We detail in this manuscript the experiment developed in the MMTF team of the Laboratoire de Physique des Lasers, whose aim is to contribute to the redefinition of the International System of units through the measurement of the Boltzmann constant, kB, by laser spectroscopy. The method used is based on the linear absorption spectroscopy of an ammonia gas maintained at constant temperature. The Boltzmann constant is deduced from the gas temperature measurement and the width of the recorded absorption profile. First and second generation experiments already led to a 38x10-6 statistical uncertainty on the measurement of kB. In this manuscript, we describe how we improved the experimental set-up to reduce the measurement uncertainty to a fewx10-6 level. We make profit of these experimental improvements and fit the data in a new way to reduce the measurement statistical uncertainty to a 6.4x10-6 level. Then, we carry out a complete study of systematic effects that could affect the Boltzmann constant measurement. We notably describe the experimental set-up built to analyze NH3 hyperfine structure. Among other things, we evaluate the impact of the gas composition, the transition saturation, and the laser beam modulation. Finally, we deeply study how to model collisions between molecules. In the end, the global uncertainty budget on systematic effects is reduced to a 2.1x10-6 level. All the obtained results and future prospects pave the way to an optical measurement of kB with an accuracy of a few 10-6.
