Analytical and Numerical Study of Superconducting Dipole and Quadrupole Performance Limits for a Muon Collider

In accordance with the guidelines set forth in the Updated European Strategy for Particle Physics, the International Muon Collider Collaboration has been initiated to assess the feasibility of a Muon Collider facility with a center-of-mass energy of 10 TeV. The primary challenges stem from the brief muon lifetime at rest, which is limited to 2.2 µs. Addressing this demanding condition necessitates the incorporation of advanced technologies in magnets, RF systems, targets, shielding, and cooling. To minimize collimated neutrino beams resulting from muon decay and reduce the impact of radiation background around the facility, the straight sections in the collider ring must be minimized. Achieving this goal involves integrating beam optics quadrupoles with bending dipoles featuring a high magnetic field ($>$10 T) and gradient ($>$100 T/m) within a large aperture (140 mm). These stringent constraints require cutting-edge technologies in material selection, mechanical layout, quench protection, shielding, and cooling. This contribution explores the performance limits of potential candidate materials for such magnets, specifically LTS (Low-Temperature Superconductors) Nb$_{3}$Sn and (High-Temperature Superconductors) ReBCO, regarding the maximum field, mechanical stress, and stored energy. We present an original approach that allows us to explore the achievable phase space of parameters using both analytical expressions and the FEM software ANSYS, handled by Python code. Using this approach, we will show the obtained results for dipoles and quadrupoles.