is shown in linear scale in the inset. A mass resolving power (FWHM) of 180 000 was achieved for the time of flight of 5.8 ms, corresponding to 128 isochronous turns.

Figure 9

Measured shift of the time of flight of ${}^{133}\mathrm{Cs}$ ions due to the MRS as a function of the number of isolation cycles. The shift is normalized by ${(m/q)}^{1/2}$ (for details, see text). The lines show a fit of Eq. (13) to the data points. The resulting fit constant is $A_{\mathrm{mrs}}=1.1\times {10}^{-11}\mathrm{s}/\sqrt{\mathrm{u}/\mathrm{e}}$.

Figure 10

Example for the effect of the correction algorithm for fitting overlapping peaks of class C. The deviations of measured and literature values for the mass-to-charge difference of isomeric and ground states determined for different nuclides are shown without the correction (red data points) and with the correction (blue data points). The literature values were taken from the AME2016 [62]. The error bars of the measurements before the correction do not include the uncertainty of the correction. The gray band centered around the horizontal axis shows the error bars of the mass-to-charge differences as obtained from literature values.

Figure 11

Statistical uncertainty normalized to the FWHM of the peak as a function of the number of measured ${}^{213}\mathrm{Fr}$ ions per spectrum of experiment I. The mean value for the FWHM of the distribution in this measurement was $1.03\times {10}^{-3}\mathrm{u}/\mathrm{e}$.

Figure 12

Measured variation of the TOF of ${}^{133}\mathrm{Cs}^{+}$ ions after two isochronous turns in dependence of the storage time of the ions in the analyzer. The lower panel shows an enlargement of the time interval, for which a mass determination can be performed. The nonideal ejection (NIE) uncertainty is calculated from that interval. For the gray range in the lower panel, an increased NIE uncertainty of 2 ns is used. The TOF deviations are the differences between the TOF for the individual measurement and the mean of all measurements in the lower panel range.

Figure 13

Time-of-flight spectrum of different atomic and polyatomic ions with TOF of about 18.8 ms. Depending on their mass-to-charge ratio, the ions undergo 603 to 628 isochronous turns ($N_{\mathrm{it}}$).

Figure 14

Deviations of measured mass excess (ME) values from literature values, obtained for ions undergoing different numbers of turns ($N_{\mathrm{it}}$). The corresponding TOF spectrum in shown in Fig. 13. The ions ${}^{12}\mathrm{C}{}^{13}\mathrm{C}{}^{19}\mathrm{F}_4^{+}$, ${}^{12}\mathrm{C}_2{}^{19}\mathrm{F}_4^{+}$, ${}^{12}\mathrm{C}_2{}^{16}\mathrm{O}{}^{19}\mathrm{F}_4^{+}$, and ${}^{12}\mathrm{C}_3{}^{19}\mathrm{F}_3^{+}$ (open diamond symbol) were used for the determination of the calibration parameters $t_0$ and $c$. TRC was performed using the ion ${}^{12}\mathrm{C}_2{}^{13}\mathrm{C}{}^{19}\mathrm{F}_3^{+}$. The ion ${}^{12}\mathrm{C}_2{}^{16}\mathrm{O}{}^{19}\mathrm{F}_3^{+}$ (filled diamond symbols) was used for the calculation of the final mass-to-charge value.

Figure 15

Mass-to-charge spectrum of the nuclides ${}^{133}\mathrm{I}$, ${}^{133m}\mathrm{I}$ (excitation energy of about 1.6 MeV) and ${}^{133}\mathrm{Te}$, ${}^{133m}\mathrm{Te}$ (excitation energy of about 300 keV) obtained with the MR-TOF-MS in experiment I. The binned data (black squares), the fits with the Hyper-EMG(1,1) function (red [gray] and blue [dark gray] curves), and the sum of all fits (thin black curve) are shown. The spectrum was calibrated using ${}^{133}\mathrm{Cs}$ ions. The mass resolving power (FWHM) amounts to 410 000. Note that the mass-to-charge spectrum is shown with a logarithmic abundance scale.

Figure 16

Mass-to-charge spectrum of ${}^{134}\mathrm{Sb}$ and ${}^{134m}\mathrm{Sb}$ ions. There are 25 counts in the spectrum. The unbinned data events are depicted above the mass-to-charge ratio axis. The curves represent a fit with Hyper-EMG(0,1) functions to the unbinned data. The curves correspond to ${}^{134}\mathrm{Sb}$ (red [gray]), ${}^{134m}\mathrm{Sb}$ (blue [dark gray]), and their sum (thin black).

Figure 17

Relative deviation of measured ground-state masses $(m_{\mathrm{FRS}-\mathrm{IC}}-m_{\mathrm{AME}16})/m_{\mathrm{FRS}-\mathrm{IC}}$ (Table 1) and isomer excitation energies $(E_{\mathrm{ex},\mathrm{FRS}-\mathrm{IC}}-E_{\mathrm{ex},\mathrm{AME}16})/m_{\mathrm{FRS}-\mathrm{IC}}$ (Table 2) from the literature values given in AME2016 [62]. All relative deviations are calculated with respect to the ground-state masses. The gray band around the horizontal axis represents the literature uncertainty. In the right-hand panel, a histogram of the relative deviations with a bin size of $2.5\times {10}^{-7}$ is shown. The weighted mean of all relative deviations is $(4.5\pm 5.3)\times {10}^{-8}$ and the standard deviation is $3.5\times {10}^{-7}$.

Figure 18

Two-neutron separation energy $S_{2\mathrm{n}}$ vs neutron number including mass excess data from AME16 (open symbols) and the ones measured in this work (red filled symbols). The closed neutron shell at $N=126$ appears as a steeper slope.

Figure 19

Slope of the two neutron separation energy ${\mathrm{\Delta }}_{1\mathrm{n}}{S}_{2\mathrm{n}}$ vs proton number. The black symbols show to the data corresponding to the AME16 and the blue symbols the measurements from this work.

Figure 20

Slope of the two neutron separation energy ${\mathrm{\Delta }}_{1\mathrm{n}}{S}_{2\mathrm{n}}$ vs proton number in different mass models, Thomas-Fermi [88], HBF21 [89], and UNDEF0 [90].