1. Accelerator Physics Particle Acceleration
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 8
G. A. Krafft
Jefferson Lab

2. Radial Equation
If go to full ¼ oscillation inside the magnetic field in the “thick” lens case, all particles end up at r = 0! Non-zero emittance spreads out perfect focusing!

3. Larmor’s Theorem
This result is a special case of a more general result. If go to frame that rotates with the local value of Larmor’s frequency, then the transverse dynamics including the magnetic field are simply those of a harmonic oscillator with frequency equal to the Larmor frequency. Any force from the magnetic field linear in the field strength is “transformed away” in the Larmor frame. And the motion in the two transverse degrees of freedom are now decoupled. Pf: The equations of motion are

4. Transfer Matrix For solenoid of length L, transfer matrix is
Decoupled matrix in rotating coordinate system (Eq )
Matrix from Rotating Coordinates (Eq , corrected)

5. Transfer Matrix For solenoid of length L, transfer matrix is
Decoupled matrix in rotating coordinate system (Eq )
Matrix from Rotating Coordinates (Eq , corrected)

6. To/from rotating coordinates

7. Fringe effect by conservation cannonical momentum
Match to Boundary Conditions at z = 0

8. Total Solenoid Transfer
Wiedemann 17.39
Thin Lens

9. Easy Calculation that Works
Wiedemann points out the following simple calculation is OK
Works because
Explanation hard to follow

10. Skew Quadrupole S N N S

11. Equations
Focusing in x + y, defocusing in x - y

12. Transfer Matrix

13. In terms of the usual variables
Thin Lens

14. Coupled Pendula
Equations of motion

15. Solutions
Eigenvalues
Eigenvectors

16. Time Dependence
For an oscillation starting in x-direction

17. Qualitatively Oscillation energy migrates x→y→x
Period for a complete cycle is
Becomes longer the weaker the coupling (→compensation)
If un-coupled oscillation periods different
Eigenvectors no longer pure symmetric and antisymmetric but “migration” is fairly generic behavior