Laserspectroscopic determination of the nuclear charge radius of ${}^{13}$C

Light nuclei, that consist of only a few nucleons, are exciting testing grounds for our understanding of fundamental interactions. Bound by the residual strong interaction acting between the quarks inside the protons and neutrons, these nuclei form interesting structures such as condensed α clusters or halo nuclei that are challenging to describe by nuclear theory. Over the last decades, ab initio nuclear structure calculations, that are rooted in quantum chromodynamics, were improved significantly. Providing precise benchmark values for these theories is essential to improve the precision of predictions on how nuclear matter emerges. The isotopes of the light element carbon (C) are highly interesting cases to study as they exhibit pronounced α clustering and are important contributors to the nucleosynthesis process in stars. Additionally, C is at the limit of what is computationally possible using higher-order nuclear structure calculations, and due to its unfavorable spectral properties, no experimental high-precision spectroscopy data is available so far. In this work, the differential nuclear charge radius of ${}^{12,13}$C is determined purely from results of ab initio nonrelativistic quantum electrodynamics atomic structure calculations and highprecision collinear laser spectroscopy measurements carried out at the Collinear Apparatus for Laser Spectroscopy and Applied Science (COALA), located at the Institute for Nuclear Physics at the Technical University Darmstadt. For this, first high-accuracy measurements of the 1s2s ${}^3$S${}_1$ → 1s2p ${}^3$P${}_{0,1,2}$ transitions in He-like ${}^{13}$C${}^{4+}$ were carried out and combined with measurements in ${}^{12}$C${}^{4+}$ from preceding work. The C${}^{4+}$ isotopes in the metastable ${}^3$S${}_1$ state are produced in an electron beam ion source and are accessible with lasers operated at a wavelength of 227.6 nm. The fluorescence detection region (FDR) of COALA at these deep-UV wavelengths was improved with a new lens-based FDR designed and built within this work. The new segment provides an improved signal-to-noise ratio compared to the previous mirror-based design. This considerably facilitated spectroscopy of the weakest transitions in ${}^{13}$C${}^{4+}$, which split into hyperfine structure (HFS). The effect of hyperfine-induced mixing on the transition frequencies is investigated and benchmark values for atomic structure calculations are provided. The new model independent δr${}^2$${}^{12,13}$ = -0.1245(66) fm${}^2$ is compared to results from elastic electron scattering, muonic atom spectroscopy and ab initio nuclear structure calculations. In combination with the existing experimental results for ${}^{12}$C, the absolute nuclear charge radius of ${}^{13}$C is determined. An elaborate analysis of the fluorescence spectra and potential systematic uncertainties is presented that is enabled by the new Python package qspec, developed within this work for simulations and data analysis surrounding laser spectroscopy. The package was extensively tested during beamtimes at GSI, CERN/ISOLDE and ANL where it significantly contributed to decision-making processes by enabling a detailed live data analysis and simulations. In addition to the analysis of ${}^{13}$C${}^{4+}$, an investigation of quantum interference effects and optical-population transfer in the HFS of ${}^{87}$Sr${}^{+}$ is presented in the appendix.