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Journal of Undergraduate Research

Keywords

specific heat-phonon spectrum inversion, SPI, vibration energies

College

Physical and Mathematical Sciences

Department

Physics and Astronomy

Abstract

Inverse problems have long provided researchers in a variety of disciplines with fruitful and challenging problems with immense applicability. However, as early investigators discovered, most of these problems are ill-posed.1 This means that the existence of the solution is not guaranteed, and even if it does exist, the solution may not be unique. Additionally, the solution may be remarkably sensitive to small perturbations of the input data. This means that small inaccuracies in data can result in unphysical solutions when inversion is carried out. However, regularization methods can often be developed for a class of problems that allow a researcher to find conditions on existence and uniqueness and to stabilize the solution with respect to noise in the data. Because of the rich applicability of inverse problems, methods for regularizing integral equations are an area of active interest. We worked on a particular inverse problem that is famous for its severe ill-posedness, the problem of inverting crystalline specific heat data to the statistical distribution of lattice atomic vibration energies (phonon spectrum). This problem is important both for its theoretical interest and because conventional experimental methods for obtaining the phonon spectrum are difficult and costly. We applied methods from Fourier analysis, data analysis techniques, numerical methods and a variety of other specialized tools to regularize the relevant integral equation and then carried out the numerical inversion of both theoretical model specific heats and for real data from a high temperature superconductor, YBCO.

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