Cyclotrons and Synchrotrons — Question 10

Question 10

The synchronous RF phase must supply both planned acceleration and per-turn losses

An ion of positive charge q receives energy $qV_0\sin\phi$ per turn from an RF system and loses energy Wloss per turn. The desired net energy increase is G per turn. Treat both G and Wloss as prescribed constants.

1. Derive the phase condition and the feasibility bound on G.

2. For qV0=10.0 keV, Wloss=2.00 keV and G=3.00 keV, find all phases between zero and pi that satisfy energy balance. State what the energy equation alone says about phase stability.

Question 10 - Solution

The synchronous RF phase must supply both planned acceleration and per-turn losses

(a) Method and calculation

$$\Delta K=qV_0\sin\phi-W_{\mathrm{loss}}=G\Rightarrow\sin\phi=\frac{G+W_{\mathrm{loss}}}{qV_0}.$$$$0\le G+W_{\mathrm{loss}}\le qV_0\quad\text{for }0\le\phi\le\pi.$$

(b) Interpretation and further calculation

$$\sin\phi=(3+2)/10=1/2\Rightarrow\phi=\pi/6\quad\text{or}\quad5\pi/6.$$$$\frac{d(\Delta K)}{d\phi}=qV_0\cos\phi\quad\text{has opposite signs at the two phases}.$$Stability additionally requires the relation between an energy error and the next arrival-phase error.

Physical check

An equilibrium energy balance identifies candidate synchronous phases but does not by itself establish stable longitudinal motion.