DUE October 14

Betatron Motion

Suppose a beam of particles entering a synchrotron has a phase space distribution (consider \(x\) only, for now) that is parameterized by the Courant-Snyder variables \(\beta\) = 25 m, \(\alpha\) = -1 and rms emittance \(\epsilon\) = 3.75 \(\pi\) mm-mrad when the particles first pass by a particular obervation point in the ring. If a matrix calculation were performed, corresponding to the magnetic elements about the circumference of this ring, we find that the one-turn matrix at this same observation point is given by \[ M_0 = \begin{pmatrix} 1.0238 & -28.970~{\rm m} \\ 0.10072{\rm /m}& -1.8732 \end{pmatrix} \] which transports the vector \(\vec{X} = (x,x')^T\) according to \(\vec{X}_{n+1} = M_0\vec{X}_n = M_0^n\vec{X}_0\) where \(n\) is the revolution number, or turn number.

  1. Create a distribution of several thousand particles that has the initial Courant-Snyder parameterization given above. Plot the phase space distribution after \(n\) = 0, 1, 2, 3, 4 and 5 revolutions, as well as histograms of the distribution in \(x\) (corresponding to possible beam width measurements at our observation point) for the same values of \(n\).

  2. Create a plot of how the beam distribution rms width varies with revolution number, for \(0<n<200\).

  3. Next, using the matrix \(M_0\), find the periodic Courant-Snyder parameters \(\beta_0\) and \(\alpha_0\) at the observation point, as well as the betatron tune \(\nu_x\) of the synchrotron.

  4. Can you relate the period of the rms width variation to the betatron tune?

  5. Create phase space plots of (\(\beta_0\;x'+\alpha_0\;x\)) vs. \(x\) of the distribution for \(n\) = 0, 1, 2, 3, 4, 5.

  6. Change \(\alpha\) and \(\beta\) of the initial distribution to be equal to \(\alpha_0\) and \(\beta_0\), thus creating a new corresponding initial particle distribution; repeat the plots of part (e) above.

The Standard FODO Cell

  1. Provide a formula for the Courant-Snyder parameters \(\beta_{max,min}\) for a thin lens FODO cell in terms of the cell phase advance, \(\mu\), and the half-cell length \(L\).

  2. Make a single plot of \(\beta_{max}/L\) and \(\beta_{min}/L\) vs. \(\mu\) for \(0 < \mu < 150^\circ\).

  3. Compute \(\beta_{max}\) and \(\beta_{min}\) for the Tevatron FODO cell, where the lenses were separated by 30 m, using a cell phase advance of 75 degrees. What focal length does this correspond to?

  4. Use MADX to perform the same calculation at the downstream end of each quadrupole in a thick lens system. Use Tevatron-like quadrupole magnets with effective lengths of 2 m. Note that the total cell length should be equivalent to that of the “thin lens” version and that the total cell phase advance should again be 75 degrees. What values for \(\alpha_x\) and \(\alpha_y\) do you find at these locations? What value of \(|K| = |B'|/B\rho\) is required in this calculation? What focal length does this value of \(K\) correspond to?
  5. Re-compute the values of \(\beta_{max}\) and \(\beta_{min}\) at the mid-point of each thick quad, using MADX. Also, what are the values of \(\alpha\) at these points?

  6. In the thin lens calculations, how would you determine CS parameters at the quad mid-points?

THE Booster Ring

The original Fermilab Booster synchrotron – still in operation today – was designed in the late 1960’s and has the following configuration:

  1. Verify that the above configuration should bend the beam through 360 degrees. What is the total circumference of the ring, and what fraction of the circumference contains bending?
  2. Create an appropriate MADX input file with a beam line that describes the Fermilab Booster. Compute and print out the values of the Courant-Snyder (Twiss) parameters at the end of each element according to the geometry described above.
  3. What are the betatron oscillation tunes \((\nu_x, \nu_y)\) of the synchrotron?
  4. Plot the periodic amplitude functions \(\beta_x\), \(\beta_y\) and the dispersion function \(D_x\) throughout a single period.
  5. Protons are injected into the Booster ring from a linear accelerator, when the proton kinetic energy is 400 MeV. Make a plot of the revolution time (in microseconds) for an ideal proton as a function of magnetic field corresponding to the ring’s range of operation.
  6. Suppose that a single BF magnet in the ring has its gradient altered by +5%. Compute the resulting change in the horizontal and vertical tunes of the Booster using appropriate modifications to the matrices found above. Estimate the change by using the simple tune shift formula and compare.