# Exploring the Interaction of Cosmic Rays with Water by Using an Old-Style Detector and Rossi’s Method

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

- Interaction mean free path: denoted by a small “λ” (cm), it is the average path of interaction, or else the average distance traveled by a particle between one interaction and another.
- Attenuation length, or radiation length: denoted by a capital “Λ” (or X
_{0}) (cm), is defined as the length needed to reduce the energy of a particle to a value 1/e of its original (0.3678, so of 36.78%). - Interaction depth or radiation depth: denoted by a capital “X”, is the path traveled by a particle in a medium and is measured in gcm
^{−2}(length times the density of the medium: cm∙g∙cm^{−3}= g∙cm^{−2}, from which it can be seen that λ = X/ρ, thus g∙cm^{−2}/g∙cm^{−3}= cm). Sometimes these quantities may confound because among scientists there is no uniform denotation; some authors express both “λ” and “Λ” multiplied by the density of the medium (in that case λ and Λ are usually denoted, respectively, by λ’ and Λ’, or by other symbols), becoming likewise gcm^{−2}. Even worse, some authors use the lowercase or uppercase Greek letter lambda indifferently. - Meter of water equivalent (mwe): it is a unit of measurement for attenuation that expresses the thickness of any material as a function of its density in relation to the thickness of a meter of water. This unit is equivalent to the length (thickness) of a medium in meters times its density: mwe = L
_{ (medium) }[m] ρ [gcm^{−3}]. For instance, 0.127 m of iron is equivalent to an mwe (in other words, a meter of water has the same effect as 12.7 cm of iron). The mwe is sometimes convenient, as it makes it possible to directly and intuitively compare the thickness of matter that cosmic rays have to pass through, like in underground laboratories. For example, given that “standard” rock has a density of 2.65 gcm^{−3}, a detector placed 380 m below the ground has an attenuation of 1000 mwe, or equal to a column of water measuring one kilometer.

^{−2}.

#### 2.1. Photon Intensity Reduction in Matter

_{0}is the initial intensity, d is the distance (cm), and α is the linear coefficient of attenuation (1/cm), which in turn is equal to:

^{2}g

^{−1}), and ρ the density. So, the Relation (2) can be written as:

#### 2.2. Probability of Interaction

#### 2.3. Development of Electromagnetic Showers in Matter

^{−2}, or 1.67 mwe). Showers of electromagnetic particles can be initiated in the first layers of water (say, in the top 100 cm) by a few penetrating particles, namely, muons, electrons, and gamma rays, while more in-depth, only hard components (muons) could start a shower. Muons traveling in water can lose energy through a number of different processes, including ionization and excitation, direct electron pair production, bremsstrahlung, and photo-nuclear interactions. The total energy loss can be described as:

- The decay of neutral pions results in the emission of two gamma rays with a combined energy of at least 140 MeV;
- Electrons from muon decay and other accelerated electrons have a 30–40% chance of producing bremsstrahlung;
- The annihilations between electrons and positrons produce a characteristic energy line at 0.51 MeV;
- Nuclear collisions yield neutrons with energies of approximately 10 MeV; these neutrons can undergo scattering or be captured by nitrogen-14 and oxygen-16, resulting in the creation of excited states that emit characteristic gamma energies;
- Gamma rays below 2 MeV degrade slowly by multiple Compton scattering;
- Photons around 30 keV have interactions through the photoelectric effect.

^{−3}cm

^{−2}s

^{−1}sr

^{−1}(Palmatier 1952, Greisen 1942, Chou 1953) [7]. High energy photons can trigger off the pair-production in metals and water:

_{0}, the cascade continues until it reaches a maximum number of particles. When the energy of the particles falls below the critical energy E

_{c}(i.e., radiative energy loss is equal to ionization energy loss), the shower is halted because bremsstrahlung no longer dominates, so that the maximum number of particles is defined as:

_{max}at which the cascade reaches its maximum size is given by an equation derived from this model:

_{max}describes the depth of the cascade maximum in terms of radiation length, and since Λ is proportional to A/Z

^{2}ρ, it follows that electromagnetic showers are deeper the lower the atomic number and the lower the material density.

_{max}crossed by a shower is a function of the radiation length traveled (Λ), its number, and the density of the medium, as follows:

^{2}ρ). However, if the thickness is measured in units of interaction length, the development of the shower is independent of the material.

## 3. Experimental Setup

#### 3.1. The Cosmic Ray Telescope and Shower Detector AMD16

#### 3.2. Geometry of the Muon Telescope and Shower Detector

^{2}at sea level 0.7∙10

^{−2}(cm

^{−2}s

^{−1}sr

^{−1}) (value taken from Particle Data Group Booklet, PDG 2022), the theoretical acceptance of the telescope for muons should be about 0.38 (cm

^{−2}min

^{−1}). Since the detector averages a flux rate of about 0.18 (cm

^{−2}min

^{−1}) (about five counts per minute, or cpm for about 27 cm

^{2}), its efficiency for muon counting (considering the ratio between the actual counts and the theoretical counts) should be approximately 0.18/0.38 = 0.47, i.e., 47%.

#### 3.3. Methodology for Detecting and Measuring Particle Showers in Water

^{−2}and a measurement under plates of iron up to about 118 gcm

^{−2}. The first trial was made both with tap water and distilled water up to 45 gcm

^{−2}, but since the results were quite similar between the two types of water, the test continued only with tap water, and all the data are related to it. Further, from 0 to 100 gcm

^{−2}, we made two distinct measurements, with and without GMT shielding, and in this case, the results were pretty much different.

## 4. Results and Analysis

#### 4.1. Electromagnetic Cascades in Iron

^{−2}, with another prominent peak at around 70 gcm

^{−2}in Figure 8a and at 86 gcm

^{−2}in Figure 8b. The first peak is expected to be the maximum of the shower and corresponds exactly to 23.62 gcm

^{−2}or 3 cm of iron (in both cases); this can be compared to the early results of Rossi (Figure 1), where he obtained the maximum in correspondence to 1.6 cm of lead, which from Equation (17) is equivalent to 18.4 gcm

^{−2}. The expected Rossi curve is better represented by the fit of the first run, where the tubes were not shielded. The second big maximum is a controversial feature already seen in the literature known as “the second maximum of the shower transition curve” [16]. As introduced before, the “background noise” is relatively high: 16.7 cph on the first run vs. 13.4 cph on the second run. In the second graph, the lower rate due to the lead shield is evident, but despite the shielding, the background still remains high.

#### 4.2. Electromagnetic Cascades in Water

^{−2}of water above the detector; every measurement lasted at least 24 h. As for the experiments with iron, the detector provides a count every minute, so we had more than 1447 counts for every measure. Then, the data were integrated to calculate the count per hour. The results are shown in the graphs in Figure 10. For every measurement, the standard error was around 5%, as shown by the error bars and from the analysis in Table 2.

^{−2}for the first run and at 60 gcm

^{−2}for the run with shielded GMTs. In both graphs, there is a flatness or inflection of the curve above, say, 80 gcm

^{−2}. This means that over this depth, the particles are going to be halted, losing all their energy, and the maximum of the showers is surely reached between 60 and 75 gcm

^{−2}. In these experiments, the effect of the lead shield is still evident; in the second run, the average count is about half that of the first run, but anyway, the trends are very similar. We used the models of Equation (16) to evaluate the energy of the particle necessary to produce a shower reaching the maximum depth; the results are shown in Figure 11 and in Table 3.

_{0}) for the initial particle should be in the range 0.5 ÷ 1 GeV (Table 3). Now, using Equation (15), we can estimate that the number of particles N

_{max}should be from 4 to 12. This is consistent with Equation (13), which forecasts a value of two for n, meaning that the number of photons and electrons from a shower in 72 cm of water initiated by a single particle should not be more than six. More precisely, if an electron initiates the cascade, to produce six particles its initial energy should be approximately 558 MeV.

#### 4.3. Muon Absorption in Water and Iron

^{−2}, muons are absorbed, but then the counting increases again. This may have some relation to the inflexion visible around 50 gcm

^{−2}in Figure 10a and around 30 gcm

^{−2}in Figure 10b. Over 60 gcm

^{−2}, in the case of GMTs shielded, the muon counting abruptly drops again; instead, without the shielding, the counting diminishes slightly.

## 5. Discussion

^{−5}counts per second), so the non-zero value recorded means that from time to time, there are shower detections even without material above the detector (apart from the building roof). The data logger records the number of events (showers and muons) every minute (cpm). We do not have a time stamp for every event, but we noticed that a shower event is present if and only if some muon events are also present. There are at least three possible explanations for this behavior:

- The simpler explanation is that muons belonging to the same atmospheric shower pass through the water and simultaneously hit all the GMTs. It would be extremely difficult to avoid this possibility, even with a more complex instrument setup and anti-coincidence circuits;
- Another possible explanation is that sometimes an energetic muon can produce a shower directly inside the instrument by scattering tertiary particles from the metallic surface of the sensors, leading to the counting of a shower event;
- The phenomenon could result from a combination of the above factors, and local natural radioactivity.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Instrument AMD16: (

**a**) Cross-section of the detector; (

**b**) An image, at the bottom a close-up of the GMTs, in this case, three of them shielded by lead.

**Figure 8.**Measurement of electromagnetic cascades in iron from 0 to 15 cm; the value on the abscise is expressed in radiation depth (gcm

^{−2}); the orange line is a kind of fit represented by the running mean: (

**a**) First run with GMT unshielded; (

**b**) Second run with GMT shielded by lead.

**Figure 10.**Measurement of electromagnetic cascades in water from 0 to 100 cm; the value on the abscise is expressed in radiation depth (gcm

^{−2}); the orange line is a kind of fit represented by the running mean: (

**a**) First run with GMT unshielded; (

**b**) Second run with GMT shielded by lead.

**Figure 12.**Measurement of muons in water and iron; the value on the abscise is expressed in radiation depth (gcm

^{−2}): (

**a**) Comparison between two sets of data from iron, shielded vs. unshielded GMTs (total events); (

**b**) Comparison between two sets of data from water, shielded vs. unshielded GMTs (total events).

Material | Density (gcm^{−3}) | Λ (cm) | E_{c} (MeV) |
---|---|---|---|

Al | 2.7 | 8.9 | 42.7 |

Fe | 7.87 | 1.76 | 21.7 |

Pb | 11.4 | 0.56 | 7.4 |

H_{2}O | 1 | 36 | 93 |

Air | 10^{−3 1} | 3.7∙10^{4 1} | ≅100 ^{1} |

^{1}At about 1500 m.

**Table 2.**The table shows 95.0% confidence intervals for the means and standard deviations (counts per minute) of each of the variables in the first run (1) and in the second run with GMT shielded by lead (2).

Water X (gcm^{−2}) | Mean (cpm) | Stnd. Error | Lower Limit | Upper Limit | ||||
---|---|---|---|---|---|---|---|---|

(1) | (2) | (1) | (2) | (1) | (2) | (1) | (2) | |

0 | 0.20 | 0.22 | 0.012 | 0.016 | 0.17 | 0.19 | 0.22 | 0.25 |

12 | 0.35 | 0.28 | 0.015 | 0.015 | 0.32 | 0.25 | 0.38 | 0.31 |

30 | 0.40 | 0.28 | 0.016 | 0.014 | 0.36 | 0.25 | 0.43 | 0.31 |

45 | 0.38 | 0.31 | 0.013 | 0.016 | 0.35 | 0.28 | 0.40 | 0.35 |

60 | 0.45 | 0.32 | 0.018 | 0.020 | 0.41 | 0.28 | 0.48 | 0.36 |

75 | 0.47 | 0.30 | 0.030 | 0.014 | 0.41 | 0.27 | 0.53 | 0.33 |

100 | 0.45 | 0.30 | 0.030 | 0.018 | 0.39 | 0.27 | 0.51 | 0.34 |

Element | E_{0} (Equation (16a)) (MeV) | E_{0} (Equation (16b)) (MeV) | E_{0} (Equation (16c)) (MeV) |
---|---|---|---|

Pb | 60 | 200 | 150 |

Fe | 80 | 200 | 150 |

H_{2}O * | 300 | 900 | 500 |

400 | 1000 | 750 |

**Table 4.**Results from a quick simple regression analysis; possible coefficients (least squares) for modeling the results of the experiment of muon absorption in iron.

Experiment | Intercept | Slope | R^{2} (%) | Std. Err. of Estimate | Mean Abs. Err. | Model |
---|---|---|---|---|---|---|

Iron GMTs unshielded | 5.6 | −0.00087 | 75 | 0.020 | 0.013 | e^{(5.6−0.00087×Fe)} |

Iron GMTs shielded | 5.3 | −0.00084 | 64 | 0.025 | 0.017 | e^{(5.3−0.00084×Fe)} |

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**MDPI and ACS Style**

Arcani, M.; Liguori, D.; Grana, A.
Exploring the Interaction of Cosmic Rays with Water by Using an Old-Style Detector and Rossi’s Method. *Particles* **2023**, *6*, 801-818.
https://doi.org/10.3390/particles6030051

**AMA Style**

Arcani M, Liguori D, Grana A.
Exploring the Interaction of Cosmic Rays with Water by Using an Old-Style Detector and Rossi’s Method. *Particles*. 2023; 6(3):801-818.
https://doi.org/10.3390/particles6030051

**Chicago/Turabian Style**

Arcani, Marco, Domenico Liguori, and Andrea Grana.
2023. "Exploring the Interaction of Cosmic Rays with Water by Using an Old-Style Detector and Rossi’s Method" *Particles* 6, no. 3: 801-818.
https://doi.org/10.3390/particles6030051