If the magnetic field is flat, nu_r-1=0, then a phase band of width 180 -
h dee_angle degrees, centred on zero, is unstable. This is shown below
for a dee_angle of 38 degrees:
The contour lines are for 2pi imaginary part of the tune = 0.0, 0.1, 0.2, 0.3, 0.4, 0.5. If the field has a radial dependence to give isochronism, then nu_r=gamma. In this case, the phase band shifts to negative (negative phases are those which enter the dee gap first); by the end of the plot, 1.1 MeV, it has slipped only 4 degrees.
But nur is not very near gamma on the first few turns. Hongjuan gave
me the nu_r vs. Energy. This goes through zero at 0.08MeV and again at
0.33MeV. Between these two values, it is negative. The stability plot
for 38 degree dees is shown below:
With 45 degree dees:
Phases with a real centring frequency that is near zero are not
guaranteed to be safe. In that case, linear growth is possible. This is
the case in the TRIUMF cyclotron. Since initial centring errors are as
large as the beam radius, and growth is sign-dependent, initial beam
ellipses are stretched linearly with turn number (as in TRIUMF). For
CYCIAE, the dominant effect is for particles with displacement
orthogonal to the dee axis, to drift along the dee axis. The rate of
drift per turn is
delta x_c= 2 y sin(phi) / turn number
So outside particles with a phase of 30 degrees in a beam of 2 mm radius can move 2 mm on turn 1, 1 mm on turn 2, 0.7 mm on turn 3, etc. This does not go on indefinitely because when nu_r-1 is large enough, the centres will rotate and the effects average out.
An effect not taken into account is the transverse (einzel lens-like) focusing that occurs in the first 1 or 2 gaps. This has the beneficial effect of augmenting nu_r.
It turns out that the formula for vertical tune is almost the same as for radial. I use the Marti-Gordon (Particle Accelerators, Vol. 11, p. 161) formula 23. The macro is here.
Using Hongjuan's provided nuz vs. E, the 38 degree dee is
and the 45 degree case is
Notice the scale of the contours: this is a much stronger instability for the wrong phases.