ANALYTICAL FORMALISM FOR THE LONGITUDINAL ACCELERATION INCLUDING PARTICLE VELOCITY CHANGING EFFECT

To calculate the longitudinal linear beam dynamics, the Panofsky equation which introduces the concept of transit time factor and average phase is widely used. The transit time factor is generally calculated under the assumption of a constant beta through the element. In the case of large beta variations or long and complex accelerating element, this approach can lead to some inaccuracies. To address this problem an analytic method taking into account the variation of the beta within the accelerating element has been developed. This method is applicable to any element by using decomposition of the electrical field into Fourier components. The average phase concept is adapted to the new formulation and the passage from the physical entrance phase to the average phase is clearly stated. The accuracy of the method is also presented through comparison with a slow and precise numerical approach. ions with only two different types of 6-cell elliptical superconducting radio frequency (SRF) cavities. For such structures, the longitudinal dynamics treatment must offer flexibility to accommodate with the high accelerating gradient, with the large phase slips induced by the difference between the beta of the particles and the geometric beta of the structure, and with the field asymmetry present in the end-cells due to the large bore radius of the cavities. This need in flexibility is combined to the need in accuracy and fast computation. In the pursuit of these three prerequisites a new set of longitudinal dynamics equations have been developed. In Sec. 2, the usual set of equations for the longitudinal dynamics are applied to a case with large beta-changing and some losses in the accuracy for the energy gain and time of flight calculations are illustrated. In Sec. 3, a more general and precise method for the longitudinal dynamics based on consecutive analytical iterations is developed. In Sec. 4 this method is applied to the previous case of large beta-variation to show the gain in the accuracy.