LITMIR.BIZ

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consider the daily and seasonal variations in the muon flux. These variations are mainly caused by the spatio‐temporal variations in the interplanetary magnetic field (IMF) and the atmospheric temperature, and affect the entire energy range of the muon spectrum. Anisotropies of the IMF are related to the solar activities (11‐year and 22‐year solar cycles and large‐scale bursts of plasma and magnetic field, such as coronal mass ejections [CME]). The 11‐year and 22‐year solar cycles affect the sub‐GeV energy region of the cosmic‐ray muons, and variations of the muon flux due to this effect are small (a few percent) (de Mendonca et al., 2016). The CME also affects only the sub‐GeV energy region, but the muon flux (E > 100 MeV) can be periodically changed by 10% within a few days (Augusto et al., 2012).

      The temperature variations cause barometric (tropospheric pressure) variations, and thus affect the muon count rate. Therefore, the muon count rate shows a negative temperature correlation within a day. Since daily barometric variations are generally less than one percent in the mid‐latitude region, the corresponding muon flux variations are up to a few percent. For example, the daily flux variations measured at 44^o 21′N, 76 m a.s.l. was ± 2.5% for muons with energies above 0.4 GeV; this corresponds to the muon range of 200 g/cm² in rock (Saftoiu et al., 2010).

A major contributor to the seasonal variations in muon flux is not only the tropospheric but also the stratospheric temperature (or pressure) variations. While the tropospheric temperature variations affect low‐energy muons, the stratospheric temperature variations affect high‐energy muons. The temperature variations in the stratosphere, where the mesons are copiously produced, change the MFP of the meson‐nucleon interactions in the stratosphere. As a consequence, high‐energy muons have a positive stratospheric temperature correlation. The Daya‐Bay detectors were located underground at three different depths (ranging from 250–860 hg/cm²) and recorded the seasonal variations in muon flux at each depth (Daya Bay Collaboration, 2018). There was a tendency for the seasonal variations to be more enhanced for deeper detectors: 0.5% at a depth of 250 hg/cm² and 1% at a depth of 860 hg/cm². At a deeper site (3,800 hg/cm²), stronger variations (~3%) were observed (Borexino Collaboration, 2019).

      In conclusion, muographers have to consider the following two factors when they conduct the time‐dependent measurements: (i) The stratospheric effect in high‐energy muons must be considered. However, currently, there are no experimental data for the horizontal muons. Such data will be necessary in the near future. (ii) When muographers need to normalize the time‐sequential muographic data to the open‐sky muon count rate for the purpose of cancellation of the factors originated in the detector configuration, such as geometrical acceptance, efficiency, etc., the spatial scale of the target objects and the timescale of the measurements have to be well considered so that the aforementioned geomagnetic and atmospheric effects become negligible in their measurements.

      1.2.5 Muon Flux Reduction through Matter

      During the ionization process, muons frequently collide with electrons, losing a very small fraction of their energy in each collision. If the energy loss of muons in matter were only due to ionization, muons of a given energy would have an almost unique range because the number of such encounters is proportional to the densimetric length they traverse. However, for high‐energy muons, the radiative processes become more predominant, and at the same time, fluctuations within the range are enhanced by the radiative processes, namely bremsstrahlung, direct pair production, and photonuclear interactions. In these processes, muons lose a large but random fraction of their energy. For high‐energy muons (> 1 TeV), the contribution of the ionization process is small compared with the other three processes. For example, the radiative process dominates above 708 GeV in SiO₂ (Groom et al., 2001).

      The muon's energy loss rate depends on materials. The electrons have a tendency to be more concentrated per unit mass in lighter materials, and thus muons lose more energy in lighter materials through the ionization process. However, larger nuclei increase the possibilities of radiative processes occurring and thus, high‐energy muons lose more energy in heavier materials. For example, the 1‐GeV muon's range is 550 g/cm² in SiO₂ but muons with the same energy can penetrate only 470 g/cm² of water (Groom et al., 2001). On the contrary, while 10‐TeV muons can penetrate 6,900 hg/cm² in SiO₂, they can penetrate 7,800 hg/cm² in water (Groom et al., 2001).

      For the calculation of the analytical range of high‐energy muons, the energy loss relationship presented by Adair and Kasha (1977) is convenient:

(1.1)

where E is the muon energy, x is the densimetric thickness (path length times average density along the muon path) along the path traversed by the muon, k(E) is the ionization energy loss, and b(E)E is the sum of the energy losses via three stochastic processes. The values for k(E) and b(E)E can be found at the reference for various materials (Groom et al. 2001). By integrating equation 1.1 over the energy range between 0 and the muon’s incident energy, the continuous slowing down approximation (CSDA) range of the muon can be calculated.

      Since the range is a function of the incident muon energy, it can be incorporated in the open‐sky muon spectrum. Once both the muon path length and the average density along the path are known, the densimetric thickness (x) can be calculated by multiplying them, and thus the minimum energy (E _c) of muons that can penetrate through a material with this thickness can be determined using equation 1.1. By integrating the open‐sky spectrum I (E, θ) from E _c to infinity, we obtain the integrated muon intensity N(E _c,), which represents the number of muons that have enough energy to escape from the target of interest:

(1.2)

where I(E,θ) is the zenith‐angle‐dependent open‐sky muon energy spectrum. The integrated muon intensities calculated with equation 1.2 are shown for various near‐horizontal angles in Fig. 1.1.

      1.2.6 Muon Scattering

      When muons travel through matter, the Coulomb force between the muons and the nuclei in the medium leads to numerous small deflections in the muon trajectories. As a result, their angular distribution becomes broader as they propagate through matter. However, owing to the steep nature of the muon spectrum (Taira & Tanaka, 2010) the angular spread is suppressed to ~12 mrad, half width at half maximum after penetrating matter thicker than 500 hg/cm², which is almost independent of the total thickness of the material that is traversed by muons. This angular spread has the effect of limiting the positioning resolution at the target, for example, 12 m at a distance of 1 km from the detector. The highly penetrating nature of high‐energy muons, coupled with their low divergence, enables the efficient muographic imaging of a distant target, making them suitable for long‐range applications.

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