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Accelerator Physics Particle Acceleration

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

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Accelerator Physics Particle Acceleration

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

  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. 17.34) • Matrix from Rotating Coordinates (Eq. 17.36, corrected)

  5. Transfer Matrix • For solenoid of length L, transfer matrix is • Decoupled matrix in rotating coordinate system (Eq. 17.34) • Matrix from Rotating Coordinates (Eq. 17.36, 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

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