Comparison of doing muons through Geant4 vs MUSUN
The muon flux at 4850 ft level, Davis Cavern, Homestake Mine, is simulated by
MUSUN [add ref to MUSUN from previous sections] and Geant4 simulation codes.
The modified Gaisser’s parameterisation [1] is adopted for muon energy spectrum
and angular distribution above ground:
−2.7
𝑑𝐼
𝑑𝐸𝜇 𝑑𝑐𝑜𝑠𝜃
= 0.14 (
𝐸𝜇
𝐺𝑒𝑉
(1 +
3.64𝐺𝑒𝑉
𝐸𝜇 [𝑐𝑜𝑠𝜃 ∗ ]1.29
))
[
1
1.1𝐸𝜇 𝑐𝑜𝑠𝜃∗
1+
115𝐺𝑒𝑉
+
0.054
1.1𝐸𝜇 𝑐𝑜𝑠𝜃∗
1+
850𝐺𝑒𝑉
]
(1)
Muon’s energy and polar angle are sampled using Eq. (1) and generated uniformly
on the surface of the earth. Both MUSUN and the new Geant4 application take
account of the surface mountain profile, then transfer the surface muons down to
the underground cavern. Figure 1 (a) shows the surface elevation map from the
combination of a geographic survey [2] and satellite data [3] whereas map (b) is a
closer look of the Homestake Mine area. The map (a) was adopted both by the
Geant4 simulation and the MUSUN code where the regions outside the map are
assumed to be flat. The present version of MUSUN for Homestake Mine uses one
degree per bin for zenith and azimuthal angles to interpret the surface profile while
the Geant4 application uses 5×5 m2 cells in an XY plane.
Figure 1. (Left): the surface mountain profile at Homestake Mine area from
the satellite data and a geographic survey. (Right): a zoom-in look of
Homestake Mine area.
There is over 1400 meters of rock overburden for the Davis Cavern. The
composition of the rock sampled from Homestake Mine has been measured in
reference [4] and a representative sample (No. 278-2[5]) is adopted in both
simulations. The average rock density applied in Geant4 is 2.82 g/cm3 and 2.70
g/cm3. The current version of MUSUN for Homestake uses 2.70 g/cm3 but can easily
be changed if required. The determination of the muons’ energy loss in the rocks is
the key for the calculation.
Geant4 tracks muons step by step. All processes of muon energy loss are
automatically registered and simulated by Geant4 itself.
MUSIC (the muon transport code whose results are used in MUSUN) [add reference
to MUSIC from the previous sections] also tracks and simulates individual muon and
processes involved in muon energy loss. MUSIC, however, does not track secondary
particles produced as a result of muon interactions. This makes the code run faster
but still reliable since transport of secondary particles does not affect muon fluxes
underground.
The absolute muon flux at the Davis Cavern is determined mainly by four factors,
the surface mountain profile, the rock density and composition, the muon energy
spectrum at sea level and muon interaction cross-sections . Due to the complexity of
the geological structure, the simulation with a homogeneous rock with a single
density and composition (which may not be very well determined) can only give an
approximate value for the muon flux. The calculated total muon flux has to be
normalized to the results from measurements. The muon energy distribution
obtained from these different approaches (GEANT4 and MUSUN) is shown in Figure
3. The absolute fluxes in the left plot are calculated to be 5.31e-9 cm-2s-1, 6.15e-9
cm-2s-1, 4.85e-9 cm-2s-1 and 4.40e-9 cm-2s-1 for MUSUN(ρ=2.70), GEANT4(ρ=2.70),
GEANT4(ρ=2.82) and Mei & Hime prediction, respectively. Since the calculation of
the absolute fluxes depend on multiple unknown parameters, the total fluxes are
normalized to 1 in order to compare the shapes of muon energy spectra (Figure 3,
right plot).
Figure 3. Muon energy spectrum at 4850 ft Davis Cavern estimated by
MUSUN, GEANT4 and Mei&Hime prediction [6]. The left plot is the
comparison for the absolute fluxes while the right plot is the comparison
with the fluxes normalized to 1.
Figure 4. Comparison of muon angular distribution at 4850 ft Davis Cavern
estimated by MUSUN and GEANT4. Both of the flux are normalized to 1.
Comparison of the muon angular distribution obtained using MUSUN and Geant4
code is shown in Figure 4. The spectrum match each other quite well regardless the
absolute normalization. The total muon flux underground appears to be very
sensitive to the average rock density assumed here. Roughly 1% of density change
would cause 5% of total flux variation, accordingly. The muon energy spectrum at
sea level is another source to affect the results. Instead of using Eq. (1), the
traditional MUSUN takes the Gaisser’s formula with the spectra index of -2.77 and
normalization factor of 0.14×1.84 for muon energies above 1.5 TeV. Such a change
makes the calculated muon flux at the Davis Cavern vary from 5.31e-9 cm-2s-1 to
5.51e-9 cm-2s-1.
Muon’s angular distribution is sensitive to the surface mountain profile and the
structure of penetrating rock. The elevation map we used here assumed an uniform
rock density which is a rough approximation from the reality. Adopting such an
assumption, the slant depths to an underground lab can be actually measured
through a moun detector which can be converted to an equivalent elevation map. An
example is set for Soudan Lab whose slant depths are measured by MINOS (Figure 5.
Left) which combined the variation of surface mountain profile and the rock
structure while the satellite data (Figure 5. Right) only takes the surface profile into
account. Two independent simulations conducted by adopting the two elevation
maps using MUSUN and GEANT4, individually. The effect on the angular distribution
is presented in Figure 6 which indicates ~15% discrepancy induced by the two
elevation maps.
Figure 5. Left: the equivalent elevation map around Soudan Mine area
converted from the slant depths measured by MINOS experiment [7]. Right:
digitized elevation map around Soudan mine area from the satellite data [3].
Figure 6. comparison of muon angular distribution at Soudan mine got from
GEANT4 and MUSUN by using different elevation maps.
[1] T. K. Gaisser and T. Stanev, “cosmic rays”, review of particle physics, Phys. Lett. B
592 (2004) 1.
[2] Geographic survey data are provided by SURF.
[3] http://eros.usgs.gov/
[4] B.T. Jordan, Geochemistry tectonic setting of the Yates unit of the Poorman
Formation (DUSEL bedrock) and other northern Black Hills amphibolites: geological
Society of Americal Abstracts with Programs 41 (7) (2009) 271
[5] D.-M. Mei et al., Astroparticle physics 34(2010)33-39
[6] D.-M. Mei, A. Hime, Phys. Rev. D 73(2006)053004
[7] http://homepages.spa.umn.edu/~schubert/far/s2rock/vdepav.data