1. Wednesday Week 1 Lecture
Jeff Eldred
Transfer Matrices, Betatron motion 1 1 1 1 1

2. Overview Transfer Matrices Courant Snyder Betatron motion2
Overview
Transfer Matrices
Courant Snyder
Betatron motion
Poincare Sections 2 2 2 2 2

3. 3
Transfer Matrices 3 3 3 3 3

4. Transfer Matrices We can solve the linear Hill’s equation:
The final position and slope is a linear combination of the initial position and initial slope. We can use matrices: 4 4 4 4

5. Common Transfer Matrices
Dipole
Drift
Defocusing Quad
Defocusing Quad, thin
Defocusing Quad, thick
Focusing Quad
Horizontal dipoles are vertical drifts.
Horizontal F (D) quads are vertical D (F) quads.
Skew quads and solenoid require 4x4 matrices. 5 5 5 5

6. Stability of Transfer Matrices
Stable means can propagate indefinitely:
Which means M can be written in the form: 6 6 6 6

7. Beta Functions in a FODO Lattice7 7 7 7

8. Beta Functions in a FODO Lattice8 8 8 8

9. 9
Betatron Motion 9 9 9 9 9

10. Betatron Motion 10 10 10 10

11. Betatron Motion
Courant Snyder ellipse 11 11 11 11

12. Betatron Phase If the tune is an integer, than errors accumulate.
There is a symmetry in field errors, errors will also accumulate if tune is a rational number. 12 12 12 12

13. Two BPM Measurement
If there is a pi/2 phase advance between X1 and X2, then X2 can be used as a measure of X1’ 13 13 13 13

14. 14
Poincare Sections 14 14 14 14 14

15. 15
Poincare Sections
Useful whenever there are two important time-scales. The (X,X’) plots are one example of this. Use to look at betatron period vs. revolution period. 15 15 15 15 15

16. Another Example: Slip-stacking16
Another Example: Slip-stacking 16 16 16 16 16

17. 17 17 17 17 17 17