# Centring in CYCIAE

## Calculations

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:

## Observations:

This calculation does not take into account phase
slipping, due either to lack of isochronism or phase
compression. Nevertheless, some qualitative features are
apparent. - Early phases (negative ones) are more stable than late phases (just as
in the TRIUMF case).
- 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