One way of producing an intense muon beam for a Neutrino Factory or Muon Collider is to collide protons with a target material to produce pions, which decay into muons. This paper examines the case of a solid tantalum rod target and how the choice of proton energy affects the pion yield, both in total and in a forward directed cone useable in later capture optics.

The newest revision of the MARS particle production code, MARS15(2004)1, was used in order to get realistic results, with 10⁵ input protons per run. Unfortunately, data on pion production from the HARP experiment2 is not yet available, so has not yet been integrated into MARS. When this happens, the simulations that follow will be rerun to see the impact of the new data.

The target rod is a cylinder of tantalum with a radius of 1cm and a length of 20cm. The proton beam hits it perpendicularly at one end and MARS was configured to record any pi+ or pi- leaving through its surface. See Appendix B for the input file used for these simulations.
It should be noted that a target of this length does not fully stop the proton beam, so some protons pass right through, albeit experiencing severe multiple scattering in the high Z material. The target also produces copious quantities of neutrons; indeed, a stopping tantalum target is sometimes used in dedicated neutron production machines.

MARS by default provides choices of a uniform rectangular beam or a 2D normal distribution. Neither of these are quite right for this application, as the rod's cross-section is circular not rectangular and the normal distribution has tails that miss the rod. A typical choice for target work might be the uniform distribution on a circle, particularly as this evens out the heating the rod would experience. However, this is not attainable with the kinds of optics available for a proton extraction line. The real proton distribution might resemble a parabolic distribution independently in the X and Y axes, i.e. ρ(x,y)∝(1−x²)(1−y²) for −1≤x,y≤1, but this in turn is unpleasant because it breaks the circular symmetry of the problem again.
Instead, the compromise solution of a radial parabolic distribution ρ(r)∝(1−r²), 0≤r≤1, was used. This also has the advantage of being the projection into the plane of a 4D "waterbag" phase-space distribution (that is, uniform within a 4-dimensional ellipsoid, in this case a sphere), which is often used as a model of proton bunches.
The beam of incoming protons was assumed to be parallel for the purposes of the simulation, as typical proton drivers give extracted beams with too small an angular divergence to greatly affect the results of a hadron shower calculation in a block of material.

Results were generated for a selection of energies, listed in the table, including those for some candidate proton drivers already under study.

The most obvious quantity to look at is the total number of pions leaving the rod for each proton energy. The raw numbers are converted into pions per proton.GeV, which is proportional to the number of pions produced per watt of proton beam (pions per proton will always increase with energy).

For this figure, the dependence on energy is moderate, with the highest yield ~60% better than the worst. The peak for pi+ is at 30GeV and the peak for pi- is at 20GeV, though the graphs (figure 1) are roughly flat down to 15GeV.

The pions will be difficult to capture in a decay channel if they are travelling sideways or backwards from the rod. Therefore it is important to measure the angular distribution for each energy, to make sure the high-yield energies are not producing an unusable beam. Figures 2a-e plot the distribution of the angle (theta) between the emission velocity of the pions and the forward axis of the rod, in 1° bins.

Note that the linear fall towards theta=0° is not a hole in the distribution but an effect of the element of area for each bin being 2pi sin theta dtheta. One could divide the distribution by sin theta to see how the true density (per steradian) varies.

We see in figure 3 that the off-axis angle of the pions decreases with proton energy; the decrease becomes slower at high energies, with the most dramatic fall before 5GeV. The pi- distribution is always less focussed than that of the pi+.

The model that MARS uses for pion calculations changes between 3 and 4 GeV, in the region of the fall in the graphs. This may be a coincidence or it may indicate that the two models produce somewhat different results. Benchmarking of these results against those produced by the GEANT4 code for an identical setup is planned in the future.

One way to combine the total pion yield and the angular distribution data into a single figure of merit is to count the pions emitted within a certain angle of the forward axis (i.e. with velocities inside a forward-pointing cone). This is a fairly crude cut, designed to eliminate only pions whose direction is so far off-axis that they would fall outside the range of dynamics considered in any conventional decay channel. The relatively large cutoff angle of 20° is therefore chosen.
For long-range acceptance into a regular focussing channel, transverse momentum might be a better figure to use, and this is done in the next section.

Figure 4 shows a far steeper falloff at the low energy end than the graph of total pion yield, while the high energy decrease is shallower. For this figure of merit, the lowest energy is 5.4× worse than the peak for pi+ and 7.2× worse for pi-. By contrast, the peaks are only respectively 38% and 25% better than the 120GeV value.
The best energy for pi+ is still 30GeV but for pi- it has become 40GeV. The upward skew is a consequence of the beam focussing with increasing proton momentum, an effect that is particularly important for pi- as they tend to be less focussed than the pi+.

While the analysis in the previous section is suitable for solid-angle type effects in the early few solenoids, a 20° spread of pions is harder to capture into a focussing channel if the pions have higher energies, since they will be less affected by the focussing system. Here we additionally apply a transverse momentum threshold of 100MeV/c, which is roughly the transverse momentum transmitted by a decay channel studied earlier3.

Although some statistical noise is starting to appear in figure 5, it is clear enough to show that the momentum cut has reduced the very steep decrease at low energy, which is now only about a factor of two or three. The noise makes it harder to identify a clear peak, though 30GeV still stands out as a good energy, with acceptable yields from 5GeV up.

Pion distributions have been generated at a selection of energies in MARS15 for a simple but well-defined target model. In all analyses, a 30GeV proton beam appears to produce a near-optimum pion yield, with a shallow dependence on either side so that halving or doubling this energy should not give a dramatic reduction in efficiency. Proton driver energies below 5GeV appear consistently bad due to a high angular spread in the pion beam, with significant numbers emitted backwards.

The pion data sets themselves can be obtained by e-mailing the author.

• Confirmation/benchmarking of this paper's results using GEANT4.
• Tracking of these distributions through various decay channel scenarios, to achieve realistic yield figures.
• Optimisation of decay channel systems3 for the suggested proton driver energy of 30GeV.
• Additional investigations of the target parameters, such as variation of rod radius.
References

[1] N.V. Mokhov, K.K. Gudima, C.C. James et al., Recent Enhancements to the MARS15 Code, Fermilab-Conf-04/053 (2004); http://www-ap.fnal.gov/MARS/
[2] M. Ellis, Status and prospects of the HARP experiment, J. Phys. G: Nucl. Part. Phys. 29 (2003) pp.1613-1620
[3] S.J. Brooks, Quantitative Optimisation Studies of the Muon Front-End for a Neutrino Factory, Proc. EPAC'04

MARS.INP Template File

The token '$energy' in the file below is replaced by a number in GeV for each energy required. MARS requires that this end in a decimal point if it is a whole number (for instance '6.' for 6GeV but '2.2' for 2.2GeV). The token '$hole' is always replaced by '0.' currently, as the rod is solid.

Tantalum Rod Pion Production [TEMPLATE] (2005-Jan-06) /home/csf/brooks/mars15/dat C GUI switch CTRL 0 C Generate 100000 events NEVT 100000 C 0.01mm boundary precision, 2cm initial trial step SMIN 0.001 2. C Beam is protons(1), parallel gaussian(2) [now modified in BEG1 subroutine] IPIB 1 2 C 0.5cm RMS transversely (so rod is 2sigma), 1ns RMS longitudinally BEAM 0.5 0.5 5=29.9792458 C Energy in GeV ENRG $energy C Two materials used: tantalum and vacuum NMAT 2 MATR 'TA' 'VAC' C Use 'R-sandwich' type geometry (default) INDX 2=F C Two R-sections: vacuum(#2) to $hole cm radius, tantalum(#1) to 1cm radius, ending at z=20cm NLTR 2 RSEC $hole 1. 51=10 10 101=2 1 ZSEC 20. 1251=200 C We record particles at three surfaces, 1cm out, z=0-20cm; and r=0-1cm, z=0cm and z=20cm to fort.83 (81..90) NSUR 3 RZTS 1. 1. 0. 20. 83. 0. 1. 0. 0. 83. 0. 1. 20. 20. 83. STOP Modification to User Subroutine 'BEG1' in m1504.f

This is the section of the long Fortran file 'm1504.f' that has been modified to generate the correct input proton distribution.

* PARAMETER (CLIGHT=29979245800.D0) PARAMETER (PI=3.141592653589793238D+00) C- - - - - - - - - - - - - - - - - - - - - - - - - - - - C+++ INSERT YOUR SOURCE TERM HERE +++ C READ()...,W1,... C W=W*W1 R=SQRT(1.D0-SQRT(RAND())) ! SJB: 1cm radius circular parabolic beam TH=RAND()*PI*2.D0 X=R*COS(TH) Y=R*SIN(TH) C++++++++++++++++++++++++++++++++++++ RETURN END