E-Cloud Map Formalism: an Analytical Expression for Quadratic Coefficient

The bunch-to-bunch evo­lu­tion of the elec­tron cloud den­si­ty can be mod­eled using a cubic map. The map ap­proach has been proved re­li­able for RHIC* and LHC**. The cofficients that pa­ram­e­ter­ize the map may be ob­tained by fitting from time con­sum­ing numer­i­cal sim­u­la­tions. In this com­mu­ni­ca­tion we de­rive a simple ap­prox­i­mate for­mu­la for the quadrat­ic coefficient, which de­ter­mines the sat­u­ra­tion of the cloud due to space charge, in the electron cloud den­si­ty map, under the as­sump­tions of round cham­bers and free-field mo­tion of the elctrons in the cloud. Results are com­pared with simula­tions for a wide range of pa­ram­e­ters gov­ern­ing the evo­lu­tion of the electron cloud.