1. Lecture 7
ACCELERATOR PHYSICS HT
E. J. N. Wilson

2. 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

3. 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

4. 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.

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

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

7. Multipole field shapes

8. Sextupole magnet

9. Normal and skew sextupoles
Fig. cas 1.24c and Lines

10. 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

11. Circle diagram

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

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

14. Stop band concept
Fig. ca 1.11

15. Third integer resonance
Substituting
Incidentally

16. Phase space trajectory for the one-third integer resonance

17. 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