1. Academic Training Lecture 2 : Beam Dynamics
Equation of motion in transverse co-ordinates
The lattice
Meaning of Twiss parameters
Twiss Matrix
Effect of a drift length and a quadrupole
Calculating the Twiss parameters
FODO Cell
Liouville’s theorem
Phase stability
Closed orbit of an ideal machine
Dispersion
Dispersion – from the “sine and cosine” trajectories
Transition
Synchrotron motion

2. The lattice

3. 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

4. Meaning of Twiss parameters
is either :
Emittance of a beam anywhere in the ring
Courant and Snyder invariant fro one particle anywhere in the ring

5. 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

6. Twiss Matrix (continued)
Writing
The matrix elements are
Above is the general case but to simplify we consider points which are separated by only one PERIOD and for which
The “period” matrix is then
If you have difficulty with the concept of a period just think of a single turn.

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

8. Effect of a drift length and a quadrupole

9. Calculating the Twiss parameters
THEORY
COMPUTATION
(multiply elements)
Real hard numbers
Solve to get Twiss parameters:

10. FODO Cell
Write down matrices from mid-F to mid-F
Since they must equal the Twiss matrix:

11. Liouville’s theorem (Rough Version)
“The area of a contour which encloses all the beam in phase space is conserved”
This area = is the “emittance”
It is the same all round the ring
NOT TRUE:
during acceleration
in an electron machine where synchrotron emission damps

12. Phase stability E t

13. Bucket and pendulum  
The “bucket” of synchrotron motion is just that of the rigid pendulum
Linear motion at small amplitude
Metastable fixed point at the top
Continuous rotation outside

14. Closed orbit of an ideal machine
In general particles executing betatron oscillations have a finite amplitude
One particle will have zero amplitude and follows an orbit which closes on itself
In an ideal machine this passes down the axis
Closed orbit
Zero betatron
amplitude

15. Dispersion Low momentum particle is bent more
It should spiral inwards but:
There is a displaced (inwards) closed orbit
Closer to axis in the D’s
Extra (outward) force balances extra bends
D(s) is the “dispersion function”
Fig. cas C

16. Dispersion in the SPS
This is the long straight section where dipoles are omitted to leave room for other equipment - RF - Injection - Extraction, etc
The pattern of missing dipoles in this region indicated by “0” is chosen to control the Fourier harmonics and make D(s) small
It doesn’t matter that it is big elsewhere

17. Dispersed beam cross sections
These are real cross-section of beam
The central and extreme momenta are shown
There is of course a continuum between
The vacuum chamber width must accommodate the full spread
Half height and half width are:

18. Dispersion – from the “sine and cosine” trajectories
The combination of diplacement, divergence and dispersion gives:
Expressed as a matrix
It can be shown that:
Fulfils the particular solution of Hill’s eqn. when forced :

19. Transition - does an accelerated particle catch up - it has further to go
Is a function of two, momentum
dependent, terms  and R.
and
Using partial differentials to define
a slip factor:
This changes from negative to positive
and is zero at ‘ transition’ when:
GAMMA TRANSITION

20. Synchrotron motion Recall Elliptical trajectory for small amplitude
Note that frequency is rate of change of phase
From definition of the slip factor 
Substitute and differentiate again
But the extra acceleration is
THUS

21. Synchrotron mtion (continued)
This is a biased rigid pendulum
For small amplitudes
Synchrotron frequency
Synchrotron “tune”

22. Summary of : Beam Dynamics
Equation of motion in transverse co-ordinates
The lattice
Meaning of Twiss parameters
Twiss Matrix
Effect of a drift length and a quadrupole
Calculating the Twiss parameters
FODO Cell
Liouville’s theorem
Phase stability
Closed orbit of an ideal machine
Dispersion
Dispersion – from the “sine and cosine” trajectories
Transition
Synchrotron motion