Yamamura analytically derived the capture threshold by integrating an equation of motion; m d(γ&beta)/dt=Eosin k(x-t). where one dimentional motion is only considered in longitudinal direction (x) . The capture condition is βf=1 at end of the integration. Therefore, Eo > mk/(1+cos &Phii), where &Phii is initial phase in (x-t) . The minimal Eo, i.e. threshold gradient, should be mk/2 . At the x-band LINAC, Eo > 61.0MV/m .
He also had a simulation study in two cases of sine-wave and "MAFIA-field" in the accelerating structure, where the accelerating gradient is 65MV/m. He calculated the final phase (&Phif) after 10ns as a function of the initial phase (&Phii) . For the sine-wave, the &Phif distribution is symmetric at &Phii=&pi.
However, the "MAFIA-field" has complicated distribution which can not be intuitively understood. So, he is requested for more detailed investigation in the "MAFIA-field" calculation.
(2) Final Focus System (S. Kuroda)
(transparencies, 1 page, pdf, 24KB)
Kuroda estimated longitudinal position (dz), field strength (dk) and rotational (d&theta) errors of sextupole magnets (SD0, SF1, SD4, SF5 and SF6) in the FF system, which blow up beam sizes by 50% in both directions at IP. i.e. 1.5&sigma*x=300nm and 1.5&sigma*y=4nm. These errors are expected to affect on the beam sizes at IP. The dz has no significant effect on the beam sizes, i.e. it can be greater than 100 um. The smallest dk error is 0.73-0.76 at SD0 and SD4, and the smallest d&theta one is 1 mrad at SD0, SF1 and SD4.