```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.
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