Centring in CYCIAE


The following calculations were made using the formulas of Schulte and Hagedoorn (NIM 171 p. 409). The Hamiltonian (45) was analyzed to determine the frequency of movement of the orbit centre. This is done as follows. For H=a y^2/2 + b yP + c P^2/2 (y is y-coordinate of orbit centre, the momentum P conjugate to y is actually the x-coordinate of orbit centre), it is easy to show that the angular frequency of y-P motion is 2pi * sqrt(ac - b^2), which equals 2pi times the imaginary part of the tune. This quantity, when imaginary is plotted in the contour plots below. The MATLAB macro for these plots is here. I also made a calculation using the formula (28) of Gordon and Marti (Particle Accelerators, Vol. 12. (1982) p.13-26); the macro is here. They give substantially the same results.

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:


This calculation does not take into account phase slipping, due either to lack of isochronism or phase compression. Nevertheless, some qualitative features are apparent.
  1. Early phases (negative ones) are more stable than late phases (just as in the TRIUMF case).
  2. Dee angle of 45 degrees is preferred, but not very strongly; a large range of phases will still have frequency near zero.

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.

Verical stability in CYCIAE

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.

Rick Baartman
Last modified: Fri Aug 25 10:38:06 PDT 2006