Final Class Project

Beam Physics for Muon g-2

As we have discussed occasionally during class, the Muon g-2 experiment (E989) taking place at Fermilab relies heavily on many aspects of accelerator and beam physics. Below is a list of topics relevant to the experiment, with questions that can be addressed from the principles discussed throughout this semester.

For the Final Project, each student must select two topics from the list below, and answer all questions under those topics. The questions should be addressed fully in text, equations, code, plots, and summaries in an R notebook format (*.nb.html) created using Rstudio. Short answers by themselves (like, “We found that \(x_{final}\) = 3.87 mm.”) are not acceptable; the process for arriving at the answer should at least be described and the code or calculation of the result must be present in the notebook document. Feel free to expand beyond the questions to be answered in order to illustrate your approach or explain your answers further.

Each student must create her/his own individual notebook, written in one’s own words. Students enrolled in the class are allowed to work together (not with individuals outside the class!!) on coding and computations, and collaborative efforts among classmates are strongly encouraged. However, proper acknowledgements and citations must be provided – If students work together on coding or computations, each student must be sure to list near the beginning of their report who they worked with, who did what part, how decisions were made to divide up the work, etc. If a student or group wishes to use the results obtained by another student or group working on a problem not being solved by the first group, then the first group MUST adequately acknowledge the second group (and, the instructor advises that they be very sure the second group did their work correctly!).

The class sessions of December 3 and December 5, 2019, will be used to discuss the topics and various approaches toward addressing the questions under each topic, as well as to answer any questions about the generation of the final notebooks. The students must submit their final R notebooks (name_Final.nb.html) to the instructor via email time-stamped no later than 11:59 p.m. CST 10 December 2019. No further course material will be accepted from a student beyond this time.

Please: Ask the instructor questions!

Topics

Topic 1: Bunch Formation

Bunches destined to the Muon g-2 target for muon production are produced in the Fermilab Recycler Synchrotron. (Yes, it’s called the Recycler, though it has never “recycled” any particles as originally envisioned.)

The Recycler is a constant-energy storage ring for protons and has the following relevant parameter values:

Parameter Symbol Value Unit
Circumference \(C\) 3319 m
proton kinetic energy \(W\) 8.0 GeV
transition gamma \(\gamma_t\) 18
RF harmonic number (inj.) \(h_0\) 588
RF harmonic number \(h\) 28
RF voltage (max.) \(V\) 120 kV
Dispersion (max.) \(D_x\) 2.1 m
Transverse Aperture \(\pm r_0\) 30 mm
initial bunch length (rms) \(\sigma_t\) 3 ns
initial energy spread (rms) \(\sigma_W/W\) 0.0025
  1. What is the maximum momentum deviation \(\Delta p/p_0\) that can be contained within the aperture of the Recycler (\(\pm\) 30 mm)? What maximum energy deviation \(\Delta W/W\) does this correspond to?

When the beam arrives from the Booster into the Recycler it has a bunch pattern maintained by an \(h=h_0\) = 588 RF system. However, not all buckets need be filled with beam; the fill pattern can be determined by the upstream accelerators.

Suppose the injector complex forms groups of 15 equal bunches in 15 consecutive \(h\) = 588 buckets, with the bunch properties described above. We wish to combine these 15 bunches into one super-bunch through a “bunch coalescing” process in the Recycler. The RF system we use operates with harmonic number \(h\) = 28 and with maximum possible effective voltage of 120 kV.

  1. Create a simulation in R that
    • generates 11 bunches of the above characteristics at the proper initial time spacing, and
    • tracks the synchrotron motion of each particle in the Recycler for a given voltage of the \(h\) = 28 system. For example, take \(V\) to be 80 kV, use \(\phi_s\) = 0, and track particles for 5000 turns.
    • provides plots of the initial and final phase space distributions.

Use 400 particles per bunch in the simulation.

  1. Rather than keeping the voltage constant, now suppose we ramp up our \(h\) = 28 RF voltage from zero to its maximum value within 100 ms, according to \[ V(n) = V_0\cdot[ a_1 (n/n_{on})^2 + a_2 (n/n_{on})^5], ~~~~ 0<n\leq n_{on} \] and \(V(n)\) = 1 for \(n>n_{on}\). The coefficients are tuned such that \(a_1+a_2\) = 1, and \(n_{on}\) corresponds to the number of turns in our 100 ms interval.

Tune the values of \(a_1\), \(a_2\) and \(V_0\) within their allowable limits in order to “coalesce” the bunches into a single bunch. What are the final values of rms time spread and rms momentum spread of the super-bunch? Make plots of both the initial and final phase space conditions.

  1. Make a movie (animated gif, for instance) of the bunching process that you have come up with, showing the evolution of the phase space starting with the 11 bunches and ending with a single super-bunch. The animation should start at \(t\)=0 and end at \(t\) = 200 ms and contain roughly 100 frames at 24 frames per second. (Hint: The R package, magick, may be of some use.)

Topic 2: Muon Beam Preparation

With the beam bunched according to the process described in Topic 1, the proton bunch is delivered to a target, from which a variety of different charged particles emerge, including protons, antiprotons, pions (\(\pi^\pm\)), electrons, positrons, muons (\(\mu^\pm\)), deuterons, etc. A magnet is placed downstream of the target which selects positive particles with momentum approximately 3.1 GeV/c and directs them into the M2/M3 beam line. The \(\pi^+\) will decay, resulting in \(\mu^+\) which are the particles desired by the experiment.

The beam of 3.1 GeV positively charged particles is transmitted 300 m to the Delivery Ring (DR) – a synchrotron with the following parameters:

Parameter Symbol Value Unit
Circumference \(C\) 505 m
Central momentum \(p_0\) 3.094 GeV/c
Transition gamma \(\gamma_t\) 6.7
Dispersion (max.) \(D_x\) 1.9 m
Transverse Aperture \(\pm r_0\) 30 mm

After a number of revolutions about the DR (which can be chosen by the experiment), the beam is extracted and further transported another 200 m to the Muon g-2 Storage Ring. Plots of Courant-Snyder parameters and dispersion along the beam lines and DR are provided in section Useful Project Information.

  1. How many revolutions about the Delivery Ring are needed in order to separate protons from the muons by 1 \(\mu\)s, which is the “rise-time” of the magnetic field of the extraction kicker? Explain your answer.
  2. The pulse of particles emanating from the target have a time spread (rms) of \(dt\) = 25 ns. After the proper number of turns (an integer) in the Delivery Ring found in part (a), what will be the time spread \(dt\) of the muons due to their relative spread in momentum, \(dp/p_0\)? If there are 10 times more muons than pions entering the Delivery Ring, what will be the ratio of pions to muons after this number of turns?

  3. The file g4beamline.out gives the result of a simulation of particles that reach the end of the beam line, 5 m in front of the inflector magnet in the Muon g-2 Storage Ring. From the data, what is the momentum spread of the muon beam? What transverse emittances result from the simulation?

  4. The file M5emittMeas.txt contains rms beam sizes at a monitor in the M5 beam line, taken for different current settings of a single quadrupole. The table lists values of \(K \equiv B'/B\rho\) of the quad, plus the measured values of \(\sigma_x\) and \(\sigma_y\). The quadrupole has an effective length of 0.61 m and its center is located 11.461 m upstream of the monitor. Treating the quadrupole as a thin lens, and assuming the dispersion to be zero trough the region, determine the horizontal emittance observed at the monitor location.
    (Hint: Make a plot of \(\sigma^2\) vs. \((1-Lq)\) and fit a curve to the result. The curve coefficients are related to the Courant-Snyder parameters and the emittance through \(\Sigma=M\Sigma_0M^T\).)

Topic 3: Storage Ring Lattice Considerations

The Muon g-2 Storage Ring is a relatively simple but precise-field storage ring, with the following parameters:

Parameter Symbol Value Unit
Radius \(R_0\) 7.112 m
Magnetic Field \(B_0\) 1.45 T
Aperture radius (circular) \(r_0\) 45 mm
Quadrupole High Voltage \(\pm V_q\) 12-20 kV
Quad plate separation \(2a_0\) 100 mm

The magnetic field is uniform and present everywhere along the beam path. There are actually sets of short and long quads in the experiment with short gaps between them, but for our purposes assume a total of four equally separated devices each extending over 40\(^\circ\) of azimuth. The quadrupoles provide vertical focusing of the beam.

  1. What “\(K\)” values (as in Hill’s Equation, \(x'' + K(s)x = 0\)) are present in the ring due to electric and magnetic fields, and what are their arrangement? (May wish to obtain further quadrupole information using results from Topic 3 below.)
  2. Create a MAD-X model representative of the Muon g-2 Storage Ring, and create (1) a table and (2) a plot of \(\beta_x(s)\), \(\beta_y(s)\), \(D_x(s)\), \(\psi_x(s)\), and \(\psi_y(s)\) about the ideal circumference of the ring.
  3. For comparison, create a matrix calculation of the same parameters using R.
  4. Using R, make a plot of the betatron tunes \(\nu_x\) and \(\nu_y\) vs. high voltage on the quadrupoles, for 12 < \(V_q\) < 20 kV.
  5. Create a list of possible strong betatron resonances \(m\nu_x + n\nu_y = k\) (to order k=8) in the \((\nu_x,\nu_y)\) tune plane that are within the tunability of the system given in part (d). For the table, look for values of \(0\le m\le\) 8, \(|n| \le\) 8 and where \(k\) as computed from the tunes is within a value \(\Delta k\) of an integer value and for which \(|m|+|n|\le\) k (the “order”). Search for these conditions over quad voltages of 12, 12.1, 12.2, … 20 kV. For the tolerance on \(\Delta k\) use the following: \[ | \Delta k | < (0.01~{\rm kV}) \times (0.003/{\rm kV})\cdot(|m|+3|n|). \] From your table, what would be the top 3 concerning resonances in the range of operation of the Storage Ring?

Topic 4: Beam Line and Storage Ring Magnet Considerations

Beam Line Magnets –

The file M45mad.out is derived from MAD-X output and contains the Courant-Snyder parameters at the center of the quadrupoles and dipole magnets for the present-day settings of this beam line. The table includes the length L of each element, along with a parameter K1L which is the value of \(K = K_1\ell =B'\ell/B\rho\) of each magnet.

  1. Read in the file data and make a single plot of \(\beta_x\) and \(\beta_y\) vs. \(s\). Also make a single plot of the quadrupole strengths (\(K\)) and bend angles vs. \(s\). (Hint: For this plot, use the option type=S in the plot statements to make “step” plots of the values.)

  2. Make an estimate of the size (rms) of expected beta function error at the end of the M4/M5 beam line, given setting errors of 1% and field gradient construction errors of 0.05% (both rms values) of typical beam line quadrupoles in this line. (Just consider one degree of freedom \(x\) or \(y\), and assume the perturbations are small enough to use linear approximations.)

  3. Perform a Monte Carlo calculation to estimate the probability that the beam size at the end of the line will be greater than 5% its design value due to the expected gradient errors presented in part (a). (Just consider one degree of freedom \(x\) or \(y\).)

Storage Ring Magnet –

  1. Create a Monte Carlo model of field quality around the circumference that generates an overall field quality of \(\Delta B/B_0\) = 10 ppm, where \(\Delta B = \int_0^C (B-B_0) ds/C\), \(C = 2\pi R_0\). Assume a “measurement” every 1 degree around the ring.

  2. Using your results from part (c) above, perform a simple estimate of the integrated gradient error that could be produced from these field errors. What shifts in the betatron tunes would you expect from these “errors”? For this estimate, assume that the values for \(\beta_x\) and \(\beta_y\) are roughly constant about the circumference and use a high voltage quadrupole setting of 20 kV. (See Topic 3.)

  3. Suppose there is a radial component of the field of magnitude \(\Delta B_r/B_0\) = 30 ppm at one location in the ring that extends through 5 degrees of azimuth due to a local magnet shimming error. What would be the resulting vertical orbit distortion around the ring due to this single error? You may wish to consult the solutions to Topic 3 above for pertinent information.

Topic 5: Storage Ring Electrostatic Quadrupoles

The Storage Ring uses electrostatic quadrupoles for providing vertical focusing. The inner and outer plates of the quadrupoles are cylindrical in shape, with the inner cylinder at \(r\) = 7.062 m and the outer cylinder at \(r\) = 7.162 m (The beam central orbit is at \(r = R_0\) = 7.112 m.). The top and bottom plates are made from two concentric circles separated by 100 mm. Each quadrupole extends over azimuthal angles of roughly 40\(^\circ\). (There are actually sets of short and long quads in the experiment with short gaps between them, but for our purposes assume a total of four equally separated devices each extending over 40\(^\circ\).)

  1. Using FEMM, create a model of the cylindrical quadrupole plates and set their voltages to \(\pm\) 20 kV to provide vertical focusing of positive muons. Make 2-D plots of the electric potential and of the electric field. Also make a plot of \(V\) vs. \(x \equiv r-R_0\) for \(y=0\).
  2. Taking the FEMM numerical results from part (a), use R or python to determine a set of “normal” field multipoles, \(v_n\), at a reference radius of \(r_0\) = 45 mm. Here, \[ V(x,y=0) = V_0 \sum_{n=0}^\infty \;v_n \; (x/r_0)^n \] describes the electric potential along the line \(y=0\) that is valid between \(-r_0<x<r_0\). (Computationally, we need only take terms up to about \(n=8\) for this exercise.)
  3. Using the coefficients found for the potential, plot \(E_x\) vs. \(x\) and find the field gradient \(E' \equiv \partial E_x/\partial x\) at \(x\)=0.
  4. If the multipoles \(v_n\) above were zero for \(n \ne 2\) (i.e., a “perfect” quadrupole), what field gradient \(E'\) would be produced if the voltages on the plates were set to \(\pm\) 20 kV? Explain why this differs from the answer (if it does) from the FEMM results.
  5. Suppose one of the plates has a failure in its circuitry and only delivers 19 kV instead of the 20 kV requested. How do the multipoles \(v_n\) and the resulting \(E_x\) become altered? What beam dynamics consequences might you expect from this situation?
  6. Suppose a single ideal quadrupole unit had its inner and outer plates misaligned, such that its inner plate was radially inside by 1.25 mm, while its outer plate was radially outside by 1.25 mm. What (first-order) effect(s) would be manifested, and approximately by how much? You may wish to consult the solutions to Topic 3 above for pertinent information.

Topic 6: Storage Ring Injection

When beam arrives from the M5 line it enters the Storage Ring from the “outside” of the ring and passes through the so-called inflector magnet. This superconducting magnet creates a field-free region for the muons to pass through en route to their desired orbits, without affecting the overall field uniformity of the final storage region of the magnet. Two problems, though, are that (a) the inflector has a very small horizontal aperture (\(\pm\) 9 mm) and (b) there is material at the two ends of the inflector which the muons must pass through.

For our analysis here, some of the solutions to Topic 2 and Topic 3 above may be useful for pertinent information.

  1. Estimate the admittance (both horizontal and vertical) of the Storage Ring for ideal-momentum muons, i.e., the maximum phase space area of the trajectories for particles that can survive long-term.

  2. As the beam passes through the inflector magnet, it passes through material at each end. If the material is composed mostly of tungsten, and is approximately 4 mm thick at each end of the inflector, estimate the emittance growth from passing through the two ends and the average energy loss. Materials information can be found here. (For this estimate, you may presume that the Courant-Snyder parameters do not vary over the length of the inflector.) For initial beam conditions in front of the inflector, use \(\epsilon_x = \epsilon_y\) = 15 \(\pi\) mm-mrad (rms) and \(\sigma_p/p_0\) = 1.2%.

  3. Use your best estimates for the final emittances and the momentum spread that will be found after the inflector, starting with the values at the end of the M4/M5 beam line (see Topic 1) to make a simple estimate of the fraction of incoming muons that might possibly be stored long-term in the ring, (ignoring that muons are decaying along the way).

  4. In addition, positrons that are created at the target and have similar momenta as the positive muons can survive all the way from the target and be injected into the Storage Ring.
    1. For a muon and an positron each with a momentum of 3.094 GeV/c, what will be the difference in time of arrival to the Storage Ring? Use a total path length of target to ring equal to 2500 m.
    2. Once in the ring, both particles will, of course, radiate due to synchrotron radiation. By what factor will the positrons radiate energy relative to the muons?
    3. How long will a 3.094 GeV/c position on the design orbit of the Storage Ring survive? The aperture of the ring is restricted to \(\pm\) 45 mm.

Topic 7: Storage Ring Beam Lifetime

Given all of the above information, …

Vacuum:

  1. What is the maximum time that muons will spend in the ring? (Assume 0.1% remain at the end.)
  2. If the average vacuum pressure in the ring is \(10^{-6}\) Torr, by how many mrad would a typical muon [let’s say, that lasts half of the maximum time found in part (a)] be scattered during its time in the ring?
  3. By how much might this particle’s transverse emittance grow during this time? You may wish to consult the solutions to Topic 3 above for pertinent information.

Aperture Enhancement

  1. Assume that a beam enters the Storage Ring with a phase space distribution that completely fills the admittances in \(x\) and \(y\) and that the momentum spread of the beam is uniform and much larger than the momentum acceptance of the Storage Ring. Perform a Monte Carlo exercise to determine the relative increase of stored beam that can possibly be obtained through using a square aperture of transverse dimensions \(2R_0\times 2R_0\) compared to that obtainable using a circular aperture of area \(\pi R_0^2\). Remember that the dispersion of the circulating orbits plays a role.

Life on the Edge

  1. Let’s maintain the circular aperture restriction of the present Muon g-2 Storage Ring, which is at radius \(r_0\) = 45 mm. Assume that the vacuum is good enough that emittance growth from scattering is not an issue. It is still possible for particles to be injected and then, at some time later, find themselves at the aperture. Consider a distribution of muons that all have the ideal momentum of 3.094 GeV/c. As an exercise, assume that each particle has initial phase space conditions such that its horizontal and vertical “amplitudes” of its betatron motion are \(a\) and \(b\), but each particle has different initial betatron phases, \(\psi_{x0}\) and \(\psi_{y0}\). The values of \(a\) and \(b\) conspire such that it is possible for the particle to eventually reach the aperture at \(\sqrt{a^2+b^2} = R_0\). Given uniform initial distributions in betatron phases for the particles in question, what is the average time that it takes for such a particle to reach within 0.1 mm of the aperture if \(b = a\)? Illustrate how this time varies (or not) with the ratio of \(b/a\) over a range of \(0<b/a\ \le 20\).