Incoherent effect due to electron cloud

A study of the electron-cloud pinch dynamics during a bunch passage under the effect of a single arbitrary-order multipole was presented earlier [1]. However, in a realistic situation, the proton beam will not be located in the center of the vacuum chamber. If the beam is offset a new pinch regime is encountered, where feed-down effects and asymmetry of pinch density render the dynamics more challenging. More recently we initiated an approach to investigate the dynamics of the field created by the pinched electrons, including transverse displacements the beam, the vacuum chamber of the surrounding magnet [2]. In the present paper we continue this study and develop a field map approach to the modeling of the proton beam dynamics under the effect of the pinched electrons. We compute the detuning with amplitude under various conditions through tracking simulations and compare the results with a static prediction based on the gradient of the electric field experienced on the beam axis.