1. Lecture 8
ACCELERATOR PHYSICS HT
E. J. N. Wilson

2. Summary of last lecture – Non Linear 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

3. Instabilities Non-Linear Dynamics II
Definition of Hamiltonian
Relativistic H of a charged particle in an electromagnetic field
Casting the Hamiltonian in the correct form
Hamiltonian for a particle in an accelerator
Multipoles in the Hamiltonian
Hills equation from the Hamiltoninan
The effect of a sextupole
Third integer phase space
Injection studies at FNAL
Example of tracking
Crossing the stochastic limit

4. Definition of Hamiltonian
Any old coordinates
Lagrangian is just kinetic - potential energy
Equation of motion
Define canonical momenta
Prescription for Hamiltonian
Equations of motion are:

5. Relativistic H of a charged particle in an electromagnetic field
Remember from special relativity:
i.e. the energy of a free particle
Add in the electromagnetic field
electrostatic energy
magnetic vector potential has same dimensions as momentum

6. Casting the Hamiltonian in the correct form
Hamiltonian of a relativistic particle in
a general electromagnetic field.
In our transverse coordinates:
Turn inside out to change from t to s

7. Hamiltonian for a particle in an accelerator
is the new Hamiltonian with s as the independent variable instead of t
By concentrating on the linear transverse terms
assumes curvature is small
assumes
assumes magnet has no ends
assumes small angles
ignores y plane
Dividing by
Finally

8. Multipoles in the Hamiltonian
We said contains x (and y) dependence
We find out how by comparing the two expressions:
We find a series of multipoles:
For a quadrupole n=2 and:

9. Hills equation from the Hamiltoninan
Equations of motion are:
Or in terms of the new independent variable:
Applying the second equation we get
And if we repeated for the vertical plane ( and keep the curvature term)
Where we allow a momentum defect dp/p

10. The effect of a sextupole
Lets look at sextupoles
In they appear as:
If you apply “canonical transformations” to eliminate the linear terms:
J and f are new canonical coordinates in which we can draw phase space trajectories

11. Third integer phase space
Inner trajectories are triangular
And beyond it is unstable
the contours of constant (trajectories!) go off to infinity trying to satisfy W82

12. INJECTION STUDIES AT FNAL
Remanent sextupole in the FNAL main ring caused serious beam loss due to non-linear resonances.
This was exacerbated by magnet ripple.
A three dimensional hill and dale model spanning the Q ( or  ) diagram
The vertical co-ordinate of this three dimensional model is the fraction of the beam which survives to be accelerated (always < 70%)
FNAL-MODELL.PCT

13. Example of tracking
Below the beam-beam limit and without tune modulation we see the many archipelagos of islands due to different multipole components in the beam-beam potential.
Increasing the beam-beam interaction will enlarge the islands so that they overlap in stochastic regions where particle may diffuse out in 4 dimensional phase space.
Increasing amplitude tune slope will merge them too

14. Crossing the stochastic limit
Increasing Q shift causes islands to merge in a stochastic sea of chaos

15. Summary Non-Linear Dynamics II
Definition of Hamiltonian
Relativistic H of a charged particle in an electromagnetic field
Casting the Hamiltonian in the correct form
Hamiltonian for a particle in an accelerator
Multipoles in the Hamiltonian
Hills equation from the Hamiltoninan
The effect of a sextupole
Third integer phase space
Injection studies at FNAL
Example of tracking
Crossing the stochastic limit