Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment
Most probable trajectory of a muon in a scattering medium, when input and output trajectories are known
Introduction
Interactions between cosmic rays and particles in the upper atmosphere generate showers of high energy charged particles, including muons. Muons are similar to electrons, but with about 207 times the mass. Due to their high energy, in the order of GeV, cosmic ray muons are highly penetrating, and can travel through hundreds of metres of solid material, albeit with energy loss and scattering.
The scattering depends on the composition of the material, and so by observing the trajectory of a muon before and after passing though an object, information about that object's internal composition can be obtained. This property has led to the development of muon tomography, where the information obtained from many muon trajectories is used to derive a 3D image of an object's interior [1], [2]. Cosmic ray showers contain both muons and antimuons, but in the context of muon tomography, the distinction between the two is unimportant, as they are scattered by neutral matter in a very similar way.
Muon tomography is particularly promising for homeland security applications, including the detection of smuggled nuclear material [2], as high atomic-mass elements such as uranium and plutonium are very strongly scattering. Furthermore, due to the penetrating nature of the muons, it is very difficult to shield against muon tomography. This long penetration distance also enables imaging of large-scale objects, such as large industrial structures [3] and geological features [4], [5].
The data processing algorithms which underlie muon tomography are still an area of active research. Current algorithms are based on the Point Of Closest Approach (POCA) model of this trajectory [6], [7], [8]. This extrapolates the ingoing trajectory of the muon in a straight line forwards, and the outgoing trajectory in a straight line backwards, and assumes that the muon is scattered from one trajectory to the other, due to a strongly scattering region at the point of closest approach. A schematic of a POCA path can be seen in Fig. 1. The density of material in that region can be estimated from the angle of deflection, and by combining lots of muon measurements, a 3D image can be built up.
However, the POCA method makes certain assumptions as to the physical nature of the system, which are not necessarily true, and so may degrade the quality of the resulting images. One such problem is that in three dimensions, two lines will not necessarily intersect, and so in general, some contrivance is required to link the input and output trajectories. This can be done by assuming that the scattering region provides a translation as well as a deflection, or by tweaking the input and output trajectories so that the two intersect.
Another problem with the POCA method, is that by insisting on straight lines through most of the object, with only one bend, it presumes that the object being observed is best understood as having a single strongly scattering region, under vacuum elsewhere. The assumption is appropriate in samples containing large amounts of empty space, but it becomes unreliable for more uniform materials, such as the rock inside a mountain. As well as being unreliable, the assumption is inflexible, as there is no way of incorporating prior knowledge, and in particular, of allowing the knowledge derived from other muon events to inform the analysis of a specific event. In fact, when multiple strong scatterers are present, a POCA reconstruction can make conflicting assumptions. For example, if a region is far away from the POCA region, but still intersected by the POCA path, it will be regarded as weakly scattering, even it has been established as strongly scattering by other muon measurements.
We have succeeded in moving beyond this assumption, using a technique which amounts to the minimization of a functional describing the curvature of the trajectory, which is described in Section 2. (Somewhat similar analysis has been done for protons [9], but using different mathematical techniques.) In Section 3, a numerical method for performing this functional minimization, given a known input and output trajectory, is described. In Section 4 the results for a variety of cases are presented. In Section 5 potential applications for the method, including improved tomographic reconstructions, are discussed.
Section snippets
Formulating path calculation as minimization of curvature functional
The probability density function (PDF) f that a muon will undergo an angular deflection of , with an azimuth relative to its path , over a small distance increment , is assumed to obey a Fokker–Planck equation of the formThe diffusion coefficient D is related to the muon's momentum, and the scattering cross-section of the intervening material [10]. This formulation, and the Gaussian probability distribution that results from it, is a good approximation to reality for most
Numerical method
The minima of the curvature functionals were calculated by discretizing the paths along the z axis, thus reducing the minimization to a finite-dimensional algebraic problem. (Another possible approach is to use the Calculus of Variations to convert the minimization problem to an equality problem. The corresponding Euler Lagrange equations, however, are a coupled pair of third-order nonlinear partial differential equations, which would be more difficult to solve.)
The first step is to represent
Results
Calculations of most probable muon trajectory were done for a variety of input and output conditions. In all cases a planar object was assumed to lie halfway between the detectors, the profile of which is given in Fig. 3. The functional form of the object, in terms of the scattering coefficient D, iswhere the coefficient specifies the ratio in D between the center of the object, and the background medium. This choice of function is not particularly important; it is just
Conclusions
A method to calculate the most probable muon trajectory through a scattering medium, for given input and output trajectories has been developed. Firstly, the problem of maximizing probability was reformulated as a problem of minimizing a functional describing the integrated curvature of the muon path. Secondly, an algorithm for finding the muon path which minimizes the functional was developed. The method returns trajectories that are sometimes similar to, but don't match POCA trajectories.
Acknowledgments
We received funding from The Atomic Weapons Establishment (AWE), under contract number 30182146.
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