1. Lecture 2 - Transverse motion
ACCELERATOR PHYSICS
Lecture 2
Transverse Motion
TT 2012
E. J. N. Wilson

2. Need for Accelerators

3. Weak focusing in a synchrotron
The Cosmotron magnet
Vertical focusing comes from the curvature of the field lines when the field falls off with radius ( positive n-value)
Horizontal focusing from the curvature of the path
The negative field gradient defocuses horizontally and must not be so strong as to cancel the path curvature effect

4. Gutter

5. Transverse ellipse

6. Cyclotron
Magnetic rigidity
Constant revolution frequency

7. Vertical Focusing People just got on with the job of building them.
Then one day someone was experimenting
Figure shows the principle of vertical focusing in a cyclotron
In fact the shims did not do what they had been expected to do
Nevertheless the cyclotron began to accelerate much higher currents

8. Equation of motion in a cyclotron
Non relativistic
Cartesian
Cylindrical

9. Cyclotron orbit equation
For non-relativistic particles (m = m0) and with an axial field Bz = -B0
The solution is a closed circular trajectory which has radius
and an angular frequency
Take into account special relativity by
And increase B with g to stay synchronous!

10. Cyclotron focusing – small deviations
See earlier equation of motion
If all particles have the same velocity:
Change independent variable and substitute for small deviations
Substitute
To give

11. Cyclotron focusing – field gradient
From previous slide
Taylor expansion of field about orbit
Define field index (focusing gradient)
To give horizontal focusing

12. Cyclotron focusing – betatron oscillations
From previous slide - horizontal focusing:
Now Maxwell’s
Determines
hence
In vertical plane
Simple harmonic motion with a number of oscillations per turn:
These are “betatron” frequencies
Note vertical plane is unstable if

13. Components of a synchrotron

14. Transverse coordinates

15. Strong focussing

16. Fields and force in a quadrupole
No field on the axis
Field strongest here
(hence is linear)
Force restores
Gradient
Normalised:
POWER OF LENS
Defocuses in
vertical plane
Fig. cas 10.8

17. SPS

18. Equation of motion in transverse co-ordinates
Hill’s equation (linear-periodic coefficients)
where at quadrupoles
like restoring constant in harmonic motion
Solution (e.g. Horizontal plane)
Condition
Property of machine
Property of the particle (beam) e
Physical meaning (H or V planes)
Envelope
Maximum excursions

19. Twiss Matrix
All such linear motion from points 1 to 2 can be described by a matrix like:
To find elements first use notation
We know
Differentiate and remember
Trace two rays one starts “cosine”
The other starts with “sine”
We just plug in the “c” and “s” expression for displacement an divergence at point 1 and the general solutions at point 2 on LHS
Matrix then yields four simultaneous equations with unknowns : a b c d which can be solved

20. Twiss concluded Can be simplified if we define the “Twiss” parameters:
Giving the matrix for a ring (or period)

21. Alternating gradients

22. The lattice

23. Beam sections
after,pct

24. Physical meaning of Q and beta

25. Example of Beam Size Calculation
Emittance at 10 GeV/c