The impressive progress that elementary particle physics made in the second half of the last century led to the formulation of a comprehensive theory, known as the Standard Model (SM), which correctly describes all fundamental interactions in nature, except for the gravitational one. Indirect discoveries have always played an important role in high energy physics scenario and indirect research can be considered to all intents and purposes complementary to the direct one, since allows to test much higher energy scales than those the current colliders are able to reach. This is very important now that electroweak precision tests and measurements on Flavour Changing Neutral Currents (FCNC) processes put very stringent constraints on physics beyond the SM, requiring it to appear at scales O(10 TeV). On the other hand, New Physics (NP) is expected already at scales O(1 TeV) in order to offer a natural explanation to the smallness of the Higgs mass. This scale is also confirmed by recent constraints on thermal dark matter [1] which show how new physics should manifest not far above the electroweak scale. Rare B decays have always played a crucial role in shaping the flavour structure of the SM and particle physics in general. Since the first measurement of rare radiative B æ Kú“ decays by the CLEO Collaboration [2] this area of particle physics has received much experimental and theoretical attention. In particular, FCNC B decays, involving the b-quark transition b æ (s, d) + “ and b æ (s, d) + ¸+¸≠(¸ = e, μ, ·, ‹), provided crucial tests for the SM at the quantum level since they proceed through loop or box diagrams, and they are highly suppressed in the SM (also by helicity). Hence, these rare B decays are characterised by their high sensitivity to NP. The B0 s æ μ+μ≠ channel is the most direct example of the b æ s ¸¸ transitions. The SM predicted branching ratio [3] can be enhanced by coupling to non-SM heavy particles, such as those predicted by the Minimal Supersymmetric Standard Model (MSSM) and other extensions. Updated measurements on the B0 s æ μ+μ≠ branching ratio have been presented by ATLAS [4], LHCb [5] and CMS [6] collaborations. In this thesis I will report all the studies I performed within the rare B decays ATLAS group, measuring the branching ratio of the B0 s æ μ+μ≠ channel on data collected during LHC Run 1. The first chapter provides a general introduction to the SM, focusing in particular on the flavour sector and the possible new physics scenarios. Chapter 2 briefly introduces the LHC collider and the ATLAS detector, detailing the muon and trigger systems, fundamental for the rare B decays measurements. In chapters 3 and 4, I will summarise the work done, during my presence at CERN, on the ATLAS semiconductor strip detector, monitoring the Lorentz angle during Run 1 and measuring the backplane resistance of the silicon modules installed in the ATLAS cavern. In chapter 5, I will review the strategy adopted to measure the B0 s æ μ+μ≠ branching ratio, reporting all the studies I performed on the combinatorial background, and the results obtained on 4.9 fb≠1 of data collected in 2011. Chapters 6 and 7 detail respectively the additional studies I performed on the 2011 datasets and all the tests I made in preparation for the analysis on 20 fb≠1 of data collected in 2012. I will show the studies on the discriminating variables for the rejection of the background, the tests on the multivariate analysis and on the possible strategies for the invariant mass fit for the extraction of the signal yield. All these studies proved to be fundamental for the 2012 measurement detailed in chapter 8.