DUE September 23

Magnets and Multipoles

Consider an iron-dominated \(2n\)-pole magnet (of “infinite length”) that to lowest-order generates a pure multipole (\(B_r \sim \sin n\phi\), \(n = integer >0\)) and whose magnetic field is described by a scalar potential, \(\vec{B} = \nabla\Phi_m\).

1. Show that the magnetic scalar potential satisfies Laplace’s equation \(\nabla^2\Phi_m = 0\) and that its solution may be written as \[ \Phi_m = C\;r^n\sin n\phi \] where \(C\) is a constant.
2. Show that the magnetic field of the 2\(n\)-pole magnet may be written in cylindrical corrdinates as \[ B_r = C\;n \;r^{n-1}\sin n\phi ~~~ ,\\ B_\phi = C\;n \;r^{n-1}\cos n\phi ~~~ . \] The constant \(C\) can now be interpreted as \[ C = \frac{1}{n!}\left( \frac{\partial^{n-1} B_\phi}{\partial r^{n-1}} \right)_{\phi=0}. \]
3. Suppose there are \(N\) turns of conductor per pole, each carrying current \(I\). If the pole faces are equipotential surfaces with minimum radius \(R\) at angles \(\phi = k\cdot(\pi/2n)\) for \(k = 1,3,5,\ldots,4n-1\), show that \[ \left( \frac{\partial^{n-1} B_\phi}{\partial r^{n-1}} \right)_{\phi=0} = \frac{\mu_0n!NI}{R^n}. \]
4. Create a plot of the cross-section of the poles of a decapole (10-pole) magnet with an inner aperture of radius \(R\) = 5 cm, out to radius 8 cm.

Doublet

Consider a beam of electrons, each with kinetic energy 200 MeV. A quadrupole lens is used to focus this beam, with a focal length of 4 m.

1. What magnetic gradient, \(B' \equiv \partial B_y/\partial x\), is required if the effective length of the magnet is 0.2 m? Is a thin lens approximation justified for use in this situation?

2. We wish to focus the beam both horizontally and vertically using a “doublet”. Consider a second quadrupole magnet with the same parameters as the one used above, but with its electrical polarity reversed creating a “defocusing” lens. Treating these two quadrupoles as thin lenses, if they are now separated by 0.6 m, center-to-center, with the beam passing through the horizontally focusing magnet first and then the horizontally defocusing magnet,

   1. what is the resulting focal length in the horizontal plane, measured from the midpoint of the two lenses?
      
   2. What is it in the vertical plane?
      
   3. How do these compare with the theoretical thin lens doublet focal length?
      
   4. Can you create a system of these two magnets that has the same focal length in both planes? (Justify your answer.)

Betatron Motion

1. Create an arbitrary sytem of 50 thin lenses of varying focal lengths and varying distances of separation. For instance, take all the lenses equally spaced by 30 m, but then add random flucutations of \(\pm\) 2 m to each spacing. Next, take the lenses to have equal focal lengths of 25 m, with every-other quad focusing or defocusing, then add a 10% random fluctuation to each quad. Next, create a distribution of 2000 particles from an arbitrary set of Courant-Snyder parameters and emittance, say \(\beta\) = 40 m and \(\alpha\) = -1, and track each particle through the system (1-D phase space only). Is the motion bounded (say, less than 50 mm maximum)? If not, vary the initial CS parameters of the problem until the maximum particle excursion is less than about 50 mm throughout most the system. Make a single plot of the 2000 trajectories (\(x\) vs. \(s\), say) through the system to verify your result.

2. From the initial set of Courant-Snyder parameters used above, “track” their values using \(K = MKM^T\) along the exact same beamline and plot \(\sqrt{\beta}\) vs. \(s\); compare with the earlier tracking result.

3. If your arbitrary system above were repeated indefinitely, would the system be stable?

A Booster Ring

Consider a system made of thin lens quadrupoles with the following arrangement: FOOFODOOOODO

Here, F represents a horizontally focusing lens with focal length \(F\) = 6 m, D represents a horizontally defocusing lens of focal length \(-F\), and O represents a drift space of length \(L\) = 2.5 m.

1. Determine if the system is stable in each plane \((x, y)\). If it is, determine the phase advance in degrees through the system for each plane.

2. For a synchrotron made up of 24 such sections, provided that the motion is stable, determine the betatron oscillation tunes \((\nu_x, \nu_y)\) of the synchrotron for each degree of freedom (\(x\) and \(y\)).

3. What are the maximum values of the amplitude functions \(\beta_x\) and \(\beta_y\)?