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A Toy Model Study of Decay Trapping, reported by Brett Parker


A group from the BNL Superconducting Magnet Division is looking at various options for dipole magnets which would be suitable for use in a muon storage ring that is used as a neutrino factory. Since the useful neutrino beams from a neutrino factory come from straight sections it is desirable to minimize the rings arc circumference, in relation to straight section length, in order to ensure that the fraction of muons which decay in the straight section is as large as possible. Therefore superconducting magnets, with higher B-fields and smaller bend radii, are reasonable to consider for this application. Unfortunately the decay electrons generated along with the neutrinos carry on average about a third of the original muons energy and when these decay electrons strike ring components they can lead to significant energy deposition.

Dealing with a significant amount of energy deposition inside a superconducting magnet presents a formidable design challenge. One approach investigated is to place inside the superconducting magnet a warm, water-cooled, tungsten absorber to catch this energy deposition before it can reach cryogenic regions of the magnet. However this approach forces one to increase the coil aperture and adds to magnet complexity.

Mike Harrison asked whether we could find a magnet configuration and lattice layout in which the decay electrons are for the most part passed completely through to the end of the magnet and are then absorbed in an external separate warm collimator. He proposed we look at a configuration that has:

  • a warm iron yoke to both reduce the cold mass which must be cooled and the impact of any energy which may be deposited in the yoke,
  • a simple flat pancake coil configuration, which both simplifies magnet construction and keeps the magnet midplane open,
  • a region of reversed dipole field that bends the decay electrons back away from the inner vacuum beam pipe wall.

The utility of the last design scenario, keeping the decay electrons away from the beam pipe by having a region of reduced or even reversed dipole field, is denoted here as "trapping" and is the focus of the present study.

Study Outline

In order to see if trapping is at all possible we track muon decay electrons through a simplified structure that represents the muon storage ring half cell, the "toy model." A useful tool for doing such tracking is the MARS13 code suite developed by Nikolai Mokhov et. al. For this study MARS provides a tool to:

  • Describe the 3-dimensional geometry of the dipole magnets, with subdivided structure for coils, beam pipe and yoke, the half cell collimator and a normal-conducting quadrupole magnet.
  • Implement a detailed field map to describe the magnetic field not only near magnet centers but throughout the full aperture and in the yoke.
  • Launch tracks according to a source term where the muon decays are distributed randomly along the closed orbit and with appropriate electron decay energy and angular distributions.
  • Save the resulting tracks and ensuing showers due to subsequent interactions in the magnets and collimators.
  • Estimate the spread of the synchrotron radiation fan generated by the muon decay electrons.
  • Visualize the results.

The questions we ask here are:

  • Is it possible to get very many decay electrons through the dipole magnet without interacting?
  • What role does magnet saggita, free beam pipe horizontal or vertical aperture, and the presence of low or reversed B-field play in trapping?
  • How are the track hits distributed on the collimator? And on the downstream quadrupole magnet?
  • Given the magnets significant saggita, do many shower tracks make it back into the aperture to hit downstream superconducting coils?
  • Are very many shower particles able to penetrate outside the warm yoke?
  • Is there a significant amount of backscatter albedo from the collimator? And does any albedo make it back to the dipole magnet coils?

At this preliminary stage we are only looking for qualitative answers.

  • First we launch decay electrons within specific energy and/or angle bands and see where they first hit (these results are not reported here). From this earlier study we have a first approximation to the required apertures and beam pipe geometry.
  • Then for the present study we launch decay electrons distributed both randomly along the muon beam envelope and statistically weighted in angle and energy so as to mock up the muon 3-body decay (the results reported here).

Only if the results look promising will it be worthwhile to do the detailed particle scoring, histogramming and energy deposition calculations need to make quantitative statements.

Toy Model Parameters

For the toy model we take:

  • The primary muon energy, E = 50 GeV. This typical of the beam energies under consideration for a neutrino factory.
  • Each arc half cell has 2 dipoles each of which have 5 m magnetic length.
  • The dipole bend radius is 33 m.
  • From the above the dipole central guide field is B = 5.045 T and deflection angle is 150 mrad for 300 mrad bend per half cell. Choosing round numbers for dipole length, bend radius and deflection angle slightly simplifies specification of the toy model geometry.
  • The clear vertical vacuum beam pipe aperture is +/- 3.6 cm and this corresponds to vertical +/- 4.5 sigma_y for the muons. Additionally we take the horizontal sigma_x = sigma_y for a round beam envelope.

For the toy model the number taken for the beam sigma is arbitrarily chosen and in particular it is not based explicitly on a particular choice of ring optics. As a practical number, for the purpose of this toy model study, 4.5 sigma is taken since a smaller number would imply significant beam loss of the primary beam (a confounding factor when we are interested in identifying trapping). The point is to have the decay electrons start off close enough to the beam pipe wall to be sensitive to the additional divergence due to the decay electrons phase space. Probably few people would be surprised that trapping works for +/- 20 sigma free aperture but it would be prohibitive to design a muon storage ring with dipoles having such large midplane gaps.

The decay electrons are generated:

  • with a random 1 mrad RMS gaussian angular spread,
  • and a statistically weighted energy distribution which approximates the 50 GeV muon 3-body decay spectrum.

At this point the electron decay energy- angle-correlations are not taken into account. Providing a complete kinematically correct electron decay spectrum is a natural extension of this work (and is in progress).

Dipole Magnet Description

Since we are investigating decay electron trapping via the combined effects of low and/or reversed magnetic fields outside the main circulating beam aperture we are forced to provide a complete field map in the regions sampled by the electron beam. [i.e. it is not good enough to take an ideal uniform dipole field as it is a poor approximation, for instance, when the particle is directly under the superconducting coil] We read the field map into MARS13 as a set of lookup tables and use interpolation to derive vertical and horizontal field components at an arbitrary point. Multiple interpolation tables are used, with finer or coarser grids, in different regions of the magnet. In general the fields maps are made fine grained in the interior vacuum regions, in the aperture and near the superconducting coils and coarse grained in the iron yoke.

The field maps are obtained by 2-dimensional finite-element modeling using the Vector Fields Opera2d code. At present the fields are simply truncated at the ends of the magnet; for the toy model we do not explicitly model the coil ends or calculate the end field distribution. In order to simplify both the Opera2d field calculation and the MARS13 special region numbering scheme we calculate using a one-quarter magnet model and obtain the field at other points via symmetry considerations (e.g. see the magnet icon below). Using symmetry is purely a calculational convenience for the toy model and is not really required. Since the decay electrons are all bent the same way, toward the ring center, we do not need to provide for any extended horizontal aperture for them on the opposite side. Since there is a small amount of synchrotron radiation, generated by the decay electrons, which goes to the ring outside, it might still appear desirable to provide an extended vacuum beam pipe profile for them (which avoids depositing energy under the superconduting coils). However, we find that these photons are easy to contain.

The very first coil and yoke configurations studied showed excessive particle loss even when the free aperture was increased to beyond 6 sigma; also the primary muon beam, when it was not allowed to decay, experienced several percent losses when just traversing 10 m of dipole. The explanation for this particle loss comes from the fact that the first magnetic models were not optimized to give good field quality and had close to a ten percent admixture of the first allowed sextupole component. As soon as we brought the harmonics back in line by pole shimming these central pole losses went away.

While doing this harmonic adjustment we realized that the horizontal field components we were typically finding at the edge of the pole and between the coils would tend to push the electrons further toward the coils or cause them to hit the top or bottom of the beam pipe sooner. Thus to be sure to have a chance to observe trapping we:

  • Adjusted the coil pancake geometry and pole trimming so that the first horizontal fields seen by the decay electrons tended to deflect them away from the coils. The electrons would then stay closer to the horizontal axis when crossing between the coils and not experience as much force pulling toward coils.
  • We flared the vacuum beam pipe in the reversed field region beyond the coils in order to make it less likely that the decay electrons would hit the top or bottom of the beam pipe in this region.

Along the way we also made small adjustments to the pancake coil geometry to reduce the peak field seen on the face of the superconductor.

Magnet design information.


The main observation from this study it that it has been found possible to find a dipole configuration for which many decay electrons make their way out of the magnet to hit a collimator. In this sense the toy model is taken as a proof of principle but is by no means claimed to be an optimized configuration. We found providing good field uniformity (toy model DeltaB/B is about 10^-3) to be an important first step for improving particle transmission.

As far as the relative importance of magnet saggita, free horizontal or vertical aperture, and the presence of low or reversed B-field, these were all found to have some impact on particle loss within the magnet; however, no quantitative conclusions can be reached with the present study. Also since we observe a correlation between decay electron energy and the position the electron strikes the vacuum chamber, in future studies it will be important not just to count particle hits but to score the energy deposition due to these hits. For the toy model we find that some low energy decay electrons do hit the outer vacuum wall but their contribution, if any, to energy deposition at the superconducting coils cannot be evaluated without such scoring.

It is entirely probably that the toy model dipole presented here is overdesigned in the sense that pole width, cutout dimensions and overall yoke size were deliberately increased to the point that a large number of decay electron tracks made it all the way to the collimator. With an eye to greater economy we may be able to back off with these dimensions if we find that a smaller yoke solution with acceptable energy deposition. Also the magnetic design is itself advancing as we look to engineering concerns as how to minimize the thermal heat leak to the superconductor while maintaining adequate coil support and this will be reflected in future studies.

Energy deposition scoring is also needed to address the significance of the track hit distribution at the collimator and on the downstream quadrupole magnet. From the toy model we see indications that backscatter albedo from the collimator will be important to control; however, this is one detail that requires greater knowledge of the coil end configuration in order to be addressed properly. At this point we can only say that collimator albedo is an issue which should not be forgotten. Also since we see very few tracks which penetrate outside the warm magnet yoke, it would seem that the collimator is also the main place where extra radiation shielding might be needed.

In order to assess the relative importance of the magnets saggita versus trapping due to the reversed field region the toy model can be run with different dipole bean radii. There is a desire to have the smallest possible bend radius as this both reduces the total ring size and increases the fraction of muons that decay in the straight sections (for higher neutrino production efficiency). The bend radius may be decreased by lowering the muon beam energy while keeping the dipole field fixed. Otherwise to decrease the bend radius we have to increase the central field.

Lowering the muon beam energy is easy to study with the toy model as this only requires a small change in model geometry and this should be done soon. At present there is also magnetic design work underway to investigate what would be needed to implement a dipole magnet configuration similar to the toy model but at higher field.

Here we note that if the muon beam energy were to be dropped to 20 GeV the bend radius would go to 13.2 m (from the 33 m toy model value). Or the bend radius could be dropped to 19 m at 50 GeV if the dipole field were to be increased to 8.75 T. With either scenario the bend radius starts to become comparable to the present toy model half cell length and the dipole saggita is even more pronounced.