Transverse-longitudinal coupling by space charge in cyclotrons

A method is presented that enables one to compute the parameters of matched beams with space charge in cyclotrons with emphasis on the effect of the transverse-longitudinal coupling. Equations describing the transverse-longitudinal coupling and corresponding tune shifts in first order are derived for the model of an azimuthally symmetric cyclotron. The eigenellipsoid of the beam is calculated and the transfer matrix is transformed into block-diagonal form. The influence of the slope of the phase curve on the transverse-longitudinal coupling is accounted for. The results are generalized and numerical procedures for the case of an azimuthally varying field cyclotron are presented. The algorithm is applied to the PSI injector II and ring cyclotron and the results are compared to TRANSPORT.

Figure 1

Transport (re)computation of a matched beam in injector II for a circulating beam of 2.2 mA at low energy. The boxes indicate the position of the sector magnets SM1–SM4. The horizontal beam width is plotted as a negative value. The longitudinal beam size is plotted as $2{\sigma }_z$.

Figure 2

Top: Phase shift per turn for the ring cyclotron computed from the measured magnetic field map. Middle: Horizontal and vertical betatron tunes of the ring cyclotron as computed by EO code (gray solid lines), the transfer matrix method over the EO (black solid lines), and with space charge (dashed lines). Bottom: Longitudinal betatron tune as induced by transverse-longitudinal coupling. The region with a negative slope of the phase shift is indicated by vertical lines. Without further field trimming, the longitudinal focusing is suppressed by the term $-h\frac{d\epsilon }{dr}$ and the longitudinal beam size may increase. The calculation has been performed for a beam current of 2.2 mA and $({\epsilon }_x,{\epsilon }_y,{\epsilon }_z)=(1.5,2.5,0.5)\pi \mathrm{mm}\mathrm{mrad}$.