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This white paper is the result of a collaboration by those that attended a workshop at the Facility for Rare Isotope Beams (FRIB), organized by the FRIB Theory Alliance (FRIB-TA), on Theoretical Justifications and Motivations for Early High-Profile FRIB Experiments. It covers a wide range of topics related to the science that will be explored at FRIB. After a brief introduction, the sections address: (II) Overview of theoretical methods, (III) Experimental capabilities, (IV) Structure, (V) Near-threshold Physics, (VI) Reaction mechanisms, (VII) Nuclear equations of state, (VIII) Nuclear astrophysics, (IX) Fundamental symmetries, and (X) Experimental design and uncertainty quantification., Comment: 227 pages, 24 figures

Fast beam collinear laser spectroscopy is the established method to investigate nuclear ground state properties such as the spin, the electromagnetic moments, and the charge radius of exotic nuclei. These are extracted with high precision from atomic observables, i.e., the hyperfine splitting and its the isotope shift, which becomes possible due to a large reduction of the Doppler broadening by compressing the velocity width of the ion beam through electrostatic acceleration. With the advancement of the experimental methods and applied devices, e.g., to measure and stabilize the laser frequency, the acceleration potential became the dominant systematic uncertainty contribution. To overcome this, we present a custom-built high-voltage divider, which was developed and tested at the German metrology institute (PTB), and a feedback loop that enabled collinear laser spectroscopy to be performed at a 100-kHz level. Furthermore, we describe the impact of field penetration into the laser-ion-interaction region. This strongly affects the determined isotope shifts and hyperfine splittings, if Doppler tuning is applied, i.e., the ion beam energy is altered instead of scanning the laser frequency. Using different laser frequencies that were referenced to a frequency comb, the field penetration was extracted laser spectroscopically. This allowed us to define an effective scanning potential to still apply the faster and easier Doppler tuning without introducing systematic deviations.

A method for calculating the field shift contribution to isotope shifts in many-electron atoms, incorporating quantum electrodynamics (QED) effects, is introduced. We also implement the model QED approach to incorporate QED contribution to the nuclear recoil effect at the high-order correlation effects treatment level. The proposed computational scheme is used to revise the value of the root-mean-square (rms) nuclear charge radius of the isomer of aluminium-26, $^{26m}$Al. This radius is important for the global analysis of the $V_{ud}$ element of the Cabibbo-Kobayashi-Maskawa matrix. The difference in mean-square nuclear charge radii of $^{27}$Al and $^{26m}$Al, obtained by combining the calculated atomic factors with recently measured isotope shift (IS) of the $3s^23p~^2P_{3/2} \to 3s^24s~^2S_{1/2}$ transition in Al, is $0.443(44)(19)~{\rm fm}^2$, where the first and second uncertainties are experimental and theoretical ones, respectively. The latter is reduced by a factor of 4 with respect to the previous study. Using this value and the known value of the rms charge radius of $^{27}$Al, the resultant value $R_c(^{26m}$Al) = 3.132(10)~fm is obtained. With the improved accuracy of the calculated IS factors the error in $R_c(^{26m}$Al) is now dominated by the experimental uncertainty. Similar revision of rms charge radii is made for the $^{28}$Al, $^{29}$Al, $^{30}$Al, $^{31}$Al and $^{32}$Al isotopes using existing IS measurements. Additionally, atomic factors are computed for the $3s^23p~^2P_{3/2} \to 3s^24s~^2S_{1/2}$, $3s^23p~^2P_{1/2} \to 3s^25s~^2S_{1/2}$ and $3s^23p~^2P_{3/2} \to 3s^25s~^2S_{1/2}$ transitions in Al, which can be used in future experimental studies.