The smallest parts of the world that we live in are described mathematically by the hugely successful Standard Model of Particle Physics. Nevertheless, during the last several decades researchers within physics have observed many phenomena that cannot be explained within this framework. Among the most noticeable observations are the fact that the Standard Model only describes 5% of the energy content of the whole Universe. There is a worldwide search ongoing for a more complete understanding, and the analysis described in this thesis is a small part of this comprehensive work.
Some theories beyond the Standard Model include new, electrically neutral, massive, force carrying particles called Z'. These might be produced at the Large Hadron Collider (LHC) at CERN in the energetic collisions between protons. Using data from the ATLAS detector, one of the four main experiments at the LHC, I have searched for Z' particles in final states with two muons (elementary particles), without finding it. In the absence of a detected signal, I set limits on the lowest possible mass of the Z' in several theories, between 3.8 and 4.4 TeV. My results are comparable to the recent published results from the ATLAS Collaboration and the CMS Collaboration.