Lecture 7 ACCELERATOR PHYSICS HT6 2005 E. J. N. Wilson

Summary of last lectures – Instabilities INSTABILITIES I 1. General Comment on Instabilities 2. Negative Mass Instability 3. Driving terms (second cornerstone) 4. A cavity-like object is excited 5. Equivalent circuit 6. Above and below resonance 9. By analogy with the negative mass INSTABILITIES Ii 1. A short cut to solving the instability 2. An imaginative leap 3. The effect of frequency shift 4. Square root of a complex Z 5. Contours of constant growth 6. Landau damping 7. Stability diagram 8. Robinson instability 9. Coupled bunch modes 10 Microwave instability

Instabilities Non-Linear Dynamics I Resonance condition Simple resonant condition Multipole field expansion (polar) Taylor series expansion Multipole field shapes Sextupole magnet Normal and skew sextupoles Dipole magnet errors Third integer resonance Phase space trajectory for the one-third integer resonance

Resonance condition Simple resonant condition This equation just defines a set of lines in the Q diagram for each order of resonance and for each value of the integer p. Figure shows these lines for the SPS.

Multipole field expansion (polar) Scalar potential obeys Laplace whose solution is Example of an octupole whose potential oscillates like sin 4around the circle

Taylor series expansion Field in polar coordinates: To get vertical field Taylor series of multipoles Fig. cas 1.2c

Multipole field shapes

Sextupole magnet

Normal and skew sextupoles Fig. cas 1.24c and Lines

The dN and dS poles superimposed Dipole magnet errors The dN and dS poles superimposed on the magnet poles give the effect of cutting the poles to a finite width. The remanent magnetomotive force : is weaker at the pole edges, and the field tends to sag into a parabolic or sextupole configuration Fig. cas 1.12c

Circle diagram

Second order resonance Substituting Incidentally If Fourier analysis of has a frquency 2Q - Resonance occurs. Fig. cas 1.16

Fourier analysis of a perturbation Basic expressions Modulation of Q etc. .....

Stop band concept Fig. ca 1.11

Third integer resonance Substituting Incidentally

Phase space trajectory for the one-third integer resonance

Summary Non-Linear Dynamics I Resonance condition Simple resonant condition Multipole field expansion (polar) Taylor series expansion Multipole field shapes Sextupole magnet Normal and skew sextupoles Dipole magnet errors Third integer resonance Phase space trajectory for the one-third integer resonance