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

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

3 Transfer Matrices 3 3 3 3 3

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

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

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

Beta Functions in a FODO Lattice 7 7 7 7

Beta Functions in a FODO Lattice 8 8 8 8

9 Betatron Motion 9 9 9 9 9

Betatron Motion 10 10 10 10

Betatron Motion Courant Snyder ellipse 11 11 11 11

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

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 Poincare Sections 14 14 14 14 14

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

Another Example: Slip-stacking 16 Another Example: Slip-stacking 16 16 16 16 16

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