Muon simulation codes MUSIC and MUSUN for underground physics
aa r X i v : . [ phy s i c s . c o m p - ph ] O c t Muon simulation codes MUSIC and MUSUN forunderground physics
V. A. Kudryavtsev , Department of Physics and Astronomy, University of Sheffield, Sheffield S3 7RH, UK
Abstract
The paper describes two Monte Carlo codes dedicated to muon simulations:MUSIC (MUon SImulation Code) and MUSUN (MUon Simulations UNderground).MUSIC is a package for muon transport through matter. It is particularly useful forpropagating muons through large thickness of rock or water, for instance from thesurface down to underground/underwater laboratory. MUSUN is designed to usethe results of muon transport through rock/water to generate muons in or aroundunderground laboratory taking into account their energy spectrum and angulardistribution.
Keywords: Muons; Muon interactions; Muon transport; Muons underground; Muon-induced backgroundPACS: 14.60.Ef, 25.30.Mr, 24.10.Lx, 95.85.Ry Corresponding author; address: Department of Physics and Astronomy, University of Sheffield,Sheffield S3 7RH, UK, e-mail: v.kudryavtsev@sheffield.ac.uk Introduction
Muon transport through matter plays an important role in many areas of particle andastroparticle physics. Cosmic-ray muons are detected at large depths underground andunderwater (here and hereafter we use the term underwater that includes also under-iceexperiments). They are used to study the energy spectrum and composition of primarycosmic rays and calculations of their fluxes, energy and angular distributions are the keyelement of this research (see, for instance, Refs. [1, 2, 3, 4, 5].Experiments with high-energy muon neutrino beams from accelerators require accuratesimulations of muon transport from the point of neutrino interaction to the detector.Similarly, neutrino telescopes are detecting (or expecting to detect) muons from atmo-spheric and astrophysical neutrinos, and three-dimensional propagation of muons fromtheir production point to the detector is crucial for the interpretation of experimentaldata [6, 7, 8].Cosmic-ray muons are also a background in experiments looking for rare events at lowand high energies deep underground or underwater. Atmospheric down-going muons canbe erroneously reconstructed as upward-going muons that mimic neutrino-induced eventsin a search for astrophysical neutrinos at GeV-TeV energies or in an atmospheric neu-trino detection for neutrino oscillation studies. Cosmic-ray muons also produce secondaryneutrons (with MeV-GeV energies) by interacting with rock. These neutrons can mimiclow-energy (keV-MeV) events in detectors looking for WIMP (Weakly Interacting MassiveParticle) dark matter, neutrinoless double-beta decay and neutrinos (solar, geophysical,supernova neutrinos, etc.) (see Ref. [9] for a review and Refs. [10, 11, 12, 13] for examplecalculations of muon-induced neutron fluxes underground) . High-energy (GeV) neutronsfrom muons can produce events with a signature similar to proton decay.There are a few more applications from different areas of science. A morphological re-construction of mountains and natural caves using atmospheric muons was suggested inRef. [14]. A search for hidden chambers in pyramids was discussed in Ref. [15]. A ‘muonradiography’ using multiple scattering of cosmic-ray muons was proposed recently [16] todiscriminate between low-A and high-A materials in cargo.All applications mentioned above require accurate calculations of muon spectra and scat-tering beyond a slab of material. Most of them involve muon transport through largethickness of matter. Hence the CPU time should be reduced to a minimum withoutcompromising the accuracy of calculations.Several Monte Carlo codes are able to transport muons through matter with high accuracy.The codes can be split in two categories: (i) multipurpose particle transport codes, such asGEANT4 [17] and FLUKA [18], and (ii) codes developed specifically for muon propagationthrough large thickness of material, such as PROPMU [19], MUSIC [20, 21], MUM [22]and MMC [23].Significant progress has recently been achieved in the development of the multipurposetransport codes for particle physics applications. The codes have become faster, more ro-bust, flexible and accurate. However, their flexibility requires a good knowledge of physicsand programming skills from a user. GEANT4, for instance, is designed as a powerfultoolkit but a good knowledge of the code including models and programming language is2eeded to use it properly. Significant efforts and time are required to become familiar withsuch a toolkit. These codes are absolutely necessary when simulating events consistingof many particles that should be produced, transported and detected practically at thesame time. Meanwhile, some tasks, for instance muon transport through a homogeneousmaterial, may be accomplished without using multipurpose codes. If a user is interested intransporting muons without following secondary particles produced by them, it is enoughto consider accurately only muon interactions and muon energy losses neglecting the fateof secondaries. This is the idea implemented in specially developed muon transport codes.In this paper we describe the three-dimensional muon propagation code MUSIC. Althoughthe first version of the code was released in 1997 [20] and several modification were re-ported since then [21], we believe that the recent developments and improvements madeto the code and the variety of applications should be described in a separate paper. Thesecond part of the paper is dedicated to the code MUSUN written to sample muons un-derground [11] or underwater [24] using the results of muon transport carried out withMUSIC.
The first version of MUSIC (MUon SImulation Code), written in FORTRAN, has beenreleased in 1997 [20]. It has been used in the interpretation of data from the LVD ex-periment in the Gran Sasso Laboratory, namely in the reconstruction of the muon energyspectrum at surface from the measured depth – vertical muon intensity relation [3] and inthe evaluation of the fraction of prompt muons in the high-energy muon flux [4]. Severalimprovements have been done to the code and new features have become available sincethen. The basic features of the code and improvements are described below.The code takes into account the energy losses of muons due to four processes: ionisation(using Bethe-Bloch formula) including knock-on electron production, bremsstrahlung (orbraking radiation), electron-positron pair production and muon-nucleus inelastic scatter-ing (or photonuclear interactions). The cross-section of bremsstrahlung was taken fromRef. [25] in the first version of MUSIC with an option to use the cross-section from Refs.[26, 27]. The effect of different cross-sections on the MUSIC results was studied in Ref.[20]. The correction to the Born approximation (Coulomb correction) for bremsstrahlungcross-section was not taken into account. It was shown [27] that this correction does notexceed 1% even for heavy nuclei. The pair production cross-section was taken from Ref.[28] in the first version of MUSIC. New parameterisation of the pair production on atomicelectrons [29] has been implemented in the second version of the code [21]. Original pa-rameterisation of muon inelastic scattering cross-section [30] has been complemented bya more accurate treatment suggested in Ref. [31]. An option has been added to calcu-late this cross-section using the deep-inelastic scattering formalism and nucleon structurefunctions suggested in Ref. [32]. The ALLM parameterisations [33] of structure functionshave been implemented. The default options in MUSIC (the recommended cross-sections)are: (i) bremsstrahlung – Ref. [25]; (ii) pair production – Refs. [28, 29]; (iii) inelasticscattering – Ref. [31]. All results presented here have been obtained with this set of3ross-sections.Muon interaction cross-sections are calculated in MUSIC at the beginning of the first run(or by the code developer) for all elements present in a material and specified by a user,and are averaged using the weights (a fraction of each element by mass) provided by theuser.There are two major versions of MUSIC existing and developed in parallel: (i) ‘standard’,dedicated for muon transport through large thickness of matter; (ii) ‘thin slab’, developedfor muon transport through thin slabs of materials.The standard version of MUSIC considers all interaction processes stochastically if thefraction of energy lost by a muon in the interaction exceeds a pre-defined value of aparameter, v cut (see Ref. [20] for the full code description and tests). The value of v cut can vary from 10 − to 1 and can be set by a user but the recommended value takenas a compromise between the accuracy of the code and its speed, is 10 − as in the firstversion of the code [20]. The program evaluates the mean free path of a muon betweentwo subsequent interactions (with v > v cut ) using the sum of integrated cross-sections,where integrals are computed between v cut and 1. Then it samples the real path of themuon to the next interaction using a random number generator from the CERN library(RANLUX). Another random number determines the type of the interaction. The thirdrandom number is used to select the fraction of muon energy lost in the interaction v . Thenthe code calculates the continuous energy losses of the muon between the two interactions,i.e. mean energy losses due to the four aforementioned processes with v < v cut . Meanenergy loss due to ionisation is computed using Bethe-Bloch formula for v < v cut . Knock-on electron production is added to stochastic processes at v ≥ v cut including correctionsto the ionisation energy loss due to e -diagrams for muon bremsstrahlung on an electronsuggested in Ref. [25]. In this process the photon is emitted by the electron and isaccompanied by the high-energy recoiling electron.Muon deflection due to multiple Coulomb scattering in the plane perpendicular to theinitial muon direction is calculated between every two interactions with v ≥ v cut [20]. Theprocess is treated in the Gaussian approximation [34] that was also used in Ref. [19]. Moreaccurate treatment of muon angular deviation in the framework of Moli`ere theory resultsin a similar distribution of muon scattering angles beyond the large thickness of rock witha small increase of the mean deflection angle [20]. However, the original Moli`ere theorywas developed for angular deviation only and does not provide the lateral displacementthat is sometimes more important from the experimental point of view (for instance, formuon bundles underground).Although multiple Coulomb scattering dominates over stochastic processes in the muondeflection [20], muon deviation due to other interactions is also taken into account inMUSIC. The angular deviation due to muon inelastic scattering is computed using double-differential cross-section [30]. The muon scattering angle due to bremsstrahlung and pairproduction is calculated following the parameterisations suggested in Ref. [35] (see alsoRef. [20] for detailed description).Note that the muon transport code MUM [22] is one-dimensional and does not take intoaccount muon deflection in the plane perpendicular to the initial direction. The codeMMC [23] considers only muon deviation due to multiple scattering and not many details4r results are given in the original paper [23]. PROPMU [19] also treats only multiplescattering. Muon transport codes MUSIC, MUM and MMC were found to agree with eachother giving similar muon energy distributions beyond large thickness of rock or waterand similar muon survival probabilities [22]. MUSIC and PROPMU are in agreement formuon transport in rock [20] but results obtained with PROPMU in water were found tobe different from those obtained with MUSIC and MUM [22].The MUSIC code allows the transport of muons with energies up to 10 GeV. All muonsare considered to be ultrarelativistic. If the total muon energy becomes smaller than themuon mass, the muon is considered to be stopped. The code consists of two files and isarranged as consecutive calls to two or three subroutines written in FORTRAN from the‘main’ user program. The call to the first subroutine is optional: it allows calculationof the muon cross-sections and energy losses and can be done at the beginning of thefirst run. The files with cross-sections and energy losses can also be supplied by theauthor. The call to the second subroutine allows reading the muon cross-sections fromthe computer disk. A single call to the third subroutine transports one muon with giveninitial parameters (energy, coordinates and direction cosines) to a specified distance ina material with previously calculated cross-sections. The code (the third subroutine)returns the muon parameters at the end of the muon path in the material. If the muonhas stopped before reaching the end of the material, the zero value for the muon energy isreturned together with approximate coordinates of the point where the muon has stopped.A special version of MUSIC (‘thin slab’) has been developed for muon transport throughthin slabs of materials. Originally it has been written for water as part of the software forthe ANTARES experiment [8]. This version has been aimed at providing muon energy,position and direction at the end of every small segment of muon path in water and atpassing the muon energy loss at the segment to another part of software that generatedCherenkov photons. Since then this version has also been used to estimate muon deflectionin high-A materials (iron, lead and uranium) for possible security applications (searchingfor hidden high-A materials, like uranium, in cargo) [36]. The ‘standard’ version ofMUSIC, in the absence of stochastic interactions on the small segment, always returns themean value for continuous energy loss at the end of the segment. In the ‘thin slab’ versionthe cut that separates stochastic and continuous parts of the energy loss is reduced to 1MeV, meaning that practically all muon interactions are stochastic. The ionisation energyloss is calculated using Landau distribution (call to a function from the CERN library).This version of the code allows the transport of muons with energies up to 10 GeV butwithout taking into account the LPM effect. Both versions of MUSIC (standard and thinslab) give consistent results for thick slabs of matter but the ‘thin slab’ version is moreCPU consuming because of the lower value of v cut . The thin slab version gives moreaccurate results for energy spectra and angular deviation beyond thin slabs of material.Although the energy loss due to muon pair production by muons is not included in thecode because of its small value compared to the electron-positron pair production, thereis a possibility to generate muon pairs along the muon path [21].A few results from muon transport through standard rock (Z=11, A=22, ρ = 2 .
65 g/cm )and pure water are shown in Figures 1-3. Muons with initial energies ranging from10 GeV to 10 GeV were transported through 15 km w. e. of standard rock and waterand their energies at different depths (distances from initial point) were recorded. 10 Z/A ≈ Z/A ≤ .
5. Ionisation energy loss is proportional to
Z/A whereas energy losses dueto pair production and bremsstrahlung are approximately proportional to Z ( Z + 1) /A .At low muon energies (below 1 TeV) and small depths ionisation energy loss dominatesover other processes, muon energy losses in water are bigger than in standard rock andmuon survival probability for a fixed initial energy is smaller in water than in standardrock. At intermediate depths (between 1 and 3 km w. e.) the muon survival probabilitiesin water are smaller for low energies and bigger for higher energies compared to standardrock. At large depths (high muon energies) the energy loss due to pair production andbremsstrahlung dominate over ionisation and the muon survival probabilities in water arebigger than those in standard rock for all energies.Muon energy spectra at vertical at different depths in standard rock (black solid curves)and water (red dashed curves) are presented in Figure 2. Numbers above the curves forstandard rock show the depth in km w. e. The spectra have been calculated by convolutingmuon energy distributions underground obtained with MUSIC, with muon energy spectraat surface taken in the form that fits the data from the LVD experiment [3, 4] (for fulldescription of the procedure see Section 3 below). Curves for water are shifted above orbelow those for standard rock due to the presence of hydrogen in water (see discussionabove).MUSIC has been extensively tested against experimental data. It has first been used inthe analysis of muon intensities measured by the LVD experiment [3, 4]. Since there areseveral factors that affect the calculation of muon intensity underground (muon cross-sections, muon energy spectrum at surface, slant depth distribution and rock composi-tion), comparison between measured and calculated muon intensities does not providean accurate test of the muon transport code. In fact the energy spectrum of muons atsurface has been reconstructed from the measured intensities assuming that other factorsare known. However, the fact that measured intensities agree with simulations over alarge range of zenith angles and slant depths, provides a strong evidence for the validityof the muon transport code.Depth – vertical muon intensity relation (muon intensity at vertical as a function of depth)is shown in Figure 3 for standard rock and water. Muon intensities have been calculatedby integrating muon energy spectra underground over energy. The simulated curves agreewell with the measurements both in rock (black triangles – compilation of data pointsfrom Ref. [37]) and water (blue open circles – [6], blue filled circles – [7]).A comprehensive comparison of calculated (using MUSIC) muon intensities undergroundwith measurements has been done in Ref. [38]. Data points were found to be scatteredsymmetrically around the calculated depth–intensity curve showing the overall consistency6f the muon transport. Large spread of data around simulations may be explained by thecomplexity of factors involved in data interpretation, such as, rock composition, procedureof data conversion to standard rock etc.Muon intensities and mean muon energies, calculated with the MUSIC transport code andthe LVD parameterisation for the muon spectrum at surface [3, 4] (for full description ofthe procedure see Section 3 below) are given in Table 1 for standard rock and water.MUSIC has been used in the analysis of SNO [39] and MACRO [5] data. The code hasalso been applied for the calculation of expected background induced by cosmic-ray muonsin deep underground experiments, such as KamLAND, Super-Kamiokande, etc.Comparison of energy losses as calculated by MUSIC, GEANT4 and FLUKA is discussedin Ref. [40]. All codes agree well in calculating energy distributions for high-energymuons transported through small and large slabs of materials. Figure 4 shows the energyspectrum of muons with initial energy of 2 TeV transported through 3 km of water us-ing MUSIC, GEANT4 [41] and FLUKA. Distributions look very similar except for smalldifference at high energies. The survival probability is equal to 0.779 (MUSIC), 0.793(GEANT4) and 0.756 (FLUKA) with a statistical error of about 0.001. The mean energyof survived muons is 323 GeV (MUSIC), 317 GeV (GEANT4) and 344 GeV (FLUKA).Figures 5 and 6 show distributions of angular deviation and lateral displacement, re-spectively, for muons with initial energy of 2 TeV transported through 3 km of water.GEANT4 predicts larger number of muons to be scattered to high angles and moved tolarge distances in the plane perpendicular to the muon direction. The mean scatteringangle is equal to 0 . ◦ (MUSIC) and 0 . ◦ (GEANT4), whereas the mean displacement inthe plane perpendicular to the initial muon direction is found to be 2.6 metres (MUSIC)and 3.3 metres (GEANT4). Unfortunately it is practically impossible to obtain data onhigh-energy muon scattering beyond very large thicknesses of matter to test codes, sincethe lateral separation of muon bundles underground is largely dominated by the scatteringangle of the muon parent in the atmosphere at the interaction point where this parent isproduced.Similar transport code has also been developed for tau-leptons: TAUSIC (TAU SimulationCode). MUSUN (MUon Simulations UNderground) is the muon generator useful for samplingmuons in underground laboratories according to their energy spectrum and angular dis-tribution. It uses the results of muon transport through matter carried out with MUSIC,convoluted with the muon energy spectrum and angular distribution at surface.At the first stage muons with various initial energies (from 100 GeV to 10 GeV with astep of ∆ log E = 0 . I µ ( E µ , X, cos θ ), arecalculated using the equation: I µ ( E µ , X, cos θ ) = Z ∞ P ( E µ , X, E µ ) dI µ ( E µ , cos θ ⋆ ) dE µ dE µ (1)where dI µ ( E µ , cos θ ⋆ ) dE µ is the muon spectrum at sea level at zenith angle θ ⋆ (zenith angle atsurface, θ ⋆ , is calculated from the zenith angle underground, θ , taking into account thecurvature of the Earth), and P ( E µ , X, E µ ) is the probability for a muon with an initialenergy at surface E µ to have an energy E µ at a depth X .The energy spectrum at sea level can be taken either according to the parameterisationproposed by Gaisser [1] (modified for large zenith angles [3]) or following the best fit tothe ‘depth – vertical muon intensity’ relation measured by the LVD experiment [3]. Thefirst parameterisation [1] has the power index of the primary all-nucleon spectrum 2.70,while the second one [3] uses the index of 2.77 with the normalisation to the absoluteflux measured by LVD. For small depths (less than 2–3 km w. e.) that correspond to lowmuon energies at surface (less than 1 TeV) it is recommended to use the original Gaisser’sparameterisation with an additional factor that takes into account muon decay in theatmosphere [42] if necessary. For larger depths the LVD parametrisation is the preferredoption since it agrees with experimental data of the LVD [3] and MACRO experiments[2].The ratio of prompt muons (from charmed particle decay) to pions is recommended tobe set to 10 − , which is well below an upper limit set by the LVD experiment [4]. Note,however, that prompt muon flux does not affect much muon intensities even at largedepths.To calculate integral muon intensity for normalisation, an integration of I µ ( E µ , X, cos θ )over dE µ and cos θ is carried out.MUSUN offers the choice of the muon energy spectrum (as described above), the fractionof prompt muons, the vertical depth of the laboratory, the range of zenith and azimuthalangles, and the range of energies. No additional muon propagation is required for differentoptions. Different types of rocks (rock compositions), however, require separate muontransport.MUSUN is organised as a set of subroutines written in FORTRAN that are called fromthe user-defined ’main’ program. The first call is made to a subroutine that calculatesdifferential and integrated muon intensities for a specific vertical depth (assuming flatsurface). The intensity as a function of energy and zenith angle is stored in the computer8emory as a two-dimensional array. Subsequent calls to a ’sampling’ subroutine returnmuon parameters (energy and direction cosines) sampled following energy and zenith angledistribution. Azimuthal angle is sampled randomly as evenly distributed between 0 and2 π since the assumption of the flat surface leads to the spherical symmetry. The muoncharge is generated according to the ratio measured for high-energy muons µ + /µ − ≈ . ◦ as measured by LVD [4, 43] (datapoints with error bars) and generated with MUSUN (dashed curve). Good agreement isseen over the whole range of angles and intensities. Similar conclusion has been achievedin Ref. [44] when comparing azimuthal distributions for the whole range of zenith angles.Figure 8 shows the energy spectrum of muons at Gran Sasso as generated with MUSUN.This spectrum looks different from Figure 2 because the number of muons is given hereper energy bin which is constant on the logarithmic scale but increases with energy onthe linear scale, whereas in Figure 2 the spectrum is given per constant energy bin on thelinear scale (1 GeV). The mean muon energy at the Gran Sasso Laboratory is calculatedas 273 GeV, in good agreement with the measured value of 270 ± ±
18 (syst.)GeV [45]. A lego plot of the number of generated muons versus zenith and azimuthalangles is presented in Figure 9.At present the versions of the MUSUN code exist for the underground sites at Gran Sasso(LNGS), Modane (LSM), Boulby and Soudan. It has been used to study muon-inducedneutron background for experiments looking for rare events, such as WIMPs (see, forinstance, [11, 12, 46, 13, 47, 48]).
The two Monte Carlo codes MUSIC and MUSUN dedicated to muon simulations havebeen described. MUSIC, a package for muon transport through matter, can be used forpropagating muons through large thickness of rock or water, for instance from the surfacedown to underground/underwater laboratory. It can also be implemented in the eventgenerators for large underwater/under-ice neutrino telescopes or other neutrino detectors.9USUN uses the results of muon transport through rock/water to generate muons in oraround underground laboratory taking into account their energy spectrum and angulardistribution. Various tests showed good agreement of the codes’ results with experimentaldata and other packages.Since there are several versions of both codes the author finds impractical to submitall of them to the code library. Any specific version can be obtained by request tov.kudryavtsev@sheffield.ac.uk. There is a possibility to adapt the codes to specific needsof a user as was done on several occasions in the past.
Many recent improvements to the codes have been carried out as part of the ILIASintegrating activity (Contract No. RII3-CT-2004-506222) in the framework of the EUFP6 programme in Astroparticle Physics. The author is grateful to Dr. M. Robinson forproviding results of GEANT4 simulations. The author wishes to thank Drs. P. Antonioli,C. Ghetti, E.V. Korolkova and G. Sartorelli who contributed to the original version of theMUSIC code and initial tests.
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Column 1 – depth, X , in kilo-metres of water equivalent, km w. e.; column 2 – vertical muon intensity in standardrock, I vertµ ; column 3 – mean muon energy for the muon flux in standard rock at vertical, E vertµ ; column 4 – global intensity (integrated over solid angle for a spherical detector)in standard rock for flat surface, I µ ; column 5 – mean muon energy for the global muonflux in standard rock, E µ ; column 6 – vertical muon intensity in water (ice); column 7– mean muon energy for the muon flux in water at vertical; column 8 – global intensity(integrated over solid angle for a spherical detector) in water (ice) for flat surface; column9 – mean muon energy for the global muon flux in water. Standard rock Water
X I vertµ E vertµ I µ E µ I vertµ E vertµ I µ E µ km w.e. cm − s − sr − GeV cm − s − GeV cm − s − sr − GeV cm − s − GeV0.5 1 . × −
69 2 . × −
93 7 . × −
80 1 . × − . × −
120 2 . × −
150 1 . × −
144 2 . × − . × −
197 1 . × −
225 1 . × −
246 1 . × − . × −
248 3 . × −
271 2 . × −
322 3 . × − . × −
284 6 . × −
301 7 . × −
379 8 . × − . × −
308 1 . × −
319 2 . × −
421 2 . × − . × −
324 4 . × −
332 8 . × −
453 8 . × − . × −
335 1 . × −
341 3 . × −
477 2 . × − . × −
344 3 . × −
347 1 . × −
495 1 . × − . × −
349 9 . × −
351 5 . × −
508 3 . × − . × −
351 2 . × −
353 2 . × −
519 1 . × − Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Muon energy at surface, GeV M u o n s u r v i va l p r o b a b ili t y Figure 1: Survival probabilities as functions of muon energy at surface for different depths(from 0.5 to 10 km w. e.) in standard rock (black solid curves) and water (red dashedcurves). Numbers to the right from each solid curve show the depths in km w. e. forstandard rock. Survival probability curves for water are shifted to the right (for smalldepths) or to the left (for depths larger than 1 km w. e.) relative to those for standardrock. 14 -20 -19 -18 -17 -16 -15 -14 -13 -12 -11 -10 -9 -8 -7
10 10 Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Muon energy, GeV V er t i c a l m u o n i n t e n s i t y , c m - s - s r - G e V - Figure 2: Muon energy spectra at vertical at different depths in standard rock (black solidcurves) and water (red dashed curves). Numbers above the curves for standard rock showthe depth in km w. e. Curves for water are shifted above or below corresponding curvesfor standard rock. 15 -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 Vertical depth, km w. e. V er t i c a l m u o n i n t e n s i t y , c m - s - s r - RockWater
Figure 3: Depth – vertical muon intensity relation (muon intensity at vertical as a functionof depth) for standard rock and water. The data points for standard rock (black triangles)are from the compilation of experimental results [37]. The data points for water are fromthe Baikal [6] (blue open circles) and AMANDA [7] (blue filled circles) experiments.16 Muon energy, GeV N u m b er o f e v e n t s Black solid - MUSICRed dashed - GEANT4Blue dotted - FLUKA
Figure 4: Energy distribution of muons with initial energy of 2 TeV transported through 3km of water using MUSIC (black solid curve), GEANT4 (red dahsed curve) and FLUKA(blue dotted curve). 17 Scattering angle, degrees N u m b er o f e v e n t s Black solid - MUSICRed dashed - GEANT4
Figure 5: Distribution of angular deviation for muons with initial energy of 2 TeV trans-ported through 3 km of water using MUSIC (black solid curve) and GEANT4 (red dahsedcurve). 18 Lateral displacement, metres N u m b er o f e v e n t s Black solid - MUSICRed dashed - GEANT4
Figure 6: Distribution of lateral displacement for muons with initial energy of 2 TeVtransported through 3 km of water using MUSIC (black solid curve) and GEANT4 (reddahsed curve). 19 -9 Azimuthal angle, degrees M u o n i n t e n s i t y , c m - s - d e g ree - Figure 7: Azimuthal distribution of single muon intensities in the underground Gran SassoLaboratory for zenith angles up to 60 ◦ as measured by LVD [4, 43] (data points with errorbars) and generated with MUSUN (dashed curve). LVD acceptance as a function of zenithand azimuthal angles has been taken into account when generating muons. Azimuthalangle is calculated in the LVD reference system [3, 4, 43].20 Muon energy, GeV N u m b er o f m u o n s Figure 8: Energy spectrum of muons as generated by MUSUN for the underground GranSasso Laboratory. 21 C o s i n e o f z e n i t h a n g l e A z i m u t h , d e g N u m b er o f e v e n t ss