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Muon Absorption Tomography of a Lead Structure

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Muon Absorption Tomography of a Lead Structure MUON ABSORPTION TOMOGRAPHY OF A LEAD STRUCTURE THROUGH THE USE OF ITERATIVE ALGORITHMS APREPRINT Guglielmo Baccani Department of Physics and Astronomy University of Florence P.zza S.Marco 4, 50121 Florence, Italy [email protected] National Institute for Nuclear Physics (INFN) Unit of Florence Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy October 27, 2020 ABSTRACT The muon radiography technique is part of the straight-ray scanning imaging methods and it is usually employed to produce two-dimensional images of the integrated density in the radial direction from the detector position. However, in the literature there are already some examples of muon tomographies made by combining several measurements from different observation points. The work presented in this article fits into this line of research: a tomography of a target composed of lead blocks arranged in a non-trivial geometry was carried out by means of muon absorption radiography measurements. To solve the inversion problem, an iterative algorithm called SART was employed, obtaining better results than using a Chi-squared minimization. Keywords Muon tomography · Muography · Computerized Tomography (CT) · Computed Radiography (CR) · Micropattern gaseous detectors (MICROMEGAS) · Particle tracking detectors (Gaseous detectors) 1 Muon radiography and scattering tomography Muon radiography (or muography) is a geophysical technique that allows to study the density and the atomic number of large dimension systems taking advantage of cosmic ray muons. The muon is an unstable lepton with a mean life of 2:2 µs and a mass 200 times greater than that of the electron 2 (mµ ≈ 106 MeV=c ). Muons are the most frequent charged particles from cosmic rays at the sea level, with a vertical flux of about 70 m−2 s−1sr−1 for energies greater than 1 GeV [1, chapter 30.3]. The integral muon flux depends on the zenith angle θ approximately as cos2 θ, while it weakly depends on the azimuth angle φ because of the interaction of the mainly positive particles in the primarily cosmic radiation with the Earth’s magnetic field (this is the so called arXiv:2010.13562v1 [physics.ins-det] 22 Oct 2020 East-West asymmetry). Going into more detail, the muons spectrum depends on the measuring site (on its altitude, latitude and longitude) and varies with time: solar activity and variations in the atmospheric pressure influence the flux of cosmic muons [2, 3]. There are mainly two muographic techniques that are based on two different interaction phenomena of muons within matter: muon transmission (and absorption) radiography and muon scattering tomography. All of these techniques relies on the detection of muon tracks by means of particle detectors called trackers. Some general reviews can be found in [4, 3]. Muon transmission and absorption radiography are based on the energy loss (and the eventual decay) of muons when they interact with matter [1, chapter 34.2]. A muon with a given impulse pmin, before being stopped, will be able to cross on average a certain opacity X defined as X = R ρ dL, where ρ is the density of the material and L is the particle A PREPRINT -OCTOBER 27, 2020 Figure 1: Transmitted muon flux as a function of the crossed opacity (in meter water equivalent) and of the elevation angle (the dependency from the zenith angle is neglected). path length. The relationship between minimum impulse and crossed opacity can be found tabulated in reference [5]. The higher the opacity of the target, the lower the number (and flux) of muons that have an impulse high enough to cross it. Fixed a certain angular direction (θ; φ) and a certain opacity, the expected value of the flux transmitted through the target can be estimated as the integral from pmin to infinity of the differential flux in that direction. In figure 1 the transmitted muon flux is depicted as a function of the opacity of the crossed object and of the elevation angle (the dependence of the muon flux from the azimuth angle is neglected). This histogram has been obtained from the knowledge of the differential muon flux on Earth measured with the ADAMO spectrometer [6]. With the previous conversion map or using appropriate calibration measurements (as in this work), the fraction of transmitted (absorbed) muons can be converted into an opacity measurement for each direction of observation. Dividing the opacity by the total target length, one eventually obtains the average density along the inspected direction. For the transmission technique a single tracker is placed downstream the target to detect the transmitted muons, while for the absorption one the object under inspection is placed between the tracker and a veto plane (respectively above and below the target) to identify those muons that have been stopped within the target. Although the two techniques are similar, transmission muon radiography is the most common as it allows to examine very large objects. Transmission muography sees its greatest development in the field of volcanology [7, 8, 9, 10, 11, 12, 13, 14]. The first studies were able to provide a description of the internal architecture of volcanoes and afterwards dynamic studies have been carried out, demonstrating the possibility of monitoring volcanoes in real time so as to prevent the risks associated with an eruptive event. Apart from volcanology there are several other applications in the fields of archaeology, geology, mining and civil engineering. Following in the footsteps of Luis Alvarez’s pioneering measure inside the Chefren pyramid from the 1970’s [15], more recently a hidden chamber inside the Cheops pyramid was discovered by the ScanPyramids collaboration [16]. In the field of geology the technique has been employed to determine the shape of the bedrock underneath alpine glaciers in Switzerland [17], while there are numerous uses in the mining sector [18, 19, 20, 21, 22]. For all of these applications a typical acquisition lasts from a few days up to a month. The scattering muon tomography technique takes advantage of another phenomena: the deflection of muons from their straight trajectory due to multiple coulomb interactions [1, chapter 34.3]. The deflection angle has null average and depends on velocity and impulse of the particle, and on opacity and atomic number Z of the crossed material. The deflection of muons is particularly relevant for low impulse muons and it can constitute a not negligible background for the transmission (absorption) radiography [3]. Scattering tomography allows to directly reconstruct the density and the atomic number of the target by measuring the deflection of muons trough it. To do that two tracking detectors, one upstream and one downstream the target, are needed, thus limiting the size of the objects that can be studied (that is limited also by the fact that the technique is no longer sensible when multiple deflections occur). Given the higher level of information obtained from these measurements, acquisition times are generally reduced (around a few minutes) 2 A PREPRINT -OCTOBER 27, 2020 compared to transmission radiography. The technique is particularly suitable to discriminate materials with an high atomic number [23] opening applications in the nuclear sector and in homeland security [24, 25, 26, 27, 28, 29, 30, 31]. 2 3D reconstruction with muon radiography 2.1 Tomographic reconstruction While muon scattering tomography is naturally a three-dimensional imaging technique, muon radiography provides bidimensional maps of density integrated on the radial direction from the point of view of the detector. Nonetheless, just like in medical CT, combining multiple muon radiography measures from different positions, it is possible to obtain the tree-dimensional distribution of density. Some tomographic reconstructions using muon radiography have already been carried out and each has its own peculiarity which depends on the system under observation [32, 22, 21, 33, 34, 20, 35, 19]: unlike the work presented in this article, some of these reconstructions involve a combination of muographic and gravimetrical measurements [34, 35, 19] and many of them use regularization terms based on a priori information to reduce the non-uniqueness typical of these problems [21, 33, 34, 20, 19]. 2.2 Inversion problem definition In all these reconstructions the inspected volume, from now on World, is modeled as a 3D grid of Voxels usually cuboids in shape (except in [22]) with unknown uniform density. Then the muographic information is organized in a series of Rays each characterized by an opacity measurement and by geometrical limits: in case the tracker can be considered point-like, each Ray will correspond to a precise solid angle. Supposing to have I Rays and J Voxels, after this discretization process, from the equation of the opacity definition we obtain the following system of equations: J X Xi = Lijρj = L~ i~ρ; i = 1; 2;:::;I; (1) j=1 where Xi is the opacity of the ith Ray, ρj is the density of the jth Voxel and Lij is the average intersection length between the ith Ray and the jth Voxel. For a given Ray, just a few Voxels will be intersected and will contribute with their density to the Ray’s opacity and therefore Lij is an element of a sparse matrix (L). The goal will be to determine the density values that solve the system of equations 1. In case of real opacity measurements with errors σi, one will have to find the density vector that minimizes the Chi-squared I ~ !2 X Xi − Li~ρ χ2 = : (2) σ i=1 i 2.3 The ART family algorithms Given the usually high values for I and J (see section 3.3 for example) the minimization of the Chi-squared defined in 2 is computationally expensive. To reduce the computational time and still achieve a good result, iterative methods can be used.
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