Separation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). As is the case with many other solution techniques for differential equations, separation of variables may be codified within the broader framework of symmetry analysis. Though the separation of variables technique is frequently used in the nuclear engineering context with various equations describing neutron transport, its connection to the symmetries of those equations has not yet been thoroughly established. It is thus the purpose of this work to establish that connection using neutron diffusion as both an initial step toward analysis of more generally applicable equations, and as a connection to previous results in related problems. Using Lie group analysis, it is found that the traditional space-time separable solution of the neutron diffusion equation (featuring a single αeigenvalue) corresponds to time translation and flux scaling symmetries. Additional solutions of this equation are also constructed using its broader symmetry set. IntroductionSeparation of variables is one of the oldest techniques for solving certain classes of partial differential equations (PDEs). Like many other such methods, it is often employed as a specific element out of a 'bag of tricks', so that given a certain equation the technique can either be expected to apply, or not. In this more general sense, it was not until the late 19th century that S. Lie began his investigation into a 'unification theory' of solution techniques for differential equations, inspired in part by the preceding successful application of group theory to algebraic equations. Lie was successful in his aim, and indeed it has since been found that a wide variety of solution techniques for differential equations can be integrated into the broader framework of symmetry analysis (as now set forth in rigorous detail by Ovsiannikov [1], Bluman and Anco [2], Ibragimov [3], Hydon [4], Olver [5], Cantwell [6], Stephani [7], and many others). Beginning in the 1970s, C Boyer, E Kalnins, and W Miller embarked upon a thorough program of analyzing the connection between Lie symmetries, separation of variables, and special functions in a variety of contexts, including Schrödinger equations, Helmholtz and Laplace equations, harmonic oscillator equations, Hamilton-Jacobi equations, wave equations, and multitudinous other structures in numerous coordinate systems and spaces. This effort is largely summarized in two books and a survey by Miller [8-10]. Following this work, a generalized attempt at demonstrating the connection in question was provided by Gegelia and Markovski [11]. Additional studies have been performed by Chou and Qu [12] for nonlinear diffusion-convection equations and Estevez et al [13] for a porous medium equation, and Polyanin and Zhurov [14] for the axisymmetric unsteady boundary-layer equations, to name a few. Modern developments intended to reconcile (or not) separation of variables within the symmetry analysis framework appear t...
Bremsstrahlung spectra produced by 5 keV electrons incident on Al2O3 and MgO targets at air pressures of 30, 50, and 100 Pa have been compared with results produced using pyPENELOPE, a program designed to simulate electron microscopy. The comparisons showed that the experimental results were in good agreement with the results simulated using pyPENELOPE, except near the Duane–Hunt limit, where the bremsstrahlung amplitudes were consistently greater than PENEPMA predicted. The discrepancies may be due to charging effects, which are not simulated by PENEPMA, the Monte Carlo code on which pyPENELOPE is based. If so, the phenomena could potentially impact the accuracy of energy dispersive X-ray spectrometry measurements.
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