Study of Conduction Cooling Effects in Long Aspect Ratio Penning-Malmberg Micro-Traps

A first order perturbation with respect to velocity has been employed to find the frictional damping force imposed on a single moving charge due to a perturbative electric field, inside a long circular cylindrical trap. We find that the electric field provides a cooling effect, has a tensorial relationship with the velocity of the charge. A mathematical expression for the tensor field has been derived and numerically estimated. The corresponding drag forces for a charge moving close to the wall in a cylindrical geometry asymptotically approaches the results for a flat surface geometry calculated in the literature. Many particle conduction cooling power dissipation is formulated using the single particle analysis. Also the cooling rate for a weakly interacting ensemble is estimated. It is suggested that a pre-trap section with relatively high electrical resistivity can be employed to cool down low density ensembles of electrons/positrons before being injected into the trap. For a micro-trap with tens of thousands of micro-tubes, hundreds of thousands of particles can be cooled down in each cooling cycle. For example, tens of particles per micro-tube in a $5 cm$ long pre-trap section with the resistivity of $0.46 \Omega m$ and the micro-tubes of radius $50 \mu m$ can be cooled down with the time constant of $106\mu s$.