key: cord-0329811-hpyaw43k authors: Karthein, Jonas; Atanasov, Dinko; Blaum, Klaus; Lunney, David; Manea, Vladimir; Mougeot, Maxime title: Analysis methods and code for very high-precision mass measurements of unstable isotopes date: 2021-02-21 journal: nan DOI: 10.1016/j.cpc.2021.108070 sha: 3bb9a8b4aad0c42d8e3d551da6281a5c3e869eed doc_id: 329811 cord_uid: hpyaw43k We present a robust analysis code developed in the Python language and incorporating libraries of the ROOT data analysis framework for the state-of-the-art mass spectrometry method called phase-imaging ion-cyclotron-resonance (PI-ICR). A step-by-step description of the dataset construction and analysis algorithm is given. The code features a new phase-determination approach that offers up to 10 times smaller statistical uncertainties. This improvement in statistical uncertainty is confirmed using extensive Monte-Carlo simulations and allows for very high-precision studies of exotic nuclear masses to test, among others, the standard model of particle physics. INTRODUCTION -One of the fundamental observables of the atomic nucleus is the nuclear binding energy. This quantity directly results from the difference of the nuclear mass and the sum of the individual nuclear constituent masses (neutrons and protons). While measurements of stable systems allow for long preparation and measurement times, the precision for short-lived nuclides is strongly limited by the production rate at the radioactive ion beam (RIB) facilities (down to a few atoms per hour), as well as the half-lives (down to a few tens of milliseconds). However, studying these exotic systems often provides exclusive insights into the nuclear structure, the weak interaction or rapid nucleosynthesis processes compared to stable atoms. This particularity motivates technical advances in the field of high-precision mass spectrometry. Over the past four decades, Penning traps have emerged as the experimental tool of choice for high-precision measurements of atomic masses [Bl06, Hu17] . The ISOLTRAP spectrometer has pioneered the technique for radioactive species [Kl13] and Penning trap facilities now exist at all radioactive ion beam facilities worldwide [Lu18] . Further developments in the measurement technique using ISOLTRAP [Bl02, Ge07] allowed us to recently set a new world record for the highest precision on a mass of a short-lived (T½ < 1 h) isotope [Ka19a] . But this experiment also demonstrated the technique's intrinsic precision limitation, requiring hundreds of ions per measurement cycle, which is not feasible for the most short-lived isotopes of interest for nuclear physics. Given an optimal beam characteristics and high statistics, a new analysis approach presented below allows for a multifold reduction of the statistical uncertainty compared to the state of the art without any intervention on the experimental apparatus. It can even be applied to all alreadyexisting datasets and opens the door for veryhigh-precision mass determination required for stringent tests of the Standard Model with uncertainties in the order of a few (tens) eV/ . Penning trap is based on the direct determination of the cyclotron frequency , In the case of an ideal Penning trap, is also equal to the sum of the frequencies of the radial eigenmotions, and , of the trapped ion [Br86] (also, see Fig. 3 ). The magnetic field strength can be eliminated with the ratio , between the cyclotron frequency of a wellknown reference ion and the cyclotron frequency the ion of interest as long as these consecutive measurements are performed within a timeframe shorter than non-linear fluctuations of the magnetic field (typically ~15 min) [Ke03] . In order to reduce the statistical uncertainty, This ratio-determination method has proven to be very robust [Na93, Ka19a, Ka19b, Fi12, El15] c 2 (1) 3. The ion is prepared on a pure motion as in step 2 and immediately converted into a pure motion by resonant RF-excitation [Bl03, Ei09] . After evolution at for precisely the same accumulation time the position is recorded as in step 1 and 2. It is important to also use the same starting point as for step 2 in order for Eq. (3) to be valid. Each of these three measurement steps is repeated multiple times per measurement cycle in order to allow for a determination of the mean position of the so-called spots on the detector (see Fig. 2 ). These steps reduce the determination of the cyclotron frequency , to the determination of the final phase , defined by the spots in steps 1, 2 and 3 instead of requiring one additional spot for each frequency at thanks to the same preparation scheme. is divided into three parts. First, the raw data is read and transformed into position information. Second, the position information is used to determine the total phase and the cyclotron frequency. Third, the mass of the ion of interest is calculated based on the temporal distribution of cyclotron frequencies. Thanks to the Jupyter notebook environment single-pulse [Gr80] or Ramsey-type application [Ge07] with no adjustments. PI-ICR analysis software can occur due to asymmetries and imperfections in the trap or transport potentials. The true phase is compared to the average phase resulting from the mean X/Y coordinates from the multivariate 2D fit and the average phase from the 1D fit. Both averages are calculated from the fit results of 1000 random MC samples for each of the three distributions. The given uncertainties correspond to the average one-sigma uncertainty for the given quantity over the 1000 simulations. Additionally, the reduction factor in statistical uncertainty from the 2D to the 1D fit is shown for the three spot scenarios. The symmetric and aligned spots agree well with the true phase value, whereas the 1D fit for the rotated elliptical spot shows a >2 sigma deviation from the true value. February 2021 J. Karthein et al. ference between the two models. Deviations will however occur in reality. If we assume an exaggerated case with a highly elliptical or asymmetric spot at a rotation of 45 degrees (as depicted in Fig. 4b) It builds on the spatial distribution based on initial buffer-gas cooling, as derived in Eq. (12) ff. in [El14] . [Br18] . This integration is planned in a future performance update, but is not essential for the analysis code at this stage. From the positions of the reference, , and spots one can determine the phase with the arctangent function "atan2" of the NumPy library [Ha20a] , which has the whole circle as image set (1) to describe the temporal distribution of all measured -values using the least squares method (see Fig. 1 ). The statistical uncertainty is calculated from the covariance matrix of the fit, which is performed for all possible polynomial degrees with the number of data points to avoid and 1000 (c) ions per spot. Each deviation represents the mean of 1000 MC simulations and is plotted with its standard deviation as error bar. The behavior is compared to a sine function (red) from which the maximal deviation (~0.1 rad) and the rotation angle for the maximal deviation (~32 degrees) were derived. For comparison, the mean individual statistical uncertainty for 1000 repeated MC simulations is shown with its range for the given spot rotation angles. Carbon clusters for absolute mass measurements at ISOLTRAP Recent developments at ISOLTRAP: towards a relative mass accuracy of exotic nuclei below 10 -8 High-accuracy mass spectrometry with stored ions JAX: composable transformations of Python+NumPy programs Geonium theory: Physics of a single electron or ion in a Penning trap PI-ICR analysis software ROOT -An Object Oriented Data Analysis Framework Position-sensitive ion detection in precision Penning trap mass spectrometry Phase-imaging ion-cyclotron-resonance measurements for short-lived nuclides A phase-imaging technique for cyclotron-frequency measurements Direct Measurement of the Mass Difference of 163 Ho and 163 Dy Solves the Q-Value Puzzle for the Neutrino Mass Determination Q-value and half-lives for the double-β-decay nuclide 110 Pd Ramsey method of separated oscillatory fields for high-precision Penning trap mass spectrometry A direct determination of the proton electron mass ratio Array programming with NumPy Superallowed 0 + → 0 + nuclear β decays : 2020 critical survey , with implications for Vud and CKM unitarity Evaluation of input data; And adjustment procedures QEC-value determination for 21 Na → 21 Ne and 23 Mg → 23 Na mirror-nuclei decays using high-precision mass spectrometry with ISOLTRAP at the CERN ISOLDE facility Direct decay-energy measurement as a route to the neutrino mass PI-ICR analysis software First direct mass measurement of a superheavy element and high-precision mass spectrometry of nobelium and lawrencium isotopes and isomers using the Phase-Imaging Ion-Cyclotron-Resonance technique at SHIPTRAP From direct to absolute mass measurements: A study of the accuracy of ISOLTRAP Penning trap mass spectrometry of radionuclides Jupyter Notebooks -a publishing format for reproducible computational workflows A quantum mechanical model of Rabi oscillations between two interacting harmonic oscillator modes and the interconversion of modes in a Penning trap Extending and refining the nuclear mass surface with ISOLTRAP PI-ICR analysis software First Glimpse of the N=82 Shell Closure below Z=50 from Masses of Neutron-Rich Cadmium Isotopes and Isomers Data structures for statistical computing in Python Precision Penning trap comparison of nondoublets: Atomic masses of H, D, and the neutron iminuit -A Python interface to MINUIT High-precision measurement of the atomic mass of the electron SciPy 1.0-fundamental algorithms for scientific computing in Python