The evolution of the electron density during electron cloud formation can be reproduced using a bunch-to-bunch iterative map formalism. The reliability of this formalism has been proved for RHIC [1] and LHC [2]. The linear coefficient has a good theoretical framework, while quadratic coefficient has been proved only by fitting the results of compute-intensive electron cloud simulations. In this communication we derive an analytic expression for the quadratic map coefficient. The comparison of the theoretical estimate with the simulations results shows a good agreement for a wide range of bunch population.
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