Linear transverse motion
This chapter begins with a discussion that deepened the introductory remarks made on betatron oscillations. It explains that in such a scheme, one can have simultaneous stability in both vertical and horizontal transverse dimensions. It introduces strong focusing based on periodic arrays of focusing and defocusing quadrupoles in order to circumvent the natural scaling of weak focusing. It notes that powerful matrix methods are employed to analyse the piece-wise periodic focusing arising from these arrays. It explores in detail these methods, along with the stroboscopic Poincaré trace space-mapping visualisation tool. It employs an alternative approach to understand periodic focusing, which uses a perturbative analytical technique, which helped in indentifying an alternating gradient focusing as a type of ponderomotive focusing. It introduces momentum dispersion as well as the general six-dimensional square matrix description of the six-dimensional phase space dynamics of a charged particle.
Keywords: betatron oscillations, focusing, stroboscopic Poincaré trace, periodic focusing, perturbative analytical technique, ponderomotive focusing, momentum dispersion, six-dimensional square matrix, six-dimensional phase space dynamics
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