Mathematical Descriptions of Axially Varying Penning Traps for the Antimatter Experiment: gravity, Interferometry, and Spectroscopy

Antimatter, though proposed in 1933, is still not well understood. AEgIS aims to study the interaction of antihydrogen with the earth's gravitational field. This information will add to our understanding of the matter-antimatter asymmetry present in our universe. This paper discusses a Penning-Malmberg with a magnetic mirror that will hold $C_{2}^{-}$ that will be used for sympathetic cooling of antiprotons before the antihydrogen is created. The trap, which is critical to the cooling process of the antihydrogen, can be characterized by the separatrix between trapped and untrapped particles. This paper applies analytical processes used to define the separatrix of pure electron plasmas to a molecular plasma. Our work is based on the desire conditions (density, particle number, field strength, trap size) of the high field region. The initial application of a semi-analytical method applied to our trap defines the trap potential difference at \~ 0.6V. The separatrix is defined in both the high and low field region, by an ellipse and a hyperbola respectively. This shape varies from the separatrix of a Penning trap with a constant axial magnetic field. Now that we have successfully applied this method to the our $C_{2}^{-}$ trap and defined the high and low field separatrices, we may move forward with a better understanding of the cooling process that the carbon molecule will undergo.