**2. Basic properties of the primary and secondary cosmic radiation**

Since its very beginning, our Earth is continuously bombarded by energetic particles—cosmic rays—which enter the Earth atmosphere from outer space. Most of them are charged nuclei, with H nuclei (protons) being dominant, and other, heavier nuclei, which reflect, with some difference, the nuclear abundance found in nature. Their energy may vary over many orders of magnitude, from a few hundred MeV to 1020eV, with an energy spectrum roughly described by a power law, dN/dE = const E−γ, with γ = 2.7 up to 3 x1015eV. At higher energies, up to 1019eV, the spectrum steepens, with γ approximately equal to 3.1 (the so-called knee), flattening again above this energy. While a small fraction of the lowest energy particles come from our Sun, most of the primary cosmic rays originate from within the Milky Way, and the most energetic ones have an extra-galactic origin. Apart from the lowest energy component, which is subjected to space and time variations, due to the Sun and to the interplanetary environment, the majority of cosmic rays exhibit a homogeneous distribution, with very small anisotropies investigated, of the order of 10−4 to 10−3.

The existence of the Earth atmosphere has a peculiar effect on the arrival of a primary cosmic ray, since an extensive air shower (EAS) is created following the first interaction of a high energy primary particle with the atmospheric nuclei. This interaction produces a cascade of secondary particles, which may produce in turn additional particles or may decay, constituting a shower with hadronic and electromagnetic components. While a single detector may reveal the passage of individual particles in the shower, the coincidence detection between several particles in the shower allows the identification and reconstruction of the primary particle. This is the way in which extended arrays of detectors are able to measure even the largest energy primary particles. The lateral profile of extensive air showers depends on the initial energy and may reach hundreds of metres or even kilometres.

of view, an important application of the muon absorption technique dates back to the work of Alvarez et al. [2], concerning the search for hidden chambers in one of the Egyptian pyramids. Nowadays, the applications of muon tomography to various aspects of everyday life include the study of large structures such as mountains and volcanoes, the inspection of large volumes to search for hidden, high-Z materials, such as for fissile illicit elements in containers, the monitoring of civil structures as large buildings or bridges, the control of nuclear reactors

It is usual to distinguish, from an experimental point of view, between those applications where the absorption of muons is employed to have information about the amount of material traversed by the particles and other applications where the multiple scattering effect is used instead, especially sensitive to the atomic number of the traversed material. Many examples of the two approaches have been given over the last two decades, and this review, although quoting many of these, is in no way a complete listing of what is available in the literature. Moreover, the field is rapidly expanding, with new detector prototypes being designed and tested, and many additional examples reported. After a brief introduction on the basic properties of the primary and secondary cosmic radiation, especially concerning the aspects which are relevant for muon tomography, a review of the problems and applications making use of the muon absorption technique is given in Section 3. Applications of the muon scattering are described in Section 4. Additional examples of applications of the muon interaction in matter are briefly reported in Section 5, while Section 6 reports a naïve discussion of the possible use of this technique outside our planet. Due to the importance of numerical algorithms for track reconstruction, and image processing, some of the relevant problems in this field are recalled

in Section 7. Some concluding remarks are finally discussed in Section 8.

anisotropies investigated, of the order of 10−4 to 10−3.

**2. Basic properties of the primary and secondary cosmic radiation**

Since its very beginning, our Earth is continuously bombarded by energetic particles—cosmic rays—which enter the Earth atmosphere from outer space. Most of them are charged nuclei, with H nuclei (protons) being dominant, and other, heavier nuclei, which reflect, with some difference, the nuclear abundance found in nature. Their energy may vary over many orders of magnitude, from a few hundred MeV to 1020eV, with an energy spectrum roughly described by a power law, dN/dE = const E−γ, with γ = 2.7 up to 3 x1015eV. At higher energies, up to 1019eV, the spectrum steepens, with γ approximately equal to 3.1 (the so-called knee), flattening again above this energy. While a small fraction of the lowest energy particles come from our Sun, most of the primary cosmic rays originate from within the Milky Way, and the most energetic ones have an extra-galactic origin. Apart from the lowest energy component, which is subjected to space and time variations, due to the Sun and to the interplanetary environment, the majority of cosmic rays exhibit a homogeneous distribution, with very small

The existence of the Earth atmosphere has a peculiar effect on the arrival of a primary cosmic ray, since an extensive air shower (EAS) is created following the first interaction

and their waste products, and many others.

42 Cosmic Rays

Apart from neutrinos, which are hardly detected, muons are the most penetrating part of the shower. Relativistic effects increase their lifetime (about 2.2 μs at rest), allowing a large fraction of them to reach the Earth surface. Even though detailed measurements of the energy, angular distributions and charge ratio of cosmic muons at different altitudes and locations on the Earth surface are still pursued, especially for the high energy component, the basic properties of the muon flux are well known and many compilations exist concerning the distributions of these particles [3].

Several parameterizations exist for the angular and energy distributions of cosmic muons at the sea level or moderate altitudes. As a result of the muon absorption in the Earth atmosphere, the dependence on the zenithal angle *θ* at sea level is often expressed as

$$\frac{dN}{d\Omega}\gamma\cos\theta^2\tag{1}$$

while the momentum distribution of vertical muons roughly follows a power law. A reasonable parametrization of the vertical muon flux as a function of the momentum is given by [4].

$$\mathbf{C} \cdot \mathbf{p}^{-\|\epsilon\_n + c\_{\hbar}\mathbf{h}\mathbf{p} + c\_{\hbar}(\mathbf{h} \cdot \mathbf{p})^{\natural} + c\_{\hbar}(\mathbf{h} \cdot \mathbf{p})^{\natural}\|}\tag{2}$$

where the values of the *ci* coefficients are given for selected ranges of the muon momentum. As an example, **Figure 1** shows a plot of the muon momentum distribution extracted from the above formula, for momenta up to about 1 TeV/c.

A semi-empirical parametrization of the muon flux at sea level as a function of both the zenithal angle and muon energy, especially valid for high energy muons (E > 100 GeV/cos*θ*) is the following:

$$\frac{dN}{dE d\Omega} = \frac{0.14 \, E^{\circ 2}}{cm^2 s \, GeV \, sr} \left( \frac{1}{1 + \frac{1.1 \, E \cos \theta}{115 \, GeV}} + \frac{0.054}{1 + \frac{1.1 \, E \cos \theta}{850 \, GeV}} \right) \tag{3}$$

The mean energy of muons arriving at the sea level is ~4 GeV and the mean number of particles traversing a horizontal detector is of the order of 1 per cm2 per minute. For detailed calculations one has to take into account the variations in these quantities which are due to the altitude and geographical location (especially latitude). On a large time scale, solar effects may also modify the numerical values of the measured flux.

very high muon energies however, radiative processes become more important than the ionization processes. In case of muons, the value of the critical energy, where the two contributions are comparable, is of the order of several hundred GeV for medium-Z materials like the Iron. Radiative processes then dominate the energy loss of highly energetic cosmic muons, and should be taken into account when considering muons which have to traverse hundred metres solid rock. As an example, **Figure 3** shows the muon energy loss in Lead as a function of the muon energy [6]. The relative contribution of the individual terms due to pair production, Bremmstrahlung and photonuclear processes depends on the muon energy, with the last

Considering a realistic momentum distribution of muons, GEANT simulations of the interaction of muons with solid rock may be performed. As an example, **Figure 4** shows the fraction of surviving muons after traversing a given thickness of volcano rock, modelled by a realistic

CaO. As it is seen from **Figure 4**, about 1% of the muon flux is still emerging after traversing

Due to the energy loss of muons in a solid material (such as the rock of a mountain), which for a thin layer is proportional to the quantity *ρ dx*, where *ρ* is the density of the material, the fraction of muons which survive after traversing a finite thickness x of material is given, to

0

**Figure 2.** Mass attenuation coefficient of X- or γ-rays of various energies in lead, derived from the NIST standard data [5].

, Al2 O3

Cosmic Ray Muons as Penetrating Probes to Explore the World around Us

http://dx.doi.org/10.5772/intechopen.75426

*<sup>L</sup> ρ*(*x*)*dx* (5)

, FeO, MgO, and

45

one being much smaller than the other two for increasing muon energies.

chemical composition of the lava from Etna, mainly including SiO2

first order, by the integrated density over the path length *L*

∫

100 m thickness.

**Figure 1.** Momentum spectrum of cosmic muons, parametrized by Eq. (2).
