12/98 HVS This is HVS's synchrotron oscillation cheat sheet. When I read through Sand's paper about electron storage rings, he talks about a linear approximation to energy variations. In considering the RF phase advance with energy coupled with the energy gained due to an RF phase advance, he comes up with a second order differential equation for energy (or alternatively phase). Solutions (undamped) are oscillatory, and the oscillation frequency is expressed in eq 3.43. If I re-arrange equation 3.43 into terms with which I a more familiar, I conclude that the SYNCHROTRON TUNE can be determined as: NuS = sqrt( alpha*h*eVg*cos(phis) / 2*pi*E0 ) Where: alpha = momentum compaction factor ==(dL/L)/(dP/P) h = harmonic number (note eVg/E0 is just == (RF freq)/(Go Around freq) the ratio of Gap Voltage e = electron charge and Beam Energy, each Vg = Total RF gap voltage expressed in volts) E0 = Beam energy phis = "synchronous phase" A little thought tells you for small deviations cos(phis) = sqrt(1-(U0/Vg)^2) where U0 is energy loss per turn.