  The creation of plasma is key for achieving fusion in Inertial Electrostatic
Confinement (IEC) fusion devices, and the conditions for such electrical
breakdown are modelled by Paschen's law, which predicts the breakdown voltage
of a system as a function of the product of pressure and gap distance (pd)
between electrodes. However, the Paschen curve only models parallel plate
configurations of electrodes and is rarely explored in the more complex
electrode geometries often seen in IEC fusion devices, including
Farnsworth-Hirsch fusors. To bridge this gap, we study the Paschen curve for
various electrode configurations by use of Monte Carlo (MC) simulations in
Mathematica and Java. We compute alpha - the rate of ionizations per unit
length - in constant E-field systems to explore differences between predicted
alpha values from the scientific literature and those from our MC simulations.
We explore a Markov chain model for 1D alphas which outperforms the literature
alpha prediction in modelling our 1D MC-derived alphas and more closely matches
parallel plate MC breakdown results at the minimum. 3D simulations and
breakdown plots are created for concentric sphere and sphere-in-cylinder
geometries. We observe a widened breakdown curve for concentric spheres and
sphere-in-cylinder geometries and note the reduced growth of breakdown voltage
at small pd values for the sphere-in-cylinder case. These findings help explain
the data collected from our own experimental fusor setup and suggest further
work in the simulation of more complex electrode configurations and the
incorporation of experimental data to verify results.
